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author:
- 'A. J. Princep'
- 'H. L. Feng'
- 'Y. F. Guo'
- 'F. Lang'
- 'H. M. Weng'
- 'P. Manuel'
- 'D. Khalyavin'
- 'A. Shenshyn'
- 'M. Rahn'
- 'Y. H. Yuan'
- 'Y. Matsushita'
- 'S. J. Blundell'
- 'K. Yamaura'
- 'A. T. Boothroyd'
bibliography:
- 'PCOO\_supp.bib'
title: 'Supplementary Material for: Magnetically driven loss of centrosymmetry in metallic Pb$_2$CoOsO$_6$'
---
Preparation and characterization
================================
Both poly- and single-crystalline Pb$_2$CoOsO$_6$ were prepared by solid-state reaction in a belt-type high-pressure apparatus [@synthesis1; @synthesis2]. A mixture of PbO$_2$ (3N, High Purity Chemicals. Co., Ltd., Japan), Os (99.95%, Heraeus Materials Technology, Germany), Co (4N, Alfa Aesar), and KClO$_4$ ($>$ 99.5%, Kishida Chemical Co., Ltd., Japan) in the molar ratio PbO$_2$/Os/Co/KClO$_4$ = 2:1:1:0.5 was placed in a platinum capsule (6.9 mm in diameter and $\sim$ 5 mm in height). The capsule was loaded into the high-pressure apparatus and treated in a pressure of 6 GPa at 1500 $^\circ$C for 1 hour (polycrystals) or at 1600 $^\circ$C for 2 hours (single crystals). After heating, the capsule was quenched to room temperature within a few seconds before releasing the pressure. The products were then rinsed in water for 5 min for several times to remove residual ingredients. Phase purity of the powder sample was examined by the synchrotron x-ray diffraction (SXRD) at room temperature using the large Debye-Scherrer camera at the BL15XU beamline in SPring-8, Japan [@spring8]. The data were collected between 2$^\circ$ and 60$^\circ$ with a 0.003$^\circ$ step in 2$\Theta$ and an incident wavelength of $\lambda$ = 0.65297 $\AA$. Fine powder of the sample was packed into a 0.1 mm Lindenmann glass capillary which was rotated during measurement. The obtained SXRD patterns were analyzed by the Rietveld method using the programs RIETAN-2000 and VESTA [@RIETAN; @VESTA]. The refinement results are presented in Fig. S1 and Table S1. Laboratory X-ray diffraction was performed using a Mo-source Oxford Diffraction Supernova diffractometer on a single crystal of Pb$_2$CoOsO$_6$ of approximate size 820$\times$310$\times$110$\mu$m$^3$ (Figure S2a). More than 96% of the detected peaks were successfully indexed by a single monoclinic domain with the space group P2$_1$/$n$. The diffraction patterns along the \[0 k l\], \[h 0 l\], and \[h k 0\] directions are shown in Figs. S2b-2c.
Specific heat was measured in a Quantum Design PPMS. The peak in the specific heat shows a slight downwards shift ($<$ 1 K) in an applied magnetic field of 90 kOe. To estimate the magnetic entropy change (SMag), we subtracted the lattice contribution (C$_{lat}$) from C$_p$ via a polynominal fit. We estimate S$_{Mag}$ to be approximately 3.65 J mol$^{-1}$ K$^{-1}$. This value is less than 20$\%$ of the expected value of 20.7 J mol$^{-1}$ K$^{-1}$ for spin-only Os and Co. The low-temperature part of Cp was analyzed separately using the Debye model with an electronic specific heat term (Fig. S2b), i.e. C$_p$/T = $\gamma$ + $\beta$T$^2$, where $\gamma$ is the Sommerfeld constant. The analysis yielded a $\gamma$ value of 33.6(6) mJ mol$^{-1}$ K$^{-2}$, which is consistent with a substantial density of states at the Fermi level. In a magnetic field of 90 kOe, a similar analysis revealed negligible change of $\gamma$.
To parameterize C$_p$, the curve was analyzed by a linear combination of the Debye and the Einstein models (see the solid line in Fig. 2c): $$\begin{split}
\frac{C_p(T)}{n_D} = 9N_Ak_B ( \frac{T}{T_D} )^3 \int_0^{T_D/T} \frac{x^4e^x}{( e^x -1 )^2} \\+ 3N_Ak_B ( \frac{T_E}{T} )^2 \frac{e^{T_E/T}}{( e^{T_E/T} -1 )^2} ,\label{eq:1}
\end{split}$$ where N$_A$ is the Avogadro’s constant, k$_B$ is the Boltzmann’s constant, and T$_E$ and T$_D$ are the Einstein and Debye temperatures, respectively. The scale factors n$_D$ and n$_E$ correspond to the number of vibrating modes per the formula unit in the Debye and the Einstein models, respectively. The fit yielded T$_D$ of 610(7) K, T$_E$ of 90(12) K, n$_D$ of 6.997(4), and n$_E$ of 3.535(5). The non-trivial Einstein term likely suggests anharmonic lattice dynamics in Pb$_2$CoOsO$_6$.
Site Wyckoff x y z B$_{iso}$ ($\AA^2$)
------ --------- ------------ ------------ ------------ ---------------------
Pb 4e 00.0032(4) 0.5057(5) 0.25071(5) 1.14252
Co 2a 0 0 0 -1.18
Os 2b 0 0 0.5 1.93
O1 4e -0.0606(5) -0.0019(4) 0.2587(11) 0.99996
O2 4e 0.2399(5) 0.2817(3) 0.0297(7) 0.99996
O3 4e 0.2797(4) 0.7610(3) 0.0327(5) 0.99996
: Structural parameters of Pb$_2$CoOsO$_6$ from synchrotron x-ray diffraction at room temperature. Space group P2$_1$/$n$ ( \# 14, setting choice 2), Z = 2, $\lambda$ = 0.65297 $\AA$; Full occupancy factors for all sites; a = 5.6116(41) $\AA$, b = 5.65743(7) $\AA$, c = 7.9049(21) $\AA$, and $\beta$ = 89.990(33)$^\circ$; Final R values are: R$_p$ = 1.095% and R$_{wp}$ = 1.778%.
\[table:2\]
Bond Bond distance ($\AA$) Bond Bond angle ($^\circ$)
-------- ----------------------- ---------- -----------------------
Co1-O1 1.9027 (6) $\times$ 2 Os-O1-Co 172.2(8)
Co1-O2 2.1084(3) $\times$ 2 Os-O2-Co 168.1(7)
Co1-O3 2.2409(2) $\times$ 2 Os-O3-Co 145.4(5)
BVS 2.19251
Os2-O1 1.8258(13) $\times$ 2
Os2-O2 1.8857(11) $\times$ 2
Os2-O3 2.1119(6) $\times$ 2
BVS 6.04595
: Selected interatomic distances, angles, and Bond Valence Sums (BVS) of Pb$_2$CoOsO$_6$ at room temperature. BVS = $\sum_{i=1}^N v_i$ $v_i = \exp{(R_0 - l_i)/B}$, N is the coordination number, B = 0.37, R$_0$(Co$^{2+}$) = 1.692, and R$_0$(Pb$^{2+}$) = 2.1124, and R$_0$(Os$^{6+}$) = 1.925.
\[table:3\]
![\[fig:S1\] Fig. S1. Rietveld refined synchrotron XRD profiles of Pb$_2$CoOsO$_6$ at room temperature. The lower ticks show the unknown impurity.](FigS1.pdf "fig:"){width="75.00000%"}\
![\[fig:S2\] Fig. S2.(a) picture of crystal used for X-ray diffraction measurement and diffraction patterns at room temperature along the (b)\[0 k l\], (c) \[h 0 l\], and (d) \[h k 0\] directions. The correct unit cell found is monoclinic with space group of P2$_1$/$n$ and lattice parameters being consistent with those refined from synchrotron XRD. ](FigS2.pdf "fig:"){width="75.00000%"}\
![\[fig:S3\] Fig. S3. (a) Temperature dependence of reciprocal magnetic susceptibility ($\chi^{-1}$ vs. T) measured at H = 10 kOe. The solid line represents the result of Curie-Weiss plot ($>$ 200 K); (b) Specific heat in a form of C$_{p}$/T vs. T$^{2}$ at H = 0 and 90 kOe. The red solid line represents the plot result by using the Debye model.](FigS3.pdf "fig:"){width="75.00000%"}\
![\[Fig4\] Total and projected spin-polarised partial density of states (DOS) for antiferromagnetic Pb$_2$CoOsO$_6$ calculated in the GGA approximation. Dotted (magenta) lines are for the Co 3$d$ orbitals and dashed (blue) lines are the Os 5$d$ orbitals. Those contributions from the crystal field split t$_{2g}$ and e$_g$ orbitals are also labeled. The Fermi energy is set at zero. Positive and negative values of the DOS represent the spin-up and spin-down parts, respectively. ](Fig4.pdf){width="75.00000%"}
DFT calculations were performed with the OpenMX software package [@openMX] using the lattice parameters and the AFM configuration obtained from the NPD refinement. The choice of pseudo-potentials, basis sets and the sampling of Brillouin zone with 6$\times$6$\times$10 grid have been carefully optimized and the exchange-correlation functional within the generalized gradient approximation (GGA) [@Perdew1996] was used. For Co 3$d$ orbitals, the spin splitting is larger than that due to the CF, so that Co$^{2+}$ is in its high spin state with configuration as t$_{2g}^{3\uparrow}$ e$_g^{2\uparrow}$ t$_{2g}^{2\downarrow}$ e$_g^{0\downarrow}$ , where the superscript number represents the number of occupied electrons and arrows indicate the spin state. On the contrary, the extended 5$d$ orbitals of Os have a larger CF splitting than spin splitting, which consequently results in a low spin state of Os$^{6+}$ (5$d^2$) with an approximate configuration of t$_{2g}^{1\uparrow}$ e$_g^{0\uparrow}$ t$_{2g}^{1\downarrow}$ e$_g^{0\downarrow}$ .
Magnetic and transport properties measurements
==============================================
DC magnetic susceptibility ($\chi$) of several crystals (total mass 21.0 mg) was measured in a Quantum Design magnetic properties measurement system (MPMS) between 2 K and 400 K in an applied magnetic field (H) of 10 kOe. The crystals were loosely gathered in a sample holder and cooled down to 2 K. The magnetic field was then applied to the crystals and the temperature was slowly raised up to 400 K (Zero-Field Cooling, ZFC), followed by cooling down to 2 K again in the field (Field Cooling, FC). The isothermal magnetization was measured in the instrument at 5 K between -50 kOe and 50 kOe, which shows linear behavior. A physical properties measurement system (PPMS) from Quantum Design was used to measure the electrical resistivity ($\rho$) of a selected crystal at temperatures between 2 K and 300 K upon cooling by a four-terminal method with an ac-gauge current of 10 mA at a frequency of 110 Hz. Silver epoxy was used to fix platinum wires on the crystal. Specific heat (Cp) of an amount of crystals (3.6 mg) was measured in the same apparatus between 2 K and 300 K at H = 0 and 70 kOe.
Neutron powder diffraction measurements were carried out on a 3.5 g powder sample at various temperatures on the Spodi high-resolution diffractometer [@SPODI] at the FRM-II facility of the Technische Universität München, Germany, and also on the WISH time-of-flight diffractometer [@WISH] at the ISIS Facility of the Rutherford Appleton Laboratory, UK. The Rietveld refinement of the crystal and magnetic structures was performed using FullProf [@fullprof]. There were only very weak impurity peaks arising from unknown impurities in the patterns (quantified as 0.9 wt%).
Muon Spin Rotation analysis
===========================
Zero-field (ZF) muon spin relaxation ($\mu^+$ SR) spectra of Pb$_2$CoOsO$_6$ were measured in a $^3$He cryostat in the General Purpose Spectrometer (GPS) at the Swiss Muon Source at the Paul Scherrer Institute, Switzerland. The muon spectra (Figure \[plot:asymsfit\], blue curve) below the transition can be well modelled with the equation $$A=A_1\cos{(\omega_1 t)} e^{-\lambda_1t}+A_2\cos{(\omega_2 t)} e^{-\lambda_2t}+A_3 e^{-\lambda_3t}
.\label{eq:asymsfit}$$ The first two terms of Eq. \[eq:asymsfit\] represent muons stopping in the sample and experiencing long-range magnetic order with precession frequencies $\omega_i=\gamma_\mu\rm{B}_i/(2\pi)$ and relaxation rates $\lambda_i$, where $\gamma_\mu=2\pi\times135.5\rm{MHzT}^{-1}$ is the muon muon gyromagnetic ratio and $\rm{B}_i$ are the local fields. The third term accounts for muons stopping outside the sample, such as in the sample holder. The results for $\lambda_i$ and $\nu_i$ are plotted in Figure \[plot:asymsfit\], together with phenomenological order parameter models of the form $\nu=\nu_0[1-(T/T_{\rm N})^\alpha]^\beta$ from which a critical temperature of $T_{\rm N}=45.5(10)$ K can be extracted. Furthermore, the ratios of the amplitudes $A_i$ averaged over the measured temperatures below $T_{\rm N}$ indicate that about 35% of the muons in our experiments experienced the smaller magnetic field, about 60% the higher field, and about 5% implanted outside the sample.
Muon sites\[sec:dft\]
=====================
In order to compare the observed muon precession frequencies with ones predicted from different magnetic structures, it is necessary to establish the potential muon stopping sites in Pb$_2$CoOsO$_6$. To this end we employed Density Functional Theory (DFT) calculations using the plane-wave program [*Quantum Espresso*]{} [@Gianozzi2009] within the generalized gradient approximation (GGA) [@Perdew1996]. We modelled the ions with ultrasoft pseudopotentials [@Rappe1990] and the muon with a norm-conserving hydrogen pseudopotential. The energy cutoffs for the wavefunction and the charge density were set to $80$Ry and $800$Ry, respectively, and a $3\!\times\! 3\! \times \!3$ Monkhorst-Pack $k$-space grid [@Monkhorst1976] was used for the integration over the Brillouin zone. Within these parameters the calculations gave well converged results and reproduced the experimentally observed atomic positions and lattice parameters within a 1% accuracy. The results shown in Figure \[plot:coulomb\] were visualised with the Vesta software [@Momma2008].
We use the converged electron density to map out the electrostatic Coulomb potential of Pb$_2$CoOsO$_6$ throughout its unit cell, as plotted in Figure \[plot:coulomb\]. The global maximum is used as the reference point, since large values of the Coulomb potential correspond to a low energy cost to add a positive charge and such regions have been found to be a reliable first-order estimate of potential muon sites [@Moller2013_DFT; @Foronda2015]. Addtionally, we performed relaxation calculations, which allow for distortions of the lattice due to the implanted muon, which lead to muon site candidates in good agreement with the sites predicted by the local maxima of the electrostatic potential. In fact, we find that there are eight potential muon site candidates, which are all fairly closely related in energy and symmetry due to the proximity of the crystal structure of Pb$_2$CoOsO$_6$ to a more symmetric cubic one. These muon sites, plotted in Figure \[plot:coulomb\] and tabulated in Table \[tab:dftmu\], are also all characterised by an O–H like bond between the muon and an oxygen with a bond length of about $\SI{1.0}{\angstrom}$, which is a typical occurence in oxygen containing compounds [@Moller2013_DFT; @Foronda2015].
The muon asymmetry, plotted in Figure \[plot:asymsandfft\](a) for three temperatures, exhibits an oscillatory beating pattern at low temperatures indicative of long-range magnetic ordering. The Fourier transform spectra of the ZF-$\mu^+$SR asymmetries, presented in Figure \[plot:asymsandfft\](b), reveals two broad peaks centered around muon precession frequencies of roughly 30 MHz and 55 MHz at the lowest measured temperatures which vanish above the transition temperature. Likely muon stopping sites in the magnetic unit cell were calculated using DFT methods, and we find 8 possible muon stopping sites with comparable energies, which could thus potentially be occupied. We label these sites Mu1-8 in order of increasing enregy cost for occupation and thus decreasing probability of occupation.
[@l @\*3c @\*2c l@]{} Site & & $\Delta$E/u.c. & O–H\
Label & x & y & z & (meV) & with\
Mu1 & 0.786 & 0.225 & 0.838 & 0 & O3\
Mu2 & 0.264 & 0.720 & 0.334 & 4.5 & O2\
Mu3 & 0.295 & 0.878 & 0.026 & 6.5 & O3\
Mu4 & 0.878 & 0.699 & 0.983 & 24.5 & O2\
Mu5 & 0.880 & 0.288 & 0.988 & 24.8 & O3\
Mu6 & 0.305 & 0.121 & 0.015 & 40.5 & O2\
Mu7 & 0.710 & 0.291 & 0.403 & 46.6 & O2\
Mu8 & 0.710 & 0.711 & 0.407 & 46.8 & O3\
Dipole fields\[sec:dipolefields\]
=================================
With the potential muon spotting sites from Table \[tab:dftmu\], we can calculate the local magnetic fields at each of these sites. Because our neutron experiments revealed an antiferromagnetic ordering the Lorentz and demagnetising fields are zero and since the muon sites are far from the Os and Co ions we also expect hyperfine field contributions to be negligible, such that we only have to focus on the dipolar fields $$\boldsymbol{{B}}_\mathrm{local}\approx\boldsymbol{{B}}_\mathrm{dip}
=\sum_i \frac{\mu_o}{4\pi |\Delta\boldsymbol{r}_i|^3}\left[\frac{3(\boldsymbol{\mu}_i\cdot\Delta\boldsymbol{r}_i)\Delta\boldsymbol{r}_i}{|\Delta\boldsymbol{r}_i|^2}-\boldsymbol{\mu}_i\right]
\nonumber.$$ Here, the $\boldsymbol{r}_i$ correspond to the relative positions of the magnetic moments $\boldsymbol{\mu}_i$ with respect to the muon. Using the total moment size, as listed in Table 1, we computed the local magnetic fields, assuming a fraction $x$ of the total moment lies on the Os sites, while a fraction $1-x$ of the total moment lies on the Co sites, for each of the muon site candidates and their 16 symmetry equivalent sites in the magnetic unit cell. The resulting muon precession frequencies are plotted in Figure \[plot:dipolefields\]. Note that each line in Figure \[plot:dipolefields\] actually represents eight nearly identical lines, such that the 16 positions symmetry equivalent to the Mu1 site actually only lead to two distinct experimentally observable frequencies (unless the Os moment fraction $x$ is very close to 0 or 1). We can further note that the local fields at some of the muon stopping sites are almost identical, which is owed to the proximity to a cubic crystallographic symmetry (see Figure \[plot:coulomb\]). Based on the muon precession frequencies shown in Figure \[plot:dipolefields\] we can identify two scenarios which can plausibly explain our experimentally observed frequencies. Either, only the two energetically most favourable sites (Mu1 and Mu2) are significantly occupied and that $x \approx 0.5$. Alternatively, all eight of the identified muon site candidates might be occupied and $x \approx 1$. Both scenarios would lead to only two experimentally distinguishable frequencies, with a ratio of the oscillation amplitudes of these two frequencies expected in the region of 1:1, very roughly in line with that observed experimentally.
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abstract: 'The article describes an experimental method that allows to estimate the inhomogeneous and homogeneous linewidths of the photoluminescence band of a point defect in an amorphous solid. We performed low temperature time-resolved luminescence measurements on two defects chosen as model systems for our analysis: extrinsic Oxygen Deficient Centers (ODC(II)) in amorphous silica and F$_3^+$ centers in crystalline Lithium Fluoride. Measurements evidence that only defects embedded in the amorphous matrix feature a dependence of the radiative decay lifetime on the emission energy and a time dependence of the first moment of the emission band. A theoretical model is developed to link these properties to the structural disorder typical of amorphous solids. Specifically, the observations on ODC(II) are interpreted by introducing a gaussian statistical distribution of the zero phonon line energy position. Comparison with the results obtained on F$_3^+$ crystalline defects strongly confirms the validity of the model. By analyzing experimental data within this frame, we obtain separate estimations of the homogenous and inhomogeneous contributions to the measured total linewidth of ODC(II), which results to be mostly inhomogeneous.'
author:
- 'Michele D’Amico'
- Fabrizio Messina
- Marco Cannas
- Maurizio Leone
- Roberto Boscaino
title: 'Homogeneous and inhomogeneous contributions to the luminescence linewidth of point defects in amorphous solids: Quantitative assessment based on time-resolved emission spectroscopy'
---
Introduction
============
The physics of color centers embedded in a solid matrix is a fundamental and interesting scientific field both from the point of view of basic physics and for their wide technological applications as modifiers of the macroscopic physical properties of solids, (e.g. optical transparency, refractive index, electrical resistance, and so on).[@stoneham; @nalwa] Several experimental evidences have led to a general agreement on the fact that the properties of point defects may be significantly different depending on the crystalline or amorphous structure of the solid they are embedded in.[@Erice] Indeed, in a crystal each member of an ensemble of identical defects experiences the same local environment. As a consequence, every spectroscopical property of the ensemble of defects, such as the lineshape of the related absorption or photoluminescence (PL) bands, can be interpreted as a property of the single center, and is referred to as *homogeneous*. The homogeneous absorption linewidth is mainly determined by the electron-phonon interaction and it is related to other important physical properties of the defect, such as the Huangh-Rhys factor and the phonon vibrational frequencies.[@nalwa; @Erice] On the other hand, defects in an amorphous solid are believed to feature site-to-site statistical distributions of the spectroscopic properties due to the disorder of the surrounding matrix. Hence, the lineshapes of their optical bands are characterized by an *inhomogeneous* broadening,[@nalwa; @Erice; @holeburning] which reflects the degree of disorder of the amorphous solid and concurs, together with the homogeneous effects, to determine the overall spectroscopic signature of the color center.
Many experimental approaches have been proposed to estimate the homogeneous and inhomogeneous contributions to the experimental linewidth of an optically active center: exciton resonant luminescence, resonant second harmonic scattering, femtosecond photon echo, spectral hole burning and site-selective spectroscopy.[@holeburning; @furumiya; @woggon; @mittlemann; @Skujaprb1995; @kuroda] However, the issue is still open since none of these techniques is applicable to the whole variety of inhomogeneous physical systems of interest. For instance, in amorphous solids site-selective spectroscopy can been successfully applied only to defects which allow the the direct observation of the zero phonon line (ZPL) by virtue of a weak coupling with the vibrational modes of the matrix.[@Erice; @Skujaprb1995]
In this paper we propose a new experimental approach to this problem, which allows to estimate the inhomogeneous and homogeneous linewidths based on mapping the variations of the radiative decay lifetime within an inhomogeneously broadened luminescence emission band by time-resolved laser-excited luminescence. To this purpose, in the next section we first describe an adapted version of the theoretical treatment of the optical properties of a point defect in a solid, which takes into account the effects of heterogeneity in amorphous systems. Next, in the experimental section we demonstrate that the predictions of our model are consistent with the results of measurements performed on two model point defects, one in a crystal solid and the other one in a glass. Finally, we use the theoretical model to estimate the inhomogeneous and homogeneous widths of the two model defects and to obtain other physical parameters of interest.
Theoretical description of optical defect properties
====================================================
We briefly review the standard theoretical description of the optical properties of a point defect in a crystal,[@stoneham; @nalwa; @Erice] in order to adapt it later to the case of amorphous systems. In addition to the crude *Born-Oppenheimer* and *Franck-Condon* approximations, we suppose the defect to be coupled with only one vibrational mode of the solid matrix of frequency $\omega_p$, assumed to be the same for ground and excited electronic states. The frequency $\omega_p$ can be regarded also as the mean frequency of the vibrational modes of the solid or can be thought as the effective phonon frequency coupled with the electronic transition.[@nalwa2]
In this frame, the absorption cross section $\Omega(E)$ of a defect as a function of the excitation energy $E$ at the absolute zero temperature is given by:[@Erice; @nalwa2] $$\Omega(E)=\beta\sum_k{|M_{0k}|^2} E \cdot \delta[E-(E_0+k \hbar \omega_p)] \label{sigmaE}$$ where $\delta$ indicates a Dirac delta function, and the summation is carried out over the vibronic transitions linking the ground electronic state with zero phonons towards different vibrational sub-levels $(k)$ of the electronic excited state, spaced by $\hbar
\omega_p$. The energy value $E_0$ (zero phonon line) is the absorption transition without emission or absorption of phonons. $M_{0k}$ is the overlap integral between nuclear wave functions associated to the ground and excited states, while $\beta$ is given by: $\beta=\frac{1}{n} \left( \frac{E_{eff}}{E_{ext}} \right)^2
\frac{4 \pi^2}{3 \hbar c}\frac{1}{g_l}|D|^2$, where $D$ is the matrix element of the electric dipole operator between the ground and excited electronic states, and $g_l$ is the degeneracy of the lower electronic state. The effective field correction $\frac{1}{n}
\left(\frac{E_{eff}}{E_{ext}} \right)^2$ accounts for the polarization effect induced by the external field on the solid.[@stoneham; @Erice] We assume here that the refraction index $n$ is constant in the electromagnetic range investigated. In the harmonic approximation for the vibrational sublevels relative to the ground and excited states, the $|M_{0k}|^2$ coefficients are given by a Poisson distribution:[@Erice] $$|M_{0k}|^2=e^{-H}\frac{H^k}{k!} \label{Poisson}$$ where $H$ is the *Huangh-Rhys factor*, expressing the number of phonons emitted by the system after absorption of a photon while relaxing to the ground vibrational substate of the excited electronic level.[@stoneham] Given a population of identical defects, the envelop of the $\delta$ functions in Eq. (\[sigmaE\]) describes their characteristic *homogeneous* absorption lineshape, with (aside from the effect of the factor E) a $E_{Abs}=E_0+H \hbar \omega_p$ first moment and a $\sigma_{ho}=\sqrt{H}\hbar \omega_p$ width.
After relaxation towards the bottom of excited electronic state, the system can relax back to the ground state by spontaneous photon emission (photo-luminescence). The following relationship of *mirror symmetry* links the absorption $\Omega(E)$ and luminescence $L(E)$ band shapes: [@Erice] $$\frac{L(E)}{E^3}\propto \frac{\Omega(2E_0-E)}{2E_0-E}\label{mirror}$$ The energy difference between absorption and emission peaks (*Stokes shift*) is linked to the Huangh-Rhys factor and results to be $2S=2H \hbar \omega_p$. Using the *mirror symmetry* Eq. (\[mirror\]) and Eq. (\[sigmaE\]) we obtain: $$L(E)\propto \beta \sum_k |M_{0k}|^2 E^3 \cdot \delta[E-(E_0-k \hbar \omega_p)] \label{PLomog}$$ which represents the *homogeneous* emission lineshape, with (aside from the effect of the factor $E^3$) a $E_{em}=E_0-H \hbar
\omega_p$ first moment and a $\sigma_{ho}$ width. Expression (\[PLomog\]) does not take into account the dependence from the excitation energy within the absorption band. This is based on experimental results and it will be discussed later.
The PL radiative lifetime $\tau$ is linked to the absorption profile by the *Forster’s equation*: [@Erice; @forster] $$1/\tau=\frac{n^2}{\pi^2c^2\hbar^3} \frac{g_l}{g_u} \int(2E_0-E)^3\frac{\Omega(E)}{E}dE \label{forster}$$ where $g_u$ is the degeneracy of the upper electronic state. Combining Eq. (\[forster\]) and Eq. (\[sigmaE\]) we obtain the decay rate $1/\tau$: $$1/\tau=\gamma \sum_k |M_{0k}|^2 (E_0-k \hbar \omega_p)^3 \label{ratetau}$$ where $\gamma=\frac{n^2}{\pi^2 c^2 \hbar^3}\frac{g_l}{g_u} \beta $. The cubic dependence appearing in the above expression is a direct consequence of the relation between Einstein coefficients for absorption and spontaneous emission, which forms the basis of Forster’s equation. Eq. (\[ratetau\]) can be approximated by neglecting the contributions far from $k\sim H$, thus obtaining: $$1/ \tau = \gamma (E_0-S)^3. \label{tauapprox}$$ This expression shows that the decay rate is proportional to $\gamma$ and approximately to the third power of the first moment of the emission band.
Summing up, the global expression for the luminescence of a population of identical point defects in a solid matrix as a function of the spectral position $E$ and time $t$ after an exciting light pulse (*homogeneous shape*) is: $$\label{rate}
L(E,t) \propto \gamma \sum_k |M_{0k}|^2 E^3 e^{-t/\tau} \cdot \delta[E-(E_0-k \hbar \omega_p)]$$ This expression assumes that non radiative channels from the excited state are absent. As we see from Eq. (\[rate\]), the shape and kinetics of the homogeneous luminescence band are completely characterized by four parameters: $E_0$ (the ZPL position), $\hbar
\omega_p$ (the phonon energy), $\gamma$ (proportional to $|D|^2$) and $H$ (the Huangh-Rhys factor). $H$ and $\hbar \omega_p$ can be expressed in terms of the half Stokes shift $S$ and of the homogeneous width $\sigma_{ho}$: $\hbar \omega_p$=$\sigma_{ho}^2/S$ and $H=S^2/\sigma_{ho}^2$. In this way, expression (\[rate\]) can be alternatively regarded as depending on the four parameters $E_0$, $S$, $\sigma_{ho}$, $\gamma$, thus being indicated by the expression: $L(E,t|E_0,S,\sigma_{ho},\gamma)$.
For defects in an amorphous matrix, we can argue the hypothesis of a population of identical defects to fail. Indeed, each point defect interacts with different environments and it is possible that this conformational heterogeneity causes a site-to-site statistical distribution of one or more of the homogeneous properties of single defects. The simplest model we can put forward to take into account the disorder effects is to introduce a gaussian distribution of the ZPL position $E_0$, peaked at $\widehat{E_0}$ and with an inhomogeneous width $\sigma_{in}$; in this scheme, $\gamma$, $S$, and $\sigma_{ho}$ are still considered as undistributed parameters. Within these hypotheses, the global PL signal $L^*(E,t)$ emitted by the ensemble of non-identical point defects can be now expressed as the convolution of the homogeneous shape $L(E,t)$ with the inhomogeneous distribution of $E_0$: $$\begin{aligned}
L^*(E,t|\widehat{E_0},\sigma_{in},S,\sigma_{ho},\gamma)\propto \nonumber \\
\int L(E,t|E_0,S,\sigma_{ho},\gamma) \cdot e^{-\frac{\left(E_0-\widehat{E_0}\right)^2}{2\sigma_{in}^2}}dE_0 \label{gaussnonomo}\end{aligned}$$
Eqs. (\[rate\]) and (\[gaussnonomo\]) lead us to predict a difference between the PL signals of defects in crystalline and amorphous solids. Indeed, when the inhomogeneous broadening $\sigma_{in}$ is almost zero, as expected for point defects in a crystalline matrix, Eq. (\[rate\]) has to be used, and the radiative lifetime $\tau$ should be independent from the spectral position at which it is measured within the emission band. In fact, $\tau$ is expressed by Eq. (\[ratetau\]), so being a function of the homogeneous parameters $E_0$, $\gamma$, $S$, and $\sigma_{ho}$, which are expected to be the same for all defects in the solid. In contrast, in an amorphous solid a PL band due to an ensemble of point defects can be thought as arising from the overlap of several bands with different $E_0$ as described by Eq. (\[gaussnonomo\]), and thus featuring different lifetimes. Hence, when $\sigma_{in}$ is comparable with $\sigma_{ho}$ it should be possible to experimentally observe a dispersion in $\tau$ by measuring the decay of the PL signal at different emission energies. Also, the shape of a band arising from the overlap of sub-bands with different lifetimes should vary in time, so that the position of its first moment $M_1(t)$, calculated by the usual expression: $$M_1(t)=\frac {\int E\ L^*(E,t) dE} {\int
L^*(E,t)dE}\label{firstmoment}$$ should depend on time. Therefore, both the dispersion of $\tau$ within the emission band and the time dependence of the first moment can be used in principle as experimental probes of inhomogeneous effects.
It is worth noting that according to Eq. (\[tauapprox\]), $\tau$ strongly depends on the first moment of the emission band, $E_{em}=E_0-S$, and more weakly on $\gamma$. This leads to $E_0$ as the parameter of choice to be distributed in our model. Moreover, a gaussian distribution of $E_0$ was experimentally demonstrated for the non-bridging oxygen hole center point defect in silica, for which the zero-phonon line can be directly observed by site-selective spectroscopy at low temperatures.[@Skujaprb1995; @vaccaro] On the other side, we acknowledge that similar predictions can be obtained by introducing a distribution of the half Stokes shift $S$ with an undistributed $E_0$. Data reported later on in this paper do not allow to discriminate between these two possibilities.
Finally, to get further insight into the meaning of Eq. (\[gaussnonomo\]) it is useful to consider the extreme case in which the homogeneous width is so narrow to be negligible with respect to the inhomogeneous one. In this case, the homogeneous lineshape $L(E,t)$ can be approximated as $\delta(E-(E_0-S))\cdot
e^{-t/\tau}$, with $\tau$ given by Eq. (\[tauapprox\]). By substituting in Eq. (\[gaussnonomo\]) we get that: $$L^*(E,t)\propto e^{-\gamma E^3t}
\cdot e^{-\frac{\left(E+S-\widehat{E_0}\right)^2}{2\sigma_{in}^2}}$$ This expression predicts an exponential decay whose $\tau$ depends cubically from the experimental observation energy $E$ within the inhomogeneous band.
In the intermediate situation of non-negligible homogeneous width, Eq. (\[gaussnonomo\]) deviates in principle from a single exponential decay, as it contains contributions with different values of $\tau$. However, we verified that the typical values of the parameters which will be used in the following to fit experimental data ($\widehat{E_0}$, $\sigma_{in}$, $S$, $\sigma_{ho}$, $\gamma$), correspond to predicted decay curves that always remain very close to a single exponential for all practical purposes. From a theoretical point of view, we can define in general $\tau(E)$ as the time in which $L^*(E,t)$ (at a fixed $E$) decreases by a 1/e factor from $L^*(E,0)$. With this definition, we can summarize the above considerations as follows: the $\tau(E)$ curve (with $E$ varying within the observed emission band) is expected to vary progressively from a constant value (for a completely homogeneous system) to a cubic dependence (for a completely inhomogeneous system) with increasing inhomogeneous/homogenous ratio. To check the validity of our model we have performed experimental measurements (described in the following) on crystalline and amorphous defects.
Materials and Experimental Methods
==================================
We chose F-type-centers in lithium fluoride (LiF) and Oxygen Deficient Centers of the second type, ODC(II), in amorphous silicon dioxide (SiO$_2$, or silica) as model point defects on which testing our approach. Both centers feature broad near-gaussian luminescence bands in the ultraviolet (UV) range with close decay lifetime values ($\sim$8 ns), and they have both been widely studied in literature because of their important technological applications. Specifically, LiF is a material traditionally employed in the production of high-quality optical elements to be used in the infrared, visible, and particularly in the ultraviolet spectral regions. F-type-centers in LiF (electron trapped in anion vacancies) are the subject of active investigation in the areas of color center lasers, radiation dosimetry and integrated optics (see Ref. and references therein). The study of point defects in silica is a fundamental technologic problem as well, because their presence compromises the optical and electrical properties of glasses in their wide uses as optical components, as insulators in MOS transistors, and for guiding or processing light signals (optical fibers and Bragg gratings).[@nalwa; @Erice] ODC(II) is a peculiar defect of the amorphous phase of SiO$_2$,[@Erice; @skujajncs98] thus being an interesting model system to investigate the characteristic properties of defects in disordered materials with respect to crystalline ones. Its microscopic structure consists in an atom bonded to two oxygen atoms of the matrix (=X$^{\bullet\bullet}$), where $X$ is an atom belonging to the isoelectronic series Si, Ge, Sn.[@Erice; @skujajncs98; @skuja1984] Previous studies have suggested that the spectroscopic properties of ODC(II) are significantly conditioned by inhomogeneous effects.[@trukhin; @leone1; @leone2; @cannizzophilos]
We report measurements performed on two samples: the first one is a crystalline Lithium Fluoride sample, hereafter denoted as LiF. Prior to any measurement this specimen, $5\times5\times1.25$ mm$^3$ sized, was irradiated at room temperature with electrons of 3 MeV energy, for a total dose of $1.5\cdot10^6$ rad. The purpose of irradiation was to induce in the sample the formation of luminescent F-type centers. The second sample is a fused silica (commercial name: Infrasil301, provided by Heraeus Quartzglas,[@heraeus] and $5\times5\times1$ mm$^3$ sized), hereafter named I301, manufactured by fusion and quenching of natural quartz, with typical concentration of impurities of $\sim$20 ppm in weight.[@heraeus] In particular, as-grown I301 contains a $\sim$1 ppm concentration of Ge impurities, due to contamination of the quartz from which the material was produced. Previous studies demonstrated that in the as-grown material most of the Ge impurities are arranged as Ge-ODC(II) defects (=Ge$^{\bullet\bullet}$); moreover comparison with sol-gel silica samples doped with Ge atoms ensures us that the contribution to PL of intrinsic ODC(II) defects in I301 sample is negligible.[@skujajncs98; @sgjncs03] The optical activity of Ge-ODC(II) at low temperature ($<$100 K) consists in an absorption band centered at $\sim5.1$ eV which excites a fast (lifetime in the ns range) emission band centered at $\sim4.3$ eV, due to the inverse transition.[@skujajncs98; @nalwa2; @agnelloprb03]
PL measurements were done in a standard back-scattering geometry, under excitation by a pulsed laser (Vibrant OPOTEK: pulsewidth of 5 ns, repetition rate of 10 Hz, energy density per pulse of 0.30$\pm$0.02 mJ/cm$^2$) tunable in the UV-Visible range. The luminescence emitted by the sample was dispersed by a spectrograph (SpectraPro 2300i, PI Acton, 300 mm focal length) equipped with three different gratings, and detected by an air-cooled intensified CCD (Charge-Coupled Device PIMAX, PI Acton). The detection system can be triggered in order to acquire the emitted light only in a given temporal window defined by its width (t$_W$) and by its delay t$_D$ from the end of the laser pulse. All measurements reported here were performed on samples kept at 25 K in high vacuum ($\sim10^{-6}$ mbar) within a He flow cryostat (Optistat CF-V, OXFORD Inst.). All luminescence signals in I301 were acquired with a 300 grooves/mm grating with a 2 nm bandwidth, while the signals in LiF were measured with a a 150 grooves/mm grating with a 2.5 nm bandwidth. All the spectra were corrected for the spectral response and for the dispersion of the detection system.
Experimental Results
====================
In Fig. \[decad3D\]-(a) we show a typical time-resolved measurement of the PL activity of Ge-ODC(II) in the I301 sample, performed at 25 K under laser excitation at 240 nm (5.17 eV). The PL decay was analyzed by performing 60 acquisitions with the same integration time t$_W$=1 ns but at different delays t$_D$, going from 0 to 60 ns from the laser pulse. Fig. \[decad3D\]-(b) shows the normalized spectra of panel (a) in a contour plot and evidences that the first moment of the band (continuous line) varies in time.
{width="12"}
![\[plblif\]Low temperature (25 K) luminescence of Ge-ODC(II) in the I301 sample (a), and of F-centers in the LiF sample (b). Both PL bands are obtained by exciting at the maximum of the respective absorption bands and acquired for $t_D=0$ and with $t_W$=1 ns. The continuous line is the result of the fitting procedure by our theoretical model. The Poissonian homogeneous shape is also shown (see discussion).](fig2){width="8"}
In Fig. \[plblif\]-(a) we report the signal acquired for t$_D$=0, corresponding to the first spectrum in Fig. \[decad3D\]-(a). The PL band of Ge-ODC(II), as acquired immediately after the end of the laser pulse, is peaked at $\sim$4.4 eV and has a $\sim$0.45 eV width (Full width at Half Maximum, FWHM) consistent with literature data.[@skujajncs98] Completely analogous time-resolved measurements were carried out on the PL activity of F-type centers in the LiF sample. This specimen was excited at 450 nm (2.76 eV) and its luminescence was collected by varying t$_D$ from 0 to 100 ns with t$_W$=1 ns. We report in Fig. \[plblif\]-(b) the luminescence signal detected in LiF at t$_D$=0. It is apparent that the PL signal of LiF comprises two contributions peaked at $\sim$2.3 eV and $\sim$1.8 eV. These signals are known to be associated to two different defects, the $F_3^+$ and $F_2$ centers respectively, both consisting in aggregates of F-type centers. [@baldacchini; @LiF1] In particular, the main $\sim$2.3 eV band with a $\sim$0.27 eV FWHM is due to $F_3^+$, consisting in two electrons localized on three adjacent anion vacancies.[@baldacchini] For each activity (Ge-ODC(II) and $F_3^+$), one can extract the time dependence of the first moment of the luminescent bands from the time-resolved measurements (e.g. those in Fig. \[decad3D\] in the case of Ge-ODC(II)). Data so-obtained are reported in Fig. \[taumm1\]-(a). The origin of the time scale corresponds to t$_D$=0.
{width="13"}
We observe that the PL activity in silica shows an approximately linear decrease of the first moment as a function of time, while this decrease is not observed in LiF, where the first moment of the $F_3^+$ centers band has a constant value within experimental sensitivity. As already discussed in the theoretical section, the progressive shift of the PL peak position observed for ODC defects can be alternatively understood as a dependence of the luminescence lifetime from the spectral position within the emission band. Hence, in Fig. \[taumm1\]-(b) we report the values $\tau(E)$ of the PL lifetime as a function of the emission energy. The lifetimes were estimated for both PL activities by least-square fitting data from time-resolved spectra (\[decad3D\]-(a)) at different emission energies with an exponential function ($I(t)=
I(0)e^{-t/\tau}$).[^1] At this temperature (25 K), the decays are purely exponential for both activities.[@agnelloprb03; @baldacchini2] Fig. \[taumm1\]-(b) shows that the lifetime of Ge-ODC(II) centers in silica strongly varies within the emission band: $\tau$ goes from $\sim$7.0 ns to $\sim$10.7 ns. A similar behavior for Ge-ODC(II) was observed also under excitation by synchrotron radiation.[@agnellopss07] On the contrary, the lifetime of $F_3^+$ centers is almost constant in the observed range of emission energies. The above results were obtained exciting at the absorption peak for both PL activities. Although we performed the same measurements for different excitation energies within the absorption band, only a very weak dependence from this parameter was evidenced, consistently with previous results.[@agnelloprb03]
Discussion
==========
The results in Fig. \[taumm1\] qualitatively confirm the predictions of our theoretical analysis, i.e. that the dependence of the lifetime on the emission energy or, equivalently, the progressive red-shift of the emission peak with time, are characteristic features of luminescent defects embedded in a glassy matrix, as opposed to “crystalline” defects. We stress that the non-radiative decay channels are almost completely quenched for both PL signals at the temperature at which the experiments were performed (25 K).[@baldacchini; @agnelloprb03] As a consequence, it is a very good approximation to consider the luminescence decay to be purely radiative. The main point of the following discussion is to fit all experimental data by our model and extract the values of the homogeneous and inhomogeneous widths of the PL emission bands and other interesting physical parameters.
For both investigated PL activities we have performed numerical integration of Eq. (\[gaussnonomo\]) to obtain a set of three theoretical curves which simultaneously fit i) the shape of the PL band at t$_D$=0, ii) the time dependence of the first moment (calculated by Eq. (\[firstmoment\])) and iii) the dependence of $\tau$ on emission energy.[^2]. To increase the reliability of the fit procedure, the half Stokes shift $S$ was fixed to the value obtained experimentally by measuring the difference between the spectral positions of the absorption and emission peaks: $S$=0.38 eV and $S$=0.24 eV in silica and LiF respectively. In this way, the fitting procedure was performed by varying only four free parameters, $\widehat{E_0}$, $\sigma_{in}$, $\sigma_{ho}$, $\gamma$. From the experimental point of view, the vibrational sub-structure of homogeneous luminescence bands cannot usually be resolved due to the bandwidth of the measuring system and to further broadening effects due for instance to the coupling with several low energy modes. To take into account this effect, the homogeneous lineshape, Eq. (\[rate\]), was convoluted with a gaussian distribution of a narrow width $\hbar\omega_p$ before being inserted into Eq. (\[gaussnonomo\]).
The continuous lines in Fig. \[plblif\] and \[taumm1\] represent the results of our fitting procedure. It is worth underlining the goodness of the fit, obtained using only four parameters, and considering especially that data in Fig. \[taumm1\] take into account simultaneously all data acquired in a time-resolved PL measurement (typically $\sim$600 spectral positions for each of the $\sim$100 temporal acquisitions of Fig. \[decad3D\]). Table \[results\] summarizes the best parameters obtained via our fitting procedure for the two investigated PL activities.
$\widehat{E_0}\ [eV]$ $\sigma_{in}\ [meV]$ $\sigma_{ho}\ [meV]$ $S\ [eV]$ $\gamma \ [10^6\ eV^{-3}s^{-1}]$
------ ----------------------- ------------------------ ------------------------- --------------- ----------------------------------
I301 4.70$\pm$0.05 177$\pm$10 93$\pm$12 0.38$\pm$0.02 1.41$\pm$0.09
LiF 2.50$\pm$0.02 20$\pm$10 109$\pm$6 0.24$\pm$0.02 10.0$\pm$0.6
$\lambda (\%)$ $\sigma_{tot} \ [meV]$ $\hbar \omega_p\ [meV]$ $H$ $f$
I301 78$\pm$5 200$\pm$10 23$\pm$6 17$\pm$5 0.073$\pm$0.010
LiF 3$\pm$2 111$\pm$6 51$\pm$7 5$\pm$1 0.32$\pm$0.04
From data in Table \[results\] we can also calculate the Huang-Rhys factor $H=S^2/\sigma_{ho}^2$, the vibrational frequency $\hbar \omega_p =\sigma_{ho}^2/S$, the total width (from $\sigma_{tot}^2=\sigma_{in}^2+\sigma_{ho}^2$),[^3] and finally the parameter $\lambda=\sigma_{in}^2/\sigma_{tot}^2$ which estimates the degree of inhomogeneity. All these quantities are reported in Table \[results\] as well. As expected, $\lambda$ is very small for the LiF defects in comparison with the amorphous ones: $\sim3\%$ against $\sim78\%$. These values correspond to $\sigma_{in}$ being about 0.2 times and 2 times $\sigma_{ho}$, in LiF and SiO$_2$ respectively. We note that the inhomogeneous broadening in the crystalline sample is not exactly zero; beside the approximations in our model, we note that a real crystal is always distorted by some dislocations, strains or other imperfections distributed at random into the matrix. The obtained value of $\lambda$ for Ge-ODC(II) shows that for a defect embedded in a glassy matrix the inhomogeneous width can be prominent with respect to the homogeneous one. This conclusion may be at variance with previous suggestions that $\sigma_{ho}$ and $\sigma_{in}$ are typically comparable.[@skujajncs98]
In Fig. \[plblif\] we also show the discrete Poissonian homogeneous lineshape of width $\sigma_{ho}$ and ZPL position $\widehat{E_0}$, as obtained by our fit procedure for both investigated activities. As already pointed out, the crystalline PL band is completely described by the homogeneous shape,[^4] whereas the silica PL band is not reproduced without taking into account inhomogeneous effects. It is also worth noting that the value $\hbar\omega_p$=23$\pm$6 meV obtained via our fitting procedure is very close to the value of $26
\pm 2$ meV found for the same defect by the analysis of the temperature dependence of the experimental absorption linewidth.[@cannizzo2003] Moreover, $\hbar\omega_p$ is in good agreement with experimental and computational works on silica glasses which predict the presence of vibrational modes of low frequency.[@galeener; @umari] These agreements further confirm the correctness of the present value of $\sigma_{ho}$ and consequently of our analysis.
To show the accuracy of our fitting procedure in determining $\lambda$, in Fig. \[fsigmai\] we compare the experimental lifetimes of Ge-ODC(II) in the I301 sample with the predictions of our model obtained for different $\lambda$ values. The theoretical $\tau(E)$ curves are obtained by keeping $\sigma_{tot}$ fixed to the value which best fits the overall experimental shape of the PL band. This analysis clearly evidences a continuous transition from constant lifetimes for $\lambda$=0 (that is a completely homogeneous PL band), to an inverse cubic dependence of $\tau$ from emission energy for $\lambda$=1 (that is a completely inhomogeneous PL band), as anticipated in the theoretical section.
![\[fsigmai\] Experimental decay lifetime at different emission energies for Ge-ODC(II) point defects (white circles). Lifetime as predicted by our model for different values of the parameter $\lambda$ (continuous lines). Dashed line represents the extreme case of 1/$E^3$ dependence (see discussion). The arrow indicates the direction of increasing $\lambda$.](fig4){width="8"}
Finally, the oscillator strength *f* reported in Table \[results\] is calculated using:[@Erice] $$f=\frac{2m_e}{3\hbar^2e^2} \frac{1}{g_l} E_{Abs}|D|^2\label{oscillator}$$ where $m_e$ and $e$ are respectively the mass and the charge of electron. We have substituted in Eq. (\[oscillator\]) the value of $|D|^2$, calculated from the fitting parameter $\gamma$, and we have used for $E_{Abs}$ the value $\widehat{E_0}+S$. In regard to the effective field correction, the term $\frac{1}{n} \left(
\frac{E_{eff}}{E_{ext}} \right)^2$ calculated within the Onsager model,[@stoneham; @Erice] results to be close to unity both in SiO$_2$ (n$\sim$1.5) and in LiF (n$\sim$1.4) in the investigated spectral range. The oscillator strength found here for Ge-ODC(II) in silica is consistent with the range of values reported in literature:[@skujajncs98] 0.03-0.07. For $F_3^+$ centers in LiF our result is close to 0.2 reported in ref. .
The main assumption of our model that all amorphous effects can be completely accounted for by a simply gaussian distribution of a single homogenous parameter (i.e zero phonon line) is strongly corroborated by the excellent agreement between theoretical curves and data. On the other side, a distribution of the emission peak $E_0-S$ is strongly suggested *a priori* by the almost Einstein-like proportionality of 1/$\tau$ on E$^3$ shown by experimental data in Fig. \[fsigmai\]. Finally, it is important to note that in this scheme $\gamma$ and thus $|D|^2$ are assumed as undistributed parameters. This means that the oscillator strength given by Eq. (\[oscillator\]) can be distributed only as a consequence of the variations of $E_{Abs}$ associated to different homogeneous absorption sub-bands.
Conclusions
===========
We have investigated the inhomogeneous properties of point defects in a glassy matrix via mapping by time-resolved PL the dependence of the radiative decay lifetime on emission energy. We propose a theoretical model, based on an extension of the standard theory of the optical properties of point defects, incorporating a statistical distribution of the zero phonon line to account for the effects of the non-equivalent environments probed by each point defects in an amorphous matrix as opposed to a crystalline one. This model enlightens a direct connection between the dispersion of the radiative decay lifetime within a luminescence band as a function of emission energy and the inhomogeneous properties of defects in a glassy environment. To confirm our prediction we have experimentally studied the luminescence of Oxygen Deficient Centers in silica and of aggregates of F-centers in a crystalline sample of LiF. The model is able to fit all experimental data and to provide an estimate of the ratio $\lambda=\sigma_{in}^2/\sigma_{tot}^2$ between the inhomogeneous and the total width, namely $\sim$78% for ODCs and $\sim$3% for F$_3^+$. Finally, our model allowed us to determine the homogeneous parameters of ODC and F$_3^+$ centers: homogeneous width, oscillator strength, Huangh-Rhys factor and the frequency of the vibrational local mode.
We acknowledge financial support received from project “P.O.R. Regione Sicilia - Misura 3.15 - Sottoazione C”. The authors would like to thank R. M. Montereali for having kindly provided the irradiated LiF sample. We also thank G. Lapis and G. Napoli for assistance in cryogenic work. Finally we are grateful to LAMP research group (http://www.fisica.unipa.it/amorphous/) for support and enlightening discussions.
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[^1]: In regard to LiF, the fits were carried out in the range $\sim$2.10-2.60 eV so as to avoid the region of the $F_3^+$ emission band possibly affected by the overlap with the signal due to $F_2$.
[^2]: The lifetimes predicted by the model were estimated by least-square fitting the decay curves (not reported) predicted by Eq. (\[gaussnonomo\]) at different emission energies with a single exponential. It is worth noting that the simulated data (as real data) feature no appreciable non-exponential behavior in the timescale of experimental data, at least when the parameters of the model are close to the best-fit ones.
[^3]: Alternatively, one can estimate $\sigma_{tot}$ directly from experimental data, so obtaining a consistent value.
[^4]: As explained above, the homogeneous shape is obtained by a convolution of the discrete Poissonian with a narrow gaussian curve of width $\hbar \omega_p$ to take into account further homogeneous broadening effects and experimental bandwidth.
|
---
abstract: 'We prove a new formula for the Hirzebruch-Milnor classes of global complete intersections with arbitrary singularities describing the difference between the Hirzebruch classes and the virtual ones. This generalizes a formula for the Chern-Milnor classes in the hypersurface case that was conjectured by S. Yokura and was proved by A. Parusinski and P. Pragacz. It also generalizes a formula of J. Seade and T. Suwa for the Chern-Milnor classes of complete intersections with isolated singularities.'
address:
- 'L. Maxim : Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison WI 53706-1388 USA'
- 'M. Saito: RIMS Kyoto University, Kyoto 606-8502 Japan'
- 'J. Schürmann : Mathematische Institut, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany'
author:
- Laurentiu Maxim
- Morihiko Saito
- Jörg Schürmann
title: 'Hirzebruch-Milnor Classes of Complete Intersections'
---
**Introduction**
For the proof of his Riemann-Roch theorem \[Hi\], F. Hirzebruch defined the $\chi_y$-genus of a compact complex manifold $X$ by $$\chi_y(X):={{\hbox}{$\sum$}}_p\,\chi(\Omega_X^p)\,y^p\in{{\mathbf Z}}[y].$$ This specializes respectively to the Euler characteristic, the arithmetic genus, and the signature of $X$ at $y=-1,0,1$. It is the highest degree part of the cohomology Hirzebruch characteristic class $T_y^*(TX)$ of the tangent bundle $TX$. Using the Chern roots $\{{\alpha}_i\}$ for $TX$, this cohomology Hirzebruch class is defined by $$\aligned T^*_y(TX)&:={{\hbox}{$\prod$}}_{i=1}^{\dim X}Q_y({\alpha}_i)\in
{{\mathbf H}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(X)[y],\raise-7pt{\hbox}{}\\
{\hbox}{with}{\quad}Q_y({\alpha})&:={{\alpha}(1+y)/{\bigl}(1-e^{-{\alpha}(1+y)}
{\bigl})}-{\alpha}y\in{{\mathbf Q}}[y][[{\alpha}]],\endaligned$$ where $Q_y({\alpha})$ is as in \[Hi\], 1.8 (see also (1.1) below), and ${{\mathbf H}}^k(X)=H^{2k}(X,{{\mathbf Q}})$ in this paper. (It is known that the theory works also for the Chow cohomology groups as defined in \[Fu1\], \[Fu2\].) Substituting $y=-1$, $0$, $1$, we see that $Q_y({\alpha})\in{{\mathbf Q}}[y][[{\alpha}]]$ becomes respectively $$1+{\alpha},{\quad}{\alpha}/(1 -e^{-{\alpha}}),{\quad}{\alpha}/\tanh{\alpha},$$ and hence $T^*_y(TX)$ specializes respectively to the Chern class $c^*(TX)$, the Todd class $td^*(TX)$, and the Thom-Hirzebruch $L$-class $L^*(TX)$, see \[HBJ\], Sect. 5.4. In the smooth case, this cohomology class $T_y^*(TX)$ is identified by Poincaré duality with the (Borel-Moore) homology class $T_y^*(TX)\cap[X]$ which will be denoted by $T_{y*}(X)$.
A generalization of $T_{y*}(X)$ to the singular case was given by \[BSY\] using the Du Bois complex in \[DB\] or ${{\mathbf Q}}_{h,X}\in D^b{{\rm MHM}}(X)$, the bounded complex of mixed Hodge modules on $X$ whose underlying ${{\mathbf Q}}$-complex is the constant sheaf ${{\mathbf Q}}_X$, where ${{\rm MHM}}(X)$ is the category of mixed Hodge modules \[Sa2\]. The Hirzebruch class $T_{y*}(X)$ is actually a special case of $T_{y*}({{\mathcal M}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}})$ defined for any ${{\mathcal M}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}\in D^b{{\rm MHM}}(X)$ in \[BSY\], see (1.2) below.
Hirzebruch \[Hi\] also introduced the notion of virtual $\chi_y$-genus (or $\chi_y$-characteristic) which is the $\chi_y$-genus of smooth complete intersections $X$ in smooth projective varieties $Y$. The [*virtual Hirzebruch characteristic class*]{} $T^{\,{{\rm vir}}}_{y*}(X)$ can be defined like the virtual genus (even in the singular case), see (1.3) below. In this paper, we show that the difference between the Hirzebruch class and the virtual one is given by the [*Hirzebruch-Milnor class*]{} supported on the singular locus of $X$, and prove an [*inductive*]{} formula to [*calculate*]{} it explicitly in the case of global complete intersections with arbitrary singularities as follows.
Let $Y^{(0)}$ be a smooth projective variety, and $L$ be a very ample line bundle on $Y^{(0)}$. We have the graded ring $$R_L:={{\hbox}{$\bigoplus$}}_{k\ge 0}\,R_{L,k}{\quad}{\hbox}{with}{\quad}R_{L,k}:={\Gamma}(Y^{(0)},L^{\otimes k}).$$ Let $s_j\in R_{L,a_j}$ with $\{a_j\}$ a decreasing sequence of positive integers. Let $r$ be a positive integer strictly smaller than $\dim Y^{(0)}$. For $j\in[1,r]$, set $$Y^{(j)}:={{\hbox}{$\bigcap$}}_{k\in[1,j]}\,s_k^{-1}(0)\subset Y^{(0)},$$ and assume ${\rm codim}_{Y^{(0)}}Y^{(j)}=j$. Define $$X:=Y^{(r)},{\quad}Y:=Y^{(r-1)}.$$ Note that $Y$ is not necessarily smooth. It is a complete intersection of codimension $r-1$ in a smooth projective variety $Y^{(0)}$. The sections $s_j\,(j\in[1,r])$ generate a graded ideal $$I_X={{\hbox}{$\bigoplus$}}_{k\ge 0}\,I_{X,k}\subset R_L.$$ Set $${{\mathbf P}}(R_{L,k}):=(R_{L,k}\setminus\{0\})/{{\mathbf C}}^*,{\quad}{{\mathbf P}}(I_{X,k}):=(I_{X,k}\setminus\{0\})/{{\mathbf C}}^*.$$ Replacing the $s_j$ without changing $I_X$, we may assume by Proposition (3.2) below $${{\rm Sing}}\,Y^{(j)}\subset Y^{(r)}=X{\quad}(j\in[1,r]).$$ This condition is satisfied (and the $s_j$ generate the ideal $I_X$) if the $s_j$ are sufficiently general, i.e. if $([s_j])$ belongs to a sufficiently small Zariski-open subset $${{\mathcal U}}_I\subset{{\hbox}{$\prod$}}_{j=1}^r\,{{\mathbf P}}(I_{X,a_j}),$$ where $[s_j]$ denotes the image of $s_j$ in ${{\mathbf P}}(I_{X,a_j})$.
Take a sufficiently general $s'_r\in R_{L,a_r}$, and set $$X':=s_r^{\prime -1}(0)\cap Y.$$ This is viewed as a complete intersection of codimension $r-1$ in the [*smooth*]{} hypersurface section $s_r^{\prime -1}(0)$ of $Y^{(0)}$. Here “sufficiently general” means that $[s'_r]$ belongs to a sufficiently small Zariski-open subset $U_r$ of ${{\mathbf P}}(R_{L,a_r})$. It satisfies at least the following two conditions:
\(1) The hypersurface section $s_r^{\prime -1}(0)$ transversally intersects all the strata of an algebraic Whitney stratification of $Y$ such that $Y\setminus X$ is a stratum.
\(2) For an embedded resolution of $X\subset Y$, the pullback of $X'$ is smooth and transversally intersects any intersections of the irreducible components of the pullback of $X$.
Conditions (1) and (2) respectively define a non-empty Zariski-open subset of ${{\mathbf P}}(R_{L,a_r})$ for each choice of an algebraic Whitney stratification of $Y$ or an embedded resolution of $(Y,X)$. Note that the intersection of finitely many non-empty Zariski-open subsets of $R_{L,a_r}$ is non-empty, and this is useful when we have to show the independence of the choices of the $s_j$, $s'_j$ later. Note also that the Thom $a_f$-condition follows from condition (1) (see \[BMM\], \[Pa\]), and condition (2) follows from (1) if there is an algebraic Whitney stratification of the embedded resolution such that the desingularization is a stratified morphism, i.e. any stratum of the resolution is smooth over a stratum of $Y$.
We have $${\Sigma}:={{\rm Sing}}\,X\supset{{\rm Sing}}\,Y\supset{\Sigma}':={{\rm Sing}}\,X'={{\rm Sing}}\,Y\cap X'.$$ Let $i_{A,B}$ denote the inclusion morphism for $A\subset B$ in general. Set $f:=(s_r/s'_r)|_{Y\setminus X'}$.
[**Theorem 1.**]{} [*With the above notation and assumptions, there is $M_y(X)\in{{\mathbf H}}_{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}({\Sigma})[y]$, called the Hirzebruch-Milnor class, where ${{\mathbf H}}_{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}({\Sigma}):=H_{2{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}^{{{\rm BM}}}({\Sigma},{{\mathbf Q}})$, and satisfying $$T^{\,{{\rm vir}}}_{y*}(X)-T_{y*}(X)=(i_{{\Sigma},X})_*M_y(X),
\leqno(0.1)$$ -20pt $$M_y(X)=T_{y*}{\bigl}((i_{{\Sigma}\setminus X',{\Sigma}})_{!\,}\varphi_f
{{\mathbf Q}}_{h,Y}{\bigl})+(i_{{\Sigma}',{\Sigma}})_*M_y(X').
\leqno(0.2)$$ More precisely, choosing sufficiently general $s'_j\in R_{L,a_j}\,(j\in[1,r])$ inductively, we have a mixed Hodge module ${{\mathcal M}}(s'_1,\dots,s'_r)$ on ${\Sigma}$ such that $$M_y(X)=(-1)^{\dim X}T_{y*}{\bigl}({{\mathcal M}}(s'_1,\dots,s'_r){\bigl}),$$ and $M_y(X)$ is independent of the choice of the $s'_j$.*]{}
Here $\varphi_f{{\mathbf Q}}_{h,Y}$ is viewed as a mixed Hodge module on ${\Sigma}\setminus X'$ up to a shift. (Note that ${{\mathbf Q}}_Y[\dim Y]$ is a perverse sheaf for any local complete intersection $Y$.) The monodromy around $X'$ of the local systems $H^j\varphi_f{{\mathbf Q}}_{h,Y}|_S$ on a stratum $S$ of a Whitney stratification coincides with the Milnor monodromy, since $f=(s_r/s'_r)|_{Y\setminus X'}$ by definition. So it is rare that the monodromy is trivial, and hence the situation is quite different from the one in \[CMSS\]. By Theorem 1, we have $T^{\,{{\rm vir}}}_{y,k}(X)=T_{y,k}(X)$ for $k>\dim{\Sigma}$ where $T^{\,{{\rm vir}}}_{y,k}(X)\in{{\mathbf H}}_k(X)[y]$ is the degree $k$ part of $T^{\,{{\rm vir}}}_{y*}(X)$. This generalizes \[CMSS\], Cor. 3.3 in the hypersurface case. Theorem 1 was proved in \[CMSS\], (1.17) in the case of hypersurfaces with isolated singularities.
The above $M_y(X)$ may depend on the choice of the $s_j$. However, it is well-defined if $([s_j])$ belongs to a sufficiently small non-empty Zariski-open subset ${{\mathcal U}}_I$ of $\prod_{j=1}^r{{\mathbf P}}(I_{X,a_j})$ (although it is not necessarily easy to write down explicitly ${{\mathcal U}}_I$), see Proposition (4.4) below. Note that $M_y(X)$ coincides with this canonical one if it is invariant by any small perturbation of the $s_j$, i.e. if it is constant for any element in a sufficiently small open neighborhood of $([s_j])\in\prod_{j=1}^r{{\mathbf P}}(I_{X,a_j})$ in [*classical*]{} topology.
We briefly explain the construction of $M_y(X)$. There is a flat family ${{\mathcal Z}}_T$ over a Zariski-open subset $T$ of ${{\mathbf C}}^r$ defined by $${{\mathcal Z}}_T:={{\mathcal Z}}\cap(Y^{(0)}\times T){\quad}{\hbox}{with}{\quad}{{\mathcal Z}}:={{\hbox}{$\bigcap$}}_{i=1}^r\{s_i=t_is'_i\}\subset Y^{(0)}\times{{\mathbf C}}^r.$$ Here $t_1,\dots,t_r$ are the coordinates of ${{\mathbf C}}^r$, and $T\subset{{\mathbf C}}^r$ is defined by the following condition: $t\in T\iff\dim{{\mathcal Z}}_t=\dim Y^{(0)}-r$ with ${{\mathcal Z}}_t:={{\mathcal Z}}\cap(Y^{(0)}\times\{t\})$ for $t\in{{\mathbf C}}^r$. (Note that ${{\mathcal Z}}_T$ is a complete intersection in $Y^{(0)}\times T$, and is flat over $T$, i.e. the $t_i-c_i$ for $i\in[1,r]$ form a regular sequence in ${{\mathcal O}}_{{{\mathcal Z}}_T,(y,c)}$ for any $(y,c)\in{{\mathcal Z}}_T$. This follows from a well-known theory of commutative algebra about regular sequences and flat morphisms, see e.g. \[Ei\].)
We have ${{\mathcal Z}}_0=Y^{(r)}=X$ by definition. Applying the iterated nearby cycle functors, we define a mixed Hodge module on $X$ by $${{\mathcal M}}'(s'_1,\dots,s'_r):=\psi_{t_r}\cdots\psi_{t_1}{{\mathbf Q}}_{h,{{\mathcal Z}}_T}[\dim X],$$ which satisfies $$T^{\,{{\rm vir}}}_{y*}(X)=(-1)^{\dim X}T_{y*}{\bigl}({{\mathcal M}}'(s'_1,\dots,s'_r){\bigl}).$$ Indeed, the nearby cycle functor of mixed Hodge modules corresponds to the Gysin morphism of Borel-Moore homology via the transformation ${{\rm DR}}_y$ in (1.2.1), see Proposition (3.3) below. (Note that the latter transformation is denoted by $mC_y$ in \[Sch2\].) We can moreover prove the injectivity of the canonical morphism of mixed Hodge modules $${{\mathbf Q}}_{h,X}[\dim X]{\hookrightarrow}{{\mathcal M}}'(s'_1,\dots,s'_r),$$ together with the equality (0.2) at the level of the Grothendieck group of mixed Hodge modules by increasing induction on $r$. Then ${{\mathcal M}}(s'_1,\dots,s'_r)$ is defined to be the cokernel of the above injection. It may depend on the $s'_i\,(i\in[1,r])$ although its image by $T_{y*}$ (and hence $M_y(X)$) does not by using a one-parameter family together with Lemma (2.6) below, see also \[Sch3\], Cor. 3.7.
By \[BSY\] and (1.3.4) below, $T_{y*}(X)$ and $T^{\,{{\rm vir}}}_{y*}(X)$ respectively specialize for $y=-1$ to the MacPherson-Chern class $c(X)$ (see \[Ma\]) and the virtual Chern class $c^{{{\rm vir}}}(X)$ (called the Fulton or Fulton-Johnson class, see \[Fu2\], \[FJ\]) with rational coefficients. Specializing Theorem 1 and using \[Sch1\], Prop. 5.21, we then get the following.
[**Corollary 1.**]{} [ *With the notation and the assumptions of Theorem $1$, there is $M(X)\in{{\mathbf H}}_{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}({\Sigma})$, called the Milnor class, and satisfying*]{}
$$\aligned&c^{{{\rm vir}}}(X)-c(X)=(i_{{\Sigma},X})_*M(X),\\
&M(X)=c{\bigl}((i_{{\Sigma}\setminus X',{\Sigma}})_{!\,}\varphi_f
{{\mathbf Q}}_Y{\bigl})+(i_{{\Sigma}',{\Sigma}})_*M(X')\raise4pt{\hbox}{}.\endaligned$$
Here $c{\bigl}((i_{{\Sigma}\setminus X',{\Sigma}})_{!\,}\varphi_f{{\mathbf Q}}_Y{\bigl})$ is the MacPherson-Chern class \[Ma\] with ${{\mathbf Q}}$-coefficients of the associated constructible function. To get Corollary 1 with ${{\mathbf Z}}$-coefficients (i.e. in ${{\mathbf H}}_{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}({\Sigma},{{\mathbf Z}})$), we have to prove some assertions in this paper (e.g. Proposition (4.1)) with ${{\mathbf Q}}$ replaced by ${{\mathbf Z}}$, and use \[Ve2\] (and \[Ke1\], \[Ke2\] if ${{\mathbf H}}_{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(X)={{\rm CH}}_{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(X)$).
As special cases of Theorem 1, we also get the following.
[**Corollary 2.**]{}
*With the notation and the assumptions of Theorem $1$, assume furthermore $r=1$, i.e. $Y$ is smooth, or $\dim{\Sigma}=0$, i.e. $X$ has only isolated singularities. Then the second term in the right-hand side of $(0.2)$ vanishes so that $$M_y(X)=T_{y*}{\bigl}((i_{{\Sigma}\setminus X',{\Sigma}})_{!\,}\varphi_f
{{\mathbf Q}}_{h,Y}{\bigl}).\leqno(0.3)$$*
If $\dim{\Sigma}=0$, then we can omit $(i_{{\Sigma}\setminus X',{\Sigma}})_!$ $($since ${\Sigma}\cap X'=\emptyset)$, and $$M_y(X)={{\hbox}{$\bigoplus$}}_{x\in{{\rm Sing}}\,X}\,\chi_y(\widetilde{H}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(F_x)).
\leqno(0.4)$$ Here $\widetilde{H}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(F_x)$ denotes the reduced Milnor cohomology of $f:(Y,x)\to({{\mathbf C}},0)$ endowed with a canonical mixed Hodge structure, and $\chi_y$ is a polynomial defined by using its Hodge numbers, see $(1.2.7)$ below.
Note that $\chi_y(\widetilde{H}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(F_x))$ is essentially the Steenbrink spectrum \[St3\] with information on the action of the monodromy forgotten (see \[CMSS\], Remark 3.7 for a more precise statement). This spectrum coincides with the spectrum of the complete intersection $(Y^{(0)},X)$ at a general point of $C_XY^{(0)}$ over $x\in{{\rm Sing}}\,X$ defined in \[DMS\], if $s_1,\dots,s_{r-1}$ and $s'_1,\dots,s'_r$ are sufficiently general. One can calculate $\chi_y(\widetilde{H}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(F_x))$ by taking an embedded resolution of singularities and using the theory of motivic Milnor fibers \[DL\] (which is explained in a slightly different situation in Proposition (5.7) below). We can also apply a construction of Steenbrink \[St2\] in the isolated singularity case or the theory of mixed Hodge modules as in \[BS\], Th. 4.3.
By using \[Sch1\], Prop. 5.21, we see that Corollary 2 specializes for $y=-1$ to a formula for the Chern classes in the hypersurface case which was conjectured by S. Yokura \[Yo2\] (see also \[Yo3\], \[Yo4\]), and proved by A. Parusiński and P. Pragacz \[PP\]. Note that the formulation itself of Theorem 0.2 in loc. cit.cannot be generalized to the case of Hirzebruch classes because of some monodromy problem. The equality (0.4) specializes for $y=-1$ to a formula of J. Seade and T. Suwa for the Chern classes in the case of complete intersections with isolated singularities (see \[SeSu\], \[Su\]).
Note finally that the Hirzebruch-Milnor class $M_y(X)$ can be [*calculated explicitly*]{} by using (0.2), see Section 5 below. This is quite important form the view point of applications of Theorem 1.
Part of this work was done during a visit of the second named author to Mathematical Institute of Münster University, and he thanks the institute for the financial support. The first named author is partially supported by NSF-1005338. The second named author is partially supported by Kakenhi 21540037. The third named author is supported by the SFB 878 “groups, geometry and actions”.
In Section 1 we review some basics of the Hirzebruch characteristic classes. In Section 2 we prove some properties of the Gysin morphisms which are used in later sections. In Section 3 we show some assertions in the global complete intersection case, e.g. Proposition (3.4). In Section 4 we prove Theorem 1 after showing Proposition (4.1). In Section 5 we show how the Hirzebruch-Milnor classes can be calculated explicitly.
[**Conventions.**]{}
1\. A variety means a complex algebraic variety which is always assumed reduced and irreducible except for the case of subvarieties (e.g. hypersurfaces) which may be non-reduced or reducible. We assume a variety is quasi-projective when we use \[Fu1\].
2\. Cohomology and homology groups are always with ${{\mathbf Q}}$-coefficients unless the coefficients are explicitly stated.
3\. In this paper, ${{\mathbf H}}_k(X)$, ${{\mathbf H}}^k(X)$ are respectively $H^{{{\rm BM}}}_{2k}(X,{{\mathbf Q}})$ and $H^{2k}(X,{{\mathbf Q}})$ in order to simplify the explanations. We have similar assertions for ${{\mathbf H}}_k(X):={{\rm CH}}_k(X)_{{{\mathbf Q}}}$ where ${{\mathbf H}}^k(X)$ is either the Chow cohomology in \[Fu1\] or Fulton-MacPherson’s operational Chow cohomology in \[FM\], \[Fu2\] (see \[To\] for the relation with $H^{2k}(X,{{\mathbf Q}})$).
4\. The nearby and vanishing cycle functors $\psi_f$, $\varphi_f$ for $D^b{{\rm MHM}}(X)$ are not shifted by $-1$ as in \[Sa1\], \[Sa2\]. They are compatible with the corresponding functors for the underlying ${{\mathbf Q}}$-complexes without a shift of complexes, but do not preserve mixed Hodge modules.
5\. We use [*left*]{} ${{\mathcal D}}$-modules, although right ${{\mathcal D}}$-modules are mainly used in \[Sa1\], \[Sa2\]. The transformation between filtered left and right ${{\mathcal D}}$-modules is given by $$(M,F)\mapsto(\Omega^{\dim X}_X,F)\otimes_{{{\mathcal O}}_X}(M,F),$$ where $(M,F)$ are left ${{\mathcal D}}_X$-modules, and ${{\rm Gr}}_p^F\Omega_X^{\dim X}=0$ for $p\ne-\dim X$.
**1. Hirzebruch characteristic classes**
In this section we review some basics of the Hirzebruch characteristic classes.
[**1.1. Cohomology Hirzebruch classes.**]{}
Let $X$ be a smooth complex algebraic variety of dimension $n$. The cohomology Hirzebruch characteristic class $T_y^*(TX)$ of the tangent bundle $TX$ is defined by $$T^*_y(TX):={{\hbox}{$\prod$}}_{i=1}^n\,Q_y({\alpha}_i)\in {{\mathbf H}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(X)[y],
\leqno(1.1.1)$$ where $Q_y({\alpha})$ is explained as below and the $\{{\alpha}_i\}$ are the (formal) Chern roots of $TX$, i.e. $${{\hbox}{$\prod$}}_{i=1}^n(1+{\alpha}_it)={{\hbox}{$\sum$}}_{j=0}^n\,c_j(TX)t^j.$$
We have normalized and unnormalized power series (see \[Hi\], 1.8 and \[HBJ\], 5.4.): $$Q_y({\alpha}):=\frac{{\alpha}(1+y)}{1-e^{-{\alpha}(1+y)}}-{\alpha}y,\,\,\,
{\widetilde{Q}}_y({\alpha}):=\frac{{\alpha}(1+ye^{-{\alpha}})}{1-e^{-{\alpha}}}\in
{{\mathbf Q}}[y][[{\alpha}]].
\leqno(1.1.2)$$ Their initial terms are as follows: $$Q_y(0)=1,{\quad}{\widetilde{Q}}_y(0)=1+y.
\leqno(1.1.3)$$ These two power series have the following relation $$Q_y({\alpha})=(1+y)^{-1}\,{\widetilde{Q}}_y({\alpha}(1+y)).
\leqno(1.1.4)$$
[**1.2. Homology Hirzebruch classes.**]{}
Let $X$ be a complex algebraic variety. Let ${{\rm MHM}}(X)$ be the category of mixed Hodge modules on $X$. For ${{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}}\in D^b{{\rm MHM}}(X)$, its homology Hirzebruch characteristic class is defined by $$\aligned T_{y*}({{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}})&:=td_{(1+y)*}{\bigl}({{\rm DR}}_y[{{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}}]{\bigl})\in
{{\mathbf H}}_{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(X){\bigl}[y,{\hbox}{$\frac{1}{y(y+1)}$}{\bigl}]{\quad}{\hbox}{with}\\
{{\rm DR}}_y[{{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}}]&:={{\hbox}{$\sum$}}_{i,p}\,(-1)^i\,{\bigl}[{{\mathcal H}}^i{{\rm Gr}}_F^p{{\rm DR}}({{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}}){\bigl}]
\,(-y)^p\in K_0(X)[y,y^{-1}],\raise12pt{\hbox}{ }\endaligned
\leqno(1.2.1)$$ setting $F^p:=F_{-p}$. Here we define $$td_{(1+y)*}:K_0(X)[y,y^{-1}]\to
{{\mathbf H}}_{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(X){\bigl}[y,{\hbox}{$\frac{1}{y(y+1)}$}{\bigl}]$$ to be the scalar extension of the Todd class transformation $$td_*:K_0(X)\to{{\mathbf H}}_{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(X)$$ (denoted by $\tau$ in \[BFM\]) together with the multiplication by $(1+y)^{-k}$ on the degree $k$ part (see \[BSY\]). The last multiplication is closely related to the identity (1.1.4), and we have actually by \[Sch1\], Prop. 5.21 $$T_{y*}({{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}})\in {{\mathbf H}}_{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(X)[y,y^{-1}].$$ In this paper, we set ${{\mathbf H}}_k(X):=H^{{{\rm BM}}}_{2k}(X,{{\mathbf Q}})$ to simplify the arguments, see Convention 3. In some other papers (as \[BSY\], \[CMSS\], etc.), ${{\rm DR}}_y[{{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}}]$ is denoted by $mC_y({{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}})$. (Note also that it is nontrivial to show that the ${{\mathcal H}}^i{{\rm Gr}}_F^p{{\rm DR}}({{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}})$ are ${{\mathcal O}}_X$-modules in the singular case, see \[Sa1\], Lemma 3.2.6.)
By definition we have for $k\in{{\mathbf Z}}$ $${{\rm DR}}_y[{{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}}(k)]={{\rm DR}}_y[{{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}}]\,(-y)^{-k},{\quad}T_{y*}({{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}}(k))=T_{y*}({{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}})\,(-y)^{-k},
\leqno(1.2.2)$$ where the Tate twist $(k)$ on the filtered ${{\mathcal D}}$-module part is given by the shift of the filtration $[k]$ which is defined by $$(F[k])^p:=F^{p+k},{\quad}(F[k])_p:=F_{p-k}.
\leqno(1.2.3)$$
The [*homology Hirzebruch characteristic class*]{} $T_{y*}(X)$ is defined by applying the above definition to the case ${{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}}={{\mathbf Q}}_{h,X}$ (see \[BSY\]), i.e. $$\aligned &T_{y*}(X):=T_{y*}({{\mathbf Q}}_{h,X})=td_{(1+y)*}{{\rm DR}}_y[X]\in
{{\mathbf H}}_{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(X)[y],\\
&{\hbox}{with}{\quad}{{\rm DR}}_y[X]:={{\rm DR}}_y[{{\mathbf Q}}_{h,X}].\endaligned$$ This coincides with the definition using the Du Bois complex \[DB\] by \[Sa3\], and it is known that $T_{y*}(X)$ belongs to ${{\mathbf H}}_{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(X)[y]$, see \[BSY\]. In case $X$ is smooth, we have $${{\rm DR}}_y[X]={\Lambda}_y[T^*X],
\leqno(1.2.4)$$ where ${\Lambda}_y[V]:=\sum_p[{\Lambda}^pV]\,y^p$ for a vector bundle $V$. Indeed, we have $${{\rm DR}}({{\mathbf Q}}_{h,X})={{\rm DR}}({{\mathcal O}}_X)[-n]=\Omega_X^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}{\quad}{\hbox}{with}\,\,\,
n:=\dim X,
\leqno(1.2.5)$$ and the Hodge filtration $F^p$ on $\Omega_X^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}$ is given by the truncation ${\sigma}_{\ge p}$ in \[D1\].
To show the coincidence with the above definition of $T_{y*}(X)$ in the smooth case, we need the relation (1.1.4) together with some calculation about Hirzebruch’s power series $Q_y({\alpha})$ as in \[HBJ\], Sect. 5.4 or in the proof of \[Yo1\], Lemma 2.3.7, which is closely related to the generalized Hirzebruch-Riemann-Roch theorem in \[Hi\], Th. 21.3.1. This coincidence is part of the characterization of the Hirzebruch characteristic classes for singular varieties, see \[BSY\], Th. 3.1.
By a generalization of the Riemann-Roch theorem \[BFM\], $td_*$ commutes with the pushforward under proper morphisms, and so does $T_{y*}$. If we apply this to $a_X:X\to pt$ with $X$ compact, then the pushforward for ${{\mathbf H}}_{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}$ is identified with the degree map or the trace morphism, and we have $$K_0(pt)={{\mathbf Z}},{\quad}{{\mathbf H}}_{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(pt)={{\mathbf Q}},{\quad}{{\rm MHM}}(pt)={{\rm MHS}},
\leqno(1.2.6)$$ where ${{\rm MHS}}$ is the category of graded-polarizable mixed ${{\mathbf Q}}$-Hodge structures in \[D1\]. By definition we have for $H^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}\in D^b{{\rm MHS}}$ $$T_{y*}(H^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}})=\chi_y(H^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}):=
{{\hbox}{$\sum$}}_{j,p}\,(-1)^j\dim_{{{\mathbf C}}}{{\rm Gr}}_F^pH_{{{\mathbf C}}}^j\,(-y)^p.
\leqno(1.2.7)$$ So the degree 0 part of $T_{y*}(X)$ is identified with $\chi_y{\bigl}(H^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(X){\bigl})$ if $X$ is compact and connected. Here the cohomology groups $H^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(X)$ (with the zero differential) is defined in $D^b{{\rm MHS}}$ by \[D1\], and this is compatible with \[Sa2\] by \[Sa3\] (using \[Ca\]).
[**1.3. Virtual Hirzebruch classes.**]{}
Hirzebruch also introduced the notion of virtual $\chi_y$-genus (or $\chi_y$-characteristic) which is the $\chi_y$-genus of general complete intersections $X$ in smooth projective varieties $Y$. Let $X$ be a complete intersection in a smooth projective variety $Y$. The virtual Hirzebruch characteristic class $T^{\,{{\rm vir}}}_{y*}(X)$ can be defined like the virtual genus by $$T^{\,{{\rm vir}}}_{y*}(X):=td_{(1+y)*}{{\rm DR}}^{{{\rm vir}}}_y[X]\in{{\mathbf H}}_{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(X)[y],
\leqno(1.3.1)$$ with ${{\rm DR}}^{{{\rm vir}}}_y[X]$ the image in $K_0(X)[[y]]$ of $${\Lambda}_y(T^*_{{{\rm vir}}}X)={\Lambda}_y[T^*Y|_X]/{\Lambda}_y[N^*_{X/Y}]\in K^0(X)[[y]],
\leqno(1.3.2)$$ and ${{\rm DR}}^{{{\rm vir}}}_y[X]$ belongs to $K_0(X)[y]$ by Proposition (3.4) below. Here $K^0(X)$, $K_0(X)$ are respectively the Grothendieck group of locally free sheaves of finite length and that of coherent sheaves. We denote respectively by $T^*Y$ and $N^*_{X/Y}$ the cotangent and conormal bundles, and the virtual cotangent bundle is defined by $$T^*_{{{\rm vir}}}X:=[T^*Y|_X]-[N^*_{X/Y}]\in K^0(X).$$ More precisely, $N^*_{X/Y}$ in the non-reduced case is defined by the locally free sheaf ${{\mathcal I}}_X/{{\mathcal I}}_X^2$ on $X$ where ${{\mathcal I}}_X\subset{{\mathcal O}}_Y$ is the ideal sheaf of the subvariety $X$ of $Y$. Here we set for a virtual vector bundle $V$ on $X$ in general $${\Lambda}_yV:={{\hbox}{$\sum$}}_{p\ge 0}\,{\Lambda}^pV\,y^p\in K^0(X)[[y]].
\leqno(1.3.3)$$
We can also define $T^{\,{{\rm vir}}}_{y*}(X)$ by using the virtual tangent bundle $$T_{{{\rm vir}}}X:=[TY|_X]-[N_{X/Y}]\in K^0(X),$$ together with the above cohomological transformation $T^*_y$ as in \[CMSS\] (in the hypersurface case) so that $$T^{\,{{\rm vir}}}_{y*}(X)=T^*_y(T_{{{\rm vir}}}X)\cap[X]{\quad}{\hbox}{in}\,\,\,
{{\mathbf H}}_{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(X)[y],
\leqno(1.3.4)$$ see Proposition (1.4) below. Here $[X]:=\sum_jm_j[X_j]$ is the fundamental class of $X$ with $m_j$ the multiplicities along the irreducible components $X_j$ of $X$. We have the equality $T_{y*}(X)=T^{\,{{\rm vir}}}_{y*}(X)$ if $X$ is smooth. The problem is then how to describe the difference in the singular case, which is called the [*Hirzebruch-Milnor class*]{}. (For the degree-zero part, i.e. on the level of Hodge polynomials, see also \[LM\].)
[**1.4. Proposition.**]{} [*The equality $(1.3.4)$ holds in ${{\mathbf H}}_{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(X)[[y]]$ under the assumption that $X$ is a locally complete intersection in a smooth variety $Y$ of pure codimension $r$.*]{}
[*Proof.*]{}
By the compatibility of $td_*$ with the cap product \[BFM\], we have $$td_*{\bigl}({{\rm DR}}^{{{\rm vir}}}_y[X]{\bigl})=ch(\Lambda_yT^*_{{{\rm vir}}}X)\cap td_*(X)
{\quad}{\hbox}{in}\,\,\,{{\mathbf H}}_{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(X)[[y]],$$ where ${{\rm DR}}^{{{\rm vir}}}_y[X]$ is as in (1.3.1), and $ch$ denotes also the scalar extension of the Chern character $ch:K^0(X)\to{{\mathbf H}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(X)$ under ${{\mathbf Q}}{\hookrightarrow}{{\mathbf Q}}[[y]]$. By \[Ve1\], Th. 7.1, we have $$td_*(X):=td_*([{{\mathcal O}}_X])=td_*(i^*[{{\mathcal O}}_Y])=
i^!td_*(Y)\cap td^*(N_{X/Y})^{-1},$$ and $$td_*(Y)=td^*(TY)\cap[Y].$$ Using the relation of $i^!$, $i^*$ with the cap product (see (2.3.4) below), we then get $$td_*{\bigl}({{\rm DR}}^{{{\rm vir}}}_y[X]{\bigl})=
ch(\Lambda_yT^*_{{{\rm vir}}}X)\cdot td^*(T_{{{\rm vir}}}X)\cap[X].$$
Let ${\alpha}_i$ and ${\beta}_j$ be respectively the (formal) Chern roots of $TY|_X$ and $N_{X/Y}$. Then $$ch(\Lambda_yT^*_{{{\rm vir}}}X)\cdot td^*(T_{{{\rm vir}}}X)=
\frac{{{\hbox}{$\prod$}}_{i=1}^m(1+ye^{-{\alpha}_i})\cdot{\alpha}_i/(1-e^{-{\alpha}_i})}
{{{\hbox}{$\prod$}}_{j=1}^r(1+ye^{-{\beta}_i})\cdot{\beta}_j/(1-e^{-{\beta}_j})},$$ where $m=\dim Y$ and $r=m-n$ with $n=\dim X$. Indeed, if $\ell(\gamma_j)$ denotes a (formal) line bundle class with the first Chern class $\gamma_j$ in some ring extension of $K^0(X)$, then $$ch{\bigl}(\ell(\gamma_1)\,\cdots\,\ell(\gamma_k){\bigl})=
ch{\bigl}(\ell(\gamma_1+\cdots+\gamma_k){\bigl})=e^{\gamma_1+\cdots+\gamma_k},$$ and this implies (see also \[Hi\]) $$ch(\Lambda_yT^*_{{{\rm vir}}}X)=
{{\hbox}{$\prod$}}_{i=1}^m(1+ye^{-{\alpha}_i})/{{\hbox}{$\prod$}}_{j=1}^r(1+ye^{-{\beta}_i}).$$
Since $td_{(1+y)*}$ is the composition of $td_*$ with the multiplication by $(1+y)^{-k}$ on ${{\mathbf H}}_k(X)$, and the last multiplication corresponds to the multiplication by $(1+y)^k$ on ${{\mathbf H}}^k(X)$ under the cap product, we then get the desired equality as follows: $$\aligned &{\quad}\,\,\,td_{(1+y)*}{\bigl}({{\rm DR}}^{{{\rm vir}}}_y[X]{\bigl})\\
&=\frac{{{\hbox}{$\prod$}}_{i=1}^m{\bigl}(1+ye^{-{\alpha}_i(1+y)}{\bigl})\cdot
{\alpha}_i(1+y)/(1-e^{-{\alpha}_i(1+y)})}
{{{\hbox}{$\prod$}}_{j=1}^r{\bigl}(1+ye^{-{\beta}_i(1+y)}{\bigl})\cdot
{\beta}_j(1+y)/(1-e^{-{\beta}_j(1+y)})}\cap(1+y)^{-n}[X]\\
&=\frac{{{\hbox}{$\prod$}}_{i=1}^m{\bigl}({\widetilde{Q}}_y({\alpha}_i(1+y))/(1+y){\bigl})}
{{{\hbox}{$\prod$}}_{j=1}^r{\bigl}({\widetilde{Q}}_y({\beta}_j(1+y))/(1+y){\bigl})}\cap[X]\\
&=T^*_y{\bigl}([TY|_X]-[N_{X/Y}]{\bigl})\cap[X]\\
&=T^*_y(T_{{{\rm vir}}}X)\cap[X],\endaligned$$ where the relation (1.1.4) between the power series $Q_y$ and ${\widetilde{Q}}_y$ is used, see also \[BSY\], (1.1) and \[Yo1\], Lemma 2.3.7. This finishes the proof of Proposition (1.4).
**2. Gysin morphisms**
In this section we prove some properties of the Gysin morphisms which are used in later sections.
[**2.1. Construction.**]{}
Let $i:X{\hookrightarrow}Y$ be a locally complete intersection morphism of pure codimension $r$. By \[Fu2\], \[Ve1\] there are Gysin morphisms $$i^!:{{\rm CH}}_k(Y)\to{{\rm CH}}_{k-r}(X),{\quad}i^!:H^{{{\rm BM}}}_{2k}(Y)\to
H^{{{\rm BM}}}_{2k-2r}(X)(r),
\leqno(2.1.1)$$ in a compatible way with the cycle map. These are constructed by using the deformation to the normal cone $q:Z\to S:={{\mathbf C}}$ such that $q^{-1}(0)=N_{X/Y}$ and $q^{-1}(S')=Y{{\hbox}{$\times$}}S'$ where $S':={{\mathbf C}}^*$. More precisely, $Z$ is the complement of the proper transform of $Y{{\hbox}{$\times$}}\{0\}$ in the blow-up of $Y{{\hbox}{$\times$}}S$ along $X{{\hbox}{$\times$}}\{0\}$, see loc. cit.
We recall here the definition of the Gysin morphism for Borel-Moore homology (see \[Ve1\]). We have the isomorphisms $$(R^kq_{!\,}{{\mathbf Q}}_Z)_0=H^k_c(N_{X/Y}),{\quad}R^kq_{!\,}{{\mathbf Q}}_Z|_{S'}=a_{S'}^{-1}H^k_c(Y),$$ where $a_{S'}:S'\to pt$ is the natural morphism. So we get the cospecialization morphism $${{\rm sp}}^*:H^k_c(N_{X/Y})\to H^k_c(Y).
\leqno(2.1.3)$$ Taking the dual, we get the Gysin morphism $$i^!:H^{{{\rm BM}}}_k(Y)\buildrel{{{\rm sp}}_*}\over\longrightarrow
H^{{{\rm BM}}}_k(N_{X/Y})\buildrel{\sim}\over\leftarrow H^{{{\rm BM}}}_{k-2r}(X)(r),
\leqno(2.1.4)$$ where the last isomorphism is induced by the pullback under the canonical morphism $\pi_N:N_{X/Y}\to X$.
The above definition implies the compatibility with the pushforward under a proper morphism $\rho:{\widetilde{Y}}\to Y$ such that ${\widetilde{X}}=X{{\hbox}{$\times$}}_Y{\widetilde{Y}}$ has codimension $r$ everywhere in ${\widetilde{Y}}$. In this case we have the cartesian diagram $$\begin{matrix}{\widetilde{X}}&\buildrel{{\tilde{i}}}\over{\hookrightarrow}&{\widetilde{Y}}\\
\,\,\,\,\downarrow\!{\scriptstyle\rho'}&&
\,\,\,\downarrow\!{\scriptstyle\rho}\\
X&\buildrel{i}\over{\hookrightarrow}&Y\end{matrix}$$ and $$i^!{\,\raise.15ex{\hbox}{${\scriptstyle\circ}$}\,}\rho_*=\rho'_*{\,\raise.15ex{\hbox}{${\scriptstyle\circ}$}\,}{\tilde{i}}^{\,!}.
\leqno(2.1.5)$$ Indeed, the specialization morphism ${{\rm sp}}_*$ in (2.1.4) is compatible with the pushforward by proper morphisms, and $N_{{\widetilde{X}}/{\widetilde{Y}}}\to{\widetilde{X}}$ is the base change of $N_{X/Y}\to X$ by the hypothesis.
[**2.2. Sheaf-theoretic description**]{}
\[Ve1\]. With the above notation, set $Z_s:=q^{-1}(s)$, and let $i_s:Z_s{\hookrightarrow}Z$ denote the inclusion for $s\in S$. We have $$Z_0=N_{X/Y},{\quad}Z_s=Y\,(s\ne 0).
\leqno(2.2.1)$$ Let $p:Z\to Y$ be the canonical projection. Set $p_s:=p{\,\raise.15ex{\hbox}{${\scriptstyle\circ}$}\,}i_s:Z_s\to Y$.
For $s\in S$ we have the canonical flasque resolutions $${{\mathbf Q}}_Z\buildrel{\rm qi}\over\longrightarrow{{\mathcal I}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}_Z,{\quad}{{\mathbf Q}}_{Z_s}\buildrel{\rm qi}\over\longrightarrow{{\mathcal I}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}_{Z_s},$$ together with the canonical morphisms $${{\mathcal I}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}_Z\to(i_s)_*{{\mathcal I}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}_{Z_s},
\leqno(2.2.2)$$ which are compatible with ${{\mathbf Q}}_Z\to(i_s)_*{{\mathbf Q}}_{Z_s}$. Indeed, $(i_s)_*{{\mathcal I}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}_{Z_s}$ is naturally isomorphic to the canonical flasque resolution of $(i_s)_*{{\mathbf Q}}_{Z_s}$, and this resolution is functorial for morphisms of sheaves. The dualizing complex ${{\mathbf D}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}_Z$ is defined by $${{\mathbf D}}^k_Z(V):={{\rm Hom}}({\Gamma}_c(V,{{\mathcal I}}^{-k}_Z),{{\mathbf Q}}){\quad}{\hbox}{for open}\,\,\,
V\subset S,$$ and similarly for ${{\mathbf D}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}_{Z_s}$. Then we have by (2.2.2) the canonical morphisms $$(i_s)_*{{\mathbf D}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}_{Z_s}\to{{\mathbf D}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}_Z.$$ They induce the canonical morphisms $${\Gamma}(Z_s,{{\mathbf D}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}_{Z_s})\to{\Gamma}_c(S,q_*{{\mathbf D}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}_Z),
\leqno(2.2.3)$$ and this is a quasi-isomorphism for $s=0$. We thus get the canonical morphisms of complexes $${\Gamma}(Y,{{\mathbf D}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}_Y)={\Gamma}(Z_1,{{\mathbf D}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}_{Z_1})\to
{\Gamma}_c(S,q_*{{\mathbf D}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}_Z)]\buildrel{\rm qi}\over\longleftarrow
{\Gamma}(Z_0,{{\mathbf D}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}_{Z_0})={\Gamma}(N_{X/Y},{{\mathbf D}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}_{N_{X/Y}}),
\leqno(2.2.4)$$ where qi means an quasi-isomorphism.
For any open subset $U\subset Y$, set $$X_U:=X\cap U,{\quad}Z_U:=p^{-1}(U),{\quad}Z_{U,s}:=Z_U\cap Z_s,{\quad}q_U:=q|_{Z_U}:Z_U\to S.$$ Then $$q_U^{-1}(0)=N_{X_U/U},{\quad}q_U^{-1}(s)=U\,\,\,(s\ne 0),$$ and (2.2.4) implies the canonical morphisms of complexes $${\Gamma}(U,{{\mathbf D}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}_U)\to{\Gamma}_c(S,(q_U)_*{{\mathbf D}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}_{Z_U})
\buildrel{\rm qi}\over\longleftarrow
{\Gamma}(N_{X_U/U},{{\mathbf D}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}_{N_{X_U/U}}).
\leqno(2.2.5)$$ Here we have presheaves defined by associating to $U$ respectively $${\Gamma}_c(S,(q_U)_*{{\mathbf D}}^k_{Z_U}),{\quad}{\Gamma}(N_{X_U/U},{{\mathbf D}}^k_{N_{X_U/U}}),$$ and these are sheaves. So we get the canonical morphisms of sheaf complexes on $Y$, and this gives the morphism in the derived category $D^b_c(Y,{{\mathbf Q}}_Y)$ $${{\mathbf D}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}_Y\to(i{\,\raise.15ex{\hbox}{${\scriptstyle\circ}$}\,}\pi_N)_*{{\mathbf D}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}_{N_{X/Y}}
\buildrel{\sim}\over\leftarrow i_*{{\mathbf D}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}_X(r)[2r],
\leqno(2.2.6)$$ where $\pi_N:N_{X/Y}\to X$ denotes the canonical projection and the last quasi-isomorphism is given by $\pi_N^*$. So we get the sheaf-theoretic description of the Gysin morphism in (2.1). (This is a detailed version of a slight modification of a sketch of an argument in \[Ve1\], Sect. 8 where a point of $S'={{\mathbf C}}\setminus\{0\}$ was not chosen as in the above argument. We may assume that this point is $1$ by using an equivariant ${{\mathbf C}}^*$-action on $q:Z\to S$.)
[**2.3. Compatibility with the cap product.**]{}
Let $\zeta\in H^j(Y,{{\mathbf Q}})$. By the canonical isomorphism $$H^j(Y,{{\mathbf Q}})={{\rm Hom}}({{\mathbf Q}}_Y,{{\mathbf Q}}_Y[j]),
\leqno(2.3.1)$$ together with ${{\mathbf D}}_Y={{\mathbf Q}}_Y\otimes{{\mathbf D}}_Y$, we get the morphism $$\zeta\,\cap:H^{{{\rm BM}}}_k(Y,{{\mathbf Q}})\to H^{{{\rm BM}}}_{k-j}(Y,{{\mathbf Q}}).
\leqno(2.3.2)$$ So the above sheaf-theoretic construction of the Gysin morphism implies the compatibility with the cap product $$\begin{matrix}H^{{{\rm BM}}}_k(Y,{{\mathbf Q}})&\buildrel{i^!}\over\to&
H^{{{\rm BM}}}_{k-2r}(X,{{\mathbf Q}})(r)\\
{\quad}\downarrow{\!\scriptstyle{\zeta\,\cap}}&&
{\quad}\downarrow{\!\scriptstyle{i^*\zeta\,\cap}}\\
H^{{{\rm BM}}}_{k-j}(Y)&\buildrel{i^!}\over\to&H^{{{\rm BM}}}_{k-j-2r}(X)(r)\end{matrix}
\leqno(2.3.3)$$ i.e. $$i^!(\zeta\cap\xi)=i^*\zeta\cap i^!\xi{\quad}{\hbox}{for}\,\,\,\zeta\in
H^j(Y,{{\mathbf Q}}),\,\,\xi\in H^{{{\rm BM}}}_k(Y).
\leqno(2.3.4)$$ Since $i^![Y]=[X]$, this implies the commutative diagram $$\begin{matrix}H^{2n-k}(Y)(n)&\buildrel{i^*}\over\to&H^{2n-k}(X)(n)\\
{\quad}\downarrow{\!\scriptstyle{\cap[Y]}}&&
{\quad}\downarrow{\!\scriptstyle{\cap[X]}}\\
H^{{{\rm BM}}}_k(Y)&\buildrel{i^!}\over\to&H^{{{\rm BM}}}_{k-2r}(X)(r)\end{matrix}
\leqno(2.3.5)$$ This is used in the proof of \[CMSS\], Lemma 3.1 in the hypersurface case (i.e. $r=1$).
[**2.4. Principal divisor case.**]{}
Assume $X$ is a globally principal divisor on $Y$, i.e. there is a function on $Y$ with $X=f^{-1}(0)$ and $r=1$. Then the Gysin morphism induces a ‘well-defined’ morphism $$i^!:{{\rm CH}}_k(Y\setminus X)\to{{\rm CH}}_{k-1}(X),
\leqno(2.4.1)$$ using the exact sequence $${{\rm CH}}_k(X)\buildrel{i_*}\over\to{{\rm CH}}_k(Y)\buildrel{j^*}\over\to
{{\rm CH}}_k(Y\setminus X)\to 0,$$ where ‘well-defined’ means that $i^!:{{\rm CH}}_k(Y)\to{{\rm CH}}_{k-1}(X)$ factors through the surjection $j^*$. Indeed, the composition $i^!{\,\raise.15ex{\hbox}{${\scriptstyle\circ}$}\,}i_*$ vanishes by the triviality of the normal bundle.
By \[Ve1\], Th. 7.1, we have $$td_*{\,\raise.15ex{\hbox}{${\scriptstyle\circ}$}\,}i^*=i^!{\,\raise.15ex{\hbox}{${\scriptstyle\circ}$}\,}td_*.
\leqno(2.4.2)$$ where $i^*:K_0(Y)\to K_0(X)$ is the pull-back of the Grothendieck groups of coherent sheaves defined by the mapping cone of the multiplication by the global defining function $f$ of $X$. Using the exact sequence $$K_0(X)\buildrel{i_*}\over\to K_0(Y)\buildrel{j^*}\over\to
K_0(Y\setminus X)\to 0,$$ the last definition implies that the composition $i^*{\,\raise.15ex{\hbox}{${\scriptstyle\circ}$}\,}i_*$ vanishes so that $i^*$ induces also a ‘well-defined’ morphism $$i^*:K_0(Y\setminus X)\to K_0(X).
\leqno(2.4.3)$$
We have a similar property for the nearby cycle functor $\psi_f$ with $f$ a global defining function of $X$, and we have the ‘well-defined’ functor $$\psi_f[-1]:{{\rm MHM}}(Y\setminus X)\to{{\rm MHM}}(X).
\leqno(2.4.4)$$ Indeed, the restriction gives the surjection ${{\rm MHM}}(Y)\to{{\rm MHM}}(Y\setminus X)$ by extendability of mixed Hodge modules in the algebraic case, and $\psi_f{{\mathcal M}}'[-1]$ is independent of the extension ${{\mathcal M}}'$ of ${{\mathcal M}}$ to $Y$ by using the functorial morphism $id\to j_*j^*$, since $\psi_fM''[-1]$ vanishes if ${\rm supp}\,M''\subset X$.
As for Borel-Moore cohomology, we have the long exact sequence $$\cdots\to H^{{{\rm BM}}}_k(X)\buildrel{i_*}\over\to H^{{{\rm BM}}}_k(Y)
\buildrel{j^*}\over\to H^{{{\rm BM}}}_k(Y\setminus X)\to H^{{{\rm BM}}}_{k-1}(X)
\to\cdots,
\leqno(2.4.5)$$ and the next proposition shows the vanishing of the composition $$H^{{{\rm BM}}}_k(X)\buildrel{i_*}\over\to H^{{{\rm BM}}}_k(Y)
\buildrel{i^!}\over\to H^{{{\rm BM}}}_{k-2}(X)(1).
\leqno(2.4.6)$$
[**2.5. Proposition.**]{} [*With the above notation and the assumptions, the composition of the two morphisms in $(2.4.6)$ vanishes.*]{}
[*Proof.*]{}
By the construction in (2.1), the restriction of ${{\rm sp}}_*$ to the image of $i_*$ is essentially the identity. More precisely, we have $${{\rm sp}}_*{\,\raise.15ex{\hbox}{${\scriptstyle\circ}$}\,}i_*=(s_0)_*:H^{{{\rm BM}}}_k(X)\to H^{{{\rm BM}}}_k(N_{X/Y}),
\leqno(2.5.1)$$ where $s_0:X{\hookrightarrow}N_{X/Y}$ is the zero-section. We have moreover $$s_0^!=(\pi_N^*)^{-1}:H^{{{\rm BM}}}_k(N_{X/Y}){\buildrel\sim\over\longrightarrow}H^{{{\rm BM}}}_{k-2}(X)(1),
\leqno(2.5.2)$$ where $\pi_N:N_{X/Y}\to X$ is the projection. Indeed, we have $$s_0^!{\,\raise.15ex{\hbox}{${\scriptstyle\circ}$}\,}\pi_N^*=id:H^{{{\rm BM}}}_k(X)\to H^{{{\rm BM}}}_k(X),
\leqno(2.5.3)$$ since the deformation to the normal cone is identified with a trivial deformation.
So the assertion is reduced to $$s_0^!{\,\raise.15ex{\hbox}{${\scriptstyle\circ}$}\,}(s_0)_*=0.
\leqno(2.5.4)$$ Since the normal bundle $N_{X/Y}$ is trivial, we have a section $s_1$ whose image is disjoint from that of $s_0$, and $s_1^!{\,\raise.15ex{\hbox}{${\scriptstyle\circ}$}\,}\pi_N^*=id$ by the same argument as above. So the assertion is further reduced to $$s_1^!{\,\raise.15ex{\hbox}{${\scriptstyle\circ}$}\,}(s_0)_*=0.
\leqno(2.5.5)$$ This is more or less trivial by the construction of $s_1^!$ since the images of $s_0$ and $s_1$ are disjoint. (The reader can also use (2.1.5) together with the fact that the codimension of an empty set can be any number.) This finishes the proof of Proposition (2.5).
By a similar argument, we have the following.
[**2.6. Lemma.**]{} [*Let $X$ be a complex algebraic variety. Let $i_t:X=X\times\{t\}{\hookrightarrow}X\times{{\mathbf C}}$ denote the inclusion for $t\in{{\mathbf C}}$. Then $i_t^!:{{\mathbf H}}_{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(X\times{{\mathbf C}})\to{{\mathbf H}}_{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}-1}(X)$ is independent of $t\in{{\mathbf C}}$.*]{}
[*Proof.*]{}
Let $Y:=X\times{{\mathbf C}}$ so that $Y=N_{X/Y}$ where $X=X\times\{t\}\subset Y$. Then the assertion follows from (2.5.2).
**3. Global complete intersections**
In this section we show some assertions in the global complete intersection case, e.g. Proposition (3.4). We first give proofs of the following lemma and proposition for the convenience of the reader. These are related to the independence of the choice of the sections $s_i$. They should be well-known to specialists.
[**3.1. Lemma.**]{} [*Let $L$ be a line bundle on a smooth variety $X$, and $s_i\in{\Gamma}(X,L)$ for $i\in[1,m]$. Assume $\bigcap_{i\in[1,m]}s_i^{-1}(0)=\emptyset$. Then there is a non-empty Zariski-open subset $U$ of ${{\mathbf P}}^{m-1}$ such that the zero locus of $\sum_{i\in[1,m]}t_is_i$ in $X$ is smooth for any $t=(t_1,\dots,t_m)\in U$.*]{}
[*Proof.*]{}
Consider $$Z:={\bigl}\{(y,t)\in X\times{{\mathbf P}}^{m-1}\,\big|\,
{{\hbox}{$\sum$}}_{i\in[1,m]}\,t_is_i(y)=0{\bigl}\}.$$ By the Sard-type theorem, it is enough to show that $Z$ is smooth. Set $$U_j:=\{t_j\ne 0\}\subset{{\mathbf P}}^{m-1},{\quad}X_i:=X\setminus s_i^{-1}(0).$$ Then $X=\bigcup_iX_i$, and $Z\cap{\bigl}(X_i{{\hbox}{$\times$}}{{\mathbf P}}^{m-1}{\bigl})$ is covered by $X_i\times U_j$ with $i\ne j$ by the definition of $Z$. Moreover, $Z\cap{\bigl}(X_i\times U_j{\bigl})$ is defined by $$t_i/t_j=-{{\hbox}{$\sum$}}_{k\ne i}\,(t_k/t_j)(s_k(y)/s_i(y))
{\quad}{\hbox}{in}\,\,\,X_i\times U_j,$$ where the $t_k/t_j\,(k\ne j)$ are the affine coordinates of $U_j$. So the assertion follows.
[**3.2. Proposition.**]{} [*Let $L$ be a very ample line bundle on a projective variety $Y$. Let $s_i\in{\Gamma}(Y,L^{\otimes a_i})\,(i=1,\dots,r)$ with $a_i$ a decreasing sequence of positive integers where $r\ge 2$. Set $Y^{(j)}:=\bigcap_{i\in[1,j]}s_i^{-1}(0)\,(j=1,\dots,r)$. Assume $Y\setminus Y^{(r)}$ is smooth and ${\rm codim}_YY^{(r)}=r$. Then, the $Y^{(j)}\setminus Y^{(r)}$ are smooth for any $j\in[1,r-1]$ by replacing $s_i$ with $s_i+{{\hbox}{$\sum$}}_{i'>i}\,s'_{i,i'}s_{i'}$ if we choose the $s'_{i,i'}\in{\Gamma}(Y,L^{\otimes(a_i-a_{i'})})$ generically.*]{}
[*Proof.*]{}
We proceed by increasing induction on $r\ge 2$. Consider the embedding $Y{\hookrightarrow}{{\mathbf P}}^N$ by the line bundle $L$. Let $\{\tau^{(i)}_k\}_{k\in[1,\nu_i]}$ be a basis of ${\Gamma}(Y,L^{\otimes i})$. We apply Lemma (3.1) to the case where $X=Y\setminus Y^{(r)}$, $m=\sum_{i=1}^r\nu_{a_1-a_i}$, and the $s_j\,(j\in[1,m])$ in Lemma (3.1) are given by the restrictions of $$\tau^{(a_1-a_i)}_ks_i{\quad}(k\in[1,\nu_{a_1-a_i}],\,i\in[1,r]).$$ Note that the intersection of the zero loci of $\tau^{(a_1-a_i)}_ks_i\,(k\in[1,\nu_{a_1-a_i}])$ coincides with $s_i^{-1}(0)$ for each $i\in[1,r]$. We may assume that the coefficient of $s_1$ does not vanish since this condition defines a non-empty Zariski-open subset of ${{\mathbf P}}^{m-1}$ (and the intersection of any two non-empty Zariski-open subsets of ${{\mathbf P}}^{m-1}$ is non-empty). So we get the assertion for $j=1$, and hence for $r=2$. Moreover, the assertion for $r\ge 3,j\ge 2$ is reduced to the assertion with $Y$ replaced by $Y^{(1)}$, and $(r,j)$ by $(r-1,j-1)$. Thus we can proceed by induction on $r$. This finishes the proof of Proposition (3.2).
The following was shown in \[Sch2\] without assuming the condition that the variety $Y$ is embeddable into a smooth variety. We give here a simplified proof assuming this condition. This proposition will be used in the proof of Proposition (3.4) below.
[**3.3. Proposition**]{}
(\[Sch2\]). [*Let $f:Y\to{{\mathbf C}}$ be a non-constant function on a complex algebraic variety. Assume $Y$ is embeddable into a smooth variety. Set $Y_0:=f^{-1}(0)$ with the inclusion $i_0:Y_0{\hookrightarrow}Y$. Then, for any bounded complex of mixed Hodge modules ${{\mathcal M}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}$ on $Y$, we have $$\aligned i_0^*{{\rm DR}}_y[{{\mathcal M}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}]&={{\rm DR}}_y[\psi_f{{\mathcal M}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}](1+y)
{\quad}{\hbox}{in}\,\,\,K_0(Y_0)[y,y^{-1}],\\
i_0^!T_{y*}({{\mathcal M}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}})&=T_{y*}(\psi_f{{\mathcal M}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}})
{\quad}{\hbox}{in}\,\,\,{{\mathbf H}}_{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}(Y_0)[y,y^{-1}],\endaligned
\leqno(3.3.1)$$ where $\psi_f{{\mathcal M}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}$ is viewed as a complex of mixed Hodge modules on $Y_0$.*]{}
[*Proof.*]{}
It is enough to show the first equality since the second equality follows from it using \[Ve1\], Th. 7.1. (Here the term $(1+y)$ disappears since $i_0^!$ send ${{\mathbf H}}_k(Y)$ to ${{\mathbf H}}_{k-1}(Y_0)$.) We may assume that ${{\mathcal M}}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}$ is a mixed Hodge module ${{\mathcal M}}$ on $Y$. By assumption there is a closed embedding $Y{\hookrightarrow}Y'$ with $Y'$ smooth. Set $$Z:=Y\times{{\mathbf C}}\subset Z':=Y'\times{{\mathbf C}}.$$ Let $i_f:Y{\hookrightarrow}Z$ be the graph embedding, and $(M,F)$ be the underlying filtered left ${{\mathcal D}}_{Z'}$-module of the direct image of ${{\mathcal M}}$ by $i_f$, see \[Sa1\]. It has an increasing filtration $V$ of Kashiwara and Malgrange such that $\partial_tt-a$ is nilpotent on ${{\rm Gr}}^a_VM$. (In this paper we use left ${{\mathcal D}}$-modules, and $V^a$ corresponds to $V_{-a}$ in loc. cit., see Convention 5.)
Setting $Z'{}^*:=Y'{{\hbox}{$\times$}}{{\mathbf C}}^*$, we have $V^aM|_{Z'{}^*}=M|_{Z'{}^*}$ for any $a\in{{\mathbf Q}}$. (Indeed, for any $m\in M$, we have $t^im\in V^aM$ for $i\gg a$ and $t$ is invertible on $Z'{}^*$). Using the well-definedness of (2.4.3), the left-hand side of (3.3.1) can be given by applying $i_0^*$ to the mapping cone $$C{\bigl}({{\rm Gr}}_F^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}{{\rm DR}}_{Z'/{{\mathbf C}}}(V^aM)\buildrel{{{\rm Gr}}\,\partial_t}\over{\longrightarrow}{{\rm Gr}}_F^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}-1}{{\rm DR}}_{Z'/{{\mathbf C}}}(V^bM){\bigl}),
\leqno(3.3.2)$$ and considering it in $K_0(Y_0)[y,y^{-1}]$ (using \[Sa1\], Lemma 3.2.6). Here $a,b$ can be any rational numbers satisfying $$a-1\ge b>0.
\leqno(3.3.3)$$ (This implies that \[Sa1\], (3.2.1.3) is not needed in \[Sch2\].) By \[Sa1\], (3.2.1.2), we have the isomorphisms $$t:V^aM{\buildrel\sim\over\longrightarrow}V^{a+1}M{\quad}{\hbox}{for any}\,\,\,a>0.
\leqno(3.3.4)$$ So the left-hand side of (3.3.1) is given by $$-{{\rm DR}}_y[(V^a/V^{a+1})(M,F)]+{{\rm DR}}_y[(V^b/V^{b+1})(M,F[-1])],
\leqno(3.3.5)$$ where $(V^a/V^{a+1})(M,F)$ and $(V^b/V^{b+1})(M,F)$ are viewed as filtered left ${{\mathcal D}}_{Y'}$-module. (For the shift of filtration $[-1]$, see (1.2.3).)
As for the right-hand side of (3.3.1), we have by definition $$\psi_t(M,F)={{\hbox}{$\bigoplus$}}_{a\in(0,1]}{{\rm Gr}}_V^a(M,F)[1],
\leqno(3.3.6)$$ where the shift of complex by 1 comes from Convention 4. So we get (3.3.1) by using (1.2.2) and (3.3.4). This finishes the proof of Proposition (3.3).
[**3.4. Proposition.**]{} [*Let ${{\mathcal M}}'(s'_1,\dots,s'_r)$ and $\Lambda_y(T^*_{{{\rm vir}}}X)$ be as in the introduction and $(1.3)$ respectively. Then $${{\rm DR}}_y[{{\mathcal M}}'(s'_1,\dots,s'_r)]=(-1)^{\dim X}\Lambda_y(T^*_{{{\rm vir}}}X)
{\quad}{\hbox}{in}\,\,\,K_0(X)[[y]][y^{-1}].$$ Hence $\Lambda_y(T^*_{{{\rm vir}}}X)\in
K_0(X)[y]\,{\bigl}(=K_0(X)[y,y^{-1}]\cap K_0(X)[[y]]{\bigl})$, and*]{}
$$T_{y*}{\bigl}({{\mathcal M}}'(s'_1,\dots,s'_r){\bigl})=
(-1)^{\dim X}T^{\,{{\rm vir}}}_{y*}(X).$$
[*Proof.*]{}
It is enough to show the first equality, since the last assertion follows from it by \[Ve1\], Th. 7.1 applied to the inclusion $X{\hookrightarrow}{{\mathcal Z}}_T$. In the notation of the introduction, set $$T_j:=\{t_k=0\,\,(k\le j)\}\subset T,{\quad}{{\mathcal Z}}_j:={{\mathcal Z}}_T\times_TT_j,$$ with the inclusions $i_j:{{\mathcal Z}}_j{\hookrightarrow}{{\mathcal Z}}_{j-1}$. Let $i_X:X{\hookrightarrow}{{\mathcal Z}}_T$ denote the composition of the $i_j$ for $j\in[1,r]$. Set $n:=\dim X$. Applying Proposition (3.3) to the $i_j$, we get $$(-1)^n{{\rm DR}}_y[{{\mathcal M}}'(s'_1,\dots,s'_r)]=
i^*_X{{\rm DR}}_y[{{\mathbf Q}}_{h,{{\mathcal Z}}_T}](1+y)^{-r}.
\leqno(3.4.1)$$
Let $U$ be a non-empty Zariski-open subset of $T$ such that ${{\mathcal Z}}_U:={{\mathcal Z}}_T\times_TU$ is smooth over $U$. Let $T^*({{\mathcal Z}}_U/U)$ denote the relative cotangent bundle. Since $T^*U$ is trivial, we have $$\aligned &{{\rm DR}}_y[{{\mathbf Q}}_{h,{{\mathcal Z}}_U}](1+y)^{-r}=
\Lambda_y[T^*{{\mathcal Z}}_U](1+y)^{-r}\\
&=\Lambda_y{\bigl}[T^*({{\mathcal Z}}_U/U){\bigl}]=
i_{{{\mathcal Z}}_U}^*{\,\raise.15ex{\hbox}{${\scriptstyle\circ}$}\,}pr_1^*{\bigl}(\Lambda_y[T^*Y^{(0)}]/
{{\hbox}{$\prod$}}_{j=1}^r\Lambda_y[L_j^*]{\bigl}),\endaligned
\leqno(3.4.2)$$ where $i_{{{\mathcal Z}}_U}:{{\mathcal Z}}_U{\hookrightarrow}Y^{(0)}\times U$ and $pr_1:Y^{(0)}\times U\to Y^{(0)}$ are natural morphisms, and $L_j^*$ is the dual of a very ample line bundle $L_j$ such that $s_j\in{\Gamma}(Y^{(0)},L_j)$. (Here it is not necessary to assume that $L_j=L^{\otimes a_j}$.)
In order to apply inductively (2.4.3) and (2.4.4), we have to take Zariski-open subsets $U'_j$, $U_j$ of $T_j\,\,(j\in[0,r-1])$ satisfying the two conditions $$U'_j\setminus T_{j+1}=U_j\setminus T_{j+1},{\quad}U'_{j+1}:=U_j\cap T_{j+1}\ne\emptyset,
\leqno(3.4.3)$$ by increasing induction on $j\in[0,r-1]$, where $U'_r=T_r=\{0\}$, and (2.4.3), (2.4.4) will be applied to the base changes of the inclusions $$U'_{j+1}=U_j\cap T_{j+1}{\hookrightarrow}U_j.$$ If $j=0$, we set $U'_0:=U$. In general, if $U'_j$ is given, then $U_j$ can be any Zariski-open subset of $T_j$ satisfying the two conditions in (3.4.3). Here a canonical choice would be the maximal one satisfying the two conditions, i.e. $$U_j:=T_j\setminus\overline{(T_j\setminus(U'_j\cup T_{j+1}))}.$$ This means that $T_j\setminus U_j$ is the union of the irreducible components of $T_j\setminus U'_j$ which are not contained in $T_{j+1}$.
Consider now the inclusions $${{\mathcal Z}}_{U'_{j+1}}{\hookrightarrow}{{\mathcal Z}}_{U_j},
\leqno(3.4.4)$$ which are obtained by the base change of the inclusions $U'_{j+1}{\hookrightarrow}U_j{\hookrightarrow}U$ by ${{\mathcal Z}}_U\to U$ for $j\in[0,r-1]$. The assertion then follows from the ‘well-definedness’ of (2.4.3) and (2.4.4) (as is explained in (2.4)) which we apply to the inclusions (3.4.4) inductively. This finishes the proof of Proposition (3.4).
**4. Proof of the main theorem**
In this section we prove Theorem 1 after showing Proposition (4.1) below. A proof of the latter has been presented in the proof of \[PP\], Prop. 5.1 in case $Y$ is smooth, where they demonstrated the contractibility of the Milnor fiber using the integration of a controlled vector field as in \[Mi\]. We give a sheaf-theoretic proof using an embedded resolution.
[**4.1. Proposition.**]{} [*Let $X$ be a hypersurface of a projective variety $Y$ such that $X$ is a very ample divisor, and $Y\setminus X$ is smooth. Let $X'$ be a sufficiently general member of the linear system associated with $X$ so that we have a one-parameter family ${{\mathcal Y}}=\coprod_{c\in{{\mathbf P}}^1}Y_c$ with $Y_0=X$, $Y_{\infty}=X'$. Here ${{\mathbf P}}^1$ is identified with ${{\mathbf C}}\cup\{\infty\}$ by the inhomogeneous coordinate $t$ of ${{\mathbf P}}^1$. Let $\pi:{{\mathcal Y}}\to{{\mathbf P}}^1$ be the projection. Set $X'':=X\cap X'$. Then*]{}
$$\varphi_{\pi^*t}{{\mathbf Q}}_{{{\mathcal Y}}}|_{X''}=0,{\quad}{\hbox}{\it i.e.}{\quad}\psi_{\pi^*t}{{\mathbf Q}}_{{{\mathcal Y}}}|_{X''}={{\mathbf Q}}_{X''}.$$
[*Proof.*]{}
We first treat the following simple case.
[**(a) Normal crossing case.**]{}
Assume that $Y$, $X'$ are smooth, $X$ is a divisor with simple normal crossings, and $X'$ transversally intersects any intersections of irreducible components of $X$. (Here it is not necessary to assume that $X$ is a very ample divisor.) Let $g$ be a local equation of $X$ around $x\in X''=X\cap X'$. Let $y_1,\dots,y_n$ be local coordinates of $Y$ such that locally $$g={{\hbox}{$\prod$}}_{i=1}^r\,y_i^{m_i},{\quad}X'=\{y_n=0\},$$ where $r<n:=\dim Y$ and $m_i\ge 1\,(i\in[1,r])$. Then we have locally $${{\mathcal Y}}=\{g=y_nt\}\subset Y{{\hbox}{$\times$}}{{\mathbf C}},$$ where $t$ is identified with the affine coordinate of ${{\mathbf C}}\subset{{\mathbf P}}^1$. So the Milnor fiber of $\pi:{{\mathcal Y}}\to{{\mathbf P}}^1$ around $x\in X\cap X'\subset\pi^{-1}(0)$ is given by $${\bigl}\{(y_1,\dots,y_n)\in{{\mathbf C}}^n\,\big|\,g=y_nt,\,{{\hbox}{$\sum$}}_{i=1}^n\,
|y_i|^2<{\varepsilon}^2{\bigl}\}.$$ where $0<|t|\ll{\varepsilon}\ll 1$. This is identified with $${\bigl}\{(y_1,\dots,y_{n-1})\in{{\mathbf C}}^{n-1}\,\big|\,\,{{\hbox}{$\sum$}}_{i=1}^{n-1}\,
|y_i|^2+|t|^{-2}{{\hbox}{$\prod$}}_{i=1}^r\,|y_i|^{2m_i}<{\varepsilon}^2{\bigl}\}.$$ This set is contractible by using the natural action of $\lambda\in[0,1]$ defined by $$\lambda(y_1,\dots,y_{n-1})=(\lambda y_1,\dots,\lambda y_{n-1}).$$ So the assertion in the normal crossing case follows.
[**(b) General case.**]{}
Let ${\sigma}:({\widetilde{Y}},{\widetilde{X}})\to(Y,X)$ be an embedded resolution of singularities such that ${\widetilde{X}}:={\sigma}^{-1}(X)$ is a divisor with simple normal crossings. By assumption, ${\widetilde{X}}':={\sigma}^{-1}(X')$ is smooth and transversally intersects any intersections of irreducible components of ${\widetilde{X}}$. So ${\widetilde{X}}'$ is the total transform of $X'$, and we get the one-parameter family $${\widetilde{\mathcal Y}}:={{\hbox}{$\coprod$}}_{c\in{{\mathbf P}}^1}\,{\widetilde{Y}}_c{\quad}{\hbox}{with}{\quad}{\widetilde{Y}}_0={\widetilde{X}},\,\,
{\widetilde{Y}}_{\infty}={\widetilde{X}}'.$$ Set $${{\mathcal Y}}_U:={{\hbox}{$\coprod$}}_{c\in U}\,Y_c,{\quad}{\widetilde{\mathcal Y}}_U:={{\hbox}{$\coprod$}}_{c\in U}\,{\widetilde{Y}}_c{\quad}{\hbox}{with}{\quad}U:={{\mathbf P}}^1\setminus\{0\}.$$ Let $\pi_U:{{\mathcal Y}}_U\to{{\mathbf P}}^{1}$, ${\widetilde{\pi}}_U:{\widetilde{\mathcal Y}}_U\to{{\mathbf P}}^{1}$ denote the canonical morphisms. Then ${\sigma}$ induces $${\widetilde{{\sigma}}}_U:{\widetilde{\mathcal Y}}_U\to{{\mathcal Y}}_U{\quad}{\hbox}{with}{\quad}{\widetilde{\pi}}_U=\pi_U{\,\raise.15ex{\hbox}{${\scriptstyle\circ}$}\,}{\widetilde{{\sigma}}}_U:{\widetilde{\mathcal Y}}_U\to{{\mathcal Y}}_U\to
U.$$ Let ${\sigma}_c:{\widetilde{Y}}_c\to Y_c\,(c\in U)$ be morphisms induced by ${\sigma}$. Set $${\widetilde{X}}'':={\widetilde{X}}\cap{\widetilde{X}}',{\quad}X'':=X\cap X'{\quad}{\hbox}{with}{\quad}{\sigma}'':=
{\sigma}|_{{\widetilde{X}}''}:{\widetilde{X}}''\to X''.$$ By the definition of ${\widetilde{\mathcal Y}}_U$, ${{\mathcal Y}}_U$, we have the inclusions $${\tilde{i}}''_U:{\widetilde{X}}''\times U{\hookrightarrow}{\widetilde{\mathcal Y}}_U,{\quad}i''_U:X''\times U{\hookrightarrow}{{\mathcal Y}}_U{\quad}{\hbox}{over}\,\,\,U,$$ and ${\widetilde{{\sigma}}}$ induces an isomorphism $${\widetilde{\mathcal Y}}_U\setminus{\tilde{i}}''_U{\bigl}({\widetilde{X}}''\times U{\bigl}){\buildrel\sim\over\longrightarrow}{{\mathcal Y}}_U\setminus i''_U{\bigl}(X''\times U{\bigl}).
\leqno(4.1.1)$$
Consider now the distinguished triangle $${{\mathbf Q}}_{{{\mathcal Y}}_U}\buildrel{{\widetilde{{\sigma}}}_U^{\#}}\over{\longrightarrow}{{\mathbf R}}({\widetilde{{\sigma}}}_U)_*{{\mathbf Q}}_{{\widetilde{\mathcal Y}}_U}
\to{\rm Cone}\,{\widetilde{{\sigma}}}_U^{\#}\buildrel{+1}\over{\longrightarrow}.
\leqno(4.1.2)$$ By the definition of ${\widetilde{{\sigma}}}_U$ and using (4.1.1), we have $${\rm Cone}\,{\widetilde{{\sigma}}}_U^{\#}\cong(i''_U)_*pr_1^*K''{\quad}{\hbox}{with}{\quad}K'':={\rm Cone}{\bigl}({{\mathbf Q}}_{X''}\to{{\mathbf R}}{\sigma}''_*{{\mathbf Q}}_{{\widetilde{X}}''}{\bigl}),
\leqno(4.1.3)$$ where $pr_1:X''\times U\to X''$ is the first projection.
Let $t$ be the affine coordinate of ${{\mathbf C}}\subset{{\mathbf P}}^1$. Since the nearby cycle functor commutes with the direct image by a proper morphism, we have a canonical isomorphism $${{\mathbf R}}({\sigma}_0)_*\psi_{{\widetilde{\pi}}^*t}{{\mathbf Q}}_{{\widetilde{\mathcal Y}}_U}=
\psi_{\pi^*t}{{\mathbf R}}{\widetilde{{\sigma}}}_*{{\mathbf Q}}_{{\widetilde{\mathcal Y}}_U}.$$ By the assertion in the normal crossing case together with the proper base change theorem, we then get $$\aligned &{{\mathbf R}}{\sigma}''_*{{\mathbf Q}}_{{\widetilde{X}}''}{\buildrel\sim\over\longrightarrow}{{\mathbf R}}{\sigma}''_*{\bigl}(\psi_{{\widetilde{\pi}}^*t}{{\mathbf Q}}_{{\widetilde{\mathcal Y}}_U}\big|_{{\widetilde{X}}''}{\bigl})\\
=\,\,&{{\mathbf R}}({\sigma}_0)_*\psi_{{\widetilde{\pi}}^*t}{{\mathbf Q}}_{{\widetilde{\mathcal Y}}_U}\big|_{X''}
=\psi_{\pi^*t}{{\mathbf R}}{\widetilde{{\sigma}}}_*{{\mathbf Q}}_{{\widetilde{\mathcal Y}}_U}\big|_{X''}.\endaligned
\leqno(4.1.4)$$ Let $i_0:Y_0{\hookrightarrow}{{\mathcal Y}}$ denote the inclusion. Apply the functorial morphism $i_0^*\to\psi_{\pi^*t}$ to the distinguished triangle $${{\mathbf Q}}_{{{\mathcal Y}}}\buildrel{{\widetilde{{\sigma}}}^{\#}}\over{\longrightarrow}{{\mathbf R}}({\widetilde{{\sigma}}})_*{{\mathbf Q}}_{{\widetilde{\mathcal Y}}}
\to{\rm Cone}\,{\widetilde{{\sigma}}}^{\#}\buildrel{+1}\over{\longrightarrow},$$ which is the extension of (4.1.2) over ${{\mathbf P}}^1$. Restricting these over $X''\subset X=Y_0$, and using the ‘well-definedness’ of the nearby cycle functor as in (2.4.4), we then get a morphism of distinguished triangles $$\begin{matrix}{{\mathbf Q}}_{X''}&\to&{{\mathbf R}}{\sigma}''_*{{\mathbf Q}}_{{\widetilde{X}}''}&\to&K''&
\buildrel{+1}\over{\longrightarrow}\\
\downarrow\,&&\downarrow&&||\,\\
\psi_{\pi^*t}{{\mathbf Q}}_{{{\mathcal Y}}_U}|_{X''}&\to&
\psi_{\pi^*t}{{\mathbf R}}{\widetilde{{\sigma}}}_*{{\mathbf Q}}_{{\widetilde{\mathcal Y}}_U}|_{X''}&\to&K''&
\buildrel{+1}\over{\longrightarrow}\end{matrix}
\leqno(4.1.5)$$ where the right vertical isomorphism follows from (4.1.3). Moreover, the middle vertical morphism is an isomorphism by (4.1.4). So the left vertical morphism is an isomorphism. This finishes the proof of Proposition (4.1).
[**4.2. Proof of Theorem 1.**]{}
We proceed by increasing induction on $r$. We fix $s_1,\dots,s_r$ such that $${\Sigma}:={{\rm Sing}}\,X\supset{{\rm Sing}}\,Y\supset{\Sigma}':={{\rm Sing}}\,X'={{\rm Sing}}\,Y\cap X',$$ in the notation of the introduction, e.g. $Y:=Y^{(r-1)}$, $X:=Y^{(r)}$, $X'=Y\cap s'_r{}^{-1}(0)$.
Let ${{\mathcal Y}}$ be the blow-up of $Y$ along $X'':=X\cap X'$ so that $${{\mathcal Y}}={{\hbox}{$\coprod$}}_{c\in{{\mathbf P}}^1}\,Y_c{\quad}{\hbox}{with}{\quad}Y_0=X,\,\,Y_{\infty}=X'.$$ Here we use $c$ to denote a point of ${{\mathbf P}}^1$, and the coordinate $t_r$ will be used to define the nearby and vanishing cycle functors.
Set $T_r:=\{t_i=0\,\,(1\le i<r)\}\subset T$. Then $${{\mathcal Y}}\setminus Y_{\infty}\cong{{\mathcal Z}}_T\cap(Y^{(0)}\times T_r).$$ Set ${{\mathcal Y}}_U:={{\mathcal Y}}\times_{T_r}U$ with $U$ a sufficiently small non-empty Zariski-open subset of $T_r$ such that the hypersurface $\{s_r=c\,s'_r\}\subset Y^{(0)}$ is smooth and intersects transversally any strata of Whitney stratifications of $Y^{(k)}\,\,(0<k<r)$ for any $c\in U$. Set $U':=U\cup\{\infty\}\subset{{\mathbf P}}^1$. Let $i_c:Y_c{\hookrightarrow}{{\mathcal Y}}_{U'}$ denote the inclusion for $c\in U'$. Set $${{\mathcal M}}'_{U'}:={{\mathcal M}}'(s'_1,\dots,s'_{r-1})|_{{{\mathcal Y}}_{U'}},{\quad}{{\mathcal M}}'_c:=i_c^*{{\mathcal M}}'_{U'}[-1]\,\,(c\in {U'}).$$ These are mixed Hodge modules on ${{\mathcal Y}}_{U'}$ and $Y_c\,\,(c\in U')$ respectively, and moreover $Y_c$ transversally intersects any strata of a Whitney stratification of ${{\mathcal Y}}_U$ compatible with ${{\mathcal M}}'_U$ by shrinking $U$ if necessary. (Here $U'$ may contain $\infty\in{{\mathbf P}}^1$, since $X'=\{s'_r=0\}\subset Y$ is assumed to be sufficiently general in the one-parameter family $Y_c=\{s_r=c\,s'_r\}\subset Y\,\, (c\in U)$.)
Set $n=\dim X$. By inductive hypothesis applied to the smooth projective variety $\{s_r=c\,s'_r\}\subset Y^{(0)}$ for $c\in U'$, we have short exact sequences of mixed Hodge modules on $Y_c\,\,(c\in U')$ $$0\to{{\mathbf Q}}_{h,Y_c}[n]\to{{\mathcal M}}'_c\to{{\mathcal M}}_c\to 0,
\leqno(4.2.1)$$ where the last term is defined by the cokernel of the injection. (Indeed, the restriction by the inclusion $\{s_r=c\,s'_r\}{\hookrightarrow}Y^{(0)}$ commutes with the nearby cycle functors by \[DMST\].) They imply a short exact sequence of mixed Hodge modules on ${{\mathcal Y}}_{U'}$ $$0\to{{\mathbf Q}}_{h,{{\mathcal Y}}_{U'}}[n+1]\to{{\mathcal M}}'_{U'}\to{{\mathcal M}}_{U'}\to 0,
\leqno(4.2.2)$$ together with the isomorphism $${{\mathcal M}}_c=i_c^*{{\mathcal M}}_{U'}[-1]\,\,(c\in U').$$ Applying the nearby cycle functor $\psi_{t_r}$ to (4.2.2), we get an exact sequence of mixed Hodge modules on $X$ $$0\to\psi_{t_r}{{\mathbf Q}}_{h,{{\mathcal Y}}_{U'}}[n]\to{{\mathcal M}}'(s'_1,\dots,s'_r)\to
\psi_{t_r}{{\mathcal M}}_{U'}[-1]\to 0.
\leqno(4.2.3)$$ Here we use the well-definedness of (2.4.4) which implies that $\psi_{t_r}$ can be defined for mixed Hodge modules defined on the complement of $\{t_r=0\}$. We have moreover the short exact sequence $$0\to{{\mathbf Q}}_{h,X}[n]\to\psi_{t_r}{{\mathbf Q}}_{h,{{\mathcal Y}}_{U'}}[n]\to
\varphi_{t_r}{{\mathbf Q}}_{h,{{\mathcal Y}}}[n]\to 0,
\leqno(4.2.4)$$ since $X$ is a complete intersection so that ${{\mathbf Q}}_X[n]$ is a perverse sheaf. These exact sequences imply the injectivity of the natural morphism $${{\mathbf Q}}_{h,X}[n]\to{{\mathcal M}}'(s'_1,\dots,s'_r).$$ Let ${{\mathcal M}}(s'_1,\dots,s'_r)$ be its cokernel. Then we have a short exact sequence of mixed Hodge modules $$0\to\varphi_{t_r}{{\mathbf Q}}_{h,{{\mathcal Y}}}[n]\to{{\mathcal M}}(s'_1,\dots,s'_r)\to
\psi_{t_r}{{\mathcal M}}_{U'}[-1]\to 0.
\leqno(4.2.5)$$
Set $f:=(s_r/s'_r)|_{Y\setminus X'}$ as in the introduction. Since ${\Sigma}:={{\rm Sing}}\,X\supset{{\rm Sing}}\,Y$, the support of $\varphi_f{{\mathbf Q}}_{h,Y}$ is contained in ${\Sigma}\setminus X'$, and $\varphi_f{{\mathbf Q}}_{h,Y}[n]$ can be viewed as a mixed Hodge module on ${\Sigma}\setminus X'$. By Proposition (4.1) we get $$\varphi_{t_r}{{\mathbf Q}}_{h,{{\mathcal Y}}}[n]=(i_{{\Sigma}\setminus X',{\Sigma}})_{!\,}
\varphi_f{{\mathbf Q}}_{h,Y}[n],
\leqno(4.2.6)$$ where $i_{X\setminus X',X}:X\setminus X'{\hookrightarrow}X$ is the inclusion, and the left-hand side is viewed as a mixed Hodge module on ${\Sigma}$. So the proof of Theorem 1 is reduced to Proposition (4.3) below, except for the independence of the $s'_j$. To show the latter, we consider the one-parameter families $$s'_{j,\lambda}:=(1-\lambda)s'_j+\lambda s''_j\,\,\,(\lambda\in{{\mathbf C}})
{\quad}{\hbox}{for}\,\,\,j\in[1,r],$$ if we are given sufficiently general $s'_j$ and $s''_j$ for $j\in[1,r]$, where $\lambda$ is independent of $j\in[1,r]$. Then the assertion follows from Lemma (2.6). (See also the proof of Proposition (4.4) below.)
[**4.3. Proposition.**]{} [*With the above notation, $\psi_{t_r}{{\mathcal M}}_{U'}[-1]$ is supported in ${\Sigma}'$, and*]{}
$$M_y(X')=(-1)^nT_{y*}{\bigl}(\psi_{t_r}{{\mathcal M}}_{U'}[-1]{\bigl}).$$
[*Proof.*]{}
By inductive hypothesis, we have $${\rm supp}\,{{\mathcal M}}_c\subset{{\rm Sing}}\,Y_c={\Sigma}'{\quad}(c\in U'),$$ where the last equality holds since $s'_r$ is assumed to be sufficiently general. So ${{\mathcal M}}_{U'}$ is viewed as a mixed Hodge module on ${\Sigma}\times U'$, and this can be extended to a mixed Hodge module on ${\Sigma}\times{{\mathbf C}}$ by extendability of mixed Hodge modules in the algebraic case, where ${{\mathbf C}}$ is the complement of some point of ${{\mathbf P}}^1$. By Proposition (3.3) together with (2.4.2) and Lemma (2.6), we then get the assertion. (Note that the non-characteristic restriction $i^*_c$ for any $c\in U'$ trivially commutes with ${{\rm DR}}_y$ up to the factor as in Proposition (3.3).) This finishes the proofs of Proposition (4.3) and Theorem 1.
[**4.4. Proposition.**]{} [*In Theorem $1$, $M_y(X)$ is independent of the choice of the $s_j$ if $([s_j])$ belongs to a sufficiently small non-empty Zariski-open subset ${{\mathcal U}}_I$ of $\prod_{j=1}^r{{\mathbf P}}(I_{X,a_j})$.*]{}
[*Proof.*]{}
Since ${{\mathcal M}}(s'_1,\dots,s'_r)$ in the introduction may depend also on $s_1,\dots,s_r$, we denote it by ${{\mathcal M}}({{\mathbf s}},{{\mathbf s}}')$ with ${{\mathbf s}}:=(s_1,\dots,s_r)$, ${{\mathbf s}}':=(s'_1,\dots,s'_r)$. By the construction of ${{\mathcal M}}({{\mathbf s}},{{\mathbf s}}')$ in the introduction, there is a non-empty Zariski-open conical subset ${{\mathcal U}}_{I,R}$ of $${{\hbox}{$\prod$}}_{j=1}^rI_{X,a_j}\times{{\hbox}{$\prod$}}_{j=1}^rR_{L,a_j},$$ together with a mixed Hodge module ${{\mathcal M}}_{{{\mathcal U}}_{I,R}}$ on ${\Sigma}\times{{\mathcal U}}_{I,R}$ such that ${{\mathcal M}}({{\mathbf s}},{{\mathbf s}}')$ is the restriction of ${{\mathcal M}}_{{{\mathcal U}}_{I,R}}$ by the inclusion ${\Sigma}\times\{{{\mathbf s}},{{\mathbf s}}'\}{\hookrightarrow}{\Sigma}\times{{\mathcal U}}_{I,R}$, and moreover this inclusion is strictly non-characteristic for the underlying perverse sheaf ${{\mathcal F}}_{{{\mathcal U}}_{I,R}}$ of ${{\mathcal M}}_{{{\mathcal U}}_{I,R}}$, i.e. ${{\mathcal F}}_{{{\mathcal U}}_{I,R}}$ is a topologically locally constant family of perverse sheaves parametrized by ${{\mathcal U}}_{I,R}$. (This can be shown for instance by using \[BMM\] together with a Whitney stratification of the total space, since its restriction to a sufficiently general fiber is a Whitney stratification.)
We then get the independence of $M_y(X)$ by the choices of ${{\mathbf s}},{{\mathbf s}}'$ using Lemma (2.6) together with a one-parameter family as in the last part of the proof of Theorem 1 in (4.2). Since the independence of ${{\mathbf s}}'$ is already shown, the assertion follows.
[**4.5. Remarks.**]{}
\(i) Let ${{\mathcal Z}}_T\to T$ be as in the introduction. If $X={{\mathcal Z}}_0$ has an isolated singular point $x$, then we have a mixed Hodge structure on the vanishing cohomology. This can be defined by restricting the morphism ${{\mathcal Z}}_T\to T$ over a generic smooth curve on $T$ passing through $0$ and using an embedded resolution of singularities as in \[St2\]. (This can be calculated also by using mixed Hodge modules as in \[BS\], Th. 4.3.) The associated mixed Hodge numbers are independent of the choice of the generic smooth curve by the theory of spectra for arbitrary varieties which uses the deformation to the normal cone (see e.g. \[DMS\], Remark (1.3)(i)). By a similar argument these mixed Hodge numbers coincide with those given by Corollary 2 assuming that $s_1,\dots,s_{r-1}$ and $s'_1,\dots,s'_r$ are sufficiently general (by using Proposition (3.2)).
\(ii) For a germ of a complete intersection with an isolated singularity $(X,0)$, there is a versal flat deformation $\rho:({{\mathcal X}},0)\to(S,0)$ in the following sense: any flat deformation $({{\mathcal X}}',0)\to(S',0)$ of $(X,0)$ is analytically isomorphic to the pull-back of $\rho$ by some complex analytic morphism $(S',0)\to(S,0)$, see \[KS\], \[Tju\]. Choosing a generic smooth curve on $S$ passing through $0$, we also get a mixed Hodge structure on the vanishing cohomology, and the associated mixed Hodge numbers are independent of the choice of the generic curve, see also \[ES\]. Moreover these numbers coincide with those given in Remark (i) above by using \[DMS\], Cor. 3.4.
**5. Explicit calculations of the Hirzebruch-Milnor classes**
In this section we show how the Hirzebruch-Milnor classes can be calculated explicitly.
[**5.1. Stratification.**]{}
The first term of the right-hand side of (0.2) in Theorem 1 can be described explicitly as follows. Let ${{\mathcal S}}$ be a complex algebraic stratification of ${\Sigma}\setminus X'$ such that the ${{\mathcal H}}^j\varphi_f{{\mathbf Q}}_Y|_S$ are local systems for any $j\in{{\mathbf N}}$ and $S\in{{\mathcal S}}$. Then ${{\mathcal H}}^j\varphi_f{{\mathbf Q}}_Y|_S$ underlies the variation of mixed Hodge structure ${{\mathcal H}}^ji_{S,{\Sigma}\setminus X'}^*\varphi_f{{\mathbf Q}}_{h,Y}$. Here it is enough to assume that each stratum $S\in{{\mathcal S}}$ is a locally closed smooth subvariety and the Whitney conditions are not needed as long as we have local systems on each stratum. We have the following.
[**5.2. Proposition.**]{} [*With the above notation and assumption, we have*]{}
$$T_{y*}{\bigl}((i_{{\Sigma}\setminus X',{\Sigma}})_{!\,}\varphi_f{{\mathbf Q}}_{h,Y}{\bigl})=
{{\hbox}{$\sum$}}_{j\in{{\mathbf N}},S\in{{\mathcal S}}}\,(-1)^j\,T_{y*}{\bigl}((i_{S,{\Sigma}})_!
({{\mathcal H}}^ji_{S,{\Sigma}\setminus X'}^*\varphi_f{{\mathbf Q}}_{h,Y}){\bigl}).
\leqno(5.2.1)$$
Indeed, this follows from the following.
[**5.3. Proposition.**]{} [*Let $X$ be a complex algebraic variety. Let ${{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}}\in D^b{{\rm MHM}}(X)$, and $K^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}$ its underlying ${{\mathbf Q}}$-complex. Let ${{\mathcal S}}=\{S\}$ be a complex algebraic stratification of $X$ such that for any $S\in{{\mathcal S}}$, $S$ is smooth, ${\,\overline{\!S}}\setminus S$ is a union of strata, and the ${{\mathcal H}}^iK^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}|_S$ are local systems on $S$ for any $i$. Let $j_S:S{\hookrightarrow}X$ denote the inclusion. Then $$T_{y*}({{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}})={{\hbox}{$\sum$}}_{S,i}\,(-1)^i\,T_{y*}{\bigl}((j_S)_!{{\mathcal H}}^i
(j_S)^*{{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}}{\bigl}),
\leqno(5.3.1)$$ where ${{\mathcal H}}^i$ for $(j_S)^*{{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}}$ is associated to the classical $t$-structure as in [\[Sa2\], 4.6.2]{} $($which coincides with the usual $t$-structure on the derived category of mixed Hodge modules up to a shift if it is restricted to a stratum of the stratification$)$ so that ${{\mathcal H}}^iK^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}|_S$ underlies variations of mixed Hodge structures ${{\mathcal H}}^i(j_S)^*{{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}}$.*]{}
[*Proof.*]{}
Set $X_k:=\bigcup_{\dim S\le k}S\subset X$. We have the canonical inclusions $$j_k:X_k\setminus X_{k-1}{\hookrightarrow}X_k,{\quad}i_k:X_k{\hookrightarrow}X,$$ together with the distinguished triangles $$(i_k)_*(j_k)_!j_k^*i_k^*{{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}}\to(i_k)_*i_k^*{{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}}\to
(i_{k-1})_*i_{k-1}^*{{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}}\buildrel{+1}\over\to.
\leqno(5.3.2)$$ These imply the following identity in the Grothendieck group of $D^b{{\rm MHM}}(X)$ $$[{{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}}]={{\hbox}{$\sum$}}_k\,{\bigl}[(i_k)_*(j_k)_!j_k^*i_k^*{{\mathcal M}^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}}{\bigl}].
\leqno(5.3.3)$$ Moreover, $j_S$ is the composition of $j'_S:S{\hookrightarrow}X_k$ with $i_k$ where $k=\dim S$. So the assertion follows.
[**5.4. Logarithmic forms.**]{}
The right-hand side of (5.2.1) can be described as follows. Let ${{\mathcal M}}$ be an admissible variation of mixed Hodge structure on a stratum $S$ more generally, where we may assume that ${{\mathcal M}}$ is a polarizable variation of Hodge structure by taking the graded pieces of the weight filtration $W$. Let $M$ be the underlying ${{\mathcal O}}_S$-module with the Hodge filtration $F$ of the variation of mixed Hodge structure. Take a smooth partial compactification $i_{S,Z}:S{\hookrightarrow}Z$ such that $D:=Z\setminus S$ is a divisor with simple normal crossings and moreover $i_{S,{\Sigma}}=\pi_Z{\,\raise.15ex{\hbox}{${\scriptstyle\circ}$}\,}i_{S,Z}$ for a proper morphism $\pi_Z:Z\to{\Sigma}$. Here $D$ cannot be empty, since we take a stratification of ${\Sigma}\setminus X'$ and $X'$ is a hyperplane section. Let $M_Z^{>0}$ be the Deligne extension with eigenvalues of the residues of the logarithmic connection contained in $(0,1]$. Then we have the following.
[**5.5. Proposition.**]{} [*With the above notation, we have*]{}
$$T_{y*}{\bigl}((i_{S,{\Sigma}})_!{{\mathcal M}}{\bigl})=
{{\hbox}{$\sum$}}_{p,q}(-1)^q(\pi_Z)_*td_{(1+y)*}{\bigl}[{{\rm Gr}}_F^pM_Z^{>0}
\otimes\Omega_Z^q(\log D){\bigl}](-y)^{p+q}.
\leqno(5.5.1)$$
[*Proof.*]{}
Let $(M_Z,F)$ be the underlying filtered ${{\mathcal D}}_Z$-module of $(i_{S,Z})_!{{\mathcal M}}$. Then we have a canonical inclusion $$(M_Z^{>0}\otimes\Omega^{{\raise.15ex{\hbox}{${\scriptscriptstyle\bullet}$}}}_Z(\log D),F)[\dim Z]{\hookrightarrow}{{\rm DR}}_Z(M_Z,F),
\leqno(5.5.2)$$ where the filtration $F$ on $(M_Z^{>0}\otimes\Omega^p_Z(\log D)$ is shifted by $-p$ as usual. Moreover, it is a filtered quasi-isomorphism by \[Sa2\], Prop. 3.11, see also \[CMSS\], 3.3. So the assertion follows from the commutativity of $td_*$ with the pushforward by proper morphisms \[BFM\], see also \[CMSS\], 3.3.
[**5.6. Motivic Milnor fibers.**]{}
It is also possible to use the theory of motivic nearby fibers \[DL\] (see also \[St1\]) to calculate the first term of the right-hand side of (0.2). Let ${\sigma}:({\widetilde{Y}},{\widetilde{X}})\to(Y,X)$ be an embedded resolution inducing the isomorphism outside the singular locus of $X_{\rm red}$ and such that ${\widetilde{X}}:={\sigma}^{-1}(X)$ and ${\sigma}^{-1}({\Sigma})$ are divisors with simple normal crossings, where simple means that the irreducible components $E_i$ of ${\widetilde{X}}$ are smooth. (Such a condition for ${\sigma}^{-1}({\Sigma})$ is satisfied if it is a divisor and the other conditions are satisfied.) Let $m_i$ be the multiplicity of ${\widetilde{X}}$ along $E_i\,(i=1,\dots,s)$. Put ${\widetilde{X}}':={\sigma}^{-1}(X')$. For $I\subset\{1,\dots,s\}$, set $$E_I:={{\hbox}{$\bigcap$}}_{i\in I}E_i,{\quad}E'_I:=E_I\setminus{\widetilde{X}}',{\quad}E^{{\prime\circ}}_I:=E'_I\setminus{\bigl}({{\hbox}{$\bigcup$}}_{i\notin I}E_i{\bigl}).$$ Let ${\tilde{f}}$ be the pull-back of $f$ to ${\widetilde{Y}}\setminus{\widetilde{X}}'$. There are smooth varieties ${\widetilde{E}}^{{\prime\circ}}_I$ together with finite étale morphisms $\gamma_I:{\widetilde{E}}^{{\prime\circ}}_I\to E^{{\prime\circ}}_I$ such that $(\gamma_I)_*{{\mathbf Q}}_{{\widetilde{E}}^{{\prime\circ}}_I}={{\mathcal H}}^0\psi_{{\tilde{f}}}{{\mathbf Q}}_{{\widetilde{Y}}}$, see \[DL\]. Note that we can get these varieties by taking the normalization of the base change of ${\tilde{f}}$ by a ramified $m$-fold covering of an open disk where $m={\rm LCM}(m_i)$, see \[St1\]. They may be called the Stein factorization of the Milnor fibers, see \[Lo\]. Let ${\,\widetilde{\!j}}_I:{\widetilde{E}}^{{\prime\circ}}_I{\hookrightarrow}{\widetilde{E}}_I$ be a smooth compactification such that $\gamma_I$ is extended to ${\widetilde{\gamma}}_I:{\widetilde{E}}_I\to E_I$. We have canonical morphisms $$i_{I,J}:{\widetilde{E}}_{I,J}{\hookrightarrow}{\widetilde{E}}_I,{\quad}{\sigma}_{{\Sigma}}:={\sigma}|_{{\sigma}^{-1}({\Sigma})}:
{\sigma}^{-1}({\Sigma})\to X.$$ Set ${\Lambda}':=\{I\subset\{1,\dots,r\}\mid E_I\subset{\sigma}^{-1}({\Sigma})\}$. Then we have the following.
[**5.7. Proposition.**]{} [*With the above notation and assumption, we have*]{}
$$\aligned T_{y*}{\bigl}((i_{{\Sigma}\setminus X',{\Sigma}})_{!\,}\varphi_f{{\mathbf Q}}_{h,Y}
{\bigl})&={{\hbox}{$\sum$}}_{I\in{\Lambda}'}\,({\sigma}_{{\Sigma}}{\,\raise.15ex{\hbox}{${\scriptstyle\circ}$}\,}{\widetilde{\gamma}}_I)_*T_{y*}
{\bigl}(({\,\widetilde{\!j}}_I)_{!\,}{{\mathbf Q}}_{h,{\widetilde{E}}^{{\prime\circ}}_I}){\bigl})(1+y)^{|I|-1}\\
&{\quad}-T_{y*}({\Sigma})+(i_{{\Sigma}\cap X',{\Sigma}})_*T_{y*}({\Sigma}\cap X').\endaligned
\leqno(5.7.1)$$
[*Proof.*]{}
Let $h:{{\mathcal Z}}'\to{{\mathbf C}}$ be the normalization of the base change of ${\tilde{f}}:{\widetilde{Y}}':={\widetilde{Y}}\setminus{\widetilde{X}}'\to{{\mathbf C}}$ by the $m$-fold ramified covering ${{\mathbf C}}\to{{\mathbf C}}$ defined by $t\mapsto t^m$ where $m={\rm LCM}(m_i)$ with $m_i$ the multiplicity of ${\widetilde{X}}$ along $E_i$. Set ${{\mathcal Z}}'_0:=h^{-1}(0)$. There is a canonical morphism $\gamma:{{\mathcal Z}}'_0\to{\widetilde{X}}\setminus{\widetilde{X}}'$. Set $${\widetilde{E}}_I^{{\prime\circ}}:=\gamma^{-1}(E_I^{{\prime\circ}}),{\quad}m_I:={\rm GCD}{\bigl}(m_i\,(i\in I){\bigl}).$$ Then the induced morphism $\gamma_I:{\widetilde{E}}_I^{{\prime\circ}}\to E_I^{{\prime\circ}}$ is finite étale of degree $m_I$. By \[St1\] we have $${{\mathcal H}}^i\psi_h{{\mathbf Q}}_{{{\mathcal Z}}'}|_{{\widetilde{E}}_I^{{\prime\circ}}}=\buildrel{\nu(I,i)}\over{{\hbox}{$\bigoplus$}}{{\mathbf Q}}_{{\widetilde{E}}_I^{{\prime\circ}}}(-i){\quad}{\hbox}{with $\,\,\nu(I,i)=\binom{|I|-1}{i}$},$$ since ${{\mathcal Z}}'_0$ is a divisor with $V$-normal crossings on a $V$-manifold ${{\mathcal Z}}'$. So we get $${{\mathcal H}}^i\psi_{{\tilde{f}}}{{\mathbf Q}}_{{\widetilde{Y}}'}|_{E_I^{{\prime\circ}}}=\buildrel{\nu(I,i)}\over{{\hbox}{$\bigoplus$}}(\gamma_I)_*{{\mathbf Q}}_{{\widetilde{E}}_I^{{\prime\circ}}}(-i).$$ This is an isomorphism as ${{\mathbf Q}}$-complexes. However, it holds as variations of Hodge structures of type $(i,i)$ since the Hodge filtration is trivial.
Let $j:X\setminus X'{\hookrightarrow}X$ denote the inclusion. By Proposition (4.1), we have $$\varphi_{\pi^*t}{{\mathbf Q}}_{h,{{\mathcal Y}}}=j_!\varphi_f{{\mathbf Q}}_{h,Y\setminus X'}=
(i_{{\Sigma}})_*\,i_{{\Sigma}}^*\,j_!{\bigl}(C({{\mathbf Q}}_{h,X\setminus X'}\to
\psi_f{{\mathbf Q}}_{h,Y\setminus X'}){\bigl}).$$ Here the second isomorphism follows from the distinguished triangle $${{\mathbf Q}}_{h,X\setminus X'}\to\psi_f{{\mathbf Q}}_{h,Y\setminus X'}\to
\varphi_f{{\mathbf Q}}_{h,Y\setminus X'}\buildrel{+1}\over\longrightarrow,$$ together with the fact that $\varphi_f{{\mathbf Q}}_{h,Y\setminus X'}$ is supported on ${\Sigma}\setminus X'$.
By the commutativity of the nearby cycle functor with the direct image by a proper morphism, we have $$\psi_f{{\mathbf Q}}_{h,Y\setminus X'}=
({\sigma}'_0)_*\psi_{{\tilde{f}}}{{\mathbf Q}}_{h,{\widetilde{Y}}\setminus{\widetilde{X}}'},$$ where ${\sigma}'_0:{\widetilde{X}}\setminus{\widetilde{X}}'\to X\setminus X'$ is the restriction of ${\sigma}$. So the assertion follows from Proposition (5.3). This finishes the proof of Proposition (5.7).
[**5.8. Remarks.**]{}
\(i) The term $(1+y)^{|I|-1}$ in (5.7.1) comes from the mixed Hodge structure on the cohomology of the irreducible components of the Milnor fiber which is homeomorphic to $({{\mathbf C}}^*)^{|I|-1}$ in the normal crossing case and corresponds to $(1-{{\mathbf L}})^{|I|-1}$ in \[DL\]. (Note that the stalks of the nearby cycle sheaves are given by the cohomology of the Milnor fibers.) Here we may assume that ${\widetilde{E}}_I\setminus{\widetilde{E}}^{{\prime\circ}}_I$ is a divisor with simple normal crossings replacing ${\widetilde{E}}_I$ if necessary. In this case, let ${\widetilde{E}}_{I,j}$ be the irreducible components of ${\widetilde{E}}_I\setminus{\widetilde{E}}^{{\prime\circ}}_I$ which are assumed smooth. Set ${\widetilde{E}}_{I,J}:={{\hbox}{$\bigcap$}}_{j\in J}{\widetilde{E}}_{I,j}$ where ${\widetilde{E}}_{I,\emptyset}={\widetilde{E}}_I$ if $J=\emptyset$. Then the usual resolution argument implies $$T_{y*}{\bigl}(({\,\widetilde{\!j}}_I)_{!\,}{{\mathbf Q}}_{h,{\widetilde{E}}^{{\prime\circ}}_I}{\bigl})=
{{\hbox}{$\sum$}}_J\,(-1)^{|J|}(i_{I,J})_*T_{y*}({\widetilde{E}}_{I,J}).
\leqno(5.8.1)$$
\(ii) In Theorem 1 and its corollaries and also in Propositions of this section, most of the formulas hold also on the level of the Grothendieck group of mixed Hodge modules and hence on that of coherent sheaves by using ${{\rm DR}}_y$ (except for (5.5.1) which is also valid, or rather meaningful, only on the level of the Grothendieck group of coherent sheaves).
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|
---
address: |
Physikalisches Institut, Universität Freiburg\
Hermann-Herder-Str. 3, 79104 Freiburg, Germany
author:
- Volker Büscher
title: Searches for Higgs Bosons and Supersymmetry at the Tevatron
---
Introduction
============
Since March 2001 the Tevatron [$p\bar{p}$]{}-collider at the Fermi National Accelerator Laboratory is operating at a center-of-mass energy of 1.96 TeV and a bunch spacing of 396 ns (Run II). After an initial commissioning period for accelerator and detectors, the machine has delivered about 450 [$\mathrm{pb^{-1}}$]{} between April 2002 and June 2004. Peak luminosities of up to 1.0 $\cdot$ 10$^{32}$$\rm{cm}^{-2}\rm{s}^{-1}$ have been achieved so far. Once the new recycler ring is fully operational the luminosity is expected to increase further, resulting in a projected Run II dataset of up to 8 [$\mathrm{fb^{-1}}$]{} by the year 2009.
Given its high energy and steadily increasing luminosity, the Tevatron collider is an ideal tool for searches for massive particles beyond the reach of LEP. The results presented in this contribution are based on about 200 [$\mathrm{pb^{-1}}$]{} of data recorded and analyzed by the two Tevatron experiments CDF and DØ. After a short description of the two detectors in Section 2, the status of searches for Higgs bosons and supersymmetric particles is summarized in Sections 3 and 4. For completeness, a brief overview of searches for other new physics is given in Section 5.
\[detectors\]CDF and DØ Detectors
=================================
The CDF and DØ detectors are described in detail in Ref. [@cdf-detector] and [@d0-detector]. Only a brief overview is presented in the following.
CDF track reconstruction relies on silicon detectors and a drift chamber situated inside a solenoid that provides a 1.4 T magnetic field coaxial with the beam. The silicon microstrip detector consists of eight cylindrical layers of mostly double-sided silicon, distributed in radius between 1.5 cm and 28 cm. The system is readout in about 700.000 channels and can provide 3D precision tracking up to pseudorapidities of 2.0. Outside of the silicon detectors and for pseudorapidities less than 1.0, charged particles are detected with up to 96 hits per track by the central outer tracker, an open-cell drift chamber with alternating axial and 2$^\circ$ stereo superlayers with 12 wires each. Just inside the solenoid, a scintillator-based time-of-flight detector allows particle identification with a timing resolution of about 100 ps.
The electromagnetic (hadronic) calorimeters are lead-scintillator (iron-scintillator) sampling calorimeters, providing coverage up to pseudorapidities of 3.6 in a segmented projective tower geometry. Proportional wire and scintillating strip detectors situated at a depth corresponding to the electromagnetic shower maximum provide measurements of the transverse shower profile. In addition, an early energy sampling is obtained using preradiator chambers positioned between the solenoid coil and the inner face of the central calorimeter. Outside of the calorimeter and behind additional steel absorbers, a multi-layer system of drift chambers and scintillation counters allows detection of muons for pseudorapidities up to 1.5.
The tracking system of the DØ detector consists of a silicon vertex detector and a scintillating fiber tracker, situated inside a superconducting coil providing a 2 T magnetic field. The DØ silicon tracker has four cylindrical layers of mostly double-sided microstrip detectors covering 2.7 cm up to 9.4 cm in radius, interspersed with twelve disk detectors in the central region and four large disks in the forward region. The full system has about 800.000 channels and provides 3D precision tracking up to pseudorapidities of 3.0. The volume between the silicon tracker and the superconducting coil is instrumented with eight cylindrical double layers of scintillating fibers. Each layer has axial and stereo fibers (stereo angle $\pm 3^{\circ}$) with a diameter of 835 $\mu$m, that are readout using solid-state photodetectors (Visible Light Photon Counters, VLPCs).
The DØ calorimeter is a Liquid Argon sampling calorimeter with Uranium absorber (Copper and Steel for the outer hadronic layers) with hermetic coverage up to pseudorapidities of 4.2. Signals are readout in cells of projective towers with four electromagnetic, at least four hadronic layers and a transverse segmentation of 0.1 in both azimuth and pseudorapidity. The granularity is increased to 0.05 for the third EM layer, roughly corresponding to the electromagnetic shower maximum. To provide additional sampling of energy lost in dead material, scintillator-based detectors are placed in front of the calorimeter cryostats (preshower detectors) and between the cryostats (intercryostat detector). The preshower detectors consist of three layers of scintillator strips with VLPC readout providing, in addition to the energy measurement, a precise 3-dimensional position measurement for electromagnetic showers.
The DØ Muon system consists of three layers of drift tubes and scintillators, with toroid magnets situated between the first and second layer to allow for a stand-alone muon momentum measurement. Scintillator pixels are used for triggering and rejection of out-of-time backgrounds in both central and forward region. Proportional drift tubes are stacked in three or four decks per layer in the central region. Tracking of muons in the forward region is accomplished using decks of mini drift tubes in each layer, allowing muons to be reconstructed up to pseudorapidities of 2.0. The muon system is protected from beam-related backgrounds by shielding around the beampipe using an iron-polyethylene-lead absorber.
Both CDF and DØ detectors are readout using a three-level trigger system which reduces the event rate from 2.5 MHz to about 50 Hz. This includes programmable hardware triggers at Level 1 that provide basic track, lepton and jet reconstruction, secondary vertex or impact parameter triggers at Level 2 as well as a PC-based quasi-offline event reconstruction at Level 3.
\[higgs\]Searches for Higgs Bosons
==================================
As the Run II luminosity increases, CDF and DØ will start reaching sensitivity to production of low-mass Higgs bosons beyond the LEP limits. For Standard Model Higgs bosons decaying to [$b\bar{b}$]{}, the production in association with $W$ or $Z$ bosons is the most promising channel. At masses up to about 180 GeV, Higgs bosons produced via gluon fusion might be observable in their decays to $WW$. For models with more than one Higgs doublet, the coupling of the Higgs boson to b-quarks can be significantly enhanced, allowing to search for Higgs bosons produced in association with b-quarks.
The following sections summarize the current status and projections for the most important channels at Tevatron Run II.
\[s:tev-vh\] Associated Production
----------------------------------
The production of Higgs bosons in association with vector bosons can be searched for in all leptonic decays of $W$ and $Z$: $W\to \ell \nu$, $Z\to{\ensuremath{\nu\bar{\nu}}}$ and $Z\to \ell\ell$ (with $\ell$=$e$,$\mu$,$\tau$). Sensitivity studies based on Monte Carlo simulation of detector performance throughout the course of Run II exist.[@tev-report; @tev-hss] For $WH$ production, first preliminary results[@tev-smhiggs] of searches in 162 [$\mathrm{pb^{-1}}$]{} (CDF) and 174 [$\mathrm{pb^{-1}}$]{} (DØ) of Run II data are described in the following.
Final states compatible with the $WH$ signature can be selected by requiring one isolated lepton and missing transverse energy as well as two b-tagged jets. After additional topological cuts, backgrounds are entirely dominated by physics backgrounds from $W/Z$[$b\bar{b}$]{}, $WZ$ and [$t\bar{t}$]{}. To improve the signal-to-background ratio further, the Higgs boson mass has to be reconstructed with the best possible resolution. Currently, a relative jet energy resolution of 13.9% has been achieved by DØ, as measured in Run II data for central jets at [$E_{\mathrm{t}}$]{}=55 GeV.[@tev-hss] It is expected that this can be improved by 30% due to more sophisticated jet reconstruction algorithms as well as refinements in jet energy calibration, including a calibration of the [$b\bar{b}$]{} mass reconstruction using the $Z\to{\ensuremath{b\bar{b}}}$ signal.
A Higgs signal is then searched for as an excess in the [$b\bar{b}$]{} mass spectrum, as shown in Fig. \[f:tev-wh\]
![Invariant [$b\bar{b}$]{} mass spectrum in search for $WH$ production in 162 $pb^{-1}$ of CDF data in comparison with signal and background expectation. []{data-label="f:tev-wh"}](plots/mass115_dist.eps){width="10cm"}
for the CDF analysis. No evidence for $WH$ production is observed in current Run II searches by CDF and DØ, allowing to set an upper limit on the product of cross-section and branching fraction $\sigma$($WH$)$\times$BR($H\to$[$b\bar{b}$]{}) of 5 pb for a Higgs boson mass of 120 GeV,[@tev-smhiggs] which is still more than an order of magnitude higher than the Standard Model expectation.
In Fig. \[f:tev-higgs\]
![The integrated luminosity needed per experiment for a 95% CL exclusion, a 3$\sigma$ and a 5$\sigma$ discovery of a Standard Model Higgs boson at the Tevatron as a function of the Higgs boson mass, after combining all searches for associated production with a vector boson. Thin lines show recent estimates based on tuned full simulation[@tev-hss], thick lines indicate results of an earlier study with fast simulation (includes searches for $H\to WW$).[@tev-report]. []{data-label="f:tev-higgs"}](plots/TeVLumiPlot.new.eps){width="10cm"}
the luminosity required to observe (or exclude) a Standard Model Higgs boson is shown as a function of mass. After combining all channels and both experiments, a sensitivity at the 95% C.L. for m$_H$=120 GeV is expected to be achieved with an integrated luminosity of 1.8 [$\mathrm{fb^{-1}}$]{} per experiment. Evidence for a signal at the 3$\sigma$ (5$\sigma$) level will require 4 [$\mathrm{fb^{-1}}$]{} (10 [$\mathrm{fb^{-1}}$]{}) for the same Higgs boson mass. These estimates assume a 30% improvement in jet energy resolution and do not include systematic errors. Given the Higgs event yields (3 $WH$ events selected per [$\mathrm{fb^{-1}}$]{}), these analyses will require precise knowledge of the backgrounds over the entire mass range. While the normalization of the background can be obtained from a fit outside of the signal region, the shape and relative normalization of the [$b\bar{b}$]{} mass spectrum of the various background components has to be known to allow extrapolation below the Higgs peak. Procedures to obtain this information from data are outlined in Ref. [@tev-hss] and typically involve a measurement of the shape of the dijet mass spectrum in background-enriched samples, which is then extrapolated to the final signal sample using a mixture of Monte Carlo and data-driven methods.
$H\to WW$
---------
Within the mass range of interest at the Tevatron, Higgs boson decays into two $W$ bosons are the dominant decay mode for Higgs boson masses above 140 GeV. The relatively clean signature of two leptonic $W$ decays allows to search for this decay in the gluon-gluon-fusion channel. While the production cross-section in this channel is higher compared to the associated production, the suppression due the branching fractions of the leptonic $W$ decays limits the event yield to only about 4 events per [$\mathrm{fb^{-1}}$]{}.
Both Tevatron collaborations have started analyzing their Run II data in search for a $H\to WW$ signal.[@tev-smhiggs] So far, efficiencies of up to 15–20% have been achieved for the dilepton plus final states with electrons or muons. The background is dominated by $WW$ production, which remains after selection cuts with a cross section of about 25 fb. Further separation of signal and $WW$ events is possible using the difference in azimuthal angle $\Delta\phi$ between the two charged leptons.[@hww-dphi] Due to spin correlations, $\Delta\phi$ tends to be small for decays of a spin-0 resonance. Both CDF and DØ observe no significant excess of events in 184 [$\mathrm{pb^{-1}}$]{} and 176 [$\mathrm{pb^{-1}}$]{} of Run II data, respectively. Limits on the production cross-section of $H\to WW$ have been set as a function of the Higgs boson mass as shown in Fig. \[f:tev-hww\]. For a mass of 160 GeV, cross-sections larger than 5.6 pb have been excluded at 95% C.L., which is still more than an order of magnitude higher than the expectation within the Standard Model.
The performance of these analyses is consistent with the expectations based on the fast simulation, which projected a total background of 30.4 fb at an efficiency of 18.5% for a Higgs boson mass of 150 GeV.[@tev-report] Based on this projection, sensitivity at the 95% C.L. to a Standard Model Higgs boson with masses between 160 and 170 GeV will be reached with an integrated luminosity of 4 [$\mathrm{fb^{-1}}$]{} per experiment (10 [$\mathrm{fb^{-1}}$]{} for a 3$\sigma$ sensitivity), as shown in Fig \[f:tev-higgs\].
In models beyond the Standard Model, the rate of $H\to WW$ events can be enhanced due to larger production cross-sections (models with heavy 4th generation quarks) or due to an increase in branching fraction (Topcolor models[@tev-report]). In the former case, the gluon-gluon fusion process is enhanced due to loop-diagrams involving heavy quarks by a factor of about 8.5 within the mass range of interest at the Tevatron, with only a mild dependence on the heavy quark mass.[@4gen] The latter class of models also predicts an enhanced branching fraction for $H\to\gamma\gamma$, which can be searched for with diphoton analyses to increase the Tevatron sensitivity at low Higgs boson masses.[@tev-report] First Run II results from DØ in this channel exist, but improve only marginally on existing limits from LEP and Run I.[@tev-diphoton]
![Upper limit (at 95% CL) on the cross section $\sigma \cdot BR(H\to
WW)$ set by the DØ Collaboration using 176 $pb^{-1}$ of Run II data, modified to include the limit set by the CDF Collaboration using 184 $pb^{-1}$ of Run II data. The limits are compared to expectations from Standard Model Higgs production and alternative models. []{data-label="f:tev-hww"}](plots/hww_limit.all.eps){width="8cm"}
Neutral Higgs Bosons in Supersymmetry
-------------------------------------
Sensitivity to a low-mass Higgs boson is of particular interest within supersymmetric extensions of the Standard Model, which predict the existence of at least one neutral Higgs boson [$\mathit{\Phi}$]{}=$h,H,A$ with a mass below 135 GeV. Searches for the Standard Model Higgs boson produced in association with a vector boson can be interpreted within SUSY parameter space. In addition, the enhancement of the Higgs coupling to [$b\bar{b}$]{} at large [$\tan\!\beta$]{} results in sizeable cross-sections for two search channels that are inaccessible within the Standard Model: the production of Higgs bosons in association with one or more b-quarks[@bh-bbh] as well as the gluon-gluon-fusion channel $gg\to {\ensuremath{\mathit{\Phi}}}$ with the subsequent decay ${\ensuremath{\mathit{\Phi}}}\to\tau\tau$.
### ${\ensuremath{\mathit{\Phi}}}b(b)\to bbb(b)$
The DØ collaboration has analyzed 131 [$\mathrm{pb^{-1}}$]{} of Run II data collected with multijet triggers optimized for the ${\ensuremath{b\bar{b}}}{\ensuremath{\mathit{\Phi}}}\to{\ensuremath{b\bar{b}}}{\ensuremath{b\bar{b}}}$ signal.[@tev-susyhiggs] Requiring two jets with transverse momenta [$p_{\perp}$]{}$>$25 GeV and a third jet with [$p_{\perp}$]{}$>$15 GeV, this trigger consumed less than 4 Hz of Level-3 bandwidth at instantaneous luminosities of , while maintaining a signal efficiency of about 70% after offline cuts. The offline analysis requires at least three b-tagged jets with [$p_{\perp}$]{}$>$15 GeV. Depending on the Higgs mass hypothesis, the [$p_{\perp}$]{} cuts for the leading two jets are tightened to values between 35 and 60 GeV to optimize for best expected sensitivity.
The background at this stage is dominated by multijet production with b-quarks. Further discrimination is possible by searching for a peak in the invariant mass spectrum of the two leading jets, which is shown in Fig. \[f:tev-bbh\]
![Invariant mass spectrum of the two leading jets in the 3-jet sample with three b-tags after final cuts in the DØ search for bh production using 131 $pb^{-1}$ of Run II data (left); Regions in ([$\tan\!\beta$]{}, m$_A$) excluded by this analysis at 95% C.L. in comparison with LEP limits (right). []{data-label="f:tev-bbh"}](plots/bbh_bbmass.eps "fig:"){width="48.00000%"} ![Invariant mass spectrum of the two leading jets in the 3-jet sample with three b-tags after final cuts in the DØ search for bh production using 131 $pb^{-1}$ of Run II data (left); Regions in ([$\tan\!\beta$]{}, m$_A$) excluded by this analysis at 95% C.L. in comparison with LEP limits (right). []{data-label="f:tev-bbh"}](plots/bbh_limit.eps "fig:"){width="48.00000%"}
in comparison with the expectation from background and a Higgs signal with m$_{\ensuremath{\mathit{\Phi}}}$=120 GeV. The shape of the dijet mass spectrum in background is obtained from a multijet sample with two b-tagged jets, which is expected to have negligible contamination from signal, by weighting events using b-tag fake rates measured in data as a function of jet [$p_{\perp}$]{} and $\eta$. The background is then normalized by fitting this shape to the mass spectrum outside the signal region.
No evidence for production of neutral Higgs bosons $h$, $H$, $A$ in association with b-jets has been observed, which allows to derive limits on [$\tan\!\beta$]{} as a function of m$_A$. In Fig. \[f:tev-bbh\] the region excluded in the plane of ([$\tan\!\beta$]{}, m$_A$) is shown in comparison with existing limits set by the LEP experiments. The DØ Run II limit is significantly worse than the limit published in 2001 by the CDF collaboration based on the analysis of 91 [$\mathrm{pb^{-1}}$]{} of Run I data.[@tev-bbh-cdf] Detailed comparisons of both results indicate that the apparent loss of sensitivity observed by DØ can be traced back to the use of more recent cross-section calculations and PDF fits, which cause a significant reduction in signal acceptance and cross-section compared to the CDF Run I analysis.[@tev-bbh-comparison]
With more luminosity and after combining results from both Tevatron experiments, the reach in [$\tan\!\beta$]{} at the 95% C.L. will be extended to about [$\tan\!\beta$]{}=25 at m$_A$=120 GeV (for 5 [$\mathrm{fb^{-1}}$]{}, within the [*mhmax*]{} scenario [@mhmax]), but deteriorates quickly with increasing m$_A$.
### ${\ensuremath{\mathit{\Phi}}}\to\tau\tau$
In addition to the usually dominant decay mode ${\ensuremath{\mathit{\Phi}}}\to{\ensuremath{b\bar{b}}}$, a light supersymmetric Higgs boson can be searched for in its decay to ${\ensuremath{\tau\tau}}$. This decay mode is of particular interest both for SUSY scenarios that favour suppressed couplings of Higgs bosons to b-quarks as well as for the large [$\tan\!\beta$]{} region, where the channels ${\ensuremath{\mathit{\Phi}}}b(b)\to \tau\tau b(b)$ and $gg\to {\ensuremath{\mathit{\Phi}}}\to\tau\tau$ provide a viable complement to the search for ${\ensuremath{\mathit{\Phi}}}b(b)\to bbb(b)$.
Both Tevatron experiments have demonstrated the ability to reconstruct hadronic tau decays in Run II data by measuring the $Z\to\tau\tau$ cross-section. The CDF collaboration has analyzed 200 [$\mathrm{pb^{-1}}$]{} of Run II data in search for $gg\to {\ensuremath{\mathit{\Phi}}}\to\tau\tau$ with one tau decaying leptonically to electron or muon and the other tau decaying into hadrons.[@tev-susyhiggs] The hadronic tau decay is reconstructed as one or more tracks pointing to a narrow energy deposition in the calorimeter. Background from jets misreconstructed as tau objects is further suppressed using cuts on track multiplicity, mass and isolation of the tau candidate. The selection then requires one such tau candidate in addition to an isolated electron or muon. After topological cuts using the transverse momenta of the lepton, the tau candidate as well as the transverse missing energy, the sample is dominated by irreducible background from $Z\to{\ensuremath{\tau\tau}}$ with a purity of 90%. Higgs events are selected with an efficiency of about 7% (5%) in the electron (muon) channel.
Separation of signal events from the $Z\to{\ensuremath{\tau\tau}}$ background is possible by reconstructing an event mass $m_{vis}$ based on the momenta of lepton and tau candidate as well as the missing transverse energy. Fig. \[f:tev-htautau\]a
![Distribution of visible mass after all cuts of the CDF search for $H\to{\ensuremath{\tau\tau}}$ in 195 $pb^{-1}$ of Run II data (left); Upper limit (at 95% CL) on the cross section $\sigma \cdot BR(h\to \tau\tau)$ in comparison with the expected limit (right). []{data-label="f:tev-htautau"}](plots/htautau.eps "fig:"){width="46.00000%"} ![Distribution of visible mass after all cuts of the CDF search for $H\to{\ensuremath{\tau\tau}}$ in 195 $pb^{-1}$ of Run II data (left); Upper limit (at 95% CL) on the cross section $\sigma \cdot BR(h\to \tau\tau)$ in comparison with the expected limit (right). []{data-label="f:tev-htautau"}](plots/limit_final.eps "fig:"){width="50.00000%"}
shows the distribution of $m_{vis}$ for data, backgrounds and a potential Higgs signal. No evidence for an excess of events with respect to the Standard Model prediction has been observed. Using a binned likelihood fit of this distribution, a limit on the production cross-section of ${\ensuremath{\mathit{\Phi}}}\to \tau\tau$ has been extracted as displayed in Fig. \[f:tev-htautau\]b as a function of the Higgs boson mass.
### Combined Reach
Combining dedicated searches for Higgs bosons at high [$\tan\!\beta$]{} with searches for production of Higgs bosons in association with vector bosons, sensitivity at 95% C.L. to MSSM Higgs bosons within the [*mhmax*]{} scenario can be achieved independent of [$\tan\!\beta$]{} with 5 [$\mathrm{fb^{-1}}$]{} per experiment, as shown in Fig. \[f:tev-susyreach\].[@tev-report] However, within this challenging scenario, a 5$\sigma$ discovery will not be possible at Tevatron Run II for most of the ([$\tan\!\beta$]{}, m$_A$) plane.
![Luminosity required for exclusion at 95% C.L. (left) or 5$\sigma$ discovery (right) of a SUSY Higgs boson as a function of m$_A$ and [$\tan\!\beta$]{} within the mhmax scenario (taken from Ref. [@tev-report] and modified to include most recent LEP2 limit[@lep-mssm]). Shaded regions indicate the reach of $WH$/$ZH$ searches, the region above the diagonal lines are accessible to searches for $hb(b)$, the dark line indicates the LEP2 limit. []{data-label="f:tev-susyreach"}](plots/fullmhmax95_final.newlep.eps "fig:"){width="48.00000%"} ![Luminosity required for exclusion at 95% C.L. (left) or 5$\sigma$ discovery (right) of a SUSY Higgs boson as a function of m$_A$ and [$\tan\!\beta$]{} within the mhmax scenario (taken from Ref. [@tev-report] and modified to include most recent LEP2 limit[@lep-mssm]). Shaded regions indicate the reach of $WH$/$ZH$ searches, the region above the diagonal lines are accessible to searches for $hb(b)$, the dark line indicates the LEP2 limit. []{data-label="f:tev-susyreach"}](plots/fullmhmax_final.newlep.eps "fig:"){width="48.00000%"}
Charged Higgs Bosons
--------------------
Models with an extended Higgs sector predict charged Higgs bosons $H^\pm$, or in the case of additional Higgs Tripletts doubly-charged Higgs bosons $H^{\pm\pm}$. Production of doubly charged Higgs bosons can provide particularly striking signatures in LR inspired models, where BR($H^{\pm\pm}\to
\ell^\pm \ell^\pm$) is expected to be 100%. CDF (DØ) have analyzed 240 [$\mathrm{pb^{-1}}$]{} (107 [$\mathrm{pb^{-1}}$]{}) of Run II data to search for $H^{\pm\pm}$ production in like-sign dilepton events.[@tev-susyhiggs] Requiring two acoplanar, isolated electrons or muons, no excess of events has been observed at high dilepton masses. For left-handed (right-handed) $H^{\pm\pm}$, CDF set a lower mass limit of 135 GeV (112 GeV) for BR($H^{\pm\pm}\to \ell^\pm \ell^\pm$)=1.
\[susy\]Searches for Supersymmetry
==================================
Supersymmetry predicts a large number of new particles, most of which could be light enough to be produced at the Tevatron. The CDF and DØcollaborations have searched their Run II data for evidence of squarks, gluinos, charginos and neutralinos. For most analyses, minimal Supergravity (mSUGRA) is used as a reference model for optimisation of the analysis and interpretation of the result, even though the resulting cross-section limits can be interpreted in a more model-independent way. Alternative models leading to different final state topologies have been considered as well, including models with gauge-mediated SUSY breaking as well as R-parity violation.
The following sections summarize a selection of current Tevatron results with relevance to Supersymmetry.
Squarks and Gluinos
-------------------
Squarks and gluinos are produced through the strong interaction, resulting in relatively large signal cross-sections and therefore providing, if kinematically accessible, a promising signature for Supersymmetry at the Tevatron. The final state contains two or more jets along with missing transverse energy carried away by the two lightest supersymmetric particles. DØ have searched for pair production of squarks, each decaying into a quark and the lighest neutralino.[@np-squarks] This decay channel is expected to be dominant if the gluino is heavier than the squark. 85 [$\mathrm{pb^{-1}}$]{} of Run II data collected with a dedicated multijet trigger have been analyzed using tight cuts on $>$175 GeV and $H_T$$>$275 GeV against the massive multijet background. Fig. \[f:squarks\]a shows the distribution after all other cuts, with a tail
![Distribution of after all cuts except for the cut on itself (left), showing contributions from multijet backgrounds (line), other standard model processes (red) and signal (yellow); point in the squark/gluino mass plane corresponding to the limit set by the DØ analysis for the mSUGRA model line with $m_0$=25 GeV, [$\tan\!\beta$]{}=3, $A_0$=0 and $\mu<0$ (right). []{data-label="f:squarks"}](plots/N05F04.epsi "fig:"){width="56.00000%"} ![Distribution of after all cuts except for the cut on itself (left), showing contributions from multijet backgrounds (line), other standard model processes (red) and signal (yellow); point in the squark/gluino mass plane corresponding to the limit set by the DØ analysis for the mSUGRA model line with $m_0$=25 GeV, [$\tan\!\beta$]{}=3, $A_0$=0 and $\mu<0$ (right). []{data-label="f:squarks"}](plots/ep-2002-026_fig6.edit.eps "fig:"){width="39.00000%"}
at high expected from standard model processes such as $Z$+jets with $Z\to\nu\bar{\nu}$. After a veto of events containing isolated leptons to supress background from leptonic $W$ decays, an expectation of 2.7$\pm$1.0 Standard Model events remains. Four events are observed in the DØ data, allowing to set upper limits on the squark production cross-section of about 2 pb. Assuming that squarks are degenerate in mass for the first two generations, the reach in mSUGRA parameter space has been evaluated for $m_0$=25 GeV, [$\tan\!\beta$]{}=3, $A_0$=0 and $\mu<0$ (see Fig. \[f:squarks\]). For this model line, squark masses below 292 GeV can be excluded, corresponding to a slight improvement over existing limits (see Fig. \[f:squarks\]b).
For the third generation, mass unification is broken in many SUSY models due to potentially large mixing effects. This can result in third generation squarks much lighter than the other squarks and the gluino. The CDF collaboration has considered this scenario by searching 156 [$\mathrm{pb^{-1}}$]{} of Run II data for pair production of gluinos which subsequently decay to b-quarks plus sbottoms, resulting in final states with four b-quarks and missing transverse energy.[@np-sbottoms] Signal events are isolated by requiring the presence of at least three jets with [$E_{\mathrm{t}}$]{}$>$15 GeV, one or two of which need to be b-tagged using a secondary vertex algorithm. After an additional cut on $>$80 GeV, 2.6$\pm$0.7 events are expected from Standard Model sources, while four events are observed in the data. Assuming a gluino branching fraction to sbottoms of 100%, regions in the gluino/sbottom mass plane can be excluded up to gluino masses of 280 GeV (see Fig. \[f:sbottom\]).
![Regions in the gluino/sbottom mass plane excluded by the CDF Run II analysis requiring one (blue line) or two (violet line) b-tagged jets. []{data-label="f:sbottom"}](plots/bl_june13_excl_limit.epsi){width="8cm"}
CDF also considered a scenario with light stop quarks, in this case assuming that stops are long-lived and decay outside of the detector.[@np-stops] Experimentally, pair production of light stable stop quarks can be detected as a pair of heavily ionizing, slow moving charged particles. The events are triggered on using standard muon triggers, and then selected in offline analysis using the time-of-flight detector. In this way backgrounds from light particles are suppressed to an expectation of 2.9$\pm$0.7(stat)$\pm$3.1(syst) in 53 [$\mathrm{pb^{-1}}$]{} of Run II data. Seven events have been observed, allowing to exclude stop masses below 107 GeV (95 GeV) for isolated (non-isolated) stops.
Charginos and Neutralinos
-------------------------
Many SUSY models expect squarks and gluinos to be the heaviest supersymmetric particles, which might well put them out of the reach of the Tevatron Run II experiments. In this case, a search for the associated production of charginos and neutralinos provides the most promising way for direct detection of supersymmetric particles at the Tevatron. Due to its striking signature, the trilepton channel ${\ensuremath{\widetilde{\chi}}}^{\pm}{\ensuremath{\tilde{\chi}^0}}_2\to 3l+\nu+{\ensuremath{\tilde{\chi}^0}}_1{\ensuremath{\tilde{\chi}^0}}_1$ is considered the most powerful analysis channel, despite its low rate due to the small cross-section and branching fraction. The DØ collaboration has searched for an excess of trilepton events in 175 [$\mathrm{pb^{-1}}$]{}of Run II data.[@np-trileptons] Three different selections have been defined to cover topologies with two electrons and a third isolated track, one electron plus one muon with a third isolated track, and two muons with the same charge. To maintain the highest-possible efficiency, the third lepton is reconstructed (if at all) as an isolated track, which is designed to be efficient for electrons, muons and hadronic tau decays. A total of 0.9 events are expected from Standard Model backgrounds, dominated by irreducible backgrounds from $WW$ and $WZ$ as well as $W\gamma$ with a converted photon. Two events are observed in data, allowing to set an upper limit of about 0.6 pb on production cross-section times branching fraction into three leptons. Fig. \[f:trileptons\] shows this limit as a function the chargino mass in comparison with the LEP limit from direct chargino searches. The new DØ limit significantly improves on the Run I limit, but for signal rates as expected within minimal SUGRA is not yet sensitive to chargino masses beyond the LEP limit.
![The DØ Run II limit (red line) and expected limit (dashed line) on cross-section times branching fraction into three leptons from the search for associated chargino/neutralino production. The shaded area is excluded by direct chargino searches at LEP, the black line shows the expectation within mSUGRA for slepton masses equal to the second lightest neutralino mass. []{data-label="f:trileptons"}](plots/N11F01.epsi){width="10cm"}
$B_s\to\mu\mu$
--------------
While not a direct search for supersymmetric particles, the measurement of the branching fraction of $B_s\to\mu\mu$ is a sensitive probe of Supersymmetry. This rare flavour-changing neutral current decay is heavily suppressed within the Standard Model, where a branching fraction of only 3.8$\times 10^{-9}$ is expected. However, within supersymmetry this decay can be significantly enhanced by loop corrections. For instance, within SUGRA the branching fraction is proportional to ([$\tan\!\beta$]{})$^6$, leading to an enhancement of up to three orders of magnitude.[@Bs-sugra] At the Tevatron, $B_s$ mesons are produced with a very large rate, and a possible decay into two muons can be identified with high efficiency. Both collaborations have reported on analyses searching about 170 [$\mathrm{pb^{-1}}$]{} of Run II data for evidence of this rare decay. The selection suppresses dimuon background from prompt production and semi-leptonic b-decays by requiring two isolated muons originating from a vertex that is significantly displaced from the primary interaction point. Both CDF and DØ estimate the sensitivity of their analysis to branching fractions of about 9$\times 10^{-7}$ at 95% C.L.[@bsmumu] While DØ have not yet quoted a result, CDF observe no significant excess of events in the search window around the $B_s$ mass (see Fig. \[f:bsmumu\]) and have set a limit of BR($B_s\to\mu\mu$)$<$7.5$\times 10^{-7}$ at 95% C.L.
![Dimuon invariant mass distribution after all cuts of the CDF Search for $B_s\to\mu\mu$. One event is found in the $B_s$ search window, consistent with the background expectation extrapolated from the side bands. []{data-label="f:bsmumu"}](plots/results_prl.epsi){width="7cm"}
Gauge-Mediated SUSY Breaking
----------------------------
The mass hierarchies expected in models with gauge-mediated SUSY breaking (GMSB) can significantly alter the signatures of supersymmetry at colliders. While the Gravitino is expected to be the lightest supersymmetric particle in these models, the NLSP can be either a neutralino or a slepton. Neutralino NLSPs decay to a photon and a gravitino, the latter escaping detection in a collider detector. Production of charginos and neutralinos therefore leads to final states containing at least two photons and missing transverse energy. Both Tevatron collaborations have searched for an excess of such events in about 200 [$\mathrm{pb^{-1}}$]{} of Run II data.[@np-gmsb] After requiring missing transverse energy larger than 40 GeV (45 GeV), DØ (CDF) observe one (zero) events compared to an expected background of 2.5$\pm$0.5 (0.6$\pm$0.5) events. Limits on the production cross-section of charginos and neutralinos have been set as a function of chargino mass (see Fig. \[f:gmsb\]). Within a particular GMSB scenario (one messenger field, M=2$\Lambda$, [$\tan\!\beta$]{}=5, $\mu>$0), these limits can be translated into lower limits on chargino (neutralino) masses of 192 GeV (105 GeV) and 168 GeV (93 GeV) for DØ and CDF, respectively.
![The distribution of missing transverse energy (left) after all cuts of the DØ search for GMSB SUSY, compared to the expectation within the Standard Model (dashed line) and the prediction for multijet events (solid line); Upper cross-section limit on GMSB SUSY with neutralino NLSP in comparison with the NLO signal cross-section (right). []{data-label="f:gmsb"}](plots/N04F01.eps "fig:"){width="50.00000%"} ![The distribution of missing transverse energy (left) after all cuts of the DØ search for GMSB SUSY, compared to the expectation within the Standard Model (dashed line) and the prediction for multijet events (solid line); Upper cross-section limit on GMSB SUSY with neutralino NLSP in comparison with the NLO signal cross-section (right). []{data-label="f:gmsb"}](plots/N04F03.epsi "fig:"){width="46.00000%"}
\[other\]Other Searches for New Physics
=======================================
A large variety of other searches have been performed at the Tevatron. While not directly aimed at the discovery of supersymmetric particles, most of the topologies covered by these searches are of relevance for SUSY models as well. This section gives a brief overview of these results.
Final states with two high-pt leptons or photons are predicted in a number of extensions of the Standard Model. Signals include new neutral gauge bosons $Z'$ as well as the production or exchange of gravitons in models with extra dimensions. Both CDF and DØ have observed no significant excess in their high-pt dilepton and diphoton data, as demonstrated by the good agreement between data and background predictions in the invariant dilepton mass distributions shown in Fig. \[f:zprime\].
![Distribution of invariant dielectron (left) and dimuon (right) mass in the high-pt dilepton searches by DØ and CDF, respectively. The left plot shows the MC expectation for a $Z'$ signal with m$_{Z'}$=600 GeV on top of the Standard Model backgrounds. []{data-label="f:zprime"}](plots/N03F01.epsi){width="\textwidth"}
![Distribution of invariant dielectron (left) and dimuon (right) mass in the high-pt dilepton searches by DØ and CDF, respectively. The left plot shows the MC expectation for a $Z'$ signal with m$_{Z'}$=600 GeV on top of the Standard Model backgrounds. []{data-label="f:zprime"}](plots/data_winter04_all_log_2.epsi){width="\textwidth"}
For given models, these results can be translated into limits on $Z'$ masses or the fundamental Planck scale as summarized in Table \[limits\].[@np-dileptons]
------------------------ ------ ----------------------------- ---------------------- ---------------------------
High-mass Dilepton DØ, 200 [$\mathrm{pb^{-1}}$]{} $Z'\to ee$ m$_{Z'}>$780 GeV$^{(1)}$
CDF, 200 [$\mathrm{pb^{-1}}$]{} $Z'\to\mu\mu$ m$_{Z'}>$735 GeV$^{(1)}$
CDF, 195 [$\mathrm{pb^{-1}}$]{} $Z'\to\tau\tau$ m$_{Z'}>$394 GeV$^{(1)}$
Leptoquarks DØ, 175 [$\mathrm{pb^{-1}}$]{} $LQ\to eq$ m$_{LQ}>$240 GeV$^{(2)}$
CDF, 198 [$\mathrm{pb^{-1}}$]{} $LQ\to$$\mu q$ m$_{LQ}>$241 GeV$^{(2)}$
CDF, 191 [$\mathrm{pb^{-1}}$]{} $LQ\to$$\nu q$ m$_{LQ}>$117 GeV$^{(2)}$
Excited Electrons CDF, 200 [$\mathrm{pb^{-1}}$]{} $e^*\to e\gamma$ m$_{e^*}>$889 GeV$^{(3)}$
Large Extra Dimensions DØ, 200 [$\mathrm{pb^{-1}}$]{} $\gamma\gamma$, $ee$ M$_S$$>$1.43 TeV$^{(4)}$
DØ, 85 [$\mathrm{pb^{-1}}$]{} jet+$G_{KK}$ M$_D$$>$685 GeV$^{(5)}$
------------------------ ------ ----------------------------- ---------------------- ---------------------------
: Selection of most stringent CDF and DØ limits on dilepton resonances, leptoquarks, excited leptons and large extra dimensions. The results are quoted for $(1)$ a sequential $Z'$ with Standard Model couplings, $(2)$ scalar leptoquarks with 100% branching fraction, $(3)$ a contact interaction scale equal to the mass of the excited electron, $(4)$ the GRW convention, $(5)$ four extra dimensions.
\[limits\]
Similar searches exist for other new high-mass particles such as leptoquarks [@np-leptoquarks], 4th generation quarks [@np-tprime] and excited leptons [@np-eleptons]. Given the high mass scale of the signal, fairly stringent cuts on transverse energy can be applied to select final states with high-pt jets, leptons or large missing transverse energy. No evidence for any excess has been reported. Table \[limits\] summarizes the mass limits that have been derived by CDF and DØ.
Acknowledgements
================
I would like to thank my colleagues at CDF and DØ for providing the material for this presentation as well as the organizers of the SUSY 2004 conference for this very well-organized event.
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|
---
abstract: 'We study a Vicsek-style model of self-propelled particles where ferromagnetic and nematic alignment compete in both the usual “metric” version and in the “metric-free” case where a particle interacts with its Voronoi neighbors. We show that the phase diagram of this out-of-equilibrium XY model is similar to that of its equilibrium counterpart: the properties of the fully-nematic model, studied before in [@Ginelli2010], are thus robust to the introduction of a modest bias of interactions towards ferromagnetic alignment. The direct transitions between polar and nematic ordered phases are shown to be discontinuous in the metric case, and continuous, belonging to the Ising universality class, in the metric-free version.'
author:
- Sandrine Ngo
- Francesco Ginelli
- Hugues Chaté
title: 'Competing ferromagnetic and nematic alignment in self-propelled polar particles'
---
Collective motion is a topic currently enjoying interest in various communities [@Sumpter; @Giardina-review; @SR-review; @ABP-review]. Within (statistical) physics, the seminal work of Vicsek [*et al.*]{} [@Vicsek1995], followed by the remarkable calculation of Toner and Tu [@Toner], has offered to view the emergence of collective motion in leaderless groups of identical individuals as the spontaneous breaking of rotational invariance. The celebrated Vicsek model, which consists of self-propelled particles aligning ferromagnetically their orientations with that of their neighbors in the presence of noise, was originally presented —and rightly so— as an out-of-equilibrium XY model where spins are forced to move. As is now well-known, the Vicsek model is endowed with properties very different from those of the XY model: in two dimensions, true long-range polar (orientational) order emerges [@Toner] from a discontinuous phase transition [@Chate2008], and the long-range correlations and anomalous fluctuations predicted by Toner and Tu for the ordered collective motion phase, although not observed in the region near onset, are indeed present in a large portion of parameter space.
Other, Vicsek-style, flocking models have been introduced which serve as key members of different universality classes for “dry active matter” [@SR-review], i.e. situations in which global momentum is not conserved and hydrodynamic interactions play no significant role. A prominent case is the “self-propelled rods" model, in which the ferromagnetic interaction of the Vicsek model is replaced by nematic alignment [@Ginelli2010], in line with the typical outcome of inelastic collisions between moving elongated objects. Switching from ferromagnetic to nematic symmetry of interactions in this other out-of-equilibrium XY model, changes the symmetry of the ordered phase (which is then nematic), in line with the symmetry change of the quasi-ordered phase of the corresponding equilibrium version [@GXY]. Numerical results [@Ginelli2010] suggest that like the original (ferromagnetic) Vicsek model, the nematic order observed is truly long-range, but no Toner-Tu-like calculation is available to confirm this at some analytical level.
![(color online) (a) Sketch of two self-propelled needle-like objects (black arrows) moving at the same speed in some overdamped dynamics, which collide with an incoming angle of exactly $\frac{\pi}{2}$. The middle of each rod is indicated by the thin red line. If the impact point is along the first half of the hit needle, (some degree of) polar alignment is expected (top panel), whereas anti-alignment will typically occur for an impact point at the rear (bottom panel). Friction forces may very well, though, lead to polar alignment, not anti-alignment, even if the impact point is slightly beyond the first half of the needle. In such a case, the nematic symmetry of interactions would be weakly broken and biased towards ferromagnetic alignment. (b) Schematic phase diagram of an equilibrium generalized XY model with Hamiltonian $H=-\sum_{\langle ij\rangle} s\cos[\theta_i-\theta_j] +(1-s)\cos[2(\theta_i-\theta_j)]$ (after [@GXY]). Interactions are purely nematic (resp. ferromagnetic) for $s=0$ (resp. 1). D, P, and N, respectively stand for disorder, polar order, and nematic order.[]{data-label="fig1"}](fig1.eps){width="\columnwidth"}
One might object that the properties of the “nematic Vicsek model” are [*not*]{} robust in the sense that the strict nematic symmetry of its interactions may be generically broken, albeit weakly, by “friction” effects during collisions between actual rods (Fig. \[fig1\]a). On the other hand, in equilibrium, it is known that the quasi-ordered phase of the XY model with [*nematic*]{} interactions resists a modest amount of ferromagnetic alignment (Fig. \[fig1\]b) [@GXY].
In this Rapid Communication, we introduce and study an out-of-equilibrium, Vicsek-style, version of such a generalized XY model where ferromagnetic and nematic alignment compete. We show that its phase diagram is similar to that of its equilibrium counterpart, although rendered complicated by density-segregated sub-phases. Thus, the fully-nematic Vicsek model is robust to the introduction of a modest bias of interactions towards ferromagnetic alignment, alleviating the concern raised above for collisions of actual rods. We also show that this conclusion holds in the case of “topological” neighbors where interactions are not limited to some metric zone but occur with those objects defining the first shell of Voronoi cells around the considered particle. In both the metric and this “metric-free” case, we study the direct polar-nematic transition present in phase diagrams like that of Fig. \[fig1\]b. We provide evidence that it seems to be discontinuous in the metric model, but continuous in the metric-free case, with critical exponents of the Ising universality class, as in equilibrium [@GXY]. We finally discuss the inherent difficulties in deriving hydrodynamic theories in the case of mixed ferromagnetic and nematic interactions.
Our starting point is a Vicsek-style model with competing ferromagnetic and nematic interactions: $N$ point particles move off-lattice at constant speed $v_0$ on a two dimensional $L \times L$ torus; particle $j$ is defined by its position $\mathbf{r}_j^t$ and orientation $\theta_j^t$, updated according to $$\begin{aligned}
\label{motion_angle}
\theta_j^{t+1}&=&
\arg \left[\sum_{k\sim j} g_s(\theta_j^t,\theta_k^t)
\right] + \eta\,\xi_{j}^{t} \\
\mathbf{r}_j^{t+1}&=& \mathbf{r}_j^{t}+v_0\, \mathbf{v}_j^{t+1}\, ,
\label{motion_pos} \end{aligned}$$ where $\mathbf{v}_j^t=\left( \cos \theta_j^t, \sin \theta_j^t\right)^T$, the sum is taken over all particles $k$ within unit distance from particle $j$ (including $j$ itself), $\xi$ is a white noise uniformly distributed in $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$, and the complex [*stochastic*]{} function $g_s(\theta,\theta')$ is: $$\label{eq:fp}
g_s(\theta, \theta')=\left\{
\begin{array}{ll}
e^{i\theta'}\;\;&{\rm with \;\; prob.}\;s\\
{\rm sign}\left[\cos(\theta -\theta') \right]e^{i\theta'}\;\;&{\rm otherwise}
\end{array}\right.$$ Note that in the second case $g_s(\theta, \theta')$ is invariant under the transformation $\theta' \to \theta' + \pi$ and thus codes nematic alignment, while the first case expresses ferromagnetic interaction. For $s=0$, this model reduces to model studied in [@Ginelli2010], while $s=1$ is fully equivalent to the standard Vicsek model. Thus, $s$ is a key parameter governing the relative weight of ferromagnetic interactions which comes in addition to the two main parameters, the density $\rho=N/L^2$ and the noise strength $\eta$. In the following, we focus on a low density system at $\rho=\frac{1}{8}$ with $v_{\rm 0}=\frac{1}{2}$, as in [@Ginelli2010], and study the $(s,\eta)$ parameter plane. [@NOTE]
Systematic scans were performed for different sizes. While $L=512$ allows for a rough determination of the main features of the phase diagram, one needs larger sizes to capture its details, mostly because the segregated banded states of the fully nematic ($s=0$) model arise clearly only for large systems [@Ginelli2010]. Polar and nematic order were characterized by means of the two time-dependent global scalar order parameters $P(t)=|\langle\exp (i\theta^t_j)\rangle_j|$ (polar) and $Q(t)=|\langle\exp (i 2\theta^t_j)\rangle_j|$ (nematic), as well as their asymptotic time averages $P=\langle P(t)\rangle_t$ and $Q=\langle Q(t)\rangle_t$. To detect the various density-segregated phases, we relied on snapshots and movies of coarse-grained density and order parameter fields.
![(color online) Top panels: Phase diagram of the generalized Vicsek model with competing ferromagnetic and nematic interactions ($\rho=\frac{1}{8}$, $L=1024$, $v_0=\frac{1}{2}$) (the top right panel is a close up of the left panel.). Results are presented in function of $\sqrt{s}$ for clarity. Symbols located where individual runs were performed, coding the resulting observed phase: red triangles for polar order, black circles for nematic order, and cyan diamonds for disorder. Full/empty symbols code for homogeneous/density segregated phases. Bottom panels: typical snapshots of the 3 density segregated phases present (system size $L=1024$) Left: ${\rm P}_{\rm waves}$ at $s=0.05$, $\eta=0.1$. Middle: ${\rm N}_{\rm band}$ at $s=0.01$, $\eta=0.14$. Right: ${\rm N}_{\rm chaos}$ at $s=0.01$, $\eta=0.166$. The thick magenta arrows indicate the direction of motion of particles.[]{data-label="fig2"}](diagram-new.eps){width="\columnwidth"}
The phase diagram is presented in Fig. \[fig2\] for $L=1024$, a size beyond which its various features do not seem to change. Close to the $s=0$ axis, one observes that the various nematic sub-phases described in [@Ginelli2010] for the purely nematic case are extended to finite but small $s$ values: the spatially-homogeneous nematically-ordered phase (denoted ${\rm N}_{\rm hom}$) can be observed up to $\sqrt{s}\approx 0.17$ for $\eta\approx 0.1$. The segregated phase with a unique, dense, nematically-ordered band occupying a finite fraction of space (${\rm N}_{\rm band}$) extends up to $\sqrt{s}\approx0.4$ for $\eta\approx 0.13$. The spectacular spatiotemporal chaos regime in which thin unstable dense bands elongate, twist, break, collide and form again (denoted ${\rm N}_{\rm chaos}$) is limited to a narrow tongue near $\eta=0.16$. As $s$ is increased, the width of this tongue decreases, making it difficult to locate numerically; our results suggest, though, that it extends all the way to the meeting point with the polar order region.
Polar order can be observed at arbitrary small $s$, provided $\eta$ is small enough; it is delimited by a smooth line (like in the equilibrium case, see Fig. \[fig2\]) starting at the origin and ending at $\eta\simeq 0.4$ for $s=1$, the transition point of the Vicsek model at this density[@NOTE4]. The polar order region is itself divided in two, as expected from our knowledge of the Vicsek model: in a large, tongue-like region bordering the onset of polar order (${\rm P}_{\rm waves}$ in Fig. \[fig2\]), the ordered phase consists of trains of solitary traveling waves, and is thus different from the Toner-Tu homogeneous phase (${\rm P}_{\rm hom}$) present at lower noise strength.
The phase diagram of our out-of-equilibrium, Vicsek-style, generalized XY model thus possesses the same general structure as its equilibrium counterpart: a small nematic triangular region is present on the small-$s$ side above a continuous line delimiting polar order. This general structure is complicated by the presence of the various density-segregated ordered subphases. At our numerical resolution the line dividing the polar region (red dashed line in Fig. \[fig2\]) and that dividing the nematic region (black dashed line) seem to meet the border of the polar region (blue solid line) at the same point [@NOTE2]. There are thus two different ways of transitioning directly from polar to nematic order, as opposed to only one at equilibrium: at small $s$ values, the P-N transition occurs between spatially-homogeneous phases, while at intermediate $s$ values, it links the two density segregated phases ${\rm P}_{\rm waves}$ and ${\rm N}_{\rm band}$ in Fig. \[fig2\].
![(color online) Direct transitions from nematic to polar order in the generalized Vicsek model. (a) transition between the segregated phases ${\rm P}_{\rm waves}$ and ${\rm N}_{\rm band}$ at $s=0.1$ occurring near $\eta=0.132$ (time series of polar (top) and nematic (bottom) order parameter for $L=256$). (b-d) transition between the homogenous phases ${\rm P}_{\rm hom}$ and ${\rm N}_{\rm hom}$ at $s=0.01$ occurring near $\eta=0.085$. (b,c): order parameter curves at sizes $L=128$, 256, 512, 1024. Panel (c) is a close-up of panel (b). (d): Binder cumulant curves at the same system sizes. The arrows indicate increasing system size.[]{data-label="fig3"}](fig3.eps){width="\columnwidth"}
We now turn our attention to the nature of the main transitions (disorder/nematic, disorder/polar, and polar/nematic). Let us first recall that the ordered phases observed all seem numerically to show true long-range order, even the nematic phases ${\rm N}_{\rm hom}$ and ${\rm N}_{\rm band}$ [@Ginelli2010]. In these last cases, however, as argued in [@Ginelli2010], one cannot exclude the possibility that nematic order is only quasi-long-range at very large scales. In the following, we assume true long-range order, in agreement with the numerical results.
The transition from disorder to nematic order actually occurs between the chaotic ${\rm N}_{\rm chaos}$ phase and the ordered, segregated ${\rm N}_{\rm band}$ phase. As reported in [@Ginelli2010], it is determined by the long-wavelength instability of the band, which leads to the ${\rm N}_{\rm chaos}$ phase. This phase being disordered, albeit with very large intrinsic scales, the nematic order parameter $Q$ shows a discontinuous jump at the transition. The disorder/polar transition occurs between the microscopically disordered phase D and the segregated phase ${\rm P}_{\rm waves}$. Like for the original Vicsek model, it is also discontinuous, as the polar order parameter takes finite, order 1 values as soon as the waves appear [@Chate2008].
We studied the two different P-N transitions present. The transition between the segregated phases ${\rm P}_{\rm waves}$ and ${\rm N}_{\rm bands}$ was studied as a function of $\eta$ at $s=0.1$. It is clearly discontinuous, as seen, e.g., in the characteristic flip-flop dynamics of the polar order parameter $P$ in the transition region, leading to a bimodal distribution testifying of phase coexistence (Fig. \[fig3\]a). The transition between the homogeneous nematic and polar phases is more difficult to characterize. We performed a finite-size scaling study at $s=0.01$, varying $\eta$ around the transition point. The behavior of the $P(\eta)$ curves (Fig. \[fig3\]b) indicates the premises of a discontinuous transition: although no discontinuity proper is present even at the largest size considered, these curves cross each other, suggesting that a jump may appear at still larger sizes. This conclusion is also borne out of the behavior of the Binder cumulant $G=1-\langle P(t)^4 \rangle_t/(3\langle P(t)^2 \rangle^2_t )$, which develops a deeper minimum as the system size is increased (Fig. \[fig3\]d) [@FSS].
We now report on the properties of the metric-free, “topological" version of our generalized Vicsek model with competing ferromagnetic and nematic alignment. This case is of theoretical interest because no density-segregated phases are present in the purely ferromagnetic or nematic cases [@TOPOVICSEK; @TOPOKINETIC; @MCM-Gopinath], opening the way to continuous phase transitions (see below). It is also relevant in the context of collective motion. Common sense and some experimental/observational evidence indicate that in groups of higher organisms moving collectively (bird flocks, fish schools, crowds, etc.), one individual, rather than interacting with neighbors located within a given metric zone around itself, takes into account those individuals forming some “angular landscape" to perform navigation decisions [@Moussaid2011]. It was suggested for instance that starlings interact with their 7-8 nearest neighbors, irrespective of the flock density [@Starlings]. Careful study of fish trajectory data showed that in some schools the stimulus/response function of fish can be modeled by interactions with their first Voronoi neighbors (those whose polygons, in a Voronoi tessellation of space, form the first shell around the focal fish) [@Gautrais2012]. Here, following a study of a metric-free version of the Vicsek model [@TOPOVICSEK], we consider these Voronoi neighbors, and we use “vectorial” noise, which means that Eq.(\[motion\_angle\]) is replaced by $\theta_j^{t+1}=
\arg \left[\sum_{k\sim j} g_s(\theta_j^t,\theta_k^t)+ \eta\,{\cal N}_j\vec\xi_{j}^{t}\right]$ where $\vec\xi$ is a randomly oriented unit vector and ${\cal N}_j$ is the current number of (Voronoi) neighbors of particle $j$.
![(color online) Metric-free version of the model where particles interact with their Voronoi neighbors. (a): phase diagram obtained for $L=128$. Legends as in Fig. \[fig2\]ab. (b-d): finite-size scaling study of the P-N transition occuring at $s=0.05$. (b) and (c): $P(\eta)$ and $G(\eta)$ for $L=24$, 32, 64, 128, and 256 (the arrows indicate increasing system size). The order parameter curves do [*not*]{} cross each other; the Binder cumulant curves show a minimum which first deepens then start receding as $L$ increases. These curves do cross each other when $G\sim\frac{2}{3}$ (not shown). (d): scaling of finite-size distance to asymptotic threshold $\Delta\eta_{\rm c}(L)=|\eta_c(L)-\eta_c^\infty|$ for estimated threshold $\eta_c^\infty=0.4530(1)$. Red squares: $\eta_{\rm c}(L)$ is the location fo the maximum of susceptibility at size $L$. Black circles: $\eta_{\rm c}(L)$ is the crossing point of two $G(\eta)$ curves at sizes $L_1$ and $L_2$ with $L=\sqrt{L_1L_2}$. The dashed blue line has slope -1. (e): scaling of order parameter at $\eta=\eta_{\rm c}^\infty$. The dashed blue line has slope $-\frac{1}{8}$.[]{data-label="fig4"}](fig4.eps){width="\columnwidth"}
As shown in [@TOPOVICSEK; @TOPOKINETIC], global density drops out of the problem in metric-free models (it can be scaled out). In practice we worked at unit density, with $v_0=0.5$. The phase diagram of this metric-free version of our generalized Vicsek model is shown in Fig. \[fig4\]a for $L=128$ (diagrams obtained at larger sizes are nearly indistinguishable). As expected, no density-segregated phases are present, leaving a diagram qualitatively similar to that of the equilibrium case. The P-D and N-D transitions are now continuous, as expected from [@TOPOVICSEK; @TOPOKINETIC] where the $s=1$ and $s=0$ cases were studied. A detailed study of the associated critical exponents will be presented elsewhere. (For the P-D transition, they are consistent with the values reported in [@TOPOVICSEK].) We studied the direct P-N transition at $s=0.05$ by finite-size scaling. The (polar) order parameter curves now do [*not*]{} cross each other (Fig. \[fig4\]b), and the Binder cumulant curves show a minimum which, after deepening at small sizes, eventually start receding at the largest sizes we could probe (Fig. \[fig4\]c). These qualitative facts point to a continuous transition. Quantitatively, our estimates of critical exponents are as follows: The crossings of the Binder cumulant curves (near $G\sim\frac{2}{3}$, not visible on Fig. \[fig4\]c) converge to an asymptotic threshold $\eta_{\rm c}=0.4530(1)$ with an exponent $1/\nu=1.00(5)$. The location of the maxima of the susceptibility also converge to the same estimated threshold with the same estimated $1/\nu$, in excellent agreement with the Ising value $\nu=1$ (Fig. \[fig4\]d). The peak values of the susceptibility diverge with system size with exponent $\gamma/\nu=1.74(2)$, the Ising value being $\frac{7}{4}$ (not shown). The order parameter decreases algebraically at the estimated critical point with exponent $\beta/\nu=0.126(3)$, in close agreement with the Ising value $\frac{1}{8}$ (Fig. \[fig4\]e) . Thus, in spite of rather strong finite-size effects as testified by the behavior of the Binder cumulant curves on the nematic side, all estimated critical exponents are very close to their Ising universality class values, as expected from studies of the equilibrium generalized XY model [@GXY].
Before summarizing, we discuss briefly the derivation of a continuous theory describing active matter systems with competing ferromagnetic and nematic alignment interactions. Such a theory must a priori be in terms of a polarity field ${\bf P}$ and a nematic tensorial field ${\bf Q}$ if it is to account for nematically-ordered phases. Baskaran and Marchetti have proposed rather complicated such equations for the case of self-propelled rods interacting via steric exclusion [@MCM-NEMA]. We have followed the somewhat simpler route of the “Boltzmann” approach used in [@BDG; @TOPOKINETIC], which is particularly adapted to dilute Vicsek-like models, for the case of competing ferromagnetic and nematic alignment[@NOTE3]. While details will be published elsewhere [@TBP], we only mention here that the obtained equations for ${\bf P}$ and ${\bf Q}$ fail to account for the phase diagrams presented here. They confirm the existence of a nematic phase at finite $s$, but are not suitable to describe the polar phase, probably since the truncation scheme used is only valid near onset of nematic order.
In conclusion, we have studied a Vicsek-style model with competing ferromagnetic and nematic alignment in both a metric and a “metric-free” version where interactions take place with Voronoi neighbors. We have shown that the fully nematic case of this out-of-equilibrium XY model (“self-propelled rods”) resists some bias toward ferromagnetic alignment, thus conferring some robustness to its nematically-ordered phases and allaying our initial concern about friction effects inducing a “polar bias” in aligning collisions of elongated objects in experiments. In the metric case, the direct polar/nematic transition has been found discontinuous, in line with the order/disorder transitions, whereas the metric-free version exhibits an Ising-class continuous transition, as in equilibrium. We have signalled that a simple derivation of a continuous theory able to account for all observed facts is not easy, and constitutes an important future step for putting these results on firmer ground.
We thank Eric Bertin and Anton Peshkov for useful discussions. This work was largely performed within the activities of the Advanced Study Group “Statistical Physics of Collective Motion” at the Max Plank Institute for the Physics of Complex Systems, Dresden, Germany. FG acknowledges support by Grant IIT-Seed Artswarm.
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We do not expect significant changes for different densities, but a full exploration of parameter space remains to be done.
There is a factor 2 between the noise strength definition used here and that used in [@Chate2008], where this threshold was found near 0.2.
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A. Peshkov, [*et al.*]{}, in preparation.
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author:
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Department of Physics, Tokai University, Hiratsuka, 259-1292, Japan\
E-mail: [email protected]
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Yuta Koshimizu, Toyokazu Fukuoka, Kenji Takagi, Hikoya Kasari\
Tokai University
title: 'Inflation, Gravitino and Reheating in Modified Modular invariant Supergravity'
---
N=1, d = 4 supergravity from d = 10 heterotic string by dimensional reduction has No-scale structure with $E_8 \times E_8$ gauge group [@01]. We would like to propose a new Modular invariant $N=1$ Supergravity, where the Kähler potential and thesuperpotential are given as: $$\!\!\!\!\!\!\!K=-\ln \!\left(S+S^\ast\right)
-3\ln \!\left(T+T^\ast-|Y|^2 \right), \qquad\quad
W=\alpha+\beta S+3bY^3\ln\left[c\>e^{S/3b}\>Y\eta^2(T)\right],$$ where $\eta$ is Dedekind’s $\eta$ function, $c$ is a free parameter in the theory and $S$ is a dilaton, $T$ is a moduli and $Y$ is a complex scalar superfield defined by the gaugino condensation $U\sim <\lambda\lambda>=Y^3$ of the $E_8$ hidden sector[@02], and $\alpha$, $\beta$ are new parameters that should be determined from observations. The renormalization group parameter $b=\frac{15}{16\pi^2}$ can correspond to the $E_8$ hidden sector gauge group.\
A modified string-inspired modular invariant supergravity is proposed here to apply it to inflationary cosmology. Because inflation is concerned with Planck scale physics, the dilaton can be one of the strong candidates for the inflaton[@03; @04]. We assume that the massless Goldstino is identified with the dilatino $\tilde{S}$ because the mass of $\tilde{S}$ satisfies $m_{SS}=0$, where $m$ is defined by $m \equiv e^{K/2} W$.\
Then the scalar potential ($V_E \equiv e^G \left[ G_i G^{ij^*} G_{j^*} - 3 \right] $) is in order: $$\begin{aligned}
&&\hspace{-30pt} V_E= \frac{1}{(S+S^*)(T+T^*-|Y|^2)^2} \left[ 3 b^2 |Y|^4 \left| 1 + 3 \ln [O] \right|^2 \right. \nonumber \\[5pt]
&&\hspace{-15pt} + \frac{1}{T+T^*-|Y|^2} \left| \strut \alpha + \beta S + 3 b Y^3 \ln [O] - (S+S^*) (Y^3 + \beta) \right|^2 \nonumber \\[5pt]
&&\hspace{-15pt} + 6 b^2 |Y|^6 \left\{ \left( 1- \frac{\alpha + \beta S^*}{b{Y^*}^3} \right) \frac{ \eta' (T)}{ \eta (T)} + \left( 1- \frac{\alpha + \beta S}{bY^3} \right) \frac{ \eta' (T^*)}{ \eta (T^*)} \right. \left. \left. + 2 (T+T^*) \left| \frac{ \eta' (T)}{ \eta (T)} \right|^2 \right\} \right],\end{aligned}$$ where $O=c\>e^{S/3b}\>Y\eta^2(T)$ and the potential is explicitly modular invariant in $T$. Instead of imposing $W_Y+K_YW=0$, we will assume $W_Y=0$ which is a rather good approximation. Then, a relation between $S$ and $Y$ is obtained as follows: $Y = \frac{1}{c \eta^2 (T) e^{\frac{1}{3}}} e^{-\frac{S}{3b}}.$\
We will here only present one case among the parameter choices $c$, $\alpha$ and $\beta$, for which the potential $V(S,Y)$ at $T=1$ has a stable minimum.\
Hereafter we fix $T=1$ ($\eta (1) = 0.768225$, $\eta^2 (1) = 0.590170$, $\eta' (1) = -0.192056$, $\eta'' (1) = -0.00925929$) and $b=\frac{15}{16\pi^2}$ corresponding to the $E_8$ gauge group. The results with the parameter choice $c=10^2$, $\alpha = 10^{-6}$, $\beta = 6 \times 10^{-5}$ are as follows: The minimum of the potential is given by $
S_{{\rm min}} = 2.23 \times 10^{-2}, \quad
Y_{{\rm min}} = 1.12 \times 10^{-2}, \quad
V(S_{{\rm min}},Y_{{\rm min}}) = 5.94 \times 10^{-12}.
$ The parameters of inflation are predicted as follows $$\begin{aligned}
&&S_{{\rm end}} = 0.7394, \quad S_* = 10.90, \quad \mathcal{P_{R^*}} = 2.438 \times 10^{-9}, \nonumber \\
&&N = 58.79, \quad\,\,\, n_{S^*} = 0.9746, \quad \alpha_{S^*} = -4.303 \times 10^{-4}.\end{aligned}$$ The Gravitino mass and the SUSY breaking scale are predicted as: $$\begin{aligned}
m^{3/2} = | M_P e^{\frac{K}{2}} W | = 8.99 \times 10^{12} \,\, {\rm GeV}, \qquad
F_S = 2.19 \times 10^{12} \,\, {\rm GeV}.\end{aligned}$$ We show the potential $V(S)$ minimized with respect to $Y$ in Fig. 1, and the evolution of the slow-roll parameters in Fig. 2. The stability of the potential minimum at $T=1$ can also be proved.\
This case seems to explain the WMAP observations well. The slow-roll condition is well satisfied, and the $\eta$-problem can be resolved. Using the slow-roll approximation, the number of $e$-folds at which a co-moving scale $k$ crosses the Hubble scale $aH$ during inflation is given by: $
N\sim -\int^{S_*}_{S_{\rm end}}\frac{V}{\partial V}dS \sim 58.72,
$ by integrating from $S_{\rm end}$ to $S_*$, fixing the parameters $c\ {\rm and}\ b$ as well as $\alpha\ {\rm and}\ \beta$. That is, our potential has the ability to produce the cosmologically plausible number of $e$-folds.\
A scalar spectral index for a scale dependence of the spectrum of density perturbation and its tilt are defined by $ n_s-1=\frac{d\ln \mathcal{P_R}}{d\ln k} $ and $ \alpha_s=\frac{dn_s}{d\ln k} $. Substituting $S_*$ into these formula, we have $n_{s*}\sim 0.9746$ and $\alpha_{s*}\sim -4.3 \times 10^{-4}$. Estimating the spectrum of the density perturbation $\mathcal{P_R}$ caused by slow-rolling dilaton, we found $\mathcal{P_R}_* \sim 2.438 \times 10^{-9}$. Finally, the ratio $r$ between the scalar power spectrum $\mathcal{P_R}_*$ and the tensor one $\mathcal{P}_T$ is predicted as $r\sim 6.755 \times 10^{-2}$, which seems to be in the range possibly observed by the Planck satellite soon. The energy scale of the potential at the minimum, moreover, is given as $V\sim 5.9 \times 10^{-12}$, which is non-negative and may be considered to be small. It is the end of inflation, when one of the slow-roll parameters $\epsilon_\alpha$ or $\eta_{\alpha\beta}$ reaches the value 1. After passing through the minimum of the potential, reheating will begin.\
Let us consider the Super Higgs mechanism in our model. The inflatino field $\tilde{S}$ with its mass $m_{\tilde{S}}=0\ {\rm{GeV}}$, which is the SUSY partner of the inflaton (dilaton) field $S$, can play the role of the Higgsino field. Because the metric elements satisfy $g_{ST}=g_{SY}=0$ in the Kähler metric $g_{ij}$, $S$ does not mix with $Y,\ T$. Then the result of Super Higgs mechanism is given as $$\begin{aligned}
{\mathcal{L}}_{\rm{SHM1}}
=ee^{\frac{G}{2}}\Big\{\Big(\psi_\mu+\frac{i}{3\sqrt{2}}G_{S^*}\bar{\tilde{S}}\bar{\sigma}_\mu\Big)\sigma^{\mu\nu}\Big(\psi_\nu-\frac{i}{3\sqrt{2}}G_{S^*}\sigma_\nu \bar{\tilde{S}}\Big)+\frac
{1}{2}(G_{S^*S^*}+\frac{1}{3}G_{S^*}G_{S^*})\bar{\tilde{S}}\bar{\tilde{S}}\Big\}\label{shm}.\end{aligned}$$ The last term of Eq.(\[shm\]) implies the mass of $\tilde{S}$, which is proved to be exactly zero in our model. The first term can be identified with the mass term of the massive gravitino field, whose mass is given by $m_{3/2}=e^{G/2}$. This is the scenario of Super Higgs mechanism in our model. The predicted value of gravitino mass is given as: $m^{3/2} = | M_P e^{\frac{K}{2}} W | = 8.99 \times 10^{12}$ GeV. The scale of SUSY breaking is $F_S = 2.19 \times 10^{12} \,\, {\rm GeV}$.\
After the scalars $S,Y,T$ are canonically normalized and the masses diagonalized, these masses are calculated as\
$M_{S''}=9.98 \times 10^{12}$ GeV, $M_{Y"}=2.61 \times 10^{16}$ GeV, $M_{T''}=2.27 \times 10^{12}$ GeV,\
where the mass eigenstates are denoted by $S'',Y'',T''$.\
The decay rate of the process $Y'' \rightarrow \psi_{3/2}+\psi_{3/2}$ is estimated as $$\begin{aligned}
\Gamma(Y'' \rightarrow \psi_{3/2}+\psi_{3/2}) = 4.78 \times 10^4 \,\, {\rm GeV}, \quad \quad
\tau (Y'' \rightarrow \psi_{3/2}+\psi_{3/2}) = 1.38 \times 10^{-29} \,\, {\rm sec}.\end{aligned}$$ This process occurs almost instantly.\
In order to estimate the reheating temperature, the decay rate of $S''$ into gauginos is calculated. By using the term ${\mathcal{L}}_{gaugino}=\kappa \int d^2\theta f_{ab}(\phi)W_{\alpha}W^{\alpha}$, $f_{ab}(\phi)=\phi\delta_{ab}$, the interaction between $S(=\phi)$ and gauginos $\lambda^a$’s is given by $$\begin{aligned}
&&{\mathcal{L}}_{gaugino}
=\frac{i}{2}f^R_{ab}(\phi)\left[\lambda^a\sigma^\mu\tilde{\mathcal{D}}_\mu\bar{\lambda}^b+\bar{\lambda}^a\sigma^\mu\tilde{\mathcal{D}}_\mu\lambda^b \right]-\frac{1}{2}f^I_{ab}(\phi)\tilde{\mathcal{D}}_\mu\left[\lambda^a\sigma^{\mu}\bar{\lambda}^b\right] \nonumber \\
&&\qquad -\frac{1}{4}\frac{\partial f_{ab}(\phi)}{\partial \phi}e^{K/2}G_{\phi\phi^*}D_{\phi^*}W^*\lambda^a\lambda^b +\frac{1}{4}\left(\frac{\partial f_{ab}(\phi)}{\partial \phi}\right)^*e^{K/2}G_{\phi\phi^*}D_{\phi}W\bar{\lambda}^a\bar{\lambda}^b. \label{gaugino_decay}\end{aligned}$$ By using the relation $F_S \sim M_Pm_{SP}$, which holds for the mass of SUSY particles, the gaugino masses can be estimated as $$m_\lambda = \frac{F_S^2}{M_P} \sim 1.97 \times 10^6 \,\, {\rm GeV}.$$ Then the decay rate of $S'' \rightarrow \lambda+\lambda$ is given by $$\begin{aligned}
\Gamma(S''\to \lambda\lambda) = 2.96 \times 10^{-3} \,\, {\rm GeV}.\end{aligned}$$ The reheating temperature $T_R({\rm gaugino})$ is derived from the Boltzmann equation by using the decay rate, and is given by $
T_R({\rm gaugino})=\left(\frac{10}{g_*}\right)^\frac{1}{4}\sqrt{M_P~\Gamma(S'' \to \lambda + \lambda) },
$ where $g_*$ is the number of the effective degrees of freedom of MSSM, i.e. $g_* =228.75$.\
By inserting the decay rate, the reheating temperature is estimated as $$\begin{aligned}
T_R({\rm gaugino}) = 3.88 \times 10^{7} \,\, {\rm GeV}. \end{aligned}$$ Because the reheating temperature is lower than the gravitino mass scale, gravitino reproduction will not occur after reheating. Though only one example of parameter choices has been discussed here, the other seven candidates of parameter choices, which are compatible with WMAP data, have already been found by the authors. Some of the examples show that the gauginos can be observed by LHC experiments[@05].
[99]{} E.Witten, Phys.Lett.B155, 151(1985).\
S. Ferrara, N. Magnoli, T. R. Taylor and G. Veneziano, Phys.Lett. B245, 409 (1990).\
M.J. Hayashi, T. Watanabe, I. Aizawa and K. Aketo, Mod. Phys. Lett. A18, 2785 (2003); M.J. Hayashi and T. Watanabe, Proceedings of ICHEP2004, Beijing, 423, eds. H. Chen, D. Du, W. Li and C. Lu, World Scientific, P.423 (2005).\
M.J. Hayashi, S. Hirai, T. Takami, Y. Okame, K. Takagi and T. Watanabe, Int. J. Mod. Phys A22 2223 (2007); M.J. Hayashi, S. Hirai, T. Takami, Y. Okame, K. Takagi and T. Watanabe, Frontiers of Fundamental and Computational Physics: 9th International Symposium, American Institute of Physics, p.74 (2008). Yuta Koshimizu, Toyokazu Fukuoka, Kenji Takagi, Hikoya Kasari and M.J. Hayashi, arXiv:1009.5171, and references therein.
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abstract: 'Color–temperature relations and bolometric corrections in the HST–NICMOS F1110W, F160W and F222M and in the WFPC2 F439W, F555W and F814W photometric systems, using two different sets of model atmospheres, have been derived. This database of homogeneous, self–consistent transformations between the theoretical and observational planes also allows combinations of visual and infrared quantities, without any further transformation between the two different photometric systems. The behavior of the inferred quantities with varying the stellar parameters, the adopted model atmospheres and the instrumental configurations are investigated. Suitable relations to transform colors and bolometric corrections from HST to ground–based photometric systems are also provided.'
author:
- Livia Origlia
- Claus Leitherer
title: 'Transformations between the theoretical and observational planes in the HST–NICMOS and WFPC2 photometric systems'
---
Introduction
============
Transformations between the theoretical and the observational planes are fundamental tools to compare stellar evolution models with observed color–magnitude diagrams. In order to calibrate suitable relations among colors, temperatures and bolometric corrections (BCs) several basic ingredients are needed:
- homogeneous and complete grids of stellar spectra (observed and/or theoretical);
- accurate filter profiles;
- reference spectra (the Sun, Vega etc.) to set the zero points of the relations;
- suitable routines to interpolate within the grids and to integrate along the spectra.
These calibrations are certainly model dependent and systematic shifts between different scales are common features. The goal of this paper is to obtain color–temperature relations and BCs in the HST–NICMOS photometric system. We also derive analogous transformations for a few selected filters in the WFPC2 system to provide homogeneous, self–consistent color–temperature relations and BCs when combinations of optical–infrared colors are used. A more complete calibration of the color–temperature transformations in the WFPC2 system can be found in the paper by Holtzman et al. (1995, hereafter H95).
In Sect. 2 we describe the code we used to derive the transformations. In Sect. 3 and 4 we discuss the transformations in the HST–NICMOS and the WFPC2 systems, respectively. In Sect. 5 we derive suitable relations to transform a few selected colors and BCs from the HST to the ground–based photometric system. In Sect. 6 we draw our conclusions.
The code
========
The synthetic colors and BCs were computed using a modified version of the evolutionary synthesis code by Leitherer et al. (1999).
We implemented two different set of model atmospheres: [*i)*]{} the compilation by Lejeune, Cuisinier & Buser (1997, hereafter LCB97), based on the ATLAS code by Kurucz and corrected to match empirical color–temperature calibrations; [*ii)*]{} the compilation by Bessell, Castelli & Plez (1998, hereafter BCP98) using only the homogeneous set of models without overshooting computed by Castelli (1997) and based on Kurucz’s ATLAS9 models. The grid of stellar parameters explored by our computations is:
- Metallicities in the range 0.01 – 1.00 solar.
- Effective temperatures $T_{\rm eff}$ in the range 3500 – 50000 K.
- Gravities log$~g$ in the range 0.0 – 5.0.
The ground–based BVIJHK filter profiles are in the Johnson’s (1966) photometric system and were taken from Buser & Kurucz (1978) (BV filters) and from Bruzual (1983) (IJHK filters) (see also Sect. 5 in Leitherer & Heckman 1995 for more details). The selected broadband filters in the HST–WFPC2 and NICMOS photometric systems are: F439W, F555W, F814W and F110W, F160W, F222M, respectively. The filter profiles have been multiplied by the wavelength dependent detector quantum efficiency of the four WFPC2 and the three NICMOS cameras. We used the most updated post–launch throughput curves and CCD quantum efficiencies, according to the WFPC2 and NICMOS documentation on the WEB. The quoted NICMOS throughput curves are already empirically adjusted to match standard star measurements (the correction factors are less than 10% in the case of the F110W and F160W filters and about 25% for the F222M one). We do not apply any further empirical adjustment to these response curves.
The colors have been normalized assuming 0.00 values in all passbands for a Vega–like star with $T_{\rm eff}$=9500K, log$~g$=4.0 and \[Fe/H\]=–0.5 (cf. e.g. BCP98). The BCs have been normalized assuming a value of –0.07 for a Sun–like star with $T_{\rm eff}$=5750 K, log$~g$=4.5 and \[Fe/H\]=0.0 (cf. e.g. Montegriffo et al. 1998). Different assumptions for the colors and bolometric corrections of Vega and Sun –like stars can be accounted for by simply scaling the inferred quantities by the corresponding amounts.
In order to quantify the influence of the adopted filter response curves on the inferred colors we also compare our B–V and V–K values with those in Table 2 of BCP98 for solar metallicity. The adopted model atmospheres are exactly the same and also similar within 0.01 mag are the reference colors for the Vega–like star. The only difference between the two sets of transformations are the adopted filter response curves. Within 0.01 mag our V–K color is fully consistent with BCP98 values, while only below 5000K our B–V becomes progressively bluer (at most 0.03 mag at 3500K).
For each model atmosphere at a given $T_{\rm eff}$ and log$~g$ we tabulate the corresponding colors in the form (V-F), where F is the selected filter in the ground–based or in the HST system, and the bolometric correction BC$_V$ in the V passband. We also provided analogous colors and BCs using simple blackbodies at a given temperature.
All the tables with the computed transformations can be retrieved from http://www.stsci.edu/science/starburst/.
In the following we analyze the behavior of the inferred quantities varying the stellar parameters and the adopted model atmospheres.
The HST–NICMOS infrared plane
=============================
In Fig. 1 we plot the (F110W–F160W) and (F110W–F222M) color–temperature transformations in the HST–NIC2 system, using the BCP98 models for three gravities (log$~g$ of 0.0, 2.0 and 4.0) and two metallicities ($Z=Z_{\odot}$ and $Z=0.1Z_{\odot}$). For temperatures hotter than 4000 K these infrared colors show a scatter within 0.1 mag with varying log$~g$ and $Z$. At lower temperatures the scatter among models with different gravities increases up to a few tenths of mag at $T_{\rm eff}$=3500 K, while the metallicity dependence is less critical.
Pure blackbodies have systematically bluer colors than model atmospheres, especially at the coolest temperatures, as expected from the omission of molecular opacities. This means that for a given color, blackbodies are cooler than the model atmospheres.
Analogous color–temperature relations using the LCB97 models have been obtained. In Fig. 2 we report the difference of the color–temperature transformations in the NIC2 system, by adopting the BCP98 or the LCB97 model atmospheres. At temperatures below $\sim$5000 K the infrared colors using the LCB97 models get progressively bluer (up to 0.2–0.3 mag) than those using the BCP98 models. This behavior mainly reflects the difference between the original and corrected grids of model atmospheres by LCB97. Comparing their Figs. 6,7 (the original synthetic colors for giants and dwarfs, respectively) with their Figs. 14,15 (the corresponding corrected models to match empirical color–temperature relations) one can see that in the range of temperatures between 4000 and 3500 K the [*corrected*]{} (J–H) ground based color is bluer (from 0.1 up to 0.3 mag) than the corresponding original value, regardless the stellar gravity. A similar comparison can be done between the original and the [*corrected*]{} (J–K) ground based color: the latter becomes bluer with decreasing temperature by about 0.2 and up to 0.3 mag for giants and dwarfs at T$_{\rm eff}$=3500 K, respectively.
The un–corrected LCB97 models, from the original grids of atmosphere spectra by Kurucz, should provide colors more similar to those obtained from the BCP98 models.
Color–temperature transformations in the NIC1 and NIC3 systems are also provided, as shown in Fig. 3. Regardess the model atmosphere used and the adopted stellar parameters, for a given temperature the (F110W–F160W) color in the NIC1 system is only $\le$0.01 mag redder than in the NIC2 system, while both the (F110W–F160W) and the (F110W–F222M) colors in the NIC3 system are slightly redder ($\le$0.03 mag below $\sim$4000 K).
In Fig. 4 we plot the BC in the NIC2 F110W, F160W and F222M passbands as a function of the temperature using the BCP98 models. Increasing the temperature of the stellar atmospheres, progressively larger corrections (that is smaller values of the BC) to the infrared fluxes have to be applied in order to get the bolometric luminosity. The scatter in the inferred quantities for different gravities and metallicities is generally small. At low temperatures the use of pure blackbodies requires larger corrections than using model atmospheres, as expected since the former have bluer spectra than the latter.
In Fig. 5 we plot the difference between the values of the BC in the NIC2 photometric system, adopting the BCP98 or the LCB97 models, as we did for the colors in Fig. 2. At temperatures below 5000 K, larger BC values in the F110W and F222M and smaller ones in the F160W passbands (up to $\le$0.2 mag at $T_{\rm eff}$=3500 K) are required when LCB97 models are used compared to the BCP98 ones.
The above trends are almost independent of the adopted metallicity.
For a given set of model atmospheres, very similar BCs within $<$0.01 mag are also obtained in the NIC1 and NIC3 systems, as shown in Fig. 6. Only in the F110W passband we find that the BC values in the NIC3 system are slightly smaller ($\le$0.02 mag) than in the NIC2 system.
Our discussion on the behavior of the inferred colors and bolometric corrections with varying the stellar parameters has been limited to the low temperature domain (T$_{\rm eff}<$5000 K) where they are more sensitive. At higher temperatures these infrared quantities become progressively less dependent on the adopted model atmospheres.
The HST–WFPC2 visual plane
==========================
In Fig. 7 we plot the (F439W–F555W) and (F555W–F814W) color–temperature transformations in the HST–PC1 system, using the BCP98 models for the three gravities and two metallicities of Fig. 1.
The scatter among models with different log$~g$ and $Z$ is $\le$0.1 mag at all temperatures in the case of the (F555W–F814W) color, while at low temperatures ($T_{\rm eff}\le$5000 K) the scatter among models with different gravities in the (F439W–F555W) color increases up to about 0.6 mag at $T_{\rm eff}$=3500 K, while the metallicity dependence is less critical. As we found for the infrared colors, pure blackbodies show bluer colors than model atmospheres.
Analogous color–temperature relations using the LCB97 models are also obtained. In Fig. 8 we report the difference in the color–temperature transformations by adopting the BCP98 or the LCB97 model atmospheres. Using the LCB97 models, at low temperatures the (F439W–F555W) color becomes progressively bluer (up to 0.1–0.2 mag) at low gravities and only slightly redder at larger ones, compared to the values obtained from the BCP98 models. The (F555W–F814W) color becomes rapidly redder (particularly at low gravities) than the corresponding quantities using the BCP98 models for T$_{\rm eff}\le$4000 K. This behavior has little metallicity dependence. As for the infrared colors, this behavior can be mainly ascribed to the corrections applied by LCB97 to the original model atmospheres. Comparing again their Figs. 6,7 (the original synthetic colors) with their Figs. 14,15 (the corresponding corrected models) one can see that below 4000 K the [*corrected*]{} ground–based (B–V) color of giants becomes bluer and slightly redder for dwarfs, while the ground–based (V–I) color becomes rapidly redder (a few tenths of mag, even larger values for giants).
In Fig. 9 we plot the BC in the F439W, F555W and F814W passbands as a function of the temperature using the BCP98 models. The required BC values using pure blackbodies are on average slightly larger than using model atmospheres, as expected since the former have bluer spectra than the latter and, contrary to the infrared case, in the visual range they tend to be brighter than the model atmospheres.
In Fig. 10 we plot the difference between the values of the BC, adopting the BCP98 or the LCB97 models, as we did in Fig. 8 for colors. At temperatures below 5000 K, smaller BC values in the F439W and F555W and larger (up to $\le$0.2 mag at $T_{\rm eff}$=3500 K) in the F814W passbands are inferred if the LCB97 models are used. As for the infrared plane, the above trends are sensitive to the adopted gravity and almost independent of metallicity.
For a given set of model atmospheres, very similar color–temperature transformations and BC (within 0.01 mag) are obtained in all four WFPC2 cameras.
Discussion
==========
Model atmospheres are a powerful tool to calibrate suitable color–temperature transformations since homogeneous and complete grids for a wide range of stellar parameters are available. Nevertheless, for some specific applications empirical scales even if they are less complete in terms of stellar parameters are preferred.
Using the BCP98 model atmospheres, we calibrated average relations to transform several representative colors and BCs from the HST to the ground–based photometric system where most of the empirical scales are calibrated (cf. Montegriffo et al. 1998 and references therein). All the BCP98 models included in our grid of stellar parameters for all the three NICMOS and four WFPC2 cameras have been used to compute the best fits. Very similar best fit relations can be obtained using the LCB97 model atmospheres.
In Fig. 11 the best model fits to the difference between the ground–based (J–H) and (J–K) colors and the NICMOS (F110W–F160W) and (F110W–F222M) ones, respectively, as a function of the NICMOS quantities are shown.
The derived transformations are practically metallicity and gravity independent and very similar relations can be obtained adopting a particular metallicity or gravity.
A cubic polynomial relation is required to transform the (F110W–F160W) into the ground–based (J–H) color, while a simple linear relation allows one to transform (F110W–F222M) into the ground–based (J–K) color. Very small ($<$0.01 mag) global [*r.m.s*]{} values have been obtained. The maximum scatter between the best fit and the model atmospheres occurs at the lowest temperatures ($T_{\rm eff}$=3500–4000 K) and is $\le$0.06 mag in both colors.
For comparison, we also plot the colors of the standard stars with measured F110W, F160W and F222M and ground–based JHK magnitudes, according to the NICMOS Photometry Update WEB page (November 25, 1998), even though the quoted values are still in the process of being updated, and of a set of red stars in the Baade’s window measured by Stephens et al. (1999).
Unfortunately, most of these stars are cooler than 3500K, that is out of the temperature range covered by the selected set of model atmospheres. Nevertheless, the observed quantities are reasonably reproduced (within $\sim$0.1 mag) by the best model fits in the temperature range covered by the models, that is $T_{\rm eff}\ge$3500 K, while at lower temperatures the scatter is larger and also depends on the adopted extrapolation.
In Fig. 12 we show the best model fits to the difference between the ground–based BC$_J$ and BC$_K$ and the corresponding NICMOS BC$_{F110W}$ and BC$_{F222M}$ values, as a function of the (F110W–F160W) and (F110W–F222M) NICMOS colors, respectively. Cubic polynomial relations with even smaller scatters than for colors are required to transform the BC from the NICMOS into the ground–based infrared photometric system.
The numerical relations to transform the selected colors and BCs from the NICMOS to the ground–based photometric system are listed below:
$(J-H)=-0.063(F110W-F160W)^3+0.172(F110W-F160W)^2+0.563(F110W-F160W)+0.007$
$(J-K)=0.803(F110W-F222M)+0.003$
$BC_J=BC_{F110W}+0.069(F110W-F160W)^3-0.181(F110W-F160W)^2+
0.443(F110W-F160W)-0.008$
$
BC_K=BC_{F222M}+0.023(F110W-F222M)^3-0.110(F110W-F222M)^2+
0.204(F110W-F222M)-0.001
$
In Fig. 13 the best model fits to the difference between the ground–based (B–V) and (V–I) colors and the corresponding WFPC2 (F439W–F555W) and (F555W–F814W) values as a function of the corresponding WFPC2 quantities are shown. Cubic polynomial relations are required to transform the (F439W–F555W) and the (F555W–F814W) colors into the corresponding ground–based (B–V) and (V–I) quantities.
The global [*r.m.s*]{} values are still very small ($\le$0.02 mag) as for infrared colors, while the maximum scatter (which occurs at the lowest temperatures and reflects a gravity dependence in the case of the (B–V) and a metallicity dependence in the case of the (V–I)) between the best fit and model atmospheres with selected metallicity or gravity is somewhat larger ($\sim$0.10–0.16 mag) than for the infrared colors.
A direct comparison between our color transformations in the (B–V) planes and those proposed by H95 using their Table 7 indicates an excellent agreement, with only a minor, systematic difference of 0.01 mag (our (B–V)–(F39W–F555W) are bluer than the corresponding H95 values).
In Fig. 14 we report the best model fits to the difference between the ground–based BC$_V$ and BC$_I$ and the corresponding WFPC2 BC$_{F555W}$ and BC$_{F814M}$ values, as a function of the (F439W–F555W) and (F555W–F814W) WFPC2 colors, respectively. As for the corresponding infrared quantities, cubic polynomial relations with very small scatters are required to transform the BC from the WFPC2 into the ground–based visual photometric system.
The numerical relations to transform the selected colors and BCs from the WFPC2 to the ground–based photometric system are listed below:
$(B-V)=0.024(F439W-F555W)^3-0.136(F439W-F555W)^2+1.060(F439W-F555W)-0.014$
$(V-I)=0.049(F555W-F814W)^3-0.077(F555W-F814W)^2+1.120(F555W-F814W)-0.005$
$BC_V=BC_{F555W}+0.016(F439W-F555W)^3-0.070(F439W-F555W)^2+
2.092(F439W-F555W)-0.001$
$BC_I=BC_{F814W}+0.052(F555W-F814W)^3-0.110(F555W-F814W)^2+
2.189(F555W-F814W)-0.007$
Conclusions
===========
Using our synthesis code we derive homogeneous, self consistent color–temperature relations and BCs in the HST infrared and visual photometric systems.
We investigated the behavior of the derived quantities for different stellar parameters and adopted set of model atmospheres. Major results are:
- For a given set of model atmospheres, scatters larger than 0.01 mag in the inferred colors are only observed at low temperatures ($\le$6000 K) when models with different gravities and, to a lower degree, with different metallicities are compared.
- For given stellar parameters, the BCP98 and LCB97 sets of model atmospheres provide significantly different colors (up to a few tenths of a magnitudes) below 5000 K. The inferred discrepancies can be mainly ascribed to the corrections applied by LCB97 to the original grids of Kurucz’s model atmosphere spectra in order to match empirical color–temperature relations.
- The inferred BCs are less dependent on the adopted stellar parameters and model atmospheres than the colors, even at low temperatures.
- Very similar colors and BCs have been obtained for the different NICMOS and WFPC2 cameras.
- Average relations, with a negligible dependence on the stellar parameters and the selected NICMOS or WFPC2 cameras, can be adopted to provide useful transformations between the HST and ground–based photometric systems. In the low temperature domain the ground–based (B–V), (J–H) and (J–K) colors are bluer than the corresponding ones in the HST photometric system, while the ground–based (V–I) color is redder than the corresponding (F555W–F814W) one in the WFPC2 system.
We acknowledge Jon Holtzman for his careful Referee Report and comments and Laura Greggio for the helpful discussions and suggestions. We thank the STScI NICMOS Staff for providing the information on the NICMOS throughput curves. L.O. acknowledges the financial support of the [*“Ministero della Università e della Ricerca Scientifica e Tecnologica”*]{} (MURST) to the project [*Stellar Evolution*]{}.
Bessel, M. S., Castelli, F., & Plez, B. 1998, , 333, 231 (BCP98)
Castelli, F., 1997, private communication
Holtzman, J. A., Burrows, J., Casertano, S., Hester J., Trauger, J. T., Watson, A. M., & Worthey, G. 1995, , 107, 1065 (H95)
Johnson, H. L. 1966, , 4, 193
Leitherer, C., & Heckman, T. M. 1995, , 96, 9
Leitherer, C., et al. 1999, , in press
Lejeune, T., Cuisinier, F., & Buser, R. 1997, , 125, 229 (LCB97)
Montegriffo, P., Ferraro, F.R., Origlia, L., & Fusi Pecci, F. 1998, , 297, 872
Stephens, W., Frogel, J.A., Ortolani, S., Davies, R., Jablonka, P., & Renzini, A., 1999, astro-ph/9909001
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abstract: 'We investigate the dynamical importance of a newly recognized possible source of significant feedback generated during structure formation; namely cosmic ray (CR) pressure. We present evidence for the existence of numerous shocks in the hot gas of galaxy clusters (GCs). We employ for the first time an explicit numerical treatment of CR acceleration and transport in hydro simulations of structure formation. According to our results, CRs provide an important fraction of the total pressure inside GCs, up to several tenths. This was true even at high redshift (z=2), meaning that such non-thermal component could affect the evolution of structure formation.'
author:
- Francesco Miniati
- Dongsu Ryu
- 'T. W. Jones'
- Hyesung Kang
title: Acceleration of CR at Large Scale Shocks and Their Cosmological Role for Structure Formation in the Universe
---
Introduction
============
During the hierarchical process of structure formation, supersonic gas infall and merging events invariably generate powerful, large and long-lived shock waves (Miniati et al. 1999). These should produce copious amounts of CRs, by way of diffusive shock acceleration (e.g. Blandford & Ostriker 1978), including both electrons and ions. In addition, the post-shock gas and diffusively trapped CRs are mostly advected into non-expanding regions, such as filaments and clusters. It turns out that the energy of most of the CR-protons is only marginally affected by radiative losses during a Hubble time. The important possibility, then, is that the latter might accumulate inside forming structures, storing up a substantial fraction of the total pressure there. In addition to cosmic shocks other sources of CRs are also possible. These include AGNs, SNe and stellar winds all of which are candidates for important contributions to the total population of CRs in cosmic structures, although they are not discussed here.
There is growing observational evidence that significant non-thermal activity takes place in GCs. This evidence is provided by extended sources of polarized radio emission, interpreted as synchrotron radiation from relativistic electrons (e.g. Hanisch 1984; Deiss et al. 1997); and by the detection of radiation in excess to what is expected from the hot, thermal X-ray emitting Intra Cluster Medium (ICM), both in the extreme ultra-violet (e.g. Lieu et al. 1996; Kaastra 1998) and in the hard X-ray band above $\sim 10$ KeV (e.g. Fusco-Femiano et al. 1999; Valinia et al. 1999). Although a coherent picture of the non-thermal status of the ICM is still lacking, a very plausible origin for these radiation excesses is inverse-Compton (IC) due to relativistic electrons (e.g. Sarazin & Lieu 1998). Based on this assumption and on the measured EUV excess in Coma cluster, Lieu et al. (1999) have estimated the existence of a CR proton component in approximate [*equipartition*]{} of energy with the thermal gas.
Lessons from Hydro Simulations of Structure Formation
=====================================================
Fig. 1a illustrates a slice of a typical cosmic structure formed in a hydro simulation of structure formation with $\Omega_m\equiv \rho_m/\rho_c = 1$, $\sigma_8=1.05$, computational box size 32 $h^{-1}$ Mpc and 256$^3$ cells. It shows contours of compression ($\nabla \cdot v$) corresponding to shock waves, superposed on a grayscale image of X-ray bremsstrahlung emission from the hot ICM (brighter regions correspond to higher emission). We can easily recognize the external, [*accretion shock waves*]{} enveloping clusters and filaments and processing for the first time the supersonic (accretion) flows. In addition, however, it is also shown that the ICM of GCs is commonly populated by a complex structure of what we call [*internal shock waves*]{}. Unlike the external accretion shocks, internal shocks propagate through gas inside formed (or forming) structures that have already being shock heated. Such shocks include not only [*merger shocks*]{} associated with merger events, but also, and more commonly, [*flow shocks*]{} that are generated because of the complexity of the supersonic accretion flows. They have similar properties (e.g. size, Mach number) to, but are more common than the largest merger shocks. For this reason they are of primary importance for production of relativistic CRs and must be considered when addressing the issues on the non-thermal activity inside GCs.
Hydro simulations allow us to estimate roughly the expected contribution of cosmic shocks in terms of CR production over cosmological time-scales. The relative importance of the CR dynamical role is usually expressed as the ratio of the CR pressure to the thermal pressure: $P_{CR}/P_{th}$. The energy stored in CR can be estimated as a fraction $\epsilon_{E_k\rightarrow CR}$ of the total kinetic energy that has been processed thorough shocks since a certain epoch, say $z=1.5$, up to now ($z=0$). Here, $\epsilon_{E_k\rightarrow CR}$ is the conversion efficiency of kinetic energy into CR energy and can conservatively be assumed to be circa 0.1. Then $P_{CR}/P_{th}$ is given by: $$\frac{P_{CR}}{P_{th}} = \frac{\epsilon_{E_k\rightarrow CR}}{2~E_{th}(z=0)}\int_{t(z=1.5)}^{t(z=0)} (\Phi_{E_k})_{shock} dt$$ where $P_{th}$ is the average pressure in the computational box. Here, $(\Phi_{E_k})_{shock}$ is the flux of kinetic energy across shocks. Our conclusions indicate that roughly $P_{CR}/P_{th} \simeq 6-8 \epsilon_{E_k\rightarrow CR} \sim 0.6-0.8$ (see Miniati et al. 1999 for more detail). Thus a large fraction of the total pressure inside GCs today could be provided by CRs, in rough agreement with Lieu et al. (1999).
Preliminary Results from Explicit CR Numerical Treatment
========================================================
We have developed unique numerical tools to treat explicitly CR ions acceleration and transport inside simulations of cosmological models (Jones et al. 1999; Ryu et al. 1999). With the CR spatial and spectral information provided by our new scheme, employed in such simulations for the first time, we can assess the question of the non-thermal dynamical contribution in GCs more accurately. In particular, we can begin to explore the ratio $P_{CR}/P_{th}$ as a function of the cluster temperature and for different redshifts. In the simulation described here we have adopted for the fraction of postshock thermal particles to be injected at the shock, the value $f_{inj}\simeq 10^{-4}$. Our results are shown in Fig. 1b. First of all, they indicate that for any $T_x$ and $z$ $P_{CR}$ is [*not*]{} a negligible fraction of $P_{th}$, in accord with our previous findings. Also, the four panels show that for any $z$ the ratio $P_{CR}/P_{th}$ tends to be larger for smaller clusters. Finally, such a ratio not only is still not negligible at high $z$, but it is actually larger at higher $z$. This indicates that the evolution of the large scale structure could be significantly affected by this dynamical component.
Discussion & Conclusions
========================
We have shown that the ICM of GCs is commonly populated by numerous internal flow shocks with similar characteristic to, but not necessarily associated with major merger events. These along with accretion shocks and merger shocks are likely to play an important role for the non-thermal activity of the ICM. We have also shown that CR pressure could provide a substantial fraction of the total pressure in GCs today, thus affecting GC mass estimates based on the hydrostatic equilibrium assumption and in turn, the baryonic fraction estimates (which end up being biased high). We have also shown that CR pressure was significant already at high $z$, therefore possibly affecting the evolution of structure formation. Since this is often used as a tools for discriminating among different cosmological models (e.g. Carlberg et al. 1997; Bahcall & Fan 1998), the role of CR pressure should be well understood in order to apply evolutionary arguments with confidence.
Support at the University of Minnesota was provided by NSF and the U of MN Supercomputing Institute. FM was supported in part by a Doctoral Dissertation Fellowship at the University of Minnesota. DR and HK were supported in part by the KOSEF grant 1999-2-113-001-5.
Bahcall, N. A. & Fan, X. 1998, , 504, 1 Blandford, R. D. & Ostriker, J. P. 1978 , 221, L29 Carlberg, R. G., Morris, S. L., Yee, H. K. C. & Ellingson, E. 1997, , 479, L19 Deiss, B. M., Reich, W., Lesch, H. & Wielebinsky, R. 1997, , 321, 55 Fusco-Femiano, R., Dal Fiume, D., Feretti, L., Giovannini, G., Grandi, P., Matt, G., Molendi, S. & Santangelo, A. 1999, , 513, L21 Hanisch, R. J. 1982, , 116, 137 Kaastra, J. 1998, in Proc. 32nd COSPAR Scientific Assembly, in press (astro-ph/9808012) Jones, T. W., Ryu, D. & Engel, A. 1999, , 512, 105 Lieu, R., Mittaz, J. P. D., Bowyer, S., Lockman, F. J., Hwang, C. -Y. & Schmitt, J. H. M. M. 1996, , 458, L5 Lieu, R., Ip, W.-H., Axford, W. I. & Bonamente, M. 1999, , 510, L25 Miniati, F., Ryu, D., Kang, H., Jones, T.W., Cen, R. & Ostriker, J. P. 1999, , submitted Ryu, D., Miniati, F., Jones, T.W., Kang, H. 1999, Astrophysical Plasmas: Codes, Models & Observations. Mexico City 25-29 October, Editors: S.J. Arthur, N. Brickhouse and J. Franco Sarazin, C. L. & Lieu, R. 1998, , 494, L177 Valinia, A., Henriksen, M., Loewenstein, M., Roettiger, K., Mushotzky, R. F. & Madejski, G. 1999, , 515, 42
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abstract: 'This paper presents 13 color CCD intermediate-band spectrophotometry of a field centered on the open cluster M48 (NGC 2548), from 400nm to nearly 1000nm, taken with Beijing-Arizona-Taiwan-Connecticut (BATC) Multi-Color Survey Photometric system. The fundamental parameters of this cluster are derived with a new method which based on the comparison between the spectral energy distributions (SEDs) of cluster stars and the theoretical SEDs of Padova models. We find that the best-fitting age of M48 is 0.32 Gyr, the distance is 780 pc and a reddening $E(B-V)=0.04$ with a solar metallicity $Z=0.019$.'
author:
- 'Zhen-Yu Wu, Xu Zhou, Jun Ma, Zhao-Ji Jiang, Jian-Sheng Chen'
title: 13 Color Photometry of Open Cluster M48
---
Introduction
============
Open clusters have long been recognized as important tools in the study of the Galactic disk. Their value lies in the improvement in accuracy for distance determination, metal content, reddening and age produced by the collective stellar sample which shares these properties. So open clusters are excellent tracers of the abundance gradient along the Galactic disk as well as many other important disk properties, such as the age-metallicity relation, abundance gradient evolution, disk age, and so on [@fr95; @tw97; @ch03].
The open cluster M48, also known as NGC 2548 is a quite conspicuous object and should be a naked-eye object under good weather conditions. It was firstly studied by @pesch who made $UBV$ photoelectric photometry of 37 stars, from which the cluster turned out to have a reddening $E(B-V) = 0.04 \pm 0.05$ and a distance 630 pc. DDO photoelectric photometry of 5 red giants was obtained by @claria. From 4 members of these red giants, he conclude that this cluster has a metallicity \[Fe/H\]=0.1, $E(B-V) = 0.06 \pm 0.02$ and a distance of 530 pc. @merm derived an age of 0.30 Gyr based on a synthetic composite color-magnitude diagram. @wu determined absolute proper motions and membership probabilities for 501 stars in the field of M48. More recently, @rid03 obtained a new result of M48 taken in the $u^{'}g^{'}r^{'}i^{'}z^{'}$ SDSS filter system. They find that a distance of 700 pc, an age of 0.40 Gyr and a metallicity of \[Fe/H\]=0.0 can fit their data best.
In this paper we present a new photometric result of M48 taken with BATC Multi-Color Survey Photometric system. The BATC filter system consists of 15 filters of band-widths 150 – 350 that cover the wavelength range 3300 – 10000, which avoids strong and variable sky emission lines [@fan]. As the first object in BATC survey, the old open cluster M67 has been studied based on color-magnitude diagram (CMD)[@fan]. Using the BATC filter system, @ch01 studied the globular cluster NGC 288 by comparing SEDs of bright stars with Kurucz models. The estimated effective temperatures and average value of \[Fe/H\] for these stars are consistent with spectroscopic determinations. Based on the BATC survey observations, the main aim of this study is to determine simultaneously the fundamental parameters of M48, such as age, distance, metallicity and reddening, by comparing observational SEDs of cluster stars with theoretical stellar evolutionary models.
The observations and reduction of the M48 data are described in Sec. 2. In Sec. 3, we derive fundamental parameters of M48. Conclusions and summary are presented in Sec. 4.
Observation and Data Reduction
==============================
Observation
-----------
The observations are done with BATC photometric system at Xinglong Station of National Astronomical Observatories, Chinese Academy of Sciences (NAOC). The 60/90 cm f/3 Schmidt telescope is used with a Ford Aerospace $2048\times2048$ CCD camera at its main focus. The field of view of the CCD is $58\arcmin\times58\arcmin$ with a plate scale of 1.7 arcsec per pixel.
The filter system of BATC project is defined by 15 intermediate-band filters which are designed specifically to avoid most of the known bright and variable night sky emission lines. The definition of magnitude for the BATC survey is in the $AB_{\nu}$ system which is a monochromatic $F_{\nu}$ system first introduced by @og: $$m_{\textrm{batc}}=-2.5\,\log\,{F_{\nu}}-48.60$$ where $F_{\nu}$ is the appropriately averaged monochromatic flux (measured in unit of $\mbox{erg}~\mbox{s}^{-1}~\mbox{cm}^{-2}~\mbox{Hz}^{-1}$) at the effective wavelength of the specific pass-band [@fuku]. In BATC system, the $F_{\nu}$ is defined as [@yan]: $$F_{\nu}= \frac{\int\textrm{d}(\log \nu)f_{\nu}R_{\nu}}{\int\textrm{d}(\log
\nu)R_{\nu}}$$ which ties directly the magnitude to input flux. The system response $R_{\nu}$ actually used to relate the spectrum energy distribution of the source $f_{\nu}$ and $F_{\nu}$, includes only the filter transmissions. Other effects, such as the quantum efficiency of the CCD, the response of the telescope’s optics, the transmission of atmosphere, etc., are ignored. This makes the BATC system filter-defined, and the effective wavelengths are affected only at the $\pm 6$ level after taking CCD quantum efficiency and aluminum reflection into account [@yan].
The flux calibration of the BATC photometric system is defined by four spectrophotometric standard stars of @og: HD 19445, HD 84937, BD+262606 and BD+ 174708. The fluxes of the 4 standard stars have been re-calibrated by @fuku. Their magnitudes in BATC system have been slightly corrected recently by cross checking with the data obtained on a number of photometric nights [@zhou01].
In the nights judged photometric by the observers, the standard stars were observed between air-masses 1.0 and 2.0 for each filter band. The observing procedure of survey program field and photometry are described in detail in @zhou01 and @yan.
Because of the very low quantum efficiency of the thick CCD used in the bluest filters, two BATC filters $a$ and $b$ are not used in the observation of M48 field. In Table \[tb1\], for each BATC filter, we list the corresponding effective wavelength, FWHM and exposure time.
Data Reduction
--------------
Preliminary reductions of the CCD frames, including bias subtraction and field flattening, were carried out with an automatic data reduction procedure called *Pipline I* which has been developed as a standard for the BATC survey in NAOC [@fan]. The astrometric plate solution is obtained by *a priori* knowing the approximate plate center position, and then using this information to register the brighter stars in each frame with the Guide Star Catalog (GSC) coordinate system [@jen].
A *Pipeline II* program which based on the DAOPHOT II stellar photometric reduction package of Stetson [@stet] was used to measure the instrumental magnitudes of point sources in BATC CCD frames. The *Pipeline II* reduction procedure was performed on each single CCD frame to get the PSF magnitude of each point source. The instrumental magnitudes were then calibrated to BATC standard system [@zhou03]. The average calibration error of each filter is less than 0.02 mag. The other sources of photometric error include photon statistics of star and sky, readout noise, random and systematic errors from bias subtraction and flat-fielding, and the PSF fitting, are considered in *Pipeline II* and the total estimated errors of each star are given in the final catalog [@zhou03]. Stars which are detected in at least 3 filters are included in the final catalog.
Fundamental Parameters Derived from SEDs
========================================
CMD is the main tool to derive the fundamental parameters of star cluster. In the BATC photometric system, 15 filters can be used to form various CMDs. However, it is difficult to derive a consistent result from various different CMDs. In the other hand, the large number of bands available in our data set provides a sort of *low resolution spectroscopy*, which defines the SED of each star quite well. So it is possible to derive the fundamental parameters of star cluster by fitting the SEDs of member stars with theoretical stellar evolutionary models.
Fitting Procedure and Results
-----------------------------
The observed SED of a star is determined by intrinsic (mass, age, metallicity) and extrinsic (distance and reddening) parameters. The member stars in a star clusters share the same age, metallicity, reddening and distance. The idea of our new method is to compare the observed SEDs of member stars with theoretical models to obtain a combination of best-fitting cluster parameters.
Our fitting procedure can be separated into two steps. First, we calculate the deviation between observed and theoretical SEDs of each member star with different sets of cluster parameters including age, metallicity, distance and reddening: $$m_{x}[E(B-V), r]=m_{x_{obs}}+5-5\log r-R_{x}\times E(B-V)$$ where $m_{x_{obs}}$ is the observed magnitude of a cluster member in filter $x$, $m_{x}[E(B-V),r]$ is the absolute magnitude corrected by distance $r$ and reddening $E(B-V)$, $R_{x}$ is extinction coefficient which transform the $E(B-V)$ to the BATC filter system and derived by @ch00 based on the procedure in Appendix B of @sch.
The criteria of fit for a cluster member $s$ between observed SED and theoretical model is defined by: $$\zeta_{s}\,[\log(t),Z,m,r,E(B-V)]=\sqrt{\frac{\sum\{m_{x}[E(B-V),r]-M_{x}\}^{2}}{n}}$$ where $M_{x}$ is the theoretical magnitude of a star in filter $x$, computed from an chosen theoretical isochrone models with age $\log(t)$, metallicity $Z$ and mass $m$, $n$ is the number of filters used in the fit.
For any set of cluster parameters combination, we can find the smallest value of $\zeta_{s}$ viz. $MIN_{\zeta_{s}\,[\log(t),Z,r,E(B-V)]}$ for the member star $s$ and give the best-matched theoretical mass of that star. Then we can get the total deviation of all member stars of cluster under this set of fundamental parameters. $$\zeta\,[\log(t),Z,r,E(B-V)]=\frac{\sum{MIN_{\zeta_{s}\,[\log(t),Z,r,E(B-V)]}}}{N}$$ where $N$ is the number of cluster members used in the fit. For a set of fundamental parameters that best match the observed and theoretical SEDs of cluster members will yield the minimum of $\zeta$.
As our request, Dr. L. Girardi has kindly calculated isochrones of our filter system using the known BATC filter transmission curves and their Padova stellar evolutionary models [@gir00; @gir02]. The Padova isochrone sets are computed with updated opacities and equations of state, and a moderate amount of convective overshoot.
The results of @wu proper motion and membership study of M48 are used to determine members in this cluster. Stars with membership probabilities greater than 0.7 are considered to be members [@wu]. All stars considered as members based on their proper motions were used in our fitting.
In our fitting procedure, the distances was chosen from 600 pc to 900 pc in intervals of 10 pc, $E(B-V)$ from 0.00 to 0.10 in intervals of 0.01. Theoretical isochrone models with metallicity $Z=$ 0.08, 0.019, 0.030 and age $\log(t)$ from 8.0 to 9.0 in intervals of 0.05 were chosen.
We find that, with a distance of 780 pc, reddening $E(B-V)=0.04$, the theoretical model with age of 0.32 Gyr and metallicity $Z=0.019$ give the smallest value of $\zeta$ and best fit the observed SEDs. In Figure \[sed\], we plot the best-fitting results for some member stars. The mass of each object in Padova models is labelled on the right of each corresponding curve in the unit of solar mass $M_\sun$. In the top panel of Figure \[sed\], SEDs of 11 main sequence (MS) stars with mass from 1.5662 $M_\sun$ to 3.1878 $M_\sun$ are plotted. In the bottom panel of Figure \[sed\], the SED of a red giant star is plotted. We can see that the observed SEDs of both MS stars and red giant stars with the derived best-fitting parameters, can fit the theoretical ones very well.
In Figure \[cmd\], we plot four representative CMDs from our data: ($c-p$) vs $c$ gives us the widest pass-band colors; ($c-e$) vs $c$ gives us the cleanest CMD; and ($f-i$) vs $f$ give us the deepest CMD. All of the CMDs have a well defined MS and MS turnoff point. All stars in the field of M48 are plotted in each diagram. Stars with known membership probabilities greater than 0.7 determined from the proper motion study of @wu, are given distinct plotting symbols. In each CMD of Figure \[cmd\], parameters derived from SED-fitting are adopted. The theoretical isochrone with the parameters derived by SED-fitting can fit star’s distribution in each CMD very well.
The uncertainties in our derived best-fitting fundamental parameters can be determined from the observational photometric errors. For each member of cluster that used in previous fitting procedure, a new magnitude in each BATC filter was generated in a Monte Carlo fashion by adding Gaussian deviates to the observed magnitude. The standard deviations of the deviates in each BATC filter were taken to be 0.05 mag which is the maximum photometric error in most filters of used sample stars. All of these stars with new artificial magnitudes were then fitted by the same procedure as previous section. We repeated this simulation 100 times. In the end, we find that the uncertainty caused by photometric errors in distance is $\pm10$ pc, in reddening $E(B-V)$ is $\pm0.01$, in age $\log(t)$ is $\pm0.05$. The obtained metallicity keeps the same value. So the effect of photometric errors on the best-fitting results is very small. At the same time, there still remains the possibility that there could be some sort of systematic calibration problem between the photometry and the models, or systematic errors in this particular set of photometry, although such errors are small, and not necessarily zero.
In Table \[tb2\] we list the fundamental parameters of M48 derived from this and previous works. The derived age of this work is consistent with those derived by previous works, the largest difference is 0.08 Gyr [@rid03]. Our derived distance is very close to these results derived in resent years [@dias02; @rid03]. Our derived metellicity is consistent with that of @rid03 and close to the value derived by @claria. The reddening $E(B-V)$ derived in this work is same as that of @pesch and very close to those derived by previous works.
Conclusions
===========
In this paper, we present and discuss new BATC multicolor photometry results for the intermediate age open cluster, M48. Comparing the observed SEDs of cluster member stars with the theoretical SEDs of Padova models, we find a set best-fitting fundamental parameters for this cluster: an age of 0.32 Gyr, a distance of 780 pc, a metallicity of $Z=0.019$ with a reddening $E(B-V)=0.04$. This SED-fitting result can also fit CMDs formed from our data very well. Our derived values are also consistent with those derived by previous authors [@merm; @rid03]. So, we can say that using the Padova theoretical isochrones, we can effectively fit our data obtained based on BATC filter system with theoretical isochrones and SEDs to extract useful cluster information.
We would like to thank the anonymous referee for his/her insightful comments and suggestions that improved this paper. We wish to thank Dr. Leo Girardi for his assistance in the theoretical models. Z.Y.W acknowledge Dr. Wen-Ping Chen for many useful suggestions. This work has been supported by the Chinese National Key Basic Research Science Foundation (NKBRSF TG199075402), in part by the Chinese National Science Foundation, No. 10473012 and 10373020.
Chen, A. Ph. D. thesis, 2000, Institute of Astronomy, National Central University, Chung-Li, Taiwan Chen, A., Tsay, W., Tsai, W., et al. 2000, , 120, 2569 Chen, L., Hou, J. L., & Wang, J. J. 2003, , 125, 1397 Clariá, J. J. 1985, , 59, 195 Dias, W. S., Alessi, B. S., Moitinho, A., & Lépine, J. R. D. 2002, , 389, 871 Fan, X., Burstein, D., Chen, J. et al. 1996, , 112, 628 Friel, E. D. 1995, , 33, 381 Fukugita, M., Ichikawa, T., Gunn, J. E., et al. 1996, , 111, 1748 Girardi, L., Bressan, A., Bertelli, G., & Chiosi, C. 2000, , 141, 371 Girardi, L., Bertelli, G., Bressan, A., et al. 2002, , 391, 195 Jenkner, H., Lasker, B. M., Sturch, C. R., et al. 1990, , 99, 2082 Mermilliod, J. C. 1981, , 97, 235 Oke, J. B., & Gunn, J. E. 1983, , 266, 713 Peshch, P. 1961, , 134, 620 Rider, C. J., Tucker, D. L., Smith, J. A., 2004, , 127, 2210 Schlegel, D. J., Finkbeiner, D. P., & Davis, M., 1998, , 500, 525 Stetson, P. B. 1987, , 99, 191 Twarog, B. A., Ashman, K. M., & Anthony-Twarog, B. J. 1997, , 114, 2556 Wu, Z. Y., Tian, K. P., Balaguer-Núñez, L., et al. 2002, , 381, 464 Yan, H., Burstein, D., Fan, X., et al. 2000, , 112, 691 Zhou, X., Jiang Z., Xue, S., et al. 2001, , 1, 372 Zhou, X., Jiang Z., Ma, J., et al. 2003, , 397, 361
[ccccc]{} 1&$c$&4180.0&310.0&38\
2&$d$&4532.0&330.0&183\
3&$e$&4916.0&370.0&98\
4&$f$&5258.0&340.0&84\
5&$g$&5785.0&290.0&64\
6&$h$&6069.0&310.0&41\
7&$i$&6646.0&490.0&44\
8&$j$&7055.0&240.0&11\
9&$k$&7545.0&190.0&21\
10&$m$&8020.0&260.0&76\
11&$n$&8483.0&170.0&88\
12&$o$&9180.0&250.0&148\
13&$p$&9736.0&280.0&134\
[ccccc]{} This work&780&0.32&0.04&0.0\
@rid03&700&0.40&0.03&0.0\
@dias02&770&0.36&0.03&0.08\
@claria&530&&0.06&0.1\
@pesch&630&&0.04&\
@merm&&0.30&&\
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author:
-
bibliography:
- 'IEEEabrv.bib'
- 'bibliography.bib'
title: 'Towards Hardware Implementation of Neural Network-based Communication Algorithms'
---
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors thank Luc Dartois for comments that greatly improved the manuscript.
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---
abstract: 'Since its launch in 2009, the *Kepler* telescope has found thousands of planets with radii between that of Earth and Neptune. Recent studies of the distribution of these planets have revealed a rift in the population near 1.5–2.0$R_\Earth$, informally dividing these planets into “super-Earths" and “sub-Neptunes". The origin of this division is not well understood, largely because the majority of planets found by Kepler orbit distant, dim stars and are not amenable to radial velocity follow-up or transit spectroscopy, making bulk density and atmospheric measurements difficult. Here, we present the discovery and validation of a newly found $2.03^{+0.08}_{-0.07}~R_{\Earth}$ planet in direct proximity to the radius gap, orbiting the bright ($J=8.32$ mag), nearby ($D=44.5$ pc) high proper motion star Wolf 503 (EPIC 212779563). We classify Wolf 503 as a K3.5V star and member of the thick disc population. We determine the possibility of a companion star and false positive detection to be extremely low using both archival images and high-contrast adaptive optics images from the Palomar observatory. The brightness of the host star makes Wolf 503b a prime target for prompt radial velocity follow-up, *HST* transit spectroscopy, as well as detailed atmospheric characterization with *JWST*. With its measured radius near the gap in the planet radius and occurrence rate distribution, Wolf 503b offers a key opportunity to better understand the origin of this radius gap as well as the nature of the intriguing populations of “super-Earths" and “sub-Neptunes" as a whole.'
author:
- 'Merrin S. Peterson'
- Björn Benneke
- 'Trevor J. David'
- 'Courtney D. Dressing'
- David Ciardi
- 'Ian J. M. Crossfield'
- 'Joshua E. Schlieder'
- 'Erik A. Petigura'
- 'Eric E. Mamajek'
- 'Jessie L. Christiansen'
- Sam Quinn
- 'Benjamin J. Fulton'
- 'Andrew W. Howard'
- Evan Sinukoff
- Charles Beichman
- 'David W. Latham'
- Liang Yu
- Nicole Arango
- Avi Shporer
- Thomas Henning
- 'Chelsea X.Huang'
- 'Molly R. Kosiarek'
- Jason Dittmann
- Howard Isaacson
bibliography:
- 'main.bib'
title: 'A 2 Earth Radius Planet Orbiting The Bright Nearby K-Dwarf Wolf 503'
---
Introduction {#sec:intro}
============
The majority of close-in planets found by NASA’s *Kepler* satellite throughout the past decade are smaller than Neptune, but larger than Earth [@Batalha2013] [@Mullally15] [@Howard13]. The *Kepler* and *K2* missions have shown us that, of the planets within our detection limits ($P>100$ days, $R_p>1.0R_{\Earth}$), these smaller planets are by far the most common in the galaxy [@Fressin13] [@Fulton2017], though there is no analog in the solar system from which this could have been predicted. A drop in the population of planets at radii larger than $4.0~R_\Earth$ (i.e., larger than Neptune) is satisfactorily explained by runaway gas accretion [@Bate2003] [@Mordasini09]. Larger planets are massive enough to accrete H and He from the protoplanetary disc, becoming puffy and increasing in radius. However, refined studies of the distribution of planets within the $1-4~R_\Earth$ range have revealed a significant drop in the population, or “Fulton gap" between $1.5-2.0~R_\Earth$ [@Fulton2017] [@Owen13], which is not yet well-understood.
{width="2.0\columnwidth"}
Photoevaporation presents a possible explanation for the gap, and is a particularly important factor for the close-in planets preferentially detected by *Kepler*. Planets with radii between $1.5$ and $2.0~R_\Earth$ could represent a relatively rare group of planets retaining thin atmospheres, while super-Earths are photoevaporated rocky bodies and the sub-Neptunes are massive enough to retain thick atmospheres [@Lopez16]. It has also been postulated that the sub-Neptunes form earlier in the evolution of the protoplanetary disc than super-Earths, when there is still more gas in the protoplanetary disc, giving them thicker atmospheres and larger radii [@Lee2014]. The gap would then represent an intermediate stage in disc evolution in which planets are not likely to form.
Explanations for the bimodal distribution of planets which invoke composition should be tested with mass (i.e., bulk density) measurements and transit spectroscopy to determine the composition and atmospheric mass fraction of planets on both sides of the rift.
However, planets favorable for these detailed follow-up characterizations are missing. Although *Kepler* has found thousands of bona fide $1-4~R_\Earth$ planets, due to the satellite’s due to Kepler’s 100 sq. deg. field-of-view, relatively few bright stars were targeted and most Kepler planet hosts are distant and dim. As is shown in Fig. \[fig:Hmag\_vs\_Rp\], Wolf 503b joins only a handful of bright targets at its size. With so few photons, the detailed spectra required to make quality mass and atmospheric composition measurements are often impossible to obtain, and although there has been much effort to constrain the density of planets in this region [@Dumusque14] [@Weiss2014] [@Rogers15], the parameter space near the Fulton gap remains relatively unexplored.
In this work, we present the detection and validation of a newly found $2.0~R_\Earth$ planet from *K2* which represents one of the best opportunities to date to conduct a detailed radial velocity and atmospheric study of a planet in the 1-4$R_{\Earth}$ range. In Sec. \[sec:photometry\] we describe the collection and calibration of the *K2* photometry, as well as our detection pipeline. In Sec. \[sec:stellarHistory\] we discuss the research history of the host star and its galactic origins. We obtain our own spectrum of Wolf 503, classify the star and determine stellar parameters in Sec. \[sec:spectroscopy\]. Our methods of target validation are described in Sec. \[sec:validation\] and the final light curve fitting and results are found in Sec. \[sec:fitting\]
[ m[2.4cm]{} m[2.2cm]{} m[2.4cm]{} ]{} Parameter & Value & Source\
&\
\[0.1ex\] $\alpha$ R.A. (hh:mm:ss) J2000 & 13:47:23.4439 &\
$\delta$ Dec. (dd:mm:ss) J2000 & -06:08:12.731 &\
EPIC ID & 212779563 &\
&\
B (mag)..... & $11.30\pm0.01$ & [@Mermilliod87]\
V (mag)..... & $10.28\pm0.01$ & [@Mermilliod87]\
G (mag)..... & $9.808\pm0.001$ & Gaia DR1\
J (mag)..... & $8.324\pm0.019$ & 2MASS\
H (mag)..... & $7.774\pm0.051$ & 2MASS\
K (mag)..... & $7.617\pm0.023$ & 2MASS\
&\
$\mu_{\alpha}$ (mas yr$^{-1}$) & $-343.833\pm0.073$ & Gaia DR2\
$\mu_{\delta}$ (mas yr$^{-1}$) & $-573.134\pm0.073$ & Gaia DR2\
Barycentric rv (km s$^-1$) & $-46.826\pm0.015$ & Gaia DR2\
Distance (pc) & $44.583\pm0.096$ & Gaia DR2\
Age (Gyr) & $11\pm2$ & This Paper\
Spectral Type & $K3.5\pm0.5$V & This Paper\
$[Fe/H]$ & $-0.47\pm0.08$ & This Paper\
$logg$ (K) & $4.62^{+0.02}_{-0.01}$ & This Paper\
${\ifmmode{T_{\rm eff}}\else $T_{\rm eff}$\fi}$ (K) & $4716 \pm 60$ & This Paper\
$M_*$ ($M_{\odot}$) & $0.688^{+0.023}_{-0.016}$ & This Paper\
$R_*$ ($R_{\odot}$) & $0.690^{+0.025}_{-0.024}$ & This Paper\
$L_*$ ($L_{\odot}$) & $0.227^{+0.009}_{-0.010}$ & This Paper\
\[tab:stellar\]
Observations and Analysis {#sec:obs}
=========================
Identified as a planet candidate from C17 of *K2*, Wolf 503 was recognized as an excellent host for follow-up study, being both bright ($K_p=9.9$) and nearby (45pc). Here we present the treatment of the photometry used to detect Wolf 503b, as well as our planet validation techniques, and derive both planetary and stellar parameters.
{width="2.0\columnwidth"}
Photometry Extraction and Transit Detection {#sec:photometry}
-------------------------------------------
The photometric extraction and transit detection methods used to identify Wolf 503b are the same as those applied to all light curves in C17 and are described in our corresponding C17 summary paper Crossfield et al. 2018 (submitted). As *K2* operates using only two of *Kepler*’s four initial reaction wheels, the telescope drifts along its roll axis by a few pixels every several days, and thruster fires are used to maintain the telescope’s pointing. The change in flux resulting from this drift is removed by fitting the flux as a function of position along the drift path, which is highly similar between thruster fires. However, data acquired during these thruster burns is not reliable and is masked out, as in the first transit of the light curve for Wolf 503, shown in Fig. \[fig:full\_curve\].
With the extracted light curve, we detected a candidate at $P=6.0$ days with $S/N=38$ having 11 transits throughout the time of observation. The candidate was marked as a particularly intriguing KOI for the properties of its host star following the manual vetting procedure of the C17 candidates.
![Contrast sensitivity and inset image of Wolf 503 in Br-$\gamma$ as observed with the Palomar Observatory Hale Telescope adaptive optics system, The $5\sigma$ contrast limit is plotted against angular separation in arcseconds (fill circles). The shaded region represents the dispersion in the sensitivity caused by the azimuthal structure in the image (inset).\[fig:AO\]](wolf503_AO_cropped.pdf){width="1.0\columnwidth"}
{width="2.0\columnwidth"}
Activity, Age and Membership {#sec:stellarHistory}
----------------------------
Wolf 503 (BD-05 3763, MCC 147, LHS 2799, G 64-24, HIP 67285, TYC 4973-1501-1, 2MASS J13472346-0608121) has been a sparsely studied nearby cool star since its discovery a century ago as a high proper motion star by @Wolf1919. The star subsequently appeared in several catalogues of high proper motion over the past century, Ci 20 806 [@Porter30], G 64-24 [@Giclas63], with Wilhelm Luyten designating the star no fewer than six times in his proper motion catalogs [^1].
The star was classified in numerous spectral surveys, as a K5V by @Upgren72 [; identified as UPG 336], and @Bidelman85 published Kuiper’s posthumous classification for the star as K4 from his 1937-1944 survey. Of the K5V stars from Upgren et al.’s (1972) survey, 89 were later classified in the CCD optical spectroscopy surveys of nearby stars by @Gray03 and @Gray06, and assigned an average type of K4.6 ($\pm$0.1 subtype s.e.m., $\pm$1.1 subtype rms). @Pickles10 found that the best fit template for the $B_T V_T J H K_s$ photometry was that for a K4V star. The combined previously published values suggest a spectral type of K4V.
Recently Gaia DR2 has provided an ultra-precise trigonometric parallax ($\varpi$ = 22.430$\pm$0.048 mas; corresponding to $d$ = 44.583$\pm$0.096 pc), as well as precise proper motion and radial velocity measurements, which are listed in Table \[tab:stellar\]. Gaia itself measured a radial velocity of -46.64$\pm$0.50 kms$^{-1}$ (2 observations), and independently, @Sperauskas16 reported radial velocity of -47.4$\pm$0.7kms$^{-1}$ based on 2 CORAVEL measurements over 98 days. Combining the Gaia DR2 position, proper motion, and parallax, and the mean Gaia DR2 ground-based radial velocity (from HARPS), we estimate barycentric space velocity of $U, V, W$ = -25.21, -116.86, -88.44 ($\pm$0.18, 0.21, 0.13) kms$^{-1}$ (total velocity 148.71$\pm$0.18 kms$^{-1}$), where $U$ is towards the Galactic center, $V$ is in the direction of Galactic rotation, and $W$ is towards the north Galactic pole [@ESA97]. Using the velocity moments and local stellar population densities from @Bensby03, this $UVW$ velocity is consistent with the following membership probabilities: $<$10$^{-5}$%, 81%, 19%, for the thin disk, thick disk, and halo, respectively, highly indicative of membership to the thick disc population.
@Mikolaitis17 analyzed high resolution high S/N HARPS spectra and found the star to be fairly metal poor (\[Fe/H\] $\simeq$ -0.37 based on two pairs of \[Fe I/H\] and \[Fe II/H\] abundances). Its combination of low metallicity, supersolar \[Mg I/Fe\] ($\sim$0.28) and \[Zn I/Fe\] (0.19), and subsolar \[Mn I/Fe\] ($\sim$-0.16), led @Mikolaitis17 to chemically classify the star as belonging to the thick disk. Hence there is both kinematic and chemical abundance data for Wolf 503 consistent with its membership to the thick disk. The thick disk shows a metallicity-age gradient [e.g. @Bensby04], and given Wolf 503’s combination of \[Fe/H\] and \[Mg/Fe\] compared to age-dated thick disk members [@Haywood13], it is likely in the age range $\sim$9-13 Gyr. Hence we adopt 11$\pm$2 Gyr for Wolf 503.
Spectroscopy and Stellar Parameters {#sec:spectroscopy}
-----------------------------------
We obtained an $R\approx2000$ infrared spectrum of Wolf 503 covering the spectra range between $0.7-2.55 \micron$ at the NASA Infrared Telescope Facility (IRTF). We use the SpeX spectrograph in SXD mode with the 0.3“ x 15” slit. The spectrum was taken June 3, 2018, on a partly cloudy night with an average seeing of $0.6\arcsec$. Reduction of the spectrum was performed with the SpeXTool [@Cushing2005] and xtellcor [@Vacca03] software packages as in @Dressing17. The sky subtraction was performed using a nearby A star, HD 122749, observed immediately after Wolf 503b, at a similar airmass. Before performing our spectral analysis, we corrected for the radial velocity of the target and barycentric velocity of IRTF. The final spectrum is shown in Fig. \[fig:spectrum\]. The best match indicates a spectral type of $K3.5\pm1$V suggesting an effective temperature of approximately $4750\pm100\,K$ for SpeX spectrum.
During the vetting of candidates from C17 of $K2$ described in Crossfield et al. 2018 (submitted), a spectrum was also obtained from the Tillinghast Reflector Echelle Spectrograph [TRES; @furesz:2008] mounted on the 1.5-m Tillinghast Reflector at Fred Lawrence Whipple Observatory on Mount Hopkins was obtained on UT 2018 May 23. TRES is a fiber-fed, cross-dispersed echelle spectrograph with a resolving power of $R {\mathord{\sim}}44,000$, a wavelength coverage of $3850$–$9100$Å, and radial-velocity stability of $10$ to $15$[ms$^{-1}$]{}. The spectrum was reduced and optimally extracted, and wavelength calibrated according to the procedure described in @buchhave:2010, and we derived stellar atmospheric parameters using the Stellar Parameter Classification code [SPC; @buchhave:2012]. We find ${\ifmmode{T_{\rm eff}}\else $T_{\rm eff}$\fi}= 4640 \pm 50$K, ${\ifmmode{\log{g}}\else $\log{g}$\fi}= 4.68 \pm 0.10$, ${\mbox{$\rm{[Fe/H]}$}}= -0.47 \pm 0.08$, and ${\ifmmode{v\sin{i_\star}}\else $v\sin{i_\star}$\fi}= 0.8 \pm 0.5$. We note that SPC determines the stellar parameters using synthetic spectra with a fixed macroturbulence of $1$[kms$^{-1}$]{}, which may bias [$v\sin{i_\star}$]{} measurements of slow rotators like this one. Regardless, Wolf 503 has a low projected rotational velocity, as is expected for an old K dwarf, which bolsters its status as a good candidate for precise radial velocity observations. We derive an absolute radial velocity of $-46.629 \pm 0.075$[kms$^{-1}$]{}.
We conclude that the SpeX spectrum and the TRES spectrum result in consistent estimates of the stellar temperature. These values are also consistent with the value from the PASTEL catalogue of 4759 K [@soubiran_pastel_2010] as well as Wolf 503’s colors ($B-V=1.02$, $V-K=2.66$), leading us to adopt the spectral type of $K3.5\pm0.5V$.
Finally, we adopt ${\ifmmode{T_{\rm eff}}\else $T_{\rm eff}$\fi}=4716\pm60$ K, the average and scatter of the three spectroscopic values, as our final value for the stellar temperature. We then calculate the stellar parameters using Isoclassify [@huber_asteroseismology_2017]. We adopt the ${\ifmmode{\log{g}}\else $\log{g}$\fi}$ and ${\mbox{$\rm{[Fe/H]}$}}$ from the TRES spectrum, as well as the K magnitude. We use the K magnitude because it is least affected by extinction and inflate the error bars of the K magnitude to account for the uncertainty extinction. We determine the best stellar radius estimate using the direct method in Isoclassify [@huber_asteroseismology_2017]. We obtain the stellar mass using the grid mode. The resulting stellar parameters are listed in Table \[tab:stellar\].
Target Validation {#sec:validation}
-----------------
By far the most pernicious false positives detected by *K2* are eclipsing binaries, which may closely resemble exoplanet transits at grazing incidence, or when the binary system is found in the background of a brighter star [@Morton12]. We used archival and adaptive optics images to investigate the possibility of a false positive detection due to a companion star or background sources, and find no source in the vicinity of Wolf 503 which could have contaminated our detection.
{width="2.0\columnwidth"}
### Adaptive Optics
Wolf 503 was observed on the night of UT 2018 June 01 UT at Palomar Observatory with the 200 Hale Telescope using the near-infrared adaptive optics (AO) system P3K and the infrared camera PHARO (Hayward et al. 2001). PHARO has a pixel scale of 0.025 per pixel with a full field of view of approximately 25. The data were obtained with a narrow-band Br-$\gamma$ filter ($\lambda_o = 2.18;\ \Delta_\lambda = 0.03\ \micron$).
The AO data were obtained in a five-point quincunx dither pattern with each dither position separated by 4. Each dither position is observed three times, each offset from the previous image by 0.5 for a total of 15 frames; the integration time per frame was 4.428 s for a total of 66 on-source integration time. We use the dithered images to remove sky background and dark current, and then align, flatfield, and stack the individual images. The final PHARO AO data have a FWHM of 0.099.
The sensitivities of the final combined AO image were determined by injecting simulated sources azimuthally around Wolf 503 every 45$^\circ$ at separations of integer multiples of the central source. The brightness of each injected source was scaled until standard aperture photometry detected it with 5$\sigma$ significance. The resulting brightness of the injected sources relative to Wolf 503 set the contrast limits at that injection location. The average 5$\sigma$ limits and associated rms dispersion caused by azimuthal asymmetries from residual speckles as a function of distance from the primary target are shown in Fig. \[fig:AO\].
The AO imaging revealed no additional stars within the limit of 0.099. For a binary system at a distance of $44.58$ pc, this limits the separation of a possible binary to less than 4.4 AU. According to the distribution of binary star systems found in @Raghavan10, only 12% of stars are found in such systems. Furthermore, we find the light curve properties (discussed in Sec. \[sec:discussion\]) inconsistent with an eclipsing binary, except in the case of a multiple star system featuring two smaller companions in a 6.0 day orbital period, in which one companion star were completely eclipsed by the other. We consider such a unique multiple-star system far less likely than a single transiting planet. Additionally, most $0.1M_{\odot}$ eclipsing binary companions orbiting within 4.4 AU would induce a radial velocity amplitude on the order of 15 km/s, of which there is no indication through years of radial velocity measurements. We determine the likelihood of a false positive due to a bound companion to be extremely low.
### Archival Images
Even in the absence of a nearby contaminant, adaptive optics cannot eliminate the possibility of a background source directly behind the target, which could be responsible for the signal itself, or would otherwise decrease the apparent transit depth. To address this, we exploit archival imaging from the Palomar Observatory Sky Surveys I, II and Guide Star Catalogue 2 surveys. Fig. \[fig:archival\_images\] shows the present-day location of Wolf 503 in each of the 3 surveys. The blue plate from POSS I (taken May 23, 1952) and the HST image from GSC2 (taken March 29, 1993 with HST) have a 1pixel scale, and the blue plate from POSS II (taken May 7, 1983) has a 0.59 pixel scale.
The nearest object detected to Wolf 503’s 2018 location is the galaxy LCRS B134447.1-055347, which is located $\approx25.1 ''$ from the target, placing it outside the aperture used in our extraction. Moreover, the galaxy has a Gaia magnitude of 19.6: being both 10 magnitudes fainter and outside the aperture, we find no background sources which may influence our photometry.
Parameter Units Value
----------------------- ----------------------- ---------------------------------------------
$T_0$ BJD$_{TBD}$ - 2457000 $1185.36087\substack{+0.00053 \\ -0.00038}$
$P$ day $6.00118\substack{+0.00008 \\ -0.00011}$
$R_p/R_{*}$ % $2.694\substack{+0.026 \\ -0.026}$
$T_{14}$ hr $1.321\substack{+0.051 \\ -0.039}$
$b$ - $0.387\substack{+0.067 \\ -0.061}$
$R_p$ $R_{\Earth}$ $2.030^{+0.076}_{-0.073}$
$a$ AU $0.0571\pm0.0020$
$S$ $S_\Earth$ $69.6\pm3$
$T_{\mathrm{eq,A=0}}$ K $805\pm9$
: Planet Parameters
\[tab:planet\]
![Final light curve fit from ExoFit for the combined 11 transits. In the top panel, the best fit is shown in black with the detrended light curves for each transit. Accounting for the 30 minute cadence of the *K2* data gives the best fit its trapezoidal shape. The residuals are plotted in the middle panel, and are binned in the bottom panel histogram by the number of $\sigma$ from the best fit, where they follow a standard normal distribution of the same area. \[fig:pretty\_plot\]](joint_fit.pdf){width="1.0\columnwidth"}
Light Curve Fitting {#sec:fitting}
-------------------
We fit the light curve of Wolf 503 using ExoFit, a modular light curve analysis tool developed for the joint analysis of data from *Kepler*, *Spitzer*, and *HST*. ExoFit jointly or individually fits transits and explores the parameter space using the Affine Invariant Markov Chain Monte Carlo (AI-MCMC) Ensemble sampler available through the emcee package in Python. Details can be found in @Benneke2017.
We performed individual transit fits in addition to fitting the transits simultaneously. For all fits, we initialize the MCMC chains with uniform priors using the best fit values from TERRA, and fit the transit start time $T_0$, duration $T_{14}$, depth $R_p/R_{*}$, impact parameter $b$, limb darkening coefficient, as well as a linear background for each transit and scatter term. For the joint fit, we also fit the period $P$. In each fit, we assign 6 walkers for each parameter and find good convergence after 3000 steps, taking the initial 60% as burn-in.
The transits were first fit individually, and the resulting fits are shown in Fig. \[fig:individual\_fits\]. Of the 11 transits observed, all are consistent in $R_p/R_{*}$ and $T_{14}$. We obtain our best fitting planet parameters from a joint fit of the 11 transits using the initialization as previously described. The parameters resulting from this fit are summarized in Table \[tab:planet\], where the error in $R_p$ and $a$ are dominated by the stellar parameters, to which we have assigned conservative error estimates. The best fit light curve is shown in Fig. \[fig:pretty\_plot\], where the combined residuals are well-behaved. The regularity in depth and duration shown in Fig. \[fig:individual\_fits\] is most consistent with a transiting planet, and we detect no even-odd variation indicative of an eclipsing binary. Furthermore, the best-fit in Fig. \[fig:pretty\_plot\] is distinctly flat-bottomed, inconsistent with the V-shaped light curves characteristic of eclipsing binaries. This diluted, flat-bottomed shape could be reproduced by two smaller companions orbiting each other with a 6 day period. However, as discussed in Sec. \[sec:validation\], in addition to being less likely than a single transiting planet, such a companion would induce a significant radial velocity which has not been detected.
Discussion {#sec:discussion}
==========
![Planet radius and stellar host magnitude of Wolf 503b (larger circle) in comparison to all planets at the NASA Exoplanet Archive (colored points). The color of the points indicates the stellar temperature. Planets in a similar size range orbiting bright stars are labeled. Wolf 503 is among the brightest systems with a planet near 2 $R_\Earth$ detected to date.\[fig:Hmag\_vs\_Rp\]](Hmag_vs_Rp_modified.pdf){width="1.0\columnwidth"}
![Model transit spectra and simulated *JWST* observations for Wolf 503b. Observations of a single transit with JWST/NIRISS (green) or JWST/NIRSpec (red) could readily detect molecular absorption for hydrogen-dominated, cloud free atmospheres (blue). The planetary mass assumed in the models is 5.3$M_{\Earth}$. Models are computed as described in @benneke_atmospheric_2012 and @benneke_strict_2015. Simulated observational uncertainties are from PandExo [@batalha_pandexo_2017].\[fig:sim\_spec\]](Wolf_503_b_70_1000xsolar.pdf){width="1.0\columnwidth"}
From our combined imaging, photometric and spectral analyses, we establish Wolf 503b as a $2.03^{+0.08}_{-0.07}~R_{\Earth}$ planet orbiting its host star with a period of 6.0012 days. Wolf 503b is truly distinguished as its radius places it directly at the edge of the radius gap near 1.5–2.0$R_\Earth$, while its bright host star (H=7.77 mag, V=10.28 mag) makes it one of the best targets for radial velocity follow-up and transit spectroscopy at its size (Fig. \[fig:Hmag\_vs\_Rp\]).
Radial velocity measurements of Wolf 503b present an excellent opportunity to probe the bulk density of a planet just outside the radius gap. The amplitude of the expected RV signal depends strongly on the planet composition and amount of gas accreted. As Wolf 503b is similar in size to 55 Cnc e, though at a lower temperature, we investigate its composition using the mass-radius relationships for rocky compositions found in @valencia_composition_2010 and @Gillon12. For the gas-poor scenario, the minimum mass required for a rocky composition (with no iron), is roughly $10\,M_{\Earth}$, with an Earth-like composition corresponding so $14\,M_{\Earth}$. These masses would result in RV amplitudes of roughly 4.5 and 6.3 m/s, higher than the RV amplitudes resulting from the gas-rich scenario. For a volatile planet with a 0.01% H/He envelope, we would expect a mass of roughly $8\,M_{\Earth}$, whereas a 20% water envelope would suggest $6\,M_{\Earth}$, and the empirical mass-radius relation by @weiss_mass_2013 would suggest $5.3\,M_{\Earth}$, giving RV amplitudes of 3.6, 2.7, and 2.4 m/s. These amplitudes are detectable with existing precision radial velocity spectrographs, particularly for a bright target such as Wolf 503, and will provide critical constraints on the bulk composition of the planet.
Wolf 503b is also an ideal target for detailed characterization with *JWST*. With $J=8.32$ mag, it is just below the saturation levels of $J>7$ mag and $J>6$ mag on the *NIRISS* and *NIRSpec* grisms. If Wolf 503b indeed harbours a thick atmosphere, it is one of the best known targets to date for transmission spectroscopy at its size. Fig. \[fig:sim\_spec\] shows two simulated transit spectra for Wolf 503b, the blue corresponding to a hydrogen-rich, Neptune-like atmosphere and the orange corresponding to an atmosphere rich in water. Simulated *NIRISS* and *NIRSpec* data for the Neptune-like atmosphere is overplotted, demonstrating the high-confidence with which we will be able to constrain the structure and abundances of atmospheric molecules on Wolf 503b.
Both radial velocity measurements and atmospheric characterization with *HST* would be valuable short-term follow-up to this work. Wolf 503b is among only a handful of planets in its size range for which this follow-up can be done efficiently today. As such, we expect Wolf 503b to play a critical role in providing near-term insights into distribution of core masses, envelope fraction, and the role of photoevaporation for planets near the Fulton gap. It can also serve as archetype to this class of small planets orbiting nearby stars in preparation for future characterization of similarly bright *TESS* systems.
[^1]: entry \#402 in @Luyten23 (stars with motions exceeding 0.5/yr), LPM 492 [@Luyten41], LFT 1037 [@Luyten55], LHS 2799 [@Luyten79], and as NLTT 35228 and LTT 5351 in @Luyten80
|
---
abstract: 'Recent advances in deep neural network demand more than millions of parameters to handle and mandate the high-performance computing resources with improved efficiency. The cross-bar array architecture has been considered as one of the promising deep learning architectures that shows a significant computing gain over the conventional processors. To investigate the feasibility of the architecture, we examine non-idealities and their impact on the performance. Specifically, we study the impact of failed cells due to the initialization process of the resistive memory based cross-bar array. Unlike the conventional memory array, individual memory elements cannot be rerouted and, thus, may have a critical impact on model accuracy. We categorize the possible failures and propose hardware implementation that minimizes catastrophic failures. Such hardware optimization bounds the possible logical value of the failed cells and gives us opportunities to compensate the loss of accuracy via off-line training. By introducing the random weight defects during the training, we show that the model becomes more resilient on the device initialization failures, therefore, less prone to degrade the inference performance due to the failed devices. Our study shed light on the hardware and software co-optimization procedure to cope with potentially catastrophic failures in the cross-bar array.'
author:
- Youngseok Kim
- Seyoung Kim
- 'Chun-chen Yeh'
- Vijay Narayanan
- Jungwook Choi
bibliography:
- 'inference\_failure\_ACM\_v4\_arxiv.bib'
title: 'Hardware and software co-optimization for the initialization failure of the ReRAM based cross-bar array'
---
<ccs2012> <concept> <concept\_id>10010583.10010600.10010607.10010610</concept\_id> <concept\_desc>Hardware Non-volatile memory</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010583.10010786.10010787.10010788</concept\_id> <concept\_desc>Hardware Emerging architectures</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012>
Introduction {#sec:intro}
============
Recent progress in algorithm and computing hardware has made it possible to train the neural network in large scale and demonstrated that neuromorphic computing is a robust and efficient way to solve various problems including the pattern recognition [@he2016deep], speech recognition [@zhang2016towards], and optimization [@villarrubia2018artificial]. The central step of training is to modify the mapping rule from one neuron layer to the other adjacent layer minimizing the assumed cost function. Such mapping is often expressed as weight matrices and optimizing the weights requires intensive matrix operations setting a bottleneck in the training. While transistor scaling and the subsequent performance gain has resolved such bottleneck last few decades, the pace of the scaling has significantly slowed due to the growing cost and diminishing returns. The technological and economical challenges drive the computing more toward to the specialized computing architectures rather than the general purpose processors [@thompson2018decline].
Recently, cross-bar array hardware has been investigated as an emerging architecture. The architecture exploits the analog memory elements for multi-state weight representations and performs in-memory multiply-accumulate (MAC) operations [@gokmen2016acceleration; @2019Haensch]. The architecture is optimized to perform the MAC operations in parallel and such parallelism shows significant advantages over a conventional hardware both in speed and power consumption [@gokmen2016acceleration; @chi2016prime; @shafiee2016isaac; @song2017pipelayer]. Moreover, the feasibility of the architecture has been demonstrated using non-volatile memory such as a phase change memory (PCM) [@ambrogio2018equivalent], a resistive random access memory (ReRAM) [@bocquet2018inmemory; @zhou2018anewhardware], and charge trapping memory [@Merrikh2018; @Guo2017] element.
As cross-bar array hardware has a fundamentally different physical realization from the conventional architecture, the optimal ways of implementing the cross-bar array in the architecture level has been investigated for the fully connected [@gokmen2016acceleration], convolutional [@chi2016prime; @shafiee2016isaac; @song2017pipelayer; @gokmen2018trainingConv], and recurrent [@gokmen2018trainingLSTM; @long2018reram] neural networks. The analog nature of the memory element, however, lacks the explicit quantization of the states and errors occurred in the analog hardware domain may be accumulative. Thus evaluating and understanding the impact of the non-idealities in the analog domain and the analog/digital interface is a key element to enable the cross-bar array technology. The device-to-device, cycle-to-cycle update variations [@gokmen2016acceleration] as well as the resistance drift [@liu2017analyzing] has been considered for the individual analog elements. The variations in the peripheral circuitry have been investigated including the error occurring at the analog/digital interface [@gokmen2016acceleration; @gokmen2018trainingConv; @gokmen2018trainingLSTM] and the sense amplify circuitry [@sun2018xnor]. In addition, general strategies to address the non-idealities in analog domain has been discussed [@Jain2019NeuralNA].
Following this trajectory, we have investigated the possible device failure scenario and its impact on the model accuracy. The impact of the failed cell has been examined in the fully-connected neural network for the PCM based architecture [@romero2019training]. However, the lack of prior study on the ReRAM hardware motivates us to study the device failure scenario which occurs during the initialization process of the ReRAM based cross-bar array. After the fabrication of the device, the forming process is a necessary step to generate the filamentary conductive path for the ReRAM. During the forming process, the memory cell may stuck on the high-resistance or low-resistance state and potentially play as a critical source of error. Unlike the conventional memory, however, the analog memory elements are hard-wired and it is not straightforward to re-route the failed cells. Instead, one may incorporate a whole redundant arrays or columns to address the failed cells [@xia2017stuck]. To provide a different perspective, we attempt to address the failed cells in cross-bar array via hardware and software co-optimization without utilizing hardware redundancies. We first evaluate the potential impact of such failed cells on the architecture and propose the optimized hardware. We discuss on the inference model accuracy and propose an off-line training strategy that further compensates the accuracy lost due to the failed cells.
Modeling of initialization failure in ReRAM cross-bar array {#sec:model}
===========================================================
Using cross-bar array, we may perform multiplication and summation in parallel and improve calculation efficiency by a following mechanism. Figure \[fig:RPU\](a) shows the individual resistive memory unit cell connected to a word-line (horizontal line) and a bit-line (vertical line). Assuming that the voltage across the word-line is $v_i$, the current read from the bit-line is $I_{ji}=g_{ji} v_i$, where $g_{ji}$ is the conductance of the resistive element. By applying a certain voltage to the available word-line at a given period of time $t_i\in[0,t_{max}]$ and integrate the output current, each bit-line reads the total accumulative charges as $$\label{eq:zj}
z_j=\sum_i \int_t dt I_{ji}=\sum_i w_{ji}\times a_i,$$ where $a_i=v_it_{max}$ and $g_{ji}=w_{ji}$. Here, the conductance of the resistive element $g_{ji}$ represents the weight value $w_{ji}$ and the applied bias $v_i$ at the given word-line maps to the activation of the previous layer or an input value. As a result, Eq. (\[eq:zj\]) effectively represents the matrix multiplication of the hidden layer. Once the matrix multiplication is done by the cross-bar array, the results are processed in peripheral circuits and converted to the digital data at the data interface in Fig. \[fig:RPU\](a) [@gokmen2016acceleration]. Then, other operations such as the batch normalization or activation functions are executed in the digital circuits before the next cross-bar array consumes the output data. For this reason, the weight elements utilized for the matrix multiplication are represented by the analog memory elements, whereas remainder weights including batch normalization parameters are assumed to be handled in the digital circuits in floating point precision.
![**The schematics of cross-bar array system.** (a) A cross-bar array is comprised of a resistive memory cell, peripheral circuits, and data interface. (b) The resistive memory cell stores weight values in a form of a resistance, which may consist of one resistor (1R). (c) The resistive memory cell may comprise one resistor and one transistor (1T1R) where the transistor plays a role to limit the current.[]{data-label="fig:RPU"}](fig_individual_cell){width="50.00000%"}
Among the various resistive memory candidates, we mainly focus on the resistive switching memory or resistive random access memory (ReRAM) whose resistance states are determined by a filamentary conductive path of a dielectric material. The filament is formed by applying large bias across the dielectric which induces a soft-breakdown of the material and this process is called *forming*. For example, the breakdown process in HfO$_x$ occurs by the oxygen vacancy movement which creates a metallic conductive path [@kim2004engineering; @pan2014recent]. Once such conductive path is *formed*, the reverse polarity bias (or RESET bias) may be applied to induce the recombination of the oxygen and the oxygen vacancy. Once such recombination disconnects the filamentary conductive path, the device is in a high resistance state (HRS). Likewise, the same polarity with the forming bias (or SET bias) may be applied to re-connect the conductive filament to set the device in a low resistance state (LRS). Figure \[fig:Grange\] shows the ideal ReRAM cell which may allow us to explore intermediate states between LRS and HRS by applying SET and RESET bias. In this study, we assume that each ReRAM cell is written by write and verify method. This means that we assume the writing process involves in multiple checking and re-writing steps until we reach the desired state for each cell. Although such method becomes a serious bottle-neck of the algorithm for training purpose, it is a valid approach for inference purpose as we need to update the weight matrices once when we copy the trained model to the ReRAM cells. Using this approach, ReRAM cell have been demonstrated to express 7 bits (128 states) with $0.39\%$ fluctuations compared with the spacings between states [@li2018analogue]. Of course, there are other source of non-idealities such as noise and an intrinsic stochasticity in ReRAM. More comprehensive studies have been done, i.e. [@gokmen2016acceleration], and have shown that the cross-bar array architecture is robust in certain degree to such non-idealities. Therefore, we assume that the ReRAM device that is not failed are ideal and the states are written exactly as it supposed to be up to the 4 bits of quantization levels.
However, the device may fail during the forming process if (i) the ReRAM cell is not formed and remained open for a given forming bias. The forming voltage is a function of dielectric thickness and area [@chen2013area]. A ReRAM cell that has process variations on these parameters may result in the forming voltage higher than the maximum voltage that is supplied by the peripheral circuit, and may not be formed. In addition, (ii) the filament may be *overformed* and cannot be disconnected by the RESET bias. To form the conduction path with a desirable filamentary thickness, it is important to control the current supplied during the forming process. The failure of the current control often results in overforming the device. A fast rate of filament formation (less than 1ns [@bersuker2019metal]) make it challenging to control the supplied current simply by removing the applied voltage timely. Therefore, a typical approach to limit the supply current is to utilize a transistor. However, the existence of the parasitic capacitance results in an overshoot of the current over the compliance current even in the presence of the transistor [@kinoshita2008reduction]. Such overshooting in current partially ascribe to form the filament thicker than the desirable value and may produce filament that is overformed and cannot be reset. Above mentioned failures result in the cell states stuck in (i) HRS or (ii) LRS.
According to Eq. (\[eq:zj\]), an extreme resistance value results in an abnormally large weight value and may cause a non-trivial impact on the model accuracy. In conventional random access memory architecture, such failed cells may be re-routed to the working redundant cells. However, the cells in cross-bar array are hard-wired and the failed cells cannot be re-routed. The goal of this study is to evaluate an impact of the two major failure mechanism on the inference model, and provide methods to minimize the loss in the model accuracy as well as an insight on the acceptable forming yield.
Individual resistive memory forming failure analysis {#sec:cellAnalysis}
----------------------------------------------------
![**The schematics of the conductance range of ReRAM.** The top schematics show the filament states and their corresponding conductance range. The normal cell dynamic range is defined by the conductance of high resistance state ($g_{HRS}$) and the low resistance state ($g_{LRS}$). The intermediate states are quantized into $N$ states and represents the relevant weight values ranging from $w_{HRS}$ to $w_{LRS}$. The conductance of the forming failed cell is close to $0$ and the resultant deviation from the $g_{HRS}$ is defined as $\delta g_{FF}$. Such deviation results in an additional error in weight value from $w_{HRS}$, which is defined as $\delta w_{FF}$. The overformed device tends to have lower resistance than the typical LRS. The resultant deviation from the desired LRS conductance is defined as $\delta g_{OF}$, and the corresponding deviation in the weight element is defined as $\delta w_{OF}$.[]{data-label="fig:Grange"}](fig_conductance_range){width="50.00000%"}
Figure \[fig:Grange\] depicts the conductance range of the forming failed cells during the forming process. The top and bottom electrodes are electrically disconnected by the insulating dielectric for the forming failed (FF) devices. As a result, the conductance of the FF device typically shows few orders of magnitude lower than the conductance of the formed device. Therefore, we may assume the conductance of the FF devices as $g_{FF}\simeq0$. This assumption is consistent with the other works such as a dead device modeled as $g=0$ in PCM array [@romero2019training], or a stuck-at-1 fail cell modeled as a minimum conductance (or $g\simeq0$) of the system in ReRAM array [@xia2017stuck]. In contrast, the resistance of the overformed (OF) devices can be as low as few hundreds of $\Omega$ to few $k\Omega$ whereas the desirable ReRAM operation dynamic range is from few hundreds $k\Omega$ to few $M\Omega$ [@gokmen2016acceleration]. Assuming the dynamic range of the working device is $g\in[g_{HRS}, g_{LRS}]$, the corresponding logical value is mapped to $w\in[w_{HRS},w_{LRS}]$. However, if the conductance of OF or FF devices deviates significantly from a typical range of $g$, such scenario may lead to a serious failure of the cross-bar array architecture. For example, if the resistance states of the OF device stuck at $2~k\Omega$ whereas the LRS of the working cell is $100~k\Omega$, a single OF device flows current $50$ times larger than the expected LRS devices for a given input bias. In this case, the current level may be above the maximum acceptable range of the current integrator at the peripheral circuits and the whole bit-line signals may be overwhelmed. To prevent such scenario, we propose to use 1 transistor + 1 resistor (1T1R) cell structure shown in Fig. \[fig:RPU\](c) rather than 1 resistor + 1 selector or 1 resistor (1R) structure described in Fig. \[fig:RPU\](b). 1T1R structure limits the maximum current by adjusting the gate bias, $v_g$. Assuming that the transistor is operating in the linear regime, the drain current is determined as $I_{ds}=(W/L)\mu C_{ox}(v_{g}-v_{th})v_{ds}$, where $W$ and $L$ is the width and length of the gate, respectively, $\mu$ is the carrier mobility, $C_{ox}$ is the gate oxide capacitance, $v_{th}$ is the threshold voltage, and $v_{ds}$ is the bias across the source and drain. Therefore, we may set the effective conductance of the transistor $g_{tr}=(W/L)\mu C_{ox}(v_{g}-v_{th})$ close to the $g_{LRS}$ by adjusting the gate bias. In this case, the unit cell conductance range becomes $$\label{eq:glist}
\begin{split}
g_{LRS}^{cell}=&\frac{g_{LRS}\cdot g_{tr}}{g_{LRS}+g_{tr}}, \\
g_{HRS}^{cell}=&\frac{g_{HRS}\cdot g_{tr}}{g_{HRS}+g_{tr}}\simeq g_{HRS}, \\
g_{OF}^{cell}=&\frac{g_{OF}\cdot g_{tr}}{g_{OF}+g_{tr}}\simeq g_{tr}, \\
g_{FF}^{cell}=&\frac{g_{FF}\cdot g_{tr}}{g_{FF}+g_{tr}}\simeq g_{FF}, \\
\end{split}$$ where we assume $g_{FF}\ll g_{HRS}\ll g_{tr}\ll g_{OF}$. With the conductance value defined in Eq. (\[eq:glist\]), we may map the conductance to the logical value, or weight value. As it is described in Eq. (\[eq:zj\]), $$\begin{split}
g_{LRS}^{cell}\rightarrow& w_{LRS}, \\
g_{HRS}^{cell}\rightarrow& w_{HRS}, \\
\end{split}$$ where $(w_{HRS},w_{LRS})\subset\{(0,2),(0,-1),(0,1)\}$ depending on the cross-bar array architecture. For the inference model, the weight value is often quantized into $N$ states without any significant accuracy loss of the model. Therefore, we may define the spacing of the conductance and the corresponding spacing in the logical value as $$\label{eq:spacing}
\Delta g=\frac{g_{LRS}^{cell}-g_{HRS}^{cell}}{N-1}\rightarrow\Delta w=\frac{w_{LRS}-w_{HRS}}{N-1}.$$ Note that the OF and FF devices are deviated from the expected maximum or minimum logical values, respectively. Figure \[fig:Grange\] describes such deviation as $\delta g_{FF}=g_{HRS}^{cell}-g_{FF}^{cell}$ for the FF device and $\delta g_{OF}=g_{OF}^{cell}-g_{LRS}^{cell}$ for the OF device. Their relative significance may be quantified by comparing with the logical value spacing defined in Eq. (\[eq:spacing\]): $$\label{eq:dwDw_min}
\begin{split}
\frac{\delta g_{FF}}{\Delta g}=\frac{\delta w_{FF}}{\Delta w}
=&(N-1)\frac{g_{HRS}^{cell}-g_{FF}^{cell}}{g_{LRS}^{cell}-g_{HRS}^{cell}} \\
\simeq&(N-1)\frac{1}{\bar{g}-1}, \\
\end{split}$$ $$\label{eq:dwDw_max}
\begin{split}
\frac{\delta g_{OF}}{\Delta g}=\frac{\delta w_{OF}}{\Delta w}
=&(N-1)\frac{g_{OF}^{cell}-g_{LRS}^{cell}}{g_{LRS}^{cell}-g_{HRS}^{cell}} \\
=&(N-1)\frac{g_{OF}^{cell}/g_{LRS}^{cell}-1}{\bar{g}-1} \\
\simeq&(N-1)\frac{g_{tr}}{g_{LRS}(\bar{g}-1)},
\end{split}$$ where $\bar{g}=g_{LRS}^{cell}/g_{HRS}^{cell}$ is the min/max conductance ratio between LRS and HRS, and the last equalities in Eqs. (\[eq:dwDw\_min\]-\[eq:dwDw\_max\]) are from Eq. (\[eq:glist\]). The logical value deviation of both OF and FF devices is minimized when the min/max ratio ($\bar{g}$) is maximized or the cell has less quantized states ($N$). In addition, setting $g_{tr}$ closer to the $g_{LRS}$ is beneficial in further reducing logical value deviation of the OF devices. However, this will reduce the overall dynamic range of the resistive unit cell due to the diminishing cell LRS conductance, $g_{LRS}^{cell}$, and eventually reduces $\bar{g}$. Prior to examine such trade-off, we need to further specify $N$. The quantized states $N$ is the number of states per individual cell, and its value may differ depending on the choice of the specific cross-bar array architecture.
Forming failure scenario analysis on resistive memory cell
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![**Cross-bar array cell structure.** (a) Individual ReRAM cell represents $w_{ji}\in[0,2]$ and shares the reference column of ReRAM cells whose conductance is set to be $g_{mid}$, or $w_{mid}=1$. By subtracting the current from the reference column, each ReRAM cell represents full range of weight values, or $w_{ji}-w_{mid}\in[-1,1]$. (b) The reference current, $I_{ref}$, can be supplied from the external circuit. (c) Individual ReRAM cell is paired with the reference cell. (d) A pair of ReRAM cell represents plus ($w^+$) and minus ($w^-$) weight values. This choice of differential reading allows to reduce the number of quantized states per cell, as each cell needs to store half of the weight range. (e) Summary on the pros and cons of the cell structures. $\;$indicates an unacceptable disadvantages over other candidates, $-$ indicates no difference with others or an acceptable level of disadvantage. $\;$indicates that the candidate is superior or equivalent to other candidates.[]{data-label="fig:CellStructure"}](fig_cell_schematics){width="50.00000%"}
Figure \[fig:CellStructure\] shows four possible resistive memory cell structures which are designed to express $w_{ji}\in[-1,1]$ logical values. The most efficient implementation in hardware resource perspective may be shown in Fig. \[fig:CellStructure\](a). In this structure, each individual cell represents $(w_{min},w_{max})=(0,2)$ and the cells in the last column serve as reference devices by fixing its conductance value to $g_{mid}=(g_{max}+g_{min})/2$, or $w_{mid}=1$. By subtracting the individual cells with the reference column, the architecture represents plus and minus values of the weights, or $w_{ji}-w_{mid}\in[-1,1]$. However, the impact of the OF or FF device exerts on the whole row if a forming process failure occurs to the reference device. To avoid such scenario, we may simply subtract current at the end of the column-wise current integration as described in Fig. \[fig:CellStructure\](b) by introducing $I_{ref,i}=\sum_i v_i\cdot g_{mid}$. While this resolves the problem, we now need to calculate $I_{ref,i}$ using extra-peripheral circuits. Instead, Fig. \[fig:CellStructure\](c) shows that we may double the hardware resources by introducing an additional resistive memory to form a resistive memory unit cell. Here, we use one cell as a weight storage and the other cell as a reference device. Note that each weight storage cell represents $(w_{min},w_{max})=(0,2)$, which is quantized into $N=2^{n_{bit}}$ levels, where $n_{bit}$ is the target model quantization bit. An alternative scheme in Fig. \[fig:CellStructure\](d) utilizes one cell for a plus logical value, or $(w_{min}^+,w_{max}^+)=(0,1)$, and the other cell for a minus logical value, or $(w_{min}^-,w_{max}^-)=(-1,0)$. As the $n_{bit}$ quantization is performed over $(-1,1)$, each cell now stores $N=2^{n_{bit}-1}$ quantized levels as each of the cell covers half of the weight range. According to the Eqs. (\[eq:dwDw\_min\]-\[eq:dwDw\_max\]), the 2T2R structure (two 1T1R) in Fig. \[fig:CellStructure\](d) is advantageous over other candidates as it reduces the number of quantized states from $N=2^{n_{bit}}$ to $N=2^{n_{bit}-1}$ which effectively reduces the logical value deviation of OF and FF devices. The above mentioned arguments are summarized in Fig. \[fig:CellStructure\](e).
![**A logical value deviation of FF and OF devices.** (a) A logical value deviation of FF device ($\delta w_{FF}$) as a function of a quantization bit ($n_{bit}$) and conductance of the transistor ($g_{tr}$). $\delta w_{FF}$ is calculated from Eq. (\[eq:dwDw\_min\]). The contour line shows $\delta w_{FF}/\Delta w=1,\;2,\;3,\;4,\;8$ and $16$. (b) A logical value deviation of OF device ($\delta w_{OF}$). $\delta w_{OF}$ is calculated from Eq. (\[eq:dwDw\_max\]). The contour line shows $\delta w_{OF}/\Delta w=1,\;2,\;3,\;4,\;8,\;16,\;32$ and $64$. We set the min/max ratio of the conductance $g_{LRS}/g_{HRS}=10$ for (a) and (b).[]{data-label="fig:wFFwOF"}](fig_wFFwOF){width="50.00000%"}
Adopting the 2T2R resistive memory cell described in Fig. \[fig:CellStructure\](d), we examine how undesirable conductance deviation in Fig. \[fig:Grange\] is affected by the physical parameter variations and quantify its significance in terms of the weight quantization step. Specifically, we vary the number of quantized conductance ($n_{bit}$) and the effective gate conductance ($g_{tr}$). Figure \[fig:wFFwOF\](a) and (b) shows the resultant logical weight deviation $\delta w_{FF}$ and $\delta w_{OF}$, respectively, with their magnitude normalized by the quantized step ($\Delta w$). We choose to vary $n_{bit}$ and $g_{tr}$ for the following reasons: (i) Eqs. (\[eq:dwDw\_min\]-\[eq:dwDw\_max\]) shows that these variables determine $\delta w_{FF}$ and $\delta w_{OF}$ and (ii) they are tunable parameters without any significant modification in the hardware. If we lower the quantization bit, the quantization step $\Delta w$ is now represented by the larger conductance window. This wider window allows the cell to tolerate a given amount of the conductance deviation. As a result, smaller $n_{bit}$ always improves the logical error of the resistive memory cell. Figure \[fig:wFFwOF\](a) shows that the logical error of FF device approaches to one quantization step when $n_{bit}\sim4$. For a given $n_{bit}$, larger $g_{tr}$ helps to reduce $\delta w_{FF}$. This is due to the fact that larger transistor conductance allows a larger dynamic range of the resistive cell, thus, the relative significance of the conductance deviation is reduced. In contrast, Fig. \[fig:wFFwOF\](b) shows that smaller $g_{tr}$ helps to reduce $\delta w_{OF}$. In this case, having $g_{tr}$ closer to $g_{LRS}$ sets a better conductance lower-bound for OF devices. The overall trade-off shows that optimizing the gate bias to adjust smaller $g_{tr}$ is desirable in reducing the logical weight value deviations rather than fully turn-on the transistor during the MAC operation. This is because $\delta w_{OF}$ is more sensitive to the parameters than $\delta w_{FF}$. For example, $\delta w_{OF}$ is as large as three quantization steps at $(n_{bit},g_{tr}/g_{LRS})=(4,1)$, whereas $\delta w_{FF}$ is equivalent or smaller than one quantization step in wider range of parameters. Note that we fix the min/max ratio of the resistive cell as $g_{LRS}/g_{HRS}=10$ in Fig. \[fig:wFFwOF\]. The logical errors are further minimized for higher min/max ratio, although improving min/max ratio requires longer term effort as it requires an improvement in materials and device structures.
-------------- ------- ---------------------- --------------------- ------------------ ----------------
$w^+$ $w^-$ possible value range best possible value logical error prob.
\[0.5ex\] FF - $[-1,0]$ $0$ $-\delta w_{FF}$ $p_{FF}(1-p)$
FF FF $0$ $0$ $\sim 0$ $p_{FF}p_{FF}$
FF OF $-1$ $-1$ $-\delta w_{OF}$ $p_{FF}p_{OF}$
OF - $[0,+1]$ $0$ $+\delta w_{OF}$ $p_{OF}(1-p)$
OF FF $+1$ $+1$ $+\delta w_{OF}$ $p_{OF}p_{FF}$
OF OF $0$ $0$ $\sim0$ $p_{OF}p_{OF}$
\[1ex\] - - $[-1,+1]$ 0 $(1-p)(1-p)$
- FF $[0,+1]$ $0$ $+\delta w_{FF}$ $(1-p)p_{FF}$
- OF $[-1,0]$ $0$ $-\delta w_{OF}$ $(1-p)p_{OF}$
\[1ex\]
-------------- ------- ---------------------- --------------------- ------------------ ----------------
: Summary of the possible forming failure types and the resultant logical error and its occurrence probability. Working cells are indicated as -.
\[tb:summary\]
Even with a zero logical value deviation, the conductance state of the OF device is fixed near $g_{max}^{cell}$. Therefore, the corresponding logical value is stuck at $\pm1$. We define such failed cells as $\pm1$ defects in the weight matrices. In contrast, the conductance of the FF device is close to $g_{min}^{cell}$ and the resultant logical value is stuck at $0$. We define these failed cells as $0$ defects. With 2T2R structure in Fig. \[fig:CellStructure\](d), we further minimize $\pm1$ defects by utilizing the following forming protocol. For example, in case the $w^+$ cell is overformed while its pair $w^-$ is properly working, we can avoid having unintentional $+1$ value by setting $w^-=-1$. In this case, the conductance of the OF device in $w^+$ is compensated by the conductance of $w^-$ and we have $0+\delta w_{OF}$ as a logical error instead of $+1+\delta w_{OF}$. In other words, by utilizing the resistive cell pairs, we effectively prune the mal-functioning weight elements instead of having $+1$ logical error. Table \[tb:summary\] summarizes all the possible scenario. In Table \[tb:summary\], we define the probability for FF and OF devices as $p_{FF}$ and $p_{OF}$, respectively, and the total forming failure probability is defined as $p=p_{FF}+p_{OF}$. The straightforward initialization protocol (referred to as a strategy A) may faithfully follow the Table \[tb:summary\]. In this case, the resistive memory cells stuck at $\pm1$ value with a probability of $p_1=2p_{FF}p_{OF}$ while $(1-p)^2$ cells are properly working. The remaining portion of the cell is forced to be $0$ and the weights are effectively pruned to minimize the impact of the failed cells. The probability of having $0$ defect is $p_0=2(1-p)p+p_{OF}p_{OF}+p_{FF}p_{FF}\simeq 2p$ if $p\ll 1$. Among all possible scenarios for $0$ defects, $(w^+,w^-)=(FF,-),\;(-,OF)$ and $(w^+,w^-)=(-,FF),\;(OF,-)$ cases are programmable and may programmed correctly if the weight value is negative or positive, respectively. If we assume that the chance of having minus or plus weight value for an arbitrary cell is $50\%$, the resistive cell still records the correct value with the probability of $p_0\simeq2p/2\simeq p$. As a result, we end up having $0$ defects with a probability of $\sim p$.
----------------- ----------------------- ----------------------
strategy A strategy B
\[0.5ex\] $p_0$ $\sim p$ $\sim2p$
$p_1$ $2p_{OF}\cdot p_{FF}$ $p_{OF}\cdot p_{FF}$
----------------- ----------------------- ----------------------
: Two forming strategies based on Table \[tb:summary\] and the resultant $0$ defect ($p_0$) and $\pm1$ defect ($p_1$) probabilities.
\[tb:summary2\]
Another possible strategy (referred to as strategy B) is to form one device first and choose not to form its pair if the device is FF device. This strategy will avoid the risk of having OF device for its pair during the forming process, and further reduce the probability of having $\pm1$ defect from $2p_{FF}p_{OF}$ to $p_{FF}p_{OF}$. Although such choice is effective in minimizing $\pm1$ defects, we no longer have $(w^+,w^-)=(FF,-)$ or $(-,FF)$ as we choose not to form the pair of FF devices. If we simply assume that we do not write the weight values to the failed cells, the probability of having $0$ defect is now $p_0\simeq 2p$.
When the OF device occurrence is low, it is more beneficial to choose the strategy A due to the lower $p_0$ probability. However, the strategy B is a good option if OF device occurrence rate is substantial compared with the occurrence of the FF device. The discussed two forming strategy A and B and their corresponding probabilities of having $\pm1$ defect ($p_1$) and $0$ defect ($p_0$) are summarized in Table \[tb:summary2\]. The numerical analysis in the following section utilizes the strategy B, but the qualitative results are consistent for both strategy A and B as they differs only by the probability combinations of $0$ and $\pm1$ defects.
Numerical experiments
=====================
The impact of $\pm1$ and $0$ defects on the inference accuracy {#sec:defectAware}
---------------------------------------------------------------
We use CIFAR-10 dataset and test the impact of the forming failed cells on the image recognition model. The model has been trained in ResNet-20 [@he2016identity]. 50,000 images has been used for training and 10,000 images are utilized for the test. According to the analysis in Section \[sec:cellAnalysis\], a deviation of the weight from the desired logical value is minimized for smaller quantized bit, or $n_{bit}$. We quantize the weight [@zhou2016dorefa] and activation [@choi2018pact] and find no significant degradation in the test accuracy for $n_{bit}\geq4$ both for weight and activation [@choi2018pact; @distiller2018]. Below $n_{bit}=4$ requires special techniques to maintain the model accuracy [@rastegari2016xnor; @courbariaux2016binarized] which is out of scope in this study, thus, we choose $n_{bit}=4$. We obtain the baseline with a test error of $7.99\%$ after 200 epochs of training with a learning rate scheduling of $0.1$, $0.01$, $0.005$ at $80$, $120$, $180$ epochs.
{width="100.00000%"}
We first assume an ideal defect by setting $\delta w_{OF,FF}\rightarrow 0$ and evaluate the impact of $0$ and $\pm1$ random defects on the trained model. Specifically, we randomly change the weight matrix elements of the trained model to $0$ or $\pm1$ with a probability of $p_0$ or $p_1$, respectively, and we set an equal probability for $+1$ and $-1$ defects for simplicity. As a first step, we isolate the impact of the $0$ defects from $\pm1$ defects by setting $p_0=0$. We then vary $p_1$ and obtain the inference results. The black dashed line in Fig. \[fig:experiment\](a) shows the inference results for $p_1=0.2\%-5\%$. The test error rapidly increases and the model shows more than $20\%$ error rate for $p_1>1\%$. However, such degradation in the test accuracy may be recovered by retraining the network if we know the exact defect configuration. The blue x symbol in Fig. \[fig:experiment\](a) shows the recovered accuracy through 200 epochs of re-training from the baseline model using a fixed defect configuration and an identical learning rate scheduling with the baseline. This approach is valid when we know the exact location of the existing defects. The defect configuration, however, likely appears in random and varies from chip to chip. Therefore, the strategy requires a prior knowledge of a specific configuration as well as computational resources to re-train for each individual chips.
Instead, we may train the network with a prior knowledge of a type of defect and its likelihood. By introducing random defects for a given probability during the training, the model may find a local minimum which minimizes the loss function even in the presence of random defects. Through this process, model may become resilient to a particular set of defects without a knowledge of a specific defect configuration, thus a single trained model may be utilized for numerous defect configurations. We examine the hypothesis by introducing randomly generated $\pm1$ defects with a probability $p_1$. After the weight matrices at each layer are quantized, certain portion of the weight elements are replaced by $+1$ in $p_1/2$ and $-1$ in $p_1/2$ probability. The modified weight matrices are utilized for a given training image to perform the forward, and backward propagation. We then repeat the above procedure by generating a new set of random defects in the same probability for the next training image and the defect-aware model has been trained with 200 epochs. Using the resultant model, we generate $50$ different random defect configurations. The corresponding inference results are plotted as a red solid line in Fig. \[fig:experiment\](a). When the defect probability is smaller than $0.2\%$, the inference results from baseline (black dotted line) and defect-aware model (red line) shows little difference. However, as we increase the defect probability, the defect-aware model produces much better inference results than the inference results from the baseline. We perform the similar analysis for the $0$ defects. Namely, we replace the quantized weight matrix elements to $0$ with a probability of $p_0$. Figure \[fig:experiment\](b) shows a consistent trend with Fig. \[fig:experiment\](a). One noticeable difference is the overall shift of the model error degradation. The model accuracy shows a significant degradation when $p_0>\sim1\%$ for $0$ defect whereas $p_1>\sim0.2\%$ for $\pm1$ defect. In other words, $\pm1$ defect exhibits more critical impact on the model accuracy. This may be understood by looking at the weight value distribution. After the 200 epoch training, most of the baseline weight values are centered at $0$ as the loss function tends to minimize the weight value. The test error is expected to increase upon an introduction of the defects as the difference between the desired weight value and the defect plays as a source of error. The model weight matrices have much less $\pm1$ values than $0$, therefore, an arbitrary change of the weight value to $\pm1$ may induce larger amount of error in higher chance. Through the defect-aware training, however, the weight value distribution is forced to shift toward $\pm1$ by intentionally populating extreme weight values in random fashion. Although the resultant model is deviated from the global minimum, the model becomes less prone to fail when similar type of defects are introduced.
To keep the test error of CIFAR-10 dataset below $10\%$, for example, the baseline results shows that one needs to keep the $\pm1$ defect probability less than $0.1\%$, whereas the defect-aware model may tolerate up to $\sim0.5\%$. In contrast, the baseline model is less susceptible to $0$ defects. As the baseline model inherently includes certain amount of randomly distributed weight values close to $0$, both baseline and defect-aware model tolerates the defect probability up to $\sim 1\%$. The baseline and defect-aware model start to show a noticeable difference once we introduce more than $2\%$ of the $0$ defects. This summarizes that the defect-aware model approach is effective when the defect type tends to manifest itself as an uncommon logical value from typical weight value distributions. Furthermore, Fig. \[fig:experiment\](a-b) shows that the baseline model is more sensitive to $\pm1$ defects. This result further justifies the 2T2R structure over 1T1R shown in Fig. \[fig:CellStructure\](d) as it reduces the probability of having $\pm1$ from $p_{OF}$ to $p_{OF}p_{FF}$.
With our understanding on the impact of $\pm1$ and $0$ defects on the inference model accuracy, we now utilize the Table \[tb:summary\] to evaluate the optimized hardware performance as a function of the probability of having OF and FF devices. When the total probability of the initialization failure scenario is $p=p_{OF}+p_{FF}$, we choose $p_{OF}=p_{FF}$ which is the worst case scenario of having maximum $\pm1$ defect. Figure \[fig:experiment\](c) shows the inference results as a function of $p$. The 2T2R hardware minimizes the $\pm1$ defects and the inference results get close to the results of the $0$ defect in Fig. \[fig:experiment\](b).
The impact of $\delta w_{OF,FF}$ on the inference accuracy
----------------------------------------------------------
![**Robustness of the defect-aware model to the non-ideality of the defects ($\delta w_{OF, FF}$) on the test error.** (a) The inference error from the baseline at $p=2\%$ is $\sim16.2\%$ with $\delta w_{OF, FF}=0$. The plot shows the relative error increase caused by the non-zero $\delta w_{OF, FF}$. (b) The inference error from the defect-aware model at $p=2\%$ is $\sim12.8\%$ with $\delta w_{OF, FF}=0$. The plot shows the relative error increase by $\delta w_{OF,FF}$ is smaller than the results from the baseline model. []{data-label="fig:nonideal"}](figure_experiments_nonideal){width="50.00000%"}
In reality, the conductance value of the OF device may be higher than the normal cells and, for example, the logical value becomes $+1+\delta w_{OF}$ instead of $+1$. Similarly, the conductance of the FF device may exhibit lower conductance than the expected value which results in $0\pm\delta w_{FF}$. Such non-ideal deviation from the expected logical value has been described in Fig. \[fig:Grange\] and formulated in Eqs. (\[eq:dwDw\_min\]-\[eq:dwDw\_max\]). In our cell design, the possibility of having $\pm1$ or $0$ defects and the corresponding non-ideal deviation has been summarized in Table \[tb:summary\]. Following the prescriptions in the table, we examine the impact of such non-ideal deviations of the logical value of $\pm1$ and $0$ defects for $\delta w_{FF}, \delta w_{OF}$. Figure \[fig:wFFwOF\] shows that $\delta w_{FF}/\Delta w\sim1-2$ and $\delta w_{FF}/\Delta w\sim3-4$ are reasonable range for $n_{bit}=4$ and $g_{LRS}/g_{HRS}=10$. Therefore, we choose a range of parameters $\delta w_{FF},\;\delta w_{OF}\in[0, 3\Delta w]$, where $\Delta w=(1-(-1))/(2^{n_{bit}}-1)$ is the quantization step of the weight elements.
Figure \[fig:nonideal\](a) shows the inference results from the baseline for $p=p_{OF}+p_{FF}=2\%$. The inference error is averaged over $15$ different defect configurations and the error is $16.2\%$ for $\delta w_{FF}=\delta w_{OF}=0$. As we increase the non-ideal deviation of FF and OF devices, the averaged inference error is computed over $15$ different defect configurations and a relative error with respect to the error at $\delta w_{FF}=\delta w_{OF}=0$ is plotted in Fig. \[fig:nonideal\](a). The error monotonically increases as $\delta w_{FF}$ and $\delta w_{OF}$ increase and we lose additional $\sim10.1\%$ accuracy for $\delta w_{FF}=\delta w_{OF}=3\Delta w$.
We perform the similar analysis on the defect-aware model for $p=p_{OF}+p_{FF}=2\%$ and the inference error has been obtained by averaging over results from $15$ different defect configurations. We first obtain the inference error of $12.8\%$ for $\delta w_{FF}=\delta w_{OF}=0$, which shows a less error than the inference results from the baseline. As we increase $\delta w_{FF}$ and $\delta w_{OF}$, the monotonically increasing error shows a consistent trend with that of the baseline inference results. However, the relative error is $\sim3.3\%$ at $\delta w_{FF}=\delta w_{OF}=3\Delta w$, which is three times smaller than the relative error observed in the baseline results. Therefore, the defect-aware model shows more robust inference capability against non-ideal deviations of the weight elements.
The impact of the defect probability distribution on the inference accuracy
---------------------------------------------------------------------------
![**Robustness of defect-aware model to the impact of the statistical variation of failure probability.** (a) Inference results from the baseline model (black solid line), defect-aware model (dashed line), and distribution-aware model (blue solid line). Each box plot represents the inference results from the $50$ different defect configuration. The left $y$-axis shows the assumed Gaussian distribution with a mean defect probability $\mu_p=3\%$ and a standard deviation $\sigma_p\simeq 0.7\%$. (b) Inference results from the same set of models in (a), but utilizes the assumed defect probability distribution shown in (a). The black-dot in box plot represents a possible inference test error for individual inference chip and their possible mean and standard deviation is presented in a box plot.[]{data-label="fig:gaussian"}](fig_experiments_gaussian){width="50.00000%"}
We may extract the initialization failure probability by collecting the statistics of the individual ReRAM data for a given die. However, such statistics may vary from die to die due to the process variation across the wafer, or from wafer to wafer due to the process drift induced by the equipment. As we have only discussed the defect-aware model at a fixed probability so far, it is worthwhile to investigate the strategy to address the statistical variations in the defect probability. For a given statistically meaningful interval of $p$, we pursue to obtain lower test error with minimal standard deviation. We proceed our discussion with an exemplary Gaussian distribution of the defect probability.
The failure analysis on 4Mb ReRAM test chip shows $p_{OF}\sim1.75\%$ and $p_{FF}\sim9.04\%$ [@chen2014rram]. After applying a dc bias to form 128Kb arrays, applying alternating set/reset pulses results in $p_{OF}\sim1.28\%$ and $p_{FF}\sim4.76\%$ [@shih2011training]. A series of optimized pulse inputs has been applied to 4Kb arrays and showed the improved forming yield, yet to have $p_{FF}\sim1\%$ [@grossi2016electrical]. Based on these results, we set a reasonable distribution of $p_{OF}$ and $p_{FF}$ to illustrate our approach. Specifically, we assume the mean values of $\mu_{p_{OF}}=\mu_{p_{FF}}=1.5\%$ with standard deviations of $\sigma_{p_{OF}}=\sigma_{p_{FF}}=0.5\%$. As a result, the total defect probability has a mean value of $\mu_p=\mu_{p_{OF}}+\mu_{p_{FF}}=3\%$ and $\sigma_{p}=\sqrt{\sigma_{p_{OF}}^2+\sigma_{p_{FF}}^2}\simeq 0.7\%$. The right-side $y$-axis of Fig. \[fig:gaussian\](a) describes the statistics of the defect probability and a relevant probability interval from $p=1\%$ ($-3\sigma_p$) to $p=5\%$ ($+3\sigma_p$) has been indicated in dashed vertical lines.
The black line in Fig. \[fig:gaussian\](a) and the left-side $y$-axis depict the inference results from the baseline model. A rapid increase of the test error has been observed from $\sim 10\%$ at $p=1\%$ to $\sim 40\%$ at $p=5\%$. The result shows that the baseline model is sensitive to the statistical variation of the defects due to its steep slope. The observed variation of the test error is unacceptably large and, therefore, a small deviation from the expected defect probability may result in a large deviation from an expected inference accuracy. The remaining plot shows the improved results from the defect-aware model. The gray dotted line in Fig. \[fig:gaussian\](a) shows the inference error from the defect-aware model trained at $p=1\%$ ($-3\sigma_p$). A minor improvement over the baseline has been observed, but still suffers a rapid degradation of the accuracy for larger $p$. The dark red dotted line in Fig. \[fig:gaussian\](a) exhibits the inference results of the defect-aware model trained at $p=5\%$ ($+3\sigma_p$). As its test error saturates for $p<3.0\%$, the variation of the inference error becomes less sensitive to the statistical variation. However, the overall test error is even higher at $p<2.0\%$ than the baseline results, which results in an overall degradation of the averaged test accuracy. The pink dashed line in Fig. \[fig:gaussian\](a) shows the inference error from the defect-aware model at $p=3\%$. The model shows a better accuracy than the results from the baseline or the defect-aware model trained at $p=1\%$ within the relevant range of $p$. The inference results at $p>4\%$ shows an inferior performance than the model with $p=5\%$, but the statistical performance is expected to be better as the test error is comparable or smaller within the most significant interval of $\mu_p-\sigma_p\leq p \leq\mu_p+\sigma_p$.
Alternatively, we may utilize the known distribution of the defect probability while the training. We randomly select $p_{OF}$ and $p_{FF}$ based on the known distribution and generate a defect configuration. The generated defect configuration is utilized for forward and backward propagation for a given image during the training procedure. The same procedure is repeated to generate a new configuration with a different defect probability for the next image training cycle. The inference results from the distribution-aware model is indicated as a blue solid line in Fig. \[fig:gaussian\](a). The test error of the distribution-aware model shows comparable or less test error compared with the defect-aware model trained at the mean value of $p=3\%$ and, therefore, is expected to perform better in terms of mean and standard deviation of the inference test error.
Figure \[fig:gaussian\](b) shows the inference results distribution from various models discussed in Fig. \[fig:gaussian\](a). The defect probability $p_{OF}$ and $p_{FF}$ are randomly selected from the assumed distribution and we generate defect configurations with the selected probability. The same procedure is repeated for $500$ different trials. Each point represents one inference chip whose defect probability follows a given Gaussian distribution and the resultant statistical inference performances are presented. Figure \[fig:gaussian\](b) clearly shows that both mean and standard deviation is the lowest for the distribution-aware model whose test error is $\mu\pm\sigma=13.5\pm 0.3\%$.
Summary and conclusion
======================
We have investigated the failure mechanism of the forming process in the resistive memory based cross-bar array. As the forming process is an essential initialization step for ReRAM devices, it is pivotal to understand the impact of the forming failure on the cross-bar array performance. Specifically, we have focused on the two failure scenario: (i) forming failure (FF) devices and (ii) overformed (OF) devices. In the cross-bar array architecture, the FF device flows minimal current and may act as $0$ defects whereas OF device allows an excessive amount of current which may cause a catastrophic error. We have discussed 1T1R structure to regulate the excessive current flow and set a lower bound for conductance. In this case, OF device acts as $\pm1$ defect. We have further discussed 2T2R structure and corresponding forming strategy that minimizes $\pm1$ defects. Having the optimized hardware that minimizes the critical initialization errors, we then evaluate the impact of the $\pm1$ and $0$ defects on the inference model accuracy. The numerical experiments show that the impact of $\pm1$ defect is more significant than that of the $0$ defects. Furthermore, we show that the trained model becomes resilient to the defects if we apply the same type of random defects on the weight matrices during the training. The defect-aware model shows smaller test error than the inference results from the baseline when $\pm1$ and $0$ defects are introduced. We also have discussed the logical value deviation of OF device ($\delta w_{OF}$) and FF device ($\delta w_{FF}$). Such deviation from the desired value occurs as both OF or FF device exhibits conductance which is deviated from the lowest/highest possible conductance value. We have discussed the reasonable range of such deviation and show that defect-aware model is also resilient to the errors occurred by $\delta w_{OF,FF}$. Lastly, we discuss a variation in the defect probability which may happen by the process variation or process drift. We have examined the impact of such variation by using an exemplary defect probability distribution. The result shows that including a known distribution of the defect probability during the training further improves the average inference performance as well as the standard deviation of the test error.
One caveat of this study is that our analysis only includes the initialization failure scenario as the hardware non-ideality. In reality, the intricate interplay between different error sources may worsen the model accuracy and demands tighter parameter control than the scenario where individual components are separately considered [@2019Haensch]. To make a further statement on the realistic accuracy, future study needs to incorporate our results with the simulation framework that is capable of addressing other types of non-idealities.
The yield improvement of the semiconductor industry is one of the most critical factors determining manufacturing cost. In this regard, understanding the major failure mechanism and co-optimize the hardware and software may provide an opportunity for sustainable incremental yield improvement on top of the device engineering and material innovations. The proposed optimization strategy provides a straightforward method to implement, and may open a new avenue to address similar problems for cross-bar array based on other types of devices (e.g. Phase change memory) or other types of neural network (e.g. recurrent network).
Acknowledgments
===============
Y. Kim thanks to Theodorus E Standaert and Robert R. Robison for their managerial support. Y. Kim thanks to Wilfried Haensch and Geoffrey W. Burr for fruitful discussions. S. Kim acknowledges useful discussions from Tayfun Gokmen and managerial support from John Rozen. J. Choi thanks to Kailash Gopalakrishnan for his managerial support.
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---
abstract: 'In this article, we utilize machine learning to dynamically determine if a point on the computational grid requires implicit numerical dissipation for large eddy simulation (LES). The decision making process is learnt through *a priori* training on quantities derived from direct numerical simulation (DNS) data. In particular, we compute eddy-viscosities obtained through the coarse graining of DNS quantities and utilize their distribution to categorize areas that require dissipation. If our learning determines that closure is necessary, an upwinded scheme is utilized for computing the non-linear Jacobian. In contrast, if it is determined that closure is unnecessary, a symmetric and second-order accurate energy and enstrophy preserving Arakawa scheme is utilized instead. This results in a closure framework that precludes the specification of any model-form for the small scale contributions of turbulence but deploys an appropriate numerical dissipation from explicit closure driven hypotheses. This methodology is deployed for the Kraichnan turbulence test-case and assessed through various statistical quantities such as angle-averaged kinetic energy spectra and vorticity structure functions. Our framework thus establishes a direct link between the use of explicit LES ideologies for closure and numerical scheme-based modeling of turbulence leading to improved statistical fidelity of *a posteriori* simulations.'
author:
- 'R. Maulik'
- 'O. San'
- 'J. D. Jacob'
bibliography:
- 'references.bib'
title: 'Connecting implicit and explicit large eddy simulations of two-dimensional turbulence through machine learning'
---
Introduction
============
Over the past decade, advances in data collection and increasing access to computational resources have led to a revolution in the use of data-driven techniques for the solution of complex inverse problems. One such problem is that of turbulence, the multiscale nature of which causes extreme computational demands for most practical systems. As a result, turbulence requires the use multiple modeling approximations for the higher wavenumbers which remain unsupported by computational degrees of freedom. One such modeling approach is that of large eddy simulation (LES) [@sagaut2006large], which attempts to simulate the evolution of the smaller wavenumbers while the unresolved frequencies are modeled by an algebraic or differential equation. As such, the basic premise of LES is extendable to many partial differential equation systems with quadratic non-linearities. The procedure of modeling these smaller scales is often denoted *closure* due to insufficient knowledge about higher-order wavenumber interactions with the coarse-grained system [@berselli2006mathematics] and remains vital for the accurate computation of many applications [@hickel2014subgrid; @yu2016dynamic; @zhou2018structural]. From an LES point of view, the closure problem may be considered to be dominated by commutative errors in the calculation of the non-linear term as well as the defects associated with commutative errors stemming from the dynamic term. In this study, we focus on the former.
There are two main schools of thought when it comes to the LES of the Navier-Stokes equations. The first of these promotes the use of explicit closures. Explicit LES argues for the utilization of closures in the form of sub-grid models specified as algebraic or differential equations for the unresolved scales. These are built on intuitive reasoning of how the losses of coarse graining the Navier-Stokes equations may be incorporated into an LES deployment. Some of the most notable sub-grid closure strategies are those given by the eddy-viscosity hypothesis. Within the context of the Navier-Stokes equations, it is generally accepted that the finer scales are dissipative at the Kolmogorov length scales [@kolmogorov1941local] and therefore, most turbulence models seek to specify a sub-grid dissipation [@frisch1995turbulence]. Most sub-grid models can be traced back to the seminal work of Smagorinsky [@smagorinsky1963general], where a model was proposed based on the concepts of an effective eddy-viscosity determined by an *a priori* specified mixing length and a $k^{-5/3}$ scaling recovery for the kinetic energy content in the wavenumber domain. Similar hypotheses have also been used for two-dimensional turbulence [@leith1968diffusion] (often utilized as a test-bed for geophysical scenarios, for instance see works by Pearson *et al.*[@pearson2018log; @pearson2017evaluation]), for approximating the $k^{-3}$ cascade in two-dimensional turbulence and generally have their roots in dimensional analysis related to the cascade of enstrophy. These models may also be classified as *functional* due to the phenomenological nature of their deployment and represent the bulk of explicit LES turbulence models used in practical deployments. Explicit LES closures may also be specified through the specification of a low-pass spatial filter to account for the unresolved scales [@bardina1980improved; @stolz1999approximate; @layton2003simple; @mathew2003explicit] where phenomenology is bypassed but ansatz are provided for the bulk dissipative nature of the smaller scales through the control of a characteristic filter-width. In either scenario, (i.e., whether structural or functional), the choice of the phenomenology (or dissipation control parameter) plays a key role in the successful calculation of accurate *a posteriori* statistics. In contrast, the implicit LES (or ILES) approach utilizes numerical dissipation to model the unresolved scales in a turbulent flow [@grinstein2007implicit; @el2017investigation; @Margolin2018]. In essence, the predominantly dissipative effects of the smallest scales are replicated through an artificial numerical dissipation via a biased discretization used in the calculation of the non-linear advective term [@thornber2007implicit; @debonis2013solutions]. The ILES approach is popular due to reduced algorithmic complexity and represents a union of turbulence modeling and shock capturing mechanisms but is often criticized due to the difficulties involved in quantifying the correct amount of dissipation in a turbulent flow evolution. This results in ILES methods often proving robust and stable but overly dissipative. In this work, we propose a machine learning algorithm to enable selective dissipation within an ILES deployment through the use of explicit LES concepts during the training of the learning framework.
The past few years have seen a rapid increase in the use of data-driven techniques for the spatio-temporal modeling of dynamical systems [@schmidt2009distilling; @bright2013compressive; @xiao2015non; @ma2015using; @gautier2015closed; @brunton2016discovering; @schaeffer2017learning; @raissi2017machine; @mohan2018deep; @raissi2018hidden; @rudy2018deep; @san2018neural; @wan2018data; @kim2018deep; @muravleva2018application; @jin2018prediction]. When it comes to turbulence, some widely used strategies for inference include symbolic regression [@weatheritt2016novel; @weatheritt2017development; @weatheritt2017hybrid], where functional model-forms for Reynolds-averaged Navier-Stokes (RANS) deployments were generated through evolutionary optimization against high-fidelity data. Other techniques incorporating Bayesian ideologies have also been used, for instance by Xiao *et al.*[@xiao2016quantifying] where an iterative ensemble Kalman method was used to assimilate prior data for quantifying model form uncertainty in RANS models. In Wang *et al.*[@wang2017physics; @wang2017comprehensive] and Wu *et al.*[@wu2018data], random-forest regressors were utilized for RANS turbulence-modeling given direct numerical simulation (DNS) data. In Singh and Duraisamy [@singh2016using] and Singh *et al.*[@singh2017machine], an ANN was utilized to predict a non-dimensional correction factor in the Spalart-Allmaras turbulence model through a field-inversion process using experimental data. Bypassing functional formulations of a turbulence model (a focus of this study) was also studied from the RANS point of view by Tracey *et al.* [@tracey2015machine]. Ling and Templeton [@ling2015evaluation] utilized support vector machines, decision trees and random forest regressors for identifying regions of high RANS uncertainty. A deep-learning framework where Reynolds-stresses would be predicted in an invariant subspace was developed by Ling *et al.* [@ling2016reynolds]. Machine learning of invariance properties has also been discussed in the context of turbulence modeling [@ling2016machine]. The reader is directed to a recent review by Duraisamy *et al.*[@duraisamy2018turbulence], for an excellent review of turbulence modeling using data-driven ideas.
As shown above, the use of data-driven ideologies and in particular artificial neural networks (ANNs) has generated significant interest in the turbulence modeling community for addressing long-standing challenges (also see [@sotgiu2018turbulent; @zhang2018machine; @zhu2019machine; @zhang2019application; @raissi2019deep] for recent progress). One motivation for the popularity of ANNs is that a multilayered ANN may be optimally trained to universally approximate any non-linear function [@hornik1989multilayer]. In addition, the deployment of ANNs is amenable to integration within existing computational frameworks. Greater accessibility to data and ever-improving computing capabilities has also motivated the development of advanced ANN architectures for large-scale learning of complicated physical phenomena such as turbulence. Within the context of LES (and associated with the scope of this paper) there are several investigations into sub-grid modeling using data-driven techniques. In one of the first studies of the feasibility of using learning from DNS based high-fidelity data, Sarghini *et al.*[@sarghini2003neural] utilized ANNs for estimating Smagorinsky model-form coefficients within a mixed sub-grid model for a turbulent channel flow. ANNs were also used for wall-modeling by Milano and Koumotsakos [@milano2002neural] where it was used to reconstruct the near wall field and compared to standard proper-orthogonal-decomposition techniques. An alternative to ANNs for sub-grid predictions was proposed by King *et al.*[@king2016autonomic] where *a priori* optimization was utilized to minimize the $L^2$-error between true and modeled sub-grid quantities in a least-squares sense using a parameter-free Volterra series. Maulik and San [@maulik2017neural] utilized an extreme-learning-machine (a variant of a single-layered ANN) to obtain maps between low-pass spatially filtered and deconvolved variables in an *a priori* sense. This had implications for the use of ANNs for turbulence modeling without model-form specification. A more in-depth investigation was recently undertaken by Fukami *et al.*[@fukami2018super] where convolutional ANNs were utilized for reconstructing from downsampled snapshots of turbulence. Maulik *et al.* [@maulik2018deconvolution] also deployed a data-driven convolutional and deconvolutional operation to obtain closure terms for two-dimensional turbulence. Gamahara and Hattori [@gamahara2017searching], utilized ANNs for identifying correlations with grid-resolved quantities for an indirect method of model-form identification in turbulent channel flow. The study by Vollant *et al.* [@vollant2017subgrid] utilized ANNs in conjuction with optimal estimator theory to obtain functional forms for sub-grid stresses. In Beck *et al.*[@beck2018neural], a variety of neural network architectures such as convolutional and recurrent neural networks are studied for predicting closure terms for decaying homogeneous isotropic turbulence. A least-squares based truncation is specified for stable deployments of their model-free closures. Model-free turbulence closures are also specified by Maulik *et al.*[@maulik2018deconvolution; @maulik2019subgrid] and Wang *et al.*[@wang2018investigations], where sub-grid scale stresses are learned directly from DNS data and deployed in *a posteriori* assessments. King *et al.*[@king2018deep] studied generative-adversarial networks and the LAT-NET [@hennigh2017lat] for *a priori* recovery of statistics such as the intermittency of turbulent fluctuations and spectral scaling. A detailed discussion of the potential benefits and challenges of deep learning for turbulence (and fluid dynamics in general) may be found in the article by Kutz [@kutz2017deep].
While a large majority of the LES-based frameworks presented above utilize a least-squares error minimization technique for constructing maps to sub-grid stresses *directly* for theoretically optimal LES [@langford1999optimal; @moser2009theoretically; @labryer2015framework], this work is novel in that it utilizes sub-grid statistics (pre-computed from DNS data) to train a classifier. This classifier determines whether a location requires dissipation or not through *a priori* experience in the learning phase. Once classified, the non-linear term at this particular point is evaluated using one of two schemes. If it is determined that the point requires no sub-grid closure, a symmetric and second-order accurate, energy and enstrophy conserving Arakawa-scheme [@arakawa1981potential] is utilized for the non-linear term computation. If dissipation is necessary, an upwinding scheme is utilized instead. Therefore this study may be interpreted as a machine learning framework for devising hybrid schemes for non-linear term computation with a view to reconstructing turbulence statistics in a superior fashion. Therefore, this study is similar to that employed by Ling and Kurzawski [@ling2017data] for adaptively determining RANS corrections. We note that the classification framework devised in this study is also deployed in an aligned work to switch between functional and structural explicit LES hypotheses spatio-temporally [@maulik2018online] thus proving that high-fidelity DNS statistics may be qualitatively utilized to inform modeling strategies through conditional probability predictions. The article shall describe how the proposed framework is effective in moderating the larger dissipation of an upwinded-scheme through assessments on the Kraichnan turbulence test-case.
Turbulence modeling equations
=============================
The governing equations for two-dimensional turbulence are given by the Navier-Stokes equations in the vorticity-stream function formulation. In this formulation, our non-dimensional governing equation for incompressible flow may be represented as $$\begin{aligned}
\label{eq1}
\frac{\partial \omega}{\partial t} + J(\omega,\psi) = \frac{1}{Re} \nabla^2 \omega,\end{aligned}$$ where $Re$ is the Reynolds number, $\omega$ and $\psi$ are the vorticity and stream function respectively connected to each other through the Poisson equation given by $$\begin{aligned}
\label{eq2}
\nabla^2 \psi = - \omega.\end{aligned}$$ It may be noted that the Poisson equation implicitly ensures a divergence-free flow evolution. The non-linear term (denoted the Jacobian) is given by $$\begin{aligned}
\label{eq3}
J(\omega,\psi) = \frac{\partial \psi}{\partial y} \frac{\partial \omega}{\partial x} - \frac{\partial \psi}{\partial x} \frac{\partial \omega}{\partial y}.\end{aligned}$$ The stream function and the two-dimensional velocity components are related as $$\begin{aligned}
\label{eq3a}
u &= \frac{\partial \psi}{\partial y}, \quad v = -\frac{\partial \psi}{\partial x}.\end{aligned}$$
A reduced-order implementation of the aforementioned governing laws (i.e., an LES) is obtained through $$\begin{aligned}
\label{eq4}
\frac{\partial \bar{\omega}}{\partial t} + J(\bar{\omega},\bar{\psi}) = \frac{1}{Re} \nabla^2 \bar{\omega},\end{aligned}$$ where the overbarred variables are now evolved on a grid with far fewer degrees of freedom. Due to the reduction in supported frequencies, the non-linear Jacobian fails to capture inter-eddy interactions at different wavenumbers. If it is assumed that the finer scales of vorticity are generally dissipative in nature for two-dimensional turbulence (based on Kraichnan’s cascade of enstrophy [@kraichnan1967inertial]), dissipative models may be embedded into the coarse-grained evolution of the vorticity evolution equation to recover some portion of the effect of the finer scales. Explicit LES closures embed dissipation into the vorticity evolution in the form of eddy-viscosity phenomenology or through structural arguments of scale-similarity. However ILES manipulates the computation of the non-linear Jacobian term to add numerical dissipation to mimic that of the unresolved frequencies. The latter framework, while numerically robust, suffers from difficulties associated with *directed* dissipation where it is often very easy to be over-dissipative in regions where sub-grid dissipation may not be as pronounced. In this article, we introduce a hybrid ILES framework that focuses upwinding at areas where high probability of sub-grid dissipation necessity is detected.
Non-linear Jacobian computation
===============================
The study utilizes two types of non-linear term computation schemes. Our first choice is symmetric, second-order accurate and conserves energy and enstrophy to minimize numerical dissipation. This is given by the well-known second-order Arakawa scheme [@arakawa1981potential] as detailed below. The non-linear term in Equation \[eq4\] may be numerically calculated on a coarse grid using $$\begin{aligned}
J^A (\bar{\omega},\bar{\psi}) = \frac{1}{3} \left( J_1 (\bar{\omega}, \bar{\psi}) + J_2 (\bar{\omega}, \bar{\psi}) + J_3 (\bar{\omega}, \bar{\psi}) \right)\end{aligned}$$ where $J^A(\bar{\omega},\bar{\psi})$ will henceforth refer to the Arakawa discretization. The individual terms on the right hand side of the above equation are given as $$\begin{aligned}
\begin{split}
J_1 (\bar{\omega},\bar{\psi}) & = \frac{1}{4 \Delta x \Delta y} \left[ (\bar{\omega}_{i+1,j}-\bar{\omega}_{i-1,j}) (\bar{\psi}_{i,j+1} - \bar{\psi}_{i,j-1}) \right. \\
& \left. - (\bar{\omega}_{i,j+1}-\bar{\omega}_{i,j-1}) (\bar{\psi}_{i+1,j} - \bar{\psi}_{i-1,j}) \right],
\end{split}\end{aligned}$$
$$\begin{aligned}
\begin{split}
& J_2 (\bar{\omega},\bar{\psi}) = \frac{1}{4 \Delta x \Delta y} \left[ \bar{\omega}_{i+1,j} (\bar{\psi}_{i+1,j+1}-\bar{\psi}_{i+1,j-1}) \right. \\
& \left. - \bar{\omega}_{i-1,j} (\bar{\psi}_{i-1,j+1}-\bar{\psi}_{i-1,j-1}) - \bar{\omega}_{i,j+1} (\bar{\psi}_{i+1,j+1}-\bar{\psi}_{i-1,j+1})\right. \\
& \left. + \bar{\omega}_{i,j-1} (\bar{\psi}_{i+1,j-1}-\bar{\psi}_{i-1,j-1}) \right],
\end{split}\end{aligned}$$
$$\begin{aligned}
\begin{split}
& J_3 (\bar{\omega},\bar{\psi}) = \frac{1}{4 \Delta x \Delta y} \left[ \bar{\omega}_{i+1,j+1} (\bar{\psi}_{i,j+1} - \bar{\psi}_{i+1,j}) \right. \\
& \left. - \bar{\omega}_{i-1,j-1} (\bar{\psi}_{i-1,j}-\bar{\psi}_{i,j-1}) - \bar{\omega}_{i-1,j+1} (\bar{\psi}_{i,j+1}-\bar{\psi}_{i-1,j}) \right. \\
& \left. + \bar{\omega}_{i+1,j-1} (\bar{\psi}_{i+1,j}-\bar{\psi}_{i,j-1}) \right].
\end{split}\end{aligned}$$
The aforementioned scheme is utilized when our proposed classifier recognizes that no dissipation is necessary.
A numerically dissipative computation of the non-linear term allows for that stabilization of noise accumulation at the grid cut-off wavenumbers. Although there are many different methodologies for upwind based dissipation with varying degrees of complexity, in this article, we utilize a conventional upwind-biased scheme as detailed in the following [@hoffmann2000computational]. Our ILES Jacobian is computed as $$\begin{aligned}
\begin{split}
J^I(\bar{\omega},\bar{\psi}) =& \bar{u}_{i,j} \frac{\bar{\omega}_{i+1,j} - \bar{\omega}_{i-1,j}}{2 \Delta x} + \frac{1}{2} (\bar{u}^{+} \bar{\omega}_x^{-} + u^{-} \bar{\omega}_x^{+}) \\
& + \bar{v}_{i,j} \frac{\bar{\omega}_{i,j+1} - \bar{\omega}_{i,j-1}}{2 \Delta y} + \frac{1}{2} (\bar{v}^{+} \bar{\omega}_y^{-} + \bar{v}^{-} \bar{\omega}_y^{+}),
\end{split}\end{aligned}$$ where $$\begin{aligned}
{2}
\bar{u}^{-} &= \min(\bar{u}_{i,j},0), \quad \bar{u}^{+} &= \max(\bar{u}_{i,j},0), \\
\bar{v}^{-} &= \min(\bar{v}_{i,j},0), \quad \bar{v}^{+} &= \max(\bar{v}_{i,j},0).\end{aligned}$$ In addition, $$\begin{aligned}
\begin{split}
\bar{\omega}_x^{-} &= \frac{\bar{\omega}_{i-2,j} - 3 \bar{\omega}_{i-1,j} + 3 \bar{\omega}_{i,j} - \bar{\omega}_{i+1,j} }{3 \Delta x}, \\
\bar{\omega}_x^{+} &= \frac{\bar{\omega}_{i-1,j} - 3 \bar{\omega}_{i,j} + 3 \bar{\omega}_{i+1,j} - \bar{\omega}_{i+2,j} }{3 \Delta x}, \\
\bar{\omega}_y^{-} &= \frac{\bar{\omega}_{i,j-2} - 3 \bar{\omega}_{i,j-1} + 3 \bar{\omega}_{i,j} - \bar{\omega}_{i,j+1} }{3 \Delta y}, \\
\bar{\omega}_y^{+} &= \frac{\bar{\omega}_{i,j-1} - 3 \bar{\omega}_{i,j} + 3 \bar{\omega}_{i,j+1} - \bar{\omega}_{i,j+2} }{3 \Delta y}.
\end{split}\end{aligned}$$ Note that velocity components are recovered using $$\begin{aligned}
\begin{split}
\bar{u}_{i,j} &= \frac{\bar{\psi}_{i,j+1}-\bar{\psi}_{i,j-1}}{2 \Delta y} \\
\bar{v}_{i,j} &= -\frac{\bar{\psi}_{i+1,j}-\bar{\psi}_{i-1,j}}{2 \Delta x},
\end{split}\end{aligned}$$ where the second-order accurate reconstruction of the velocity leads to overall second-order accuracy for non-linear Jacobian reconstruction using the upwinded procedure outlined above. We also note that our Poisson equation given by Equation \[eq2\] is solved using a spectrally-accurate scheme.
With the choice of one of the two aforementioned schemes, a point in space-time may or may not have an artificial dissipation imparted to it numerically. However, we mention the caveat that switching between these two schemes would mean that the kinetic energy and enstrophy preserving property of the Arakawa scheme is lost.
Machine learning for scheme selection
=====================================
We now discuss the procedure of utilizing DNS data for learning to classify one of the two dissipation scenarios. Of these two options, one is given by the choice of the Arakawa scheme and the other by our upwinded computation of the Jacobian (i.e., when the classification framework has determined that the point does not require sub-grid dissipation or vice-versa respectively). This switching between scenarios is spatio-temporally dynamic. We proceed by outlining our training strategy through the utilization of DNS data. Five equidistant snapshots of DNS data at $Re=32000$ (i.e., at $t=0,1,2,3,4$) and at $N^2 = 2048^2$ degrees of freedom (from 40000 available snapshots) are utilized to compute the grid-filtered variables (denoted FDNS) (at $N^2 = 256^2$ degrees of freedom) through the application of a spectral cut-off filter. Perfect closure values $$\begin{aligned}
\Pi = J(\bar{\omega},\bar{\psi})-\overline{J(\omega,\psi)}\end{aligned}$$ are then obtained (the reader is directed to [@maulik2019subgrid] for details related to the calculation of these quantities). Note here, that the Kraichnan turbulence problem is transient with the evolution of vorticity represented in Figure \[Fig1\] representing different closure needs over time evolution.
We proceed by introducing the *a priori* eddy-viscosity given by $$\begin{aligned}
\nu_e^a = \frac{\Pi}{\nabla^2 \bar{\omega}}\end{aligned}$$ where the right-hand side of the above equation may be calculated from DNS snapshots. The *a priori* eddy-viscosity is centered at zero (corresponding to where closure modeling is unnecessary) and spreads out in the negative and positive directions (a hallmark of isotropic turbulence). We segregate this *a priori* estimate of sub-grid effects into three categories as follows. The *a priori* eddy-viscosities calculated from the DNS data are compared with a Gaussian distribution where values lying less than a distance of 1% of the standard-deviation from the mean (which is zero) are labeled as those requiring no dissipation (due to the low strength of the *a priori* eddy-viscosity). For posterity, we label these points as $k=1$. Positive values lying beyond this range are labeled as those requiring sub-grid dissipation and are labeled $k=2$. Negative values less than 1% of the standard-deviation are also considered to require no dissipation and are labeled $k=3$. This three-category segregation stems from a learning hypothesis that seeks to identify regions in a flow evolution that require structural, functional or no-closure modeling hypothesis. We link labels of negative or nearly-zero eddy-viscosities to the Arakawa classification and positive eddy-viscosities to the upwinded classification. The positive eddy-viscosity prediction would indicate that the sub-grid term at a point is predominantly dissipative in nature at which point the numerical dissipation of the upwinded scheme would be utilized. We note here that the concept of an *a priori* eddy-viscosity lies firmly within the explicit LES hypothesis. The classifier is therefore instrumental in moderating ILES deployments through a decision making process that recognizes the dissipative (or forcing) nature of the sub-grid quantities.
We note that the choice of 1% as the decision parameter for switching between hypothesis is motivated by a sensitivity study that showed the highest classification accuracy for the ANN framework. Larger choices of this hyper-parameter would result in a classifier that would be prone to classify most points in the ‘no-model’ zone. However, we clarify that the choice of this value is also correlated with the architecture of the ANN. A potential extension of the proposed hypothesis is to combine architecture search algorithms with varying value of the decision hyper-parameters for larger classification accuracies. In addition, the three-category framework is derived from an aligned study [@maulik2018online] where sub-grid models are determined according to negative, positive and nearly-zero *a priori* eddy-viscosities and utilizes the same learning. This enables use to determine a unified framework for switching between turbulence model hypotheses as well as numerical dissipation scenarios. However, we would like to emphasize that, for the purpose of switching between the Arakawa and upwinded Jacobian computation, a simple two-class framework would also suffice.
{width="\textwidth"}
A one-hot labeling of our eddy-viscosity classes is utilized for a classification deployment and a schematic for this hypothesis segregation and labeling is shown in Figure \[Segregation\]. The labels indicate the conditional probability of a point belonging to each possible class. As such, the training labels are given by a value of 1 for the particular class that a point belongs to and zeros for other choices. This is because there is no ambiguity in the class a training sample belongs to. Each label for the *a priori* eddy-viscosity is also associated with a corresponding input kernel of grid-resolved quantities. This kernel is given by a local stencil of vorticity and stream function. There are 9 inputs each for vorticity and stream function given by a query of the field quantity at a point on the coarse grid, 4 adjacent points in each dimension ($x,y$) and the 4 diagonally adjacent points. Each sample of our training data thus consists of 18 inputs of vorticity and stream function and outputs given by one-hot labels for the choice of closure modeling strategy. We then utilize an ANN to establish a relationship between these inputs and outputs. Mathematically, if our input vector $\mathcal{P}$ resides in a $P$-dimensional space and our desired output $\mathcal{Q}$ resides in a $Q$-dimensional space, this framework establishes a map $\mathbb{M}$ as follows: $$\begin{aligned}
\label{eq6}
\mathbb{M} : \{ \mathcal{P}_1, \mathcal{P}_2, \hdots, \mathcal{P}_P\} \in \mathbb{R}^P \rightarrow \{ \mathcal{Q}_1, \mathcal{Q}_2, \hdots, \mathcal{Q}_Q\} \in \mathbb{R}^Q.\end{aligned}$$ Accordingly, the framework utilized in this article leads to the following relation: $$\begin{aligned}
\label{eq7}
\mathbb{M} : \{ \textbf{p} \} \in \mathbb{R}^{18} \rightarrow \{ P(\textbf{q}|\textbf{p})\} \in \mathbb{R}^3,\end{aligned}$$ where $$\begin{aligned}
\begin{gathered}
\textbf{p}_{i,j} = \{ \bar{\omega}_{i,j}, \bar{\omega}_{i,j+1}, \bar{\omega}_{i,j-1}, \hdots, \bar{\omega}_{i-1,j-1}, \\ \bar{\psi}_{i,j}, \bar{\psi}_{i,j+1}, \bar{\psi}_{i,j-1}, \hdots, \bar{\psi}_{i-1,j-1} \}
\end{gathered}\end{aligned}$$ is our input vector for each query of the machine learning framework and where $$\begin{aligned}
P(\textbf{q}|\textbf{p})_{i,j} = \{ P(J^k(\bar{\omega},\bar{\psi})_{i,j}| \textbf{p}_{i,j})\},\end{aligned}$$ is the conditional probability of a Jacobian computation (given by a connection to the explicit closure hypothesis). Note that $i,j$ refer to the spatial indices on the coarse-grid (i.e., the point of deployment). The indices $k=1$ and $k=3$ refer to the Arakawa non-linear Jacobian computation and $k=2$ refers to the upwinded computation instead (see Figure \[Segregation\]). Our optimal map $\mathbb{M}$ is then trained by minimizing the categorical cross-entropy loss-function $$\begin{aligned}
E(\textbf{w}) = -\sum_{n=1}^{N} \sum_{k=1}^{K} \{ t_{nk} \log(y_{nk}) + (1-t_{nk})\log(1-y_{nk})\},\end{aligned}$$ where $\textbf{w}$ are the variable weight and bias parameters of the network, $N$ refers to the total number of samples and $K=3$ is the total number of classification scenarios (i.e., negative, positive or nearly-zero *a priori* eddy-viscosities). Here, $t_{nk}$ refers to the true label of class $k$ and sample $n$ and $y_{nk}$ refers to a corresponding prediction of the learning framework. One-hot encoding ensures that $t_{nk}$ values are always binary [@Bishop:2006:PRM:1162264] and the outputs of the ANN may be interpreted as conditional-probabilities. Our optimal architecture is given by five 40-neuron hidden layers (obtained via grid-search hyper-parameter tuning). All hidden layers utilize ReLU units to impart non-linearity to the layer-wise transformations. For reference, our architecture is trained using the open-source deep learning software Tensorflow and is optimized with the use of ADAM, a popular gradient-descent based optimizer [@kingma2014adam]. Figure \[Loss\_history\] shows the progress to convergence for our framework with our optimally trained network displaying approximately 79% accuracy in classifying points to their correct labels. To summarize this section, we train a deep ANN to estimate probabilities of negative, positive or nearly-zero eddy-viscosities which are utilized to decide the choice of the Jacobian computation. We clarify that the decision to deploy a particular hypothesis is obtained by utilization of the classification scenario which has the highest conditional probability.
![Learning rate and convergence of our classification framework training. 2000 epochs were sufficient for converged validation loss.[]{data-label="Loss_history"}](Loss_history-eps-converted-to.pdf){width="\columnwidth"}
Results
=======
*A posteriori* deployment
-------------------------
In this section, we detail the results from an *a posteriori* deployment of the classification framework (denoted ML henceforth) for the Kraichnan test-case. In the LES evolution of the problem, a considerably coarser grid is used (at $N^2=256^2$). We remark that the forward deployment of our framework needs to overcome the challenge of numerical errors and is a robust test of the generalizability and robustness of our learning. Our LES results are assessed using angle-averaged kinetic energy spectra and through structure functions of vorticity. In addition, qualitative comparisons are also provided through visual examinations of the vorticity contours. We remark that the LES deployment is performed from $t=0$ to $t=4$ which spans the training regime data obtained from DNS. In what follows we note that DNS refers to a high-fidelity evolution of the governing equations (i.e., at $N^2=2048^2$ degrees of freedom), UNS refers to results obtained using the Arakawa scheme alone and ILES refers to a simulation where the non-linear Jacobian at all points in space and time are upwinded. Figure \[Fig2\] shows the *a posteriori* performance of the proposed framework at $Re=32000$ in terms of energy spectra predictions. The reader may find an exact definition of the kinetic-energy spectra in Maulik and San [@maulik2017stable]. We note that the training data was obtained for the same Reynolds number as well. The prediction of the proposed framework is seen to agree remarkably well with DNS. It is apparent that the switching of schemes using the classifier has obtained an optimal balance between both techniques.
![The *a posteriori* performance of proposed framework (ML) for $Re=32000$ and at $t=4$ in terms of angle-averaged kinetic energy spectra. Comparisons with DNS, the Arakawa scheme (UNS) and the upwinded scheme (ILES) show that ML provides directed dissipation adequately.[]{data-label="Fig2"}](Figure_2-eps-converted-to.pdf){width="\columnwidth"}
Vorticity contours for LES resolution assessments are shown in Figure \[Fig3\], where it is apparent that the proposed framework optimally balances the energy-conserving and dissipative natures of the Arakawa and upwinded schemes respectively. This is verified by qualitative examination with FDNS contours obtained by spectrally filtering the DNS snapshot for $Re=32000$ at $t=4$.
{width="\textwidth"}
A second statistically significant quantity of interest studied in this investigation is the vorticity structure function [@grossmann1992structure] given by $$\begin{aligned}
S_\omega^x = \langle |\omega(x+r,y) - \omega(x,y)|^2 \rangle \\
S_\omega^y = \langle |\omega(x,y+r) - \omega(x,y)|^2 \rangle,\end{aligned}$$ where the angle-brackets indicate ensemble averaging and $x,y$ indicate a position on the grid with $r$ being a certain distance from this location. Figures \[Fig4\] and \[Fig5\] show the structure functions obtained from *a posteriori* deployments of the UNS, ILES and ML frameworks compared against those obtained from the final time FDNS snapshot. It is clear that the proposed framework balances between UNS and ILES deployments well to recover appropriate trends. We can thus claim that our learning is appropriate for hybrid deployments of dissipative and conservative frameworks for two-dimensional turbulence. Before moving on, we would like to point out to the reader here that the proposed methodology for closure does not require any post-processing prior to deployment in the forward simulation as utilized in several data-driven turbulence modeling studies[@beck2018neural; @maulik2019subgrid].
![*A posteriori* vorticity structure functions in $x$ direction of our proposed framework (ML), the Arakawa scheme (UNS) and the upwind scheme (ILES) with statistics obtained from an FDNS snapshot at $t=4$. It is apparent that the ML method stabilizes the UNS result optimally.[]{data-label="Fig4"}](Figure_4-eps-converted-to.pdf){width="\columnwidth"}
![*A posteriori* vorticity structure functions in $y$ direction of our proposed framework (ML), the Arakawa scheme (UNS) and the upwind scheme (ILES) with statistics obtained from an FDNS snapshot at $t=4$. It is apparent that the ML method stabilizes the UNS result optimally.[]{data-label="Fig5"}](Figure_5-eps-converted-to.pdf){width="\columnwidth"}
Validation of learning
----------------------
In this section, we proceed with a rigorous validation of our learning for deployment in regimes that are not a part of the training data. This is to ensure that the framework has truly learnt a classification based on the underlying physical hypothesis used for data segregation and is not memorizing data. This ensures that our classifier can be used in a more generalizable fashion. Figure \[Fig6\] shows kinetic energy spectra obtained from the forward deployment of the ML framework for a $Re=64000$ which represents a classification task that the framework has not previously seen (although the physics of the test-case remains similar). As observed, the proposed method performs quite well in this out-of-training data range as well. We note that a similar resolution ($N^2=256^2$) is utilized for this deployment. In contrast, Figure \[Fig7\] shows the performance of the ML technique for a reduced resolution of $N^2=128^2$ but utilizing the same Reynolds number of 32000. The kinetic energy spectra show a successful stabilization of the flow evolution at this reduced resolution although some forcing to the large scales is observed. This suggests that the classification framework may be improved by sampling from different resolutions.
![The *a posteriori* performance of proposed framework (ML) for $Re=64000$ and at $t=4$ in terms of energy spectra. This represents deployment of our learning at a different Reynolds number than that used for generating training data.[]{data-label="Fig6"}](Figure_6-eps-converted-to.pdf){width="\columnwidth"}
![The *a posteriori* performance of proposed framework (ML) for $Re=32000$, $t=4$ and at $N^2 = 128^2$ in terms of energy spectra. This represents deployment of our learning at a different resolution than that used for generating training data.[]{data-label="Fig7"}](Figure_7-eps-converted-to.pdf){width="\columnwidth"}
Concluding remarks
==================
In this article, we have proposed a neural network based classifier that enables us to take decisions on the choice of non-linear term computation in the LES evolution of the Kraichnan turbulence test-case. The classifier outputs conditional probabilities for the presence (or absence) of eddy-viscosities within three different ranges during deployment and is used to switch between the Arakawa and upwind computation of the non-linear Jacobian for a hybrid upwinded deployment that optimally directs dissipation on the coarse-grained flow field. Our machine learning framework is trained by calculating *a priori* eddy-viscosities which are projected onto a Gaussian distribution and segregated into three categories. Each category is devised to capture a unique behavior of the underlying sub-grid terms with negative and nearly-zero eddy-viscosity classes signifying absence of sub-grid dissipation. An optimally trained classifier is then utilized to identify if a point requires sub-grid dissipation based on if it is placed in the positive eddy-viscosity category. If so, the upwind Jacobian is calculated for imparting numerical dissipation.
We perform *a posteriori* assessments on the Kraichnan turbulence test-case through statistical quantities such as the angle-averaged kinetic energy spectra and the vorticity structure functions. It is observed that the proposed framework is successful in balancing the dissipative nature of the upwind scheme and the energy-conserving Arakawa scheme to give excellent agreement with DNS statistics. Validation for out-of-training regimes also indicate that the framework is able to learn the link between grid-resolved quantities at a coarse resolution and the nature of the sub-grid forcing.
Our conclusions therefore point toward the possibility of using classifiers for the unified deployment of numerical schemes with varying dissipation through the decision making process described above. A key strength of our hypothesis stems from the fact that an ILES deployment is moderated by concepts drawn from the explicit LES ideology (i.e., that of an *a priori* eddy-viscosity). The successful deployment of our method thus points towards the possibility of deploying directed numerical dissipation that preserves the statistics of turbulence without sacrificing the shock-capturing ability of many non-oscillatory schemes. Our future work lies in that particular direction.
This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research under Award Number DE-SC0019290. OS gratefully acknowledges their support. Disclaimer: This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.
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abstract: 'We report about the multiwavelength campaign on the Narrow-Line Seyfert 1 (NLS1) Galaxy PMN J0948$+$0022 ($z = 0.5846$) performed in 2010 July-September and triggered by high activity as measured by *Fermi*/LAT. The peak luminosity in the $0.1-100$ GeV energy band exceeded, for the first time in this type of source, the value of $10^{48}$ erg/s, a level comparable to the most powerful blazars. The comparison of the spectral energy distribution of the NLS1 PMN J0948$+$0022 with that of a typical blazar – like 3C 273 – shows that the power emitted at gamma rays is extreme.'
author:
- 'L. Foschini, G. Ghisellini, L. Maraschi, G. Tagliaferri, F. Tavecchio'
- 'Y. Y. Kovalev, Yu. A. Kovalev'
- 'M. L. Lister, J. L. Richards'
- 'F. D’Ammando'
- 'D. J. Thompson, D. Donato'
- 'A. Tramacere'
- 'E. Angelakis, L. Fuhrmann, I. Nestoras'
- 'A. Falcone'
- 'M. Hauser, S. Wagner'
- 'K. Mannheim, O. Tibolla'
- 'W. Max-Moerbeck, V. Pavlidou, A. C. S. Readhead, M. A. Stevenson'
- 'A. B. Pushkarev'
title: The July 2010 outburst of the NLS1 PMN J0948+0022
---
INTRODUCTION
============
Narrow-Line Seyfert 1 Galaxies (NLS1s) make a very peculiar class of active galactic nuclei (AGN). They were discovered in eighties (see [@POGGE] for a recent review) because of their difference in the optical spectra with respect to the classical Seyfert active nuclei. Basically, the full-width half maximum (FWHM) of their broad permitted emission lines (e.g. H$\beta$) has values smaller than those of Seyfert 1s, with a drop in the distribution of the values of FWHM(H$\beta$) above $\sim 2000$ km/s. These “narrower” broad-lines – yes, it is an oxymoron! – are not due to some obscuration, as indicated by the detection of the FeII bump in the optical spectra of NLS1s. The FeII bump is an indicator of the direct view of the broad-line region (BLR), because it is observed only in Seyfert 1s and not in Seyfert 2s (e.g. [@DONG]). Among the typical characteristics of NLS1s, the most interesting are the relatively low mass of the central spacetime singularity ($\sim 10^{6-8}M_{\odot}$) and the high rate of the accretion disc (up to the Eddington limit). To date, it is not yet clear if these – as well as other peculiar observational characteristics – are the symptoms of a central engine intrinsically different from the other Seyferts or NLS1s are just the low black hole mass tail of the Seyfert distribution[^1].
As the other Seyferts, NLS1s are generally radio quiet, but $\sim 7$% of them exhibit a very compact radio core with high brightness temperature and flat or even inverted spectrum, hints of the presence of a relativistic jet [@DOI; @ZHOU; @KOMOSSA]. These hints were strengthened by the detection in 1H 0323$+$342 ($z=0.061$) of a hard tail in the X-ray spectrum extending to hard X-rays and emerging during periods of high optical/UV fluxes [@FOSCHINI4]. An early attempt to detect high-energy $\gamma$ rays from radio-loud NLS1s was performed in 2004 with the [*Whipple*]{} Cerenkov telescope [@FALCONE], but without success.
\
The definitive confirmation of the presence of powerful relativistic jets in NLS1s came with the Large Area Telescope (LAT) onboard the *Fermi* satellite. After a few months of operations, LAT observed highly significant ($17\sigma$) emission of GeV photons from the NLS1 PMN J0948$+$0022 ($z=0.5846$) [@DISCOVERY; @FOSCHINI1], which in turn was already known as strong radio-loud NLS1 [@ZHOU].
Soon after the early association of the $\gamma$-ray source with PMN J0948$+$0022, made on a probabilistic basis, we started a multiwavelength campaign, which was performed between the end of 2009 March and the beginning of 2009 July [@MW2009]. The source displayed some activity at $\gamma$ rays with a peak of $\sim 4\times 10^{-7}$ ph cm$^{-2}$ s$^{-1}$ ($0.1-100$ GeV) measured in one day on 2009 April 1. Then, the drop in the $\gamma$-ray emission was followed by a decrease of X-rays-to-optical flux and an increase of the radio flux after about less than two months. Particularly, [*Swift*]{}/UVOT recorded a significant change in the spectral slope of the optical emission during the decrease of the continuum. The observed coordinated broad-band variability confirmed that:
1. the $\gamma$-ray source discovered by [*Fermi*]{}/LAT is associated with the NLS1 PMN J0948$+$0022;
2. the continuum of the $\gamma$-NLS1 is dominated by the emission of a powerful relativistic jet viewed at small angles.
Later, the observation of optical (V) polarization at $19$% level with the KANATA telescope confirmed once more the above inferences [@KANATA].
Other NLS1s have been detected in the GeV energy range by [*Fermi*]{} [@NLS1CLASS; @FOSCHINI3], thus indicating that a new class of $\gamma$-ray emitting AGN is emerging. One of the most important consequences of this discovery in our knowledge of relativistic jets is that it is now possible to study an unexplored range of black hole masses and accretion disc rates, which in turn is opening new horizons in our understanding of jets at all scales (see [@FOSCHINI3; @FOSCHINI5; @FOSCHINI6]).
It is important to continue the monitoring of these sources in order to understand their nature and to extend the sample of $\gamma$-ray emitting candidates. Therefore, several monitoring programs were started and are currently ongoing. Again in 2010 July, PMN J0948$+$0022 exploded in an intense outburst at $\gamma$ rays, reaching a peak luminosity of $\sim 10^{48}$ erg/s [@DONATO; @FOSCHINI7]. The source was already under monitoring, but the high activity at $\gamma$ rays triggered more observations. Here we report about that campaign. Some early results were presented in [@FOSCHINI2] and a complete paper – where more details are available – has been recently published [@MW2010].
![Multifrequency light curves from Effelsberg 100 m radiotelescope ([*F-Gamma*]{} program). The different symbols indicate the measurements at different frequencies: red triangles, 2.64 GHz; green squares, 4.85 GHz; blue circles, 8.35 GHz; orange stars, 10.45 GHz. The reference time (MJD 55377) is 2010 June 30.[]{data-label="fig:effelsberg"}](lcurves_eff.ps)
FACILITIES INVOLVED
===================
After the $\gamma$-ray detection, PMN J0948$+$0022 has been monitored by some facilities and, therefore, it was possible to reconstruct the multiwavelength behavior before and after the 2010 July outburst. There were some problems after the end of June, because the source position projected in the sky was apparently very close to the Sun, making it difficult the optical-to-X-ray observations. Anyway, we succeeded to collect sufficient data to do some important inferences on the nature of this source.
The data gathered to study the 2010 July outburst were from:
- [*Fermi*]{}/LAT, $0.1-100$ GeV energy band; the satellites continuously scans the $\gamma$ ray sky every two orbits (three hours).
- [*Swift*]{}, with its X-ray Telescope (XRT) operating in the $0.3-10$ keV energy band and its UltraViolet Optical Telescope (UVOT), equipped with the filters V, B, U, UVW1, UVM2, and UVW2. Only one short snapshot was possible and it was performed on 2010 July 3 (ObsID 00038394002), in the framework of the project [*Swift/XRT Monitoring of Fermi/LAT sources of interest*]{}[^2] at the Penn State University. The exposure on XRT was $\sim 1.6$ ks and the data from all the six UVOT filters were available.
- Optical data in the R filter were taken with the Automatic Telescope for Optical Monitoring for HESS (ATOM), located in Namibia.
- Multifrequency radio observations ($2-43$ GHz, Fig. \[fig:effelsberg\]) were performed with the single-dish 100 m radiotelescope of Effelsberg ([*F-Gamma*]{} program[^3], [@FGAMMA; @FGAMMA2]). PMN J0948$+$0022 is continuously monitored on monthly basis since the discovery of GeV emission with [*Fermi*]{}.
- Multifrequency radio observations ($5-22$ GHz) were performed in the period 2010 August 13-26 by using the RATAN-600 telescope [@KOVALEV].
- Radio observations at 15 GHz are continuously performed with the Owens Valley Radio Observatory (OVRO) within a program of monitoring of [*Fermi*]{} sources [@OVRO].
- VLBA high-resolution ($\sim$ mas scale) radio observations at 15 GHz within the [*MOJAVE*]{} Project[^4] [@MOJAVE]. The source is monitored since the discovery of $\gamma$ rays.
DISCUSSION
==========
Figure \[fig:0948lc\] displays four panels containing light curves at different frequencies, from radio to $\gamma$ rays. As in the 2009 campaign, the coordinated broad-band variability was dominated by the relativistic jet radiation.
It is possible to note that as the $\gamma$ ray emission became detectable (although with one month integration of data) about two months before the burst, there was an increase of the optical and radio flux, together with an inversion of the radio spectral index (see Fig. \[fig:0948lc\] and \[fig:effelsberg\]). It is worth adding that VLBA osbervations indicated a $\sim 90^{\circ}$ swing in the electric vector position angle (EVPA) at some time between 2009 July 23 and 2009 December 10 (see Fig. 5 in [@MW2010]). A similar behavior has been already observed in high power blazars, such as PKS 1502$+$106 ($z=1.839$) [@1502]. Perhaps, it is an indication of some arrangement of the jet structure to favor the radiative dissipation of the kinetic energy.
Another interesting feature observed during this outburst was the hardening of the $\gamma$ ray spectrum [@MW2010]. The photon index $\Gamma$ before and after the outburst was quite steep $\sim 2.7$, as expected from relativistic jets powered by external Compton (EC) processes. The EC needs of a nearby environment rich of photons, which in turn has the drawback to enhance the pair production probability and cutting the very high energy emission. At the peak of the emission, the $\gamma$-ray spectrum was harder, with $\Gamma \sim 2.5$.
A reanalysis of the LAT data with the most recent software ([LAT Science Tools 9.23.1]{}), Instrument Response Function ([P7SOURCE\_V6]{}), and background files ([iso\_p7v6source.txt]{} and [gal\_2yearp7v6\_v0.fits]{}), confirmed the hard spectrum, although not the change in the slope. In 2010 June (30 days of data), before the burst, the $0.1-100$ GeV flux and photon index $\Gamma$ were $(1.6\pm 0.3)\times 10^{-7}$ ph cm$^{-2}$ s$^{-1}$ and $2.5\pm 0.1$, respectively. During the day of the burst (2010 July 8), the flux reached the value of $(1.3\pm 0.3)\times 10^{-6}$ ph cm$^{-2}$ s$^{-1}$ with a photon index $2.4\pm 0.2$. The measurements done after the burst, by integrating data in the period August 1 - September 15 (45 days of data), resulted in the values of flux $(1.5\pm 0.2)\times 10^{-7}$ ph cm$^{-2}$ s$^{-1}$ and photon index $2.4\pm 0.1$. What is important is the confirmation of the hardness of the spectrum, which can have important consequences on the possibility to detect these sources at even greater energies with ground-based Cherenkov telescopes (e.g. Cherenkov Telescope Array – CTA – [@CTA], see Fig. \[fig:cta\]). Obviously, this depends on the possibility to have a spectral break at tens of GeV, but, if it is confirmed the similarity of behavior of $\gamma$-NLS1s with flat-spectrum radio quasars, it is reasonable to expect the occurrence of events like the case of PKS B1222$+$216, detected at hundreds of GeV during a strong outburst [@FERMIPROC].
![$\gamma$-ray spectra during the outburst of PMN J0948$+$0022 and perspectives of detection with CTA. The sensivity of CTA is plotted with grey dashed lines (with 0.5, 5, and 50 hours of exposure, from [@CTA]). The detection at the peak (2010 July 8) is indicated with blue continuous lines. The red dot-dashed lines refer to the spectrum measured in 2010 June and in the period August 1 and September 15, which are basically similar.[]{data-label="fig:cta"}](0948_burst.ps)
More interesting inferences can be derived by comparing the spectral energy distribution (SED) of PMN J0948$+$0022 with that of the archetypical blazar PKS 1226$+$023 (a.k.a. 3C 273, Fig. \[fig:sed\]). The radio-to-X-rays SEDs can be matched by multiplying the luminosities of PMN J0948$+$0022 by a factor $\sim 6$, which in turn corresponds to the difference in mass of the two central black holes: $\sim 1.5\times 10^{8}M_{\odot}$ for the $\gamma$-NLS1 and $\sim 8\times 10^{8}M_{\odot}$ for the blazar. This rescaling, however, increases the separation at $\gamma$ rays, with the $\gamma$-NLS1 having the greatest power. This gap can be explained by a difference of the viewing angles, being that of PMN J0948$+$0022 smaller than that of 3C 273.
Later works supported these findings. The scaling in mass is justified by the fact that both sources have discs with high accretion rates ($\rightarrow$ radiation-pressure dominated) and hence the scaling of the Blandford-Znajek power is dependent only on the mass of the central black hole [@GOSH; @MODERSKI]. The presence of two regimes (radiation- and gas-pressure dominated) is evident in the distribution of the jet power as a function of the accretion rate (see Fig. 3 of [@FOSCHINI5]) and both high-power blazars and $\gamma$-NLS1s occupy the region of sources with radiation-pressure dominated discs.
FINAL REMARKS
=============
The 2010 July outburst of the $\gamma$-ray NLS1 PMN J0948$+$0022 showed once more the importance of multiwavelength campaigns in the understanding of sources with powerful relativistic jets. In this specific case, we noted:
1. another similarity of the relativistic jet in PMN J0948$+$0022 with those in blazars: the swing of the EVPA at radio frequencies before the outburst;
2. the extreme efficiency of the jet power with respect to the mass of the central black hole ($\gamma$-NLS1s are small, but nasty).
3. the hardness of the $\gamma$-ray spectrum together with high flux during the outburst opens the possibility of a detection at hundreds of GeV with Cerenkov telescopes, which in turn will likely to be possible with future facilities, such as – for example – CTA.
This research has made use of data obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA’s Goddard Space Flight Center.
This work has been partially supported by PRIN-MiUR 2007 and ASI Grant I/088/06/0.
The OVRO 40 m monitoring program is supported in part by NASA (NNX08AW31G) and the NSF (AST-0808050).
Based on observations with the 100-m telescope of the MPIfR (Max-Planck-Institut für Radioastronomie) at Effelsberg. Ioannis Nestoras is member of the International Max Planck Research School (IMPRS) for Astronomy and Astrophysics at the Universities of Bonn and Cologne.
The *Fermi* LAT Collaboration acknowledges support from a number of agencies and institutes for both development and the operation of the LAT as well as scientific data analysis. These include NASA and DOE in the United States, CEA/Irfu and IN2P3/CNRS in France, ASI and INFN in Italy, MEXT, KEK, and JAXA in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and the National Space Board in Sweden. Additional support from INAF in Italy and CNES in France for science analysis during the operations phase is also gratefully acknowledged.
This research has made use of data from the MOJAVE database that is maintained by the MOJAVE team (Lister et al. 2009). The MOJAVE project is supported under NSF grant AST-0807860 and NASA [*Fermi*]{} grant NNX08AV67G.
RATAN-600 observations were supported in part by the Russian Foundation for Basic Research grant 08-02-00545. Y. Y. Kovalev was supported in part by the return fellowship of Alexander von Humboldt Foundation.
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[^1]: The state of the art of the researches in this field can be found in the proceedings of the workshop [*Narrow-Line Seyfert 1 Galaxies and Their Place in the Universe*]{}, Milano, Italy, 4-6 April 2011: http://pos.sissa.it/cgi-bin/reader/conf.cgi?confid=126)
[^2]: http://www.swift.psu.edu/monitoring/
[^3]: http://www.mpifr-bonn.mpg.de/div/vlbi/fgamma/fgamma.html
[^4]: http://www.physics.purdue.edu/astro/MOJAVE/index.html
|
14.5cm 0.0cm 2.0cm
[Cylindrical ideal magnetohydrodynamic equilibria\
with incompressible flows ]{}\
[G. N. Throumoulopoulos$^{\dag}$ and H. Tasso$^{\star}$\
]{}
[**Abstract**]{}
It is proved that (a) the solutions of the ideal magnetohydrodynamic equation, which describe the equlibrium states of a cylindrical plasma with purely poloidal flow and arbitrary cross sectional shape \[G. N. Throumoulopoulos and G. Pantis, Plasma Phys. and Contr. Fusion [**38**]{}, 1817 (1996)\] are also valid for incompressible equlibrium flows with the axial velocity component being a free surface quantity and (b) for the case of isothermal incompressible equilibria the magnetic surfaces have necessarily circular cross section.
[**I. Introduction**]{}
In a recent paper [@ThPa] it is proved that, if the ideal MHD stationary flows of a cylindrical plasma with arbitrary cross sectional shape are purely poloidal, they must be incompressible. This property simplifies considerably the equilibrium problem, i.e. it turns out that the equlibrium is governed by an elliptic partial differential equation for the poloidal magnetic flux function $\psi$ which is amenable to several classes of analytic solutions. For an arbitrary flow, i.e. when the velocity has non vanishing axial and poloidal components, the equilibrium becomes much more complicated. With the adoption of a specific equation of state, e. g. isentropic magnetic surfaces [@MoSo], the symmetric equilibrium states in a two dimensional geometry are governed by a partial differential equation for $\psi$, which contains five surface quantities (i.e. quantities solely dependent on $\psi$), in conjuction with a nonlinear algebraic Bernoulli equation. The derivation of analytic solutions of this set of equations is difficult.
In the present note we study the equlibrium of a cylindrical plasma with incompressible flows and show that the incompressibility condition makes it possible to construct analytic equilibria, which constitute a generalization of the ones obtained in Ref. [@ThPa]. This is the subject of Sec. II. The special class of incompressible equilibria with isothermal magnetic surfaces is examined in Sec. III. Section IV summarizes our conclusions.
[**II. Equilibrium equations and analytic solutions**]{}
The ideal MHD equilibrium states of plasma flows are governed by the following set of equations, written in standard notations and convenient units: $${\bf\nabla} \cdot (\rho {\bf v}) = 0
\label{1}$$ $$\rho ({\bf v} \cdot {\bf\nabla}) {\bf v} = {\bf j}
\times {\bf B} - {\bf\nabla} P
\label{2}$$ $${\bf\nabla} \times {\bf E} = 0
\label{3}$$ $${\bf\nabla}\times {\bf B} = {\bf j }
\label{4}$$ $${\bf\nabla} \cdot {\bf B} = 0
\label{5}$$ $${\bf E} +{\bf v} \times {\bf B} = 0.
\label{6}$$ The system under consideration is a cylindrical plasma with flow and arbitrary cross sectional shape. For this configuration convenient coordinates are $\xi$, $\eta$ and $z$ with unit basis vectors $ {\bf e}_\xi$, ${\bf e}_\eta$, ${\bf e}_z$, where ${\bf e}_z$ is parallel to the axis of symmetry and $\xi$, $\eta$ are generalized coordinates pertaining to the poloidal cross section. The equilibrium quantities do not depend on $z$. The divergence free fields, i.e. the magnetic field $\bf B$, the current density density ${\bf j}$ and the mass flow $\rho{\bf v}$, can be expressed in terms of the stream functions $\psi(\xi, \eta)$, $F(\xi, \eta)$, $B_z(\xi, \eta)$ and $v_z(\xi, \eta)$ as $${\bf B} = B_z {\bf e}_z + {\bf e}_z \times
{\bf\nabla} \psi
\label{7}$$ $${\bf j} = \nabla^2\psi {\bf e}_z - {\bf e}_z \times
\nabla B_z
\label{8}$$ and $$\rho {\bf v} =\rho v_z{\bf e}_z + {\bf e}_z \times
{\bf\nabla} F.
\label{9}$$ Constant $\psi$ surfaces are the magnetic surfaces. Eqs. (\[1\])-(\[6\]) can be reduced by means of certain integrals of the system, wich are shown to be surface quantities. To identify two of these quantities, the time independent electric field is expressed by ${\bf E} = - {\bf \nabla} \Phi$ and the Ohm’s law (\[6\]) is projected along ${\bf e}_z$ and $\bf B$, respectively, yielding $${\bf e}_z \cdot\left({\bf e}_z\times \nabla F\right)\times
\left({\bf e}_z\times \nabla \psi\right) = 0
\label{10}$$ and $${\bf B}\cdot\nabla \Phi=0.
\label{11}$$ Eqs. (\[10\]) and (\[11\]) imply that $F=F(\psi)$ and $\Phi=\Phi(\psi)$. Two additional surface quantities are found from the component of Eq. (\[6\]) perpendicular to a magnetic surface: $$\frac{B_z F^\prime}{\rho} - v_z = \Phi^\prime,
\label{12}$$ and from the component of the momentum conservation equation (\[2\]) along ${\bf e}_z$: $$B_z-F^\prime v_z\equiv X(\psi).
\label{13}$$ (The prime denotes differentiation with respect to $\psi$). Solving the set of Eqs. (\[12\]) and (\[13\]) for $B_z$ and $v_z$, one obtains $$B_z = \frac{X(\psi)\rho -F^\prime(\psi)\Phi^\prime(\psi)}
{\rho - (F^\prime(\psi))^2}
\label{12a}$$ and $$v_z = \frac{F^\prime(\psi) X(\psi) -\Phi^\prime(\psi)}
{\rho - (F^\prime(\psi))^2}.
\label{13a}$$ With the aid of Eqs. (\[10\])-(\[13\]), the components of Eq. (\[2\]) along $\bf B$ and perpendicular to a magnetic surface, respectively, are put in the form $${\bf B} \cdot \left[{\bf \nabla} \left(\frac{v^2}{2}
+ v_z\Phi^\prime\right)
+ \frac{\nabla P}{\rho}\right] = 0
\label{14}$$ and $$\begin{aligned}
{\bf \nabla} \cdot
\left[\left(1- \frac{(F^\prime)^2}{\rho}\right)
{\bf \nabla}\psi\right]
+ \frac{F^{\prime\prime}F^\prime |\nabla\psi|^2}{\rho}
+ \frac{B_z\nabla B_z\cdot\nabla \psi}
{|\nabla \psi|^2} & & \nonumber \\
+ \rho \frac{{\bf \nabla} \psi}{|{\bf \nabla} \psi|^2 }\cdot
\left[{\bf \nabla} \left(\frac{(F^{\prime})^2 |\nabla\psi|^2}{2\rho^2}\right)
+\frac{{\bf \nabla} P}{\rho}\right]=0.
\label{15} \end{aligned}$$ It is pointed out here that Eqs. (\[14\]) and (\[15\]) are valid for any equation of state for the plasma.
In order to reduce further the equilibrium equations, we employ the incompressibiliy condition $$\nabla\cdot {\bf v} = 0.
\label{6a}$$ Then Eq. (1) implies that the density is a surface quantity, $$\rho=\rho(\psi),
\label{16}$$ and, consequently, Eqs. (\[12a\]) and (\[13a\]) yield $$B_z=B_z(\psi), \ \ \ v_z=v_z(\psi).
\label{17}$$ With the use of Eqs. (\[16\]) and (\[17\]), Eq. (\[14\]) can be integrated yielding an expression for the pressure, i.e. $$P = P_s
(\psi) - \frac{F'^2}{2\rho} |{\bf \nabla} \psi|^2.
\label{19}$$ We note here that, unlike in static equilibria, in the presence of flow magnetic surfaces do not coincide with isobaric surfaces because Eq. (\[2\]) implies that ${\bf B} \cdot {\bf \nabla} P$ in general differs from zero. In this respect, the term $P_s(\psi)$ is the static part of the pressure which does not vanish when $F^\prime$ is set to zero; Eqs. (\[12a\]), (\[13a\]) and (\[15\]) have a singularity when $$\frac{\left(F^\prime \right)^2}{\rho}=1.
\label{20}$$ On the basis of Eq. (\[9\]) for $\rho {\bf v}$ and the definitions $v_{Ap}^2\equiv\frac{\textstyle |\nabla \psi|^2}
{\textstyle \rho}$ for the Alfvén velocity associated with the poloidal magnetic field and the Mach number $M^2\equiv\frac{\textstyle v^2}
{\textstyle v_{Ap}^2}$, Eq. (\[20\]) can be written as $
M^2= 1.
$
Assuming now $\frac{\textstyle (F^\prime)^2}{\textstyle \rho}\neq 1$, and inserting Eq. (\[19\]) into Eq. (\[15\]), the latter reduces to the [*elliptic*]{} differential equation $$\left[1 - \frac{(F^\prime)^2}{\rho}\right] {\bf \nabla}^2 \psi +
\frac{F^\prime}{\rho} \left(\frac{F^\prime}{2}
\frac{\rho^\prime}{\rho}
-F^{\prime\prime}\right) |{\bf \nabla} \psi|^2
+ \left(P_s +\frac{ B_z^2}{2}\right)^\prime = 0.
\label{21}$$ The absence of any hyperbolic regime in Eq. (\[21\]) can be understood by noting that, as is well known from the gas dynamics, the flow must be compressible to allow the equilibrium differential equation to depart from ellipticity. Eq. (\[21\]) [*does not contain the axial velocity $v_z$*]{} and is identical to the equation governing cylindrical equilibria with purely poloidal flow [@ThPa]. With the use of the ansatz $
\frac{\textstyle\rho^\prime}{\textstyle\rho} =
2 \frac{\textstyle F^{\prime\prime}}{\textstyle F^\prime}
$, which implies that $\frac{\textstyle (F^\prime)^2}{\textstyle \rho}\equiv M_c^2 =
\mbox{const.}$, Eq. (\[21\]) reduces to $${\bf \nabla}^2 \psi +
\frac{1}{1-M_c^2} \left( P_s + \frac{B_z^2}{2}\right)^\prime = 0.
\label{22}$$ This is similar in form to the equation governing static equilibria; the only explicit reminiscence of flow is the presence of $M_c$. Eq. (\[22\]) can be linearized for several choices of $P_s + \frac{\textstyle B_z^2}{\textstyle 2}$ and a variety of analytic solutions of the linearized equation can be derived. In particular, the exact solutions for a circular cylindrical plasma obtained in Ref. [@ThPa] are also valid for incompressible equilibrium flows with a free axial velocity $v_z(\psi)$.
The singularity $M_c^2 = 1$ is the limit at which the confinement can be assured by the axial current $\nabla^2\psi$ alone. For $M^2_c>1$ the derivative of $B_z^2/2$ must partly compensate for the pressure gradient.
[**III. Equilibria with isothermal magnetic surfaces**]{}
For fusion plasmas the thermal conduction along $\bf B$ is fast compared to the heat transport perpendicular to a magnetic surface and therefore equilibria with isothermal magnetic surfaces are of particular interest. The plasma is also assumed to obey the ideal gas law $P = R \rho T$. For this kind of equilibria, Eq. (\[19\]) implies that $|\nabla \psi|$ is a surface quantity and consequently from Eq. (\[15\]) it turns out that $\nabla^2\psi$ is a surface quantity as well. Thus, the incompressible, $T=T(\psi)$ equilibria satisfy the set of equations $$|\nabla \psi|^2 =( g(\psi))^2
\label{23}$$ and $$\nabla^2 \psi = f(\psi).
\label{24}$$ Eqs. (\[23\]) and (\[24\]) imply that, on a magnetic surface the modulus of the vector $\nabla \psi$, which is perpendicular to this (arbitrary) magnetic surface, and $\nabla^2\psi$, related to the variation of $|\nabla \psi|$, are constants. Therefore, one could speculate that magnetic surfaces are restricted to be circular. This conjecture can be proved as follows.
The coordinates $\xi$, $\eta$ and $z$ are specified to be the Cartesian coordinates $x$, $y$, $z$. With the introduction of the quantities $p=\partial \psi/\partial x$, $q=\partial \psi/\partial y$, $r=\partial^2 \psi/\partial x^2$ and $t=\partial^2 \psi/\partial y^2$ , Eqs. (\[23\]) and (\[24\]) are written in the form $$p^2 + q^2 = g^2
\label{25}$$ and $$r + t = f.
\label{26}$$ The set of Eqs. (\[25\]) and (\[26\]) can be integrated by applying a procedure suggested by Palumbo [@Pa]. Accordingly, considering the functions $p$ and $q$ which are functions of $x$ and $y$ as functions of $x$ and $\psi(x,y)$ one has $$r=\left.\frac{\partial p}{\partial x}\right|_y=
\left.\frac{\partial p}{\partial x}
+ p \frac{\partial p}{\partial\psi}\right|_y
\label{27}$$ and $$t=q\frac{\partial q}{\partial \psi}.
\label{28}$$ (It is noted here that a surface function $\zeta=\zeta(x,y)\equiv\zeta(\psi)$ can be employed instead of $\psi$). With the aid of Eqs. (\[25\]), (\[27\]) and (\[28\]), Eq. (\[26\]) reduces to $
\left.\frac{\textstyle\partial p}{\textstyle\partial x}\right|_\psi
=f-gg^\prime
$ and consequently $$p = x\left(f-gg^\prime\right) + h(\psi).
\label{30}$$ On a magnetic surface it holds that $
d\psi=\frac{\textstyle\partial \psi}{\textstyle \partial x}dx
+ \frac{\textstyle\partial \psi}{\textstyle\partial y}dy\equiv 0,
$ and therefore $$\left(\left.\frac{dy}{dx}\right|_\psi\right)^2
= \frac{p^2}{q^2}
= \frac{\left[x\left(f-gg^\prime\right) + h\right]^2}
{g^2-\left[x\left(f-gg^\prime\right) + h\right]^2}.
\label{31}$$ Introducing the new quantities $a(\psi)\equiv f-gg^\prime$, $X\equiv ax+h$ and $Y\equiv ay$, Eq. (\[31\]) is put in the form $$\left(\frac{dY}{dX}\right)^2 = \frac{X^2}{g^2-X^2}.
\label{32}$$ Eq. (\[32\]) describes a circle on the $(x,y)$ plane with radius $| g|$ centred at $(-h/a, 0)$.
[**IV. Conclusions**]{}
It was proved that the ideal MHD equilibrium states of a cylindrical plasma with incompressible flows and arbitrary cross section shape satisfy an elliptic partial differential equation \[Eq. (\[21\])\], which is identical to the equation governing cylindrical equilibria with purely poloidal flow; the axial flow velocity is a free surface quantity. This equation permits the construction of several classes of analytic solutions. In particular, the exact equlibrium solutions for a circular cylindrical plasma and purely poloidal flow [@ThPa] are also valid for the present case. In addition, it was proved that the magnetic surfaces of isothermal incompressible equilibria must have circular cross section.
It is intersting to investigate symmetric incompressible equlibria in geometries representing more realistically the magnetic confinement systems, e.g. axisymmetric and straigth helically symmetric configurations. In this respect it may be noted here that, as proved in Ref. [@Ta], the special class of axially symmetric, incompressible, $ \beta_p=1$, MHD equilibria with purely poloidal velocity does not exist; the only possible stationary equilibria of this kind are of cylindrical shape.
[**Acknowledgments**]{}
This work was conducted during a visit by one of the authors (G.N.T.) to Max-Planck Institute für Plasmaphysik, Garching. The hospitality provided at the said institute is appreciated. G.N.T. acknowledges support by EURATOM (Mobility Contract No 131-83-7 FUSC). One of the authors (H.T.) would like to thank Prof. D. Pfirsch for a useful discussion
[99]{} G. N. Throumoulopoulos and G. Pantis, Plasma Phys. and Contr. Fusion [**38**]{}, 1817 (1996). A. I. Morozov and L. S. Solov’ev, Reviews of Plasma Physics [**8**]{}, 1 (1980), ed. M. A. Leontovich (New York: Consultants Bureau). D. Palumbo, Nuovo Cimento B [**53**]{}, 507 (1968). H. Tasso, Phys. Fluids [**13**]{}, 1874 (1970).
|
---
author:
- 'F. Wyrowski, S. Heyminck, R. Güsten, K.M. Menten'
date: 'Received / Accepted'
title: 'Mid- and high-$J$ CO observations towards ultracompact HII regions'
---
[A study of 12 ultracompact HII regions was conducted to probe the physical conditions and kinematics in the inner envelopes of the molecular clumps harboring them.]{} [The APEX telescope was used to observe the sources in the CO (4–3) and (8–7) lines. Line intensities were modeled with the RATRAN radiative transfer code using power laws for the density and temperature to describe the physical structure of the clumps.]{} [All sources were detected in both lines. The optically thick CO (4–3) line shows predominantly blue skewed profiles reminiscent of infall.]{} [Line intensities can be reproduced well using the physical structure of the clumps taken from the literature. The optically thick line profiles show that CO is a sensitive tracer of ongoing infall in the outer envelopes of clumps harboring ultracompact HII regions and hot molecular cores.]{}
\[intro\]Introduction
=====================
The earliest phases of massive star formation are still poorly understood. We know that massive stars are being born in dense clumps within giant molecular cloud complexes. Ultracompact HII regions (UCHIIRs) embedded within these clumps represent a key phase in the early lives of massive stars (see review by Hoare et al. 2005). UCHIIRs were defined by Wood & Churchwell (1989) to have sizes $\le 0.1$ pc, densities $\ge 10^4$ , and emission measures $\ge 10^7$ pc $\rm
cm^{-6}$. In their environs, often hot ($T>100$ K), compact ($<0.1$ pc), and dense ($n({\rm H_2})>10^7$ ) cores are found, some of which are believed to be in a stage prior to the formation of UCHIIRs (Kurtz et al. 2000).
A better understanding of these clumps is a crucial prerequisite for models of high-mass star formation. Several studies have attacked this topic in the past. In particular, Hofner et al. (2000) conducted a survey of 16 UCHIIRs and found typical sizes of 1 pc for the clumps and average densities and temperatures of $10^5$ and 25 K. These values, obtained under simple assumptions, are an important first approximation but the high luminosities and densities of the embedded hot cores show that a proper representation of the physical conditions of the clumps requires density and temperature gradients. Hatchell & van der Tak (2003) present models with power laws for the density and temperatures that were constrained by the spectral energy distributions (SEDs) of the sources and single-dish continuum images and CS line data. One drawback of using CS is the abundance variation of this molecule with changing physical environments. Toward the sources studied by Hatchell & van der Tak (2003), the abundance varies by almost three orders of magnitudes which suggests that within an individual envelope the abundance might also vary considerably. Therefore, in the inner regions of the clumps, mid- and high-$J$ transitions of CO with a stable abundance in warm regions are better probes of the physical conditions.
One of the important results in the study of low-mass star formation has been the observation of infall motions (e.g. Belloche et al. 2002), which give direct evidence of accretion. Toward high-mass star-forming cores, the observational evidence of infall is still very scarce (e.g. Wu & Evans 2004). In addition to probing density and temperature, mid- and high-$J$ lines of CO allow the kinematics of the cores to be probed, hence to search for further evidence of infall.
Here, we present a study of 12 UCHIIRs, mostly with associated hot cores, in the CO (4–3) and (8–7) lines to probe the physical conditions and kinematics of the inner envelopes of the clumps harboring the UCHIIRs.
=
\[obs\]Observations
===================
Source R.A. (J2000) Dec. (J2000)
------------- -------------- --------------
G5.89–0.39 18:00:30.376 -24:04:00.48
G9.62+0.19 18:06:15.000 -20:31:42.10
G10.47+0.03 18:08:38.218 -19:51:49.71
G10.62–0.38 18:10:28.661 -19:55:49.77
G12.21–0.10 18:12:39.700 -18:24:20.00
G29.96–0.02 18:46:03.950 -02:39:21.40
G31.41+0.31 18:47:34.401 -01:12:45.95
G34.26+0.15 18:53:18.499 01:14:58.66
G35.20–1.74 19:01:46.440 01:13:23.50
W51D 19:23:39.946 14:31:08.13
W51E1E2 19:23:43.762 14:30:26.40
G45.12+0.13 19:13:27.808 10:53:36.72
: Source list
\[tab:sources\]
The observations were made with the Atacama Pathfinder Experiment (APEX[^1]) in June 2005. The frontend used was the MPIfR dual channel (460 and 810 GHz) FLASH receiver (Heyminck et al. 2006, this volume), which enables simultaneous observations of the (4–3) and (8–7) lines. As backends, the MPIfR Fast Fourier Transform Spectrometers (FFTS, Klein et al. 2006, this volume) were used for the line observations. The lines were covered with 2048 channels within a bandwidth of 1 GHz, resulting in spectral resolutions of 0.32 and 0.17 . The spectra were converted from $T_{\rm A}^*$ to $T_{\rm MB}$ units using forward efficiencies of 0.95 and beam efficiencies of 0.6 and 0.43 at 461 and 881 GHz, respectively. The CO (4–3) line was observed for several sources on two different days and the observed line temperatures agree within 20%. The beam sizes at the observing frequencies are 13 and 7. All observations were done in position switching mode on the UCHIIR positions given in Table \[tab:sources\] using off-positions 250 to the east. These sources were selected from the sample studied by Hofner et al. (2000) with the addition of the two sources in W51. Pointing was done regularly on strong submm continuum sources in the sample (G10.47 and G34.26) and should be accurate within 2.
\[results\]Results
==================
=
Figure \[fig:13co\] shows the observed (8–7) spectra compared with (2–1) spectra from Hofner et al. (2000) and line parameters from Gaussian fits to the spectra are given in Table \[tab:13co\]. All sources are clearly detected and show line shapes similar to . Notable deviation from Gaussian shapes are seen in G10.62, with a strong red wing, and in G5.89–0.39, where the high velocity outflow wings are prominent. The linewidths are in all cases typically larger by 2 . While line broadening can be caused by high optical depths, the absence of complicated self-absorbed spectral shapes suggests rather moderate optical depths, hence a true increase in motions. For most sources, the ratio of the (8–7) and (2–1) line temperatures is the same within a factor 2, with G35.20–1.7 and W51D being the exceptions with much stronger lines.
--------- -------------- ------------------ ------------------ --
Source $T_{\rm MB}$ $v_{\rm LSR}$ FWHM
(K) (km$\,$s$^{-1}$) (km$\,$s$^{-1}$)
G5.89 20.8 ( 2.1 ) 12.0 ( 0.4 ) 20.9 (1.2 )
G9.62 16.3 ( 1.0 ) 6.2 ( 0.1 ) 7.0 (0.3 )
G10.47 11.5 ( 1.9 ) 67.7 ( 0.5 ) 12.4 (1.1 )
G10.62 27.7 ( 1.3 ) -1.1 ( 0.1 ) 10.8 (0.3 )
G12.21 6.9 ( 1.5 ) 23.9 ( 0.5 ) 9.7 (1.2 )
G29.96 26.9 ( 1.0 ) 99.9 ( 0.1 ) 5.6 (0.2 )
G31.41 14.0 ( 1.3 ) 98.3 ( 0.2 ) 6.6 (0.5 )
G34.26 23.1 ( 2.1 ) 58.8 ( 0.2 ) 7.2 (0.6 )
G35.20 27.3 ( 1.3 ) 44.6 ( 0.1 ) 6.0 (0.2 )
W51D 43.9 ( 1.7 ) 61.6 ( 0.1 ) 8.7 (0.2 )
W51E1E2 21.5 ( 1.3 ) 59.4 ( 0.2 ) 10.2 (0.4 )
G45.12 17.3 ( 1.3 ) 60.2 ( 0.1 ) 4.1 (0.3 )
--------- -------------- ------------------ ------------------ --
: APEX (8–7) line parameters
\[tab:13co\]
The observed (4–3) spectra are shown in Fig. \[fig:12co\] compared with CS (7–6) spectra from Olmi et al. (1999). All sources show complicated line profiles with self-absorption and outflow wings. The self-absorption is in most cases redshifted compared to the velocity of the optically thin lines, with only G9.62 and G35.20 showing a blueshifted self-absorption. The line shapes are either double peaked with a stronger blue peak or skewed to the blue. G9.62 is again an exception and G45.12 shows a rather symmetric profile. G5.89 is dominated by the strong outflow and absorbing foreground clouds (Klaassen et al.2006). Also in the red line wing of G9.62, absorption due to foreground clouds is seen, consistent with the HCO$^+$ observations by Hofner et al. (2001). The asymmetries in the line shapes of the CS (7–6) line are mostly consistent with those we find in (4–3) but the self absorption is weaker. To quantify the asymmetries seen in the profiles to check for infall signatures (Table \[tab:12co\]), we determined the ratio of blue-to-red peak intensity for the 8 sources with 2 peaks. Seven sources show significantly stronger blue peaks, and the average ratio of the sample is 3.3, thus indicative of infall. For the remaining 3 sources with only one peak (G5.89 was omitted due to its strong outflow), we determined $ \delta v = (v_{\rm thick}-v_{\rm thin})/\Delta v_{\rm thin}$, the difference between optically thick () and thin () line peaks over the optically thin line widths, which can be used as an infall indicator (Mardones et al. 1997). For 2 sources $\delta v$ is $-0.5$, hence clearly blue-shifted, and for 1 it is close to 0.
Source $T_{\rm blue}/T_{\rm red}$ $\delta v$ Profile
--------- ---------------------------- ------------ --------- --
G9.62 0.44 R
G10.47 -0.50 B
G10.62 1.38 B
G12.21 -0.51 B
G29.96 4.21 B
G31.41 1.40 B
G34.26 2.03 B
G35.20 1.42 B
W51D 4.67 B
W51E1E2 7.87 B
G45.12 0.04
: APEX (4–3) collapse indicators. For two-peaked profiles the ratio of blue and red peak and for the single-peaked lines the skewness parameter $ \delta v$ are given. The profile column denotes blue (B) and red (R) line profiles.
\[tab:12co\]
\[discussion\]Discussion
========================
The CO emission of the sources has been modeled with the Monte Carlo radiative transfer program of Hogerheijde & van der Tak (2000). Besides collisional excitation, dust radiation is taken into account using grain properties from Ossenkopf & Henning (1994), Model 5. The radial profile of the density, as well as inner and outer radii, were taken from the best-fit DUSTY models of Hatchell & van der Tak (2003, RATRAN code) based on single-dish maps of dust continuum and CS molecular line emission. They determined power laws for the density structure with power law exponents between -1.5 and -2.0 for our sample. RATRAN and DUSTY were used with the same dust properties. The temperature structure was then solved by DUSTY (Ivezi[' c]{} et al. 1997) in a self-consistent way. An isotopic ratio of 40 and 2000 was used for and , respectively, relative to . To simplify the modeling, no variable velocity field was used but only constant, turbulent line widths. This is a reasonable approximation of the integrated intensities of lines that are optically thin or have only moderate optical depths; but for optically thick lines like the (4–3) line, the resulting integrated intensities will depend crucially on the assumed velocity fields. Therefore, in Fig. \[fig:compare\] only the integrated (8–7) and (2–1) line intensities from the observations and the modeling are compared. For this comparison, the modeled intensities were determined from the RATRAN output images by convolving with the APEX and 30 m observing beams. The agreement between the modeled intensities using the Hatchell & van der Tak fit results and the observations is remarkable. The only large deviations are the predictions of the intensities for G10.47 and G31.41 for which the outer radius of the clumps might be overestimated, and this mostly affects the (2–1) lines and not (8–7). Since the modeled lines probe very different excitation regimes and are completely independent of the data used by Hatchell & van der Tak, this agreement is a strong validation of the power law structure in $T$ and $n$ of the clumps.
The observed profiles are very reminiscent of infall asymmetries routinely detected toward low-mass star-forming cores (e.g. Myers et al. 2000). For individual sources, there is still the possibility that some of the absorption might be caused by colder foreground clouds, but this would not explain the general trend of redshifted absorption and most of the blue peaks being stronger. Also, in several cases the observed infall signature is consistent with results from interferometric observations where the infall is observed mostly in absorption using different molecules. Examples are G10.62 (NH$_3$, Keto et al. 1988), W51E (HCO$^+$, Rudolph et al.1990) and G29.96 (HCO$^+$, Maxia et al. 2001). We can infer a total of 9 out of 11 sources with infall evidence from the line profiles discussed in Sect. \[results\]. The excess parameter $E=(N_{\rm blue}-N_{\rm red})/N_{\rm tot}$, introduced by Mardones et al. (1997) to quantify the statistics of infall surveys, is for our sample 0.7, which is larger than for low-mass samples or the massive star-forming region survey by Wu & Evans (2003). This might be a selection effect, since mostly UCHIIRs with associated hot cores, hence early stages of massive star formation, were targeted. It might also be related to using CO as an infall tracer. The excess parameter using CS for our sample would have been smaller. The high optical depths of the CO (4–3) line make it a very sensitive probe of infalling motions in the outer envelopes of the massive star-forming clumps. Therefore, while observations of much higher-$J$ optically thick CO lines are needed to probe the kinematics in the inner part of the clumps, the CO (4–3) observations clearly show that in the outer parts (a remnant) infall is still going on.
In the near future, the upcoming CHAMP+ array – a 2x7 pixel heterodyne array for simultaneous observations in the 350 and 450 $\mu$m atmospheric windows to be installed at the APEX telescope in the middle of 2006 – will allow us to image high-$J$ CO lines and then, together with proper models of the density and temperature structure, put further and tighter constraints on the velocity structure of the massive star-forming clumps.
We thank Riccardo Cesaroni for providing the CS (7–6) spectra in CLASS format.
Belloche, A., Andr[é]{}, P., Despois, D., & Blinder, S. 2002, , 393, 927
Hatchell, J., & van der Tak, F. F. S. 2003, , 409, 589
Güsten et al., this volume
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Hoare, M. G., Kurtz, S. E., Lizano, S., Keto, E., Hofner, P. 2006, to appear in Protostars and Planets V, Edited by B. Reipurth, D. Jewitt, and K. Keil, University of Arizona Press, Tucson
Hofner, P., Wyrowski, F., Walmsley, C. M., & Churchwell, E. 2000, , 536, 393
Hofner, P., Wiesemeyer, H., & Henning, T. 2001, , 549, 425
Hogerheijde, M. R., & van der Tak, F. F. S. 2000, , 362, 697
Ivezi[' c]{}, Z., Nenkova, M. & Elitzur, M., 1999, User Manual for Dusty, University of Kentucky Internal Report, accessible at [http://www.pa.uky.edu/$\sim$moshe/dusty]{}
Keto, E. R., Ho, P. T. P., & Haschick, A. D. 1988, , 324, 920
Klaassen, P. D., Plume, R., Ouyed, R., von Benda-Beckmann, A. M., Di Francesco, J. 2006, ApJ in press
Klein et al., this volume
Kurtz, S., Cesaroni, R., Churchwell, E., Hofner, P., & Walmsley, C. M. 2000, Protostars and Planets IV, 299
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Maxia, C., Testi, L., Cesaroni, R., & Walmsley, C. M. 2001, , 371, 287
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[^1]: This publication is based on data acquired with the Atacama Pathfinder Experiment (APEX). APEX is a collaboration between the Max-Planck-Institut für Radioastronomie, the European Southern Observatory, and the Onsala Space Observatory.
|
[**Variation Principle for Calculation of Many-Particle Effects in Crystals** ]{}\
[*Halina V. Grushevskaya$^{1}$ and Leonid I. Gurskii$^{2}$* ]{}\
[*$^{1}$ Physics Department, Belarusian State University,\
4 Fr. Skorina Av., 220050 Minsk, BELARUS\
$^{2}$ Belarusian State University of Informatics and Radio electronics,\
6 P. Brovka Str., 220027 Minsk, BELARUS*]{}\
[E-mail: [email protected]]{}
> ****
>
> Abstract
>
> Variation principle has been developed to calculate many-particle effects in crystals. Within the framework of quasi-particle concept the variation principle has been used to find one-electron states with taking into account of effects due to non-locality of electronic density functional in electromagnetic fields. A secondary quantized density matrix was used to find the Green function of a quasiparticle and changes of its effective mass due to correlated motion of interacting electrons.
Introduction
============
Because of high dimensionality the complexity of many-electron problems all problems of a solid state physics of this type are much higher than of one-electron problem. Results of a solution of the one-electron problem can be utilized in many-electron problems in solid state physics if one supposes that motion of an electron happens in a self-consistent one-particle potential $V
(\vec r) $. The self-consistent potential is yielded by the solution of the Poisson equation, for which the density of one-electron states is a self-consistent solution of the one-particle Hartree - Fock equations [@Hartry] - [@Froese]. These equations have been written originally to calculate one-particle states of many-electron atom. Equations of the Hartree - Fock type for band calculations have been proposed by Kohn and Sham [@Kon_SHam]. It is practically impossible to look for their solution without additional assumptions if many-particle effects are circumscribed by distributed in space electron density functional. One of them is that one-particle solutions describe a crystal as a set of interacting quasiparticle excitations [@Ary]. In a given paper we offer an approach which allows to determine one-electron states with taking into account of the effects owing to a non-locality of electron density functional in electromagnetic fields and which is based on usage of the variation principle.
A variational principle for band calculations
=============================================
Let us write a one-electron Hartree - Fock Hamiltonian for a system of $N$ electrons, $N\to \infty $, moving in a field originated by atoms nucleus’ of a crystal $$\begin{aligned}
\left[{1\over 2} \hat p^2_i + U(r_i)
+ \hat V^{sc} (\vec r_i, \sigma_i) - \hat \Sigma ^x(\vec r_i, \sigma_i)
\right] \psi_n (k_i r_i)=
\nonumber \\
=(\epsilon_n(0)+ \epsilon_n(k_i)) \psi_n (k_i r_i)
\label{hartry-fock-eqs}\end{aligned}$$ with a unit choice $\hbar =1, \ m=1$, where $\hat p^2_i /2 = -{1\over 2} \triangle (\vec r_i)$ is a kinetic energy in system of atomic units, $\triangle (\vec r_i)$ is the Laplacian operator written in a given point with a radius-vector $\vec r_i$ in which $i$-th electron having a spin $\sigma_i$ is situated; $U(r_i)$ is potential energy of $i$-th electron in nucleus field of the crystal, $\hat V^{sc}$ and $ \hat \Sigma ^x $ are operators of Coulomb and exchange interactions, respectively [@Vesel-Labzov]: $$\begin{aligned}
\hat V^{sc}(\vec r_i, \sigma_i)\psi_n (k_i r_i)=\sum_{m=1}^N
\int \psi_m^* (k_i r'_i) v(|\vec r_i - \vec r'_i|)\psi_m (k_i r'_i)
\ d r'_i \psi_n (k_i r_i)
,\nonumber \\
\label{coulon}\\
\hat \Sigma ^x(\vec r_i, \sigma_i) \psi_n (k_i r_i)
=\sum_{m=1}^N
\int \psi_m^* (k_i r'_i) v(|\vec r_i - \vec r'_i|)\psi_n (k_i r'_i)
\ d r'_i \psi_m (k_i r_i); \nonumber \\
\label{exchange}\end{aligned}$$ $r_i\equiv \{\vec{r}_i, \sigma_i\}$, $\psi_n (k_i r_i)$ is a wave function including spin and coordinate parts, $v(|\vec r_i - \vec r'_i|)$ is potential energy of electron interaction. The physical sense of the operators (\[coulon\], \[exchange\]) becomes obvious if one rewrites them in terms of spin-zero electronic density $ \rho (\vec r, \vec r ') $ and assumes that the interaction $v $ is the Coulomb one: $$\begin{aligned}
\rho(\vec r, \vec r')={1\over 2}\sum_{m=1}^{N-1}\left(
\psi_m^* (\vec k \cdot \vec r ,\sigma )
\psi_m (\vec k \cdot \vec r' , -\sigma )+
\psi_m^* (\vec k \cdot \vec r ,-\sigma)
\psi_m (\vec k \cdot \vec r', \sigma )
\right)
\nonumber\\
=\sum_{m=1}^{(N-1)/2}
\psi_m^* (\vec k \cdot \vec r)
\psi_m (\vec k \cdot \vec r'),\quad v =e^2/|\vec r - \vec r'|.\end{aligned}$$ It follows from here that the operator $ \hat V ^ {sc} $ gives electrostatic interaction of one electron with an electron density created by residual $N-1 $ electrons, with an electrostatic self-action (s.a.): $$\begin{aligned}
\hat V^{sc}(\vec r_i, \sigma_i)\psi_n (k_i r_i)=
\sum_\sigma \sum_{m=1}^{(N-1)/2}
\int \psi_m^* (\vec k_i \cdot \vec r'_i) v(|\vec r_i - \vec r'_i|)
\psi_m (\vec k_i \cdot \vec r'_i)
\ d \vec r'_i \psi_n (\vec k_i \cdot \vec r_i)
+\mbox{s.a.}
\nonumber \\
=2 \int \ d \vec r'_i
{e^2 \rho(\vec r'_i, \vec r'_i)\over |\vec r_i - \vec r'_i|}
\psi_n (k_i r_i)
+\mbox{s.a.}\end{aligned}$$ Analogously we get that the operator $ \hat \Sigma ^x $ gives a quantum exchange with an exchange self-action (s.a.): $$\begin{aligned}
\hat \Sigma ^x(\vec r_i, \sigma_i) \psi_n (k_i r_i)
=\sum_{m=1}^{N-1}
\int \int d r_j d \sigma_j \psi_m^* (\vec k_i \cdot \vec r_j, \sigma_j)
v(|\vec r_i - \vec r_j|)\psi_n (k_i\cdot \vec r_j, \sigma_j)
\psi_m (\vec k_i \cdot \vec r_i, \sigma_i) \nonumber \\
={1\over 2}\sum_{m=1}^{N-1}\int \int d r_j d \sigma_j
\left(
\psi_m^* (\vec k \cdot \vec r_j ,\sigma_j )
\psi_m (\vec k \cdot \vec r_i , -\sigma _j) \delta(\sigma_j - \sigma_i )\right.\nonumber \\
%\times
\left. +
\psi_m^* (\vec k \cdot \vec r_i ,-\sigma_j)
\psi_m (\vec k \cdot \vec r_j, \sigma_j)\delta(\sigma_j - \sigma_i )
\right)
v(|\vec r_i - \vec r_j|)\psi_n (k_i\cdot \vec r_j, \sigma_j)+\mbox{s.a.}
\nonumber \\
=\int \ d \vec r_j
{e^2 \rho(\vec r _j, \vec r _i)\over |\vec r_i - \vec r _j|}
\psi_n (k_i r_j)
+\mbox{s.a.}
\label{exchange1}\end{aligned}$$ Since operators $ \hat V ^ {sc} $ and $ \hat \Sigma ^x $ in expression (\[hartry-fock-eqs\]) are subtracted one from another the self-acting terms vanish.
A quantity $ \epsilon_n (k_i) $ entering in expansion $E_n (k_i)
=f (\epsilon_n (0)) + \epsilon_n (k_i) $ being $n $-th eigenvalue $E_n (k_i) $ of the Hamiltonian for $N $ particles system in Hartree -Fock approximation is the energy of $n $-th band. A quantity $ \epsilon_n (0) $ determines a reference point $ \mbox
{Extr} E_n (k_i) =f (\epsilon_n (0)) $ of $n $-th bands, where $f
$ is an unknown function. Within the framework of quasiparticle concept the physical sense of $ \epsilon_n (k_i) $ is the energy of quasiparticle excitation.
Let us take a solution of one-electron problem for an atomic area in a cell potentials approximation as $ \epsilon_n (0) $ and a basic set on which one constructs an expansion of a trial function $ \widetilde \psi_n (k_i r_i) $ of $n $-th band. Then one can realize a variational principle as the following expression: $$\begin{aligned}
\delta \epsilon_n[\widetilde \psi_n] =0 ,
\label{exitation-variation-princip}\end{aligned}$$ which is determined not strictly as it is used to find excited states for which, as a rule, the basic set of functions is unknown [@Vesel-Labzov]. Let us show, that the variational principle for excited states in the form (\[exitation-variation-princip\]) can be made strictly determined one. Since, by definition, the energy of a nonexcited state coincides with an extremum of a functional $E_n [\widetilde \psi_n] $ we have the following equality of variations: $$\begin{aligned}
\delta \epsilon_n[\widetilde \psi_n]= \delta E_n[\widetilde \psi_n] =0 .
\label{variation-princip}\end{aligned}$$ The ambiguity of a variation (\[variation-princip\]) consists only in an arbitrariness of a position of a reference point for an energy band as the variation procedure for an one-electron problem is strictly determined. If a symmetry of atomic areas is definite and a cell partition of the crystalline space is unambiguous then, in principle, this arbitrariness is removed easy by utilizing the quasi-particles concept, according to which $$\begin{aligned}
\epsilon_n(k_i) = \epsilon_n(-k_i).
\label{variation-princip1}\end{aligned}$$ It follows from the expressions (\[variation-princip\]) and (\[variation-princip1\]), that the additional variation of a reference point of a band gives a coincidence of this point with a centre of the energy band for a crystal and allows to transform the variational principle (\[variation-princip\]) finding noninteracting quasiparticles states to strictly determined expression $$\begin{aligned}
\delta E_n[\widetilde \psi_n] = 0,
\quad \delta f[\widetilde \psi_n]=0.
\label{variation-princip2}\end{aligned}$$ Let us find the reference point $ \mbox {Extr} E_n (k_i) $ of energy band using a method of density matrix functional.
Secondary quantized reduced density matrix
===========================================
As is known [@faddeev], equations in quantum mechanics can be written not for a wave function, but for a density matrix $ \rho
$. The operator $ \rho $ is a projective operator for pure states and can be presented in terms of Dirac ket(bra)-vectors as $
\rho = | \psi \rangle \langle \psi | $. If the operator $ \rho $ is known then we can find, by definition, energy $E $ of system of $N $ particles, described by a Hamiltonian $ \hat H (\vec
r_1, \ldots, \vec r_N) $ and a wave function $ | \psi \rangle $ as $$\begin{aligned}
E= \mbox{Sp }\rho \hat H . \label{mean-energy}\end{aligned}$$
Let us introduce $n$-dimensional density matrix $\rho_n$ by $$\begin{aligned}
\rho _n(\vec r_1',\ldots, \vec r_n'; \vec r_1,\ldots, \vec r_n) =
{N!\over (N-n)!}%\nonumber \\ \times
\int \ d\vec r_{n+1}\ldots \ d\vec r_N
\nonumber \\
\times \langle
\vec r_1',\ldots, \vec r_n', \vec r_{n+1},\ldots,\vec r_N|\psi \rangle
\langle \psi |\vec r_1,\ldots, \vec r_n , \vec r_{n+1},\ldots, \vec r_N\rangle
,
\label{n-particle-function}\end{aligned}$$ which, by implication, is a reduced coordinate distribution $n$-particle function [@Lovdin]. By definition, it has a normalization $$\begin{aligned}
\mbox{Sp} \rho_n ={ N!\over (N-n)!} \quad n=1,\ldots ,N .\end{aligned}$$
Let us examine the system consisted of $N$ with a pairwise interaction $$\sum _{i>j=1}^N v(|\vec r_i - \vec r'_i|).$$ Then, since projective operators possess the following properties: $ \rho^2= \rho$ and $ \rho^*= \rho$ the expression (\[mean-energy\]) is transformed to the form: $$\begin{aligned}
E=\mbox{Sp }\rho \left(\hat H_0 +\sum _{i>j=1}^N v(|\vec r_i - \vec r_j|)\right)
=\mbox{Sp }\rho \sum _{i=1}^N\hat h(\vec r_i) %+
+
\mbox{Sp }\rho ^2\sum _{i>j=1}^N v(\vec r_i, \vec r_j)=
\nonumber \\
=\mbox{Sp }\rho \sum _{i=1}^N\hat h(\vec r_i) %+
+
\mbox{Sp }|\rho |^2\sum _{i>j=1}^N v(\vec r_i, \vec r_j)
, \label{mean-energy1-0}\end{aligned}$$ where $\hat h(\vec r_i)={1\over 2} \hat p^2_i + U(r_i)$. Since the non-perturbed hamiltonian $\hat H_0$ in Eq. (\[mean-energy1-0\]) consists of independent one-particle summands and interaction of particles occurs pairwise, the operator of trace appears in Eq. (\[mean-energy1-0\]) over one variable $r_1$ and two variables $r_1,r_2$: $$\begin{aligned}
E=\mbox{Sp }\sum_{i=1}^N \hat h(\vec r_1)
\int \ d\vec r_{2}\ldots \ d\vec r_N
\nonumber \\
\times \langle
\vec r_1',\vec r_{2},\ldots,\vec r_N|\psi \rangle
\langle \psi |\vec r_1,\ldots, \vec r_N\rangle
%+\nonumber\\
+
\mbox{Sp } \sum_{i>j=1}^N v(|\vec r_1 - \vec r_2|)
\nonumber \\
\times {1\over 2}\left[
\int \ d\vec r_{1}d\vec r_{3}\ldots \ d\vec r_N \left|\langle
\vec r_1,\vec r'_{2},\vec r_3, \ldots,\vec r_N|\psi \rangle
\langle \psi |\vec r_1,\ldots, \vec r_N\rangle \right|^2
\right.
\nonumber \\
+ \left.
\int \ d\vec r_{2}\ldots \ d\vec r_N \left|\langle
\vec r_1',\vec r_{2},\vec r_3, \ldots,\vec r_N|\psi
\rangle
\langle \psi |\vec r_1,\ldots, \vec r_N\rangle \right| ^2
\right]
, \label{mean-energy1}\end{aligned}$$ Using the equality: $\int \vec k \cdot \vec k d\theta =2 |k|^2$ one can transform Eq. (\[mean-energy1\]) and obtain: $$\begin{aligned}
E=
\mbox{Sp }\hat h(\vec r_1)
N\int \ d\vec r_{2}\ldots \ d\vec r_N
\nonumber \\
\times \langle
\vec r_1',\vec r_{2},\ldots,\vec r_N|\psi \rangle
\langle \psi |\vec r_1,\ldots, \vec r_N\rangle
%+\nonumber\\
+{1\over 4}\hspace{4mm}
\mbox{Sp } v(|\vec r_1 - \vec r_2|)N(N-1)
\nonumber \\
\times \left[
\int \ d\vec r_{1}d\vec r_{3}\ldots \ d\vec r_N \langle
\vec r_1,\vec r'_{2},\vec r_3, \ldots,\vec r_N|\psi \rangle
\right.
\times \nonumber \\
\times
\int \ d\vec r_{2}\ldots \ d\vec r_N \langle \psi
|\vec r_1',\vec r_{2},\vec r_3, \ldots,\vec r_N
\rangle
\langle \vec r_1,\ldots, \vec r_N|\psi \rangle
\langle \psi |\vec r_1,\ldots, \vec r_N\rangle
\nonumber \\
+
\int \ d\vec r_{2}\ldots \ d\vec r_N \langle
\vec r_1',\vec r_{2},\vec r_3, \ldots,\vec r_N|\psi
\rangle \times
\nonumber \\
\left.
\times \int \ d\vec r_{1}d\vec r_{3}\ldots \ d\vec r_N \langle
\psi | \vec r_1,\vec r'_{2},\vec r_3, \ldots,\vec r_N \rangle
\langle \vec r_1,\ldots, \vec r_N|\psi \rangle
\langle \psi |\vec r_1,\ldots, \vec r_N\rangle
\right] . \nonumber \\
\label{mean-energy2}\end{aligned}$$ It is easy to see, that the first summand from the right-hand side of Eq. (\[mean-energy2\]) contains the reduced one-particle density matrix $ \rho_1 $ as a multiplier. Therefore after some obvious transformations Eq. (\[mean-energy2\]) can be rewritten as: $$\begin{aligned}
E=
\mbox{Sp }\hat h(\vec r_1)
\rho_1(\vec r_1',\vec r_1)
+{1\over 2}
\mbox{Sp } v(|\vec r_1 - \vec r_2|)N(N-1)
\nonumber \\
\times
\int \ d\vec r_{3}\ldots \ d\vec r_N \langle
\vec r'_1,\vec r'_{2},\vec r_3, \ldots,\vec r_N|\psi \rangle
\langle \psi
|\vec r_1,\vec r_{2},\vec r_3, \ldots,\vec r_N
\rangle
\times \nonumber \\
\times
\int \ d\vec r_{1} d\vec r_{2}\ldots \ d\vec r_N
\langle \vec r_1,\ldots, \vec r_N|\psi \rangle
\langle \psi |\vec r_1,\ldots, \vec r_N\rangle
\nonumber \\
=
\mbox{Sp }\hat h(\vec r_1)
\rho_1(\vec r_1',\vec r_1)
+{1\over 2}\mbox{Sp } v(|\vec r_1 - \vec r_2|)N(N-1)
\nonumber \\
\times
\int \ d\vec r_{3}\ldots \ d\vec r_N \langle
\vec r'_1,\vec r'_{2},\vec r_3, \ldots,\vec r_N|\psi \rangle
\langle \psi
|\vec r_1,\vec r_{2},\vec r_3, \ldots,\vec r_N
\rangle .
\nonumber \\
\label{mean-energy3}\end{aligned}$$ Here one takes into account the normalization of wave function $|\psi \rangle $: $\int \langle \psi |\psi \rangle \ d\vec r_{1}
d\vec r_{2}\ldots \ d\vec r_N=1$.
If we observe that in the right side of the equation (\[mean-energy3\]) the second summand contains the reduced two-particle density matrix $ \rho_2 $ as a multiplier then it is possible to transform this equation to the expression which has the reduced density matrixes $ \rho_1 $ and $ \rho_2 $: $$\begin{aligned}
E=
\mbox{Sp }\hat h(\vec r_1)
\rho_1(\vec r_1',\vec r_1)
+{1\over 2}\mbox{Sp }
v(|\vec r_1 - \vec r_2|)
\rho_2(\vec r'_1,\vec r'_{2};\vec r_{1}, \vec r_{2})= \nonumber \\
=
\epsilon^{(0)} N +{1\over 2}\mbox{Sp }
v(|\vec r_1 - \vec r_2|)
\rho_2(\vec r'_1,\vec r'_{2};\vec r_{1}, \vec r_{2}) \label{mean-energy4},\end{aligned}$$ where $\epsilon^{(0)}$ is the one-electron state, the summand $ %\begin{eqnarray}
E^{HF}={1\over 2}\mbox{Sp }
v(|\vec r_1 - \vec r_2|)$ $
\rho_2(\vec r'_1,\vec r'_{2};\vec r_{1}, \vec r_{2})
%\end{eqnarray}
$ is excitation energy of the system under the interaction. Since in the Hartree - Fock approximation the reduced density matrix $ \rho_2 $ is factorized, at first, the energy $\epsilon ^ {(0)} N $ equal to energy of $N $ one-electron states, including kinetic energy of an electron, energy of an electron in a self-consistent scalar potential, and exchange energy, and, secondly, energy $E ^ {HF} $ of the excitation yield, as it follows from the equations (\[hartry-fock-eqs\]) and (\[mean-energy4\]), a contribution to the electronic energy $E $ of a crystal. Further we shall show, that the Hartree - Fock approximation is an one-particle approximation in the sense that in this approximation the excitation energy $E ^ {HF} $ is represented as the energy of quasiparticle states. Now, we rewrite Eq. (\[hartry-fock-eqs\]) in the representation of Dirac ket(bra)- vectors: $$\begin{aligned}
\hat h(k) | n; k \rangle + \sum_{m=1, m\neq n}^N
\int \delta(k-k') \ dk' (| n; k \rangle \langle m; k' | v(kk')| m; k'\rangle
+\nonumber \\
+ (| n; k' \rangle \delta_{nm})\langle m; k' | v(kk')
(| m; k\rangle \delta_{mn} ))
\nonumber \\
-
\sum_{m=1}^N
\int | m; k \rangle \langle m; k' | v(kk')| n; k'\rangle \delta(k-k') \ dk'
=
\nonumber \\
= | n; k \rangle (\epsilon_n(0)+ \epsilon_n(k))
\label{moment-Hartry-Fock-eq}\end{aligned}$$ where $k_i\equiv \{\vec{k}_i, \sigma_i\}$, $\hat h(k)$ is a momentum representation of the non-perturbed hamiltonian, $v(kk')=\int d \vec r d\vec r'
|\vec r\rangle \langle \vec k\cdot \vec r |
v(|\vec r - \vec r'|)
| \vec k'\cdot \vec r '\rangle \langle \vec r' |$ is a momentum representation of the Coulomb interaction operator, $\delta(k-k')$ is the Dirac $\delta $-function manifesting the presence of the law of conservation of momentum. Let us introduce projective operators $\hat\rho ^{mn}_{kk'}$ $$\begin{aligned}
\hat \rho ^{mn}_{kk'} \equiv | m; k'\rangle \langle n; k|
\label{secondary-quant-projector}\end{aligned}$$ and express Eq. (\[moment-Hartry-Fock-eq\]) via these operators $\hat \rho ^{mn}_{kk'}$. For this purpose, Eq. (\[moment-Hartry-Fock-eq\]) is multiplied on the right by bra-vector $ \langle n; k | $. Then, additional summating over $n $ and integrating over $dk $ one get the equation: $$\begin{aligned}
\sum_{n=1}^N \int dk\ \hat h(k) | n; k \rangle \langle n; k|+ \sum_{n=1}^N
\int \int \delta(k-k')\ dk\ dk'
\times \nonumber \\
\times
| n; k \rangle \sum_{m=1}^N \langle m; k' | v(k,k')| m; k'\rangle
\langle n; k|
+ \int \int \delta(k-k')\ dk\ dk' \times \nonumber \\
\times
\left(\sum_{m=1}^N| n; k '\rangle \delta_{nm}\langle m; k' |\right) v(kk')
\left(\sum_{ n=1}^N| m; k\rangle \delta_{mn}\langle n; k| \right)
-\nonumber \\
-
\int \int \sum_{m=1}^N | m; k \rangle \langle m; k' | v(kk')
\sum_{n=1}^N| n; k'\rangle \langle n; k|
\delta(k'-k) d(-k) d(-k')
\nonumber \\
= \int dk \sum_{n=1}^N \langle n; k|n; k \rangle (\epsilon_n(0)
+ \epsilon_n(k)).
\nonumber \\
\label{moment-Hartry-Fock-eq1}\end{aligned}$$ We see that the first and second terms on the left of the equation (\[moment-Hartry-Fock-eq1\]) are traces of a matrix representation of operators $ \hat \rho \hat h $ and $ \hat \rho
^* \hat \rho v $, and the third and fourth terms on the left of equation (\[moment-Hartry-Fock-eq1\]) are mutually cancelled. Hence, it means that using normability of function $
|n; k \rangle $: $ \int dk\langle n; k|n; k \rangle =1 $, we obtain the following equation: $$\begin{aligned}
\mbox{Sp} \hat \rho \hat h
+ \mbox{Sp} \hat \rho ^* \hat \rho v
=\epsilon_n(0)N +\int dk \sum_{n=1}^N \langle n; k|n; k \rangle
\epsilon_n(k).
%\nonumber \\
\label{moment-Hartry-Fock-eq2}\end{aligned}$$ Using properties of the projective operators $\hat\rho ^{mn}_{kk'}$: $\left(\hat\rho ^{mn}_{kk'}\right)^*=\hat\rho ^{mn}_{kk'}$ and $\left(\hat\rho ^{mn}_{kk'}\right)^2=\hat\rho ^{mn}_{kk'}$ one can transform Eq. (\[moment-Hartry-Fock-eq2\]) to the form: $$\begin{aligned}
\mbox{Sp} \hat \rho (\hat h + v)
=\epsilon_n(0)N +\int dk \sum_{n=1}^N \langle n; k|n; k \rangle
\epsilon_n(k)= \epsilon_n(0)N + \epsilon.
%\nonumber \\
\label{moment-Hartry-Fock-eq3}\end{aligned}$$
Let us elucidate a physical sense of introduced projective operators $ \hat\rho ^ {mn} _ {kk '} $. It follows from comparison (\[mean-energy4\]) and (\[moment-Hartry-Fock-eq3\]) that the energy of a quasi-particle $ \epsilon $ is on the right of Eq. (\[moment-Hartry-Fock-eq3\]) accurate to the constant $
\epsilon_n {(0)} N $. It follows from here that the operator $
\hat\rho ^ {mn} _ {kk '} $ allows to calculate energy $
\epsilon $ of quasiparticle excitations. It means that the expression (\[moment-Hartry-Fock-eq3\]) is nothing else but a procedure of average on density matrix. Since the averaging with the help of the operator $ \hat\rho ^ {mn} _ {kk '} $ yields energy $ \epsilon $ of quasi-particle this operator is a secondary quantized density matrix.
It follows from the comparison of right sides of the equations (\[hartry-fock-eqs\]), (\[mean-energy4\]), and (\[moment-Hartry-Fock-eq3\]) that the reference point $ \mbox {Extr}
E_n (k_i) $ of energy band determines the solution $ \epsilon_n
{(0)} $ of the one-electron problem $$\begin{aligned}
\epsilon_n{(0)} =\mbox{Extr } E_n(k_i)/N \label{reference-point}.\end{aligned}$$
Thus, one has proved that the equation (\[hartry-fock-eqs\]) can be considered as an equation describing a state of quasi-particle and determining its energy accurate to the constant $ \epsilon_n {(0)} N $.
Green function for one-particle state
======================================
It follows from Eq. (\[moment-Hartry-Fock-eq3\]) also that the quantity $ \epsilon_n {(k_i)} $ can be interpreted as an eigenvalue of the Hamiltonian for the quasi-particle excitation without taking into account interaction of quasi-particles. Therefore, the equation (\[moment-Hartry-Fock-eq3\]), written in the formalism of density matrix $ \hat\rho ^ {(0)} _ {nn '; kk
'} \equiv \hat\rho ^ {mn} _ {kk '} $ can be rewritten in the formalism of wave functions in coordinate representation and in a limit of large $N $, $N\to \infty $ in the following way: $$\begin{aligned}
\left(\imath {\partial \over \partial t }-(\hat h + \Sigma^{x}+V^{sc} )\right)
\sum_n \hat\rho ^{(0)}_{nn'; rr'}
=\lim_{N\to\infty} (-\epsilon_n(0))N \delta_{rr'},
%\nonumber \\
\label{moment-Hartry-Fock-eq4}\end{aligned}$$ where property $ \int dk\langle n; kr|n; kr'
\rangle =\delta_{rr'}$ has been used; $\delta_{rr'}$ is a delta symbol. Since the energy $ \epsilon_n (0) $ of the bound one-electron state is negative: $ \epsilon_n (0) <0 $, the right side of Eq. (\[moment-Hartry-Fock-eq4\]) represents itself a Dirac $\delta$-function $ \delta(r-r')$. It allows to write Eq. (\[moment-Hartry-Fock-eq4\]) as: $$\begin{aligned}
\left(\imath {\partial \over \partial t }-\hat h^{HF} \right)
\sum_n \hat\rho ^{(0)}_{nn'; r,r'}
=\delta(r-r'),
%\nonumber \\
\label{moment-Hartry-Fock-eq5}\end{aligned}$$ where $\hat
h^{HF}=(\hat h + \overline\Sigma^{x}+V^{sc})$, $\overline
\Sigma^{x}=-\Sigma^{x}$. Eq. (\[moment-Hartry-Fock-eq5\]) is the equation for the Green function. It means that in the secondary quantized representation an operator $$\begin{aligned}
\hat G_1^{(0)}(n'; r,r') = \sum_n \hat\rho ^{(0)}_{nn'; r,r'}\end{aligned}$$ possesses properties of non-perturbed Green function.
So, the quasi-particle excitation determined by the Hamiltonian $
\hat h ^ {HF} $ can be considered as a free particle whose equation of motion is the equation (\[moment-Hartry-Fock-eq5\]).
In the many-body problem, in particular, in calculations of an energy-band crystal structure a contribution given by interaction of electromagnetic field with matter is played the essential role. To take into account many-particle effects due to a correlated motion of electron we should describe the system by self-consistent solutions of the non-stationary equation $$\begin{aligned}
\imath {\partial\Psi (t) \over \partial t }= \hat H \Psi (t)
\label{non-stationary-quant-motion-eq}\end{aligned}$$ where $\hat H$ is a Schrödinger hamiltonian in a non-relativistic case or a Dirac hamiltonian in a relativistic case. It turns the variational principle (\[variation-princip2\]) into a variational principle for the excited states, not being strictly definite one. Further, we shall show that within the framework of the concept of quasiparticle excitations this ambiguity can be removed by means of the account of interaction as, at first, a change of a quasi-particle mass and, second, as a changing of the location of reference point of energy band.
We have proved that for the secondary quantized representation the operator $ \rho $ looks as $ \hat \rho = | \hat \psi \rangle
\langle \hat\psi | $ and possesses properties of the Green function $G_1 $. Therefore the sum $ \hat G_1 $ over $n $ from elements of the matrix $ \hat\rho ^ {nn '} _ {kk '} $ for the secondary quantized density matrix $ \hat \rho $ describing an interacting particle satisfies a Dyson equation in a nonrelativistic case or to a Schwinger - Dyson equation in a relativistic case: $$\begin{aligned}
G_1(1;2)= G_1^{(0)}(1;2)+\int d3\ d4\ G_1^{(0)}(1;3) \hat \Sigma (3,4) G_1(4;2)
\label{Shwinger-Dayson-eq}\end{aligned}$$ $G_1^{(0)}(1;2)$ is a free Green function, $\hat \Sigma (3,4)$ is a self-energy operator: $\hat \Sigma = {\overline \Sigma^{x}}+\hat \Sigma^{c}$, $\hat
\Sigma^{c}$ is a correlation interactions, representing itself a part of the self-energy which describes the many-particle effects. Here numerical labels for the arguments are used: $ \{r_1, t_1 \}
= x_1\equiv 1 $, etc. Acting on Eq. (\[Shwinger-Dayson-eq\]) by the operator $ \imath {\partial \over \partial t}-\hat h ^ {HF} $ and using the equation of motion for the free particle (\[moment-Hartry-Fock-eq5\]) we get the equation for the perturbed Green function as $$\begin{aligned}
\left[ \imath {\partial \over \partial t }-\hat h^{HF}(r_1)\right]G_1(n';1,2)-
\int d3 \hat \Sigma^{c} (n';1,3) G_1(n';3,2)= \nonumber \\
=
(-\epsilon_n(0))N \delta_{r_1r_2}.
\label{Perturbed-Green-func-eq}\end{aligned}$$ Rewriting Eq. (\[Perturbed-Green-func-eq\]) in the formalism of wave functions one gets $$\begin{aligned}
\left[\imath {\partial \over \partial t }-\hat h^{HF}(r_1)\right]\psi_{n}
(k_1r_1) -
\int d \vec r_2 \hat \Sigma^{c} (n;1,2) \psi_{n}(k_1r_2)=
\nonumber \\
(-\epsilon_n(0)) \psi_{n}
(k_1r_1).
\label{Perturbed-wave-func-eq1}\end{aligned}$$ Since the expression: $\imath {\partial \psi_{n}\over \partial t} =\epsilon_n(k_1)$ takes place, then Eq. (\[Perturbed-wave-func-eq1\]) yields the Hartree - Fock taking into account of interacting quasi-particles $$\begin{aligned}
\hat h^{HF}(r_1)\psi_{n}
(k_1r_1) + \int d \vec r_2 \hat \Sigma^c (n;1,2) \psi_{n}(k_1r_2)
=\nonumber \\
=(\epsilon_n(0)+\epsilon_n(k_1)) \psi_{n}
(k_1r_1).
\label{Perturbed-Hartry-Fock-eqs}\end{aligned}$$
Let us define a mass operator $\widehat {\Delta M} $ as: $$\begin{aligned}
\widehat {\Delta M}\psi_n (k_1 r_1)
%\nonumber \\
= \int d \vec r_2 \hat \Sigma^c (n;1,2) \psi_{n}(k_1r_2).\end{aligned}$$
Within the framework of the concept quasiparticle excitations it is possible to represent the operator $ \widehat {\Delta M} $ in the diagonal form: $$\begin{aligned}
\widehat {\Delta M}\psi_n (k_i r_i)
%\nonumber \\
=(\Delta M_n(0)+ \Delta M_n(k_i)) \psi_n (k_i r_i).\end{aligned}$$ It is known that the eigenvalue of mass operator possesses the property: $\Delta M_n(k_i)=\Delta M_n(-k_i)$. Here $\Delta M_n(0)$ is an eigenvalue of mass operator $\widehat {\Delta M}$ in the limit $\vec k \to 0$.
It follows from here the physical sense of $ \widehat {\Delta M} $. It determines an effective mass of the quasi-particle and an efficient reference point of the energy band: $$\begin{aligned}
\hat h^{HF}(r_1)\psi_{n}(k_1r_1) =
(\tilde \epsilon_n(0)+\tilde \epsilon_n(k_1))\psi_{n}(k_1r_1)\equiv \nonumber \\
\equiv
\left[(\epsilon_n(0)+\Delta M_n(0))+(\epsilon_n(k_1)+\Delta M_n(k_1))\right]
\psi_{n}(k_1r_1).
\label{Perturbed-Hartry-Fock-eqs1}\end{aligned}$$
Since the change of the mass of quasi-particle determined by the operator $ \widehat{\Delta M}$ maintains the condition (\[variation-princip1\]) then if to take into account the change of the reference point of band at interaction, the variational principle for the interacting system becomes a strictly definite one and takes the form: $$\begin{aligned}
\delta E_n[\widetilde \psi_n] = 0,
\quad \epsilon_n(0)= {1\over N}\mbox{Extr} E_n(k_i)- \Delta M_n(0).
\label{variation-princip3}\end{aligned}$$
Let us consider a Green function normalized per unit volume $V=1 $ so that an average energy in $V $ is equal to the energy of the one-particle state and $N=1 $. If $-\epsilon_n (0) \to \infty $ then Eq. (\[Perturbed-Green-func-eq\]) describes a propagation of one particle and should be rewritten as $$\begin{aligned}
\left[ \imath {\partial \over \partial t }-\hat h^{HF}(r_1)\right]G_1(n';1,2)-
\int d3 \hat \Sigma^c (n';1,3) G_1(n';3,2)= \nonumber \\
=
(-\epsilon_n(0)) \delta_{r_1r_2}.
\label{Particle-Green-func-eq}\end{aligned}$$ It follows from here that according to the definition of Green functions we have the following expression for the energy $
\epsilon_n (0) $: $$\begin{aligned}
-\epsilon_n(0)=C - a_n, \quad C \to \infty; \label{Particle-energy}\end{aligned}$$ where $a_n$ is a finite quantity. Hence, since the energy is counted off from an arbitrary value, Eq. (\[variation-princip3\]) yields the following expression for reference points $
\epsilon (0) ^ {\pm}_n $ of a quasi-particle energy and a antiquasi-particle energy $$\begin{aligned}
\epsilon(0)^{\pm}_n\equiv \pm a_n
= \left( \mbox{Extr} \tilde{E}(k_1)\mp \Delta {M}_n(0) \right )/2.
\label{zone-reference}\end{aligned}$$ Here one took into account that $N=1$; an extremum of zone is redefined as $\mbox{Extr}\tilde{E}(k_1)= \mbox{Extr}{E}_n(k_1)-C_n$, the sign $\{\pm\}$ in left-hand side denotes a case of quasiparticles and antiquasiparticles, respectively; and the energy of particles in the pair is counted off from zero level. One gets from the expression (\[zone-reference\]) that $a_n$ is the energy which is required to create a pair from quasiparticle and antiquasiparticle when $k_1=0$ because $$\begin{aligned}
a_n=(\epsilon(0)^{+} _n- \epsilon(0)^{-}_n)/2. \label{gap}\end{aligned}$$
Because of an additional term $ \tilde \epsilon_n (0) $ in a right-hand side of Eq. (\[Perturbed-Hartry-Fock-eqs1\]) we, generally speaking, cannot examine the left-hand side as a Hamiltonian operator of the quasi particles system acting on a corresponding wave function and as a consequence, can not construct a basis set of one-particle states of the problem. However, further we show, that $ \hat h ^ {HF} $ is a Hamiltonian of an electron - hole pair.
[*Non-relativistic case* ]{}
One can examine in non-relativistic limit quantum systems which are characterized by a small value of $\Delta {M}_n(0)$: $$\Delta {M}_n (0)\to 0. \label{light-electron}$$ It means that weak many-particle effects occur and, accordingly, we can speak about a “light” electron. The equality (\[zone-reference\]) occurs under condition of (\[light-electron\])only in the case if $a_n=0 $. From here it follows, that the energy $a_n $ of a pair is equal to zero. In other words, the energy is not expended to create an electron - hole pair. Substituting Eqs. (\[Particle-energy\]), (\[light-electron\]) into Eq. (\[Perturbed-Hartry-Fock-eqs1\]) and taking into account the condition $a_n=0$, one gets the Schrödinger equation as $$\begin{aligned}
\hat h^{HF}(r_1)\psi_{n}(r_1) =
\tilde {\tilde \epsilon}_n \psi_{n}(r_1),
\label{Perturbed-Hartry-Fock-eqs2}\end{aligned}$$ which describes the quasiparticle - antiquasiparticle pair (a non-relativistic electron - hole pair). Here $\tilde {\tilde
\epsilon}_n={\tilde \epsilon}_n -C$.
Since the energy $a_n $, expended on creation of a pair, equals to zero we have proved that the variable $ \tilde {\tilde
\epsilon} _n $ can be understood as the energy of an electron - hole pair. Therefore, Eq. (\[Perturbed-Hartry-Fock-eqs2\]) has a group of dynamic symmetry, which algebra is $ \sim $ if to neglect an exchange interaction. As is known, a nonrelativistic hydrogen-like atom possesses such symmetry. Hence, we have proved that to calculate quasiparticle states in the non-relativistic case it is possible to use a basis set of states of a nonrelativistic hydrogen-like atom.
However, for a heavy electron $ \Delta M_n (0) \ge 1 $ according to the formula (\[gap\]) we always have $$a_n=-\Delta {M}_n (0)/2 \label{heavy-electron}$$ and, hence, there does not exist equation such as Schrödinger one for its describing. From here we conclude that the heavy electron can not be examined in a nonrelativistic limit.
[*Relativistic case*]{}
Let us generalize the proposed approach to relativistic case. To do it we substitute Eqs. (\[Particle-energy\]) and (\[heavy-electron\]) into (\[Perturbed-Hartry-Fock-eqs1\]) and let $ n$ tends to $ n\to \infty $: $$\begin{aligned}
\hat h^{HF}(r_1)\psi_{n}(k_1r_1) =
\left({\Delta {M}_n (0)\over 2}+\tilde {\tilde \epsilon}_n(k_1)
\right)\psi_{n}(k_1r_1),
\quad n\to \infty.
\label{Quasirelativistic-eq}\end{aligned}$$ Then, one can assume that the operator ${\partial \over \partial
t}- \hat h^{HF}$ in Eq. (\[Quasirelativistic-eq\]) is a quasirelativistic hamiltonian written in the implicit form in the Hartee - Fock approximation. From consideration carried out above it follows that the desired relativistic equation of motion should describe a charged composite system from a pair of particles and have the dynamic symmetry . A spin of given quantum system should be equal 1 as motion of a hole is a motion of an electron in many-particle positively charged matrix. In the equation of motion of a relativistic charged vector boson has been found and shown, that it describes a relativistic hydrogen-like atom. The relativistic charged vector-boson appears a composite system with a corresponding spectrum of masses and in quasirelativistic limit $n\to \infty $ its energy $E_1 $ is determined by the expression: $$\begin{aligned}
E_1\approx {m\over 2} -{m\gamma ^2 \over 2{n }^2} - {m \gamma ^4\over 8 n ^3}
{ \left({4\over |k| }-{3\over n }\right)}-
{m \gamma ^6\over 8 n ^4}
{ \left({3\over n^2 }-{8\over n |k| }+{4\over k^2 }\right)}
+O(\gamma^8).\nonumber \\
\ \label{boson-energy}\end{aligned}$$ Comparison of right-hand sides of formulas (\[Quasirelativistic-eq\]) and (\[boson-energy\]) yields that $
\Delta M _ {\infty} \equiv \lim _ {n\to \infty} \Delta M_n (0) =m
$ is a rest mass $m $ of an electron. Hence, Eq. (\[Quasirelativistic-eq\]) is an equation of motion for a relativistic electron - hole pair with a reduced mass $\Delta M _
{\infty}/2=m/2 $ which, apparently, is the relativistic charged vector-boson considered in quasirelativistic limit $n\to \infty $.
Conclusion
==========
So, the variation method to find interacting quasiparticle states in crystals was developed. The quasiparticle propagator was constructed by summation over elements of secondary quantized density matrix. This approach allows us to find motion equations of one-electron states of excited atom in crystals.
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---
abstract: 'We study the $\bhull$ of a planar point set, a generalization of the Orthogonal Convex Hull where the coordinate axes form an angle $\beta$. Given a set $P$ of $n$ points in the plane, we show how to maintain the $\bhull$ of $P$ while $\beta$ runs from $0$ to $\pi$ in $\Theta(n\log n)$ time and $O(n)$ space. With the same complexity, we also find the values of $\beta$ that maximize the area and the perimeter of the $\bhull$ and, furthermore, we find the value of $\beta$ achieving the best fitting of the point set $P$ with a two-joint chain of alternate interior angle $\beta$.'
author:
- 'Carlos Alegría-Galicia [^1]'
- 'David Orden [^2]'
- 'Carlos Seara [^3]'
- 'Jorge Urrutia [^4]'
bibliography:
- 'references.bib'
title: 'On the $\bhull$ of a planar point set[^5]'
---
Introduction {#sec:intro}
============
Let $\bset$ be a set of two lines with slopes $0$ and $\tan(\beta)$, where $0 < \beta < \pi$. A region in the plane is said to be *$\bset$-convex*, if its intersections with all translations of any line in $\bset$ are either empty or connected. An *$\bset$-quadrant* is a translation of one of the ($\bset$-convex) open regions that result from subtracting the lines in $\bset$ from the plane. We call the quadrants of $\bset$ as *top-right*, *top left*, *bottom-right*, and *bottom-left* according to their position with respect to the elements of $\bset$, see Figure \[intro:fig:bhull\](a). Let $P$ be a set of $n$ points, and $\mathcal{Q}$ the set of all $\bset$-quadrants that are *$P$-free*; i.e., that contain no elements of $P$. The *$\bhull$* of $P$ is the set
$$\displaystyle
\mbox{\ensuremath{\bhullp}}=\mathbb{R}^{2}-\underset{q\in\mathcal{Q}}{\bigcup}q$$ of points in the plane belonging to all connected supersets of $P$ which are $\bset$-convex [@alegria_2014; @ottmann_1984]. See (b).
The concept of $\bset$-convexity stemmed from the notion of *restricted orientations* [@guting_thesis_1983], where geometric objects comply with a property (or a set of properties) related to some fixed set of lines. Researchers have extensively studied this notion by considering restricted-oriented polygons [@guting_thesis_1983], proximity [@widmayer_1987], visibility [@schuierer_thesis_1991], and both restrictions and generalizations of $\bset$-convexity. The particular case of *orthogonal convexity* [@rawlins_1988] considers $\beta$ to be fixed at $\frac{\pi}{2}$. In the more general *$\mathcal{O}$-convexity* [@rawlins_1987; @rawlins_1988], $\bset$ is replaced by a (possibly infinite) set of lines with arbitrary orientations. Other restricted-oriented notions of convexity include *$D$-convexity* [@franek_2009] and *$\mathcal{O}$-convexity* [@rawlins_thesis_1987]. The former is based in a functional (rather than set-theoretical) definition, while the latter (unlike $\bset$-convexity) always leads to connected sets. For a comprehensive compilation of studies on the area please refer to @fink_2004. Some recent computational results can be found in [@minimum-area_2012; @alegria_2013; @alegria_2014; @pelaez_2013].
In this paper, we solve the problem of maintaining the combinatorial structure of $\bhullp$ while $\beta$ goes from $0$ to $\pi$, and apply this result to some optimization problems. Following the lines of @bae_2009, we find the values of $\beta$ that maximize the area and the perimeter of $\bhullp$. In addition, we include an appendix extending the results from @fitting_2011 to fit a two-joint not-necessarily orthogonal polygonal chain to a point set. See .
In all cases, our general approach is to perform an angular sweep. We first discretize the set $\{\beta:\beta \in (0,\pi)\}$ into a linear sequence of increasing angles $\{\beta_{1},\beta_{2},\ldots,\beta_{O(n)}\}$. While $\beta$ runs from $0$ to $\pi$, each $\beta_i$ corresponds to an angle where there is a change in the combinatorial structure of $\bhullp$. We then solve the particular problem for any $\beta \in [\beta_{1}, \beta_{2})$ in $O(n\log n)$ time, and show how to update our solution in logarithmic time in the subsequent intervals $[\beta_{i},\beta_{i+1})$. All our algorithms run in $O(n\log n)$ time and $O(n)$ space.
#### Outline of the paper.
In we show how to maintain the $\bhull$ of $P$ while $\beta$ goes from $0$ to $\pi$. In we extend this result to solve the optimization problems we mentioned above. We end in with our concluding remarks.
The $\bhull$ of $P$ {#sec:bhull}
===================
In this section we introduce definitions that are central to our results. We also show how to compute $\bhullp$ for a fixed value of $\beta$, and how to maintain its combinatorial structure while $\beta$ runs from $0$ to $\pi$.
Preliminaries {#sec:bhull:preliminaries}
-------------
For the sake of simplicity, we will assume $P$ to have no three colinear points, and no pair of points on a horizontal line. Consider the region $\mathcal{R}$ obtained by removing from the plane all top-right $\bset$-quadrants free of elements of $P$. The *top-right $\bset$-staircase* of $P$ is the directed polygonal chain formed by the segment of the boundary of $\mathcal{R}$ that starts at the rightmost and ends at the topmost vertex (element of $P$ that lies over the boundary) of $\bhullp$, with respect to the coordinate system defined by the lines in $\bset$. We further define the *top-left*, *bottom-left*, and *bottom-right* $\bset$-staircases in a similar way. See .
\[bhull:obs:maximal\] A point in $P$ is a vertex of $\bhullp$ if, and only if, it is the apex of at least one $P$-free $\bset$-quadrant free of elements of $P$. Conversely, a point in the plane lies in the interior of $\bhullp$ if, and only if, every $\bset$-quadrant with apex on it contains at least one point in $P$.
We say that an $\bset$-quadrant is *maximal* if its boundary joins two consecutive elements in the sequence of vertices found while traversing an $\bset$-staircase in its corresponding direction. Two $\bset$-quadrants are *opposite* to each other if, after placing their apices over a common point, their rays bound opposite angles. Similarly, we say that two $\bset$-staircases are opposite to each other, if they were constructed using opposite $\bset$-quadrants. It is easy to see that $\bhullp$ is disconnected when the intersection of two opposite maximal $\bset$-quadrants is not empty. In such case we say that both $\bset$-quadrants *overlap*, and refer to their intersection as an *overlapping region*. See the regions bounded by dashed lines in .
\[bhull:obs:staircases\] Non-opposite $\bset$-staircases cannot generate overlapping regions. Moreover, only one pair of $\bset$-staircases can intersect at the same time.
We will specify $\bhullp$ in terms of its vertices and its overlapping regions. From , the set of vertices of $\bhullp$ is the set of maximal elements of $P$ under vector dominance [@theta-maxima_1999]. Thus they can be computed for a fixed value of $\beta$ in $\Theta(n \log n)$ time and $O(n)$ space [@kung_1975; @preparata_1985]. Note that $\bset$-staircases are monotone with respect to both lines in $\bset$ (they could not bound $\bset$-convex regions otherwise), so any pair of them intersect with each other at most a linear number of times. From , in a fixed value of $\beta$ there is at most a linear number of overlapping regions. Thus, if the vertices of $\bhullp$ are sorted according to either the $x$- or the $y$-axis, we can compute from them the set of overlapping regions in linear time. We get then the following theorem where the $\Omega(n\log n)$ time lower bound comes from the fact that from $\bhullp$ we can compute the convex hull of $P$ in linear time.
\[intro:thm:fixed\_computation\] For a fixed value of $\beta$, the sets of vertices and overlapping regions of $\bhullp$ can be computed in $\Theta(n \log n)$ time and $O(n)$ space.
The angular sweep {#sec:bhull:sweep}
-----------------
The $\bhull$ of $P$ is shown in at the *initial increasing configuration*, that is, where $\beta$ is equal to an angle $\beta_I = 0 + \varepsilon$ for a small enough $\varepsilon$. Note that every point in $P$ is the apex of a $P$-free $\bset$-quadrant, and is thus contained in at least one $\bset$-staircase: both top-right and bottom-left $\bset$-staircases contain the whole set $P$, and the top-left and bottom-right $\bset$-staircases are formed respectively, by the topmost and bottom-most points in $P$. Also, the intersection between the top-right and bottom-left $\bset$-staircases generate a linear number of overlapping regions.
![The initial increasing configuration.[]{data-label="bhull:fig:initial_config"}](initial_config)
By performing an *increasing sweep* (where $\beta$ goes from $0$ to $\pi$), the initial increasing configuration is gradually transformed to the *initial decreasing configuration*, where $\beta$ is equal to a value $\beta_D = \pi - \varepsilon$ for a small enough $\varepsilon$ (see ). At this configuration, the top-left and bottom-right $\bset$-staircases contain $P$ and generate a linear number of overlapping regions, and the top-right and bottom-left $\bset$-staircases contain respectively, the topmost and bottom-most points in $P$. Clearly, the converse of the above discussion holds: from the initial decreasing configuration, a *decreasing sweep* (where $\beta$ goes from $\pi$ to $0$) will gradually transform $\bhullp[P][\beta_D]$ into $\bhullp[P][\beta_I]$.
![The initial decreasing configuration.[]{data-label="bhull:fig:final_config"}](final_config)
During the transition between initial configurations, we recognize four types of events that modify the set of vertices and overlapping regions of $\bhullp$. An *insertion* (resp. *deletion*) event occurs when a vertex joins (resp. leaves) a $\bset$-staircase. At *overlap* (resp. *release*) events, an overlapping region is created (resp. destroyed).
Note that a vertex leaves (resp. joins) the same $\bset$-staircase at most once, and thus, there is in total a linear number of insertion (resp. deletion) events. From Observation \[bhull:obs:staircases\], between $\beta_I$ and $\beta_D$ there is always an interval $\phi = \left[ \beta_{1}, \beta_{2} \right]$ such that, for any $\beta \in \phi$, the $\bhull$ of $P$ contains no overlapping regions. Let us consider the angular intervals $\phi_I = [\beta_I,\beta_{N_1}]$ and $\phi_D = [\beta_{N_2},\beta_D]$. An angular sweep in $\phi_I$ results in a linear number of releasing events caused by the deletion of all overlapping regions present at the initial increasing configuration. As any vertex supports at most two maximal $\bset$-quadrants, an additional linear number of region events are generated by vertex events and, therefore, region events in $\phi_I$ add up to $O(n)$. Using the same argument on $\phi_D$, we can count a linear number of these events during an angular sweep.
\[bhull:lemma:linear\_events\] There are $O(n)$ events during an angular sweep.
We now show how to compute the sequence of increasing angles that mark vertex and overlapping events during an angular sweep.
#### Insertion and deletion events.
The set of vertices of $\bhullp$ on the top-right $\bset$-staircase has a total ordering that, at any value of $\beta$ is given by traversing the staircase along its direction. At the initial configuration, the order is also given by the sequence $p_1,\ldots,p_n$ of points in $P$ labeled in ascending vertical order.
Let us consider the set $\alpha(P) = \{ \alpha_1, \ldots, \alpha_{n-1} \}$ where for each $\alpha_i$, the slope of the line through $p_i$ and $p_{i+1}$ equals $\tan(\alpha_i)$. In an increasing sweep, the first point leaving the top-right $\bset$-staircase is $p_i$. Indeed, for any $\beta > \alpha_i$, a top-right $\bset$-quadrant with apex over $p_i$ is not $P$-free. This is not the case for points corresponding to any $\alpha_j$ such that $\alpha_j > \alpha_i$ and $\alpha_j > \beta$. See .
To compute the next value of $\beta$ where a point will leave the top-right $\bset$-staircase, we must remove $\alpha_i$ from $\alpha(P)$, update $\alpha_{i-1}$ to the angle where the slope of the line through $p_{i-1}$ and $p_{i+1}$ equals $\tan(\alpha_{i-1})$, and compute the new smallest element of $\alpha(P)$. A recursive repetition of this computation allows us to obtain all deletion events corresponding to the top-right $\bset$-staircase.
\[bhull:lemma:point\_events\] All insertion and deletion events can be computed in $O(n\log n)$ time and $O(n)$ space.
Store the points in $P$ in a balanced search tree ordered according to the $y$-axis, and the set $\alpha(P)$ in a priority queue. From , the algorithm described above requires $O(n \log n)$ time and $O(n)$ space to compute the sets of insertion and deletion events, associated with the top-right $\bset$-staircase. Considering the angles shown in , a similar algorithm can be used to obtain the corresponding events for the remaining $\bset$-staircases in the same time and space complexity.
#### Overlap and release events.
Let $Q_{r}$ and $Q_{l}$ be respectively, a pair of overlapping top-right and bottom-left maximal $\bset$-quadrants. Consider that $Q_r$ is supported by the vertices $p_j,p_{j+1}$, and $Q_l$ by the vertices $p_k,p_{k+1}$. Also, assume the supporting points are labeled according to the total ordering of their corresponding staircases (see Figure \[bhull:fig:eventsw\]).
![An overlapping region (bounded by dashed lines) generated by the intersection between a top-right and a bottom-left maximal $\bset$-quadrants.[]{data-label="bhull:fig:eventsw"}](bhull_overlaps)
The *full overlap event* for the overlapping region defined by $Q_{r}$ and $Q_{l}$ is the angle $\omega$ for which the slope of the line through $p_{j+1}$ and $p_{k+1}$ equals $\tan(\omega)$. If the supporting points do not leave their corresponding staircases, this event marks the value of $\beta$ where the overlapping region disappears.
Let $\omega(P)$ be the set of full overlap events for all the overlapping regions at the initial increasing configuration, and $\alpha_d(P)$ the set of all deletion events corresponding to the vertices over the top-right and bottom-left $\bset$-staircases. Let $\omega_{m}$ and $\alpha_m$ be the smallest values in $\omega(P)$ and $\alpha_d(P)$, respectively. Performing an increasing sweep, to obtain the first release event, we need to deal with the following cases:
1. \[bhull:step\_1\] $\alpha_{m}$ corresponds to a supporting point, and $\alpha_{m} \leq \omega_{m}$. In this case, $\alpha_{m}$ needs to be processed and $\omega(P)$ needs to be updated. By removing a supporting point, at most two overlapping regions are terminated (two release events are added to $\omega(P)$), and at most one new overlapping region is generated (one overlapping event and one full overlap event are added to $\omega(P)$). After updating $\omega(P)$, $\omega_{m}$ and $\alpha_{m}$ are recomputed and the test is repeated.
2. $\alpha_{m}$ does not correspond to a supporting point. In this case, $\omega_{m}$ is the first release event.
To compute the next release event, we must remove the current release event from $\omega(P)$, and recompute $\omega_m$ as described above. A recursive repetition of these steps allow us to obtain all release events caused by intersections between the top-right and bottom-left $\bset$-staircases.
\[bhull:lemma:overlap\_events\] All overlap and release events can be computed in $O(n \log n)$ time and $O(n)$ space.
Store the points in $P$ in a balanced search tree ordered according to the $y$-axis, and the sets $\alpha_d(P),\omega(P)$ in priority queues. From , the algorithm described above requires $O(n \log n)$ time and $O(n)$ space to compute the sets of overlap and release events associated with the top-right and bottom-left $\bset$-staircases. A similar algorithm can be used to obtain the events associated to the top-right and bottom-left $\mathcal{O}_{\beta}$-staircases, with the same time and space upper bounds.
#### Maintaining $\bhullp$.
Considering the previous results, the maintenance of $\bhullp$ is straightforward:
1. \[bhull:maintain:step\_1\] Compute all vertex and overlap events, and store them in a list sorted by appearance during an increasing sweep.
2. \[bhull:maintain:step\_2\] Compute $\bhullp[P][\beta_I]$. Store in height balanced trees the total orders of the sets of vertices lying over the four $\bset$-staircases. Store the set of overlapping regions in any constant-time access data structure (such as a hash table).
3. \[bhull:maintain:step\_3\] Simulate the angular sweep by traversing the list of events. At each insertion and deletion event, update the corresponding set of vertices. At each overlap and release event, update the set of overlapping regions.
From , to compute the sets of vertex and overlap events, we require $O(n \log n)$ time and $O(n)$ space. As we have a linear number of elements on each set, we can merge them into a single ordered set using $O(n \log n)$ time. Thus, \[bhull:maintain:step\_1\] requires $O(n \log n)$ time and $O(n)$ space.
From , computing $\bhullp$ for any fixed value of $\beta$ takes $O(n \log n)$ time and $O(n)$ space. Every $\bset$-staircase contains at most $n$ elements and therefore, to store their total order in a height balanced tree we require $O(n \log n)$ time. Using a hash table, we can initialize the set of overlapping regions in $O(n)$ time. Therefore, \[bhull:maintain:step\_2\] requires $O(n \log n)$ time and $O(n)$ space.
At each insertion and deletion event, updating the corresponding set of $\bset$-maximal elements requires $O(\log n)$ time per operation. Updates on the set of overlapping regions takes constant time, so \[bhull:maintain:step\_3\] takes $O(n \log n)$ time. From this analysis we get that, in total, we can compute and maintain $\bhullp$ through an angular sweep in $O(n \log n)$ time and $O(n)$ space. From , this time complexity is optimal.
Computing and maintaining $\bhullp$ through an angular sweep requires $\Theta(n \log n)$ time and $O(n)$ space.
Application problems {#sec:applications}
====================
In this section we extend the results from Section \[sec:bhull\] to the solution of related optimization problems. We deal with the problem of maximizing the area and the perimeter of $\bhullp$ (Sections \[sec:apps:area\] and \[sec:apps:perimeter\], respectively). As an extra application, in \[sec:apps:fitting\] we deal with the problem of fitting a two-joint polygonal chain to a point set.
Area optimization. {#sec:apps:area}
------------------
In this section we solve the following problem:
Given a set $P$ of $n$ points in the plane, compute the value of $\beta$ for which $\bhullp$ has maximum area.
Let $\{ \beta_1, \ldots, \beta_{O(n)} \}$ be the sequence of (vertex and overlapping) events, ordered by appearance during an increasing sweep. Following the lines of @bae_2009 (see also ), we express the area of $\bhullp$ for any $\beta \in [\beta_i,\beta_{i+1})$ as $$\label{apps:area:eqn:area}
{\operatorname{area}}(\bhullp) = {\operatorname{area}}(\polygon)
- \sum_i {\operatorname{area}}(\triangles)
+ \sum_j {\operatorname{area}}(\parallelograms),$$ where $\mathcal{P}(\beta)$ denotes the (simple) polygon having the same vertices as $\mathcal{O}_{\beta}\mathcal{H}(P)$ and an edge connecting two vertices if they are consecutive in a $\mathcal{O}_{\beta}$-staircase. The term $\triangle_i(\beta)$ is the $i$-th triangle defined by two consecutive vertices in a $\mathcal{O}_{\beta}$-staircase, and ${ \pgfpicture\pgfsetroundjoin
\pgftransformxslant{.6} \pgfpathrectangle{\pgfpointorigin}{\pgfpoint{.60em}{.65em}} \pgfusepath{stroke,} \endpgfpicture}_j(\beta)$ is the $j$-th overlapping region defined by the intersection of two opposite $\mathcal{O}_{\beta}$-staircases.
![The area of $\bhullp$. The polygon $\polygon$ is bounded by dotted lines. A triangle $\triangles$ and two parallelograms $\parallelograms$ are filled in blue.[]{data-label="apps:area:fig:area"}](bhull_area)
Our general approach is to maintain the terms of during a complete angular sweep. We first compute the optimal value of $\beta$ for $[\beta_1,\beta_2)$. We then traverse the event sequence, updating the affected terms in at each event. At the same time, we compute the local angle of maximum area for each $[\beta_i,\beta_{i+1})$. With any new computation, we keep the local optimal angle only if the previous maximum area is improved.
#### The polygon $\polygon$.
At any fixed value of $\beta$, the polygon can be constructed from the vertices of $\bhullp$ in linear time. Once constructed, it takes a second linear run to compute its area. During an interval between events the area does not change. As $\polygon$ only depends on the vertices of $\bhullp$, it is only modified by insertion and deletion events. Each event can be handled in constant time: the area of a triangle needs to be added (deletion event) or subtracted (insertion event) from the previous value of the area of $\polygon$. See .
#### The triangles $\triangles$.
A triangle is defined by a pair of consecutive vertices of $\polygon$. If we consider a top-right $\bset$-staircase, the area of $\triangles$ is bounded by a line through $p_i$ and $p_{i+1}$, an horizontal line through $p_i$, and a line with slope $\tan(\beta)$ through $p_{i+1}$. In this context, the area of $\triangles$ is given by
$$\begin{aligned}
\label{apps:area:eqn:triangle_area}
{\operatorname{area}}(\triangles) &= \left| (x_i-x_{i+1})(y_{i+1}-y_i)+(y_{i+1}-y_i)^2\cot(\beta) \right| \nonumber \\
&= \left| a_i \pm b_i \cot(\beta) \right|,\end{aligned}$$
with $a_i, b_i$ constants, where $\left( x_i,y_i \right)$ and $\left( x_{i+1},y_{i+1} \right)$ are respectively, the coordinates of the points $p_i$ and $p_{i+1}$.
The term $\sum_i {\operatorname{area}}(\triangles)$ is impacted by insertion and deletion events and, at each event, it needs to be modified a constant number of times. As any vertex of $\bhullp$ supports at most two maximal $\bset$-quadrants, at a deletion event two triangles are removed and one triangle is added. The converse occurs for insertion events. See .
#### The overlapping regions $\parallelograms$.
An overlapping region is defined by two pairs of consecutive vertices of $\bhullp$ belonging to opposite $\bset$-staircases. Overlapping regions are bounded by parallelograms with sides parallel to the lines in $\bset$. If we consider top-right and bottom-left $\bset$-staircases intersecting as shown in , the area of a parallelogram is given by $$\begin{aligned}
\label{apps:area:eqn:or_area}
{\operatorname{area}}(\parallelograms) &= \left| (x_{k+1} - x_{i+1})(y_{k} - y_{i})
+ (y_{k+1} - y_{i+1})(y_{k} - y_{i})\cot(\beta) \right| \nonumber \\
&= \left| a_j \pm b_j \cot(\beta) \right|,\end{aligned}$$ with $a_i, b_i$ constants, where $p_i=(x_i,y_i),p_{i+1}=(x_{i+1},y_{i+1})$ and $p_k=(x_k,y_k),p_{k+1}=(x_{k+1},y_{k+1})$ are respectively, the supporting vertices of the overlapping maximal opposite $\bset$-quadrants.
The term $\sum_j {\operatorname{area}}(\parallelograms)$ is impacted by all types of events. Overlap and release events require a single overlapping region to be added or deleted. For insertion and deletion events, at most two new overlaps are created, or destroyed.
#### Characterization.
Before describing our algorithm, in the following lemmas we answer some basic questions about the behavior of ${\operatorname{area}}(\bhullp)$. imply that it seems not possible to restrict the number of candidate angles of maximum area. On the other hand, Lemma \[apps:area:lemma:events\] shows that the angle of maximum area is actually located at an event.
\[apps:area:lemma:max\_angle\] For any $\beta_0 \in (0,\pi)$ there exists a point set $P$ such that $$\max_{\beta} \operatorname{area}(\mathcal{O}_\beta\mathcal{H}(P))\neq\operatorname{area}(\mathcal{O}_{\beta_0}\mathcal{H}(P)).$$
Consider the coordinate system formed by $\bset[\beta_0]$. Place one point over the $y^+$-, $y^-$-, and $x^+$-semiaxes, and a point over the second quadrant (see ). From this position, note that ${\operatorname{area}}(\bhullp) = 0$ for any $\beta \leq \beta_0$ (), and there exists at least one $\beta_1 > \beta_0$ such that ${\operatorname{area}}(\bhullp[P][\beta_1]) \neq 0$ (). Hence $\beta_0$ cannot be the angle of maximum area.
ht
\[apps:area:lemma:bimodal\] For any $\beta_0,\beta_1 \in (0,\pi)$, there exists a point set $P$ for which ${\operatorname{area}}(\mathcal{O}_{\beta}(P))$ has local maxima in $\beta_0$ and $\beta_1$.
Let $\ell_0$ be a line with slope $\tan(\beta_0)$, $\ell_1$ a line with slope $\tan(\beta_1)$, and without loss of generality, let us assume that $\beta_0 < \beta_1$. We define $p_l, p_r,p_t,$ and $p_c$ to be the points located respectively, at the left corner, right corner, top corner, and the interior of the triangle bounded by the $x$-axis, $\ell_0$, and $\ell_1$. See .
Consider the angles $\beta_{lc}, \beta_{ct}$, and $\beta_{rc}$ as in . Note that $\beta_{lc} < \beta_{0} < \beta_{ct} < \beta_{1} < \beta_{rc}$. Using an increasing sweep from the initial increasing configuration the first release event is $\beta_{lc}$. From there, the area of $\bhullp$ is given by a parallelogram ${ \pgfpicture\pgfsetroundjoin
\pgftransformxslant{.6} \pgfpathrectangle{\pgfpointorigin}{\pgfpoint{.60em}{.65em}} \pgfusepath{stroke,} \endpgfpicture}_{lc}$ of constant height, so both the base of ${ \pgfpicture\pgfsetroundjoin
\pgftransformxslant{.6} \pgfpathrectangle{\pgfpointorigin}{\pgfpoint{.60em}{.65em}} \pgfusepath{stroke,} \endpgfpicture}_{lc}$ and the area of $\bhullp$ increase or decrease together as $\beta$ changes. As $\beta$ goes from $\beta_{lc}$ to $\beta_{0}$, the base of ${ \pgfpicture\pgfsetroundjoin
\pgftransformxslant{.6} \pgfpathrectangle{\pgfpointorigin}{\pgfpoint{.60em}{.65em}} \pgfusepath{stroke,} \endpgfpicture}_{lc}$ increases up to $\beta_0$, there exist a local maximum. The base of ${ \pgfpicture\pgfsetroundjoin
\pgftransformxslant{.6} \pgfpathrectangle{\pgfpointorigin}{\pgfpoint{.60em}{.65em}} \pgfusepath{stroke,} \endpgfpicture}_{lc}$ then decreases from $\beta_0$ to $\beta_{ct}$, to increase again from $\beta_{ct}$ to $\beta_{1}$. At $\beta_{1}$ there is a second local maximum, as the base of ${ \pgfpicture\pgfsetroundjoin
\pgftransformxslant{.6} \pgfpathrectangle{\pgfpointorigin}{\pgfpoint{.60em}{.65em}} \pgfusepath{stroke,} \endpgfpicture}_{lc}$ starts decreasing again after $\beta_{1}$ up to the last construction event at $\beta_{rc}$, where the area of $\bhullp$ is zero. See .
\[apps:area:lemma:events\] The area of $\bhullp$ reaches its maximum at values of $\beta$ belonging to the sequence of events.
Let us consider the area of $\bhullp$ given by . From , the area of $\bhullp$ can be rewritten as
$$\begin{aligned}
\label{apps:area:eqn:event_1}
{\operatorname{area}}(\bhullp) &= {\operatorname{area}}(\polygon)
- \sum_i {\operatorname{area}}(\triangles)
+ \sum_j {\operatorname{area}}(\parallelograms) \nonumber \\
&= {\operatorname{area}}(\polygon)
- \sum_i \left| a_i \pm b_i \cot(\beta) \right|
+ \sum_j \left| a_j \pm b_j \cot(\beta) \right|.
\end{aligned}$$
If we consider the different point configurations that define a triangle (see ), we can express $|a_i \pm b_i \cot(\beta)|$ as $a_i + b_i \cot(\beta)$ or $a_i - b_i \cot(\beta)$, according to the specific configuration. Thus, we have $$\begin{aligned}
\sum_i {\operatorname{area}}(\triangles) &= \sum_i |a_i \pm b_i\cot(\beta)| \\
&= \sum_{i_0} \left( a_{i_0} + b_{i_0}\cot(\beta) \right)
+ \sum_{i_1} \left( a_{i_1} - b_{i_1}\cot(\beta)\right)= a + b\cot(\beta).
\end{aligned}$$
It is possible to make a similar case-by-case analysis for the overlapping regions, to obtain from an expression with the form $c + d\cot(\beta)$. Within an interval between events $P$ does not change, and its area remains constant. Therefore, in an interval $[\beta_i,\beta_{i+1})$ we can rewrite:
$$\begin{aligned}
\label{apps:area:eqn:event_2}
{\operatorname{area}}(\bhullp) &= {\operatorname{area}}(\polygon)
- \sum_i |a_j \pm b_j \cot(\beta)|
+ \sum_j |a_j \pm b_j \cot(\beta)| \\
&= {\operatorname{area}}(\polygon)
- \left( a + b\cot(\beta) \right)
+ \left( c + d\cot(\beta) \right) \nonumber \\
&= {\operatorname{area}}(\polygon)
+ (c-a) +(d-b)\cot(\beta) \nonumber \\
\label{apps:area:eqn:event_3}
&= A + B\cot(\beta),
\end{aligned}$$
where $A$ and $B$ contain the sum of all constants from the terms in . Note that is monotone at any interval $[\beta_i,\beta_{i+1})$, as it is monotone in $(0,\pi)$. Depending on the particular values of $A$ and $B$, ${\operatorname{area}}(\bhullp)$ might be non-decreasing or non-increasing. Thus, the local maximum is given either by $\beta_i$ or $\beta_{i+1}$.
#### The search algorithm.
The algorithm to compute the angle of optimum area is outlined as follows.
1. \[apps:area:step\_1\]Traverse the sequence of events to identify the first release event $\beta_d$, and the last overlap event $\beta_c$. Restrict the sequence to start with $\beta_d$ and finish with $\beta_c$, so that $\mathcal{O}_{\beta}(P)$ has at least one connected component in every interval. Ignored events have no effect in the result, as they belong to an initial (increasing or decreasing) configuration, where $\operatorname{area}(\mathcal{O}_\beta\mathcal{H}(P))=0$.
2. \[apps:area:step\_2\] At the first interval, compute $\mathcal{O}_\beta\mathcal{H}(P)$ and using compute $\operatorname{area}(\mathcal{O}_\beta\mathcal{H}(P))$, keeping the angle $\beta_m$ of maximum area.
3. \[apps:area:step\_3\]Traverse the sequence of events. At each event:
1. \[apps:area:step\_3\_1\]Update the set of vertices and overlapping regions of $\bhullp$.
2. \[apps:area:step\_3\_2\]Handle each event updating as explained above.
3. \[apps:area:step\_3\_3\]Compute the local angle of maximum area. Replace $\beta_m$ only if the area of $\bhullp$ is improved.
There is a linear number of events in total, so step \[apps:area:step\_1\] requires $O(n)$ time. contains at most a linear number of terms, as there is at most a linear number of vertices and overlapping regions. Thus, from and previous discussions, step \[apps:area:step\_2\] requires $\Theta(n \log n)$ time and $O(n)$ space.
From Section \[sec:bhull:sweep\], the updates on step \[apps:area:step\_3\_1\] require logarithmic time. Every event results in a constant number of modifications to , as we described previously in this section. From we can obtain the angle of maximum area in constant time. As there is a linear number of events, step \[apps:area:step\_3\] requires a total of $O(n\log n)$ time. From this analysis we obtain the following Theorem, where the lower bound comes from the maintenance of $\bhullp$.
Computing the value(s) of $\beta \in (0, \pi)$ for which $\bhullp$ has maximum area, requires $\Theta(n \log n)$ time and $O(n)$ space.
Perimeter optimization. {#sec:apps:perimeter}
-----------------------
In this section we solve the following problem:
Given a set $P$ of $n$ points in the plane, compute the value of $\beta$ for which $\bhullp$ has maximum perimeter.
The perimeter of $\bhullp$ is given by $$\label{apps:perim:eqn:perim}
{\operatorname{perim}}(\bhullp) = \sum_i {\operatorname{perim}}(\angle_i(\beta))
- \sum_j {\operatorname{perim}}(\parallelograms)
- \sum_k {\operatorname{perim}}(\diagdown_k(\beta)),$$ where the $\angle_i(\beta)$ and the $\parallelograms$ denote the *steps* and parallelograms, respectively, defined by the staircases, and $\diagdown_k$ denotes one of the (at most four) *antennas* of $\bhullp$, that is, a segment of an $\bset$-staircase bounding a zero-area region of $\bhullp$. See again .
The same approach, and most of the arguments we used to maximize the area can be applied here. Following the same ideas, we will first analyze the computation and maintenance of , we then present adaptations of , and finalize outlining the search algorithm.
#### The steps $\angle_i(\beta)$.
Considering a top-right $\bset$-staircase (see again ), the perimeter of $\angle_i(\beta)$ is given by , where $p_i=(x_i,y_i)$ and $p_{i+1}=(x_{i+1},y_{i+1})$ are the points supporting the $i$-th step. Vertices over the staircase have non-decreasing $y$ coordinates, so $a_i$ is always positive. Event handling is done in the same way as we did with triangles in the previous section.
$$\begin{aligned}
\label{apps:perim:eqn:step}
{\operatorname{perim}}(\angle_i(\beta)) &= \left|
\left( y_{i+1}-y_i \right)\cot(\beta)
+ \left( y_{i+1}-y_i \right)\csc(\beta)
+ \left( x_i-x_{i+1} \right)
\right| \nonumber \\
&= \left|
a_i \left( \cot(\beta) + \csc(\beta) \right)
\pm b_i
\right|.\end{aligned}$$
#### The overlapping regions $\parallelograms$.
If we consider top-right and bottom-left $\bset$-stair-cases intersecting as shown in , the perimeter of an overlapping region is given by . The constants $c_j$ and $d_j$ are always positive. Event handling is done in the same way as we handled overlapping regions to optimize the area of $\bhullp$.
$$\begin{aligned}
\label{apps:perim:eqn:or}
{\operatorname{perim}}(\parallelograms) &= \left| 2(y_{i+1} - y_{k+1}) \cot(\beta)
+ 2(y_{k} - y_{i})\csc(\beta)
- (x_{i+1} - x_{k+1}) \right| \nonumber \\
&= \left| c_j \cot(\beta)
+ d_j \csc(\beta)
\pm e_j \right|.\end{aligned}$$
#### The antennas $\diagdown_k(\beta)$.
An antenna is a semistep at one of the extremes of an $\bset$-staircase. Just as steps and triangles, an antenna is defined by two consecutive $\bset$-maximal points. If we consider a top-right $\bset$-staircase, the perimeter of an antenna is given by if it is the first semistep of the staircase, and by if it is the last one (see ). In both equations we consider $p_i=(x_i,y_i)$ to be the point supporting the corresponding semistep. The constant $f_k$ is always positive.
$$\begin{aligned}
\label{apps:perim:eqn:antenna:1}
{\operatorname{perim}}_f(\diagdown_k) &= \left|
\left( y_{i+1}-y_i \right)\cot(\beta)
+ \left( x_i-x_{i+1} \right)
\right| \nonumber \\
&= \left|
f_k \cot(\beta) \pm g_k
\right|\\[1em]
\label{apps:perim:eqn:antenna:2}
{\operatorname{perim}}_l(\diagdown_k) &= \left( y_{i+1}-y_i \right)\csc(\beta) \nonumber \\
&= f_k \cot(\beta)\end{aligned}$$
Considering the case-by-case analysis we did in the previous section, we can rewrite Equations \[apps:perim:eqn:step\] to \[apps:perim:eqn:antenna:2\] as
$$\begin{aligned}
\label{apps:perim:eqn:antenna:3}
\sum_i {\operatorname{perim}}(\angle_i(\beta)) &= \sum_i \left|
a_i \cot(\beta) + a_i \csc(\beta)
\pm b_i
\right| \nonumber \\
&= \sum_{i_0}
a_{i_0} \cot(\beta) + a_{i_0} \csc(\beta) + b_{i_0}
+ \sum_{i_1}
a_{i_1} \cot(\beta) + a_{i_1} \csc(\beta) - b_{i_1} \nonumber \\
&= a \cot(\beta) + a \csc(\beta) + b\end{aligned}$$
$$\begin{aligned}
\label{apps:perim:eqn:antenna:4}
\sum_j {\operatorname{perim}}(\parallelograms) &= \sum_j \left|
c_j \cot(\beta)
+ d_j \csc(\beta)
\pm e_j \right| \nonumber \\
&= \sum_{j_0}
c_{j_0} \cot(\beta) + d_{j_0} \csc(\beta) + e_{j_0}
+ \sum_{j_1}
c_{j_1} \cot(\beta) + d_{j_1} \csc(\beta) - e_{j_1} \nonumber \\
&= c \cot(\beta) + d \csc(\beta) + e\end{aligned}$$
$$\begin{aligned}
\label{apps:perim:eqn:antenna:5}
\sum_k {\operatorname{perim}}_l(\diagdown_k) &= \left|
f_k \cot(\beta) \pm g_k
\right| \nonumber \\
&= \sum_{k_0}
f_{k_0} \cot(\beta) + g_{k_0} + \sum_{k_1}
f_{k_1} \cot(\beta) - g_{k_1} \nonumber \\
&= f \cot(\beta) + g,
\end{aligned}$$
and use to rewrite as
$$\begin{aligned}
\label{apps:perim:eqn:antenna:6}
{\operatorname{perim}}(\bhullp) &= \sum_i {\operatorname{perim}}(\angle_i(\beta))
- \sum_j {\operatorname{perim}}(\parallelograms)
- \sum_k {\operatorname{perim}}(\diagdown_k(\beta)) \nonumber \\
&= \left( a + c + f \right) \cot(\beta)
+ \left( a + d \right) \csc(\beta)
+ \left( b + e + g \right) \nonumber \\
&= A \cot(\beta) + B \csc(\beta) + C.
\end{aligned}$$
Note that all the constants in adding up to $A$ and $B$ are always positive, so $A,B > 0$. Moreover, within an interval there are at most four antennas, as they contain one of the left-most, right-most, top-most, and bottom-most points in $P$ (see again ). Therefore, the number of terms contributed by antennas to is constant and, except for $C$, they do not modify the original signs of any other term.
For simplicity, we will avoid antennas in the optimization of the perimeter of $\bhullp$, by using a version of not containing the term $\sum_k {\operatorname{perim}}(\diagdown_k(\beta))$. From the discussion above, both expressions have maxima at the same values of $\beta$.
#### Characterization.
We next answer questions about the behavior of ${\operatorname{perim}}(\bhullp)$, similar to the ones answered with in . Specifically, we show that the angle of maximum perimeter corresponds to an event () and, other than that, no restriction on the candidate angles seems to be possible (Lemmas \[apps:perim:lemma:max\_angle\] and \[apps:perim:lemma:bimodal\]).
\[apps:perim:lemma:max\_angle\] For any $\beta_0 \in (0,\pi)$, there exists a point set such that $$\max_{\beta} {\operatorname{perim}}(\bhullp)\neq{\operatorname{perim}}(\bhullp[P][\beta_0]).$$
Consider the coordinate system formed by $\bset[\beta_0]$. Place one point on the origin, and a point on the second and fourth quadrants (). As the set of points is monotone with respect of the $x$- and $y$-axes, $\bhullp[P][\beta_0] = P$. Therefore, ${\operatorname{perim}}(\bhullp[P][\beta_0])$ is equal to zero.
From this position, note that ${\operatorname{perim}}(\bhullp) = 0$ for any $\beta \leq \beta_0$ (), and there exists at least one $\beta_1 > \beta_0$ such that ${\operatorname{perim}}(\bhullp[P][\beta_1]) \neq 0$ (). Clearly, $\beta_0$ is not the angle of maximum perimeter.
\[apps:perim:lemma:bimodal\] For any $\beta_0,\beta_1 \in (0,\pi)$, there exists a point set $P$ for which ${\operatorname{area}}(\mathcal{O}_{\beta}(P))$ has local maxima in $\beta_0$ and $\beta_1$.
Let ${\triangle_t}$ be an acute triangle bounded by the $x$-axis, and two lines $\ell_{tl},\ell_{tr}$ with slopes $\tan({\beta_{tl}})$ and $\tan({\beta_{tr}})$, respectively. Without loss of generality, we assume that ${\beta_{tl}}< {\beta_{tr}}$, and the intersection point between $\ell_{tl}$ and $\ell_{tr}$ lies on the $y^+$-semiplane.
Let us consider the set $P' = \{ p_l,p_r,p_t \}$ of points located respectively, over the left, right, and top vertices of ${\triangle_t}$. Note that, at any starting position, the perimeter of $\bhullp[P']$ is constant and equal to the base of ${\triangle_t}$. Using an increasing sweep, from ${\beta_{tl}}$ to ${\beta_{tr}}$ the perimeter is formed additionally by a line segment $\ell_{t,b}$ joining $p_t$, and a point $p_b$ traversing the base of ${\triangle_t}$ from $p_l$ to $p_r$. During this interval, both $\ell_{t,b}$ and the perimeter of $\bhullp[P']$ increase or decrease together as $\beta$ changes. On this conditions, the perimeter of $\bhullp[P']$ has a local minimum on $\beta = \frac{\pi}{2}$ and thus, a local maximum on ${\beta_{tl}}$ (${\operatorname{perim}}(\bhullp[P'][{\beta_{tl}}]) >
{\operatorname{perim}}(\bhullp[P'][{\beta_{tl}}-\varepsilon])$), and a second local maximum on ${\beta_{tr}}$ (${\operatorname{perim}}(\bhullp[P'][{\beta_{tr}}]) > {\operatorname{perim}}(\bhullp[P'][{\beta_{tr}}+
\varepsilon])$). See .
Let ${\triangle_b}$ be a second acute triangle bounded by the $x$-axis, and a second pair of lines $\ell_{bl},\ell_{br}$ with slopes $\tan({\beta_{bl}})$ and $\tan({\beta_{br}})$ that pass through $p_l$ and $p_r$, respectively. The angles are such that ${\beta_{bl}}> {\beta_{br}}$, and the intersection point $p_b$ between $\ell_{bl}$ and $\ell_{br}$ lie on the $Y^-$ semiplane. Note that, if we add $p_b$ to the set $P'$, the arguments from the above discussion hold for both ${\triangle_t}$ and ${\triangle_b}$, so the perimeter of $\bhullp[P']$ has now local maxima on ${\beta_{tl}}, {\beta_{tr}}, {\beta_{bl}}$, and ${\beta_{br}}$. See .
![The angles with local maxima.[]{data-label="apps:perim:fig:bimodal_2"}](perim_thm_24)
Given the angles $\beta_0$ and $\beta_1$, construct the previous point set as explained. In this construction, ${\beta_{tl}}\leq \frac{\pi}{2} < {\beta_{tr}}$ and ${\beta_{bl}}> \frac{\pi}{2} \geq {\beta_{br}}$. Set two of ${\beta_{tl}}, {\beta_{tr}}, {\beta_{bl}}, {\beta_{br}}$ to the values of $\beta_0$ and $\beta_1$ appropriately, according to the cases
$\beta_0,\beta_1 < \frac{\pi}{2}$,
$\beta_0,\beta_1 > \frac{\pi}{2}$, and
$\beta_0 \leq \frac{\pi}{2} < \beta_1$ or viceversa
. The perimeter of $\bhullp[P']$ will have local maxima at $\beta_0$ and $\beta_1$.
\[apps:perim:lemma:events\] The perimeter of $\bhullp$ reaches its maximum at values of $\beta$ corresponding to sequence events.
From we know that the perimeter of $\bhullp$ is given by
$${\operatorname{perim}}(\bhullp) = A \cot(\beta) + B \csc(\beta) + C,$$
where $A,B \geq 0$. Looking for critical points in this expression, we arrive to $$\label{apps:perim:eqn:derivative}
\cos (\beta) = - \frac{A}{B},$$ where $\beta \neq 0,\pi$. By analyzing the possible roots in , we deal with the following cases:
1. $A > B$. There are no roots in this case, as $\frac{A}{B} > 1$. The length of the perimeter is monotonic in an interval between events.
2. $A = B$. There are again no roots in this case, as $\beta$ cannot be $0$ nor $\pi$. The length of the perimeter is again monotonic in an interval between events.
3. $A < B$. There is one root at $\beta = \cos^{-1}(-\frac{A}{B})$, as $A$ and $B$ are always positive and different from zero. In an interval between events we have one inflection point, so there are either two local maxima or two local minima, located at the endpoints of the interval.
#### The search algorithm.
We look for the maximum perimeter angles in the same way as we obtained the values for maximum area. We first compute the list of events, obtain the maximum perimeter angle for the first interval between events, and repeat the procedure for the remaining events. While traversing the event list, we update the optimum value angle only if the previous value is improved. A similar complexity analysis is also valid.
Computing the value(s) of $\beta \in (0, \pi)$ for which $\bhullp$ has maximum perimeter takes $O(n\log n)$ time and $O(n)$ space.
Concluding remarks {#sec:conclusions}
==================
We presented an algorithm to maintain the $\bhull$ of a planar point set while $\beta$ runs from $0$ to $\pi$ and extended this result to solve related optimization problems. We considered the maximization of the area and the perimeter of $\bhullp$, and presented a variation of the $2$-fitting problem studied in [@fitting_2011]. In our version, the fitting curve is an alternating polygonal chain with segments forming an angle $\beta$.
A natural extension of this work is to replace $\bset$ with a set $\mathcal{O}$ containing more than two lines. Different variations can be obtained by restricting the orientations and (or) the number of lines in $\mathcal{O}$. In particular, the characterization of the area and perimeter functions on each variation, seems an interesting and non-trivial problem.
As the Orthogonal Convex Hull, the $\bhull$ is suitable to be used as a separator or an enclosing shape. As it is always contained in the standard convex hull (and therefore, in several other traditional enclosing shapes), it is relevant in applications where the separator or enclosing shape is required to have minimum area. Finally, note that we can easily extend the results from to optimize the number of vertices of $\bhullp$, by keeping track of the vertex count at each interval between events. Without much effort, the approach and arguments from @alegria_2013 can be extended to $\bset$-convexity, and applied to problems related to containment relations between $\bhull$s of colored point sets.
The oriented $\left( 2,\beta \right)$-fitting problem {#sec:apps:fitting}
=====================================================
For $k \geq 1$, $\theta \in [0, \pi)$, and $\beta \in (0, \pi)$, a *$\cset$-polygonal chain with orientation $\theta$*, $\csetc$, is a chain with $2k - 1$ consecutive alternating links with slopes $\tan(\theta)$ and $\tan(\theta + \beta)$ such that the extreme links are half-lines with orientation $\tan(\theta)$. Let us define $\ell_{i,\beta}(\theta)$ as the line passing through $p_i \in P$ with slope $\tan(\theta + \beta)$. The *fitting distance* between $p_i$ and $\csetc$ is given by
$$d_f(p_i,\csetc) = \underset{p \in \ell_{i,\beta}(\theta) \cap
\csetc}{\text{min}} d(p_i,p),$$ where $d(p_i,p)$ represents the Euclidean distance between $p_i$ and $p$. The *error tolerance* of $\csetc$ with respect to $P$ is the maximum fitting distance between $\csetc$ and the elements in $P$, that is
$$\mu(\csetc,P) = \underset{p_i \in P} \max \hspace{0.3cm} d_f(p_i,\csetc).$$ The $\cset$-fitting problem for $P$ with the Min-Max criterion, consists on finding a polygonal chain $\csetc$ with minimum error tolerance $\mu(\csetc,P)$. See Figure \[apps:fitting:fig:fitting\].
\[apps:fitting:thm:2-fitting\] The $\cset[2][\frac{\pi}{2}]$-fitting problem can be solved in $\Theta(n \log n)$ time and $O(n)$ space.
We consider here the case where $\theta$ has a constant value, namely $0$, and we want to find the chain $\csetc[2][][0] = \mathcal{C}_{2,\beta}$ of optimal error tolerance. More formally, we solve the following problem.
Given a set $P$ of $n$ points in the plane, compute a polygonal chain $\mathcal{C}_{2,\beta}$ such that $\mu(\mathcal{C}_{2,\beta},P)$ has minimum value.
Consider the algorithm used in [@fitting_2011] to obtain the $O(n \log n)$ time bound for the $\cset[2][\frac{\pi}{2}]$-fitting problem used to prove . The $\bhull[\frac{\pi}{2}]$ of $P$ is used as a tool to solve the problem in $O(\log n)$ time for a fixed value of $\theta$ in a closed orientation interval $[\theta_i,\theta_{i+1}]$. An event sequence of a linear number of orientation intervals is created to maintain $\bhullp[P][\frac{\pi}{2}]$ as $\theta$ grows from $0$ to $2\pi$.
To solve Problem 3 we can follow exactly the same techniques. We refer the reader to reference [@fitting_2011] just to see the evident changes coming from the use of a different structure. More concretely, the structure $\bhullp[P][\frac{\pi}{2}]$ is replaced by $\bhullp$ which needs also a linear number of *interval events* $[\beta_i,\beta_{i+1}]$ to be maintained, and where the angular sweep is performed over $\beta$. Thus, Lemmas 3 and 4 in [@fitting_2011] can be now stated as follows:\
(i) *Given a value $\beta\in [\beta_i,\beta_{i+1}]$, an optimal solution of the $\cset[2]$-fitting problem for $\beta$ is defined by a line $\ell_{i,\beta}$ with slope $\tan(\beta)$ passing through a point $p_i$ of $P$ which gives the bipartition of $P$*.\
(ii) *The optimal solution of the $\cset[2]$-fitting problem for an interval event $[\beta_i,\beta_{i+1}]$ occurs either at an endpoint of the interval, i.e., at $\beta_i$ or $\beta_{i+1}$, or at a value $\beta_0 \in [\beta_i,\beta_{i+1}]$ when the left and right error tolerance are equal*.\
Using the properties (i),(ii) and following the maintenance of $\bhullp$, the problem is solved as follows:
1. Compute $\bhullp$ and the optimal error tolerance for the first interval between events.
2. Traverse the event sequence, obtaining the optimal error tolerance at each interval between events.
3. Update the previous solution only when it is improved.
Thus, the approach and arguments used in Theorem \[apps:fitting:thm:2-fitting\] hold in the case of the $\cset[2]$-fitting problem See Figure \[apps:fitting:fig:fitting\]. As a consequence, we get the following theorem.
\[apps:fitting:thm:fitting\] The $\cset[2]$-fitting problem can be solved in $O(n\log n)$ time and $O(n)$ space.
[^1]: Posgrado en Ciencia e Ingeniería de la Computación, Universidad Nacional Autónoma de México, [ alegria\[email protected]]{}. Research supported by H2020-MSCA-RISE project 73499 - CONNECT.
[^2]: Departamento de Física y Matemáticas, Universidad de Alcalá, Spain, [[email protected]]{}. Research supported by MINECO Projects MTM2014-54207 and TIN2014-61627-EXP, TIGRE5-CM Comunidad de Madrid Project S2013/ICE-2919, and H2020-MSCA-RISE project 73499 - CONNECT.
[^3]: Departament de Matemàtiques, Universitat Politècnica de Catalunya, Spain, [ [email protected]]{}. Research supported by projects Gen. Cat. DGR 2014SGR46, MINECO MTM2015-63791-R, and H2020-MSCA-RISE project 73499 - CONNECT.
[^4]: Instituto de Matemáticas, Universidad Nacional Autónoma de México, [[email protected]]{}. Research supported by SEP-CONACYT 80268, PAPPIIT IN102117 Programa de Apoyo a la Investigación e Innovación Tecnológica UNAM, and H2020-MSCA-RISE project 73499 - CONNECT.
[^5]: In memorial of professor Ferran Hurtado, inspirational friend and colleague, acknowledging his key contribution to the development of Computational Geometry.
|
---
abstract: 'We consider a linearised model of incompressible inviscid flow. Using a regularisation based on the Hodge Laplacian we prove existence and uniqueness of weak solutions for smooth domains. The model problem is then discretised using H(div)-conforming finite element methods, for which we prove error estimates for the velocity approximation in the $L^2$-norm of order $O(h^{k+\frac12})$. We also prove error estimates for the pressure error in the $L^2$-norm.'
address:
- 'Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow, G1 1XH United Kingdom'
- 'Department of Mathematics, University College London, London, UK-WC1E 6BT, United Kingdom'
- 'Division of Applied Mathematics Brown University Box F 182 George Street Providence, RI 02912'
author:
- Gabriel Barrenechea
- Erik Burman
- Johnny Guzman
bibliography:
- 'references.bib'
title: 'Well-posedness and H(div)-conforming finite element approximation of a linearised model for inviscid incompressible flow'
---
Introduction
============
The use of H(div)-conforming finite element methods for the approximation of incompressible flow at high Reynolds number has been receiving increasing attention from the research community recently [@GSS17; @NC18; @SL18]. By construction such methods can satisfy the divergence-free condition exactly. The lack of $H^1$-conformity is handled using techniques drawing on ideas from discontinuous Galerkin methods [@GRW05], resulting in several possible different choices for the discretisation of the transport term and the viscous term. For the former one may either design an energy conserving method using central fluxes, or one may opt for a dissipative alternative in the form of upwind fluxes. The latter were shown in [@GSS17] to be more robust than the former, as is the case for discontinuous Galerkin (DG) methods. For DG-methods applied to scalar problems it is well known that thanks to the dissipative properties of the upwind flux one may prove an error estimate in the $L^2$-norm, of the form (see, e.g., [@JP86]) $$\label{eq:L2error}
\|u - u_h\|_{L^2(\Omega)} \leq C h^{k+\frac12} |u|_{H^{k+1}(\Omega)},$$ where $u$ is the exact solution, $u_h$ its DG-approximation, $\Omega
\subset \mathbb{R}^d$, $d=2,3$. is the computational domain, $h$ the mesh parameter, and finally $k$ the polynomial degree of the approximation space. On special meshes one can in fact prove optimal estimates with rate $h^{k+1}$ for upwind DG methods applied to scalar problems [@cockburn2008optimal; @richter1988optimal]. However, as it is shown in [@peterson1991note], the result is sharp on general meshes.
Estimates of the type are also the best that are known for either stabilised conforming finite element approximations, or fully DG methods, of laminar solutions of the Navier-Stokes’ equations in the high Reynolds number regime [@BF07; @HS90], or the incompressible Euler equations [@JS86; @Bu15]. The robustness of the H(div)-conforming elements in the case of vanishing viscosity was shown in [@KS11] for the case of the Brinkman problem, i.e. without the convection terms. Despite all the work quoted above, there seems to be no proof of an error estimate of the type for finite element methods using H(div)-conforming elements applied to incompressible flow problems (see the discussion in [@SL18; @NC18]).
The purpose of this work is to fill the gap mentioned in the last paragraph. That is, proving an estimate of the type for finite element methods approximating a stationary linearised model of inviscid flow and using H(div)-conforming approximation spaces for the velocity approximation. Both the spaces designed by Raviart and Thomas [@RT] and by Brezzi, Douglas and Marini [@BDM] enter the framework. As stabilising fluxes, these need to be either upwind, or, in case of central fluxes, an additional penalty term on the jump of the tangential component of the velocity needs to be added. In the particular case in which the velocity is approximated using the Raviart-Thomas space we also prove a convergence result for the pressure error, showing that the approximate pressure converges to the exact pressure in the $L^2$-norm also with the rate $O(h^{k+\frac12})$. For the BDM space the rate $O(h^{k+\frac12})$ is obtained for the projection of the error onto the pressure space, but since in this case the pressure space is of polynomial degree $k-1$, this is a superconvergence result.
Linear model problem {#sec:linear}
--------------------
To keep the discussion as simple as possible we consider the following linear model problem.
Find a velocity ${{\boldsymbol{u}}}$ and a pressure $p$ satisfying
\[pde\] $$\begin{aligned}
{2}
{{\ensuremath\mathop{\mathrm{div}\,}}}({{\boldsymbol{u}}}\otimes {{\bm \beta}}) + \sigma {{\boldsymbol{u}}}+ \nabla p= &{{\boldsymbol{f}}}\quad && \text{ in } \Omega\,, \\
{{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{u}}}=&0 \quad && \text{ in } \Omega\,, \\
{{\boldsymbol{u}}}\cdot {{\boldsymbol{n}}}=& 0 \quad && \text{ on } \Gamma.\end{aligned}$$
We think of ${{\boldsymbol{u}}}$ and ${{\bm \beta}}$ as column vectors and we set ${{\boldsymbol{u}}}\otimes {{\bm \beta}}= {{\boldsymbol{u}}}{{\bm \beta}}^t$. We assume that ${{\ensuremath\mathop{\mathrm{div}\,}}}{{\bm \beta}}=0$ and that $\sigma\in L^{\infty}(\Omega)$ with $\sigma({{\boldsymbol{x}}})\ge \sigma_0^{} > 0$ almost everywhere in $\Omega$. We assume that ${{\bm \beta}}\cdot {{\boldsymbol{n}}}=0$ on $\Gamma$. In spite of it being the natural candidate for a model problem for the development and analysis of numerical methods for inviscid flow this model does not seem to have been considered in the literature. Below we will first discuss the flow modelling leading to the system .
To obtain the stationary linear model problem from the incompressible Euler equations, assume that a stationary solution to the latter ${{\bm \beta}}$, is subject to a smooth, exponentially growing perturbation of the right hand side of the momentum equation of the form: $$\tilde {{{\boldsymbol{f}}}}({{\boldsymbol{x}}},t) := {{{\boldsymbol{f}}}}({{\boldsymbol{x}}}) \exp (\sigma t), \quad \sigma \in
\mathbb{R}\setminus 0.$$ Writing the perturbed solution ${{\bm \beta}}+ \tilde {{\boldsymbol{u}}}$ where $\tilde {{\boldsymbol{u}}}(x,t)$ is the perturbation resulting from the pertubation of the right hand side and neglecting quadratic terms in the perturbation $\tilde {{\boldsymbol{u}}}$, we may write the linearised momentum equation $$\label{eq:momentum}
\partial_t \tilde {{\boldsymbol{u}}}+ {{\ensuremath\mathop{\mathrm{div}\,}}}(\tilde {{\boldsymbol{u}}}\otimes {{\bm \beta}}) + {{\ensuremath\mathop{\mathrm{div}\,}}}({{\bm \beta}}\otimes \tilde{{\boldsymbol{u}}}) + \nabla \tilde p
=\tilde {{{\boldsymbol{f}}}}({{\boldsymbol{x}}},t).$$ With the above choice of perturbation we may write the solution on the separated form $$\tilde {{\boldsymbol{u}}}({{\boldsymbol{x}}},t) = {{\boldsymbol{u}}}({{\boldsymbol{x}}}) \exp(\sigma t).$$ Injecting this expression in we arrive at the following stationary form for the space varying part of the perturbation $$\label{eq:stat_momentum}
\sigma {{\boldsymbol{u}}}+ {{\ensuremath\mathop{\mathrm{div}\,}}}({{\boldsymbol{u}}}\otimes {{\bm \beta}}) + {{\ensuremath\mathop{\mathrm{div}\,}}}({{\bm \beta}}\otimes {{\boldsymbol{u}}}) +\nabla p = {{{\boldsymbol{f}}}}({{\boldsymbol{x}}}).$$ To further simplify the model problem we finally drop the second term in the left hand side of . Since ${{\ensuremath\mathop{\mathrm{div}\,}}}({{\bm \beta}}\otimes {{\boldsymbol{u}}}) = {{\boldsymbol{u}}}\cdot \nabla {{\bm \beta}}$, this is a non-essential term which can be absorbed in the reaction term under suitable assumptions on $\sigma$ and ${{\bm \beta}}$.
It is easy to construct solutions to the system . Examples of such solutions in the unit square are
1. x-independent solution.\
Let ${{\bm \beta}}\cdot {{\boldsymbol{n}}}= 0$ on $y=0$ and $y=1$ and ${{\bm \beta}}$ is defined to be periodic at $x=0$ and $x=1$. Then for any function $\varphi:{\mathbb{R}}\mapsto {\mathbb{R}}$, $\varphi \in [C^1({\mathbb{R}})]^2$ a solution is given by: $${{\bm \beta}}:= \left(
\begin{array}{c}
\varphi(y) \\
0
\end{array}
\right).$$ The associated pressure is $p=0$.
2. Stationary vortex sheet.\
Let ${{\bm \beta}}\cdot {{\boldsymbol{n}}}= 0$ on the boundaries of the square and define the streamfunction $\varphi
(x,y):= \sin (n \pi x) \sin (n \pi y)$, corresponding to the vorticity $\omega := \Delta \varphi = - 2 n^2 \pi^2
\sin (n \pi x) \sin (n \pi y) = -2 n^2 \pi^2 \varphi$ with $n$ a positive integer. Then define: $$\label{eq:vel_vortex}
{{\bm \beta}}:= \left(
\begin{array}{c}
\partial_y \varphi(x,y) \\
-\partial_x \varphi(x,y)
\end{array}
\right).$$ Since ${{\bm \beta}}\cdot \nabla \omega = -2 n^2 \pi^2 (\partial_y
\varphi(x,y) \partial_x \varphi(x,y)- \partial_x \varphi(x,y) \partial_y
\varphi(x,y)) = 0$ we see that ${{\bm \beta}}$ is a solution to the two-dimensional stationary equations of inviscid flow. It is straightforward to verify that the velocity pressure formulation is satisfied for the pressure, $$\label{eq:press_vortex}
p = n^2 \pi^2 (\cos^2 (n \pi x)-\sin^2 (n \pi y))/2.$$
In both examples (1) and (2) we achieve a problem on the form by taking ${{{\boldsymbol{f}}}}= \sigma
{{\bm \beta}}$ and the solution is then ${{\boldsymbol{u}}}= {{\bm \beta}}$.
Outline of paper
----------------
We prove existence of solutions of the model problem and uniqueness for $\sigma$ large enough, on smooth domains, in section \[sec:wp\]. The H(div)-conforming upwind finite element methods are introduced and analysed in section \[sec:upw\]. Finally in section \[sec:num\] we illustrate the theory by computing approximations to the example (2) above.
Notation and preliminary results
================================
The partial differential equation will be posed on an open polyhedral domain $\Omega\subseteq\mathbb{R}^d, d=2,3$ with Lipschitz boundary $\Gamma$. For some of the theoretical results we will assume a smoother boundary. We adopt standard notation for Sobolev and Lebesgue spaces. In particular, for $D\subset\Omega$ we denote by $(\cdot,\cdot)_D^{}$ the $L^2(D)$ inner product (without making a distinction between scalar and vector and tensor-valued functions). For $D=\Omega$ we drop the subindex in the above notation. The norm in $L^2(D)$ will be denoted by $\|\cdot\|_D^{}$. By $W^{m,p}(D), m\ge 0, 1\le p\le \infty$ we will denote the functions in $L^p(D)$, with distributional derivatives up to order $m$ belonging to $L^p(D)$, with norm (seminorm) $\|\cdot\|_{m,p,D}^{}$ ($|\cdot|_{m,p,D}^{}$). For $p=2$ we denote $H^m(D)=W^{m,2}(D)$, and the corresponding norm is denoted $\|\cdot\|_{m,D}$. As usual, $H^m_0(D)$ denotes the closure of $C^\infty_0(D)$ in the $\|\cdot\|_{m,D}^{}$-norm. We also denote by $L^2_0(D)$ the space of $L^2(D)$ functions with zero mean value in $D$. All spaces for vector-valued functions will be denoted by boldface notation, e.g., ${{\boldsymbol{H}}}^1(D)=[H^1(D)]^d$, hence we denote by ${{\boldsymbol{H}}}({{\ensuremath\mathop{\mathrm{div}\,}}},D)$ the space of ${{\boldsymbol{L}}}^2(D)$ functions with distributional divergence in ${{\boldsymbol{L}}}^2(D)$, ${{\boldsymbol{H}}}_0^{}({{\ensuremath\mathop{\mathrm{div}\,}}},D)=\{{{\boldsymbol{v}}}\in {{\boldsymbol{H}}}({{\ensuremath\mathop{\mathrm{div}\,}}},D):{{\boldsymbol{v}}}\cdot{{\boldsymbol{n}}}=0\;\textrm{on}\;\partial D\}$, and ${{\boldsymbol{H}}}({{\ensuremath\mathop{\mathrm{curl}\,}}},D)$ denotes the space of ${{\boldsymbol{L}}}^2(D)$ functions with distribution curl in ${{\boldsymbol{L}}}^2(D)$.
Below we will make use of the following preliminary result (for its proof, see, e.g., [@girault2012finite]).
\[infsup\] There exists a constant $C>0$ such that for every $q \in L_0^2(\Omega)$ there exists ${{\boldsymbol{v}}}\in {{\boldsymbol{H}}}_0^1(\Omega)$ satisfying $${{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{v}}}=q \qquad \text{ in } \Omega,$$ $$\|\nabla {{\boldsymbol{v}}}\|_{\Omega} \le C \|q\|_{\Omega}.$$
Also in [@girault2012finite] the proof of the following result can be found.
The following bound holds $$\label{l2}
\|{{\boldsymbol{v}}}\|_{{\Omega}} \le C (\| {{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{v}}}\|_{{\Omega}} +\| {{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{v}}}\|_{{\Omega}}) \qquad \forall {{\boldsymbol{v}}}\in {{\boldsymbol{H}}}_0({{\ensuremath\mathop{\mathrm{div}\,}}},\Omega) \cap {{\boldsymbol{H}}}({{\ensuremath\mathop{\mathrm{curl}\,}}},\Omega).$$ If we assume that $\partial {\Omega}$ is $C^{1,1}$ $$\label{h1}
\|\nabla {{\boldsymbol{v}}}\|_{{\Omega}} \le K(\| {{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{v}}}\|_{{\Omega}} +\| {{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{v}}}\|_{{\Omega}}) \qquad \forall {{\boldsymbol{v}}}\in {{\boldsymbol{H}}}_0({{\ensuremath\mathop{\mathrm{div}\,}}},\Omega) \cap {{\boldsymbol{H}}}({{\ensuremath\mathop{\mathrm{curl}\,}}},\Omega).$$ Finally, if ${\Omega}$ is a convex Lipschitz polyhedron [@ABDG98], or a convex more regular domain, then $$\label{h1convex}
\|\nabla {{\boldsymbol{v}}}\|_{{\Omega}}^2 \le \| {{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{v}}}\|_{{\Omega}}^2 +\| {{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{v}}}\|_{{\Omega}}^2 \qquad \forall {{\boldsymbol{v}}}\in {{\boldsymbol{H}}}_0({{\ensuremath\mathop{\mathrm{div}\,}}},\Omega) \cap {{\boldsymbol{H}}}({{\ensuremath\mathop{\mathrm{curl}\,}}},\Omega).$$
Finally, for two $3 \times 3$ matrices $A$ and $B$ with rows $A_i$ and $B_i$ ($i=1,2,3$) we define $C:=A \times B$ with $C_1= A_2 \cdot B_3-A_3 \cdot B_2$, $C_2= -(A_1 \cdot B_3-A_3 \cdot B_1)$ $C_3= A_1 \cdot B_2-A_2 \cdot B_1)$, and a simple calculation gives the following identity.
\[derivative-identity\] It holds $${{\ensuremath\mathop{\mathrm{curl}\,}}}( {{\bm \beta}}\cdot \nabla {{\boldsymbol{v}}})= {{\bm \beta}}\cdot \nabla ({{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{v}}})+ ((\nabla {{\bm \beta}})^t \times \nabla {{\boldsymbol{v}}}) .$$
Well-Posedness of the model problem {#sec:wp}
===================================
It appears that the linear inviscid model has not been analysed mathematically. Hence, will here first study its well-posedness before proceeding with the finite element analysis. Transport problems have been studied by several authors (e.g. [@girault2010lp; @fichera1963unified; @da1986stationary]). However, the incompressibility constraint seems to add new challenges to the analysis and we cannot apply the techniques of the above mentioned papers directly. The weak formulation of is given by:
Find ${{\boldsymbol{u}}}\in {{\boldsymbol{H}}}_0({{\ensuremath\mathop{\mathrm{div}\,}}},\Omega)$ and $p \in L_0^2(\Omega)$ that satisfy
\[weak1\] $$\begin{aligned}
{2}
-({{\boldsymbol{u}}}, {{\bm \beta}}\cdot \nabla {{\boldsymbol{v}}})+(\sigma {{\boldsymbol{u}}}, {{\boldsymbol{v}}})-(p, {{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{v}}})=&({{\boldsymbol{f}}}, {{\boldsymbol{v}}}) \quad &&\text{ for all } {{\boldsymbol{v}}}\in {{\boldsymbol{H}}}^1(\Omega) \cap {{\boldsymbol{H}}}_0({{\ensuremath\mathop{\mathrm{div}\,}}},\Omega), \label{weak1_1}\\
({{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{u}}}, q)=&0 \quad && \text{ for all } q \in L_0^2(\Omega) \label{weak1_2}.\end{aligned}$$
Existence of weak solutions
----------------------------
In order to prove existence of the problem we will regularize it. Consider the following problem: Find a velocity ${{\boldsymbol{u}}}_{{\varepsilon}}$ and a pressure $p_{{\varepsilon}}$ satisfying
\[pdeeps\] $$\begin{aligned}
{2}
-{\varepsilon}\Delta {{\boldsymbol{u}}}_{{\varepsilon}}+ {{\ensuremath\mathop{\mathrm{div}\,}}}({{\boldsymbol{u}}}_{\varepsilon}\otimes {{\bm \beta}}) + \sigma {{\boldsymbol{u}}}_{{\varepsilon}}+ \nabla p_{{\varepsilon}}= &{{\boldsymbol{f}}}\quad && \text{ in } \Omega\,, \\
{{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}=&0 \quad && \text{ in } \Omega\,, \\
{{\boldsymbol{u}}}_{{\varepsilon}}=& 0 \qquad && \text{ on } \Gamma.\end{aligned}$$
The weak formulation of is as follows: Find $({{\boldsymbol{u}}}_{{\varepsilon}},p_{{\varepsilon}}) \in {{\boldsymbol{H}}}_0^1(\Omega)\times L_0^2(\Omega)$ such that
\[weak1eps-Oseen\] $$\begin{aligned}
{2}
{\varepsilon}(\nabla {{\boldsymbol{u}}}_{{\varepsilon}}, \nabla {{\boldsymbol{v}}})-({{\boldsymbol{u}}}_{{\varepsilon}}, {{\bm \beta}}\cdot \nabla {{\boldsymbol{v}}})+(\sigma {{\boldsymbol{u}}}_{{\varepsilon}}, {{\boldsymbol{v}}})-(p_{{\varepsilon}}, {{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{v}}})=&({{\boldsymbol{f}}}, {{\boldsymbol{v}}}) \quad &&\text{ for all } {{\boldsymbol{v}}}\in {{\boldsymbol{H}}}_0^1(\Omega), \label{weak1eps1-0}\\
({{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}, q)=&0 \quad && \text{ for all } q \in L_0^2(\Omega)\,.\end{aligned}$$
\[Lem2.1\] There exists a unique solution ${{\boldsymbol{u}}}_{{\varepsilon}} \in {{\boldsymbol{H}}}_0^1(\Omega)$ and $p_{{\varepsilon}} \in L_0^2(\Omega)$ to the problem . In addition, if ${{\bm \beta}}\in {{\boldsymbol{L}}}^\infty(\Omega)$, then the following bound holds $$\label{bound-Oseen}
\|p_{{\varepsilon}}\|_{\Omega}+ \|\sqrt{\sigma}\,{{\boldsymbol{u}}}_{{\varepsilon}}\|_{\Omega} +\sqrt{{\varepsilon}} \| \nabla {{\boldsymbol{u}}}_{{\varepsilon}}\|_{\Omega} \le C\, \|{{\boldsymbol{f}}}\|_{\Omega}\,,$$ where the constant $C$ depends on $\sigma$ and $\|{{\bm \beta}}\|_{\infty,\Omega}^{}$, but not on negative powers of $\varepsilon$.
Existence and uniquness of a solution of follows from the Babuska-Brezzi theory [@BBF13]. Testing the equation with ${{\boldsymbol{u}}}_{{\varepsilon}}$ we get $${\varepsilon}\|\nabla {{\boldsymbol{u}}}_{{\varepsilon}}\|_{\Omega}^2 +\|\sqrt{\sigma}\,{{\boldsymbol{u}}}_{{\varepsilon}}\|_{\Omega}^2 =({{\boldsymbol{f}}}, {{\boldsymbol{u}}}_{{\varepsilon}}).$$ Therefore, we have the bound $$\label{bound-1}
{\varepsilon}\|\nabla {{\boldsymbol{u}}}_{{\varepsilon}}\|_{\Omega}^2 + \frac{1}{2} \|\sqrt{\sigma}\,{{\boldsymbol{u}}}_{{\varepsilon}}\|_{\Omega}^2 \le \frac{1}{2\sigma_0^{}} \|{{\boldsymbol{f}}}\|_{\Omega}^2.$$ Moreover, using Lemma \[infsup\] and we have that $$\|p_{{\varepsilon}}\|_{\Omega} \le C \,\Big({\varepsilon}\|\nabla {{\boldsymbol{u}}}_{{\varepsilon}}\|_{\Omega}+ \|{{\bm \beta}}\|_{\infty,\Omega}^{} \|{{\boldsymbol{u}}}_{{\varepsilon}}\|_{\Omega}
+ \|\sqrt{\sigma}\|_{\infty,\Omega}\,\|\sqrt{\sigma}\,{{\boldsymbol{u}}}_{{\varepsilon}}\|_\Omega+\|{{\boldsymbol{f}}}\|_\Omega
\Big)\,,$$ and the proof is finished using .
There exists a solution ${{\boldsymbol{u}}}\in {{\boldsymbol{L}}}^2(\Omega)$ and $p \in L^2(\Omega)$ to .
Since $\{{{\boldsymbol{u}}}_{{\varepsilon}}\}$ and $\{p_{{\varepsilon}}\}$ are uniformly bounded in ${{\boldsymbol{H}}}_0^{}({{\ensuremath\mathop{\mathrm{div}\,}}},\Omega)$ and $L^2_0(\Omega)$, respectively, there exists a subsequence such that ${{\boldsymbol{u}}}_{{\varepsilon}} \rightharpoonup {{\boldsymbol{u}}}$ and $p_{{\varepsilon}} \rightharpoonup p$ with ${{\boldsymbol{u}}}\in {{\boldsymbol{H}}}_0^{}({{\ensuremath\mathop{\mathrm{div}\,}}},\Omega)$ and $p \in L_0^2(\Omega)$. Moreover, since ${{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{u}}}_{\varepsilon}=0$, for all $\phi \in H_0^1(\Omega)$ we have $({{\boldsymbol{u}}}, \nabla \phi) =\lim_{{\varepsilon}\rightarrow 0} ({{\boldsymbol{u}}}_{{\varepsilon}}, \nabla \phi) =\lim_{{\varepsilon}\rightarrow 0} -({{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}, \phi) =0$, thus showing that ${{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{u}}}=0$ in $\Omega$. We then see that from and the fact that ${\varepsilon}\|\nabla {{\boldsymbol{u}}}_{{\varepsilon}}\|_{\Omega} \rightarrow 0 $ as ${\varepsilon}\to 0$ that ${{\boldsymbol{u}}}$ and $p$ satisfy .
Uniqueness of weak solutions
----------------------------
In general we cannot prove uniqueness of weak solutions . However, we will be to prove existence and uniqueness of solutions in the space ${{\boldsymbol{H}}}^1(\Omega) \cap {{\boldsymbol{H}}}_0({{\ensuremath\mathop{\mathrm{div}\,}}},\Omega)$ by making more stringent requirements on the coefficients and the boundary $\Gamma$. To achieve this goal, it is necessary to introduce a different regularised (as compared to ) problem to prove existence of smoother solutions to . The idea consists in considering the folllowing regularised Hodge-Oseen problem: Find a velocity ${{\boldsymbol{u}}}_{{\varepsilon}}$ and a pressure $p_{{\varepsilon}}$ satisfying
\[pdeeps2\] $$\begin{aligned}
{2}
{\varepsilon}\, {{\ensuremath\mathop{\mathrm{curl}\,}}}{{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}+ {{\ensuremath\mathop{\mathrm{div}\,}}}({{\boldsymbol{u}}}_{\varepsilon}\otimes {{\bm \beta}}) + \sigma {{\boldsymbol{u}}}_{{\varepsilon}}+ \nabla p_{{\varepsilon}}= &{{\boldsymbol{f}}}\quad && \text{ in } \Omega\,, \\
{{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}=&0 \quad && \text{ in } \Omega\,, \\
{{\boldsymbol{u}}}_{{\varepsilon}}\cdot {{\boldsymbol{n}}}=& 0 \qquad && \text{ on } \Gamma, \\
{{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}} \times {{\boldsymbol{n}}}=& 0 \qquad && \text{ on } \Gamma.\end{aligned}$$
The weak formulation of reads as follows: Find ${{\boldsymbol{u}}}_{{\varepsilon}} \in {{\boldsymbol{V}}}:= {{\boldsymbol{H}}}^1(\Omega) \cap {{\boldsymbol{H}}}_0({{\ensuremath\mathop{\mathrm{div}\,}}},\Omega)$ and $p_{{\varepsilon}} \in L_0^2(\Omega)$ that satisfy
\[weak1eps\] $$\begin{aligned}
{2}
{\varepsilon}({{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}, {{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{v}}})-({{\boldsymbol{u}}}_{{\varepsilon}}, {{\bm \beta}}\cdot \nabla {{\boldsymbol{v}}})+(\sigma {{\boldsymbol{u}}}_{{\varepsilon}}, {{\boldsymbol{v}}})-(p_{{\varepsilon}}, {{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{v}}})=&({{\boldsymbol{f}}}, {{\boldsymbol{v}}}) \quad &&\text{ for all } {{\boldsymbol{v}}}\in {{\boldsymbol{V}}}, \label{weak1eps1}\\
({{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}, q)=&0 \quad && \text{ for all } q \in L_0^2(\Omega).\end{aligned}$$
\[regcurl\] Assume that ${{\boldsymbol{f}}}\in {{\boldsymbol{L}}}^2({\Omega})$ and that $\Gamma$ is $C^{1,1}$, or $\Omega$ is a convex Lipschitz polyhedron. Then, there exists a unique solution of . In addition, it satisfies $$\label{213}
\sqrt{{\varepsilon}} \|{{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}\|_{{\Omega}}+ \|\sqrt{\sigma}\,{{\boldsymbol{u}}}_{{\varepsilon}}\|_{{\Omega}}+ \|p_{{\varepsilon}}\|_{{\Omega}} \le C \|{{\boldsymbol{f}}}\|_{\Omega}.$$ Moreover, suppose that ${{\boldsymbol{f}}}\in {{\boldsymbol{H}}}^1({\Omega}), {{\bm \beta}}\in
\boldsymbol{W}^{1,\infty}(\Omega)$, $\sigma \in W^{1,\infty}(\Omega)$ and $\Gamma$ is $C^{3}$. If $\Omega$ is convex, let $\mathcal{C}= \|\nabla {{\bm \beta}}\|_{L^\infty(\Omega)}$, or otherwise $\mathcal{C}= K \|\nabla {{\bm \beta}}\|_{\infty,\Omega}$ where $K$ is from . Then, assuming $\sigma_0^{} > \mathcal{C}$ we have $$\| {{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}\|_{{\Omega}} \le C\,\| {{\boldsymbol{f}}}\|_{{{\ensuremath\mathop{\mathrm{curl}\,}}},{\Omega}}\,,$$ where $C>0$ depends on $\sigma, {{\bm \beta}}$, and $K$, but not on negative powers of ${\varepsilon}$.
The existence and uniqueness of this solution follows from the Babuska-Brezzi theory [@BBF13] by noting that as proven in [@girault2012finite], the norm in ${{\boldsymbol{H}}}^1(\Omega)$ is equivalent to the one in ${{\boldsymbol{H}}}({{\ensuremath\mathop{\mathrm{curl}\,}}},\Omega) \cap {{\boldsymbol{H}}}_0({{\ensuremath\mathop{\mathrm{div}\,}}},\Omega)$, thanks to the hypotheses on $\Gamma$. The bound follows taking ${{\boldsymbol{v}}}={{\boldsymbol{u}}}_{\varepsilon}$ in , and the inf-sup conditions provides the stability for $p_{{\varepsilon}}$.
Next, whenever we suppose that $\Gamma$ is of class $C^3$ and ${{\boldsymbol{f}}}\in {{\boldsymbol{H}}}^1(\Omega)$, using the results in [@sil2017regularity] (see Thereom 12 and Remark 16) we have the regularity ${{\boldsymbol{u}}}_{{\varepsilon}} \in {{\boldsymbol{H}}}^3(\Omega)$ and $p_{{\varepsilon}} \in H^2(\Omega)$. Noting that ${{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}} \times {{\boldsymbol{n}}}=
0$ on $\Gamma$ it follows that ${{\ensuremath\mathop{\mathrm{curl}\,}}}{{\ensuremath\mathop{\mathrm{curl}\,}}}({{\boldsymbol{u}}}_{{\varepsilon}}) \cdot {{\boldsymbol{n}}}= 0$, so, $\tilde{{{\boldsymbol{v}}}}:= {{\ensuremath\mathop{\mathrm{curl}\,}}}{{\ensuremath\mathop{\mathrm{curl}\,}}}({{\boldsymbol{u}}}_{{\varepsilon}}) \in {{\boldsymbol{H}}}^1(\Omega) \cap {{\boldsymbol{H}}}_0({{\ensuremath\mathop{\mathrm{div}\,}}},\Omega)$, and then it is a valid test function to be used in . Thus, taking $\tilde{{{\boldsymbol{v}}}}$ as test function in and integrating by parts we obtain $$\label{uniq-1}
{\varepsilon}\|{{\ensuremath\mathop{\mathrm{curl}\,}}}{{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}\|_{{\Omega}}^2 -({{\boldsymbol{u}}}_{{\varepsilon}}, {{\bm \beta}}\cdot \nabla ({{\ensuremath\mathop{\mathrm{curl}\,}}}{{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}))+
\|\sqrt{\sigma}\,{{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}\|_{{\Omega}}^2 + (\nabla\sigma\times {{\boldsymbol{u}}}_{\varepsilon}, {{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{\varepsilon}) =({{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{f}}}, {{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}})\,.$$ The second term in the left can be written as $$-({{\boldsymbol{u}}}_{{\varepsilon}}, {{\bm \beta}}\cdot \nabla ({{\ensuremath\mathop{\mathrm{curl}\,}}}{{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}))=({{\bm \beta}}\cdot \nabla{{\boldsymbol{u}}}_{{\varepsilon}}, {{\ensuremath\mathop{\mathrm{curl}\,}}}{{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}})=({{\ensuremath\mathop{\mathrm{curl}\,}}}({{\bm \beta}}\cdot \nabla {{\boldsymbol{u}}}_{{\varepsilon}}), {{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}})\,.$$ However, using Lemma \[derivative-identity\] and the antisymmetry of the convective term $$({{\ensuremath\mathop{\mathrm{curl}\,}}}({{\bm \beta}}\cdot \nabla {{\boldsymbol{u}}}_{{\varepsilon}}), {{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}})= ({{\bm \beta}}\cdot \nabla ({{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}), {{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}})+ ((\nabla {{\bm \beta}})^t \times \nabla {{\boldsymbol{u}}}_{{\varepsilon}}, {{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}})= ((\nabla {{\bm \beta}})^t \times \nabla {{\boldsymbol{u}}}_{{\varepsilon}}, {{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}})\,,$$ and then $$-({{\boldsymbol{u}}}_{{\varepsilon}}, {{\bm \beta}}\cdot \nabla ({{\ensuremath\mathop{\mathrm{curl}\,}}}{{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}))=((\nabla {{\bm \beta}})^t \times \nabla {{\boldsymbol{u}}}_{{\varepsilon}}, {{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}).$$ Therefore, replacing the last identity in we have $${\varepsilon}\|{{\ensuremath\mathop{\mathrm{curl}\,}}}{{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}\|_{{\Omega}}^2 + \|\sqrt{\sigma}\,{{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}\|_{{\Omega}}^2=
({{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{f}}}-\nabla\sigma\times {{\boldsymbol{u}}}_{\varepsilon}, {{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{\varepsilon})-((\nabla {{\bm \beta}})^t \times \nabla {{\boldsymbol{u}}}_{{\varepsilon}}, {{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}).$$ Using the Cauchy Schwarz inequality, one of the inequalities or , and the fact that ${{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}=0$ we have $$\|\sqrt{\sigma}\,{{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}\|_{{\Omega}}^2 \le \| {{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{f}}}-\nabla\sigma\times {{\boldsymbol{u}}}_{\varepsilon}\|_{\Omega}\|{{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}\|_{\Omega} + \mathcal{C} \|{{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}\|_{\Omega}^2\,.$$ Hence, $$(\sigma_0- \mathcal{C})\,\|{{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}\|_{{\Omega}}^2 \le \|{{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{f}}}-\nabla\sigma\times {{\boldsymbol{u}}}_{\varepsilon}\|_{\Omega} \|{{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}\|_{\Omega}\,,$$ and the proof follows dividing by $\|{{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}\|_{{\Omega}}$ and applying .
\[uniqueness\] Let us assume the hypotheses Theorem \[regcurl\] . Then, there exists a unique solution of such that ${{\boldsymbol{u}}}\in {{\boldsymbol{H}}}^1({\Omega}) \cap {{\boldsymbol{H}}}_0({{\ensuremath\mathop{\mathrm{div}\,}}},\Omega)$ and $p \in H^1(\Omega)$.
Let $({{\boldsymbol{u}}}_{{\varepsilon}}, p_{{\varepsilon}})$ be the solution of . Then, by Theorem \[regcurl\] $\{({{\boldsymbol{u}}}_{{\varepsilon}},p_{{\varepsilon}})\}$ is uniformly bounded in ${{\boldsymbol{H}}}^1({\Omega})\times L^2_0(\Omega)$. Hence, there exists a subsequence such that ${{\boldsymbol{u}}}_{{\varepsilon}} \rightharpoonup {{\boldsymbol{u}}}\in {{\boldsymbol{H}}}^1(\Omega)$ and $p_{{\varepsilon}} \rightharpoonup p \in L_0^2(\Omega)$ weakly. Moreover, since ${{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}=0$ and ${{\boldsymbol{u}}}_{{\varepsilon}} \in {{\boldsymbol{H}}}_0({{\ensuremath\mathop{\mathrm{div}\,}}},\Omega)$, then ${{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{u}}}=0$ and ${{\boldsymbol{u}}}\in {{\boldsymbol{H}}}_0({{\ensuremath\mathop{\mathrm{div}\,}}},\Omega)$. This proves that ${{\boldsymbol{u}}}$ satisfies the second equation in and the boundary conditions. Since ${{\boldsymbol{u}}}_{{\varepsilon}} \rightharpoonup {{\boldsymbol{u}}}\in {{\boldsymbol{H}}}^1(\Omega)$ weakly, then ${{\boldsymbol{u}}}_{{\varepsilon}} \to {{\boldsymbol{u}}}$ strongly in ${{\boldsymbol{L}}}^2(\Omega)$. In addition, since ${\varepsilon}\|{{\ensuremath\mathop{\mathrm{curl}\,}}}{{\boldsymbol{u}}}_{{\varepsilon}}\|_{{\Omega}}\to 0$ as ${\varepsilon}\to 0$, then using the weak convergence of $p_{\varepsilon}$ to $p$ in $L^2(\Omega)$ we can take the limit as ${\varepsilon}\to 0$ in and conclude that $({{\boldsymbol{u}}}, p)$ also satisfies the first equation in . Finally, from the first equation in we have $\nabla p ={{\boldsymbol{f}}}-\sigma\,{{\boldsymbol{u}}}- {{\ensuremath\mathop{\mathrm{div}\,}}}({{\bm \beta}}\otimes{{\boldsymbol{u}}})\in {{\boldsymbol{L}}}^2(\Omega)$, and then $p\in H^1(\Omega)$.
To prove uniqueness, assume that ${{\boldsymbol{f}}}=\boldsymbol{0}$. If we test with ${{\boldsymbol{u}}}\in {{\boldsymbol{H}}}^1(\Omega) \cap {{\boldsymbol{H}}}_0({{\ensuremath\mathop{\mathrm{div}\,}}},\Omega)$ we immediately get that $\|\sqrt{\sigma}{{\boldsymbol{u}}}\|_{{\Omega}}^2=0$ which gives that ${{\boldsymbol{u}}}=\boldsymbol{0}$. It easily follows that $p=0$.
We finish this section by stating the following result that, in essence, casts the problem as the limit of the Oseen problem .
Under the same hypotheses from Theorem \[uniqueness\] the solution $({{\boldsymbol{u}}},p)$ of is the limit of the solutions of the Oseen problem in the following sense $$\label{limit}
\lim_{{\varepsilon}\to 0}\Big( \|{{\boldsymbol{u}}}-{{\boldsymbol{u}}}_{\varepsilon}\|_{{{\ensuremath\mathop{\mathrm{div}\,}}},\Omega}+\|p_{\varepsilon}-p\|_\Omega\Big) = 0\,.$$
The error $({{\boldsymbol{u}}}-{{\boldsymbol{u}}}_{\varepsilon},p-p_{\varepsilon})$ satisfies the following error equation $$\label{error-1}
(\sigma({{\boldsymbol{u}}}-{{\boldsymbol{u}}}_{\varepsilon}),{{\boldsymbol{v}}})-{\varepsilon}(\nabla{{\boldsymbol{u}}}_{\varepsilon},\nabla{{\boldsymbol{v}}})+({{\bm \beta}}\otimes({{\boldsymbol{u}}}-{{\boldsymbol{u}}}_{\varepsilon}),\nabla{{\boldsymbol{v}}}) -(p-p_{\varepsilon},{{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{v}}}) = 0\,,$$ for all ${{\boldsymbol{v}}}\in {{\boldsymbol{H}}}^1(\Omega)\cap {{\boldsymbol{H}}}_0({{\ensuremath\mathop{\mathrm{div}\,}}},\Omega)$. Since ${{\boldsymbol{u}}}\in {{\boldsymbol{H}}}^1(\Omega)\cap {{\boldsymbol{H}}}_0({{\ensuremath\mathop{\mathrm{div}\,}}},\Omega)$, $\hat{{{\boldsymbol{v}}}}:={{\boldsymbol{u}}}-{{\boldsymbol{u}}}_{\varepsilon}$ is a valid test function for . So, using $\hat{{{\boldsymbol{v}}}}$ in , the fact that both ${{\boldsymbol{u}}}$ and ${{\boldsymbol{u}}}_{\varepsilon}$ are divergence-free, the Cauchy-Schwarz inequality, and we get $$\begin{aligned}
\|\sqrt{\sigma}({{\boldsymbol{u}}}-{{\boldsymbol{u}}}_{\varepsilon})\|^2_\Omega +{\varepsilon}\|\nabla({{\boldsymbol{u}}}-{{\boldsymbol{u}}}_{\varepsilon})\|^2_\Omega &= {\varepsilon}(\nabla{{\boldsymbol{u}}},\nabla({{\boldsymbol{u}}}-{{\boldsymbol{u}}}_{\varepsilon})) \nonumber\\
&\le
\sqrt{{\varepsilon}}\|\nabla{{\boldsymbol{u}}}\|_\Omega\,\sqrt{{\varepsilon}}\|\nabla({{\boldsymbol{u}}}-{{\boldsymbol{u}}}_{\varepsilon})\|_\Omega\to 0\,,\end{aligned}$$ as ${\varepsilon}\to 0$, which proves the convergence of ${{\boldsymbol{u}}}_{\varepsilon}$ to ${{\boldsymbol{u}}}$ in ${{\boldsymbol{L}}}^2(\Omega)$. The convergence of ${{\boldsymbol{u}}}_{\varepsilon}$ to ${{\boldsymbol{u}}}$ in ${{\boldsymbol{H}}}_0({{\ensuremath\mathop{\mathrm{div}\,}}},\Omega)$ follows from the fact that both ${{\boldsymbol{u}}}_{\varepsilon}$ and ${{\boldsymbol{u}}}$ are divergence-free.\
[ ]{}\
To prove the convergence of the pressure, using Lemma \[infsup\] there exists ${{\boldsymbol{w}}}\in {{\boldsymbol{H}}}^1_0(\Omega)$ such that $|{{\boldsymbol{w}}}|_{1,\Omega}\le C\|p-p_{\varepsilon}\|_\Omega$ and ${{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{w}}}=p-p_{\varepsilon}$. Then, using , the Cauchy-Schwarz inequality, and the convergence of ${{\boldsymbol{u}}}_{\varepsilon}$ to ${{\boldsymbol{u}}}$, $$\begin{aligned}
\|p-p_{\varepsilon}\|^2_\Omega &= (p-p_{\varepsilon},{{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{w}}}) = (\sigma({{\boldsymbol{u}}}-{{\boldsymbol{u}}}_{\varepsilon}),{{\boldsymbol{w}}})+{\varepsilon}(\nabla{{\boldsymbol{u}}}_{\varepsilon},\nabla {{\boldsymbol{w}}}) + ({{\bm \beta}}\otimes ({{\boldsymbol{u}}}_{\varepsilon}-{{\boldsymbol{u}}}),\nabla{{\boldsymbol{w}}}) \nonumber\\
&\le C\,\Big(\|\sqrt{\sigma}\|_{\infty,\Omega}\,\|\sqrt{\sigma}\,({{\boldsymbol{u}}}-{{\boldsymbol{u}}}_{\varepsilon})\|_\Omega+\sqrt{{\varepsilon}}\,\sqrt{{\varepsilon}}\|\nabla{{\boldsymbol{u}}}_{\varepsilon}\|_\Omega+\|{{\bm \beta}}\|_{\infty,\Omega}\|{{\boldsymbol{u}}}-{{\boldsymbol{u}}}_{\varepsilon}\|_\Omega\Big)\,\|p-p_{\varepsilon}\|_\Omega\,,\end{aligned}$$ and the proof follows by dividing by $\|p-p_{\varepsilon}\|_\Omega$ and noticing that, thanks to the term within parentheses tends to zero as ${\varepsilon}\to 0$.
Upwind H(div) method {#sec:upw}
====================
Preliminaries
-------------
We denote by $\{ {\mathcal{T}_h}\}_{h>0}^{}$ a family of shape-regular simplicial triangulations of $\Omega$. The elements of ${\mathcal{T}_h}$ are denoted by $T$, with diameter $h_T$, and $h:=\max\{h_T:T\in{\mathcal{T}_h}\}$. The set of its facets (edges for $d=2$, faces for $d=3$) is denoted by $\mathcal{E}_h$. To cater for the nonconforming character of the approximation we also introduce the following broken versions of the scalar product $$\begin{aligned}
{1}
({{\boldsymbol{v}}}, {{\boldsymbol{w}}})_h=&\sum_{T \in {\mathcal{T}_h}} \int_T {{\boldsymbol{v}}}\cdot {{\boldsymbol{w}}}\, dx, \\
\langle {{\boldsymbol{v}}}, {{\boldsymbol{w}}}\rangle_h=&\sum_{T \in {\mathcal{T}_h}} \int_{\partial T} {{\boldsymbol{v}}}\cdot {{\boldsymbol{w}}}\, ds. \end{aligned}$$ In addition, we introduce the broken space $H({\mathcal{T}_h})$, of functions in $L^2(\Omega)$ whose restriction to every $T\in{\mathcal{T}_h}$ belongs to $H(T)$.
Let $T \in {\mathcal{T}_h}$ and let ${{\boldsymbol{x}}}\in \partial T$ then we define $${{\boldsymbol{v}}}_{{{\bm \beta}}}^\pm({{\boldsymbol{x}}})=\lim_{\epsilon \rightarrow 0} {{\boldsymbol{v}}}\big({{\boldsymbol{x}}}\pm \epsilon ({{\bm \beta}}({{\boldsymbol{x}}}) \cdot {{\boldsymbol{n}}}({{\boldsymbol{x}}})) {{\boldsymbol{n}}}({{\boldsymbol{x}}})\big).$$ and $$\hat{{{\boldsymbol{v}}}}({{\boldsymbol{x}}})= \, {{\boldsymbol{v}}}_{{{\bm \beta}}}^-({{\boldsymbol{x}}})$$ For $F \in \mathcal{E}_h$ and $F= \partial T_1 \cap \partial T_2$ for $T_1, T_2, \in {\mathcal{T}_h}$ we define the jumps $${\left[\hspace{-0.025in}\left[{{\boldsymbol{v}}}\otimes {{\boldsymbol{n}}}\right]\hspace{-0.025in}\right]}|_{F}= {{\boldsymbol{v}}}|_{T_1} \otimes {{\boldsymbol{n}}}_1+ {{\boldsymbol{v}}}|_{T_2} \otimes {{\boldsymbol{n}}}_2\,,$$ and for $F \in \mathcal{E}_h$ and $F \subset \Gamma$ we define $${\left[\hspace{-0.025in}\left[{{\boldsymbol{v}}}\otimes {{\boldsymbol{n}}}\right]\hspace{-0.025in}\right]}|_{F}= {{\boldsymbol{v}}}\otimes {{\boldsymbol{n}}}.$$
We then define the semi-norm on the jumps of the solution over element boundaries to be $$|{{\boldsymbol{v}}}|_{{{\bm \beta}}}^2=\sum_{F \in \mathcal{E}_h} \|\sqrt{|{{\bm \beta}}\cdot {{\boldsymbol{n}}}|} {\left[\hspace{-0.025in}\left[{{\boldsymbol{v}}}\otimes {{\boldsymbol{n}}}\right]\hspace{-0.025in}\right]}\|_{0,F}^2.$$ With these definitions we can state the following important identity [@GRW05 Lemma 6.1]
\[proposition1\] For all ${{\boldsymbol{v}}}\in {{\boldsymbol{H}}}^1({\mathcal{T}_h})$, the following holds $$\label{aux1}
({{\boldsymbol{v}}}\otimes {{\bm \beta}}, \nabla {{\boldsymbol{v}}})_h-\langle {{\bm \beta}}\cdot {{\boldsymbol{n}}}\hat{{{\boldsymbol{v}}}}, {{\boldsymbol{v}}}\rangle_h= -\frac{1}{2} |{{\boldsymbol{v}}}|_{{{\bm \beta}}}^2.$$
Let us define the Raviart-Thomas [@RT] and BDM spaces [@BDM]. The space of polynomials of degree at most $k$ defined in $T$ is denoted by ${\EuScript{P}}_k(T)$, and we denote ${\boldsymbol{{\EuScript{P}}}}_k(T)=[{\EuScript{P}}_k(T)]^d$. For every $T\in {\mathcal{T}_h}$, let $\text{RT}_k(T)= {\boldsymbol{{\EuScript{P}}}}_k(T)+
({\EuScript{P}}_k(T) \setminus {\EuScript{P}}_{k-1}(T)) {{\boldsymbol{x}}}$. We define, for $k\ge 0$, the spaces $$\begin{aligned}
{1}
{{\boldsymbol{V}}}_{h,k}^{\text{RT}}=&\{ {{\boldsymbol{v}}}\in {{\boldsymbol{H}}}_0({{\ensuremath\mathop{\mathrm{div}\,}}}, \Omega): {{\boldsymbol{v}}}|_T \in \text{RT}_k(T) \text{ for all } T \in {\mathcal{T}_h}\}, \\
{{\boldsymbol{V}}}_{h,k}^{\text{BDM}}=&\{ {{\boldsymbol{v}}}\in {{\boldsymbol{H}}}_0({{\ensuremath\mathop{\mathrm{div}\,}}}, \Omega): {{\boldsymbol{v}}}|_T \in {\boldsymbol{{\EuScript{P}}}}_k(T) \text{ for all } T \in {\mathcal{T}_h}\}, \\
M_{h,k}=&\{ q \in L_0^2(\Omega): q|_T \in {\EuScript{P}}_k(T) \text{ for all } T \in {\mathcal{T}_h}\}.\end{aligned}$$
A well-known property linking these two spaces is stated now (for a proof see [@CG04 Lemma 4.3]).
\[auxlemma\] Let ${{\boldsymbol{v}}}\in {{\boldsymbol{V}}}_{h,k}^{\text{RT}}$ with ${{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{v}}}=0$ on $\Omega$ then ${{\boldsymbol{v}}}\in {{\boldsymbol{V}}}_{h,k}^{\text{BDM}}$.
We next introduce the standard $L^2$-projection on polynomials on an element $T$, $P_k^T:L^2(T)
\rightarrow {\EuScript{P}}_{k}(T)$. Its global equivalent will be denoted $P_k:L^2(\Omega)
\rightarrow M_{h,k}$. We recall the standard estimates for the $L^2$-projection (see, e.g., [@EG04]) $$\begin{aligned}
\|P_k q - q \|_\Omega+ h \|\nabla (P_k q - q )\|_\Omega \le C \, h_T^{k+1} |q|_{k+1,\Omega}\,, \label{eq:L2approx}\\
\|q -P_0 q\|_{\infty,T} \le C \, h_T \| q\|_{1,\infty,T}\,.\label{eq:Linftyapprox}\end{aligned}$$
The Raviart-Thomas interpolation operator will be used in the sequel. It is defined as follows: ${{\boldsymbol{\Pi}}}: {{\boldsymbol{H}}}^1(\Omega) \cap {{\boldsymbol{H}}}_0(\text{div}; \Omega) \rightarrow {{\boldsymbol{V}}}_{h,k}^{\text{RT}}$ where ${{\boldsymbol{\Pi}}}{{\boldsymbol{v}}}$ is the only function of ${{\boldsymbol{V}}}_{h,k}^{RT}$ satisfying $$\begin{aligned}
\int_T ({{\boldsymbol{\Pi}}}{{\boldsymbol{v}}}- {{\boldsymbol{v}}}) \cdot {{\boldsymbol{w}}}\,dx=&\,0 \quad \text{ for all } {{\boldsymbol{w}}}\in {\boldsymbol{{\EuScript{P}}}}_{k-1}(T), \textrm{and all}\; T \in {\mathcal{T}_h}, \label{RT-1a}\\
\int_F ({{\boldsymbol{\Pi}}}{{\boldsymbol{v}}}-{{\boldsymbol{v}}}) \cdot {{\boldsymbol{n}}}w \, ds=&\,0 \quad \text{ for all } w \in {\EuScript{P}}_{k}(F), \textrm{and all}\; F \in \mathcal{E}_h.\label{RT-1b}\end{aligned}$$ This operator satisfies the following classical properties (see, e.g., [@BBF13]).
\[errorproj\] Let $k\ge 0$. The mapping $\Pi$ satisfies the following commutative property $$\label{commutative}
{{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{\Pi}}}{{\boldsymbol{v}}}= P_k {{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{v}}}\,.$$ Let ${{\boldsymbol{v}}}\in {{\boldsymbol{H}}}^{k+1}(\Omega)$ then we have $$\|{{\boldsymbol{\Pi}}}{{\boldsymbol{v}}}-{{\boldsymbol{v}}}\|_{T}+ h_T \|\nabla ({{\boldsymbol{\Pi}}}{{\boldsymbol{v}}}-{{\boldsymbol{v}}})\|_{T} \le C \, h_T^{k+1} |{{\boldsymbol{v}}}|_{k+1,T} \quad \text {for all } T \in {\mathcal{T}_h}.$$
We end this section recalling the following classical inverse and local trace inequalities that hold for every $T\in{\mathcal{T}_h}$ $$\begin{aligned}
|v_h^{}|_{1,T}^{} &\le Ch^{-1}\|v_h^{}\|_{T}^{}\qquad\forall\, v_h^{}\in {\EuScript{P}}_k(T)\,, \label{inverse}\\
\|v\|_{\partial T} &\le C\big( h_T^{-\frac{1}{2}}\|v\|_{T}+h_T^{\frac{1}{2}}|v|_{1,T}\big)\qquad\forall\, v\in H^1(T)\,. \label{local-trace}\end{aligned}$$
The finite element method and the error estimates for the velocity
------------------------------------------------------------------
Throughout, the velocity and pressure will be approximated using the spaces ${{\boldsymbol{V}}}_h$ and $M_h$, respectively. In this work we will consider the following choices: $${{\boldsymbol{V}}}_h={{\boldsymbol{V}}}_{h,k}^{\text{RT}} \quad \text{ and } M_h=M_{h,k}, \text{ for } k \ge 0,$$ or $${{\boldsymbol{V}}}_h={{\boldsymbol{V}}}_{h,k}^{\text{BDM}} \quad \text{ and } M_h=M_{h,k-1}, \text{ for } k \ge 1.$$
The numerical method analysed here reads: Find ${{\boldsymbol{u}}}\in {{\boldsymbol{V}}}_h$ and $p_h \in M_h$ such that
\[fem\] $$\begin{aligned}
{2}
-({{\boldsymbol{u}}}_h, {{\bm \beta}}\cdot \nabla {{\boldsymbol{v}}}_h)_h+ \langle ({{\bm \beta}}\cdot {{\boldsymbol{n}}}) \widehat{{{\boldsymbol{u}}}_h}, {{\boldsymbol{v}}}_h \rangle_h+(\sigma {{\boldsymbol{u}}}_h, {{\boldsymbol{v}}}_h) -(p_h, {{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{v}}}_h )=&\, ({{\boldsymbol{f}}},{{\boldsymbol{v}}}_h) \quad && \text{ for all } {{\boldsymbol{v}}}_h \in {{\boldsymbol{V}}}_h,\\
({{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{u}}}_h, q_h)=&\, 0 \quad && \text{ for all } q_h \in M_h. \end{aligned}$$
Thanks to the inf-sup stability of the pair ${{\boldsymbol{V}}}_h\times M_h$ (see [@BBF13]), and Proposition \[proposition1\], problem has a unique solution. Moreover, the method is consistent; in fact, for $({{\boldsymbol{u}}},p) \in {{\boldsymbol{H}}}^1(\Omega) \times L^2_0(\Omega)$ solving we have
\[weak\] $$\begin{aligned}
{2}
-({{\boldsymbol{u}}}, {{\bm \beta}}\cdot \nabla {{\boldsymbol{v}}}_h)_h+ \langle ({{\bm \beta}}\cdot {{\boldsymbol{n}}}) {{\boldsymbol{u}}}, {{\boldsymbol{v}}}_h \rangle_h+(\sigma {{\boldsymbol{u}}}, {{\boldsymbol{v}}}_h) -(p, {{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{v}}}_h )=&\, ({{\boldsymbol{f}}},{{\boldsymbol{v}}}_h) \quad && \text{ for all } {{\boldsymbol{v}}}_h \in {{\boldsymbol{V}}}_h,\\
({{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{u}}}, q_h)=&\, 0 \quad && \text{ for all } q_h \in M_h. \end{aligned}$$
A consequence of Lemma \[auxlemma\] is that the finite element method produces the same velocity approximation for ${{\boldsymbol{u}}}_h \in
{{\boldsymbol{V}}}_{h,k}^{\text{RT}}$ and ${{\boldsymbol{u}}}_h \in {{\boldsymbol{V}}}_{h,k}^{\text{BDM}}$. We show that in the following proposition.
\[eq:RTeqBDM\] Let $({{\boldsymbol{u}}}_h,p_h)$ be the solution of for the spaces ${{\boldsymbol{V}}}_h\times M_h ={{\boldsymbol{V}}}_{h,k}^{\text{RT}} \times M_{h,k}$ and $(\tilde {{\boldsymbol{u}}}_h, \tilde p_h)$ the solution of for the spaces ${{\boldsymbol{V}}}_h\times M_h ={{\boldsymbol{V}}}_{h,k}^{\text{BDM}} \times
M_{h,k-1}$. Then ${{\boldsymbol{u}}}_h=\tilde {{\boldsymbol{u}}}_h $.
Let ${\bm e}_h := \tilde {{\boldsymbol{u}}}_h - {{\boldsymbol{u}}}_h$, $\eta_h = \tilde p_h - p_h$ then using we see that $$\label{eq:peRTBDM}
-({\bm e}_h, {{\bm \beta}}\cdot \nabla {{\boldsymbol{v}}}_h)_h+ \langle ({{\bm \beta}}\cdot {{\boldsymbol{n}}}) \widehat{{\bm e}_h}, {{\boldsymbol{v}}}_h \rangle_h+(\sigma {\bm e}_h, {{\boldsymbol{v}}}_h) -(\eta_h, {{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{v}}}_h )=0 \quad \text{ for all } {{\boldsymbol{v}}}_h \in {{\boldsymbol{V}}}_{h,k}^{\text{BDM}}.$$ Since ${{\ensuremath\mathop{\mathrm{div}\,}}}{\bm e}_h = 0$ by Lemma \[auxlemma\]there holds ${\bm e}_h \in
{{\boldsymbol{V}}}_{h,k}^{\text{BDM}}$, which is a valid test function. Taking ${{\boldsymbol{v}}}_h =
{\bm e}_h$ in and applying Proposition \[proposition1\] we obtain $$\|\sqrt{\sigma} {\bm e}_h\|_\Omega = 0,$$ which proves the claim.
We can now derive an error estimate for the velocity. We let ${\bm e}_h={{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}-{{\boldsymbol{u}}}_h$ and start by noticing that $$\label{divfree}
{{\ensuremath\mathop{\mathrm{div}\,}}}{\bm e}_h=0.$$ Hence, by Lemma \[auxlemma\] we have ${\bm e}_h \in {{\boldsymbol{V}}}_{h,k}^{\text{BDM}}$ and in particular $$\label{aux2}
\nabla {\bm e}_h|_{T} \in [{{\EuScript{P}}}_{k-1}(T)]^{d \times d} \qquad \text{ for all } T \in {\mathcal{T}_h}.$$
\[thm:main\] Let ${{\boldsymbol{u}}}\in [H^1(\Omega)]^d$ solve and let ${{\boldsymbol{u}}}_h \in {{\boldsymbol{V}}}_h$ solve . Then, the following error estimate holds $$\begin{aligned}
{1}
\|\sqrt{\sigma} ({{\boldsymbol{u}}}-{{\boldsymbol{u}}}_h)\|_\Omega &+ |{{\boldsymbol{u}}}-{{\boldsymbol{u}}}_h|_{{{\bm \beta}}} \le\, C \left(1+\frac{\| {{\bm \beta}}\|_{1,\infty,T}}{\sigma_0}\right) \|\sqrt{\sigma} ({{\boldsymbol{u}}}-{{\boldsymbol{\Pi}}}{{\boldsymbol{u}}})\|_\Omega \\
& + C \|{{\bm \beta}}\|_{\infty,\Omega}^{1/2} \left( \sum_{T\in {\mathcal{T}_h}} \left(\frac{1}{h_T} \| {{\boldsymbol{u}}}-{{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}\|_{T}^2+ h_T \| \nabla({{\boldsymbol{u}}}-{{\boldsymbol{\Pi}}}{{\boldsymbol{u}}})\|_{T}^2\right) \right)^{\frac{1}{2}}\,,\end{aligned}$$ where the constant $C$ does not depend on $h$, or any physical parameter of the equation.
Using , , , and we get $$\begin{aligned}
\|\sqrt{\sigma}{\bm e}_h\|_\Omega^2=&(\sigma ({{\boldsymbol{u}}}-{{\boldsymbol{u}}}_h), {\bm e}_h)+(\sigma ({{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}-{{\boldsymbol{u}}}), {\bm e}_h) \\
=& (({{\boldsymbol{u}}}-{{\boldsymbol{u}}}_h), {{\bm \beta}}\cdot \nabla {\bm e}_h)_h- \langle {{\bm \beta}}\cdot {{\boldsymbol{n}}}({{\boldsymbol{u}}}-\widehat{{{\boldsymbol{u}}}_h}) , {\bm e}_h \rangle_h +(\sigma ({{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}-{{\boldsymbol{u}}}), {\bm e}_h) \\
=& (({{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}-{{\boldsymbol{u}}}_h), {{\bm \beta}}\cdot \nabla {\bm e}_h)_h- \langle {{\bm \beta}}\cdot {{\boldsymbol{n}}}(\widehat{{{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}}-\widehat{{{\boldsymbol{u}}}_h}) , {\bm e}_h \rangle_h \\
&+(({{\boldsymbol{u}}}-{{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}), {{\bm \beta}}\cdot \nabla {\bm e}_h)_h- \langle {{\bm \beta}}\cdot {{\boldsymbol{n}}}({{\boldsymbol{u}}}-\widehat{{{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}}), {\bm e}_h\rangle_h +(\sigma ({{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}-{{\boldsymbol{u}}}), {\bm e}_h) \\
=&-\frac{1}{2} |{\bm e}_h|_{{{\bm \beta}}}^2 +(({{\boldsymbol{u}}}-{{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}), {{\bm \beta}}\cdot \nabla {\bm e}_h)_h- \langle {{\bm \beta}}\cdot {{\boldsymbol{n}}}({{\boldsymbol{u}}}-\widehat{{{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}}), {\bm e}_h \rangle_h +(\sigma ({{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}-{{\boldsymbol{u}}}), {\bm e}_h)\,.\end{aligned}$$ Hence, we have $$\begin{aligned}
&\|\sqrt{\sigma}{\bm e}_h \|_\Omega^2+\frac{1}{2} |{\bm e}_h|_{{{\bm \beta}}}^2 \nonumber \\
&=({{\boldsymbol{u}}}-{{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}, {{\bm \beta}}\cdot \nabla {\bm e}_h)_h- \langle {{\bm \beta}}\cdot {{\boldsymbol{n}}}({{\boldsymbol{u}}}-\widehat{{{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}}), {\bm e}_h \rangle_h +(\sigma ({{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}-{{\boldsymbol{u}}}), {\bm e}_h )\,. \label{uno}\end{aligned}$$ We bound each term separately. Using , the definition of ${{\boldsymbol{\Pi}}}$ -, , and , we have $$\label{dos}
({{\boldsymbol{u}}}-{{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}, {{\bm \beta}}\cdot \nabla {\bm e}_h)_h= ({{\boldsymbol{u}}}-{{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}, ({{\bm \beta}}-P_0{{\bm \beta}}) \cdot \nabla {\bm e}_h)_h \le C \| {{\bm \beta}}\|_{1,\infty,\Omega} \|{{\boldsymbol{u}}}-{{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}\|_\Omega \|{\bm e}_h\|_\Omega\,.$$ Using the contributions from neighbouring elements on the face to express the discrete error on the faces in terms of jumps, the normal continuity of ${{\boldsymbol{u}}}$ and ${{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}$, and using the local trace inequality it is easy to show that $$\label{tres}
- \langle {{\bm \beta}}\cdot {{\boldsymbol{n}}}({{\boldsymbol{u}}}-\widehat{{{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}}), {\bm e}_h \rangle_h \le C \,\|{{\bm \beta}}\|_{\infty,\Omega}^{\frac{1}{2}} |{\bm e}_h|_{{{\bm \beta}}} \left( \sum_{T\in {\mathcal{T}_h}} \left(\frac{1}{h_T} \| {{\boldsymbol{u}}}-{{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}\|_{T}^2+ h_T \| \nabla({{\boldsymbol{u}}}-{{\boldsymbol{\Pi}}}{{\boldsymbol{u}}})\|_{T}^2\right) \right)^{\frac{1}{2}}.$$ Finally, $$\label{cuatro}
(\sigma ({{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}-{{\boldsymbol{u}}}), {\bm e}_h) \le \| \sqrt{\sigma} ({{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}-{{\boldsymbol{u}}})\|_\Omega \| \sqrt{\sigma} {\bm e}_h\|_\Omega.$$ Therefore, inserting - into we arrive at $$\begin{aligned}
{1}
\|\sqrt{\sigma} {\bm e}_h\|_\Omega+ |{\bm e}_h|_{{{\bm \beta}}} \le & C \,\Big(1+\frac{\| {{\bm \beta}}\|_{1,\infty,\Omega}}{\sigma_0}\Big) \|\sqrt{\sigma} ({{\boldsymbol{u}}}-{{\boldsymbol{\Pi}}}{{\boldsymbol{u}}})\|_\Omega \\
& + C \|{{\bm \beta}}\|_{\infty,\Omega}^{\frac{1}{2}} \left( \sum_{T\in {\mathcal{T}_h}} \left(\frac{1}{h_T} \| {{\boldsymbol{u}}}-{{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}\|_{T}^2+ h_T \| \nabla({{\boldsymbol{u}}}-{{\boldsymbol{\Pi}}}{{\boldsymbol{u}}})\|_{T}^2\right) \right)^{\frac{1}{2}}.\end{aligned}$$ The result follows after applying the triangle inequality.
The following result appears as a corollary of the last theorem and Lemma \[errorproj\].
\[cor:conv\] Let ${{\boldsymbol{u}}}\in [H^{k+1}(\Omega)]^d$ solve and let ${{\boldsymbol{u}}}_h \in {{\boldsymbol{V}}}_h$ solve . Then, the following error estimate holds $$\begin{aligned}
{1}
\|\sqrt{\sigma} ({{\boldsymbol{u}}}-{{\boldsymbol{u}}}_h)\|_\Omega+ |{{\boldsymbol{u}}}-{{\boldsymbol{u}}}_h|_{{{\bm \beta}}} \le & C \left(\left[1+\frac{\| {{\bm \beta}}\|_{1,\infty, T}}{\sigma_0}\right] \|\sqrt{\sigma}\|_{\infty,\Omega} h^{\frac{1}{2}}+\|{{\bm \beta}}\|_{\infty,\Omega}^{\frac{1}{2}}\right) h^{k+\frac{1}{2}} \|{{\boldsymbol{u}}}\|_{k+1,\Omega)}.\end{aligned}$$
The arguments of Theorem \[thm:main\] and Corollary \[cor:conv\] may be used to improve the order obtained Theorem 2.2 of [@GSS17] to $O(h^{k+\frac12})$, if an upwind flux is used. Following the ideas above, use integration by parts in the first term of $I_1$ in the equation after (2.12). Then add and subtract the exact solution to the approximate solution in term $I_3$ and recombine terms, so that one may use continuity on the norm augmented with $L^2$-control on the faces the jumps of the approximate velocity.
$L^2$-error estimates for the pressure approximation
----------------------------------------------------
Since the pressure space is of polynomial degree $k$ for the method using the $RT$ space for velocity approximation and $k-1$ for the method using the $BDM$ space, the optimal order that can be obtained for the error of the pressure approximation in the $L^2$-norm is $O(h^{k+1})$ and $O(h^k)$, respectively. Here we will prove the following orders for the pressure error :
1. in the first case (RT), $O(h^{k+\frac12})$; this is, the same suboptimality of $O(h^{\frac12})$ as for the velocity approximation.
2. in the second case (BDM) we get the optimal convergence $O(h^k)$; considering that the pressure space is of degree $k-1$. For the discrete error, i.e. the projection of the error on the space $M_h$, we get an $O(h^{k+\frac12})$ estimate, this is a superconvergence of $O(h^{\frac12})$ compared with the approximation property of the space of constant functions.
\[thm:pres\] Let $({{\boldsymbol{u}}},p) \in {{\boldsymbol{H}}}^1(\Omega)\times L^2_0(\Omega)$ solve and let $({{\boldsymbol{u}}}_h,p_h) \in {{\boldsymbol{V}}}_h \times M_h$ solve . Let $\ell$ denote the polynomial degree of the space $M_h$. Then, the following error estimate holds $$\begin{aligned}
{1}
\|P_\ell p - p_h\|_\Omega \leq\, & C (\|{{\bm \beta}}\|_{\infty,\Omega} \sigma_0^{-\frac12} + \sigma^{\frac12}) \|\sqrt{\sigma} ({{\boldsymbol{u}}}-{{\boldsymbol{u}}}_h)\|_\Omega \\
& + C \,\|{{\bm \beta}}\|_{\infty,\Omega} \left( \sum_{T\in {\mathcal{T}_h}} \left(\| {{\boldsymbol{u}}}-{{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}\|_{T}^2+ h_T^2 \| \nabla({{\boldsymbol{u}}}-{{\boldsymbol{\Pi}}}{{\boldsymbol{u}}})\|_{T}^2\right) \right)^{\frac{1}{2}}\,.\end{aligned}$$
Using the surjectivity of the divergence operator as a mapping from ${{\boldsymbol{H}}}^1_0(\Omega)$ to $L^2_0(\Omega)$ there exists ${{\boldsymbol{v}}}_p
\in {{\boldsymbol{H}}}^1_0(\Omega)$ such that ${{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{v}}}_p = P_\ell p - p_h$ and $$\label{aux567}
\|{{\boldsymbol{v}}}_p\|_{1,\Omega} \leq C \|P_\ell p - p_h\|_\Omega.$$ It follows from and that $$\|P_\ell p - p_h\|_\Omega^2 = (P_\ell p - p_h,{{\ensuremath\mathop{\mathrm{div}\,}}}{{\boldsymbol{v}}}_p) =
(P_\ell p - p_h,{{\ensuremath\mathop{\mathrm{div}\,}}}\tilde {{\boldsymbol{\Pi}}}{{\boldsymbol{v}}}_p) =
(p - p_h,{{\ensuremath\mathop{\mathrm{div}\,}}}\tilde {{\boldsymbol{\Pi}}}{{\boldsymbol{v}}}_p).$$ If ${{\boldsymbol{V}}}_h \equiv {{\boldsymbol{V}}}_{h,k}^{\text{RT}}$ then choose $\tilde {{\boldsymbol{\Pi}}}{{\boldsymbol{v}}}_p \in
{{\boldsymbol{V}}}_{h,k}^{\text{RT}}$ and if ${{\boldsymbol{V}}}_h \equiv {{\boldsymbol{V}}}_{h,k}^{\text{BDM}}$ choose $\tilde {{\boldsymbol{\Pi}}}{{\boldsymbol{v}}}_p \in
{{\boldsymbol{V}}}_{h,{k-1}}^{\text{RT}} \subset {{\boldsymbol{V}}}_{h,k}^{\text{BDM}}$. Using and we find that $$(p - p_h,{{\ensuremath\mathop{\mathrm{div}\,}}}\tilde {{\boldsymbol{\Pi}}}{{\boldsymbol{v}}}_p) = -({{\boldsymbol{u}}}- {{\boldsymbol{u}}}_h, {{\bm \beta}}\cdot \nabla
\tilde {{\boldsymbol{\Pi}}}{{\boldsymbol{v}}}_p)_h+ \langle ({{\bm \beta}}\cdot {{\boldsymbol{n}}}) ({{\boldsymbol{u}}}- \widehat{{{\boldsymbol{u}}}_h} ), \tilde {{\boldsymbol{\Pi}}}{{\boldsymbol{v}}}_p
\rangle_h+(\sigma ({{\boldsymbol{u}}}- {{\boldsymbol{u}}}_h), \tilde {{\boldsymbol{\Pi}}}{{\boldsymbol{v}}}_p).$$ Applying the Cauchy-Schwarz inequality and the stability of the RT interpolant and of ${{\boldsymbol{v}}}_p$ we have $$-({{\boldsymbol{u}}}- {{\boldsymbol{u}}}_h, {{\bm \beta}}\cdot \nabla
\tilde{{\boldsymbol{\Pi}}}{{\boldsymbol{v}}}_p)_h + (\sigma ({{\boldsymbol{u}}}- {{\boldsymbol{u}}}_h), \tilde {{\boldsymbol{\Pi}}}{{\boldsymbol{v}}}_p) \leq
(\|{{\bm \beta}}\|_{\infty,\Omega} \sigma_0^{-\frac12} + \sigma^{\frac12})
\|\sqrt{\sigma} ({{\boldsymbol{u}}}- {{\boldsymbol{u}}}_h)\|_\Omega \| {{\boldsymbol{v}}}_p\|_{1,\Omega}.$$ For the remaining term observe that, by the definition of $\langle \cdot, \cdot
\rangle_h$, the fact that ${{\bm \beta}}\cdot {{\boldsymbol{n}}}$ changes sign on neighbouring elements and that $({{\boldsymbol{u}}}- \widehat{{{\boldsymbol{u}}}_h} )$ is single valued on the faces of the triangulation, $$\langle ({{\bm \beta}}\cdot {{\boldsymbol{n}}}) ({{\boldsymbol{u}}}- \widehat{{{\boldsymbol{u}}}_h} ), \tilde {{\boldsymbol{\Pi}}}{{\boldsymbol{v}}}_p
\rangle_h = \langle ({{\bm \beta}}\cdot {{\boldsymbol{n}}}) ({{\boldsymbol{u}}}- \widehat{{{\boldsymbol{u}}}_h}
), (\tilde {{\boldsymbol{\Pi}}}{{\boldsymbol{v}}}_p - {{\boldsymbol{v}}}_p)
\rangle_h.$$ The right hand side of this equality is bounded using the Cauchy-Schwarz inequality, the trace inequality and the interpolation properties of the RT-interpolant of Lemma \[errorproj\] as follows $$\begin{aligned}
{1}
& \langle ({{\bm \beta}}\cdot {{\boldsymbol{n}}}) ({{\boldsymbol{u}}}- \widehat{{{\boldsymbol{u}}}_h}
), (\tilde {{\boldsymbol{\Pi}}}{{\boldsymbol{v}}}_p - {{\boldsymbol{v}}}_p)
\rangle_h \\
& \leq C \|{{\bm \beta}}\|_{\infty,\Omega} \sum_{T \in\mathcal{T}_h} (h_T^{-\frac12} \|{{\boldsymbol{u}}}-
{{\boldsymbol{u}}}_h\|_{T}+ h_T^{\frac12} \|\nabla ({{\boldsymbol{u}}}-
{{\boldsymbol{u}}}_h)\|_{T}) h_T^{\frac12}
\|{{\boldsymbol{v}}}_p\|_{1,T} \\
& \le C \|{{\bm \beta}}\|_{\infty,\Omega} \sum_{T \in\mathcal{T}_h} (\|{{\boldsymbol{u}}}-
{{\boldsymbol{u}}}_h\|_{T}+\|{{\boldsymbol{u}}}-{{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}\|_{T}+ h_T\|\nabla ({{\boldsymbol{u}}}-
{{\boldsymbol{\Pi}}}{{\boldsymbol{u}}})\|_{T})
\|{{\boldsymbol{v}}}_p\|_{1,T},\end{aligned}$$ where in the last step we added and subtracted ${{\boldsymbol{\Pi}}}{{\boldsymbol{u}}}$, used the triangle inequality and the inverse inequality . We conclude by using .
The following result is an immediate consequence of Theorem \[thm:pres\] and Corollary \[cor:conv\] and the approximation properties of the $L^2$-projection,
\[cor:pres\_order\] Assume that ${{\boldsymbol{V}}}_h={{\boldsymbol{V}}}_{h,k}^{\text{RT}}$ and $M_h=M_{h,k}$. Then, there exists $\tilde C_{{{\bm \beta}},\sigma}>0$ that depends only on the constants in the bounds of Theorems \[thm:pres\] and Corollary \[cor:conv\] such that $$\|p - p_h\|_\Omega
\leq \tilde C_{{{\bm \beta}},\sigma} h^{k+\frac12} \|{{\boldsymbol{u}}}\|_{k+1,\Omega}
+ C h^{k+1}|p|_{k+1,\Omega}\,.$$ For the case in which ${{\boldsymbol{V}}}_h={{\boldsymbol{V}}}_{h,k}^{\text{BDM}}$ and $M_h=M_{h,k-1}$, the following error estimate holds $$\|P_{k-1} p - p_h\|_\Omega
\leq \hat C_{{{\bm \beta}},\sigma} h^{k+\frac12} \|{{\boldsymbol{u}}}\|_{k+1,\Omega}$$ and $$\|p - p_h\|_\Omega \leq \hat C_{{{\bm \beta}},\sigma} h^{k+\frac12}
\|{{\boldsymbol{u}}}\|_{k+1,\Omega} + C h^k |p|_{k,\Omega},$$ where $\hat C_{{{\bm \beta}},\sigma}$ depends on the constants in the bounds of Theorems \[thm:pres\] and Corollary \[cor:conv\].
A numerical example {#sec:num}
===================
Here we will show some illustrations of the theory developed above using the analytical solution of example (2) in section \[sec:linear\]. For ample qualitative numerical evidence of the performance of this type of method on physically relevant problems we refer to the references [@SL18; @SLLL18].
We consider the domain $\Omega=(0,1) \times (0,1)$ and the solution - of example (2). We used the package FreeFEM++ [@He12] to implement the formulation with either the BDM(1) element and piecewise constant pressures or the RT(1) element with piecewise affine, discontinuous, pressures. The linear systems were solved using UMFPACK and the meshes were of Union Jack type. In Tables \[tab:BDM\]-\[tab:RT\] we report the errors of velocities and pressures in the (relative) $L^2$-norm. We also report the CPU time. We see that the velocity approximations have identical errors in the two cases as predicted by Proposition \[eq:RTeqBDM\], whereas as expected the BDM(1) approximation has poorer convergence of the pressure. The RT(1) computation however is more costly by almost a factor three.
In Table \[tab:n\] we report the variation of the error on a fixed mesh with $h=1/40$ and $\sigma=100$. The variable $n$, controlling the number of vortices, and hence influencing both $\|{{\bm \beta}}\|_{W^{1,\infty}(\Omega)}$ and $\|{{\boldsymbol{u}}}\|_{H^2(\Omega)}$ is taken in the set $n \in \{1,2,4,8\}$. We observe (approximately) linear growth in both velocity and pressures, except for the pressure for the method using the RT element, where the growth is stronger. For the highest value $n=8$, all errors are above $15\%$ on this mesh. In Table \[tab:sig\] we vary the coefficient $\sigma$ and see that also here the error growth for decreasing $\sigma$ is by and large linear for the velocities, as predicted by theory (Corollary \[cor:conv\]) and the RT pressure (Corollary \[cor:pres\_order\]). The BDM pressure on the other hand is very robust with respect to variations in $\sigma$, but much larger than the RT-pressure. It starts increasing only for the smallest value of the parameter, when the pressure errors of the two approximation spaces are comparable. It follows that for small values of $\sigma$ the pressure approximation is of similar quality for the BDM and RT methods.
$h$ $\|{{\boldsymbol{u}}}- {{\boldsymbol{u}}}_h\|_{L^2(\Omega)}$ $\|p - p_h\|_{L^2(\Omega)}$ CPU
-------- -------------------------------------------------------------- ----------------------------- ----------
$1/10$ $0.011\; (-)$ $0.15\; (-)$ $0.073s$
$1/20$ $0.0030\; (1.9)$ $ 0.074\; (1.0)$ $0.47s$
$1/40$ $0.00087\; (1.8)$ $ 0.037\; (1.0)$ $4.7s$
$1/80$ $0.00031\; (1.5)$ $0.019\; (1.0)$ $62.9s$
: Errors for the BDM1/P0 element. $\sigma=100$ $n=1$.[]{data-label="tab:BDM"}
$h$ $\|{{\boldsymbol{u}}}- {{\boldsymbol{u}}}_h\|_{L^2(\Omega)}$ $\|p - p_h\|_{L^2(\Omega)}$ CPU
-------- -------------------------------------------------------------- ----------------------------- ---------
$1/10$ $0.011\; (-)$ $0.026\; (-)$ $0.17s$
$1/20$ $0.0030\; (1.9)$ $ 0.0060\; (2.1)$ $1.2s$
$1/40$ $0.00087\; (1.8)$ $ 0.0018\; (1.7)$ $12s$
$1/80$ $0.00031\; (1.5)$ $0.00073\; (1.3)$ $165s$
: Errors for the RT1/P1dc element. $\sigma=100$ $n=1$.[]{data-label="tab:RT"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$n$ BDM $\|{{\boldsymbol{u}}}- {{\boldsymbol{u}}}_h\|_{L^2(\Omega)}$ BDM $\|p - RT $\|{{\boldsymbol{u}}}- RT $\|p - p_h\|_{L^2(\Omega)}$
p_h\|_{L^2(\Omega)}$ {{\boldsymbol{u}}}_h\|_{L^2(\Omega)}$
----- ------------------------------------------------------------------ ----------------------------------------------------------------- --------------------------------------------------------------------------------------------------------- --------------------------------
$1$ $0.00087$ $ 0.037$ $0.00087$ $ 0.0018$
$2$ $0.0048$ $0.074$ $0.0048$ $0.0058$
$4$ $0.031$ $0.14$ $0.031$ $0.026$
$8$ $0.21$ $0.34$ $0.21$ $0.18$
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Errors for the BDM1/P0 element (columns 2 and 3) and RT1/P1dc element (columns 4 and 5), $h=1/40$, $\sigma=100$, varying $n$.[]{data-label="tab:n"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$\sigma$ BDM $\|{{\boldsymbol{u}}}- {{\boldsymbol{u}}}_h\|_{L^2(\Omega)}$ BDM $\|p - RT $\|{{\boldsymbol{u}}}- RT $\|p - p_h\|_{L^2(\Omega)}$
p_h\|_{L^2(\Omega)}$ {{\boldsymbol{u}}}_h\|_{L^2(\Omega)}$
---------- ------------------------------------------------------------------ ----------------------------------------------------------------- --------------------------------------------------------------------------------------------------------- --------------------------------
$10^6$ $0.00061$ $ 0.037$ $0.00061$ $ 0.015$
$100$ $0.00087$ $ 0.037$ $0.00087$ $ 0.0018$
$50$ $0.0012$ $0.037$ $0.0012$ $0.0019$
$25$ $0.0021$ $0.037$ $0.0021$ $0.0022$
$10$ $0.0051$ $0.037$ $0.0051$ $0.0045$
$1$ $0.048$ $0.058$ $0.048$ $0.045$
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Errors for the BDM1/P0 element (columns 2 and 3) and RT1/P1dc element (columns 4 and 5), $h=1/40$, $n=1$, varying $\sigma$.[]{data-label="tab:sig"}
Acknowledgments {#acknowledgments .unnumbered}
===============
The work of Gabriel R. Barrenechea has been funded by the Leverhulme Trust through the Research Fellowship No. RF-2019-510. Erik Burman was partially supported by the grant: EP/P01576X/1. Johnny Guzman was partially supported by the grant: NSF, DMS \# 1620100.
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abstract: 'In supersymmetric models where the superpotential $\mu$ term is generated with $\mu\ll m_{soft}$ ([*e.g.*]{} from radiative Peccei-Quinn symmetry breaking or compactified string models with sequestration and stabilized moduli), and where the string landscape 1. favors soft supersymmetry (SUSY) breaking terms as large as possible and 2. where the anthropic condition that electroweak symmetry is properly broken with a weak scale $m_{W,Z,h}\sim 100$ GeV ([*i.e.*]{} not too weak of weak interactions), then these combined landscape/anthropic requirements act as an [*attractor*]{} pulling the soft SUSY breaking terms towards values required by models with radiatively-driven naturalness: near the line of criticality where electroweak symmetry is barely broken and the Higgs mass is $\sim 125$ GeV. The pull on the soft terms serves to ameliorate the SUSY flavor and CP problems. The resulting sparticle mass spectrum may barely be accessible at high-luminosity LHC while the required light higgsinos should be visible at a linear $e^+e^-$ collider with $\sqrt{s}>2m(higgsino)$.'
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OU-HEP-160225
[**The Higgs mass and natural supersymmetric spectrum\
from the landscape** ]{}\
[Howard Baer$^1$, Vernon Barger$^2$, Michael Savoy$^1$ and Hasan Serce$^1$ ]{}\
[ *$^1$Dept. of Physics and Astronomy, University of Oklahoma, Norman, OK 73019, USA\
*]{} [ *$^2$Dept. of Physics, University of Wisconsin, Madison, WI 53706 USA\
*]{}
The Standard Model is afflicted with several naturalness problems:
1. in the electroweak sector, why is the Higgs mass $m_h\simeq 125$ GeV so light when quadratic divergences seemingly destabilize its mass[@susskind] and
2. why is the QCD Lagrangian term $\frac{\bar{\theta}}{32\pi^2}G_{A\mu\nu}\tilde{G}_A^{\mu\nu}$ so tiny ($\bar{\theta}\alt 10^{-10}$ from measurements of the neutron electric dipole moment) when its existence seems a necessary consequence of the $\theta$ vacuum solution to the $U(1)_A$ problem (the strong $CP$ problem)[@peccei]?
3. A third naturalness problem emerges when gravity is included into the picture: why is the cosmological constant $\Lambda\simeq 10^{-47}$ GeV$^4\ll M_P^4$ so small when there is no known mechanism for its suppression[@weinberg]?
Each of these problems requires an exquisite fine-tuning of parameters to maintain accord with experimental data. Such fine-tuning is thought to represent some pathology with or missing element within the underlying theory and cries out for a “natural” solution in each case.
The most compelling solution to problem \#1 is to extend the spacetime symmetry structure which underlies quantum field theory to include its most general structure: the super-Poincare group which includes supersymmetry (SUSY) transformations[@witten; @wss]. The extended symmetry implies a Fermi-Bose correspondence which guarantees cancellation of quadratic divergences to all orders in perturbation theory. Supersymmetrization of the SM implies the existence of superpartner matter states with masses of order $M_S\sim 1$ TeV[@susyreviews; @wss]. Searches are underway at the CERN LHC for evidence of the superpartner matter states.
The most compelling solution to problem \#2 is to postulate an additional spontaneously broken global Peccei-Quinn (PQ) symmetry and its concommitant axion field $a$ which induces additional potential contributions that allow the offending $CP$ violating term to dynamically settle to a tiny value[@pqww; @ksvz; @dfsz]. Searches for the physical axion field are proceeding at experiments like ADMX[@admx] but so far sensitivity has barely reached parameter values needed to solve the strong $CP$ problem.
At present the leading solution to problem \#3 is the hypothesis of the landscape: a vast number of string theory vacua states each with different physical constants[@landscape]. In this case, the cosmological constant ought to be present, but if it is too large, then the universe would expand too quickly to allow for galaxy condensation and there would be no observers present to measure $\Lambda$. This “anthropic” explanation for the magnitude of $\Lambda$ met with great success by Weinberg[@weinberg2] who was able to predict its value to within a factor of a few even well before it was measured[@Lambda].
While the SUSY solution to the scalar mass problem seems convincing at the level of quadratic divergences, there is a high level of concern that the fine-tuning problem has re-arisen in light of 1. the apparently severe LHC bounds on sparticle masses and 2. the rather high measured value of $m_h$. This perception arises from two viewpoints on measuring naturalness.
- Log-divergent contributions to the Higgs mass $\delta m_h^2\sim \frac{-3f_t^2}{16\pi^2}m_{\tst}^2\log\left(\Lambda^2/m_{\tst}^2\right)$ become large for TeV-scale top squark masses $m_{\tst}$ and $\Lambda$ as high as $m_{GUT}\simeq 2\times 10^{16}$ GeV[@harnik]. This argument has been challenged in that a variety of inter-dependent log terms, some positive and some negative, contribute to the Higgs mass. Evaluation of the combined log terms via renormalization group equations reveals the possibility of large cancellations in evaluation of the Higgs mass[@comp3; @seige].
- The EENZ/BG fine-tuning measure[@eenz] $\Delta_{\rm BG}=max_i|\frac{\partial\log m_Z^2}{\partial\log p_i}|$ (where $p_i$ are fundamental parameters of the theory) is traditionally evaluated using the various soft SUSY breaking terms as fundamental parameters. In this case, low $\Delta_{\rm BG}$ favors sparticle masses in the 100 GeV range. These evaluations have been challenged in that in more fundamental theories, the soft terms are not independent, but are derived in terms of more fundamental quantities, for instance the gravitino mass $m_{3/2}$ in supergravity theories. Evaluation of $\Delta_{\rm BG}$ instead in terms of $\mu^2$ and $m_{3/2}^2$ allows for just $\mu$ and $m_{H_u}$ to be $\sim 100$ GeV while the other sparticles can safely lie at or beyond the TeV scale[@comp3; @seige].
A more conservative measure which is in accord with the above (corrected) measures is to evaluate just the weak scale contributions to the $Z$ mass. The minimization condition for the Higgs potential $V_{\rm tree} + \Delta V$ in the minimal supersymmetric Standard Model (MSSM) reads = -\^2 . \[eq:mzs\] The radiative corrections $\Sigma_u^u$ and $\Sigma_d^d$ include contributions from various particles and sparticles with sizeable Yukawa and/or gauge couplings to the Higgs sector. Expressions for the $\Sigma_u^u$ and $\Sigma_d^d$ are given in the Appendix of Ref. [@rns].
A naturalness measure $\Delta_{\rm EW}$ has been introduced[@ltr; @rns] which compares the largest contribution on the right-hand-side of Eq. \[eq:mzs\] to the value of $m_Z^2/2$. If they are comparable ($\Delta_{\rm EW}\alt 10-30$), then no unnatural fine-tunings are required to generate $m_Z=91.2$ GeV. The main requirement for low fine-tuning is then that
$|\mu |\sim m_Z$[@Chan:1997bi; @Barbieri:2009ew; @hgsno] (with $\mu \agt 100$ GeV to accommodate LEP2 limits from chargino pair production searches) and also that
$m_{H_u}^2$ is driven radiatively to small, and not large, negative values [@ltr; @rns]. Also,
the top squark contributions to the radiative corrections $\Sigma_u^u(\tst_{1,2})$ are minimized for TeV-scale highly mixed top squarks[@ltr]. This latter condition also lifts the Higgs mass to $m_h\sim 125$ GeV.
First and second generation squark and slepton masses may range as high as 10-20 TeV with little cost to naturalness[@rns; @upper].
The typical low $\Delta_{\rm EW}$ SUSY mass spectra is characterized by 1. a set of light higgsinos $\tw_1^\pm$ and $\tz_{1,2}$ with masses $\sim 100-200$ GeV, 2. gluinos with mass $m_{\tg}\sim 1.5-4$ TeV, 3. highly mixed stops with mass $m_{\tst_1}\alt 3$ TeV and $m_{\tst_2}\alt 8$ TeV. Several versions of supergravity GUT models have been found to generate such “natural” spectra[@guts]. For instance, the two-extra-parameter non-universal Higgs mass model[@nuhm2] (NUHM2) with matter scalars $m_0\sim 3-10$ TeV, $m_{1/2}\sim 0.5-2$ TeV, $A_0\sim\pm(1-2)m_0$ and $\tan\beta\sim 10-30$ with $m_{H_u}\sim (1.3-2)m_0$ and $m_{H_d}\sim m_A\sim 1-8$ TeV produces spectra with $\Delta_{\rm EW}\alt 30$. In particular, the up-Higgs soft mass is as large as possible such that the RG running of $m_{H_u}^2$ nearly cancels out its GUT-scale boundary value $m_{H_u}^2(\Lambda )$, [*i.e.*]{} $m_{H_u}^2$ runs to small weak scale values $\sim -(100-200)^2$ GeV$^2$ so that electroweak symmetry is [*barely broken*]{}. The soft terms, especially $m_{H_u}^2$, lie on the edge of [*criticality*]{}: if $m_{H_u}^2$ is much bigger, then EW symmetry does not get broken while if $m_{H_u}^2(\Lambda )$ is much smaller, then it would likely generate a value of $m_Z$ far beyond its measured value of 91.2 GeV.
While such effective theory parameters can successfully generate natural SUSY mass spectra, the question arises: is there some mechanism which favors parameters which barely break EW symmetry, and which generate a weak scale $m_{W,Z,h}\sim 100$ GeV rather than say in the TeV range? In this letter, we argue that the string landscape– which provides some understanding for the small but non-zero cosmological constant– also favors soft SUSY breaking terms as large as possible such that they generate a universe which is habitable for observers: if the soft parameters were much larger, then they would lead to a vacuum state with color breaking minima, or unbroken EW symmetry or if they were much smaller they would generate a weak scale characterized by the TeV regime. In the latter case, with $m_{W,Z,h}\sim 1-10$ TeV, then weak interactions would be far weaker than in our universe: then for instance nuclear fusion reactions would be sufficiently suppressed so that heavy element production in stars and in the early universe would be far different from that of our universe, likely leading to a universe with chemistry unsuitable for life forms as we known them.
This topic of anthropic selection of soft SUSY breaking terms has been addressed previously by Giudice and Rattazzi[@GR] with some follow-up work in Ref’s [@NP] (for mixed moduli-anomaly mediated SUSY breaking models) and [@DM] (for mSUGRA/CMSSM model). One of the main differences of our work here is in the treatment of the superpotential $\mu$ parameter and the so-called SUSY $\mu$ problem. Under the Giudice-Masiero mechanism[@GM], where $\mu$ arises from Higgs doublet couplings to the hidden sector via the Kahler potential, then $\mu$ is expected to have magnitude of order the other soft terms: $|\mu |\sim m_{3/2}$. Alternatively, in the Kim-Nilles mechanism[@KN]– which is assumed here as an axionic solution to the strong CP problem– $\mu$ is initially forbidden by the requirement of Peccei-Quinn symmetry, but is then re-generated upon spontaneous PQ symmetry breaking at a scale $f_a\sim 10^{11}$ GeV with a value $\mu\sim f_a^2/M_P\ll m_{3/2}$. In models where PQ symmetry breaking is induced radiatively, then values of $m_{3/2}\sim 10$ TeV easily produce $\mu$ values around 100-200 GeV[@msy; @radpq]. In classes of compactified string models with sequestration between the visible sector and the SUSY breaking sector and with stabilized moduli fields[@quevedo], it is also found that $\mu\ll M_S$ where $M_S$ stands for the approximate scale of the collective soft SUSY breaking terms. In this letter we will assume
the superpotential $\mu$ term has been generated by some mechanism such as Ref. [@radpq] or Ref. [@quevedo] to be small, comparable to $m_h= 125$ GeV. Then, instead of fixing $m_Z$ at its measured value, we will invert the usual usage of Eq. \[eq:mzs\] to calculate $m_{W,Z,h}\sim m_{weak}$ as an [*output*]{} depending on high scale values of the soft terms and a small value of $\mu$.[^1] In the following, we will assume gravity-mediated supersymmetry breaking[@sugra]. Gravity-mediation is supported by the large value of $m_h\sim 125$ GeV which requires a large trilinear $A_0$ term (generic in gravity-mediation) to provide substantial mixing in the stop sector and consequently a boost in the radiative corrections to the light Higgs mass[@mhiggs; @h125]. Gravity-mediated SUSY breaking can be parametrized by the presence of a spurion superfield $X=1+\theta^2 F_X$ where the auxiliary field $F_X$ obtains a vev which we also denote by $F_X$ (here $\theta$ are anti-commuting superspace coordinates). Under SUSY breaking via the superHiggs mechanism, then the gravitino gains a mass $m_{3/2}\sim F_X/M_P$ where $M_P=2.4\times 10^{18}$ GeV is the reduced Planck mass. The soft SUSY breaking terms are then all calculable as multiples of $m_{3/2}$[@sw]. Motivated by supergravity grand unified theories (SUSY GUTs), here we assume the soft breaking terms valid at $Q=m_{GUT}\simeq 2\times 10^{16}$ GeV include $m_0$ (a common matter scalar mass term), $m_{1/2}$ (a common gaugino mass), $A_0$ (a common trilinear soft term) and $B$. The latter soft term can be traded for the more common ratio of Higgs vevs $\tan\beta\equiv v_u/v_d$ via the electroweak minimization conditions. We also assume separate Higgs scalar soft terms $m_{H_u}^2$ and $m_{H_d}^2$ since the Higgs superfields live in different GUT representations than matter superfields[@nuhm2]. It is convenient to denote collectively the superpartner mass scale $M_S$ as the generic scale of soft terms.
It is reasonable to assume in the landscape that [*any*]{} value of the complex-valued field $F_X$ is equally likely. In this case, one expects the magnitude of soft breaking terms to statistically scale linearly in $M_S$ (the likelihood of a given value of $M_S$ is proportional to the area of an annulus $2\pi F_X\delta F_X$ in the complex $F_X$ plane). This is important because then we see a statistical draw of soft terms towards their largest values possible (while $\mu$ remains far smaller). In Ref. [@GR], additional arguments are presented that the likelihood of soft terms $M_S$ scale as a power of $M_S$; for our purposes here, we merely rely on a likely statistical draw by the landscape of vacua towards higher values of soft terms. This draw is to be balanced by the anthropic requirements that 1. electroweak symmetry is appropriately broken (no charge or color breaking minima of the Higgs potential) and 2. that the weak scale is typified by the values of $m_{weak}\sim m_{W,Z,h}\sim 100$ GeV. Rates for nuclear fusion reactions and beta decays all scale as $1/m_{weak}^4$ so that heavy element production in BBN and in stars would be severely altered for too large a value of $m_{weak}$; see Ref’s [@kribs] for discussion.
Armed with a notion of both the statistical and anthropic pull from the landscape, we may examine the soft SUSY breaking terms. First, we expect the matter scalar mass $m_0$ as large as possible while maintaining $m_{weak}\sim 100$ GeV. If $m_0$ gets much beyond the 10 TeV scale, then the weak scale top squark masses $m_{\tst_{1,2}}$ become too large, increasing the radiative corrections $\Sigma_u^u(\tst_{1,2})$ in Eq. \[eq:mzs\]. For fixed $\mu\sim 100-200$ GeV, then this increases the resultant weak scale well beyond the anthropic target 100-200 GeV. Re-interpreting the limits on $m_0$ from Ref’s [@rns; @upper] requires $m_0\alt 10$ TeV for $m_{weak}\sim 100$ GeV. Such large values of $m_0$ go a long ways towards solving the SUSY flavor and CP problems via a decoupling solution[@dine].
Likewise, we expect the gaugino mass $m_{1/2}$ as large as possible whilst maintaining $m_{weak}\sim 100-200$ GeV. If the gaugino masses are too large, then they feed into the stop masses via RG running and again the $\Sigma_u^u(\tst_{1,2})$ become too large. For $m_{weak}\sim 100$ GeV, then typically $m_{1/2}\alt 2$ TeV leading to a gluino mass bound $m_{\tg}\alt 4-5$ TeV: well above the reach of LHC14[@andre].
![Contours of $m_{weak}$ in the $A_0$ vs. $m_0$ plane for $m_{1/2}=1$ TeV, $m_{H_u}=1.3 m_0$, $\tan\beta =10$ and $m_{H_d}=1$ TeV. The arrows show the direction of statistical/anthropic pull on soft SUSY breaking terms. \[fig:A0\_m0\]](nuhm_alt){height="0.3\textheight"}
What of the trilinear soft term $A_0$? In Fig. \[fig:A0\_m0\] we show the $A_0$ vs. $m_0$ plane for the NUHM2 model with $m_{1/2}$ fixed at 1 TeV, $\tan\beta =10$ and $m_{H_d}=1$ TeV. We take $m_{H_u}=1.3 m_0$. The plane is qualitatively similar for different reasonable parameter choices. We expect $A_0$ and $m_0$ statistically to be drawn as large as possible while also being anthropically drawn towards $m_{weak}\sim 100-200$ GeV, labelled as the red region where $m_{weak}<500$ GeV. The blue region has $m_{weak}>1.9$ TeV and the green contour labels $m_{weak}=1$ TeV. The arrows denote the combined statistical/anthropic pull on the soft terms: towards large soft terms but low $m_{weak}$. The black contour denotes $m_h=123$ GeV with the regions to the upper left (or upper right, barely visible) containing larger values of $m_h$. We see that the combined pull on soft terms brings us to the region where $m_h\sim 125$ GeV is generated. This region is characterized by highly mixed TeV-scale top squarks[@mhiggs; @h125]. If instead $A_0$ is pulled too large, then the stop soft term $m_{U_3}^2$ is driven tachyonic resulting in charge and color breaking minima in the scalar potential (labelled CCB). If $m_0$ is pulled too high for fixed $A_0$, then electroweak symmetry isn’t even broken.
![Contours of $m_{weak}$ (blue) in the $m_{H_u}$ vs. $m_{1/2}$ plane for $m_0=5$ TeV, $A_0=-8$ TeV, $\tan\beta =10$ and $m_{H_d}=1$ TeV. Above the black dashed contour is where $m_h>124$ GeV. The red region has $m_{weak}<0.5$ TeV. The arrows show the direction of the statistical/anthropic pull on soft SUSY breaking terms. \[fig:mHu\_mhf\]](mHu_mhf){height="0.3\textheight"}
In Fig. \[fig:mHu\_mhf\], we show contours of $m_{weak}$ in the $m_{H_u}$ vs. $m_{1/2}$ plane for $m_0=5$ TeV, $A_0=-8$ TeV, $\tan\beta =10$ and $m_{H_d}=1$ TeV. The statistical flow is to large values of soft terms but the anthropic flow is towards the red region where $m_{weak}<0.5$ TeV. While $m_{1/2}$ is statistically drawn to large values, if it is too large then, as before, the $\tst_{1,2}$ become too heavy and the $\Sigma_u^u(\tst_{1,2})$ become too large so that $m_{weak}$ becomes huge. The arrows denote the direction of the combined statistical/anthropic flow. The region above the black dashed contour has $m_h>124$ GeV. The value of $m_{H_u}(GUT)$ would like to be statistically as large as possible but if it is too large then EW symmetry will not break. Likewise, if $m_{H_u}(GUT)$ is not large enough, then it is driven to large negative values so that $m_{weak}\sim$ the TeV regime and weak interactions are too weak. The situation is shown in Fig. \[fig:mHuQ\] where we show the running of $sign(m_{H_u}^2)\sqrt{|m_{H_u}^2|}$ versus energy scale $Q$ for several values of $m_{H_u}^2(GUT)$ for $m_{1/2}=1$ TeV and with other parameters the same as Fig. \[fig:mHu\_mhf\]. Too small a value of $m_{H_u}^2(GUT)$ leads to too large a weak scale while too large a value results in no EWSB. The combined statistical/anthropic pull is for barely-broken EW symmetry where soft terms teeter on the edge of criticality: between breaking and not breaking EW symmetry. This yields the other naturalness condition that $m_{H_u}$ is driven small negative: then the weak interactions are of the necessary strength. These are just the same conditions for supersymmetric models with radiatively-driven natural SUSY (RNS)[@ltr; @rns].
![Evolution of the soft SUSY breaking mass squared term $sign(m_{H_u}^2)\sqrt{|m_{H_u}^2|}$ vs. $Q$ for the case of no EWSB (upper), criticality (middle) as in radiatively-driven natural SUSY (RNS) and $m_{weak}\sim 3$ TeV (lower). Most parameters are the same as in Fig. \[fig:mHu\_mhf\]. \[fig:mHuQ\]](mHuQ){height="0.4\textheight"}
[*Summary:*]{} The naturalness condition of no large unnatural cancellations in $m_{Z,h}$ requires small higgsino mass $\mu\sim 100-200$ GeV, $m_{H_u}^2$ driven small rather than large negative and not-too-large radiative corrections $\Sigma_u^u (i)$. There are mechanisms where $\mu\ll M_S$– such as radiative PQ breaking– but is it merely luck that the soft terms are poised to be just large enough to guarantee also that $m_{weak}\sim 100$ GeV? Here, we argue that the statistical landscape pull towards large soft terms coupled with the anthropic pull towards the Goldilocks condition– small enough to break EW symmetry but not so small as to suppress weak interactions– gives the required conditions for SUSY with radiatively-driven naturalness and barely broken EW symmetry. While sparticles may barely be accessible to LHC, the required light higgsinos should be accessible to an $e^+e^-$ collider with $\sqrt{s}>2 m(higgsino)$. We also expect ultimately detection of a higgsino-like WIMP[@wimp] along with the axion.
[*Acknowledgements:*]{} We thank Jake Baer for an inspiring essay on the landscape and Xerxes Tata for comments on the manuscript. This work was supported in part by the US Department of Energy, Office of High Energy Physics.
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[^1]: In this case, low values of $\Delta_{\rm EW}$ can be re-interpreted as the likelihood to generate the weak scale $m_{weak}\sim 100$ GeV: [*i.e.*]{} $m_{weak}=\sqrt{\Delta_{\rm EW}m_Z^2/2}$.
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abstract: 'The existence and regularity of the classical plurisubharmonic solution for complex Monge-Ampère equations subject to the semilinear oblique boundary condition which is $C^1$ perturbation of the Neumann boundary condition, are proved in the certain strictly pseudoconvex domain in $\bf C^n$.'
address:
- 'Faculty of Mathematics $\&$ Computer Science, Hu Bei University, P.R. China'
- 'School of Science, Nanjing University of Sci. $\&$ Tech., Nanjing, P.R.China 210094'
author:
- Ni Xiang
- 'Xiao-Ping Yang'
title: 'Complex Monge-Ampère Equations with Oblique Boundary Value'
---
[^1]
Introduction and main results
==============================
Let $\Omega$ be a bounded strictly pseudoconvex domain in $\bf C^n$ with $C^\infty$ boundary, $f(z)$ be a positive function on $\Omega$, and $\varphi(z,u)$ be a function on $\partial \Omega\times \bf R$. We shall study the existence and regularity of plurisubharmonic solutions to the complex Monge-Ampère equations: $$\label{cma}
\det\frac{\partial^2u}{\partial z_i\partial\overline{z_j}}=f(z), \ \
\ \ z\in\Omega,$$ with the oblique boundary condition: $$\label{ob1}
D_\beta u=\varphi(z,u),\ \ \ \ \ z\in
\partial\Omega .$$
Assume $\beta$ is a strictly oblique unit vector field, satisfying $$\label{ob2'}\beta(z)\cdot\nu(z)\geq\beta_0>0,\ $$ where $ \nu(z) $ is the unit outer normal vector field to $\partial\Omega$. In order to apply the method of continuity for the existence of solutions, it is necessary to obtain a priori estimates. We also need to assume some conditions on $\varphi$ to derive the maximum modulus estimates. We assume the conditions in [@LTU; @U2] namely $$\label{ob4}\varphi(z,u)<0\ for\ all \ z\in \partial\Omega\ \ \ and \ \ all \ \ u\geq N_1,$$ for some constant $N_1$, and $$\label{ob5}\varphi(z,u)\rightarrow \infty \ \ as \ \
u\rightarrow-\infty \ \ uniformly \ \ for \ \ z\in
\partial\Omega.$$Conditions (\[ob4\]) and (\[ob5\]) are used for the upper bound and the lower bound of the solution, respectively. We also assume that $$\label{ob6} \inf
f(\overline{\Omega})>0.$$
The Dirichlet problem for complex Monge-Ampère equations (\[cma\]) has been an object of intensive research. In 1976, Bedford and Taylor in [@BT] considered weak plurisubharmonic solutions for complex Monge-Ampère equations: $\det(\frac{\partial^2u}{\partial z_i\partial \overline
{z_j}})dV=\mu$, where $\mu$ was a bounded non-negative Borel measure. Cheng and Yau in [@CY] solved the Dirichlet problem for the equations $\det(\frac{\partial^2 u}{\partial z_i\partial
\overline{z_ j}})=f(z,u,\nabla u)$, $z\in\Omega$ with $f=e^u$, $u=\infty$ on $\partial \Omega$, obtaining a solution $u\in
C^\infty(\Omega)$. In the non-degenerate case $(f>0)$, the existence and uniqueness of a classical plurisubharmonic solution subject to the Dirichlet boundary condition for complex Monge-Ampère equations, under suitable restrictions on $f$ and $\varphi$, have been proved by Caffarelli, Kohn, Nirenberg and Spruck in [@CKNS].
The methods to deal with the oblique boundary value problems for real Monge-Ampère equations originated from the paper of Lions, Trudinger and Urbas in [@LTU] dealing with the Neumann boundary value problems. In 1994, Li Song-Ying in [@L1] considered the Neumann problem for complex Monge-Ampère equations in strictly pseudoconvex domain in $\bf C^n$ with smooth boundary such that, $$\inf_{\partial\Omega}(\gamma_0+2\lambda_2(z))>0,$$ where $\varphi(z,u)=-\gamma_0u+\varphi(z)$, $\lambda_{2}(z)$ denotes the smallest principal curvature of $\partial \Omega$ at $z\in
\partial\Omega$. He obtained the existence and uniqueness of classical plurisubharmonic solutions. The existence of generalized solutions for $$\label{cma1}\det D^2u=f(x,u,Du)\ \ in \ \
\Omega$$ and $$\label{obl}D_\beta u+\phi(x,u)=0\ \ on \ \
\partial\Omega$$ has been established in [@W] by Wang Xu-jia for general strictly oblique vector field $\beta$ under relatively weak regularity hypotheses on the data. In [@U1; @W] examples were constructed showing that one cannot generally expect smoothness of solutions of (\[cma\]) and (\[ob1\]), no matter how smooth $\beta,\phi,f$ and $\partial \Omega$ are. Some structural condition on $\beta$ is necessary. The examples in [@U1; @W] are constructed in such a way that $\beta$ can be made arbitrarily close to $\nu$ in the $C^0$ norm, but not $C^1$ norm. Therefore, Urbas in [@U2] proved the existence of classical solutions for real Monge-Ampère type equations subject to the semilinear oblique boundary condition which is a $C^1$ perturbation of the Neumann boundary condition. In 1999, Li Song-Ying in [@L2] used different methods than Urbas in [@U2] to prove the existence, uniqueness and regularity for oblique boundary value problem to real Monge-Ampère equations in a smooth bounded strictly convex domain.
In this article we prove the existence and regularity of the strictly plurisubharmonic solution for the oblique problem (\[cma\]) and (\[ob1\]) in strictly pseudoconvex domains under suitable conditions.
In order to prove the existence of plurisubharmonic solutions, we have to deduce a priori estimates for the derivative of such solutions up to second order. In the real case, Lions, Trudinger and Urbas in [@LTU] used the convexity of the solution to obtain the first order derivative bound for the oblique boundary value problems. Unfortunately, there is no convexity for strictly plurisubharmonic function. Inspired by the method in [@L1] and [@U2], we employ similar argument to obtain a priori estimates.
Now we can state our theorem as follows:
\[thm\] Assume $\Omega$ is a bounded strictly pseudoconvex domain in $\bf
C^n$ with $\partial\Omega\in C^4$. Suppose $f\in
C^2(\overline{\Omega})$, $\varphi$ and $\beta$ satisfy the conditions (\[ob2’\])-(\[ob6\]). In addition, we assume that $\varphi\in C^{2,1}(\partial\Omega\times \bf R)$ with $$\label{ob3}\varphi_u(z,u)\leq -\gamma_0<0 \ on\
\partial \Omega\times \bf R,\ \ \inf_{\partial\Omega}(\gamma_0+2\lambda_2(z))>0,$$ where $\lambda_{2}(z)$ denotes the smallest principal curvature of $\partial \Omega$ at $z\in \partial\Omega$. Then there is a positive constant $\epsilon_0 >0$ such that $$\label{ob2}
\|\beta-\nu \|_{C^1(\partial \Omega)} \le \epsilon_0,$$ thus the oblique problem for complex Monge-Ampère equations (\[cma\]) and (\[ob1\]) has a unique plurisubharmonic solution $u\in C^{2,\alpha}(\overline\Omega) $ for any $\alpha<1$.
Note that the regularity for the solution $u$ in Theorem \[thm\] can be improved by the linear elliptic theory in [@GT] if the datas are sufficiently smooth.
The paper is organized as follows: in Section 2, we introduce some terminology, then derive $C^0$ estimates for the solutions. In Section 3, we study the first order derivative estimates. In section 4, we complete the second order derivative estimates. In Section 5, we give the proof of our main theorem by using the method of continuity.
Maximum modulus estimates
=========================
In this section, we first introduce some terminology. Then we derive the maximum modulus estimate of the solution $u$ for the oblique boundary value problem (\[cma\]) and (\[ob1\]) under some structural conditions on $\varphi(z,u)$.
Let $z=(z_1,...,z_n)\in \bf C^n$, where $z_i=x_i+\sqrt{-1}y_i,\ (i=1,\cdots,n)$. We may write z in real coordinates as $z=(t_1,...,t_{2n})$. Given $\xi\in \bf R^{2n}$, $D_\xi$ denotes the directional derivative of $u$ along $\xi$. In particular, $D_k=\frac{\partial}{\partial t_k}$. For the complex variables, we shall use notations: $\partial_k=\frac{\partial}{\partial z_k}$, $\partial_{\overline k}=\frac{\partial}{\partial {\overline {z_k}}}$ and $\partial_{i \bar j}=\frac{\partial^2}{\partial z_i \partial
\overline{z_j}}$. $u^{i\bar j}$ denotes the inverse matrix of $u_{i\bar j}$. For a compact set $X$, we let $|u|_{k,X}$ denote the $C^k$ norm on $X$.
Let $f>0$ and $g=\log f$. Then $\log[\det(u_{i\bar j})]$ is a concave function in $u_{i\bar j}$. We use Einstein convention. If $u$ is a solution of (\[cma\]), and $\xi\in \bf R^{2n}$, then $$\label{2.1}u^{i \bar j}\partial_{i\bar j}D_\xi u =D_\xi \log [\det
(u_{i\bar j})]=D_\xi g,$$ $$\label{2.2}
u^{i\bar j}\partial_{i\bar j}D_{\xi\xi}u\geq D_{\xi\xi}g
.$$ Moreover, if we let $\widetilde{f}=f^{\frac{1}{n}}$ and $F^{i\bar
j}=\frac{1}{n}\widetilde{f}u^{i\bar j}$, then $$\label{2.3}
F^{i\bar j}\partial _{i\bar j}D_\xi u= D_\xi
\widetilde{f}, F^{i\bar j}\partial_{i\bar j}D_{\xi\xi}u\geq
D_{\xi\xi}\widetilde{f},$$ $$\label{2.4}tr(F^{i\bar{j}})=\frac{1}{n}\widetilde{f}tr(u^{i\bar j})\geq n\frac{1}{n}\widetilde{f}\widetilde{f}^{-1}=
1.$$
Although the argument in this part is rather standard, for completeness, we give the maximum principle without proof:\
\[l1\] Suppose that $L=F^{i\bar j}\partial_{i\bar j}$ is elliptic, and $LH\geq 0 (\leq 0)\ \ in \ \Omega$ with $H\in C^2(\Omega)\cap
C^0(\overline \Omega)$. Then the maximum (minimum) of $H$ in $\overline\Omega$ is achieved on $\partial\Omega$, that is ,$$\sup_\Omega H=\sup _{\partial\Omega}H\ \
(\inf_\Omega H=\inf_{\partial\Omega}H).$$
\[l3\] Let $\Omega$ be a bounded domain in $\bf C^n$ with $C^1$ boundary, and $u\in C^2(\Omega)\bigcap C^1(\overline \Omega)$ be a plurisubharmonic solution of (\[cma\]) and (\[ob1\]). Assume $\varphi$ satisfies assumption (\[ob4\]), then $u(z)\leq N_1 $ on $\overline\Omega$.
Since $u$ is plurisubharmonic, $u$ attains its maximum over $\overline\Omega$ on $\partial\Omega$, say at $z_0\in
\partial\Omega$, then $D_\nu u(z_0)\geq0$. On the other hand, $\beta$ can be decomposed into normal and tangential part, from (\[ob2’\]) and the vanishing of the tangential derivative at $z_0$, we obtain $ D_\beta u(z_0)\geq 0.$ If $u(z_0)\geq N_1,$ by (\[ob4\]) we have $\varphi(z_0,u(z_0))<0$, this is contradiction. So $u(z_0)< N_1$ and the proof is complete.
\[l4\] Let $u\in C^2(\Omega)\cap C^1(\overline\Omega)$ be a plurisubharmonic solution of (\[cma\]) and (\[ob1\]). Assume that $\varphi$ satisfies (\[ob5\]), then $u(z)\geq N_2>-\infty,$ $z\in \overline \Omega$, where $N_2$ is a constant depending only on $|f|_{0,\overline\Omega},\gamma_0,\varphi$.
We use the auxiliary function $$h(z)=u(z)-\eta |z|^2,
\eta=|f|^{\frac{1}{n}}_{0,\overline\Omega}+1.$$ We assume that $h$ attains its minimum over $\overline\Omega$ at $z_0\in \overline\Omega$. Now we want to show $z_0\in
\partial\Omega$. If $z_0\in \Omega$, then $$\begin{array}{ll}
0&\leq Lh(z_0)\\
&= F^{i\bar j}(z_0)\partial_{i\bar j}h(z_0)\\
&=F^{i\bar j}(z_0)\partial_{i\bar j}u(z_0)-
\eta F^{i\bar j}(z_0)\delta_{ij}\\
&=f^{\frac{1}{n}}-\eta tr(F^{i\bar j}(z_0))\\
&\leq f^{\frac{1}{n}}-\eta \\
&\leq-1\\
&<0.
\end{array}$$ This inequality leads to a contradiction, so we have shown $z_0\in
\partial \Omega$. Without loss of generality, we can assume $u(z_0)\leq 0$, otherwise we obtain the result. Therefore, by the oblique boundary condition (\[ob1\]), $$\begin{array}{ll}
0 &\geq D_\beta h(z_0)\\
&=D_\beta
u(z_0)-D_\beta(\eta|z|^2)|_{z=z_0}\\
&=\varphi(z_0,u(z_0))-D_\beta(\eta|z|^2)|_{z=z_0}.
\end{array}$$ Then $$\varphi(z_0,u(z_0))\leq D_\beta(\eta|z|^2)|_{z=z_0}\leq C.$$ Then by the condition (\[ob5\]), we have $u(z_0)>\widetilde{N}$, where $\widetilde{N}>-\infty$ is a constant depending on $n,\
\Omega,\ \varphi, |f|_{0,\overline{\Omega}}^{\frac{1}{n}}$. For all $z\in\Omega$, $h(z)\geq h(z_0)$, which leads to, $$\begin{array}{ll}
u(z)
&=h(z)+\eta|z|^2\\
&\geq h(z_0)+\eta|z|^2\\
&=u(z_0)+\eta|z|^2-\eta|z_0|^2\\
&\geq \widetilde{N}+\eta|z|^2-\eta|z_0|^2\\
&\geq N_2\\
&>-\infty,
\end{array}$$ where $N_2$ depends on $n,\ \Omega,\ \varphi,
|f|_{0,\overline{\Omega}}^{\frac{1}{n}}$. The proof of Lemma \[l4\] is complete.
\[thm2\] Let $\Omega$ be a bounded strictly pseudoconvex domain with $C^1$ boundary. Assume that $f$ is non-negative, and $\varphi
$ satisfies (\[ob4\])-(\[ob5\]). If $u\in C^2(\Omega)\cap
C^1(\overline\Omega)$ is a plurisubharmonic solution of (\[cma\]) and (\[ob1\]), then $|u|_{0,\overline \Omega}\leq C$, where $C$ is a constant depending only on $n,\Omega,|f|_{0,\overline \Omega},
\gamma_0,\varphi$.
In the proof of Lemma \[l4\], the estimate on $|u|_{0,\overline\Omega}$ is independent of the lower bound of $f$. So when $f\geq 0$, we can obtain the same estimate by considering $f_\epsilon=f+\epsilon, \epsilon>0$ first, then let $\epsilon\rightarrow 0^+$ to complete the proof of theorem \[thm2\]. If a similar argument is needed in the following sections, we will not repeat again.
We remark that $\widetilde{C_l}$ and $C_l$ $(l=1,2,\cdots)$denote the constants depending on the known datas. As usual, constants may change from line to line in the context.
Gradient estimates
==================
In this section, we follow the idea in [@L1] and [@U2] to derive gradient estimates.
\[ge\] Let $\Omega$ be a bounded strictly pseudoconvex domain in $\bf C^n$ with $C^3$ boundary. Assume $\beta$, $\varphi\in C^{1,1}(\partial
\Omega\times \bf R)$ and $f\in C^1(\overline{\Omega})$ satisfy (\[ob2’\])-(\[ob6\]). In addition $\varphi$ satisfies (\[ob3\]). Then $$\label{3.1'}|u|_{1,\overline{\Omega}}\leq
C,$$ where $C$ is a constant depending only on $\
|f^{\frac{1}{n}}|_{1,\overline\Omega},\ \gamma_0,\ \Omega,\
\beta\ and \ |\varphi|_{C^{1,1}(\partial\Omega)}$.
In order to prove $|u|_{1,\overline\Omega}\leq C$, from Theorem \[thm2\] it suffices to prove $$\label{g1} D_\xi
u(z)\leq C,\ \ \forall\xi\in S^{2n-1},$$ where $S^{2n-1}$ is the unit sphere in $R^{2n}$. We still use $\varphi$ to denote a $C^1$ extension of $\varphi$ from $\partial\Omega$ to $\overline\Omega.$
First, we reduce the gradient estimates to the boundary by choosing the auxiliary function $R(z,\xi)=D_{\xi}u(z)+\eta_1|z|^2$, where $\eta_1=|f^{\frac{1}{n}}|_{1,\overline{\Omega}}+1$. By calculation, we have $LR>0$. By the maximum principle, $$\label{3.3}
\begin{array}{ll}
\max_{\overline{\Omega}}D_\xi u(z)&\leq \max_{\overline{\Omega}}(D_\xi u(z)+\eta_1|z|^2)\\
&\leq\max_{\partial\Omega}(D_\xi u(z)+\eta_1|z|^2)\\
&\leq\max_{\partial\Omega}D_\xi u(z)+\eta_1 diam(\Omega)^2.
\end{array}$$
So the rest task is the estimation of $D_\xi u(z)$ on $\partial\Omega$. From the condition (\[ob1\]), when $z\in\partial\Omega$, $$\label{go}
|D_{\beta} u(z)|=|\varphi(z,u(z))|\leq C_1.$$
At any boundary point, any direction $\xi$ can be written in terms of a tangential component $\tau(\xi)$ and a component in the direction $\beta$, namely $$\xi=\tau(\xi)+\frac{(\nu\cdot\xi)}{(\beta\cdot\nu)}\beta,$$ where $\tau(\xi)=\xi-(\nu\cdot\xi)\nu - \frac{(\nu\cdot\xi)}{(\beta\cdot\nu)} \beta^T$ and $\beta^T=\beta-(\beta\cdot\nu)\nu$. We compute under the condition (\[ob2’\]), $$\label{3.6}
\begin{array}{ll}
|\tau(\xi)|^2&=1-(1-\frac{|\beta^T|^2}{|(\beta\cdot\nu)|^2})(\nu\cdot\xi)^2-2\frac{(\nu\cdot\xi)(\beta^T\cdot\xi)}{(\beta\cdot\nu)}\\
&\leq 1+\frac{2}{\beta_0}|\beta^T|.
\end{array}$$ Thus, if we get the tangential derivative estimate of $u$ on $\partial\Omega$, from (\[go\]), for any $\xi\in S^{2n-1}$, we have $$\label{3.7}
\begin{array}{ll}
D_{\xi}u(z)&=D_{\tau(\xi)}u(z)+\frac{(\nu\cdot\xi)}{(\beta\cdot\nu)}D_\beta u(z)\\
&\leq D_{\tau(\xi)}u(z)+C_2\\
&=|\tau(\xi)|D_{\frac{\tau(\xi)}{|\tau(\xi)|}}u(z)+C_2\\
&\leq \sqrt{(1+\frac{2}{\beta_0}|\beta^T|)}C_3+C_2\\
&\leq C_4.
\end{array}$$
Next the key task is to get the bound for the tangential derivative of $u$ on $\partial \Omega$. We assume that the maximum tangential first order derivative on $\partial\Omega$ is attained at a boundary point which may take to be the origin, in a tangential direction which may take to be $e_1=(1,\underbrace{0,\cdots,0}_{2n-1})$, $x_n$ is the inner normal vector at 0. Thus $$\label{3.8}
D_1u(0)=\sup_{z\in \partial\Omega,\tau is\ unit\ tangential\ at z }D_\tau u(z).$$ And without loss of generality, we can assume $D_1u(0)>0$. Otherwise, we finish the proof.
We choose the auxiliary function $$Q(z)=\frac{D_{1}u(z)}{D_1u(0)}+G|z|^2-Ax_n-1,$$ and the domain $S_\mu=\{z\in\Omega:x_n\leq \mu\}$, where $G,A $ and $\mu$ are constants to be fixed.
From (\[2.3\]) we have $F^{i\bar j}\partial_{i\bar
j}\frac{D_1 u(z)}{D_1 u(0)}=\frac{D_1 f^{\frac{1}{n}}}{D_1u(0)}\geq -\frac{C_5}{D_1u(0)},$ and $$F^{i\bar j}\partial_{i\bar j}G|z|^2=G tr(F^{i\bar j})\geq G.\\$$ Thus, if we take $$\label{c1}G\geq\frac{C_5}{D_1u(0)},$$ we have $$LQ= F^{i\bar j}Q_{i\bar j}\geq0\ \ \ in\ S_\mu.$$
Then we consider the estimates of $Q(z)$ on the boundary of $S\mu$. On $\partial\Omega\cap\overline{S_\mu}$ near $0$, there is a constant $a>0$ such that(see in [@CKNS], Lemma 1.3) $$x_n\geq a|z|^2.$$ Thus $$\begin{array}{ll}
Q(z)&\leq 1+\frac{C_6|z|^2}{D_1u(0)}+G|z|^2-Ax_n-1\\
&\leq (G+\frac{C_6}{D_1u(0)}-Aa)|z|^2.
\end{array}$$ If we take $$\label{c2}Aa\geq G+\frac{C_6}{D_1u(0)},$$ then $Q(z)\leq 0$ on $\partial\Omega\cap\overline{S_\mu}$ near $0$.
On the other hand, from (\[3.3\]),(\[3.7\]) and (\[3.8\]) we have $$\begin{array}{ll}
D_1u(z)&\leq \max_{\partial\Omega}D_1 u(z)+\eta_1 diam(\Omega)^2\\
&\leq\max_{\partial\Omega}\sqrt{(1+\frac{2}{\beta_0}|\beta^T|)}D_1u(0)+C_7.
\end{array}$$ Then on $\{x_n=\mu\}\cap \overline{S_\mu}$, $$\begin{array}{ll}
Q(z)&\leq \sqrt{(1+\frac{2}{\beta_0}|\beta^T|)}+\frac{C_7}{D_1u(0)}+G|z|^2-A\mu-1\\
&\leq \sqrt{(1+\frac{2}{\beta_0}|\beta^T|)}+\frac{C_7}{D_1u(0)}+G|z|^2-A\mu-1\\
&\leq (1+\frac{2}{\beta_0}|\beta^T|)+\frac{C_7}{D_1u(0)}+G|z|^2-A\mu-1\\
&\leq \frac{2}{\beta_0}|\beta^T|+\frac{C_7}{D_1u(0)}+GC\mu-A\mu.
\end{array}$$ If $$\label{c3}\frac{2}{\beta_0}|\beta^T|+GC\mu+\frac{C_7}{D_1u(0)}\leq A\mu,$$ then $Q\leq 0$ on $\{x_n=\mu\}\cap \overline{S_\mu}$.
We now proceed to fix $G$ and $\mu$, depending on $A$ which will be fixed later. First we fix $G>0$ so small that $$CG\leq \frac{1}{2}A\ and \ G\leq\frac{1}{2}Aa,$$ then fix $\mu\in(0,1)$. Then (\[c1\]), (\[c2\]) and (\[c3\]) will be hold whenever $$\label{gzc}
\begin{array}{ll}
&\frac{C_5}{D_1u(0)}\leq G,\\
& \frac{C_6}{D_1u(0)}\leq\frac{Aa}{2},\\
& \frac{2}{\beta_0}|\beta^T|+\frac{C_7}{D_1u(0)}\leq \frac{1}{2}A\mu.
\end{array}$$ By the maximum principle we have $Q\leq 0$ in $S_\mu$, and since $Q(0)=0$, then $$D_\beta Q(0)\geq 0.$$
From (\[ob3\]) $$\begin{array}{ll}
&D_\beta D_1 u(0)\\
&=D_1D_\beta u(0)-\sum_{k=1}^{2n}(D_1\beta_k)D_ku(0)\\
&=D_1\varphi(z,u)(0)-\sum_{k=1}^{2n}(D_1\beta_k)D_ku(0)\\
&=\varphi_1(0,u(0))+\varphi_u(0,u(0))D_1u(0)-\sum_{k=1}^{2n}(D_1\beta_k)D_ku(0)\\
&\leq -\gamma_0D_1u(0)-\sum_{k=1}^{2n}(D_1\beta_k)D_ku(0)+C,
\end{array}$$ where $\beta_k$ is the component of $\beta$. Because $\Omega$ is a bounded strictly pseudoconvex domain, there is a strongly plurisubharmonic defining function $r(z)$ for $\Omega$ satisfying $|\nabla r|=1$ on $\partial\Omega=\{z\in {\bf
C^n}:r(z)=0\}$. And there is an orthogonal matrix $U$ over $R^{2n-1}$ such that $$\label{U}
U^t [\frac{\partial^2r(0)}{\partial t_k \partial
t_l}]_{(2n-1)\times(2n-1)}
U=diag(\lambda_2(0),\cdots,\lambda_{2n}(0)),$$where $\lambda_{2n}(0)\geq\cdots\geq\lambda_2(0)$. Since $\Omega$ is a bounded domain with $C^2$ boundary, there are constants $\Lambda$ and $\lambda$ such that $\Lambda>\sup\{\lambda_{2n}(z):z\in\partial\Omega\}$ and $\lambda<\inf\{\lambda_2(z):z\in\partial\Omega\}$.
To handle the $\sum_{k=1}^{2n}(D_1\beta_k)D_ku(0)$, we express $\nu$ in terms of $\beta$ and tangential components, $$D_{\nu}u(0)=\sum_{k=1}^{2n-1}(-\frac{\beta_k}{\beta_{2n}})D_ku(0)+\frac{1}{\beta_{2n}}D_\beta
u(0),$$ and $\nu_k(z)=\frac{\partial r}{\partial t_k}(z)$, $z\in\partial\Omega$, $\nu_k$ is the component of $\nu$. Then $$\begin{array}{ll}
\sum_{k=1}^{2n}(D_1\beta_k)D_ku(0)&=\sum_{k=1}^{2n}(D_1\beta_k-D_1\nu_k)D_k
u(0)+\sum_{k=1}^{2n}(D_1\nu_k)D_ku(0)\\
&=\sum_{k=1}^{2n-1}\frac{\partial^2 r}{\partial t_k\partial
t_1}D_ku(0)+\sum_{k=1}^{2n-1}D_1(\beta_k-\nu_k)D_k u(0)\\
&+[\frac{\partial^2 r}{\partial t_{2n}\partial
t_1}+D_1(\beta_{2n}-\nu_{2n})]D_\nu u(0)\\
&=\sum_{k=1}^{2n-1}\frac{\partial^2 r}{\partial t_k\partial
t_1}D_ku(0)+\sum_{k=1}^{2n-1}D_1(\beta_k-\nu_k)D_k u(0)\\
&+[\frac{\partial^2 r}{\partial t_{2n}\partial
t_1}+D_1(\beta_{2n}-\nu_{2n})][\sum_{k=1}^{2n-1}(-\frac{\beta_k}{\beta_{2n}})D_ku(0)+\frac{1}{\beta_{2n}}D_\beta
u(0)].
\end{array}$$ Thus $$\label{b}
\begin{array}{ll}
& D_\beta D_1u(0)\\
&\leq C-\gamma_0D_{1}u(0)-\sum_{k=1}^{2n-1}\frac{\partial^2 r}{\partial t_k\partial
t_1}D_ku(0)-\sum_{k=1}^{2n-1}D_1(\beta_k-\nu_k)D_ku(0)\\
&-[\frac{\partial^2 r}{\partial t_{2n}\partial
t_1}+D_1(\beta_{2n}-\nu_{2n})][\sum_{k=1}^{2n-1}(-\frac{\beta_k}{\beta_{2n}})D_{k}u(0)+\frac{1}{\beta_{2n}}D_{\beta}u(0)]\\
&\leq C-\gamma_0D_{1}u(0)-[\sum_{k=1}^{2n-1}\frac{\partial^2 r}{\partial t_k\partial
t_1}+\sum_{k=1}^{2n-1}D_1(\beta_k-\nu_k)\\
&+\sum_{k=1}^{2n-1}(\frac{\partial^2 r}{\partial t_{2n}\partial
t_1}+D_1(\beta_{2n}-\nu_{2n}))(-\frac{\beta_k}{\beta_{2n}})]D_ku(0)\\
&=C-\gamma_0D_{1}u(0)-3\Lambda
D_{1}u(0)+\sum_{k=1}^{2n-1}[\Lambda\delta_{k1}-\frac{\partial^2 r}{\partial t_k\partial
t_1}]D_ku(0)\\
&+\sum_{k=1}^{2n-1}[\Lambda\delta_{k1}-D_1(\beta_k-\nu_k)]D_ku(0)\\
&+\sum_{k=1}^{2n-1}[\Lambda\delta_{k1}+
(\frac{\partial^2 r}{\partial t_{2n}\partial
t_1}+D_1(\beta_{2n}-\nu_{2n}))(\frac{\beta_k}{\beta_{2n}})]D_ku(0).
\end{array}$$ Let $$A_1=\sum_{k=1}^{2n-1}[\Lambda\delta_{k1}-\frac{\partial^2
r}{\partial t_k\partial
t_1}]D_ku(0),$$ $$A_2=\sum_{k=1}^{2n-1}[\Lambda\delta_{k1}-D_1(\beta_k-\nu_k)]D_ku(0),$$ $$A_3=\sum_{k=1}^{2n-1}[\Lambda\delta_{k1}+
(\frac{\partial^2 r}{\partial t_{2n}\partial
t_1}+D_1(\beta_{2n}-\nu_{2n}))(\frac{\beta_k}{\beta_{2n}})]D_ku(0).$$ By using an argument similar to that given in the proof of Theorem 3.1 in [@L1], we want to obtain the relationship between $A_1,\ A_2 $ and $A_3$ with $D_{1}u(0)$.
First, by the conclusion in [@L1] directly, we set $$\hat{\tau_1}=(e'_1[\Lambda I_{2n-1}-(\frac{\partial^2 r(0)}{\partial
t_k\partial t_l})^{2n-1}_{k,l=1}],0),$$ where $e_1'=(1,\underbrace{0,\cdots,0}_{2n-2})$. Since the matrix $[\Lambda I_{2n-1}-(\frac{\partial^2 r(0)}{\partial t_k\partial
t_l})^{2n-1}_{k,l=1}]$ is non-negative definite with maximum eigenvalue $\Lambda-\lambda_2(0)$, we have $|\hat{\tau_1}|\leq
\Lambda-\lambda_2(0)$. Thus by (\[3.8\]) and $\hat{\tau_1}$ is the tangential direction at 0, $$\label{b1}
\begin{array}{ll}
A_1&=D_{\hat{\tau_1}}u(0)\\
&=|\hat{\tau_1}|D_{\frac{\hat{\tau_1}}{|\hat{\tau_1}|}}u(0)\\
&\leq (\Lambda-\lambda_2(0))D_1u(0).
\end{array}$$ Then, set $$M_{2n-1}=\left(
\begin{array}{cccc}
D_1(\beta_1-\nu_1) & D_1(\beta_2-\nu_2)&\cdots & D_1(\beta_{2n-1}-\nu_{2n-1})\\
D_1(\beta_2-\nu_2) & 0 &\cdots & 0 \\
\vdots &\vdots &\ddots&0\\
D_1(\beta_{2n-1}-\nu_{2n-1})& 0&\cdots & 0\\
\end{array}
\right),$$ $$\hat{\tau_2}=(e'_1[\Lambda I_{2n-1}-M_{2n-1}],0).$$ Thus $$A_2=D_{\hat{\tau_2}}
u(0)=|\hat{\tau_2}|D_{\frac{\hat{\tau_2}}{|\hat{\tau_2}|}}u(0).$$ By calculation, the eigenvalues of matrix $M_{2n-1}$ are $$\pi_1=0,\ \
\pi_{2,3}=\frac{D_1(\beta_1-\nu_1)\pm\sqrt{(D_1(\beta_1-\nu_1))^2+4[\sum_{i=2}^{2n-1}(D_1(\beta_i-\nu_i))^2]}}{2},$$ where $\pi_1$ is (2n-3) multiple eigenvalues. In the condition (\[ob2\]), we can take $\epsilon_0\leq\frac{\Lambda}{1+\sqrt{8n-7}}$ so that the matrix $\Lambda I_{2n-1}-M_{2n-1}$ is non-negative definite with $|\hat{\tau_2}|\leq (\Lambda+\frac{1+\sqrt{8n-7}}{2}\epsilon_0)$. Thus $$\label{b2'}
A_2\leq (\Lambda+\frac{1+\sqrt{8n-7}}{2}\epsilon_0)D_{1}u(0).$$ At last, we consider $A_3$. $$A_3=\sum_{k=1}^{2n-1}[\Lambda\delta_{k1}+
(\frac{\partial^2 r}{\partial t_{2n}\partial
t_1}+D_1(\beta_{2n}-\nu_{2n}))(\frac{\beta_k}{\beta_{2n}})]D_ku(0).$$ Set $$N(0)=[\frac{\partial^2r}{\partial t_{2n}\partial
t_1}+D_1(\beta_{2n}-\nu_{2n})](0),$$ here $N(0)$ has uniform upper bound, i.e. $|N(0)|\leq N$, where $N$ is a constant independent of $0$. Set $$G_{2n-1}=\left(
\begin{array}{cccc}
\beta_1 & \beta_2 & \cdots & \beta_{2n-1} \\
\beta_2& 0 & \cdots & 0 \\
\vdots & \vdots & \ddots &\vdots\\
\beta_{2n-1} & 0 &\cdots & 0\\
\end{array}
\right),$$ $$\hat{\tau_3}=(e_1'[\Lambda I_{2n-1}+\frac{N(z_0)}{\beta_{2n}}G_{2n-1}],0).$$ So, if we take $\epsilon_0\leq\frac{\Lambda|\beta_{2n}|}{N(1+\sqrt{8n-7})}$, we have $$\label{b3'}
\begin{array}{ll}
A_3=D_{\hat{\tau_3}} u(0) =|\hat{\tau_3}|D_{\frac{\hat{\tau_3}}{|\hat{\tau_3}|}}u(0)\leq
(\Lambda+|\frac{N}{\beta_{2n}}|\frac{1+\sqrt{8n-7}}{2}\epsilon_0)D_{1}u(0).
\end{array}$$ Thus by inserting (\[b1\]), (\[b2’\]) and (\[b3’\]) into (\[b\]) we get $$\label{3.37}
\begin{array}{ll}
&D_\beta D_1u(0)\\
&\leq C-\gamma_0D_{1}u(0) -3\Lambda
D_{1}u(0)+(\Lambda-\lambda_2(0))D_{1}u(0)\\
&+(\Lambda+\frac{1+\sqrt{8n-7}}{2}\epsilon_0)D_{1}u(0)
+(\Lambda+|\frac{N}{\beta_{2n}}|\frac{1+\sqrt{8n-7}}{2}\epsilon_0)D_{1}u(0)\\
&=C-[\gamma_0+\lambda_2(0)-\frac{(1+\sqrt{8n-7})\epsilon_0}{2}-\frac{(1+\sqrt{8n-7})|\frac{N}{\beta_{2n}}|\epsilon_0}{2}]D_{1}u(0).
\end{array}$$ Again, we take $\epsilon_0\leq
\frac{(\gamma_0+\lambda_2(0))}{[(1+\sqrt{8n-7})(1+|\frac{N}{\beta_{2n}}|)]}$, then $${\gamma_0+\lambda_2(0)}\geq
(1+\sqrt{8n-7})\epsilon_0+(1+\sqrt{8n-7})|\frac{N}{\beta_{2n}}|\epsilon_0.$$ Ultimately, by (\[ob2’\]), $\beta_{2n}(0)=\beta(0)\cdot\nu(0)\geq \beta_0$, we choose $$\begin{array}{ll}
\epsilon_0&\leq
\min\{\frac{\Lambda}{1+\sqrt{8n-7}},\frac{\Lambda\beta_0}{N(1+\sqrt{8n-7})},\frac{(\gamma_0+\lambda_2(z_0))}{[(1+\sqrt{8n-7})(1+\frac{N}{\beta_0})]}\}\\
&\leq\min\{\frac{\Lambda}{1+\sqrt{8n-7}},\frac{\Lambda|\beta_{2n}|}{N(1+\sqrt{8n-7})},
\frac{(\gamma_0+\lambda_2(z_0))}{[(1+\sqrt{8n-7})(1+|\frac{N}{\beta_{2n}}|)]}\},
\end{array}$$ then $$\label{3.40} [\gamma_0+\lambda_2(0)-\frac{(1+\sqrt{8n-7})\epsilon_0}{2}-\frac{(1+\sqrt{8n-7})|\frac{N}{\beta_{2n}}|\epsilon_0}{2}]\geq \frac{\sigma_0}{2},$$ where $\sigma_0=\inf_{\partial\Omega}(\gamma_0+\lambda_2)$. Then by inserting (\[3.40\]) into (\[3.37\]), $$\begin{array}{ll}
0&\leq D_\beta Q(0)\\
&\leq \frac{C-\frac{\sigma_0}{2}D_1u(0)}{D_1u(0)}-AD_\beta x_n(0).
\end{array}$$ $$[\frac{\sigma_0}{2}-A\beta\cdot\nu(0)]D_1u(0)\leq C.$$ Finally we fix $A$ so small that $$A\beta\cdot\nu\leq\frac{\sigma_0}{4}, on\ \partial\Omega,$$ we have for any tangential vector $D_1
u(0)\leq \frac{\sigma_0}{4}.$ We finish the proof of Theorem \[ge\].
Second order derivative estimates
==================================
In this section, we aim to derive the second order derivative estimates. First, we reduce the interior second order derivatives to the boundary. Then, we derive the second order derivative estimates on the boundary. Finally, from these estimates we have Theorem \[thm4\].
\[thm3\] Let $\Omega$ be a bounded strictly pseudoconvex domain in $\bf C^n$ with $C^4$ boundary. Assume $\beta$, $\varphi\in
C^{2,1}(\partial \Omega\times \bf R)$ and $f\in
C^2(\overline{\Omega})$ satisfy (\[ob2’\])-(\[ob6\]). In addition $\varphi$ satisfies (\[ob3\]). Then $$\label{3.1}|D^2 u|_{0,\overline{\Omega}}\leq
\widetilde{C},$$ where $\widetilde{C}$ is a constant depending only on $|u|_{1,\overline\Omega},\
|f^{\frac{1}{n}}|_{C^{2}(\overline\Omega)},\ \gamma_0,\ \Omega,
\beta\ and \ |\varphi|_{C^{2,1}(\partial\Omega)}$.
First, we reduce the second order derivative estimates to the boundary by choosing the auxiliary function $R_1(z,\xi)=D_{\xi\xi}u(z)+\eta_2|z|^2$, where $\eta_2=|f^{\frac{1}{n}}|_{C^{2}(\overline{\Omega})}+1$.
We have $$F^{i\bar j}\partial_{i\bar j}D_{\xi\xi}u
\geq D_{\xi\xi}\widetilde{f}\geq -|f^{\frac{1}{n}}|_{C^{2}(\overline\Omega)},$$ thus $LR_1=F^{i\bar j}\partial_{i\bar j}R_1>0$. By the maximum principle, $$\label{4.3}
\begin{array}{ll}
\max_{\overline{\Omega}}D_{\xi\xi} u(z)&\leq \max_{\overline{\Omega}}(D_{\xi\xi} u(z)+\eta_2|z|^2)\\
&\leq\max_{\partial\Omega}(D_{\xi\xi} u(z)+\eta_2|z|^2)\\
&\leq\max_{\partial\Omega}D_{\xi\xi} u(z)+\eta_2 diam(\Omega)^2.
\end{array}$$
Next, we give some lemmas below.
\[l31\] We reformulate (\[3.1\]) as follow, $$\label{3.2}D_{\xi\xi}u(z)\leq C,\ \ \ \xi\in \bf R^{2n},$$whenever $u$ is subharmonic.
\[l32\] $|D_{\beta\tau }u|\leq C$ on $\partial \Omega$ for any unit tangential vector field $\tau$.
By applying the tangential gradient operator to the boundary condition we obtain $|D_{\beta\tau }u|\leq C$ on $\partial \Omega$ for any tangential vector $\tau$.
\[l33\]$| D_{\beta\nu}u|\leq C$ on $\partial
\Omega$.
Without loss of generality, we set $0\in
\partial \Omega$, $x_n$ is inner normal vector at 0. Case 1: Suppose that $Du(0)=0$.
Now we use the auxiliary function $$W=\pm D_\beta u\mp
\varphi(z,u)+|Du|^2+\widetilde{K}|z|^2-\widetilde{G}x_n.$$ Now we compute $LW$. $$\begin{array}{ll}
LD_\beta u &=F^{i\bar j}\partial _i \partial _{\bar
j}\beta_k D_k u+F^{i\bar j}(\partial _{\bar j}\beta_k)(\partial _i
D_k u)+ F^{i\bar j}(\partial _i \beta_k)(\partial _{\bar j}D_k u)+
F^{i\bar j}\beta_k\partial_i\partial_{\bar j}D_k u\\
&=F^{i\bar j}\partial _i \partial _{\bar j}\beta_k D_k u+F^{i\bar
j}(\partial _{\bar j}\beta_k)(\partial _i D_k u)+ F^{i\bar
j}(\partial _i \beta_k)(\partial _{\bar j}D_k
u)+\beta_kD_k\widetilde{f}.
\end{array}$$ From the Cauchy inequality and the positivity of the matric $(F^{i\bar j})$, we have $$|F^{i\bar j}(\partial
_{\bar j}\beta_k)(\partial _i D_k u)|\leq(F^{i\bar
j}(\partial_i\beta_k\partial _{\bar
j}\beta_k))^{\frac{1}{2}}(F^{i\bar j}(\partial _i D_k
u\partial_{\bar j}D_k u))^{\frac{1}{2}},$$ $$|F^{i\bar j}(\partial _i\beta_k)(\partial _{\bar j} D_k
u)|\leq(F^{i\bar j}(\partial_i\beta_k\partial _{\bar
j}\beta_k))^{\frac{1}{2}}(F^{i\bar j}(\partial _i D_k
u\partial_{\bar j}D_k u))^{\frac{1}{2}}.$$Hence $$|LD_\beta u|\leq \widetilde{C_1}tr (F^{i\bar
j})+F^{i\bar j}(\partial_i D_k u)(\partial _{\bar j}D_k
u)+C,$$ where $\widetilde{C_1}$ depends on $|u|_{1,\overline\Omega}$, $\beta$ and $f^{\frac{1}{n}}$. We also have $$\begin{array}{ll}
L\varphi(z,u)&=F^{i\bar j}\partial _i \partial _{\bar
j}\varphi(z,u)\\
&=F^{i\bar j}[\varphi_{i\bar j}+\varphi_{u\bar
j}u_i+\varphi_{ui}u_{\bar j}+\varphi_{uu}u_iu_{\bar
j}+\varphi_uu_{i\bar j}]\\
&=F^{i\bar j}[\varphi_{i\bar j}+\varphi_{u\bar
j}u_i+\varphi_{ui}u_{\bar j}+\varphi_{uu}u_iu_{\bar
j}]+\widetilde{f}\varphi_u\\
&\leq \widetilde{C_2}tr(F^{i\bar j})+C,
\end{array}$$ where $\widetilde{C_2}$ depends on $\varphi$, $f^{\frac{1}{n}}$ and $|u|_{1,\overline\Omega}$. $$\begin{array}{ll}
L|Du|^2 &=2F^{i\bar j}(\partial _i D_k u\partial_{\bar
j}D_k u+D_k u\partial_{i\bar
j}D_ku)\\
&=2F^{i\bar j}(\partial _iD_k u)(\partial_{\bar j}D_k u)+2D_k uD_k
\widetilde{f}\\
&\geq 2F^{i\bar j}(\partial _iD_k u)(\partial_{\bar j}D_k u)-C,
\end{array}$$ where $C$ depends on $|u|_{1,\overline\Omega}$ and $f^{\frac{1}{n}}$. $$L|z|^2=tr(F^{i\bar j}).$$ Therefore, we obtain $$LW\geq(\widetilde{K}-\widetilde{C_1}-\widetilde{C_2})tr(F^{i\bar j})-\widetilde{C_3}\geq\widetilde{K}-\widetilde{C_4},$$ we can choose $\widetilde{K}>\widetilde{C_4}$ such that $LW\geq 0.$
Finally, we use a standard barrier argument for $D_\nu W$ on $\partial\Omega$(see in [@CKNS], Lemma 1.3 or [@GT], Corollary 14.5). Let $S_{\mu_1}=\{z\in \Omega|x_n \leq \mu_1\}. $ On $\partial S_{\mu_1}\cap\Omega $ if $\widetilde{G}$ is sufficiently large, we get $W\leq0$. On $\partial
S_{\mu_1}\cap\partial\Omega$, we have for some $a>0$, $x_n\geq
a|z|^2$. Then we can choose $\widetilde{G}$ large enough such that $W\leq 0 $ on $\partial S_{\mu_1}\cap\partial\Omega$. Since $W(0)=0$, by the maximum principle, we obtain $W_{x_n}(0)\leq 0$, then $|D_{\beta
x_n}u|\leq \widetilde{G}+|\varphi|_{C^{1,1}(\partial \Omega)}.$
Case 2: If $Du(0)\neq0$, we can take $$W=\pm D_\beta u\mp
\varphi(z,u)+|Du(z)-Du(0)|^2+\widetilde{K}|z|^2-\widetilde{G}x_n,$$ the proof above is still valid.
Combining Lemma \[l32\] with Lemma \[l33\] , for any direction $\xi$, we obtain $|D_{\beta\xi}u|\leq C$ on $
\partial \Omega$. In particular $|D_{\beta\beta}u|\leq C$ on $\partial \Omega.$
Since we have the bounds for $|D_{\tau\beta}u|$ and $|D_{\beta\beta}u|$ on $\partial\Omega$, in order to finish the proof of Theorem \[thm3\], the remaining task is to get the bounds for the tangential second derivatives of $u$ on $\partial\Omega$. We now assume that the maximum tangential second order derivative on $\partial\Omega$ is attained at a boundary point which may take to be the origin, in tangential direction which we may take to be $e_1$, and $x_n$ is the inner normal vector at 0. Thus $$\label{4.15}
D_{11}u(0)=\sup_{z\in\partial\Omega,\tau\ is\ unit\ tangential\ at\ z}D_{\tau\tau}u(z).$$ Without loss of generality, we can take $D_{11}u(0)>0$. Otherwise, we finish the proof.
As for any direction $\xi$, on $\partial\Omega$, $$\label{4.16}
\begin{array}{ll}
D_{\xi\xi}u&=D_{\tau(\xi)\tau(\xi)}u+2\frac{(\nu\cdot\xi)}{(\beta\cdot\nu)}D_{\tau(\xi)\beta}u
+\frac{(\nu\cdot\xi)^2}{(\beta\cdot\nu)^2}D_{\beta\beta}u, \\
&\leq (1+\frac{2}{\beta_0})D_{11}u(0)+C\ \ on \
\
\partial\Omega.
\end{array}$$
We introduce the tangential gradient operator $\delta=(\delta_1,\cdots, \delta_{2n-1})$, where $\delta_i=(\delta_{ij}-\nu_i\nu_j)D_j$. Applying this tangential operator to the boundary condition (\[ob1\]), we have $$D_ku\delta_i \beta_k + \beta_k \delta_i D_ku=\delta_i\varphi,\quad
{\rm on} \ \partial \Omega,$$ then $$D_{\tau\beta}u=-D_ku(\delta_i\beta_k)\tau_i+\delta_i\varphi\tau_i.$$
For the case $\xi=e_1$ in (\[4.16\]), we have $$\label{4.19}
D_{11}u(z)\leq (1-\frac{2\nu_1\beta_1^T}{\beta\cdot\nu})D_{11}u(0)+\frac{2\nu_1}{\beta\cdot\nu}D_{\tau(e_1)\beta}+\frac{\nu^2_1}{(\beta\cdot\nu)^2}D_{\beta\beta}u(z).$$
Similarly to the real case in [@U2], let $S_{\mu_2}=\{z\in\Omega:x_n\leq \mu_2\}$. We construct a function $$H(z)=\frac{D_{11}u(z)-V(z)}{D_{11}u(0)}+\frac{2\nu_1\beta_1^T}{\beta\cdot\nu}+\widehat{B}|Du(z)|^2+\widehat{G}|z|^2-\widehat{A}x_n-1,$$ $\widehat{G}$, $\widehat{A}$, $\widehat{B}$ and $\mu_2$ are constants to be fixed. $V(z)$ is a linear function with respect to $Du$, such that $$V(z)=a_k(z)D_ku+b(z),\ \ in\ \ \Omega,$$ where $a_k(z)$, $b(z)$ are smooth functions and $$a_k(z)=-2\frac{<\nu,e_1>}{<\beta,\nu>}(\delta_i\beta_k)\tau_i(e_1),\ \ b(z)=2\frac{<\nu,e_1>}{<\beta,\nu>}\delta_i\varphi\tau_i(e_1),\
\ \ on\ \ \partial\Omega.$$ So $V(0)=0$ on $\partial\Omega.$
Assume $Du(0)=0$, or we let $$H(z)=\frac{D_{11}u(z)-V(z)}{D_{11}u(0)}+\frac{2\nu_1\beta_1^T}{\beta\cdot\nu}+\widehat{B}|Du(z)-Du(0)|^2+\widehat{G}|z|^2-\widehat{A}x_n-1.$$
According to (\[2.3\]) and (\[2.4\]), we have $LD_{11}u(z)\geq -\widetilde{C_5}$. $$\begin{array}{ll}
&-LV\\
&=-L(a_kD_ku+b)\\
&=-F^{i\bar j}\partial_i(\partial_{\bar j}a_kD_ku+a_k\partial_{\bar j}D_ku+\partial_{\bar j}b)\\
&=-F^{i\bar j}[\partial_{i\bar j}a_kD_ku+\partial_{\bar j}a_k\partial_iD_ku\\
& \ \ \ \ \ \ \ \ +\partial_ia_k\partial_{\bar j}D_ku+a_k\partial_{i\bar j}D_ku+
\partial_{i\bar j}b]\\
&\geq -F^{i\bar j}(\partial_{\bar j}a_k\partial_i D_ku+\partial_ia_k\partial_{\bar j}D_ku)
-Ctr(F^{i\bar j}) -a_kD_k\widetilde{f}-C\\
&\geq -\widetilde{C_6}tr(F^{i\bar j})-F^{i\bar j}\partial_iD_ku\partial_{\bar
j}D_ku-\widetilde{C_7},
\end{array}$$ where $\widetilde{C_6}$ and $\widetilde{C_7}$ depend on $a_k$, $b$, $f^{\frac{1}{n}}$and $|u|_{1,\overline\Omega}$. $$\begin{array}{ll}
L(|Du|^2)
&=F^{i\bar j}(2\partial_iD_ku\partial_{\bar j}D_ku+2D_ku\partial_{i\bar j}D_ku)\\
&=2F^{i\bar j}\partial _iD_ku\partial_{\bar j}D_ku+2D_k\widetilde{f}D_ku\\
&\geq2F^{i\bar j}\partial _iD_ku\partial_{\bar j}D_ku-\widetilde{C_8},
\end{array}$$ where $\widetilde{C_8}$ depends on $|u|_{1,\overline\Omega}$ and $f^{\frac{1}{n}}$. $$L\widehat{G}|z|^2=\widehat{G}tr(F^{i\bar j}).$$ And see (2.26) in [@U2], $$L(\frac{2\nu_1\beta^T}{\beta\cdot\nu})\geq-(\widetilde{C_{9}}\sqrt{\mu_2}|\beta|_{2,\overline{\Omega}}+
\widetilde{C_{10}}|\beta^T|_{1,\overline{\Omega}})tr( F^{i\bar j})\ \ in \ S_{\mu_2}.$$ Therefore, $$\begin{array}{ll}
LH&\geq\frac{-\widetilde{C_6}tr(F^{i\bar j})-\widetilde{C_5}-\widetilde{C_7}-F^{i\bar j}\partial _iD_ku\partial_{\bar j}D_ku}{D_{11}u(0)}+2\widehat{B}F^{i\bar j}\partial _iD_ku\partial_{\bar j}D_ku-\widehat{B}\widetilde{C_8}+\widehat{G}tr(F^{i\bar j})\\
&-(\widetilde{C_{9}}\sqrt{\mu_2}|\beta|_{2,\overline{\Omega}}+
\widetilde{C_{10}}|\beta^T|_{1,\overline{\Omega}})tr( F^{i\bar j})\\
&=(\frac{\widehat{G}}{2}tr(F^{i\bar j})-\widehat{B}\widetilde{C_8}-\frac{\widetilde{C_5}+\widetilde{C_7}}{D_{11}u(0)})\\&+(\frac{\widetilde{G}}{2}
-\frac{\widetilde{C_6}}{D_{11}u(0)}-\widetilde{C_{9}}\sqrt{\mu_2}|\beta|_{2,\overline{\Omega}}-
\widetilde{C_{10}}|\beta^T|_{1,\overline{\Omega}})tr(F^{i\bar j})\\&+(2\widehat{B}-\frac{1}{D_{11}u(0)})F^{i\bar j}\partial _iD_ku\partial_{\bar j}D_ku\\
&\geq (\frac{\widehat{G}}{2}-\widehat{B}\widetilde{C_8}-\frac{\widetilde{C_5}+\widetilde{C_7}}{D_{11}u(0)})+(\frac{\widetilde{G}}{2}
-\frac{\widetilde{C_6}}{D_{11}u(0)}-\widetilde{C_{9}}\sqrt{\mu_2}|\beta|_{2,\overline{\Omega}}-
\widetilde{C_{10}}|\beta^T|_{1,\overline{\Omega}})tr(F^{i\bar j})\\
&+(2\widehat{B}-\frac{1}{D_{11}u(0)})F^{i\bar j}\partial _iD_ku\partial_{\bar j}D_ku,\ in \ S_{\mu_2}.
\end{array}$$ So if $$\label{cs1}
\frac{\widehat{G}}{2}-\widehat{B}\widetilde{C_8}-\frac{\widetilde{C_5}+\widetilde{C_7}}{D_{11}u(0)}\geq0,\ \frac{\widetilde{G}}{2}
-\frac{\widetilde{C_6}}{D_{11}u(0)}-\widetilde{C_{9}}\sqrt{\mu_2}|\beta|_{2,\overline{\Omega}}-\widetilde{C_{10}}|\beta^T|_{1,\overline{\Omega}}\geq0,\ 2\widehat{B}\geq\frac{1}{D_{11}u(0)},$$ we have $LH\geq0$ in $S_{\mu_2}$.
On $\partial\Omega\cap\overline{S_{\mu_2}}$ near $0$, from (\[4.19\]), Lemma 4.3 and Lemma 4.4, $$\begin{array}{ll}
H(z)&\leq 1+\frac{\widetilde{C_{12}}|z|^2}{D_{11}u(0)}+(\widehat{G}-\widehat{A}a)|z|^2-1\\
&\leq[\frac{\widetilde{C_{12}}}{D_{11}u(0)}+\widehat{G}-\widehat{A}a]|z|^2,
\end{array}$$ if $$\label{cs2}\frac{\widetilde{C_{12}}}{D_{11}u(0)}+\widehat{G}\leq\widehat{A}a,$$ we have $H(z)\leq 0$ on $\partial\Omega\cap\overline{S_\mu}$ near $0$.
On the other hand, from (\[3.6\]), (\[4.3\]), and Theorem \[ge\], $$\begin{array}{ll}
H(z)&\leq (1+\frac{2}{\beta_0}|\beta^T|)+\frac{\widetilde{C_{13}}}{D_{11}u(0)}+\widetilde{C_{14}}\sqrt{\mu_2}|\beta^T|+\widehat{B}C+\widehat{G}|z|^2-\widehat{A}\mu_2-1,\\
&\leq \frac{2}{\beta_0}|\beta^T|+\frac{\widetilde{C_{13}}}{D_{11}u(0)}+\widetilde{C_{14}}\sqrt{\mu_2}|\beta^T|+\widehat{G}\widetilde{C_{16}}\mu_2-\widehat{A}\mu_2.
\end{array}$$ if $$\label{cs3}\frac{2}{\beta_0}|\beta^T|+\frac{\widetilde{C_{13}}}{D_{11}u(0)}+\widetilde{C_{14}}\sqrt{\mu_2}|\beta^T|+\widehat{B}C
+\widehat{G}\widetilde{C_{16}}\mu_2\leq \widehat{A}\mu_2,$$ we have $H(z)\leq 0$ on $\{x_n=\mu\}\cap \overline{S_{\mu_2}}$.
We now proceed to fix $\widehat{G}$, $\mu_2$ and $\widehat{B}$, depending on $\widehat{A}$ which will be fixed later. We first fix $\widehat{G}>0$ so small that $$\widehat{G}\leq\frac{\widehat{A}a}{2}\ \ and \ \ \widehat{G}\widehat{C_{16}}\leq \frac{\widehat{A}}{2},$$ and then fix $\mu_2\in(0,1)$ so that $$\widetilde{C_9}\sqrt{\mu_2}|\beta|_{2,\overline{\Omega}}\leq\frac{\widehat{G}}{4},$$ and then fix $\widehat{B}$ so that $$\widehat{B}\widetilde{C_8}\leq\frac{\widehat{G}}{4}\ \ and\ \widehat{B}C\leq\frac{\widehat{A}\mu_2}{4}.$$ Then (\[cs1\]), (\[cs2\]) and (\[cs3\]) will be hold whenever $$\label{cs4}
\begin{array}{ll}
&\frac{\widetilde{C_5}+\widetilde{C_7}}{D_{11}u(0)}\leq\frac{\widetilde{G}}{4},\\
&\frac{\widetilde{C_9}}{D_{11}u(0)}+\widetilde{C_{10}}|\beta^T|_{1,\overline{\Omega}}\leq\frac{\widehat{G}}{4},\\
&\frac{1}{D_{11}u(0)}\leq 2\widehat{B},\\
&\frac{\widetilde{C_{12}}}{D_{11}u(0)}\leq \frac{\widehat{A}a}{2},\\
&(\frac{2}{\beta_0}+\widetilde{C_{14}})|\beta^T|+\frac{\widetilde{C_{13}}}{D_{11}u(0)}\leq \frac{\widehat{A}\mu_2}{4}.
\end{array}$$ When these conditions (\[cs4\]) are satisfied we have $Q\leq 0$ in $S_{\mu_2}$ by the maximum principle, and since $H(0)=0$, then $D_\beta H(0)\geq 0$.
We have $$\label{q1}
\begin{array}{ll}
D_\beta D_{11}u(0)&\leq D_{11}\varphi(0,u(0))-\sum_{k=1}^{2n}(D_{11}\beta_k)
D_ku(0)-2\sum_{k=1}^{2n}(D_{1}\beta_k)D_kD_1u(0)+C\\
&\leq\varphi_uD_{11}u(0)-\sum_{k=1}^{2n}(D_{11}\beta_k)
D_ku(0)-2\sum_{k=1}^{2n}(D_{1}\beta_k)D_kD_1u(0)+C\\
&\leq -\gamma_0D_{11}u(0)-2\sum_{k=1}^{2n}(D_1\beta_k)D_kD_1u(0)+C\\
&=C-\gamma_0D_{11}u(0)-2\sum_{k=1}^{2n}(D_1\nu_k)D_kD_1u(0)-2\sum_{k=1}^{2n}D_1(\beta_k-\nu_k)D_kD_1u(0)\\
&=C-\gamma_0D_{11}u(0)-2\sum_{k=1}^{2n-1}\frac{\partial^2 r}{\partial t_k\partial
t_1}D_kD_1u(0)\\&-2\sum_{k=1}^{2n-1}D_1(\beta_k-\nu_k)D_kD_1u(0)
-2[\frac{\partial^2 r}{\partial t_{2n}\partial
t_1}+D_1(\beta_{2n}-\nu_{2n})]D_\nu D_1u(0).
\end{array}$$ To handle the last term above, we express $\nu$ in terms of $\beta$ and tangential components, $$\label{q2}
D_{\nu}D_1u(0)=\sum_{k=1}^{2n-1}(-\frac{\beta_k}{\beta_{2n}})D_kD_1u(0)+\frac{1}{\beta_{2n}}D_\beta D_1u(0).$$ We take (\[q2\]) into the last term of (\[q1\]), $$\label{a}
\begin{array}{ll}
D_\beta D_{11}u(0)&\leq C-\gamma_0D_{11}u(0)-2\sum_{k=1}^{2n-1}\frac{\partial^2 r}{\partial t_k\partial
t_1}D_kD_1u(0)-2\sum_{k=1}^{2n-1}D_1(\beta_k-\nu_k)D_kD_1u(0)\\
&-2[\frac{\partial^2 r}{\partial t_{2n}\partial
t_1}+D_1(\beta_{2n}-\nu_{2n})][\sum_{k=1}^{2n-1}(-\frac{\beta_k}{\beta_{2n}})D_{k1}u(0)+\frac{1}{\beta_{2n}}D_{1\beta}u(0)]\\
&\leq C-\gamma_0D_{11}u(0)-2[\sum_{k=1}^{2n-1}\frac{\partial^2 r}{\partial t_k\partial
t_1}+\sum_{k=1}^{2n-1}D_1(\beta_k-\nu_k)\\
&+\sum_{k=1}^{2n-1}(\frac{\partial^2 r}{\partial t_{2n}\partial
t_1}+D_1(\beta_{2n}-\nu_{2n}))(-\frac{\beta_k}{\beta_{2n}})]D_kD_1u(0)\\
&=C-\gamma_0D_{11}u(0)-6\Lambda
D_{11}u(0)+\sum_{k=1}^{2n-1}2[\Lambda\delta_{k1}-\frac{\partial^2 r}{\partial t_k\partial
t_1}]D_kD_1u(0)\\
&+\sum_{k=1}^{2n-1}2[\Lambda\delta_{k1}-D_1(\beta_k-\nu_k)]D_kD_1u(0)\\
&+\sum_{k=1}^{2n-1}2[\Lambda\delta_{k1}+
(\frac{\partial^2 r}{\partial t_{2n}\partial
t_1}+D_1(\beta_{2n}-\nu_{2n}))(\frac{\beta_k}{\beta_{2n}})]D_kD_1u(0).
\end{array}$$ Let $$A_1'=\sum_{k=1}^{2n-1}2[\Lambda\delta_{k1}-\frac{\partial^2
r}{\partial t_k\partial
t_1}]D_kD_1u(0),$$ $$A_2'=\sum_{k=1}^{2n-1}2[\Lambda\delta_{k1}-D_1(\beta_k-\nu_k)]D_kD_1u(0),$$ $$A_3'=\sum_{k=1}^{2n-1}2[\Lambda\delta_{k1}+
(\frac{\partial^2 r}{\partial t_{2n}\partial
t_1}+D_1(\beta_{2n}-\nu_{2n}))(\frac{\beta_k}{\beta_{2n}})]D_kD_1u(0).$$ By using an argument similar to that given in the proof of Theorem \[ge\], we want to obtain the relationship between $A_1',\ A_2' $ and $A_3'$ with $D_{11}u(0)$.
First, by the conclusion in [@L1] directly, we have $$\label{a1}
\begin{array}{ll}
A_1'&=(2e'_1[\Lambda I_{2n-1}-(\frac{\partial^2 r(0)}{\partial t_k\partial t_l})^{2n-1}_{k,l=1}])\cdot(D_1D_1 u(0),\cdots,D_{2n-1}D_1u(0))\\
&=(2e'_1 UU^t[\Lambda I_{2n-1}-(\frac{\partial^2 r(0)}{\partial t_k\partial t_l})^{2n-1}_{k,l=1}]U)\cdot((D_1D_1 u(0),\cdots,D_{2n-1}D_1u(0))U)\\
&\leq 2(\Lambda-\lambda_2(0))D_{11}u(0)+C,
\end{array}$$ where $U$ is defined in (\[U\]).
Because the orthogonal transformation does not change the distance between two points, the second identity of (\[a1\]) holds. One can find the detail for the inequality of (\[a1\]) in [@L]( Theorem 3.5 and Theorem 4.2).
An argument similar to that given in the proof of Theorem \[ge\] gives, $$\label{a2'}
A_2'\leq 2(\Lambda+\frac{1+\sqrt{8n-7}}{2}\epsilon_0)D_{11}u(0)+C,$$when $\epsilon_0\leq\frac{\Lambda}{1+\sqrt{8n-7}}$. And, if we take $\epsilon_0\leq\frac{\Lambda|\beta_{2n}|}{M(1+\sqrt{8n-7})}$, we have $$\label{a3'}
\begin{array}{ll}
A_3'\leq
2(\Lambda+|\frac{M}{\beta_{2n}}|\frac{1+\sqrt{8n-7}}{2}\epsilon_0)D_{11}u(0)+C,
\end{array}$$ where $$M(0)=\frac{\partial^2r}{\partial t_{2n}\partial
t_1}+D_1(\beta_{2n}-\nu_{2n}),$$ and $|M(0)|\leq M$, $M$ is a constant independent of $0$.
Thus by inserting (\[a1\]), (\[a2’\]) and (\[a3’\]) into (\[a\]) we get $$\begin{array}{ll}
D_\beta D_{11}u(0)&\leq C-\gamma_0D_{11}u(0) -6\Lambda
D_{11}u(0)+2(\Lambda-\lambda_2(0))D_{11}u(0)\\
&+2(\Lambda+\frac{1+\sqrt{8n-7}}{2}\epsilon_0)D_{11}u(0)
+2(\Lambda+|\frac{M}{\beta_{2n}}|\frac{1+\sqrt{8n-7}}{2}\epsilon_0)D_{11}u(0)\\
&=C-[\gamma_0+2\lambda_2(0)-(1+\sqrt{8n-7})\epsilon_0-(1+\sqrt{8n-7})|\frac{M}{\beta_{2n}}|\epsilon_0]D_{11}u(0).
\end{array}$$ Again, we take $\epsilon_0\leq
\frac{\gamma_0+2\lambda_2(0)}{2[(1+\sqrt{8n-7})(1+|\frac{M}{\beta_{2n}}|)]}$, then $$\frac{\gamma_0+2\lambda_2(z_0)}{2}\geq
(1+\sqrt{8n-7})\epsilon_0+(1+\sqrt{8n-7})|\frac{M}{\beta_{2n}}|\epsilon_0.$$ Ultimately, we choose $$\begin{array}{ll}
\epsilon_0&\leq
\min\{\frac{\Lambda}{1+\sqrt{8n-7}},\frac{\Lambda\beta_0}{M(1+\sqrt{8n-7})},\frac{\gamma_0+2\lambda_2(z_0)}{2[(1+\sqrt{8n-7})(1+\frac{M}{\beta_0})]}\}\\
&\leq\min\{\frac{\Lambda}{1+\sqrt{8n-7}},\frac{\Lambda|\beta_{2n}|}{M(1+\sqrt{8n-7})},\frac{\gamma_0+2\lambda_2(z_0)}{2[(1+\sqrt{8n-7})(1+|\frac{M}{\beta_{2n}}|)]}\},
\end{array}$$ then $$\gamma_0+2\lambda_2(0)-(1+\sqrt{8n-7})\epsilon_0-(1+\sqrt{8n-7})|\frac{M}{\beta_{2n}}|\epsilon_0\geq\frac{\sigma_1}{2},$$ where $\sigma_1=\inf_{\partial\Omega}(\gamma_0+2\lambda_2)$. Then $$\begin{array}{ll}
0&\leq D_\beta H(0)\\
&\leq \frac{C-\frac{\sigma_1}{2}D_{11}u(0)}{D_{11}u(0)}-\widehat{A}D_\beta x_n(0).
\end{array}$$ $$[\frac{\sigma_1}{2}-\widehat{A}\beta\cdot\nu(0)]D_{11}u(0)\leq C.$$ Finally we fix $\widehat{A}$ so small that $$\widehat{A}\beta\cdot\nu\leq\frac{\sigma_1}{4}, on\ \partial\Omega,$$ we have for any tangential vector $D_{11}u(0)\leq \frac{\sigma_1}{4}.$ We finish the proof of Theorem 4.1.
We remark that the condition (\[ob2’\]) is necessary in the above proof. It can not be extended to the degenerate oblique case $(\beta\cdot\nu)\geq0$.
If $\gamma_0$ is sufficiently large, then we do not need any structural assumptions on $\beta$ and the principal curvature of $\partial \Omega$, for example, the condition (\[ob2\]) can be omitted.
\[thm4\]Let $\Omega$ be a bounded strictly pseudoconvex domain in $\bf C^n$ with $C^4$ boundary. Let $\varphi\in C^{3,1}(\partial
\Omega\times R)$, and $f\in C^2(\overline{\Omega})$, so that they satisfy (\[ob2’\])-(\[ob6\]). In addition $\varphi$ satisfies (\[ob3\]). Then $$\label{3.12}|u|_{2,\overline{\Omega}}\leq C,$$ where $C$ is a constant depending on $\
|f^{\frac{1}{n}}|_{C^{2}(\overline\Omega)},\ \gamma_0,\ \Omega,
\beta \ and \ |\varphi|_{C^{3,1}(\partial\Omega)}$.
The Proof of Theorem
====================
In this subsection, we shall prove Theorem\[thm\]. Although, the argument in this part is rather standard, we present its sketch here for completeness.
With the $C^0$, $C^1$ and $C^2$ bounds for the solution $u$ formulated in previous sections, the complex Monge-Ampère equation (\[cma\]) is uniformly elliptic. Since the second derivatives are bounded, the bounds of their H[ö]{}lder norms follow from the uniformly elliptic theory developed by Lieberman and Trudinger [@LieTru1986], that is $\|u\|_{C^{2,\alpha}(\bar \Omega)}\leq C$ for some $\alpha\in (0,1)$. Such a [*priori*]{} $C^{2,\alpha}$ estimation enables us to carry out the method of continuity in [@GT] and [@LTU], thus we obtain the existence of classical solutions. This completes the proof of Theorem \[thm\].
We remark that the proof for the uniqueness of the solutions is similar to that in [@L1] for the Neumann boundary case, here we need to use the condition $\varphi_u<0$.
[s2]{} S.Agmon, A.Douglis, L.Nirenberg. Estimates near the boundary for elliptic partial differential equations satisfying general boundary conditions, I\[J\]. Comm.Pure.Appl.Math.,1959,12:623-727 E.Bedford, B.A.Taylor. The Dirichlet problem for a complex Monge-Ampère equation\[J\]. Invent.Math.,1976,37:1-44 L.Caffarelli, J.J.Kohn, L.Nirenberg, J.Spruck. The Dirichlet problem for nonlinear second-order elliptic equations,II:Complex Monge-Ampère and uniformly elliptic equations\[J\]. Comm.Pure.Appl.Math.,1985:209-252 S.Y.Cheng, S.T.Yau. On the existence of a complex Kahler metric on non-compact complex manifolds and the regularity of Fefferman’s equation\[J\]. Comm.Pure.Appl.Math., 1980,33:507-544 D.Gilbarg,N.S.Trudinger. Elliptic partial differential equations of second order\[M\]. 2nd ed. Berlin,Heidelberg,New York,Tokyo:Springer-Verlag, 1983 G.M.Lieberman, N.S.Trudinger. Nonlinear oblique boundary value problems for nonlinear elliptic equations. Trans. Amer. Math. Soc.,1986,295:509-546 Song-Ying Li. On the Neumann problems for complex Monge-Ampère equations\[J\]. Indiana Univ.Math.J., 1994,43:1099-1122 Song-Ying Li. On the oblique boundary value problems for Monge-Ampère equations\[J\]. Pacific J.Math.,1999,190,1:155-172 Song-Ying Li. Boundary value problems for complex Monge-Ampère type, Ph.D. Thesis, University of Pittsburgh, 1992 P.L.Lions, N.S.Trudinger. Linear oblique derivative problems for the uniformly elliptic Hamilton-Jacobi-Bellmann equation\[J\]. Math.Zeit.,1986,191:1-15 P.L.Lions, N.S.Trudinger, J.Urbas. The Neumann problem for equations of Monge-Ampère Type\[J\]. Comm.Pure.Appl.Math.,1986,39:539-563 J.Urbas. Nonlinear oblique boundary value problems for Hessian equations in two dimensions\[J\]. Ann.Inst.Henri Poincare-Analyse Non Linear, 1995,12:507-575 J.Urbas. Oblique boundary value problems for equations of Monge-Ampère type\[J\]. Calc.Var.,1998,7:19-39 X-J.Wang. Oblique derivative problems for the equations of Monge-Ampère type\[J\]. Chinese J.Contemp.Math.,1992,13:13-22
[^1]: The first author was supported by the National Natural Science Foundation of China (No.11101132) and foundation of Hubei Provincial department of education(No.Q20120105).
|
---
abstract: |
We obtain intertwining dilation theorems for noncommutative regular domains ${{\mathcal D}}_f$ and noncommutative varieties ${{\mathcal V}}_J$ in $B({{\mathcal H}})^n$, which generalize Sarason [@S] and Sz.-Nagy–Foiaş [@SzF] commutant lifting theorem for commuting contractions. We present several applications including a new proof for the commutant lifting theorem for pure elements in the domain ${{\mathcal D}}_f$ (resp. variety ${{\mathcal V}}_J$) as well as a Schur type representation for the unit ball of the Hardy algebra associated with the variety ${{\mathcal V}}_J$.
We provide And\^ o type dilations and inequalities for bi-domains ${{\mathcal D}}_f\times_c {{\mathcal D}}_f$ which consist of all pairs $({\bf X}, {\bf Y})$ of tuples ${\bf X}:=(X_{1},\ldots, X_{n_1})\in {{\mathcal D}}_f$ and ${\bf Y}:=(Y_{1},\ldots, Y_{n_2})\in {{\mathcal D}}_g$ which commute, i.e. each entry of ${\bf X}$ commutes with each entry of ${\bf Y}$. The results are new even when $n_1=n_2=1$. In this particular case, we obtain extensions of And\^ o’s results [@An] and Agler-McCarthy’s inequality [@AM] for commuting contractions to larger classes of commuting operators.
All the results are extended to bi-varieties ${{\mathcal V}}_{J_1}\times_c {{\mathcal V}}_{J_2}$, where ${{\mathcal V}}_{J_1}$ and ${{\mathcal V}}_{J_2}$ are noncommutative varieties generated by WOT-closed two-sided ideals in noncommutative Hardy algebras. The commutative case as well as the matrix case when $n_1=n_2=1$ are also discussed.
address: |
Department of Mathematics, The University of Texas at San Antonio\
San Antonio, TX 78249, USA
author:
- Gelu Popescu
date: 'November 14, 2016'
title: ' And\^ o dilations and inequalities on noncommutative domains '
---
[^1]
Introduction {#introduction .unnumbered}
============
Extending von Neumann [@vN] inequality for one contraction and Sz.-Nagy proof using dilation theory [@SzFBK-book], And\^ o [@An] proved a dilation result that implies his celebrated inequality which says that if $T_1$ and $T_2$ are commuting contractions on a Hilbert space, then for any polynomial $p$ in two variables, $$\|p(T_1, T_2)\|\leq \|p\|_{{{\mathbb D}}^2},$$ where ${{\mathbb D}}^2$ is the bidisk in ${{\mathbb C}}^2$. For a nice survey and further generalizations of these inequalities we refer to Pisier’s book [@Pi-book] (see also [@vN], [@SzFBK-book], [@NV], [@Varo], [@Po-von], [@Pa-book], [@CW1], [@CW2], [@AM], and [@DS]). Inspired by the work of Agler-McCarthy [@AM] and Das-Sarkar [@DS] on distinguished varieties and Ando’s inequality for two commuting contractions, we found, in a very recent paper [@Po-Ando], analogues of And\^ o’s results for the elements of the bi-ball ${\bf P}_{n_1,n_2}$ which consists of all pairs $({\bf X}, {\bf Y})$ of row contractions ${\bf X}:=(X_{1},\ldots, X_{n_1})$ and ${\bf Y}:=(Y_{1},\ldots, Y_{n_2})$ which commute, i.e. each entry of ${\bf X}$ commutes with each entry of ${\bf Y}$. The results were obtained in a more general setting, namely, when ${\bf X}$ and ${\bf Y}$ belong to noncommutative varieties ${{\mathcal V}}_1$ and ${{\mathcal V}}_2$ determined by row contractions subject to constraints such as $$q(X_1,\ldots, X_{n_1})=0 \quad \text{and} \quad r(Y_1,\ldots, Y_{n_2})=0, \qquad q\in {{\mathcal P}}, r\in {{\mathcal R}},$$ respectively, where ${{\mathcal P}}$ and ${{\mathcal R}}$ are sets of noncommutative polynomials. This led to one of the main results of the paper, an And\^ o type inequality on noncommutative varieties, which, in the particular case when $n_1=n_2=1$ and $T_1$ and $T_2$ are commuting contractive matrices with spectrum in the open unit disk ${{\mathbb D}}:=\{z\in {{\mathbb C}}:\ |z|<1\}$, takes the form $$\|p(T_1, T_2)\|\leq \min\left\{ \|p(B_1\otimes I_{{{\mathbb C}}^{d_1}},\varphi_1(B_1))\|, \|p(\varphi_2(B_2), B_2\otimes I_{{{\mathbb C}}^{d_2}})\|\right\},$$ where $(B_1\otimes I_{{{\mathbb C}}^{d_1}},\varphi_1(B_1))$ and $(\varphi_2(B_2), B_2\otimes I_{{{\mathbb C}}^{d_2}})$ are analytic dilations of $(T_1, T_2)$ while $B_1$ and $B_2$ are the universal models associated with $T_1$ and $T_2$, respectively. In this setting, the inequality is sharper than And\^ o’s inequality and Agler-McCarthy’s inequality [@AM]. We obtained more general inequalities for arbitrary commuting contractive matrices and improve And\^ o’s inequality for commuting contractions when at least one of them is of class ${{\mathcal C}}_0$. In this setting, it would be interesting to find good analogues for [*distinguished varieties*]{} in the sense of [@AM].
Let ${{\mathbb F}}_n^+$ be the unital free semigroup on $n$ generators $g_1,\ldots, g_n$ and the identity $g_0$. The length of $\alpha\in
{{\mathbb F}}_n^+$ is defined by $|\alpha|:=0$ if $\alpha=g_0$ and $|\alpha|:=k$ if $\alpha=g_{i_1}\cdots g_{i_k}$, where $i_1,\ldots, i_k\in \{1,\ldots, n\}$. If ${\bf Z}:=\left< Z_1,\ldots, Z_n\right>$ is an $n$-tuple of noncommutative indeterminates, we use the notation $Z_\alpha:= Z_{i_1}\cdots Z_{i_k}$ and $Z_{g_0}:=1$. We denote by ${{\mathbb C}}\left<{\bf Z}\right>$ the complex algebra of all polynomials in $Z_1,\ldots, Z_n$. A polynomial $f:=\sum_{\alpha\in{{\mathbb F}}_n^+} a_\alpha Z_\alpha$ in ${{\mathbb C}}\left<{\bf Z}\right>$ is called [*positive regular*]{} if the coefficients satisfy the conditions: $a_\alpha\geq 0$ for any $\alpha\in {{\mathbb F}}_n^+$, $a_{g_0}=0$, and $a_{g_i}>0$ if $i=1,\ldots, n$. Define the [*noncommutative regular domain*]{} $${{\mathcal D}}_f({{\mathcal H}}):=\left\{ {\bf X}:=(X_1,\ldots, X_n)\in B({{\mathcal H}})^n: \
\sum_{|\alpha|\geq 1} a_\alpha X_\alpha
X_\alpha^* \leq I_{{\mathcal H}}\right\}$$ and the [*noncommutative ellipsoid*]{} ${{\mathcal E}}_f({{\mathcal H}})\supseteq {{\mathcal D}}_f({{\mathcal H}})$ by setting $${{\mathcal E}}_f({{\mathcal H}}):=\left\{ {\bf X}:=(X_1,\ldots, X_{n})\in B({{\mathcal H}})^n: \ \sum_{|\beta|=1} a_\beta X_\beta X_\beta^*\leq I_{{\mathcal H}}\right\},$$ where $B({{\mathcal H}})$ stands for the algebra of all bounded linear operators on a Hilbert space ${{\mathcal H}}$. Given $n_1, n_2 \in{{\mathbb N}}:=\{1,2,\ldots\}$ and $\Omega_j\subseteq B({{\mathcal H}})^{n_j}$, $j=1,2$, we denote by $\Omega_1\times_c\Omega_2$ the set of all pairs $ ({\bf X},{ \bf Y})\in \Omega_1\times \Omega_2$ with the property that the entries of ${\bf X}:=(X_{1},\ldots, X_{n_1})$ are commuting with the entries of ${\bf Y}:=(Y_{1},\ldots, Y_{n_2})$.
The main goal of the present paper is to extend the results from [@Po-Ando] for bi-balls and obtain And\^ o type dilations and inequalities for bi-domains and noncommutative varieties: $${{\mathcal D}}_f({{\mathcal H}})\times_c {{\mathcal D}}_g({{\mathcal H}})\quad \text{ and } \quad {{\mathcal V}}_{J_1}({{\mathcal H}})\times_c{{\mathcal V}}_{J_2}({{\mathcal H}}),$$ where $f\in {{\mathbb C}}\left<{\bf Z}\right>$ and $g\in {{\mathbb C}}\left<{\bf Z}'\right>$ are positive regular noncommutative polynomials while ${{\mathcal V}}_{J_1}$ and ${{\mathcal V}}_{J_2}$ are varieties generated by WOT-closed two-sided ideals in certain noncommutative Hardy algebras.
In Section 2, we obtain an intertwining dilation theorem for bi-domains which generalizes Sarason [@S] and Sz.-Nagy–Foiaş [@SzF] commutant lifting theorem for commuting contractions in the framework of noncommutative regular domains and Poisson kernels on weighted Fock spaces (see [@Po-domains]). As a consequence, we obtain a new proof for the commutant lifting theorem for pure elements in ${{\mathcal D}}_f$.
These results are extended, in Section 3, to noncommutative varieties ${{\mathcal V}}_J\subseteq {{\mathcal D}}_f$ which are generated by WOT-closed two-sided ideals $J$ in the Hardy algebra $F_n^\infty({{\mathcal D}}_f)$, a noncommutative multivariable version of the classical Hardy algebra $H^\infty ({{\mathbb D}})$. More precisely, the noncommutative variety ${{\mathcal V}}_{J}({{\mathcal H}})$ is defined as the set of all [*pure*]{} $n$-tuples ${\bf X}:=(X_1,\ldots, X_n)\in {{\mathcal D}}_f({{\mathcal H}})$ with the property that $$\varphi(X_1,\ldots, X_n)=0\quad \text{for any } \ \varphi\in J,$$ where $\varphi(X_1,\ldots, X_n)$ is defined using the $F_n^\infty({{\mathcal D}}_f)$-functional calculus for pure elements in ${{\mathcal D}}_f({{\mathcal H}})$ (see [@Po-domains]). Each variety ${{\mathcal V}}_J$ is associated with certain universal models ${\bf B}=(B_1,\ldots, B_n)$ and ${\bf C}=(C_1,\ldots, C_n)$ of constrained creation operators acting on a subspace ${{\mathcal N}}_J$ of the full Fock space with $n$ generators $F^2(H_n)$. The noncommutative Hardy algebras $F_n^\infty({{\mathcal V}}_{J})$ and $R_n^\infty({{\mathcal V}}_{J})$ are the $WOT$-closed algebras generated by $I, B_1,\ldots, B_n$ and $I, C_1,\ldots, C_n$, respectively. Using our intertwining dilation theorem on noncommutative varieties, we obtain a Schur [@Sc] type representation for the unit ball of ${{\mathcal R}}_{n}^\infty({{\mathcal V}}_J)\bar \otimes B({{\mathcal H}}', {{\mathcal H}})$.
In Section 4, we obtain And\^ o type dilations and inequalities for noncommutative varieties $${{\mathcal V}}_{J_1}({{\mathcal H}})\times_c{{\mathcal V}}_{J_2}({{\mathcal H}}),$$ where ${{\mathcal V}}_{J_1}({{\mathcal H}})\subseteq {{\mathcal D}}_f({{\mathcal H}})$ and ${{\mathcal V}}_{J_2}({{\mathcal H}})\subseteq {{\mathcal D}}_g({{\mathcal H}})$. We prove that any pair $({\bf T}_1, {\bf T}_2)$ in ${{\mathcal V}}_{J_1}({{\mathcal H}})\times_c{{\mathcal V}}_{J_2}({{\mathcal H}})$ has [*analytic dilations*]{} $$({\bf B}_1\otimes I_{\ell^2}, \varphi_1({\bf C}_1))\quad \text{and} \quad
(\varphi_2({\bf C}_2), {\bf B}_2\otimes I_{\ell^2})$$ where $\varphi_1({\bf C}_1)$ and $\varphi_2({\bf C}_2)$ are some multi-analytic operators with respect to the universal models ${\bf B}_1$ and ${\bf B}_2$ of the varieties ${{\mathcal V}}_{J_1}$ and ${{\mathcal V}}_{J_2}$, respectively. As a consequence, we show that the inequality $$\|[p_{rs}({\bf T}_1,{\bf T}_2)]_{k}\|\leq \min \left\{ \|[p_{rs}({\bf B}_1\otimes I_{\ell^2}, \varphi_1({\bf C}_1))]_{k}\|, \|[p_{rs}({\varphi_2({\bf C}_2), \bf B}_2\otimes I_{\ell^2})]_{k}\|\right\}$$ holds for any $[p_{rs}]_k\in M_k({{\mathbb C}}\left<{\bf Z}, {\bf Z}'\right>)$ and $k\in {{\mathbb N}}$. Here, ${{\mathbb C}}\left<{\bf Z}, {\bf Z}'\right>$ denotes the complex algebra of all polynomials in noncommutative indeterminates ${\bf Z}:=\left< Z_1,\ldots, Z_{n_1}\right>$ and ${\bf Z}':=\left< Z_1',\ldots, Z_{n_2}'\right>$, where we assume that $Z_iZ_j'=Z_j'Z_i$ for any $i\in \{1,\ldots, n_1\}$ and $j\in \{1,\ldots, n_2\}$.
On the other hand, we prove that the abstract bi-domain $${{\mathcal D}}_f\times_c {{\mathcal E}}_g:=\{{{\mathcal D}}_f({{\mathcal H}})\times_c {{\mathcal E}}_g({{\mathcal H}}): \ {{\mathcal H}}\text{ is a Hilbert space}\}$$ has a universal [*analytic model*]{} $({\bf W}_1\otimes I_{\ell^2}, \psi({\bf \Lambda}_1))$, where the tuples ${\bf W}_1=(W_{1,1},\ldots, W_{1,n_1})$ and ${\bf \Lambda}_1=(\Lambda_{1,1},\ldots, \Lambda_{1,n_1})$ are the weighted left and right creation operators on the full Fock space $F^2(H_{n_1})$, respectively, associated with the regular domain ${{\mathcal D}}_f$. More precisely, we show that the inequality $$\|[p_{rs}({\bf T}_1,{\bf T}_2)]_{k}\|\leq \|[p_{rs}({\bf W}_1\otimes I_{\ell^2}, \psi({\bf \Lambda}_1))]_{k}\|, \qquad [p_{rs}]_k\in M_k({{\mathbb C}}\left<{\bf Z}, {\bf Z}'\right>),$$ holds for any $({\bf T}_1, {\bf T}_2)\in {{\mathcal D}}_f({{\mathcal H}})\times_c {{\mathcal E}}_g({{\mathcal H}})$ and any $k\in {{\mathbb N}}$. A similar result holds for the abstract variety ${{\mathcal V}}_J\times_c {{\mathcal E}}_g$.
We will see, in Section 4, that all the results of the present paper concerning And\^ o type dilations and inequalities can be written in the commutative multivariable setting in terms of analytic multipliers of certain Hilbert spaces of holomorphic functions. These results are new even when $n_1=n_2=1$. In this particular case, we obtain extensions of And\^ o’s results for commuting contractions [@An], Agler-McCarthy’s inequality [@AM], and Das-Sarkar extension [@DS], to larger classes of commuting operators.
Finally, we would like to thank the referee for helpful comments and suggestions on the paper.
Preliminaries on noncommutative regular domains and universal models
====================================================================
In this section, we recall from [@Po-domains] basic facts concerning the noncommutative regular domains ${{\mathcal D}}_f({{\mathcal H}})\subset B({{\mathcal H}})^n$ generated by positive regular formal power series, their universal models, and the Hardy algebras they generate. We mention that commutative domains generated by positive regular polynomials were first introduced in [@Pot] and further elaborated in [@BS] and in a series of papers by the author (see [@Po-domains] and the references there in).
Let $H_n$ be an $n$-dimensional complex Hilbert space with orthonormal basis $e_1$, $e_2$, $\dots,e_n$, where $n\in\{1,2,\dots\}$. We consider the full Fock space of $H_n$ defined by $$F^2(H_n):=\bigoplus_{k\geq 0} H_n^{\otimes k},$$ where $H_n^{\otimes 0}:={{\mathbb C}}1$ and $H_n^{\otimes k}$ is the (Hilbert) tensor product of $k$ copies of $H_n$. Define the left creation operators $S_i:F^2(H_n)\to F^2(H_n), \ i=1,\dots, n$, by $$S_i\varphi:=e_i\otimes\varphi, \quad \varphi\in F^2(H_n),$$ and the right creation operators $R_i:F^2(H_n)\to F^2(H_n)$ by $
R_i\varphi:=\varphi\otimes e_i$, $ \varphi\in F^2(H_n)$. The noncommutative analytic Toeplitz algebra $F_n^\infty$ and its norm closed version, the noncommutative disc algebra ${{\mathcal A}}_n$, were introduced by the author (see [@Po-von], [@Po-funct], [@Po-analytic]) in connection with a multivariable noncommutative von Neumann inequality. $F_n^\infty$ is the algebra of left multipliers of $F^2(H_n)$ and can be identified with the weakly closed (or $w^*$-closed) algebra generated by the left creation operators $S_1,\dots, S_n$ acting on $F^2(H_n)$, and the identity. The noncommutative disc algebra ${{\mathcal A}}_n$ is the norm closed algebra generated by $S_1,\dots, S_n$, and the identity.
A formal power series $f:=\sum_{\alpha\in{{\mathbb F}}_n^+} a_\alpha Z_\alpha$ in noncommutative indeterminates $Z_1,\ldots, Z_n$ is called [*positive regular*]{} if the coefficients satisfy the conditions: $a_\alpha\geq 0$ for any $\alpha\in {{\mathbb F}}_n^+$, $a_{g_0}=0$, $a_{g_i}>0$ if $i=1,\ldots, n$, and $
\limsup_{k\to\infty} \left( \sum_{|\alpha|=k} |a_\alpha|^2\right)^{1/2k}<\infty.
$ If ${\bf X}:=(X_1,\ldots, X_n)\in B({{\mathcal H}})^n$, we set $X_\alpha:= X_{i_1}\cdots X_{i_k}$ if $\alpha=g_{i_1}\cdots g_{i_k}$, where $i_1,\ldots, i_k\in \{1,\ldots, n\}$, and $X_{g_0}:=I_{{\mathcal H}}$. Define the noncommutative regular domain $${{\mathcal D}}_f({{\mathcal H}}):=\left\{ {\bf X}:=(X_1,\ldots, X_n)\in B({{\mathcal H}})^n: \
\sum_{|\alpha|\geq 1} a_\alpha X_\alpha
X_\alpha^* \leq I_{{\mathcal H}}\right\},$$ where the convergence of the series is in the weak operator topology. The power series $1-f $ is invertible with its inverse $g= \sum_{\alpha\in {{\mathbb F}}_n^+} b_\alpha
X_\alpha $, $b_\alpha\in {{\mathbb C}}$, satisfies the relation $$\begin{split}
g &= 1+ f + f^2+\cdots
=1+\sum_{m=1}^\infty \sum_{|\alpha|=m}\left(\sum_{j=1}^{|\alpha|}
\sum_{{\gamma_1\cdots \gamma_j=\alpha }\atop {|\gamma_1|\geq
1,\ldots,
|\gamma_j|\geq 1}} a_{\gamma_1}\cdots a_{\gamma_j} \right)
X_\alpha.
\end{split}$$ Consequently, we have $$\label{b_alpha}
b_{g_0}=1 \quad \text{ and }\quad b_\alpha= \sum_{j=1}^{|\alpha|}
\sum_{{\gamma_1\cdots \gamma_j=\alpha }\atop {|\gamma_1|\geq
1,\ldots, |\gamma_j|\geq 1}} a_{\gamma_1}\cdots a_{\gamma_j} \quad
\text{ if } \ |\alpha|\geq 1.$$
The [*weighted left creation operators*]{} $W_i:F^2(H_n)\to
F^2(H_n)$, $i=1,\ldots, n$, associated with the noncommutative domain ${{\mathcal D}}_f$ are defined by setting $W_i:=S_iD_i$, where $S_1,\ldots, S_n$ are the left creation operators on the full Fock space $F^2(H_n)$ and each diagonal operator $D_i:F^2(H_n)\to F^2(H_n)$, is given by $$D_ie_\alpha:=\sqrt{\frac{b_\alpha}{b_{g_i \alpha}}} e_\alpha,\qquad
\alpha\in {{\mathbb F}}_n^+.$$ Note that $$W_\beta e_\gamma= \frac {\sqrt{b_\gamma}}{\sqrt{b_{\beta \gamma}}}
e_{\beta \gamma} \quad \text{ and }\quad W_\beta^* e_\alpha
=\begin{cases} \frac {\sqrt{b_\gamma}}{\sqrt{b_{\alpha}}}e_\gamma&
\text{ if }
\alpha=\beta\gamma \\
0& \text{ otherwise }
\end{cases}$$ for any $\alpha, \beta \in {{\mathbb F}}_n^+$. We prove in [@Po-domains] that ${\bf W}:=(W_1,\ldots, W_n)$ is a pure $n$-tuple in $ {{\mathcal D}}_f(F^2(H_n))$, i.e. $\Phi_{f,{\bf W}}(I)\leq I$ and $\Phi_{f,{\bf W}}^k(I)\to 0$ in the strong operator topology, as $k\to\infty$, where $\Phi_{f,{\bf W}}(Y):=\sum\limits_{|\alpha|\geq 1} a_\alpha W_\alpha
YW_\alpha^*$ for $Y\in B(F^2(H_n))$ and the convergence is in the weak operator topology. The $n$-tuple ${\bf W}$ plays the role of universal model for the noncommutative domain ${{\mathcal D}}_f$.
We also define the [*weighted right creation operators*]{} $\Lambda_i:F^2(H_n)\to F^2(H_n)$ by setting $\Lambda_i:= R_i G_i$, $i=1,\ldots, n$, where $R_1,\ldots, R_n$ are the right creation operators on the full Fock space $F^2(H_n)$ and each diagonal operator $G_i$ is defined by $$G_ie_\alpha:=\sqrt{\frac{b_\alpha}{b_{ \alpha g_i}}} e_\alpha,\qquad
\alpha\in {{\mathbb F}}_n^+,$$ where the coefficients $b_\alpha$, $\alpha\in {{\mathbb F}}_n^+$, are given by relation . In this case, we have $$\Lambda_\beta e_\gamma=
\frac {\sqrt{b_\gamma}}{\sqrt{b_{ \gamma \tilde\beta}}}
e_{ \gamma \tilde \beta} \quad
\text{ and }\quad
\Lambda_\beta^* e_\alpha =\begin{cases}
\frac {\sqrt{b_\gamma}}{\sqrt{b_{\alpha}}}e_\gamma& \text{ if }
\alpha=\gamma \tilde \beta \\
0& \text{ otherwise }
\end{cases}$$ for any $\alpha, \beta \in {{\mathbb F}}_n^+$, where $\tilde \beta$ denotes the reverse of $\beta=g_{i_1}\cdots g_{i_k}$, i.e. $\tilde \beta:=g_{i_k}\cdots g_{i_1}$. We remark that if $f :=\sum_{|\alpha|\geq 1} a_\alpha
Z_\alpha$ is a positive regular power series, then so is $\tilde{f} :=\sum_{|\alpha|\geq 1} a_{\tilde \alpha} Z_\alpha$. Moreover, ${\bf \Lambda}:=(\Lambda_1,\ldots, \Lambda_n)\in {{\mathcal D}}_{\tilde f}(F^2(H_n))$ and $W_i=U^*\Lambda_i U$, where $U\in B(F^2(H_n))$ is the unitary operator defined by $U e_\alpha:=e_{\tilde \alpha}$, $\alpha\in {{\mathbb F}}_n^+$. Throughout this paper, we will refer to the $n$-tuples ${\bf W}:=(W_1,\ldots, W_n)$ and ${\bf \Lambda}:=(\Lambda_1,\ldots, \Lambda_n)$ as the weighted creation operators associated with the regular domain ${{\mathcal D}}_f$.
In [@Po-domains], we introduced the domain algebra ${{\mathcal A}}_n({{\mathcal D}}_f)$ associated with the noncommutative domain ${{\mathcal D}}_f$ to be the norm closure of all polynomials in the weighted left creation operators $W_1,\ldots,
W_n$ and the identity. Using the weighted right creation operators associated with ${{\mathcal D}}_f$, one can define the corresponding domain algebra ${{{\mathcal R}}}_n({{\mathcal D}}_f)$. The Hardy algebra $F_n^\infty({{\mathcal D}}_f)$ (resp. $R_n^\infty({{\mathcal D}}_f)$) is the $w^*$- (or WOT-, SOT-) closure of all polynomials in $W_1,\ldots, W_n$ (resp. $\Lambda_1,\ldots, \Lambda_n$) and the identity. We proved that $F_n^\infty({{\mathcal D}}_f)'= R_n^\infty({{\mathcal D}}_f)$ and $R_n^\infty({{\mathcal D}}_f)'=F_n^\infty({{\mathcal D}}_f)$, where $'$ stands for the commutant.
Now, we recall ([@Po-poisson], [@Po-domains]) some basic facts concerning the noncommutative Poisson kernels associated with the regular domains. Let ${\bf T}:=(T_1,\ldots, T_n)$ be an $n$-tuple of operators in the noncommutative domain $ {{\mathcal D}}_f({{\mathcal H}})$, i.e. $\sum\limits_{|\alpha|\geq 1} a_\alpha T_\alpha T_\alpha^*\leq
I_{{\mathcal H}}$. Define the positive linear mapping $$\Phi_{f,{\bf T}}:B({{\mathcal H}})\to
B({{\mathcal H}}) \quad \text{ by } \quad
\Phi_{f,{\bf T}}(X)=\sum\limits_{|\alpha|\geq 1} a_\alpha T_\alpha
XT_\alpha^*,$$ where the convergence is in the weak operator topology. We use the notation $\Phi_{f,{\bf T}}^m$ for the composition $\Phi_{f,{\bf T}}\circ \cdots \circ \Phi_{f,{\bf T}}$ of $\Phi_{f,{\bf T}}$ by itself $m$ times. Since $\Phi_{f,{\bf T}}(I)\leq I$ and $\Phi_{f,{\bf T}}(\cdot)$ is a positive linear map, it is easy to see that $\{\Phi_{f,{\bf T}}^m(I)\}_{m=1}^\infty$ is a decreasing sequence of positive operators and, consequently, $Q_{f,{\bf T}}:=\text{\rm SOT-}\lim\limits_{m\to\infty} \Phi_{f,{\bf T}}^m(I)$ exists. We say that ${\bf T}$ is a [*pure*]{} $n$-tuple in ${{\mathcal D}}_f({{\mathcal H}})$ if $\text{\rm SOT-}\lim \limits_{m\to\infty} \Phi_{f,{\bf T}}^m(I)=0$. Note that, for any ${\bf T}:=(T_1,\ldots, T_n)\in {{\mathcal D}}_f({{\mathcal H}})$ and $0\leq
r<1$, the $n$-tuple $r{\bf T}:=(rT_1,\ldots, rT_n)\in {{\mathcal D}}_f({{\mathcal H}})$ is pure. Indeed, it is enough to see that $\Phi_{f,r{\bf T}}^m(I)\leq r^m \Phi_{f,{\bf T}}^m(I)\leq r^m I$ for any $m\in
{{\mathbb N}}$. Note also that if $\|\Phi_{f,{\bf T}}(I)\|<1$, then $T$ is pure. This is due to the fact that $\|\Phi_{f,{\bf T}}^m(I)\|\leq \|\Phi_{f,{\bf T}}(I)\|^m$. We define the noncommutative Poisson kernel associated with the $n$-tuple ${\bf T} \in {{\mathcal D}}_f({{\mathcal H}})$ to be the operator $K_{f,{\bf T}}:{{\mathcal H}}\to
F^2(H_n)\otimes {{\mathcal D}}_{\bf T}$ defined by $$K_{f,{\bf T}}h=\sum_{\alpha\in {{\mathbb F}}_n^+} \sqrt{b_\alpha}
e_\alpha\otimes \Delta_{f,{\bf T}} T_\alpha^* h,\qquad h\in {{\mathcal H}},$$ where $\Delta_{f,{\bf T}}:=\left(I- \Phi_{f,{\bf T}}(I) \right)^{1/2}$ is the defect operator associated with ${\bf T}$ and ${{\mathcal D}}_{\bf T}:=\overline{\Delta_{f,{\bf T}}({{\mathcal H}})}$ is the corresponding defect space. The operator $K_{f,{\bf T}}$ is a contraction satisfying relation $
K_{f,{\bf T}}^* K_{f,{\bf T}}=I_{{\mathcal H}}-Q_{f,{\bf T}}
$ and $$\label{ker-inter}
K_{f,{\bf T}} T_i^*=(W_i^*\otimes I_{{{\mathcal D}}_{\bf T}})K_{f,{\bf T}},\quad i=1,\ldots, n,$$ where ${\bf W}:=(W_1,\ldots, W_n)$ is the universal model associated with the noncommutative regular domain ${{\mathcal D}}_f$. Moreover, $K_{f,{\bf T}}$ is an isometry if and only if ${\bf T} $ is pure element of $ {{\mathcal D}}_f({{\mathcal H}})$.
Intertwining dilation theorem on noncommutative bi-domains
==========================================================
In this section, we obtain an intertwining dilation theorem which generalizes Sarason and Sz.-Nagy–Foiaş commutant lifting theorem for commuting contractions in the framework of noncommutative regular domains and Poisson kernels on weighted Fock spaces. As a consequence, we obtain a new proof for the commutant lifting theorem for pure elements in ${{\mathcal D}}_f$. More applications of this result will be considered in the next sections.
Unless otherwise specified, we assume, throughout this paper, that $f$ and $g$ are two positive regular polynomials in noncommutative indeterminates ${\bf Z}:=\left< Z_1,\ldots, Z_{n_1}\right>$ and ${\bf Z}':=\left< Z_1',\ldots, Z_{n_2}'\right>$, respectively, of the form $$f:=\sum_{{\alpha\in {{\mathbb F}}_{n_1}^+}, { 1\leq |\alpha|\leq k_1}}a_\alpha Z_\alpha\quad \text{ and } \quad
g:=\sum_{{\beta\in {{\mathbb F}}_{n_2}^+},{ 1\leq |\beta|\leq k_2}}c_\beta Z_\beta'.$$ Fix two tuples of operators ${\bf T}_1=(T_{1,1},\ldots, T_{1,n_1})\in {{\mathcal D}}_{f}({{\mathcal H}})$ and ${\bf T}'_1=(T_{1,1}',\ldots, T_{1,n_1}')\in{{\mathcal D}}_{f}({{\mathcal H}}')$ and let ${\bf T}_2:=(T_{2,1},\ldots, T_{2,n_2})$, with $T_{2,j}:{{\mathcal H}}'\to {{\mathcal H}}$, be such that ${\bf T}_2\in {{\mathcal D}}_{g}({{\mathcal H}}', {{\mathcal H}})$ and intertwines ${\bf T}_1$ with ${\bf T}_1'$, i.e. $$T_{2,j}T_{1,i}'=T_{1,i}T_{2,j}$$ for any $i\in \{1,\ldots, n_1\}$ and $j\in \{1,\ldots, n_2\}$. We denote by ${{\mathcal I}}({\bf T}_1,{\bf T}_1')$ the set of all intertwining tuples ${\bf T}_2$ of ${\bf T}_1$ and ${\bf T}_1'$. A straightforward calculation reveals that $$\Delta_{f,{\bf T}_1}^2+ \Phi_{f, {\bf T}_1}(\Delta_{g,{\bf T}_2}^2)=\Phi_{g,{\bf T}_2}(\Delta_{f,{\bf T}_1'}^2)+\Delta_{g,{\bf T}_2}^2.$$ If the defect spaces ${{\mathcal D}}_{{\bf T}_1}$, ${{\mathcal D}}_{{\bf T}_1'}$, and ${{\mathcal D}}_{{\bf T}_2}$ are finite dimensional with dimensions $d_1:=\dim {{\mathcal D}}_{{\bf T}_1}$, $d_1':=\dim {{\mathcal D}}_{{\bf T}_1'}$, and $d_2:=\dim {{\mathcal D}}_{{\bf T}_2}$, and such that $
d_1+m_1d_2=m_2 d_1'+d_2,
$ where $$m_i:=\text{\rm card} \{\alpha\in {{\mathbb F}}_{n_j}^+: \ 1\leq|\alpha|\leq k_j\},\qquad j=1,2,$$ then there are unitary extensions $U:{{\mathcal D}}_{{\bf T}_1}\oplus \bigoplus_{\alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq k_1}{{\mathcal D}}_{{\bf T}_2}\to \bigoplus_{\beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2}{{\mathcal D}}_{{\bf T}_1'}\oplus {{\mathcal D}}_{{\bf T}_2}
$ of the isometry $$\label{iso1}
U\left(\Delta_{{\bf T}_1}h\oplus \bigoplus_{{\alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq k_1}} \sqrt{a_\alpha}\Delta_{{\bf T}_2}T_{1,\alpha}^*h \right):=
\bigoplus_{\beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2}\sqrt{c_\beta}\Delta_{{\bf T}_1'}T_{2,\beta}^*h ,\qquad h\in {{\mathcal H}}.$$ We denote by ${{\mathcal U}}_{\bf T}$ the set of all unitary extensions of the isometry given by relation . In case the above-mentioned dimensional conditions are not satisfied, then let ${{\mathcal K}}$ be an infinite dimensional Hilbert space and note that the operator defined by $$\label{iso2}
U\left(\Delta_{{\bf T}_1}h\oplus \bigoplus_{{\alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq k_1}} [\sqrt{a_\alpha}\Delta_{{\bf T}_2}T_{1,\alpha}^*h\oplus 0] \right):=
\left(\bigoplus_{\beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2}\sqrt{c_\beta}\Delta_{{\bf T}_1'}T_{2,\beta}^*h\right) \oplus 0$$ is an isometry which can be extended to a unitary operator $$U:{{\mathcal D}}_{{\bf T}_1}\oplus \bigoplus_{\alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq k_1}({{\mathcal D}}_{{\bf T}_2}\oplus {{\mathcal K}})\to \bigoplus_{\beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2}{{\mathcal D}}_{{\bf T}_1'}\oplus ({{\mathcal D}}_{{\bf T}_2}\oplus {{\mathcal K}}).$$ In this setting, we denote by ${{\mathcal U}}_{\bf T}^{{\mathcal K}}$ the set of all unitary extensions of the isometry defined by . Let $ U=\left[ \begin{matrix} A&B\\C&D \end{matrix}\right]
$ be the operator matrix representation of $U\in {{\mathcal U}}_{\bf T}^{{\mathcal K}}$, where $$\label{ABCD}\begin{split}
A&:{{\mathcal D}}_{{\bf T}_1}\to \bigoplus_{{\beta\in {{\mathbb F}}_{n_2}^+}, {1\leq |\beta|\leq k_2}}{{\mathcal D}}_{{\bf T}_1'},\\
B&:\bigoplus_{\alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq k_1}({{\mathcal D}}_{{\bf T}_2}\oplus {{\mathcal K}})\to \bigoplus_{\beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2}{{\mathcal D}}_{{\bf T}_1'},\\
C&: {{\mathcal D}}_{{\bf T}_1}\to {{\mathcal D}}_{{\bf T}_2}\oplus {{\mathcal K}}, \text{\rm and }\\
D&: \bigoplus_{\alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq k_1}({{\mathcal D}}_{{\bf T}_2}\oplus {{\mathcal K}})\to {{\mathcal D}}_{{\bf T}_2}\oplus {{\mathcal K}}.
\end{split}$$ Given an operator $Z:{{\mathcal N}}\to {{\mathcal M}}$ and $n\in {{\mathbb N}}$, we introduce the ampliation $$\text{\rm diag}_{n}(Z):=\left(\begin{matrix}
Z&\cdots &0\\
\vdots&\ddots& \vdots\\
0&\cdots&Z
\end{matrix} \right):\bigoplus_{s=1}^n {{\mathcal N}}\to \bigoplus_{s=1}^n {{\mathcal M}}.$$ In what follows, we consider the lexicographic order for the free semigroup ${{\mathbb F}}_{n_1}^+$, that is $$g_0<g_1<\cdots <g_{n_1}<
g_1g_1 <\cdots < g_1g_{n_1}< \cdots< g_{n_1}g_1<\cdots < g_{n_1}g_{n_1}< \cdots$$ and so on. For the direct product ${{\mathbb F}}_{n_1}^+\times \cdots \times {{\mathbb F}}_{n_1}^+$ of $p$ copies of ${{\mathbb F}}_{n_1}^+$, we say that $(\alpha_p,\ldots, \alpha_1)<(\beta_p,\ldots, \beta_1)$ if $\alpha_p<\beta_p$ or there is $i\in \{2,\ldots, p\}$ such that $\alpha_p=\beta_p,\ldots, \alpha_i=\beta_i$, and $\alpha_{i-1}<\beta_{i-1}$. We also use the operator column notation $\left[\begin{matrix}Y_{(\alpha_p,\ldots, \alpha_1)}\\
\vdots\\
\alpha_i\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha_i|\leq k\end{matrix}\right]$, where the entries $Y_{(\alpha_p,\ldots, \alpha_1)}$ are arranged in the order mentioned above. For simplicity, $[X_1,\ldots, X_n]$ denotes either the $n$-tuple $(X_1,\ldots, X_n)\in B({{\mathcal H}})^n$ or the operator row matrix $[X_1\cdots X_n]$ acting from ${{\mathcal H}}^{(n)}$, the direct sum of $n$ copies of the Hilbert space ${{\mathcal H}}$, to ${{\mathcal H}}$.
\[le1\] If ${\bf X}_1:=[\sqrt{a_\alpha} T_{1,\alpha}:\ \alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq k_1 ]$ and $m_1:=\text{ \rm card} \{\alpha\in {{\mathbb F}}_{n_1}^+: \ 1\leq |\alpha|\leq k_1\}$, then $$\begin{split}
&\text{\rm diag}_{m_1}\left(D\text{\rm diag}_{m_1}\left(\cdots
D\text{\rm diag}_{m_1}\left(\widehat\Delta_{{\bf T}_2}\right)
{\bf X}_1^*\cdots\right){\bf X}_1^*\right){\bf X}_1^*\\
&\qquad\qquad\qquad =
\text{\rm diag}_{m_1}\left(D\text{\rm diag}_{m_1}\left(\cdots
D\text{\rm diag}_{m_1}\left(\widehat\Delta_{{\bf T}_2}\right)
\cdots\right)\right)\left[\begin{matrix}\sqrt{a_{\alpha_1}\cdots a_{\alpha_p}}(T_{1,\alpha_p}\cdots T_{1,\alpha_1})^*\\
\vdots\\
\alpha_i\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha_i|\leq k_1 \end{matrix}\right],
\end{split}$$ where $\text{\rm diag}_{m_1}$ appears $p$ times on each side of the equality, and $\widehat\Delta_{{\bf T}_2}:=\left[\begin{matrix} \Delta_{{\bf T}_2}\\0\end{matrix}\right]:{{\mathcal H}}\to {{\mathcal D}}_{{\bf T}_2}\oplus {{\mathcal K}}$.
Let $D=[D_{(\alpha)} : \ \alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq k_1]$ with $D_{(\alpha)}\in B({{\mathcal D}}_{{\bf T}_2}\oplus {{\mathcal K}})$ and note that $$D\text{\rm diag}_{m_1}\left(\widehat\Delta_{{\bf T}_2}\right)
{\bf X}_1^*=\sum_{\alpha_1\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha_1|\leq k_1} D_{(\alpha_1)}\widehat\Delta_{{\bf T}_2} \sqrt{a_{\alpha_1}}T_{1,\alpha_1}^*$$ and $$\text{\rm diag}_{m_1}\left(D\text{\rm diag}_{m_1}\left(\widehat\Delta_{{\bf T}_2}\right){\bf X}_1^*\right) {\bf X}_1^*
=
\left[\begin{matrix}\sum_{{\alpha_1\in {{\mathbb F}}_{n_1}^+}\atop { 1\leq |\alpha_1|\leq k_1}} D_{(\alpha_1)}\widehat\Delta_{{\bf T}_2} \sqrt{a_{\alpha_1}}\sqrt{a_{\alpha_2}}(T_{1,\alpha_2}T_{1,\alpha_1})^*\\
\vdots\\
\alpha_2\in {{\mathbb F}}_{n_1}^+, 1\leq|\alpha_2|\leq k_1
\end{matrix}\right].$$ An inductive argument shows that $$\begin{split}
&\text{\rm diag}_{m_1}\left(D\text{\rm diag}_{m_1}\left(\cdots
D\text{\rm diag}_{m_1}\left(\widehat\Delta_{{\bf T}_2}\right)
{\bf X}_1^*\cdots\right){\bf X}_1^*\right){\bf X}_1^*\\
&\qquad
=
\left[\begin{matrix}\sum_{{\alpha_1,\ldots, \alpha_{p-1}\in {{\mathbb F}}_{n_1}^+}\atop { 1\leq |\alpha_i|\leq k_1}} D_{(\alpha_{p-1})}\cdots D_{(\alpha_1)}\widehat\Delta_{{\bf T}_2} \sqrt{a_{\alpha_1} \cdots a_{\alpha_{p-1}}a_{\alpha_{p}}}(T_{1,\alpha_p \alpha_{p-1}\cdots \alpha_1})^*\\
\vdots\\
\alpha_p\in {{\mathbb F}}_{n_1}^+, 1\leq|\alpha_p|\leq k_1
\end{matrix}\right],
\end{split}$$ where $\text{\rm diag}_{m_1}$ appears $p$ times. On the other hand, one can easily prove by induction that $$\begin{split}
& \text{\rm diag}_{m_1}\left(D\text{\rm diag}_{m_1}\left(\cdots
D\text{\rm diag}_{m_1}\left(\widehat\Delta_{{\bf T}_2}\right)
\cdots\right)\right)\left[\begin{matrix}\sqrt{a_{\alpha_1}\cdots a_{\alpha_p}}(T_{1,\alpha_p}\cdots T_{1,\alpha_1})^*\\
\vdots\\
\alpha_i\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha_i|\leq p \end{matrix}\right]\\
&
=\text{\rm diag}_{m_1}\left([D_{(\alpha_{p-1})}\cdots D_{(\alpha_{1})} \widehat\Delta_{{\bf T}_2}: \ \alpha_1,\ldots, \alpha_{p-1}\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha_i|\leq k_1]\right) \\
&\qquad \qquad\times
\left[\begin{matrix}\left[\begin{matrix} \sqrt{a_{\alpha_1} \cdots a_{\alpha_{p-1}}a_{\alpha_{p}}}(T_{1,\alpha_p \alpha_{p-1}\cdots \alpha_1})^*\\
\vdots\\
\alpha_1,\ldots, \alpha_{p-1}\in {{\mathbb F}}_{n_1}^+, 1\leq|\alpha_i|\leq k_1
\end{matrix}\right]\\
\vdots\\
\alpha_p\in {{\mathbb F}}_{n_1}^+, 1\leq|\alpha_p|\leq k_1
\end{matrix}\right].
\end{split}$$ The proof is complete.
\[series\] Let ${\bf T}_2\in {{\mathcal I}}({\bf T}_1,{\bf T}_1')$ and let $ U=\left[ \begin{matrix} A&B\\C&D \end{matrix}\right]
$ be the matrix representation of a unitary extension $U\in {{\mathcal U}}_{\bf T}^{{\mathcal K}}$ (see relation ). If $$\begin{split}
{\bf X}_1&:=[\sqrt{a_\alpha} T_{1,\alpha}:\ \alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq k_1 ],\\
m_j&:=\text{ \rm card} \{\alpha\in {{\mathbb F}}_{n_j}^+: \ 1\leq |\alpha|\leq k_j\}, \quad j=1,2,
\end{split}$$ and ${\bf T}_1$ is a pure element in ${{\mathcal D}}_f({{\mathcal H}})$, then $$\begin{split}
\text{\rm diag}_{m_2}(\Delta_{{\bf T}_1'})&\left[\begin{matrix}
\sqrt{c_\beta}T_{2,\beta}^*\\
\vdots\\
\beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2\end{matrix}
\right]\\
&
=A\Delta_{{\bf T}_1}h
+B\sum_{p=0}^\infty \text{\rm diag}_{m_1}\left(D\text{\rm diag}_{m_1}\left(\cdots
D\text{\rm diag}_{m_1}\left(C\Delta_{{\bf T}_1}\right) {\bf X}_1^*\cdots\right){\bf X}_1^*\right){\bf X}_1^*h,
\end{split}$$ for any $h\in {{\mathcal H}}$, where $\text{\rm diag}_{m_1}$ appears $p+1$ times in the general term of the series.
Due to relation , we have $$\label{A}
A\Delta_{{\bf T}_1}h +B\left[\begin{matrix}\left[\begin{matrix}
\Delta_{{\bf T}_2}\sqrt{a_\alpha} T_{1,\alpha}^*h\\0
\end{matrix}\right]\\
\vdots \\
\alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq k_1 \end{matrix}\right]
=
\left[\begin{matrix}
\Delta_{{\bf T}_1'}\sqrt{c_\beta}T_{2,\beta}^*\\
\vdots\\
\beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2\end{matrix}
\right]$$ and $$\label{C}
C\Delta_{{\bf T}_1}h +D\left[\begin{matrix}\left[\begin{matrix}
\Delta_{{\bf T}_2}\sqrt{a_\alpha} T_{1,\alpha}^*h\\0
\end{matrix}\right]\\
\vdots \\
\alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq k_1 \end{matrix}\right]
=
\left[\begin{matrix}
\Delta_{{\bf T}_2} h\\0
\end{matrix}\right]$$ for any $h\in {{\mathcal H}}$. Since $\widehat\Delta_{{\bf T}_2}:=\left[\begin{matrix} \Delta_{{\bf T}_2}\\0\end{matrix}\right]:{{\mathcal H}}\to {{\mathcal D}}_{{\bf T}_2}\oplus {{\mathcal K}}$, we can rewrite relations and as $$\label{AA}
A\Delta_{{\bf T}_1}+B\text{\rm diag}_{m_1}(\widehat\Delta_{{\bf T}_2}) {\bf X}_1^*
=
\text{\rm diag}_{m_2}(\Delta_{{\bf T}_1'}) {\bf X}_2^*,$$ where ${\bf X}_2:=[\sqrt{c_\beta} T_{2,\beta}:\ \beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2 ]$, and $$\label{CC}
C\Delta_{{\bf T}_1}+D\text{\rm diag}_{m_1}(\widehat\Delta_{{\bf T}_2}) {\bf X}_1^*
=
\widehat\Delta_{{\bf T}_2},$$ respectively. Note that using relation we deduce that $$\label{Di}
\text{\rm diag}_{m_1}\left(\widehat\Delta_{{\bf T}_2}\right){\bf X}_1^*=\text{\rm diag}_{m_1}\left(C\Delta_{{\bf T}_1}\right){\bf X}_1^*
+\text{\rm diag}_{m_1}\left(D\text{\rm diag}_{m_1}(\widehat\Delta_{{\bf T}_2}) {\bf X}_1^*\right){\bf X}_1^*,$$ which combined with relation yields
$$\begin{split}
\text{\rm diag}_{m_2}(\Delta_{{\bf T}_1'}) {\bf X}_2^*
=A\Delta_{{\bf T}_1}+B \text{\rm diag}_{m_1}\left(C\Delta_{{\bf T}_1}\right){\bf X}_1^*
+B\text{\rm diag}_{m_1}\left(D\text{\rm diag}_{m_1}(\widehat\Delta_{{\bf T}_2}) {\bf X}_1^*\right){\bf X}_1^*.
\end{split}$$
Continuing to use relation in the latter relation and the resulting ones, an induction argument leads to the identity $$\label{rel}
\begin{split}
\text{\rm diag}_{m_2}(\Delta_{{\bf T}_1'}) {\bf X}_2^*
=A\Delta_{{\bf T}_1}&+B \text{\rm diag}_{m_1}\left(C\Delta_{{\bf T}_1}\right){\bf X}_1^*\\
&+B\sum_{p=1}^m \text{\rm diag}_{m_1}\left(D\text{\rm diag}_{m_1}\left(\cdots
D\text{\rm diag}_{m_1}\left(C\Delta_{{\bf T}_1}\right) {\bf X}_1^*\cdots\right){\bf X}_1^*\right){\bf X}_1^* \\
&+B\text{\rm diag}_{m_1}\left(D\text{\rm diag}_{m_1}\left(\cdots
D\text{\rm diag}_{m_1}\left(\widehat\Delta_{{\bf T}_2}\right)
{\bf X}_1^*\cdots\right){\bf X}_1^*\right){\bf X}_1^*,
\end{split}$$ where $\text{\rm diag}_{m_1}$ appears $p+1$ times in the general term of the sum above and $m+2$ times in the last term. Since $\Delta_{{\bf T}_2}$ and $D$ are contractions and due to Lemma \[le1\], one can easily see that $$\begin{split}
&\left\|
B\text{\rm diag}_{m_1}\left(D\text{\rm diag}_{m_1}\left(\cdots
D\text{\rm diag}_{m_1}\left(\widehat\Delta_{{\bf T}_2}\right)
{\bf T}_1^*\cdots\right){\bf T}_1^*\right){\bf T}_1^*h\right\|\\
&\qquad
\leq \|B\|\left(\sum_{{\alpha_1,\ldots, \alpha_{m+2}\in {{\mathbb F}}_{n_1}^+}\atop { 1\leq |\alpha_i|\leq k_1}}\|\sqrt{a_{\alpha_1}\cdots a_{\alpha_{m+2}}}(T_{1,\alpha_{m+2}}\cdots T_{1,\alpha_1})^*h\|^2\right)^{1/2}=\left< \Phi_{f,{\bf T}_1}^{m+2}(I)h, h\right>
\end{split}$$ for any $h\in {{\mathcal H}}$. Since ${\bf T}_1$ is pure in ${{\mathcal D}}_f({{\mathcal H}})$, we have $\lim_{m\to \infty}\Phi_{f,{\bf T}_1}^{m+2}(I)h=0$ for any $h\in {{\mathcal H}}$. Consequently, relation implies $$\begin{split}
\text{\rm diag}_{m_2}(\Delta_{{\bf T}_1'}) {\bf X}_2^*h
=A\Delta_{{\bf T}_1}h
+B\sum_{p=0}^\infty \text{\rm diag}_{m_1}\left(D\text{\rm diag}_{m_1}\left(\cdots
D\text{\rm diag}_{m_1}\left(C\Delta_{{\bf T}_1}\right) {\bf X}_1^*\cdots\right){\bf X}_1^*\right){\bf X}_1^*h,
\end{split}$$ for any $h\in {{\mathcal H}}$, where $\text{\rm diag}_{m_1}$ appears $p+1$ times in the general term of the series. The proof is complete.
We recall ([@Po-analytic], [@Po-domains]) a few facts concerning multi-analytic operators on Fock spaces. We say that a bounded linear operator $M$ acting from $F^2(H_n)\otimes {{\mathcal K}}$ to $ F^2(H_n)\otimes {{\mathcal K}}'$ is multi-analytic with respect to the universal model ${\bf W}:=(W_1,\ldots, W_n)$ if $
M(W_i\otimes I_{{\mathcal K}})= (W_i\otimes I_{{{\mathcal K}}'}) M\quad
\text{\rm for any }\ i=1,\dots, n.
$ We can associate with $M$ a unique formal Fourier expansion $ \sum_{\alpha \in {{\mathbb F}}_n^+}
\Lambda_\alpha \otimes \theta_{(\alpha)}$ where $\theta_{(\alpha)}\in B({{\mathcal K}}, {{\mathcal K}}')$. We know that $M =\text{\rm SOT-}\lim_{r\to 1}\sum_{k=0}^\infty
\sum_{|\alpha|=k}
r^{|\alpha|} \Lambda_\alpha\otimes \theta_{(\alpha)}
$ where, for each $r\in [0,1)$, the series converges in the uniform norm. Moreover, the set of all multi-analytic operators in $B(F^2(H_n)\otimes {{\mathcal K}},
F^2(H_n)\otimes {{\mathcal K}}')$ coincides with ${{\mathcal R}}_n^\infty({{\mathcal D}}_f)\bar \otimes B({{\mathcal K}},{{\mathcal K}}')$, the WOT-closed operator space generated by the spatial tensor product.
Let ${{\mathcal H}}$, ${{\mathcal H}}'$, and ${{\mathcal E}}$ be Hilbert spaces and consider $$U=\left[ \begin{matrix} A&B\\C&D \end{matrix}\right]:\begin{matrix}{{\mathcal H}}\\\oplus \\ \bigoplus\limits_{{\alpha\in {{\mathbb F}}_{n_1}^+},{ 1\leq|\alpha|\leq k_1}}{{\mathcal E}}\end{matrix}\to \begin{matrix} \bigoplus\limits_{{\beta\in {{\mathbb F}}_{n_2}^+},{ 1\leq|\beta|\leq k_2}}{{\mathcal H}}'\\\oplus \\{{\mathcal E}}\end{matrix}$$ to be a unitary operator. Setting $D=[D_{(\alpha)}: \quad \alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq 1]: \bigoplus\limits_{{\alpha\in {{\mathbb F}}_{n_1}^+},{ 1\leq|\alpha|\leq k_1}}{{\mathcal E}}\to {{\mathcal E}},
$ we associate with $U^*$ and any $r\in [0,1)$ the operator $\varphi_{U^*}(r{\bf \Lambda}_1)$ defined by $$\begin{split}
\varphi_{U^*}(r{\bf \Lambda}_1):= I_{F^2(H_{n_1})}\otimes A^*&+\left(I_{F^2(H_{n_1})}\otimes C^*\right) \left(I_{F^2(H_{n_1})\otimes {{\mathcal E}}}-\sum\limits_{{\alpha\in {{\mathbb F}}_{n_1}^+}\atop{ 1\leq |\alpha|\leq k_1}}
r^{|\alpha|}\sqrt{a_{ \alpha}} \Lambda_{1,\tilde\alpha}\otimes D_{(\alpha)}^*\right)^{-1}\\
& \times \left[\sqrt{a_{ \alpha}} \Lambda_{1,\tilde\alpha}\otimes I_{{\mathcal H}}:\ \alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq k_1\right]\left(I_{F^2(H_{n_1})}\otimes B^*\right),
\end{split}$$ where ${\bf \Lambda}_1:=[\Lambda_{1,1},\ldots, \Lambda_{1,n_1}]$ is the tuple of weighted right creation operators on $F^2(H_{n_1})$ associated with the regular domain ${{\mathcal D}}_f$. In what follows, we use the notations: ${\bf A}:=I_{F^2(H_{n_1})}\otimes A$, ${\bf B}:=I_{F^2(H_{n_1})}\otimes B$, ${\bf C}:=I_{F^2(H_{n_1})}\otimes C$, ${\bf D}:=I_{F^2(H_{n_1})}\otimes D$, and ${\bf \Gamma}(r):=\left[\sqrt{a_{ \alpha}} r^{|\alpha|} \Lambda_{1,\tilde\alpha}\otimes I_{{\mathcal E}}:\ \alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq k_1\right]$.
\[strong-limit\] The strong operator topology limit $\varphi_{U^*}({\bf \Lambda}_1):=\text{\rm SOT-}\lim_{r\to 1}\varphi_{U^*}(r{\bf \Lambda}_1)
$ exists and defines a contractive multi-analytic operator with respect to ${\bf W}_1$ having the row matrix representation $$\varphi_{U^*}({\bf \Lambda}_1)=[\varphi_{(\beta)}({\bf \Lambda}_1): \ \beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2],$$ with $\varphi_{(\beta)}({\bf \Lambda}_1)\in {{\mathcal R}}_{n_1}^\infty({{\mathcal D}}_{f})\bar\otimes B\left( {{\mathcal H}}',{{\mathcal H}}\right)$, where ${{\mathcal R}}_{n_1}^\infty({{\mathcal D}}_{f}) $ is the noncommutative Hardy algebra generated by the weighted right creation operators $\Lambda_{1,1},\ldots, \Lambda_{1,n_1}$ and the identity.
Since ${\bf D}$ and ${\bf \Gamma}(r)$ are contractions, we have $\left\|\sum\limits_{{\alpha\in {{\mathbb F}}_{n_1}^+},{ 1\leq |\alpha|\leq k_1}}
r^{|\alpha|}\sqrt{a_{ \alpha}} \Lambda_{1,\tilde\alpha}\otimes D_{(\alpha)}^*\right\|\leq r<1$. Consequently, the operator $\varphi_{U^*}(r{\bf \Lambda}_1)$ makes sense and $\varphi_{U^*}(r{\bf \Lambda}_1)=[\varphi_{(\beta)}(r{\bf \Lambda}_1): \ \beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2 ]$ with $\varphi_{(\beta)}(r{\bf \Lambda}_1)\in {{\mathcal R}}_{n_1} ({{\mathcal D}}_{f})\bar\otimes B\left( {{\mathcal H}}',{{\mathcal H}}\right)$, where ${{\mathcal R}}_{n_1} ({{\mathcal D}}_{f})$ is the noncommutative disk algebra generated by $\Lambda_{1,1},\ldots, \Lambda_{1,n_1}$ and the identity. Consequently, $$\label{fi-rep}
\varphi_{(\beta)}(r{\bf \Lambda}_1)=\sum_{k=0}^\infty \sum_{\gamma\in {{\mathbb F}}_{n_1}^+, |\gamma|=k} r^{|\gamma|} {\bf \Lambda}_{1,\gamma}\otimes \Theta_{(\gamma)}^{(\beta)}$$ for some operators $\Theta_{(\gamma)}^{(\beta)}\in B({{\mathcal H}}',{{\mathcal H}})$, where the convergence is in the operator norm topology. On the other hand, since $ \left[ \begin{matrix} {\bf A}^*&{\bf C}^*\\{\bf B}^*&{\bf D}^* \end{matrix}\right]
$ is a unitary operator, standard calculations (see e.g. [@Po-varieties]) show that $$\begin{split}
I&-\varphi_{U^*}(r{\bf \Lambda}_1)\varphi_{U^*}(r{\bf \Lambda}_1)^*\\
&= {\bf C}^*(I-{\bf \Gamma}(r) {\bf D}^*)^{-1}\left[\left( I-\sum_{\alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq k_1} r^{2|\alpha|}a_{ \alpha} \Lambda_{1,\tilde\alpha} \Lambda_{1,\tilde\alpha}^*\right)\otimes I\right] (I- {\bf D}{\bf \Gamma}(r)^*)^{-1}{\bf C}.
\end{split}$$ This shows that $\varphi_{U^*}(r{\bf \Lambda}_1)$ is a contraction for any $r\in [0,1)$ having the row matrix representation $\varphi_{U^*}(r{\bf \Lambda}_1)=[\varphi_{(\beta)}(r{\bf \Lambda}_1): \ \beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2 ]$ with $\varphi_{(\beta)}(r{\bf \Lambda}_1)\in {{\mathcal R}}_{n_1} ({{\mathcal D}}_{f})\bar\otimes B\left( {{\mathcal H}}',{{\mathcal H}}\right)$. Since $\varphi_{(\beta)}(r{\bf \Lambda}_1)$ has the Fourier representation , one can see that $\varphi_{(\beta)}({\bf \Lambda}_1):=\text{\rm SOT-}\lim_{r\to 1}\varphi_{(\beta)}(r{\bf \Lambda}_1)$ exists, $\varphi_{(\beta)}({\bf \Lambda}_1)\in {{\mathcal R}}_{n_1}^\infty\bar\otimes B\left({{\mathcal H}}',{{\mathcal H}}\right)$, and $\|\varphi_{U^*}({\bf \Lambda}_1)\|\leq 1$. The proof is complete.
We recall that ${{\mathcal I}}({\bf T}_1,{\bf T}_1')$ is the set of all tuples ${\bf T}_2:=(T_{2,1},\ldots, T_{2,n_2})$, with $T_{2,j}:{{\mathcal H}}'\to {{\mathcal H}}$, such that ${\bf T}_2\in {{\mathcal D}}_{g}({{\mathcal H}}', {{\mathcal H}})$ intertwines ${\bf T}_1$ with ${\bf T}_1'$, i.e. $
T_{2,j}T_{1,i}'=T_{1,i}T_{2,j}
$ for any $i\in \{1,\ldots, n_1\}$ and $j\in \{1,\ldots, n_2\}$. The main result of this section is the following intertwining dilation theorem for the elements of ${{\mathcal I}}({\bf T}_1,{\bf T}_1')$.
\[dil\] Let ${\bf T}_1:=(T_{1,1},\ldots, T_{1,n_1})\in {{\mathcal D}}_f({{\mathcal H}})$ and ${\bf T}_1':=(T_{1,1}',\ldots, T_{1,n_1}')\in {{\mathcal D}}_f({{\mathcal H}}')$, and let ${\bf T}_2:=(T_{2,1},\ldots, T_{2,n_2})\in {{\mathcal D}}_g({{\mathcal H}}', {{\mathcal H}})$ be such that ${\bf T}_2\in {{\mathcal I}}({\bf T}_1,{\bf T}_1')$. Let ${\bf W}_1:=(W_{1,1},\ldots, W_{1,n_1})$ and ${\bf \Lambda}_1:=(\Lambda_{1,1},\ldots, \Lambda_{1, n_1})$ be the weighted creation operators associated with the noncommutative domain ${{\mathcal D}}_f$. If $\varphi_{U^*}({\bf \Lambda}_1)=[\varphi_{(\beta)}({\bf \Lambda}_1): \ \beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2 ]$ is the contractive multi-analytic operator associated with $U\in {{\mathcal U}}_{\bf T}^{{\mathcal K}}$ and ${\bf T}_1$ is a pure element of the noncommutative regular domain ${{\mathcal D}}_f({{\mathcal H}})$, then the following relations hold: $$K_{f,{\bf T}_1'} T_{2,\beta}^* =\frac{1}{\sqrt{c_{\beta}}}\varphi_{(\beta)} ({\bf \Lambda}_1)^*K_{f,{\bf T}_1}, \qquad \beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2,$$ and $$K_{f,{\bf T}_1}T_{1,i}^*=
\left(W_{1,i}^*\otimes I_{{{\mathcal D}}_{{\bf T}_1}}\right) K_{f,{\bf T}_1},\quad
K_{f,{\bf T}_1'}(T_{1,i}')^*=
\left(W_{1,i}^*\otimes I_{{{\mathcal D}}_{{\bf T}_1'}}\right) K_{f,{\bf T}_1'}, \qquad i\in \{1,\ldots, n_1\},$$ where $K_{f,{\bf T}_1}$ and $K_{f,{\bf T}_1'}$ are the noncommutative Poisson kernels associated with ${\bf T}_1$ and ${\bf T}_1'$, respectively.
Fix $U=\left[ \begin{matrix} A&B\\C&D \end{matrix}\right]\in {{\mathcal U}}_{\bf T}^{{\mathcal K}}$ and set $$D:=[D_{(\alpha)}: \quad \alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq 1]: \bigoplus_{\alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq k_1}({{\mathcal D}}_{{\bf T}_2}\oplus {{\mathcal K}})\to {{\mathcal D}}_{{\bf T}_2}\oplus {{\mathcal K}}.$$ In what follows, we use the notations: ${\bf A}:=I_{F^2(H_{n_1})}\otimes A$, ${\bf B}:=I_{F^2(H_{n_1})}\otimes B$, ${\bf C}:=I_{F^2(H_{n_1})}\otimes C$, $$\begin{split}
{\bf Q}&:=\sum_{\alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq k_1} a_{ \alpha} \Lambda_{1,{\tilde\alpha}}\otimes D_{(\alpha)}^* \text{ and }
{\bf \Gamma}:=\left[\sqrt{a_{ \sigma}}\Lambda_{1,{\tilde\sigma}}\otimes I_{{{\mathcal D}}_{{\bf T}_2}\oplus {{\mathcal K}}}: \sigma\in {{\mathbb F}}_{n_1}^+, 1\leq |\sigma|\leq k_1\right].
\end{split}$$ As in Lemma \[le1\], an induction argument over $q$ shows that $$\label{diag-q}
\begin{split}
\text{\rm diag}_{m_1}\left(\text{\rm diag}_{m_1}\cdots\left(
\text{\rm diag}_{m_1}\left(C^*\right)
D^*\right)\cdots D^*\right) =
\text{\rm diag}_{m_1}\left(\left[\begin{matrix}C^*(D_{ (\gamma_1)}\cdots D_{ (\gamma_q)})^*\\
\vdots\\
\gamma_i\in {{\mathbb F}}_{n_1}^+, 1\leq |\gamma_i|\leq k_1\end{matrix}\right]\right),
\end{split}$$ where $\text{\rm diag}_{m_1}$ appears $q+1$ times on the left-hand side of the equality.
We associate with $U^*$ the multi-analytic operator $\varphi_{U^*}({\bf \Lambda}_1)\in {{\mathcal R}}_{n_1}^\infty({{\mathcal D}}_f)\bar\otimes B\left( \bigoplus_{\beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2}{{\mathcal D}}_{{\bf T}_1'}, {{\mathcal D}}_{{\bf T}_1}\right)$, as in Lemma \[strong-limit\], in the particular case when ${{\mathcal H}}={{\mathcal D}}_{{\bf T}_1}$, ${{\mathcal H}}'={{\mathcal D}}_{{\bf T}_1'}$, and ${{\mathcal E}}={{\mathcal D}}_{{\bf T}_2}\oplus {{\mathcal K}}$. Note that, for each $y\in \bigoplus_{\beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2}{{\mathcal D}}_{{\bf T}_1'}$ and $\alpha\in {{\mathbb F}}_{n_1}^+$ with $|\alpha|=n$, we have
$$\varphi_{U^*}({\bf \Lambda}_1)(e_\alpha\otimes y)=e_\alpha\otimes A^*y+
\sum_{q=0}^\infty {\bf C}^*{\bf Q}^q {\bf \Gamma}{\bf B}^*(e_\alpha\otimes y)$$ and ${\bf C}^*{\bf Q}^q {\bf \Gamma}{\bf B}^*(e_\alpha\otimes y)$ is in the closed linear span of all the vectors $e_{\alpha\alpha_1\cdots \alpha_{q+1}}\otimes z$, where $\alpha_1,\ldots, \alpha_{q+1}\in {{\mathbb F}}_{n_1}^+, 1\leq|\alpha_i|\leq k_1$ and $z\in {{\mathcal D}}_{{\bf T}_1}$. Consequently, using the noncommutative Poisson kernel $K_{{\bf T}_1}$, we deduce that $$\label{three}
\begin{split}
&\left<\varphi_{U^*}({\bf \Lambda}_1)^*K_{f,{\bf T}_1}h, e_\alpha\otimes y\right>\\
&=\left< \sum_{k=0}^\infty \sum_{\beta\in {{\mathbb F}}_{n_1}^+,|\beta|=k}
\sqrt{b_\beta} e_\beta\otimes \Delta_{{\bf T}_1}T_{1,\beta}^*h, \varphi_{U^*}({\bf \Lambda}_1)(e_\alpha\otimes y)\right>\\
&=
\left<\sqrt{b_\alpha}\Delta_{{\bf T}_1}T_{1,\alpha}^*h, A^*y\right>
+\sum_{q=0}^\infty \left< \sum_{{\gamma\in {{\mathbb F}}_{n_1}^+} }\sqrt{b_{\alpha \gamma}}e_{\alpha\gamma}\otimes \Delta_{{\bf T}_1}T_{1,\gamma}^* T_{1,\alpha}^*h, {\bf C}^*{\bf Q}^q {\bf \Gamma}{\bf B}^*(e_\alpha\otimes y)
\right>,
\end{split}$$ for any $h\in {{\mathcal H}}$, $y\in \bigoplus_{\beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2}{{\mathcal D}}_{{\bf T}_1'}$, and $\alpha\in {{\mathbb F}}_{n_1}^+$ with $|\alpha|=n$. Setting $$B:=[B_{(\alpha)}: \quad \alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq 1]:\bigoplus_{\alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq k_1}({{\mathcal D}}_{{\bf T}_2}\oplus {{\mathcal K}})\to \bigoplus_{\beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2}{{\mathcal D}}_{{\bf T}_1'},$$ we obtain $$\begin{split}
&{\bf C}^*{\bf Q}^q {\bf \Gamma}{\bf B}^*(e_\alpha\otimes y)\\
&=
(I_{F^2(H_{n_1})}\otimes C^*)\left(\sum_{{\alpha_1,\ldots, \alpha_q\in {{\mathbb F}}_{n_1}^+}\atop{1\leq |\alpha_i|\leq k_1}}\sqrt{a_{\alpha_1}\cdots a_{\alpha_q}}\Lambda_{1,{\tilde\alpha_1}}\cdots \Lambda_{1,{\tilde\alpha_q}}\otimes
D^*_{(\alpha_1)}\cdots D^*_{(\alpha_q)}\right)\\
&
\left[\sqrt{a_{ \sigma}}\Lambda_{1,{\tilde\sigma}}\otimes I_{{{\mathcal D}}_{{\bf T}_2}\oplus {{\mathcal K}}}: \sigma\in {{\mathbb F}}_{n_1}^+, 1\leq |\sigma|\leq k_1\right]
\left[\begin{matrix} I_{F^2(H_{n_1})}\otimes B_{(\sigma)}^*\\
\vdots\\\sigma\in {{\mathbb F}}_{n_1}^+, 1\leq |\sigma|\leq k_1\end{matrix}\right](e_\alpha\otimes y)\\
&=
\sum_{{\sigma\in {{\mathbb F}}_{n_1}^+}\atop {1\leq |\sigma|\leq k_1}}
\left(\sum_{{\alpha_1,\ldots, \alpha_q\in {{\mathbb F}}_{n_1}^+}\atop{1\leq |\alpha_i|\leq k_1}}\sqrt{a_{\alpha_1}\cdots a_{\alpha_q}a_{\sigma}}\Lambda_{1,{\tilde\alpha_1}}\cdots \Lambda_{1,{\tilde\alpha_q}}\Lambda_{1,{\tilde \sigma}}e_\alpha\otimes
C^*D^*_{(\alpha_1)}\cdots D^*_{(\alpha_q)}B_{(\sigma)}^*y\right)
\\
&=
\sum_{{\sigma\in {{\mathbb F}}_{n_1}^+}\atop {1\leq |\sigma|\leq k_1}}
\left(\sum_{{\alpha_1,\ldots, \alpha_q\in {{\mathbb F}}_{n_1}^+}\atop{1\leq |\alpha_i|\leq k_1}}\sqrt{a_{\alpha_1}\cdots a_{\alpha_q}a_{\sigma}}
\frac{\sqrt{b_\alpha}}{\sqrt{b_{\alpha\sigma\alpha_q\cdots \alpha_1}}}
e_{\alpha\sigma\alpha_q\cdots \alpha_1}
C^*D^*_{(\alpha_1)}\cdots D^*_{(\alpha_q)}B_{(\sigma)}^*y\right),
\end{split}$$ where $\widetilde\sigma$ is the reverse of $\sigma\in {{\mathbb F}}_{n_1}^+$. Consequently, using relation , we deduce that $$\begin{split}
&\sum_{q=0}^\infty \left< \sum_{{\gamma\in {{\mathbb F}}_{n_1}^+} }\sqrt{b_{\alpha \gamma}}e_{\alpha\gamma}\otimes \Delta_{{\bf T}_1}T_{1,\gamma}^* T_{1,\alpha}^*h, {\bf C}^*{\bf Q}^q {\bf \Gamma}{\bf B}^*(e_\alpha\otimes y)
\right>\\
&=
\sum_{q=0}^\infty \left< \sum_{{\gamma\in {{\mathbb F}}_{n_1}^+} }\sqrt{b_{\alpha \gamma}}e_{\alpha\gamma}\otimes \Delta_{{\bf T}_1}T_{1,\gamma}^* T_{1,\alpha}^*h,\right.\\
&\qquad \qquad \left.\sum_{{\sigma\in {{\mathbb F}}_{n_1}^+}\atop {1\leq |\sigma|\leq k_1}}
\left(\sum_{{\alpha_1,\ldots, \alpha_q\in {{\mathbb F}}_{n_1}^+}\atop{1\leq |\alpha_i|\leq k_1}}\sqrt{a_{\alpha_1}\cdots a_{\alpha_q}a_{\sigma}}
\frac{\sqrt{b_\alpha}}{\sqrt{b_{\alpha\sigma\alpha_q\cdots \alpha_1}}}
e_{\alpha\sigma\alpha_q\cdots \alpha_1}
C^*D^*_{(\alpha_1)}\cdots D^*_{(\alpha_q)}B_{(\sigma)}^*y\right)
\right>\\
&=
\sum_{q=0}^\infty
\sum_{{\sigma\in {{\mathbb F}}_{n_1}^+}\atop {1\leq |\sigma|\leq k_1}}
\sum_{{\alpha_1,\ldots, \alpha_q\in {{\mathbb F}}_{n_1}^+}\atop{1\leq |\alpha_i|\leq k_1}}
\left<
\sqrt{b_{\alpha\sigma\alpha_q\cdots \alpha_1}}e_{\alpha\sigma\alpha_q\cdots \alpha_1}\otimes \Delta_{{\bf T}_1}T_{1,\sigma\alpha_q\cdots \alpha_1}^* T_{1,\alpha}^*h,\right.\\
&\qquad \qquad\qquad \qquad\left.\sqrt{a_{\alpha_1}\cdots a_{\alpha_q}a_{\sigma}}
\frac{\sqrt{b_\alpha}}{\sqrt{b_{\alpha\sigma\alpha_q\cdots \alpha_1}}}
e_{\alpha\sigma\alpha_q\cdots \alpha_1}
C^*D^*_{(\alpha_1)}\cdots D^*_{(\alpha_q)}B_{(\sigma)}^*y
\right>\\
&=\sum_{q=0}^\infty
\sum_{{\sigma\in {{\mathbb F}}_{n_1}^+}\atop {1\leq |\sigma|\leq k_1}}
\sum_{{\alpha_1,\ldots, \alpha_q\in {{\mathbb F}}_{n_1}^+}\atop{1\leq |\alpha_i|\leq k_1}}
\left<\sqrt{b_\alpha}\sqrt{a_{\alpha_1}\cdots a_{\alpha_q}a_{\sigma}}\Delta_{{\bf T}_1}T_{1,\sigma\alpha_q\cdots \alpha_1}^* T_{1,\alpha}^*h, C^*D^*_{(\alpha_1)}\cdots D^*_{(\alpha_q)}B_{(\sigma)}^*y
\right>\\
&=
\sum_{q=0}^\infty
\left<
\left[ \begin{matrix}
\left[ \begin{matrix} \sqrt{a_{\alpha_1}\cdots a_{\alpha_q}a_{\sigma}}\Delta_{{\bf T}_1}T_{1,\sigma\alpha_q\cdots \alpha_1}^*\\
\vdots\\
{\alpha_1,\ldots, \alpha_q\in {{\mathbb F}}_{n_1}^+}\atop{1\leq |\alpha_i|\leq k_1}
\end{matrix}\right]\\
\vdots\\
\sigma\in {{\mathbb F}}_{n_1}^+, 1\leq |\sigma|\leq k_1
\end{matrix}\right]\sqrt{b_\alpha} T_{1,\alpha}^*h,
\left[ \begin{matrix}
\left[ \begin{matrix} C^*D^*_{(\alpha_1)}\cdots D^*_{(\alpha_q)}B_{(\sigma)}^*y \\
\vdots\\
{\alpha_1,\ldots, \alpha_q\in {{\mathbb F}}_{n_1}^+}\atop{1\leq |\alpha_i|\leq k_1}
\end{matrix}\right]\\
\vdots\\
\sigma\in {{\mathbb F}}_{n_1}^+, 1\leq |\sigma|\leq k_1
\end{matrix}\right]
\right>\\
&=\sum_{q=0}^\infty\left<B \text{\rm diag}_{m_1}\left(D\text{\rm diag}_{m_1}\left(\cdots
D\text{\rm diag}_{m_1}\left(C\Delta_{{\bf T}_1}\right) {\bf X}_1^*\cdots\right){\bf X}_1^*\right){\bf X}_1^*\left(\sqrt{b_\alpha} T_{1,\alpha}^*\right)h, y\right>.
\end{split}$$ Hence and using relation , we obtain $$\begin{split}
&\left<\varphi_{U^*}({\bf \Lambda}_1)^*K_{f,{\bf T}_1}h, e_\alpha\otimes y\right>\\
&=\left<\sqrt{b_\alpha}\Delta_{{\bf T}_1}T_{1,\alpha}^*h, A^*y\right>
+\sum_{q=0}^\infty\left<B \text{\rm diag}_{m_1}\left(D\text{\rm diag}_{m_1}\left(\cdots
D\text{\rm diag}_{m_1}\left(C\Delta_{{\bf T}_1}\right) {\bf X}_1^*\cdots\right){\bf X}_1^*\right){\bf X}_1^*\left(\sqrt{b_\alpha} T_{1,\alpha}^*\right)h, y\right>.
\end{split}$$ Now, using Lemma \[series\], we deduce that $$\begin{split}
\left<\varphi_{U^*}({\bf \Lambda}_1)^*K_{f,{\bf T}_1}h, e_\alpha\otimes y\right>=
\left<\text{\rm diag}_{m_2}(\Delta_{{\bf T}_1'})\left[\begin{matrix}
\sqrt{c_\beta}T_{2,\beta}^*\\
\vdots\\
\beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2\end{matrix}
\right] \left(\sqrt{b_\alpha} T_{1,\alpha}^*\right)h, y\right>
\end{split}$$ for any $h\in {{\mathcal H}}$. Hence, using the definition of the noncommutative Poisson kernel and the fact that ${\bf T}_2\in {{\mathcal I}}({\bf T}_1,{\bf T}_1')$, we deduce that, for any $\beta\in {{\mathbb F}}_{n_2}^+$ with $1\leq |\beta|\leq k_1$, $h\in {{\mathcal H}}$, and $z\in {{\mathcal D}}_{{\bf T}_1'}$, $$\begin{split}
\left<K_{f,{\bf T}_1'}\sqrt{c_\beta} T_{2,\beta}^*h, e_\alpha\otimes z\right>
&=\left<
\sum_{k=0}^\infty \sum_{\sigma\in {{\mathbb F}}_{n_1}^+|\sigma|=k} e_\sigma\otimes
\sqrt{b_\sigma}\Delta_{{\bf T}_1'} (T_{1,\sigma}')^*\sqrt{c_\beta} T_{2,\beta}^*h,e_\alpha\otimes z\right>\\
&=
\left<\sqrt{b_\alpha}\Delta_{{\bf T}_1'} (T_{1,\alpha}')^*\sqrt{c_\beta}T_{2,\beta}^*h,z\right>
=
\left<\sqrt{b_\alpha}\Delta_{{\bf T}_1'} \sqrt{c_\beta}T_{2,\beta}^*T_{1,\alpha}^*h,z\right>\\
&=
\left<\varphi_{U^*}({\bf R})^*K_{{\bf T}_1}h, e_\alpha\otimes y \right>=
\left<\varphi_{(\beta)}({\bf R})^*K_{{\bf T}_1}h, e_\alpha\otimes z\right>,
\end{split}$$ where $y=\bigoplus_{{\gamma\in {{\mathbb F}}_{n_2}^+}\atop{ 1\leq|\gamma|\leq k_2} } y_{(\gamma)}$ with $y_{(\gamma)}=0$ if $\gamma\neq \beta$ and $y_{(\beta)}=z$. Consequently, $$K_{f,{\bf T}_1'} T_{2,\beta}^* =\frac{1}{\sqrt{c_{\beta}}}\varphi_{(\beta)} ({\bf \Lambda}_1)^*K_{f,{\bf T}_1}, \qquad \beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2.$$ The last two relations in the theorem are due to relation applied to ${\bf T}_1\in {{\mathcal D}}_f({{\mathcal H}})$ and ${\bf T}_1'\in {{\mathcal D}}_f({{\mathcal H}}')$, respectively. The proof is complete.
As a consequence of Theorem \[dil\], we obtain a new proof for the commutant lifting theorem for the pure elements of the noncommutative domain ${{\mathcal D}}_f$ (see [@Po-domains]) as well as a constructive method to obtain the lifting.
\[CLT\] Let ${\bf T}_1:=(T_{1, 1},\ldots, T_{1,n_1})\in {{\mathcal D}}_f({{\mathcal H}})$ and ${\bf T}_1'=(T_{1,1}',\ldots, T_{1,n_1}')\in {{\mathcal D}}_f({{\mathcal H}}')$ be pure tuples of operators and let ${\bf W}_1:=[W_{1,1},\ldots, W_{1,n_1}]$ be the universal model associated with the noncommutative domain ${{\mathcal D}}_f$. If $A:{{\mathcal H}}'\to {{\mathcal H}}$ is an operator such that $$AT_{1,i}'=T_{1,i}A, \qquad i\in\{1,\ldots, n_1\},$$ then there is an operator $D:F^2(H_{n_1})\otimes {{\mathcal D}}_{{\bf T}_1'}\to F^2(H_{n_1})\otimes {{\mathcal D}}_{{\bf T}_1}$ such that $$D(W_{1,i}\otimes I_{{{\mathcal D}}_{{\bf T}'_1}})=(W_{1,i}\otimes I_{{{\mathcal D}}_{{\bf T}_1}})D, \qquad
i\in \{1,\ldots, n_1\},$$ $D^*|_{{\mathcal H}}=A^*$, and $\|B\|=\|A\|$, where ${{\mathcal H}}$ and ${{\mathcal H}}'$ are identified with co-invariant subspaces of $\{W_{1,i}\otimes I_{{{\mathcal D}}_{\bf T}}\}_{i=1}^{n_1}$ and $\{W_{1,i}\otimes I_{{{\mathcal D}}_{{\bf T}'}}\}_{i=1}^{n_1}$, respectively.
Without loss of generality, we can assume that $\|A\|=1$. Since $A\in {{\mathcal I}}({\bf T}_1', {\bf T}_1)$, we can apply Theorem \[dil\] in the particular case when $n_2=1$, ${\bf T}_2:=A$, and $g=X$. Consequently, there is a contractive multi-analytic operator $\varphi({\bf \Lambda}_1)\in {{\mathcal R}}_{n_1}^\infty({{\mathcal D}}_f)\bar \otimes B({{\mathcal D}}_{{\bf T}_1'},{{\mathcal D}}_{{\bf T}_1})$ such that $K_{f,{\bf T}_1'} A^*=\varphi({\bf \Lambda}_1)^* K_{f,{\bf T}_1}$. Since ${\bf T}_1$ and ${\bf T}_1'$ are pure elements, the noncommutative Poisson kernels $K_{f,{\bf T}_1}$ and $K_{f,{\bf T}_1'}$ are isometries. Under the identifications of ${{\mathcal H}}$ and ${{\mathcal H}}'$ with $K_{f,{\bf T}_1}{{\mathcal H}}$ and $K_{f,{\bf T}'_1}{{\mathcal H}}'$, respectively, we have $A^*=\varphi({\bf \Lambda}_1)^*|_{{\mathcal H}}$. Since $1=\|A^*\|\leq \|\varphi({\bf \Lambda}_1)^*\|\leq 1$, we deduce that $\|A\|=\|\varphi({\bf \Lambda}_1)\|$. Since $D:=\varphi({\bf \Lambda}_1)$ intertwines $W_{1,i}\otimes I_{{{\mathcal D}}_{{\bf T}_1'}}$ with $W_{1,i}\otimes I_{{{\mathcal D}}_{{\bf T}_1}}$ for each $i\in \{1,\ldots, n_1\}$, the proof is complete.
More applications of Theorem \[dil\] will be considered in the next sections.
Noncommutative varieties, dilations, and Schur representations
==============================================================
In this section, we obtain an intertwining dilation theorem on noncommutative varieties in regular domains and a Schur type representation for the unit ball of ${{\mathcal R}}_{n}^\infty({{\mathcal V}}_J)\bar \otimes B({{\mathcal H}}', {{\mathcal H}})$.
First, we recall from [@Po-varieties] and [@Po-domains] basic facts concerning noncommutative varieties generated by $WOT$-closed two-sided ideals of the Hardy algebra $F^\infty_n({{\mathcal D}}_f)$, their universal models, and the Hardy algebras they generate. Let $J$ be a $WOT$-closed two-sided ideal of $F^\infty_n({{\mathcal D}}_f)$ such that $J\neq F^\infty_n({{\mathcal D}}_f)$. We introduce the noncommutative variety ${{\mathcal V}}_{J}({{\mathcal H}})$ to be the set of all pure $n$-tuples ${\bf T}:=(T_1,\ldots, T_n)\in {{\mathcal D}}_f({{\mathcal H}})$ with the property that $$\varphi(T_1,\ldots, T_n)=0\quad \text{for any } \ \varphi\in J,$$ where $\varphi(T_1,\ldots, T_n)$ is defined using the $F_n^\infty({{\mathcal D}}_f)$-functional calculus for pure elements in ${{\mathcal D}}_f({{\mathcal H}})$. Define the subspaces of $F^2(H_n)$ by $${{\mathcal M}}_J:=\overline{JF^2(H_n)}\quad \text{and}\quad
{{\mathcal N}}_J:=F^2(H_n)\ominus {{\mathcal M}}_J.$$ The subspace ${{\mathcal N}}_J$ is invariant under the operators $W_1^*,\ldots, W_n^*$ and $\Lambda_1^*,\ldots, \Lambda_n^*$, and ${{\mathcal N}}_J\neq 0$ if and only if $ J\neq
F_n^\infty({{\mathcal D}}_f)$. Define the [*constrained weighted left*]{} (resp. [*right*]{}) [*creation operators*]{} associated with the noncommutative variety ${{\mathcal V}}_{J}$ by setting $$B_i:=P_{{{\mathcal N}}_J} W_i|_{{{\mathcal N}}_J} \quad \text{and}\quad C_i:=P_{{{\mathcal N}}_J} \Lambda_i|_{{{\mathcal N}}_J},\quad i=1,\ldots, n.$$ We remark that ${\bf B}:=(B_1,\ldots, B_n)$ is in ${{\mathcal V}}_{J}({{\mathcal N}}_J)$ and plays the role of universal model for the noncommutative variety ${{\mathcal V}}_{J}$. We will refer to the $n$-tuples ${\bf B}:=(B_1,\ldots, B_n)$ and ${\bf C}:=(C_1,\ldots, C_n)$ as the constrained weighted creation operators associated with ${{\mathcal V}}_J$. Note that if $J=\{0\}$, then ${{\mathcal N}}_{\{0\}}=F^2(H_n)$ and ${{\mathcal V}}_{\{0\}}({{\mathcal H}})$ is the set of all pure elements of ${{\mathcal D}}_f({{\mathcal H}})$.
Let $F_n^\infty({{\mathcal V}}_{J})$ be the $WOT$-closed algebra generated by $B_1,\ldots, B_n$ and the identity and let $R_n^\infty({{\mathcal V}}_{J})$ be the $WOT$-closed algebra generated by $C_1,\ldots, C_n$ and the identity. We proved in [@Po-domains] that $$F_n^\infty({{\mathcal V}}_{J})^\prime=R_n^\infty({{\mathcal V}}_{J})\ \text{ and } \
R_n^\infty({{\mathcal V}}_{J})^\prime=F_n^\infty({{\mathcal V}}_{J}),$$ where $^\prime$ stands for the commutant. An operator $M\in
B({{\mathcal N}}_J\otimes {{\mathcal K}},{{\mathcal N}}_J\otimes {{\mathcal K}}')$ is called multi-analytic with respect to the universal model ${\bf B}:=(B_1,\ldots, B_n)$ if $
M(B_i\otimes I_{{{\mathcal K}}})=(B_i\otimes I_{{{\mathcal K}}'})M$ for $i=1,\ldots, n.
$ We recall that the set of all multi-analytic operators with respect to ${\bf B}$ coincides with $$R_n^\infty({{\mathcal V}}_{J})\bar\otimes B({{\mathcal K}},{{\mathcal K}}')=P_{{{\mathcal N}}_J\otimes
{{\mathcal K}}'}[R_n^\infty({{\mathcal D}}_f)\bar\otimes B({{\mathcal K}},{{\mathcal K}}')]|_{{{\mathcal N}}_J\otimes {{\mathcal K}}}.$$ A similar result holds for the Hardy algebra $F_n^\infty({{\mathcal V}}_{J})$. Given a noncommutative variety ${{\mathcal V}}_J({{\mathcal H}})$ and ${\bf T}\in {{\mathcal V}}_J({{\mathcal H}})$, we define the [*constrained Poisson kernel*]{} $K_{J,{\bf T}}:{{\mathcal H}}\to {{\mathcal N}}_J\otimes {{\mathcal D}}_{\bf T}$ by $$K_{J,{\bf T}}:=(P_{{{\mathcal N}}_J}\otimes I_{{{\mathcal D}}_{\bf T}})K_{J,{\bf T}}.$$ We recall that $K_{J,{\bf T}}$ is an isometry and satisfies the relation $$\label{KJK}
K_{J,{\bf T}}T_\alpha^*=(B_\alpha^*\otimes I_{{{\mathcal D}}_{\bf T}})K_{J,{\bf T}},\quad \alpha\in {{\mathbb F}}_n^+.$$
We remark that as a consequence of Theorem \[CLT\], we deduce the commutant lifting theorem for the elements of the noncommutative varieties ${{\mathcal V}}_J$ ([@Po-domains]). More precisely, we can obtain the following result.
Let $J\neq F_{n_1}^\infty({{\mathcal D}}_f)$ be a WOT-closed two-sided ideal of the noncommutative Hardy algebra $F_{n_1}^\infty({{\mathcal D}}_f)$ and let ${\bf B}_1:=(B_{1,1},\ldots, B_{1,n_1})$ and ${\bf C}_1:=(C_{1,1},\ldots, C_{1,n_1})$ be the corresponding constrained shifts acting on ${{\mathcal N}}_J$. For each $j=1,2$, let ${{\mathcal K}}_j$ be a Hilbert space and ${{\mathcal E}}_j\subseteq {{\mathcal N}}_J\otimes {{\mathcal K}}_j$ be a co-invariant subspace under each operator $B_{1,i}\otimes I_{{{\mathcal K}}_j}$, $i=1,\ldots, n_1$. If $X:{{\mathcal E}}_1\to {{\mathcal E}}_2$ is a bounded operator such that $$X[P_{{{\mathcal E}}_1}(B_{1,i}\otimes I_{{{\mathcal K}}_1})|_{{{\mathcal E}}_1}]=[P_{{{\mathcal E}}_2}(B_{1,i}\otimes I_{{{\mathcal K}}_2})]|_{{{\mathcal E}}_2}X,\quad i=1,\ldots,n_1,$$ then there exists $G({\bf C}_1)\in {{\mathcal R}}_{n_1}^\infty({{\mathcal V}}_J)\bar\otimes B({{\mathcal K}}_1,{{\mathcal K}}_2)$ such that $$G({\bf C}_1)^*|_{{{\mathcal E}}_2}=X^* \quad \text{ and }\quad \|G({\bf C}_1)\|=\|X\|.$$
The analogue of Theorem \[dil\] on noncommutative varieties ${{\mathcal V}}_J({{\mathcal H}})$ in the domain ${{\mathcal D}}_f({{\mathcal H}})$ is the following. Recall that ${{\mathcal U}}_{\bf T}^{{\mathcal K}}$ is the set of all unitary extensions of the isometry defined by relation .
\[dil-com\] Let ${\bf T}_1:=(T_{1,1},\ldots, T_{1,n_1})$ and ${\bf T}_1':=(T_{1,1}',\ldots, T_{1,n_1}')$ be elements of the noncommutative varieties ${{\mathcal V}}_J({{\mathcal H}})$ and ${{\mathcal V}}_J({{\mathcal H}}')$, respectively, and let ${\bf T}_2:=(T_{2,1},\ldots, T_{2,n_2})\in {{\mathcal D}}_g({{\mathcal H}}', {{\mathcal H}})$ be such that ${\bf T}_2\in {{\mathcal I}}({\bf T}_1,{\bf T}_1')$. Let ${\bf B}_1:=(B_{1,1},\ldots, B_{1,n_1})$ and ${\bf C}_1:=(C_{1,1},\ldots, C_{1, n_1})$ be the weighted creation operators associated with the noncommutative variety ${{\mathcal V}}_J$. If $$\varphi_{U^*}({\bf \Lambda}_1)=[\varphi_{(\beta)}({\bf \Lambda}_1): \ \beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2 ]$$ is the contractive multi-analytic operator associated with $U\in {{\mathcal U}}_{\bf T}^{{\mathcal K}}$ and ${\bf T}_1$ is pure in ${{\mathcal D}}_f({{\mathcal H}})$, then the following relations hold: $$K_{J,{\bf T}_1'} T_{2,\beta}^* =\frac{1}{\sqrt{c_{\beta}}}\varphi_{(\beta)} ({\bf C}_1)^*K_{J,{\bf T}_1}, \qquad \beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2,$$ $$K_{J,{\bf T}_1}T_{1,i}^*=
\left(B_{1,i}^*\otimes I_{{{\mathcal D}}_{{\bf T}_1}}\right) K_{J,{\bf T}_1},\quad
K_{J,{\bf T}_1'}(T_{1,i}')^*=
\left(B_{1,i}^*\otimes I_{{{\mathcal D}}_{{\bf T}_1'}}\right) K_{J,{\bf T}_1'}, \qquad i\in \{1,\ldots, n_1\},$$ where $K_{J,{\bf T}_1}$ and $K_{J,{\bf T}_1'}$ are the constrained Poisson kernels associated with ${\bf T}_1$ and ${\bf T}_1'$, respectively.
Since ${\bf T}_1\in {{\mathcal V}}_J({{\mathcal H}})$ and ${\bf T}_1'\in {{\mathcal V}}_J({{\mathcal H}}')$, the noncommutative Poisson kernels $K_{f,{\bf T}_1}$ and $K_{f,{\bf T}_1'}$ have ranges in ${{\mathcal N}}_J\otimes {{\mathcal D}}_{{\bf T}_1}$ and ${{\mathcal N}}_J\otimes {{\mathcal D}}_{{\bf T}_1'}$, respectively. Due to Theorem \[dil\], we have $$\label{INT}
K_{f,{\bf T}_1'} T_{2,\beta}^* =\frac{1}{\sqrt{c_{\beta}}}\varphi_{(\beta)} ({\bf \Lambda}_1)^*K_{f,{\bf T}_1}, \qquad \beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2.$$ Since ${{\mathcal N}}_J$ is co-invariant under $\Lambda_{1,1},\ldots, \Lambda_{1, n_1}$, we have $$\varphi_{(\beta)}({\bf \Lambda}_1)^*({{\mathcal N}}_J\otimes {{\mathcal D}}_{{\bf T}_1})\subset {{\mathcal N}}_J\otimes {{\mathcal D}}_{{\bf T}_1'}, \qquad \beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2,$$ and $$\varphi_{(\beta)}({\bf C}_1)=P_{{{\mathcal N}}_J\otimes {{\mathcal D}}_{{\bf T}_1}} \varphi_j({\bf \Lambda}_1)|_{{{\mathcal N}}_J\otimes {{\mathcal D}}_{{\bf T}_1'}}, \qquad \qquad \beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2,$$ is a multi-analytic operator with respect to the universal model ${\bf B}$. Note that relation implies $$P_{{{\mathcal N}}_J\otimes {{\mathcal D}}_{{\bf T}_1'}}K_{f,{\bf T}_1'} T_{2,\beta}^* =\frac{1}{\sqrt{c_{\beta}}}P_{{{\mathcal N}}_J\otimes {{\mathcal D}}_{{\bf T}_1'}}\varphi_{(\beta)}({\bf \Lambda}_1)^*P_{{{\mathcal N}}_J\otimes {{\mathcal D}}_{{\bf T}_1}}K_{f,{\bf T}_1}, \qquad \beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2,$$ which proves that $$K_{J,{\bf T}_1'} T_{2,\beta}^* =\frac{1}{\sqrt{c_{\beta}}}\varphi_{(\beta)} ({\bf C}_1)^*K_{J,{\bf T}_1}, \qquad \beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2.$$ Since the other relations in the theorem are due to . The proof is complete.
As a consequence of Theorem \[dil-com\] we obtain the following Schur [@Sc] type representation for the unit ball of ${{\mathcal R}}_{n_1}^\infty({{\mathcal V}}_J)\bar \otimes B({{\mathcal H}}', {{\mathcal H}})$.
\[transfer\] An operator $\Gamma:{{\mathcal N}}_J\otimes {{\mathcal H}}'\to {{\mathcal N}}_J\otimes {{\mathcal H}}$ is in the closed unit ball of ${{\mathcal R}}_{n_1}^\infty({{\mathcal V}}_J)\bar \otimes B({{\mathcal H}}', {{\mathcal H}})$ if and only if there is a Hilbert space ${{\mathcal E}}$ and a unitary operator $$\Omega=\left[ \begin{matrix} E&F\\G&H \end{matrix}\right]: \begin{matrix} {{\mathcal H}}'\\ \oplus \\ {{\mathcal E}}\end{matrix}\to \begin{matrix} {{\mathcal H}}\\ \oplus \\ \bigoplus\limits_{{\alpha\in {{\mathbb F}}_{n_1}^+} \atop { 1\leq |\alpha|\leq k_1 }}{{\mathcal E}}\end{matrix}$$ such that $\Gamma=\text{\rm SOT-}\lim_{r\to 1}\varphi_{\Omega}(r{\bf C}_1)$, where
$$\begin{split}
\varphi_{\Omega}(r{\bf C}_1)&:= I_{{{\mathcal N}}_J}\otimes E+\left(I_{{{\mathcal N}}_J}\otimes F\right) \left(I_{{{\mathcal N}}_J\otimes {{\mathcal H}}}-\sum\limits_{{\alpha\in {{\mathbb F}}_{n_1}^+}\atop{ 1\leq |\alpha|\leq k_1}}
r^{|\alpha|}\sqrt{a_{ \alpha}} C_{1,\tilde\alpha}\otimes H_{(\alpha)}\right)^{-1}\\
& \qquad \qquad \times \left[\sqrt{a_{ \alpha}} C_{1,\tilde\alpha}\otimes I_{{\mathcal H}}:\ \alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq k_1\right]\left(I_{{{\mathcal N}}_J}\otimes G\right),
\end{split}$$
where ${\bf C}_1:=(C_{1,1},\ldots, C_{1,n_1})$ is the tuple of weighted right creation operators on $F^2(H_{n_1})$ and $H$ has the operator row matrix representation $$H=\left[ \begin{matrix} H_{(\alpha)}\\ \vdots\\ \alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq k_1 \end{matrix} \right]:{{\mathcal E}}\to \bigoplus_{{\alpha\in {{\mathbb F}}_{n_1}^+},{ 1\leq |\alpha|\leq k_1 }} {{\mathcal E}}.$$
Assume that $\Gamma:{{\mathcal N}}_J\otimes {{\mathcal H}}'\to {{\mathcal N}}_J\otimes {{\mathcal H}}$ is a contractive multi-analytic operator with respect to the universal model ${\bf B}_1:=(B_{1,1},\ldots, B_{1,n_1})$, i.e. $\Gamma(B_{1,i}\otimes I_{{{\mathcal H}}'})=(B_{1,i}\otimes I_{{{\mathcal H}}})\Gamma$ for any $i\in \{1,\ldots, n_1\}$. Due to the commutant lifting theorem for pure elements in ${{\mathcal D}}_f$, there exists a contractive multi-analytic operator $\Psi:F^2(H_n)\otimes {{\mathcal H}}'\to F^2(H_n)\otimes {{\mathcal H}}$ with respect to the universal model ${\bf W}_1:=(W_{1,1},\ldots, W_{1,n_1})$, i.e. $\Psi(W_{1,i}\otimes I_{{{\mathcal H}}'})=(W_{1,i}\otimes I_{{{\mathcal H}}})\Psi$ for any $i\in \{1,\ldots, n_1\}$, such that $\|\Gamma\|=\|\Psi\|$ and $\Psi^*|_{{{\mathcal N}}_J\otimes {{\mathcal H}}}=\Gamma^*$.
Set ${\bf T}_1:=(W_{1,1}\otimes I_{{\mathcal H}},\ldots, W_{1,n_1}\otimes I_{{{\mathcal H}}})$, ${\bf T}_1':=(W_{1,1}\otimes I_{{{\mathcal H}}'},\ldots, W_{1,n_1}\otimes I_{{{\mathcal H}}'})$, $n_2=1$, and ${\bf T}_2:=\Psi$. Since $\Psi\in {{\mathcal I}}({\bf T}_1, {\bf T}'_1)$, Theorem \[dil\] and Lemma \[strong-limit\] show that there is a unitary operator $$\Omega=\left[ \begin{matrix} E&F\\G&H \end{matrix}\right]: \begin{matrix} {{\mathcal D}}_{{\bf T}_1'}\\ \oplus \\ {{\mathcal E}}\end{matrix}\to \begin{matrix} {{\mathcal D}}_{{\bf T}_1}\\ \oplus \\ \bigoplus\limits_{{\alpha\in {{\mathbb F}}_{n_1}^+}, { 1\leq |\alpha|\leq k_1 }}{{\mathcal E}}\end{matrix}$$ such that $\varphi_{\Omega}({\bf \Lambda}_1):=\text{\rm SOT-}\lim_{r\to 1}\varphi_{\Omega}(r{\bf \Lambda}_1)$ is a multi-analytic operator in ${{\mathcal R}}_{{n}_1}^\infty({{\mathcal D}}_f)\bar\otimes B({{\mathcal D}}_{{\bf T}_1'},{{\mathcal D}}_{{\bf T}_1})$, where $\varphi_{\Omega}(r{\bf \Lambda}_1)$ is defined as in the theorem and such that $K_{f,{\bf T}_1} \Psi^*=\varphi_{\Omega}({\bf \Lambda}_1)^*K_{f,{\bf T}_1}$. Due to relation $I-\sum_{ |\beta|\geq 1} a_\beta W_\beta W_\beta^*=P_{{\mathbb C}}$, we deduce that ${{\mathcal D}}_{{\bf T}_1}={{\mathcal H}}$ and ${{\mathcal D}}_{{\bf T}_1'}={{\mathcal H}}'$. On the other hand, since $$P_{{\mathbb C}}W_\beta^* e_\alpha =\begin{cases}
\frac {1}{\sqrt{b_{\beta}}} & \text{ if } \alpha=\beta\\
0& \text{ otherwise},
\end{cases}$$ one can easily see that the noncommutative Poisson kernel $K_{f,{\bf T}_1}$ is the identity on $F^2(H_{n_1})\otimes {{\mathcal H}}$. Consequently, $\Psi=\varphi_{\Omega}({\bf \Lambda}_1)$. Since $\Psi^*|_{{{\mathcal N}}_J\otimes {{\mathcal H}}}=\Gamma^*$, we deduce that $\Gamma=\varphi_\Omega({\bf C}_1)$.
To prove the converse, note that, in the particular case when $n_2=1$ and $U^*=\Omega$, Lemma \[strong-limit\] shows that $\Psi:=\text{\rm SOT-}\lim_{r\to 1}\varphi_{\Omega}(r{\bf \Lambda}_1)$, where $$\begin{split}
\varphi_{\Omega}(r{\bf \Lambda}_1):= I_{F^2(H_{n_1})}\otimes E&+\left(I_{F^2(H_{n_1})}\otimes F\right) \left(I_{F^2(H_{n_1})\otimes {{\mathcal E}}}-\sum\limits_{{\alpha\in {{\mathbb F}}_{n_1}^+}\atop{ 1\leq |\alpha|\leq k_1}}
r^{|\alpha|}\sqrt{a_{ \alpha}} \Lambda_{1,\tilde\alpha}\otimes H_{(\alpha)}\right)^{-1}\\
& \times \left[\sqrt{a_{ \alpha}} \Lambda_{1,\tilde\alpha}\otimes I_{{\mathcal H}}:\ \alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq k_1\right]\left(I_{F^2(H_{n_1})}\otimes G\right)
\end{split}$$ and $H=\left[ \begin{matrix} H_{(\alpha)}\\ \vdots\\ \alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq k_1 \end{matrix} \right]:{{\mathcal E}}\to \bigoplus_{{\alpha\in {{\mathbb F}}_{n_1}^+} \atop { 1\leq |\alpha|\leq k_1 }} {{\mathcal E}}$, is a contractive multi-analytic operator with respect to ${\bf W}_1$. Since ${{\mathcal N}}_J$ is a co-invariant subspace under $\Lambda_{1,1},\ldots, \Lambda_{1,n_1}$, we deduce that $\Gamma:=\varphi_\Omega({\bf C}_1)=P_{{{\mathcal N}}_J\otimes {{\mathcal D}}_{{\bf T}_1'}}\Psi|_{{{\mathcal N}}_J\otimes {{\mathcal D}}_{{\bf T}_1}}$ is a contractive multi-analytic operator with respect to ${\bf B}_1$. The proof is complete.
And\^ o type dilations and inequalities on noncommutative bi-domains and varieties
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In this section, we obtain And\^ o type dilations and inequalities for the elements of the bi-domain ${\bf D}_{(f,g)}$ and a class of noncommutative varieties. The commutative case as well the matrix case are also discussed.
We recall that, given a positive regular formal power series $g=\sum\limits_{{\beta\in {{\mathbb F}}_{n_2}^+},{ 1\leq |\beta|\leq k_2}} c_\beta X_\beta$, the noncommutative ellipsoid ${{\mathcal E}}_g({{\mathcal H}})\supseteq {{\mathcal D}}_g({{\mathcal H}})$ is defined by $
{{\mathcal E}}_g({{\mathcal H}}):=\left\{ {\bf X}:=(X_1,\ldots, X_{n_2}): \ \sum_{|\beta|=1} c_\beta X_\beta X_\beta^*\leq I\right\}.
$
One of the most important consequences of the results from Section 2 is the following And\^ o type dilation for the bi-domain ${\bf D}_{(f,g)}({{\mathcal H}}):={{\mathcal D}}_f({{\mathcal H}})\times_c {{\mathcal D}}_g({{\mathcal H}})$, where $f$ and $g$ are positive regular noncommutative polynomials, and for the noncommutative variety $${\bf D}_{(f,g)}^J({{\mathcal H}}):=\left\{({\bf T}_1, {\bf T}_2)\in {\bf D}_{(f,g)}({{\mathcal H}}): {\bf T}_1\in {{\mathcal V}}_J({{\mathcal H}})\right\}.$$ We recall that ${{\mathcal U}}_{\bf T}^{{\mathcal K}}$ is the set of all unitary extensions of the isometry defined by relation . According to Lemma \[strong-limit\], for each $U\in {{\mathcal U}}_{\bf T}^{{\mathcal K}}$, the strong operator topology limit $\varphi_{U^*}({\bf \Lambda}_1):=\text{\rm SOT-}\lim_{r\to 1}\varphi_{U^*}(r{\bf \Lambda_1})
$ exists and defines a contractive multi-analytic operator.
\[dil2\] Let ${\bf T}=({\bf T}_1, {\bf T}_2)\in {\bf D}^J_{(f,g)}({{\mathcal H}})$ with ${\bf T}_1=(T_{1,1},\ldots, T_{1,n_1})$ and ${\bf T}_2:=(T_{2,1},\ldots, T_{2,n_2})$. If $$\varphi_{U^*}({\bf \Lambda}_1)=(\varphi_{(\beta)}({\bf \Lambda}_1): \ \beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2 )$$ is the contractive multi-analytic operator associated with $U\in {{\mathcal U}}_{\bf T}^{{\mathcal K}}$, then $$K_{J,{\bf T}_1} T_{1,\alpha}^* T_{2,\beta}^*=\left(B_{1,\alpha}^*\otimes I_{{{\mathcal D}}_{{\bf T}_1}}\right)\psi_\beta({\bf C}_1)^* K_{J,{\bf T}_1}, \qquad \alpha\in {{\mathbb F}}_{n_1}^+, \beta\in {{\mathbb F}}_{n_2}^+,$$ where
1. ${\bf B}_1:=(B_{1,1},\ldots, B_{1,n_1})$ and ${\bf C}_1:=(C_{1,1},\ldots, C_{1,n_1})$ are the constrained creation operators associated with the variety ${{\mathcal V}}_J$;
2. $K_{J,{\bf T}_1}$ is the constrained Poisson kernel associated ${{\mathcal V}}_J$;
3. $\psi({\bf C}_1):=(\psi_1({\bf C}_1),\ldots, \psi_{n_2}({\bf C}_1))\in {{\mathcal E}}_g({{\mathcal N}}_J\otimes {{\mathcal D}}_{{\bf T}_1})$, where $$\psi_j({\bf C}_1):=\frac{1}{\sqrt{c_{g_j}}}\varphi_{(g_j)}({\bf C}_1), \qquad j\in \{1,\ldots, n_2\}.$$
In the particular case when ${\bf T}_1={\bf T}_1'$, Theorem \[dil-com\] shows that $
K_{J,{\bf T}_1} T_{2,j}^* =\psi_j({\bf C}_1)^*K_{J,{\bf T}_1}$ for $j\in\{1,\ldots, n_2\}
$ and $
K_{J,{\bf T}_1}T_{1,i}^*=
\left(B_{1,i}^*\otimes I_{{{\mathcal D}}_{{\bf T}_1}}\right) K_{J,{\bf T}_1}$ for $ i\in \{1,\ldots, n_1\}.
$ Hence, the relation in the theorem follows.
We remark that Theorem \[dil2\] provides a model and a characterization of the elements $({\bf T}_1, {\bf T}_2)\in {{\mathcal D}}_f({{\mathcal H}})\times_c {{\mathcal E}}_g({{\mathcal H}})$ with ${\bf T}_1\in {{\mathcal V}}_J({{\mathcal H}})$ . Indeed, if ${\bf T}=({\bf T}_1, {\bf T}_2)\in B({{\mathcal H}})^{n_1}\times_c B({{\mathcal H}})^{n_2}$, then ${\bf T}\in {{\mathcal D}}_f({{\mathcal H}})\times_c {{\mathcal E}}_g({{\mathcal H}})$ with ${\bf T}_1\in {{\mathcal V}}_J({{\mathcal H}})$ if and only if there is a Hilbert space ${{\mathcal D}}$, a multi-analytic operator (with respect to ${\bf B}_1$) $$\psi({\bf C}_1)=(\psi_1({\bf C}_1),\ldots, \psi_{n_2}({\bf C}_1))\in {{\mathcal E}}_g({{\mathcal N}}_J\otimes {{\mathcal D}}),$$ and a co-invariant subspace ${{\mathcal M}}\subset {{\mathcal N}}_J\otimes {{\mathcal D}}$ under each of the operators $B_{1,i}\otimes I_{{\mathcal D}}$ and $\varphi_j({\bf C}_1)$, where $i\in \{1,\ldots, n_1\}$ and $j\in \{1,\ldots, n_2\}$, such that ${{\mathcal M}}$ can be identified with ${{\mathcal H}}$, $$(B_{1,i}^*\otimes I_{{\mathcal D}})|_{{\mathcal H}}=T_{1,i}^*,\quad \text{and} \quad \varphi_j({\bf C}_1)^*|_{{\mathcal H}}=T_{2,j}^*.$$ Note that the direct implication is due to Theorem \[dil2\] under the identification of ${{\mathcal H}}$ with $K_{J,{\bf T}_1}{{\mathcal H}}$. The converse is obvious.
In what follows, we obtain And\^ o type inequalities for the bi-domain ${\bf D}_{(f,g)}({{\mathcal H}})$ and the noncommutative variety ${\bf D}_{(f,g)}^J({{\mathcal H}})$. First, we consider the case when ${\bf T}_1=(T_{1,1},\ldots, T_{1,n_1})$ and ${\bf T}_2:=(T_{2,1},\ldots, T_{2,n_2})$ have the property that ${\bf T}=({\bf T}_1, {\bf T}_2)\in {\bf D}_{(f,g)}^J({{\mathcal H}})$ with $d_i:=\dim {{\mathcal D}}_{{\bf T}_i}<\infty$ and $
d_1+m_1d_2=m_2 d_1+d_2,
$ where $$m_i:=\text{\rm card} \{\alpha\in {{\mathbb F}}_{n_j}^+: \ 1\leq|\alpha|\leq k_j\},\qquad j=1,2.$$ The set ${{\mathcal U}}_{\bf T}$ consists of unitary extensions $U:{{\mathcal D}}_{{\bf T}_1}\oplus \bigoplus_{\alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq k_1}{{\mathcal D}}_{{\bf T}_2}\to \bigoplus_{\beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2}{{\mathcal D}}_{{\bf T}_1}\oplus {{\mathcal D}}_{{\bf T}_2}
$ of the isometry $$\label{UU}
U\left(\Delta_{{\bf T}_1}h\oplus \bigoplus_{{\alpha\in {{\mathbb F}}_{n_1}^+, 1\leq |\alpha|\leq k_1}} \sqrt{a_\alpha}\Delta_{{\bf T}_2}T_{1,\alpha}^*h \right):=
\bigoplus_{\beta\in {{\mathbb F}}_{n_2}^+, 1\leq |\beta|\leq k_2}\sqrt{c_\beta}\Delta_{{\bf T}_1}T_{2,\beta}^*h ,\qquad h\in {{\mathcal H}}.$$ Let ${\bf Z}:=\left< Z_1,\ldots, Z_{n_1}\right>$ and ${\bf Z}':=\left< Z_1',\ldots, Z_{n_2}'\right>$ be noncommutative indeterminates and assume that $Z_iZ_j'=Z_j'Z_i$ for any $i\in \{1,\ldots, n_1\}$ and $j\in \{1,\ldots, n_2\}$. We denote by ${{\mathbb C}}\left<{\bf Z}, {\bf Z}'\right>$ the complex algebra of all polynomials in indeterminates $Z_{1},\ldots, Z_{n_1}$ and $Z_{1}',\ldots, Z_{n_2}'$. Note that when $n_1=n_2=1$, then ${{\mathbb C}}\left<{\bf Z}, {\bf Z}'\right>$ coincides with the algebra ${{\mathbb C}}[z,w]$ of complex polynomials in two variable.
\[ando\] Let ${\bf T}=({\bf T}_1, {\bf T}_2)\in {\bf D}_{(f,g)}^J({{\mathcal H}})$ with ${\bf T}_1=(T_{1,1},\ldots, T_{1,n_1})$ and ${\bf T}_2:=(T_{2,1},\ldots, T_{2,n_2})$ such that $$d_i:=\dim {{\mathcal D}}_{{\bf T}_i}<\infty\ \text{ and } \ d_1+m_1d_2=d_2+m_2d_1,$$ and let ${\bf B}_1:=(B_{1,1},\ldots, B_{1,n_1})$ and ${\bf C}_1:=(C_{1,1},\ldots, C_{1,n_1})$ be the constrained weighted creation operators associated with the noncommutative variety ${{\mathcal V}}_J$. If $U\in {{\mathcal U}}_{\bf T}$, then $$\|[p_{rs}({\bf T}_1,{\bf T}_2)]_{k}\|\leq \|[p_{rs}({\bf B}_1\otimes I_{{{\mathbb C}}^{d_1}}, \psi ({\bf C}_1))]_{k}\|, \qquad [p_{rs}]_{ k}\in M_k({{\mathbb C}}\left<{\bf Z}, {\bf Z}'\right>), k\in {{\mathbb N}},$$ where $\psi({\bf C}_1)=(\psi_1({\bf C}_1),\ldots, \psi_{n_2}({\bf C}_1))$ is uniquely determined by $U$ as in Theorem \[dil2\] and each $\psi_j({\bf C}_1) $ is a $d_1\times d_1$-matrix with entries in the Hardy algebra ${{\mathcal R}}_{n_1}^\infty({{\mathcal V}}_J)$.
Since $d_1+m_1d_2=d_2+m_2d_1$, the set ${{\mathcal U}}_{\bf T}$ of all unitary extensions of the isometry $U$ defined by relation is non-empty. Fix any $U\in{{\mathcal U}}_{\bf T}$ and apply Theorem \[dil2\] to ${\bf T}=({\bf T}_1, {\bf T}_2)\in {\bf D}_{(f,g)}^J({{\mathcal H}})$ and $U\in {{\mathcal U}}_{\bf T}$. Then we deduce that $$K_{J,{\bf T}_1} T_{1,\alpha}^* T_{2,\beta}^*=\left(B_{1,\alpha}^*\otimes I_{{{\mathcal D}}_{{\bf T}_1}}\right)\psi_\beta({\bf C}_1)^* K_{J,{\bf T}_1}$$ for any $\alpha\in {{\mathbb F}}_{n_1}^+$ and $\beta\in {{\mathbb F}}_{n_2}^+$. Consequently, if $p$ is any polynomial in ${{\mathbb C}}\left<{\bf Z}, {\bf Z}'\right>$, we obtain $$K_{J,{\bf T}_1} p({\bf T}_1,{\bf T}_2)=p({\bf B}_1\otimes I_{{{\mathbb C}}^{d_1}}, \psi ({\bf C}_1))K_{J,{\bf T}_1}.$$ Since ${\bf T}_1\in {{\mathcal V}}_J({{\mathcal H}})$, the noncommutative Poisson kernel $K_{J,{\bf T}_1}$ is an isometry, which implies $$p({\bf T}_1,{\bf T}_2)=K_{J,{\bf T}_1}^*p({\bf B}_1\otimes I_{{{\mathbb C}}^{d_1}}, \psi ({\bf C}_1))K_{J,{\bf T}_1}.$$ Now, it is clear that $$\|[p_{rs}({\bf T}_1,{\bf T}_2)]_{k\times k}\|\leq \|[p_{rs}({\bf B}_1\otimes I_{{{\mathbb C}}^{d_1}}, \psi ({\bf C}_1))]_{k\times k}\|, \qquad [p_{rs}]_{k\times k}\in M_k({{\mathbb C}}\left<{\bf Z}, {\bf Z}'\right>), k\in {{\mathbb N}}.$$ The proof is complete.
Denote by ${{\mathcal Q}}_{\bf n}^*$ the set of all formal polynomials of the form $q({\bf Z}, {\bf Z}')=\sum a_{\alpha,\beta, \gamma, \sigma} Z_\alpha Z_\beta' (Z_\sigma')^*Z_\gamma^* $, with complex coefficients, where ${\bf Z}:=\left< Z_1,\ldots, Z_{n_1}\right>$ and ${\bf Z}':=\left< Z_1',\ldots, Z_{n_2}'\right>$. In what follows, we show that if we drop the conditions $d_i:=\dim {{\mathcal D}}_{{\bf T}_i}<\infty$ and $d_1+m_1d_2=d_2+m_2d_1$, in Theorem \[ando\], we can obtain the following And\^ o type inequality.
\[ando1\] Let ${\bf T}=({\bf T}_1, {\bf T}_2)\in {\bf D}_{(f,g)}^J({{\mathcal H}})$ with ${\bf T}_1=(T_{1,1},\ldots, T_{1,n_1})$ and ${\bf T}_2:=(T_{2,1},\ldots, T_{2,n_2})$. If $U\in {{\mathcal U}}_{{\bf T}}^{{\mathcal K}}$, then $$\|[q_{rs}({\bf T}_1,{\bf T}_2)]_{k}\|\leq \|[q_{rs}({\bf V}_1,{\bf V}_2)]_{k}\|, \qquad [p_{rs}]_{k\times k}\in M_k({{\mathcal Q}}_{\bf n}^*),$$ where ${\bf V}_1:={\bf B}_1\otimes I_{{{\mathcal D}}_{{\bf T}_1}}$ and ${\bf V}_2:= \psi({\bf C}_1)$ is uniquely determined by $U$ as in Theorem \[dil2\].
The proof uses Theorem \[dil2\] and is similar to the proof of Theorem \[ando\]. We shall omit it.
For any polynomial $p\in {{\mathbb C}}\left<{\bf Z},{\bf Z}\right>$, define $
\|p\|_u:=\sup \|p({\bf T}_1, {\bf T}_2)\|,
$ where the supremum is taken over all pairs $({\bf T}_1, {\bf T}_2)\in {{\mathcal D}}_f({{\mathcal H}})\times_c {{\mathcal E}}_g({{\mathcal H}})$ and any Hilbert space ${{\mathcal H}}$. Then $\|\cdot\|_u$ defines an algebra norm on ${{\mathbb C}}\left<{\bf Z},{\bf Z}'\right>$. Since the proof is very similar to that of Lemma 2.4 from [@Po-Ando], we omit it. If for $[p_{ij}]_k\in M_k({{\mathbb C}}\left<{\bf Z},{\bf Z}\right>)$, we set $$\|[p_{ij}]\|_{u,k}:=\|[p_{ij}]_k\|_u:=\sup \|[p_{ij}({\bf T}_1, {\bf T}_2)]\|,$$ where the supremum is taken over all pairs $({\bf T}_1, {\bf T}_2)\in {{\mathcal D}}_f({{\mathcal H}})\times_c {{\mathcal E}}_g({{\mathcal H}})$ and any Hilbert space ${{\mathcal H}}$, we obtain a sequence of norms on the matrices over ${{\mathbb C}}\left<{\bf Z},{\bf Z}'\right>$. We call $\left({{\mathbb C}}\left<{\bf Z},{\bf Z}'\right>, \|\cdot\|_{u,k}\right)$ the universal operator algebra for the bi-domain $ {{\mathcal D}}_f({{\mathcal H}})\times_c {{\mathcal E}}_g({{\mathcal H}})$.
In what follows, we prove that the abstract bi-domain $${{\mathcal D}}_f\times_c {{\mathcal E}}_g:=\{{{\mathcal D}}_f({{\mathcal H}})\times_c {{\mathcal E}}_g({{\mathcal H}}): \ {{\mathcal H}}\text{ is a Hilbert space}\}$$ has a universal model $({\bf W}_1\otimes I_{\ell^2}, \psi({\bf \Lambda}_1))$, where ${\bf W}_1=(W_{1,1},\ldots, W_{1,n_1})$ and ${\bf \Lambda}_1=(\Lambda_{1,1},\ldots, \Lambda_{1,n_1})$ are the weighted left and right creation operators associated with the regular domain ${{\mathcal D}}_f$, respectively, and $$\psi({\bf \Lambda}_1)=(\psi_1({\bf \Lambda}_1),\ldots, \psi_{n_2}({\bf \Lambda}_1))\in {{\mathcal E}}_g(F^2(H_{n_1})\otimes \ell^2)$$ is a certain multi-analytic operator with respect to ${\bf W}_1$.
\[ando11\] There is a multi-analytic operator $ \psi({\bf \Lambda}_1)=(\psi_1({\bf \Lambda}_1),\ldots, \psi_{n_2}({\bf \Lambda}_1))\in {{\mathcal E}}_g(F^2(H_{n_1})\otimes \ell^2)$ such that $$\|[p_{rs}({\bf T}_1,{\bf T}_2)]_{k}\|\leq \|[p_{rs}({\bf W}_1\otimes I_{\ell^2}, \psi({\bf \Lambda}_1))]_{k}\|, \qquad p_{rs}\in {{\mathbb C}}\left<{\bf Z}, {\bf Z}'\right>,$$ for any $({\bf T}_1, {\bf T}_2)\in {{\mathcal D}}_f({{\mathcal H}})\times_c {{\mathcal E}}_g({{\mathcal H}})$ and any $k\in {{\mathbb N}}$.
Given a matrix $[p_{ij}]_k\in M_k({{\mathbb C}}\left<{\bf Z},{\bf Z}'\right>)$, we have $
\|[p_{ij}]_k\|_u:=\sup \|[p_{ij}({\bf T}_1, {\bf T}_2)]_k\|,
$ where the supremum is taken over all pairs $({\bf T}_1, {\bf T}_2)\in {{\mathcal D}}_f({{\mathcal H}})\times_c {{\mathcal E}}_g({{\mathcal H}})$ and any Hilbert space ${{\mathcal H}}$. Using a standard argument, one can prove that the supremum is the same if we consider only infinite dimensional separable Hilbert spaces. Since $r{\bf T}_1$ is a pure element in ${{\mathcal D}}_f({{\mathcal H}})$ for any $r\in [0,1)$, it is clear that $$\|[p_{ij}]_k\|_u=\sup_{{({\bf T}_1, {\bf T}_2)\in {{\mathcal D}}_f\times_c {{\mathcal E}}_g}\atop{ {\bf T}_1 \text{ pure}}} \|[p_{ij}({\bf T}_1, {\bf T}_2)]_k\|.$$ Fix $[p_{ij}]_k\in M_k({{\mathbb C}}\left<{\bf X}, {\bf Y}\right>)$ and choose a sequence $\left\{({\bf T}_1^{(m)}, {\bf T}_2^{(m)})\right\}_{m=1}^\infty$ in $ {{\mathcal D}}_f({{\mathcal H}})\times_c {{\mathcal E}}_g({{\mathcal H}})$ with ${{\mathcal H}}$ separable and ${\bf T}_1^{(m)}$ pure element in ${{\mathcal D}}_f({{\mathcal H}})$, and such that $$\label{sup2}
\|[p_{ij}]_k\|_u=\sup_{m}\|[p_{ij}({\bf T}_1^{(m)}, {\bf T}_2^{(m)})]_k\|.$$ Using Theorem \[ando1\], in the particular case when $J=\{0\}$, we find, for each $m\in {{\mathbb N}}$, a multi-analytic operator $\psi^{(m)}({\bf \Lambda}_1):=(\psi_1^{(m)}({\bf \Lambda}_1),\ldots, \psi_{n_2}^{(m)}({\bf \Lambda}_1))$ with respect to ${\bf W}_1$, which belongs to the ellipsoid ${{\mathcal E}}_g(F^2(H_{n_1})\otimes {{\mathbb C}}^{d(m)})$, where $d(m):={{\mathcal D}}_{{\bf T}_1^{(m)}}$, such that $$\|[p_{ij}({\bf T}_1^{(m)},{\bf T}_2^{(m)})]_k\|\leq \|[p_{ij}({\bf W}_1\otimes I_{{{\mathbb C}}^{d(m)}},\psi^{(m)}({\bf \Lambda}_1))]_{k }\|.$$ Consequently, setting $\oplus_{m=1}^\infty {\bf T}_1^{(m)}:=\left(\oplus_{m=1}^\infty {T}_{1,1}^{(m)},\ldots,\oplus_{m=1}^\infty { T}_{1,n_1}^{(m)}\right)\in {{\mathcal D}}_f(\oplus_{m=1}^\infty {{\mathcal H}})$, relation implies $$\begin{split}
\|[p_{ij}]_k\|_u&=\|[p_{ij}(\oplus_{m=1}^\infty {\bf T}_1^{(m)}, \oplus_{m=1}^\infty {\bf T}_2^{(m)})]_k\|\\
&\leq
\|[p_{ij}(\oplus_{m=1}^\infty ({\bf W}_1\otimes I_{{{\mathbb C}}^{d(m)}}), \oplus_{m=1}^\infty \psi^{(m)}({\bf \Lambda}_1))]_k\|\leq \|[p_{ij}]_k\|_u.
\end{split}$$ This shows that $$\label{ma22}
\|[p_{ij}]_k\|_u= \|[p_{ij}({\bf W}_1\otimes I_{ \ell^2},\zeta({\bf \Lambda}_1))]_{k }\|,$$ where $\zeta({\bf \Lambda}_1)=(\zeta_1({\bf \Lambda}_1),\ldots, \zeta_{n_2}({\bf \Lambda}_1)):=\oplus_{m=1}^\infty \psi^{(m)}({\bf \Lambda}_1))\in {{\mathcal E}}_g(F^2(H_{n_1})\otimes \ell^2)$ is a multi-analytic operator with respect to ${\bf W}_1$. Let ${{\mathbb C}}_{{\mathbb Q}}\left<{\bf Z}, {\bf Z}'\right>$ be the set of all polynomials with coefficients in ${{\mathbb Q}}+i{{\mathbb Q}}$, and let $
[p^{(1)}_{ij}]_k, [p^{(2)}_{ij}]_k, \ldots$ be an enumeration of the set $\{[p_{ij}]_k: \ p_{ij}\in {{\mathbb C}}_{{\mathbb Q}}\left<{\bf Z}, {\bf Z}'\right>\}$. Due to relation , for each $s\in {{\mathbb N}}$, there is a multi-analytic operator $\zeta^{(s)}({\bf \Lambda}_1)=(\zeta^{(s)}_1({\bf \Lambda}_1),\ldots, \zeta^{(s)}_{n_2}({\bf \Lambda}_1))\in {{\mathcal E}}_g(F^2(H_{n_1})\otimes \ell^2)$ such that $$\label{ma222}
\|[p^{(s)}_{ij}]_k\|_u= \|[p_{ij}^{(s)}({\bf W}_1\otimes I_{ \ell^2},\zeta^{(s)}({\bf \Lambda}_1))]_{k }\|,\qquad s\in {{\mathbb N}}.$$ Define the multi-analytic operator $\Omega_k({\bf \Lambda}_1):=\oplus_{s=1}^\infty \zeta^{(s)}({\bf \Lambda}_1)\in {{\mathcal E}}_g(F^2(H_{n_1}\otimes \ell^2)$ and let us prove that $$\label{qqij}
\|[q_{ij}]_k\|_u= \|[q_{ij}({\bf W}_1\otimes I_{ \ell^2},\Omega_k({\bf \Lambda}_1))]_{k }\|$$ for any $[q_{ij}]_k\in M_k({{\mathbb C}}\left<{\bf Z}, {\bf Z}'\right>)$. Note that relation implies $$\label{Ga2}
\|[p^{(s)}_{ij}]_k\|_u= \|[p_{ij}^{(s)}({\bf W}_1\otimes I_{ \ell^2},\Omega_k({\bf \Lambda}_1))]_{k }\|\qquad \text{for any } s\in {{\mathbb N}}.$$ Fix $[q_{ij}]_k\in M_k({{\mathbb C}}\left<{\bf Z}, {\bf Z}'\right>)$ and $\epsilon >0$, and choose $[p^{(s_0)}_{ij}]_k$ such that $$\label{u2}
\left\|[q_{ij}]_k-[p^{(s_0)}_{ij}]_k\right\|_u<\epsilon.$$ Using relations , , and , we deduce that there is $\zeta^{(q)}:=(\zeta_1^{(q)}({\bf \Lambda}_1),\ldots, \zeta_{n_2}^{(q)}({\bf \Lambda}_1))$ in the ellipsoid $ {{\mathcal E}}_g(F^2(H_{n_1})\otimes \ell^2)$ such that $$\begin{split}
\|[q_{ij}]_k\|_u&=\|[q_{ij}({\bf W}_1\otimes I_{ \ell^2},\zeta^{(q)}({\bf \Lambda}_1))]_{k }\|
\leq\|[p_{ij}^{(s_0)}({\bf W}_1\otimes I_{ \ell^2},\zeta^{(q)}({\bf \Lambda}_1))]_{k }\| +\epsilon\\
&\leq \|[p_{ij}^{(s_0)}]_k\|_u+\epsilon
=
\|[p_{ij}^{(s_0)}({\bf W}_1\otimes I_{ \ell^2},\Omega_k({\bf \Lambda}_1))]_{k }\|+\epsilon\\
&\leq
\|[q_{ij}({\bf W}_1\otimes I_{ \ell^2},\Omega_k({\bf \Lambda}_1))]_{k }\|+2\epsilon
\end{split}$$ for any $\epsilon>0$, which proves relation . Note that $\psi({\bf \Lambda}_1):=\oplus_{k=1}^\infty \Omega_k({\bf \Lambda}_1)$ is a multi-analytic operator which belongs to the ellipsoid ${{\mathcal E}}_g(F^2(H_{n_1})\otimes \ell^2)$ and $
\|[q_{ij}]_k\|_u= \|[q_{ij}({\bf W}_1\otimes I_{ \ell^2},\psi({\bf \Lambda}_1))]_{k }\|
$ for any $[q_{ij}]_k\in M_k({{\mathbb C}}\left<{\bf Z}, {\bf Z}'\right>)$ and any $k\in {{\mathbb N}}$. The proof is complete.
Theorem \[ando11\] shows that $\left({{\mathbb C}}\left<{\bf Z},{\bf Z}'\right>, \|\cdot\|_{u,k}\right)$ can be realized completely isometrically isomorphic as a concrete algebra of operators. The closed non-self-adjoint algebra generated by the operators $ W_{1,1}\otimes I_{\ell^2},\ldots, W_{1,n_1}\otimes I_{\ell^2}, \psi_1({\bf \Lambda}_1),\ldots, \psi_{n_2}({\bf \Lambda}_1)$ and the identity is denoted by ${{\mathcal A}}({{\mathcal D}}_f\times_c{{\mathcal E}}_g)$ and can be seen as the universal operator algebra of the bi-domain ${{\mathcal D}}_f\times_c{{\mathcal E}}_g$.
We remark that the noncommutative variety ${{\mathcal V}}_J\times_c {{\mathcal E}}_g$ also has a universal model. Similarly to the proof of Theorem \[ando11\], one can show that there is a multi-analytic operator $\psi({\bf C}_1)=(\psi_1({\bf C}_1),\ldots, \psi_{n_2}({\bf C}_1))$, with respect to ${\bf B}_1$, in $ {{\mathcal E}}_g({{\mathcal N}}_J\otimes \ell^2)$ such that $$\|[p_{rs}({\bf T}_1,{\bf T}_2)]_{k}\|\leq \|[p_{rs}({\bf B}_1\otimes I_{\ell^2}, \psi({\bf C}_1))]_{k}\|, \qquad p_{rs}\in {{\mathbb C}}\left<{\bf Z}, {\bf Z}'\right>,$$ for any $({\bf T}_1, {\bf T}_2)\in {{\mathcal V}}_J({{\mathcal H}})\times_c {{\mathcal E}}_g({{\mathcal H}})$ and any $k\in {{\mathbb N}}$.
In the end of this section, we discuss the commutative case. Let $J_c$ be the $WOT$-closed two-sided ideal of the Hardy algebra $F_{n_1}^\infty({{\mathcal D}}_f)$ generated by the commutators $W_j W_i-W_iW_j$ for $i,j\in \{1,\ldots, n_1\}$. Note that the variety ${{\mathcal V}}_{J_c}({{\mathcal H}})$ consists of all pure tuples $(X_1,\ldots, X_{n_1})\in {{\mathcal D}}_f({{\mathcal H}})$ with commuting entries. The Hardy algebra $F_{n_1}^\infty({{\mathcal V}}_{J_c})$ is the $WOT$-closed algebra generated by the compressions $L_i:=P_{F_s^2({{\mathcal D}}_f)} W_i|_{F_s^2({{\mathcal D}}_f)}$, $i=1,\ldots, n_1$, and the identity, where $F_s^2({{\mathcal D}}_f)={{\mathcal N}}_{J_c}$ is the symmetric weighted Fock space associated with the noncommutative domain ${{\mathcal D}}_f$. In [@Po-domains], we prove that $F_s^2({{\mathcal D}}_f)$ can be identified with a Hilbert space $H^2({{\mathcal D}}_f^\circ({{\mathbb C}}))$ of holomorphic functions defined on the scalar domain $${{\mathcal D}}_f^\circ({{\mathbb C}}):=\left\{ (\lambda_1,\ldots, \lambda_{n_1})\in {{\mathbb C}}^{n_1}: \
\sum_{|\alpha|\geq 1} a_\alpha |\lambda_\alpha|^2<1\right\},$$ namely, the reproducing kernel Hilbert space with reproducing kernel $\kappa_f:{{\mathcal D}}_f^\circ({{\mathbb C}})\times
{{\mathcal D}}_f^\circ({{\mathbb C}})$ defined by $
\kappa_f(\mu,\lambda):=\frac{1}{1-\sum_{|\alpha|\geq 1} a_\alpha
\mu_\alpha \overline{\lambda}_\alpha}$ for $\mu,\lambda\in
C$. We also identified the algebra of all multipliers of the Hilbert space $H^2({{\mathcal D}}_f^\circ({{\mathbb C}}))$ with the Hardy algebra $F_{n_1}^\infty({{\mathcal V}}_{J_c})$. Under this identification, $L_i$ is the multiplier $M_{\lambda_i}$ by the coordinate function. We denote ${\bf M}_{\lambda,n_1}:=(M_{\lambda_1},\ldots, M_{\lambda_{n_1}})$. Similarly, one can identify the Hardy algebra $R_{n_1}^\infty({{\mathcal V}}_{J_c})$ with the algebra of all multipliers of the Hilbert space $H^2({{\mathcal D}}_{\tilde f}^\circ({{\mathbb C}}))$, where $\tilde{f} :=\sum_{|\alpha|\geq 1} a_{\tilde \alpha} Z_\alpha$. Note also that ${{\mathcal D}}_f^\circ({{\mathbb C}})={{\mathcal D}}_{\tilde f}^\circ({{\mathbb C}})$.
\[commutative\] Let $f\in {{\mathbb C}}\left<{\bf Z}\right>$ and $g\in {{\mathbb C}}\left<{\bf Z}'\right>$ be two positive regular noncommutative polynomials and let $({\bf T}_1, {\bf T}_2)\in {{\mathcal D}}_f({{\mathcal H}})\times_c {{\mathcal D}}_g({{\mathcal H}})$ be such that each tuple ${\bf T}_j=(T_{j,1},\ldots, T_{j,n_j})$ has commuting entries and $d_j:={\hbox{\rm{rank}}\,}\Delta_{{\bf T}_j}$, $j=1,2$. Then there exist multipliers $M_{\Phi_f}$ and $M_{\Phi_g}$ of $H^2({{\mathcal D}}_f^\circ)\otimes {{\mathbb C}}^{d_1}$ and $H^2({{\mathcal D}}_g^\circ)\otimes {{\mathbb C}}^{d_2}$, respectively, such that $M_{\Phi_f}\in {{\mathcal E}}_f(H^2({{\mathcal D}}_f^\circ))$, $M_{\Phi_g}\in {{\mathcal E}}_g(H^2({{\mathcal D}}_g^\circ))$, and $$\|[p_{rs}({\bf T}_1,{\bf T}_2)]_{k}\|\leq \min \left\{ \|[p_{rs}({\bf M}_{\lambda, n_1}\otimes I_{{{\mathbb C}}^{d_1}}, M_{\Phi_f})]_{k}\|, \|[p_{rs}(M_{\Phi_g}, {\bf M}_{\lambda, n_2}\otimes I_{{{\mathbb C}}^{d_2}})]_{k}\|\right\}$$ for any matrix $ [p_{rs}]_k\in M_k({{\mathbb C}}\left<{\bf Z}, {\bf Z}'\right>)$ and any $k\in {{\mathbb N}}$.
Applying Theorem \[ando1\] to the pairs $({\bf T}_1, {\bf T}_2)\in {\bf D}_{(f,g)}^{J_c}({{\mathcal H}})$ and $({\bf T}_2, {\bf T}_1)\in {\bf D}_{(g,f)}^{J_c}({{\mathcal H}})$ and using the the identifications preceding this theorem, one can easily complete the proof.
We should mention that all the results of the present paper concerning And\^ o type dilations and inequalities can be written in the commutative multivariable setting of Theorem \[commutative\]. Moreover, in the particular case when $n_1=n_2=1$, we obtain extensions of And\^ o’s results [@An], Agler-McCarthy’s inequality [@AM], and Das-Sarkar extension [@DS], to larger classes of commuting operators.
A few remarks concerning the matrix case when $n_1=n_2=1$ are necessary. If $T_1$ and $T_2$ are commuting contractive matrices with no eigenvalues of modulus 1, Agler and McCarthy proved, in their remarkable paper [@AM], that the pair $(T_1, T_2)$ has a co-isometric extension $(M_z^*, M_\Phi^*)$ on $H^2\otimes {{\mathbb C}}^d$ and, for any polynomial $p$ in two variables, $$\|p(T_1, T_2)\|\leq \|p(M_z\otimes I_{{{\mathbb C}}^{d}}, M_\Phi)\|\leq \|p\|_V,$$ where $V$ is a distinguished variety in ${{\mathbb D}}^2$ depending on $T_1$ and $T_2$.
Let $f\in {{\mathbb C}}\left[z\right]$ be a positive regular polynomial in one variable and let $T\in {{\mathcal D}}_f({{\mathbb C}}^n)$ be an $n\times n$ matrix which is pure with respect to the regular domain ${{\mathcal D}}_f$, i.e. $\text{\rm SOT-}\lim \limits_{m\to\infty} \Phi_{f,{\bf T}}^m(I)=0$. Let $m_{T}(z)=(z-\lambda_1)^{n_1}\cdots (z-\lambda_k)^{n_k}$ be the minimal polynomial of $T$ and let $J_{m_T}$ be the $WOT$-closed two sided ideal of the Hardy algebra $F_1^\infty({{\mathcal D}}_f)$ generated by $m_{T}({\bf S})$, where ${\bf S}$ is the weighted shift associated with the domain ${{\mathcal D}}_f$. Note that the variety ${{\mathcal V}}_{J_{m_T}}({{\mathbb C}}^n)$ consists of all $n\times n$ matrices $X$ such that $m_T(X)=0$. On the other hand, $B:=P_{{{\mathcal N}}_{J_{m_T}}} {\bf S}|_{{{\mathcal N}}_{J_{m_T}}}$ is the universal model of the variety ${{\mathcal V}}_{J_{m_T}}$, and the ellipsoid ${{\mathcal E}}_f({{\mathbb C}}^n)$ is a matrix-valued ball. In this case, the analytic operators with respect to $B$ are the elements $\varphi(B)$ of the Hardy algebra $R_1^\infty({{\mathcal V}}_{J_{m_T}})$.
\[last\] Let $T_1$ and $T_2$ be commuting matrices which are pure elements in ${{\mathcal D}}_f({{\mathbb C}}^n)$ and ${{\mathcal D}}_g({{\mathbb C}}^n)$ and let $B_1$ and $B_2$ be their universal models, respectively. If $d_j:=\dim (I-T_jT_j^*)^{1/2}{{\mathcal H}}$, $j=1,2$, then there exist matrix-valued analytic operators $\varphi_1(B_1)\in {{\mathcal E}}_f({{\mathcal N}}_{J_{m_{T_1}}}\otimes {{\mathbb C}}^{d_1})$ with respect to $B_1$ and $\varphi_2(B_2)\in {{\mathcal E}}_g({{\mathcal N}}_{J_{m_{T_2}}}\otimes {{\mathbb C}}^{d_2})$ with respect to $B_2$, such that $$\|[p_{rs}(T_1, T_2)]_k\|\leq \min\left\{ \|[p_{rs}(B_1\otimes I_{{{\mathbb C}}^{d_1}},\varphi_1(B_1))]_k\|, \|[p_{rs}(\varphi_2(B_2), B_2\otimes I_{{{\mathbb C}}^{d_2}})]_k\|\right\}, \qquad$$ for any $[p_{rs}]_k\in M_k({{\mathbb C}}[z, w])$.
Applying Theorem \[ando1\] to the pairs $({T}_1, {T}_2)\in {\bf D}_{(f,g)}^{{J_{m_{T_1}}}}({{\mathcal H}})$ and $({ T}_2, { T}_1)\in {\bf D}_{(g,f)}^{{J_{m_{T_2}}}}({{\mathcal H}})$, the result follows.
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[^1]: Research supported in part by NSF grant DMS 1500922
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abstract: |
A measurement of the rate for the “wrong-sign” decay $D^{0}\rightarrow K^{+}\pi^{-}\pi^{+}\pi^{-}$ relative to that for the “right-sign” decay $D^{0}\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$ is presented. Using 791 fb$^{-1}$ of data collected with the Belle detector, we obtain a branching fraction ratio of $\rws = [0.324 \pm 0.008 (\textrm{stat.}) \pm 0.007 (\textrm{sys.})]\%$. Multiplying this ratio by the world average value for the branching fraction ${\cal B}(D^0\rightarrow K^-\pi^+\pi^+\pi^-)$ gives a branching fraction ${\cal B}(D^{0}\rightarrow K^{+}\pi^{-}\pi^{+}\pi^{-}) =
(2.61 \pm 0.06\,^{+0.09}_{-0.08}) \times 10^{-4}$.
title: |
[KEK Preprint 2013-27]{}\
---
Studies of mixing in neutral meson systems have had an important impact on the development of the Standard Model. Historically, mixing was first observed in the system [@kkbar_mixing], then later in the system [@bbbar_mixing], and most recently in the [@bsbsbar_mixing] and [@belle_kk; @babar_kp; @LHCb_kp] systems. Mixing in the system is strongly suppressed due to Cabibbo-Kobayashi-Maskawa (CKM) [@ckm_matrix] matrix elements and the GIM mechanism [@gim]. It has been measured using several methods [@HFAG_preprint], one of which compares the time-dependence of “wrong-sign” $D^{0}\rightarrow K^{+}\pi^{-}(X)$ decays to that of “right-sign” $D^{0}\rightarrow K^{-}\pi^{+}(X)$ decays [@babar_kp; @belle_kp; @babar_kpp; @cdf_kp; @LHCb_kp]. Wrong-sign decays can occur either via a doubly Cabibbo-suppressed (DCS) amplitude such as $D^{0}\rightarrow K^{+}\pi^{-}(X)$ or via $D^{0}$ mixing to $\dbar$, followed by a Cabibbo-favored (CF) decay such as $\dbar \rightarrow K^{+}\pi^{-}(X)$.
In this report we present a measurement for the rate of the wrong-sign (WS) decay $\dcs$ relative to that of the right-sign (RS) decay $\cf$ using a data sample of 791 fb$^{-1}$ [@cc]. Assuming negligible $CP$ violation, the ratio of decay rates can be expressed as [@Bergmann] $$\begin{aligned}
\label{eq:rws}
\rws & \equiv & \frac{\Gamma(\dcs)}{\Gamma(\cf)} \nonumber \\
& = & R_{D} + \alpha y^{\prime}\sqrt{R_{D}} + \frac{1}{2}(x^{\prime 2} + y^{\prime 2})\,,\end{aligned}$$ where $R_{D}$ is the squared magnitude of the ratio of the DCS to CF amplitudes, $\alpha$ is a suppression factor that accounts for strong-phase variation over the phase space ($0 \le \alpha \le 1$) [@babar_kpp], and $x^{\prime}$ and $y^{\prime}$ are the mixing parameters $x \equiv \Delta m/\overline{\Gamma}$ and $y \equiv \Delta \Gamma/2\overline{\Gamma}$ rotated by the effective strong phase difference $\delta$ between DCS and CF amplitudes: $x^{\prime} = x\cos\delta + y\sin\delta$ and $y^{\prime} = y\cos\delta - x\sin\delta$. The parameters $x$ and $y$ depend only on the mass difference ($\Delta M$) and decay width difference ($\Delta\Gamma$) between the mass eigenstates, and the mean decay width ($\overline\Gamma$). The Belle collaboration has previously measured $\rws = [0.320 \pm 0.018 (\textrm{stat.})^{+0.018}_{-0.013}(\textrm{sys.})]\%$ [@belle_tian]. The measurement presented here supersedes this previous result. We use an improved reconstruction code that has a higher reconstruction efficiency for low momentum tracks. The data used in this analysis corresponds to an integrated luminosity of 791 fb$^{-1}$ collected at or near the $\Upsilon(4S)$ resonance.
The data sample was collected by the Belle detector [@belle_detector] located at the KEKB asymmetric-energy $e^+e^-$ collider [@kekb]. The Belle detector is a large-solid-angle magnetic spectrometer consisting of a silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov counters (ACC), a barrel-like arrangement of time-of-flight scintillation counters (TOF), and an electromagnetic calorimeter (ECL) based on CsI(Tl) crystals. These detector elements are located inside a superconducting solenoid coil that provides a 1.5 T magnetic field. Muon identification is provided by an array of resistive plate chambers (KLM) interspersed with iron shielding that is used as the magnetic flux return. For charged hadron identification, a likelihood ratio $\mathcal{L}^{}_{K} \equiv \mathcal{L}(K)/(\mathcal{L}(K) + \mathcal{L}(\pi))$ is formed based on $dE/dx$ measured in the CDC and the response of the ACC and TOF. Charged kaons are identified using a likelihood requirement that is about 86% efficient for $K^\pm$ and has a $\pi^\pm$ misidentification rate of about 8%.
We reconstruct the decay $D^{*\pm}\!\rightarrow\! D^{0}\pi^{\pm}_{s}$, $D^0\!\rightarrow\! K^\pm\pi^\mp\pi^+\pi^-$, in which the charge of the low-momentum (or “slow”) pion $\pi^{\pm}_{s}$ identifies the flavor of the neutral $D$ candidate. For each event, the $D^0\rightarrow K^\pm\pi^\mp\pi^+\pi^-$ candidate is formed from combinations of four charged tracks. We require that the likelihood ratio $\mathcal{L}^{}_{K}$ be greater than 0.7 for kaons and less than 0.4 for pions. All track candidates are required to have a distance-of-closest-approach of less than 5.0 cm along the $z$ axis. In the transverse $r$-$\phi$ plane, we require a distance-of-closest-approach of less than 2.0 cm for pion candidates and less than 1.0 cm for kaon candidates. To suppress backgrounds from semileptonic decays, we reject tracks satisfying electron or muon identification criteria based on information from the ECL and KLM detectors. This veto has an efficiency of 95% for signal events and reduces the number of electron (muon) background events by 93% (95%). We require that each track used to reconstruct the $D^{0}$ have at least two SVD hits in both the $r$-$\phi$ and $z$ coordinates. We retain events having a $K\pi\pi\pi$ invariant mass ($\mk3p$) satisfying $1.81$$< \mk3p < 1.92$.
For $D^0\rightarrow K^+\pi^-\pi^+\pi^-$, when the momenta of a daughter kaon and pion are similar, their masses can be exchanged without a significant effect upon $\mk3p$. This misidentification leads to “feed-through” background from RS $D^0\rightarrow K^-\pi^+\pi^+\pi^-$ decays in the WS sample. To suppress this background, we recalculate $\mk3p$ of WS candidates after swapping the kaon and pion mass assignments and reject events in which $|\mk3p {\rm (swapped)} - m^{}_{D^0}| < 20$. From Monte Carlo (MC) simulation, we find that this veto has a signal efficiency of 92% while rejecting 94% of this background.
To suppress backgrounds from the singly Cabibbo-suppressed decay $D^{0}\rightarrow K^0_S\,K^{+}\pi^{-}$ followed by $K^0_S\rightarrow \pi^{+}\pi^{-}$, we veto events in which either of the $\pi^{+}\pi^{-}$ daughter combinations has an invariant mass within 20 ($3.3\sigma$ in resolution) of the $K^{0}_{S}$ mass. This veto has an efficiency of 97% for signal events and reduces the number of $K^{0}_{S}$ background events in Monte Carlo by 90%.
To suppress background from random combinations of tracks, the daughter tracks from the $D^{0}$ candidate are required to originate from a common vertex. To reconstruct the $D^*$ candidate, we perform a vertex fit that constrains the $D^0$ and the $\pi^{}_s$ candidate to the interaction point (IP) of the beams. The resolution on the mass difference $Q \equiv M^{}_{\pi_{s}K3\pi} - \mk3p - m^{}_\pi$ is significantly improved by this requirement. We require that the $\chi^{2}$ probability for each vertex fit be greater than 0.1% and that $Q < 10$. To eliminate $D$ mesons produced in $B\overline B$ events, we require that the momentum of the $D^*$ candidate be greater than 2.5 in the center-of-mass (CM) frame. After all selection requirements, the fraction of events containing multiple candidates is 8.6%. For these events, we select the candidate that minimizes the sum of $\chi^{2}$ values divided by the sum of degrees of freedom (d.o.f.), where each sum extends over both vertex fits.
We measure RS and WS signal yields by performing a two-dimensional binned maximum likelihood fit to the $\mk3p$ and $Q$ distributions. The signal and background probability density functions (PDFs) are determined from MC samples having sizes four times that of the data set. Background PDF shapes are determined separately for RS and WS distributions and fixed in the fit. The backgrounds are divided into four categories: (1) “random $\pi_{s}$,” in which a CF $D^0\rightarrow K^-\pi^+\pi^+\pi^-$ decay is correctly reconstructed but is subsequently combined with a random slow pion having the WS charge; (2) “broken charm,” in which a true $D^{*+}\rightarrow D^0\pi^+_s$ decay is combined with a misreconstructed $D^{0}$; (3) “combinatoric,” consisting of remaining backgrounds from $e^{+}e^{-}\rightarrow c\bar{c}$ production; and (4) “$uds$,” consisting of combinatorial backgrounds from continuum $e^{+}e^{-}\rightarrow u\bar{u}, d\bar{d}, s\bar{s}$ production. As no significant correlations are found between $\mk3p$ and $Q$ for the signal or backgrounds, we model each PDF as the product of one-dimensional functions. Background PDF shapes are parametrized in $Q$ using a threshold function of the form $Q^{1/2} + aQ^{3/2}$ for the random $\pi_{s}$, combinatoric, and $uds$ components, and a broad Gaussian for the broken charm component. For $\mk3p$ a second-order Chebyshev polynomial is used for the combinatoric and $uds$ components, and an ARGUS function [@ARGUS] is used for the broken charm component. The random $\pi_{s}$ background is parametrized in $\mk3p$ using the same shape as used for the signal (see below). We compare data and MC events in the sideband regions $|Q-5.865$ $| > 2.0$ for numerous kinematic distributions and find good agreement. These distributions include the $D^{*}$ momentum, the $\chi^2$ value of the vertex fits, particle identification likelihoods, the cosine of the angle between the $D^{0}$ and each of its daughter particles, and the momentum of each final state particle. The background normalizations are floated in the fit.
The RS signal PDF is parametrized in $\mk3p$ as the sum of one Gaussian and two bifurcated Gaussians with a common mean, and in $Q$ as the sum of a bifurcated Student’s $t$-distribution and a bifurcated Gaussian with common mean. For both distributions, the relative fraction between the single Gaussian and the remaining function(s) is fixed to the value obtained from the MC while all other parameters are floated in the fit. The RS signal PDF is used also for the WS signal PDF. Since the WS and RS samples are fitted simultaneously, the ratio of WS to RS signal yields is extracted directly from the fit. We obtain a RS yield of $990594\pm 1901$ events and a “raw” ratio of WS to RS yields of $\rawrws = (0.339 \pm 0.008)\%$. This value must be corrected for the ratio of overall efficiencies of RS and WS decays. Projections of the fit are shown in Fig. \[fig:fit\_results\]. The fitted RS yield and $\rawrws$ value correspond to a WS yield of $3358\pm 79$ events. The goodness of fit is satisfactory: for WS (RS) decays, $\chi^2/{\rm d.o.f.}= 1.17\,(1.89)$ for $\mk3p$ and 0.90(1.43) for $Q$.
-0.30in
As $\dcs$ and $\cf$ decays proceed largely through intermediate resonances, RS and WS events are expected to have different distributions across the phase space. If the detector acceptance and reconstruction efficiencies vary over phase space, the overall efficiencies for RS and WS decays will differ from each other. The ratio of these efficiencies is needed to correct $\rawrws$.
To obtain the ratio of efficiencies, we divide RS and WS events into 576 bins in a five-dimensional phase space. These dimensions consist of the invariant mass combinations for $K^{\pm}\pi^{\pm}$, $K^{\pm}\pi^{\mp}_{1}$, $K^{\pm}\pi^{\mp}_{2}$, $\pi^{\pm}\pi^{\mp}_{1}$, and $\pi^{\pm}\pi^{\mp}_{2}$, where $\pi^{}_{1}$ and $\pi^{}_{2}$ label the pions with same sign charge, and $|p_{\pi_{1}}| > |p_{\pi_{2}}|$. The binning is chosen to correspond to the structure present in these variables. The efficiency for each bin ($\epsilon^{}_{i}$) is obtained using MC. We estimate background in the data for bin $i$ by multiplying the total background yield ($N_{\rm bkg}$) by the fraction of background events in that bin ($f^{}_i$) as obtained from MC simulation. The total background yield is determined from a two-dimensional fit to the $\mk3p$-$Q$ distribution of data. The total signal yield is calculated as $$\begin{aligned}
N'(K\pi\pi\pi) & = & \sum_{i=1}^{576} \frac{N^{}_{i} -
N_{\rm bkg} \cdot f_{i}}{\epsilon_{i}}\,,
\label{eq:eff_correct_yield}\end{aligned}$$ where $N^{}_{i}$ is the number of candidate events in bin $i$. The reconstruction efficiency for either $\dcs$ or $\cf$ decays is calculated as $$\begin{aligned}
\epsilon (K\pi\pi\pi) & = & \frac{1}{N'}\sum_{i=1}^{576}
\left(N^{}_{i} - N_{\rm bkg} \cdot f_{i}\right)\,,
\label{eq:efficiency}\end{aligned}$$ and thus $$\rws = \rawrws\cdot\frac{\epsilon(K^-\pi^+\pi^+\pi^-)}{\epsilon(K^+\pi^-\pi^+\pi^-)}
= \frac{N'(K^+\pi^-\pi^+\pi^-)}{N'(K^-\pi^+\pi^+\pi^-)}\,.
\label{eq:effic_correct_rws}$$ Only events located within a signal region $|m_{K3\pi}-m_{D^{0}}| < 0.01$ GeV and $|Q - Q_{0}| < 0.002$ GeV/$c^2$ are used to detemine the efficiency correction. The efficiency-corrected yields are $N'(K^+\pi^-\pi^+\pi^-) = 37297 \pm 881$ and $N'(K^-\pi^+\pi^+\pi^-) = (1.151 \pm 0.002) \times10^{7} $; thus $\rws = (0.324 \pm 0.008)$%.
We consider various sources of systematic uncertainty as listed in Table \[tab:systematics\]. Since we measure the ratio of topologically similar RS and WS decays, many systematic uncertainties cancel.
To determine the systematic uncertainty associated with the ratio of efficiencies, we propagate the statistical errors for $\epsilon_{i}$ and $f_{i}$ via a Monte Carlo method as follows. We generate values for $\epsilon_{i}$ and $f_{i}$ in all 576 bins. These values are sampled from Gaussian distributions having mean values equal to the nominal parameter values and standard deviations equal to their uncertainties. We then recalculate $\rws$ using these sampled values. We repeat this procedure $10^5$ times and plot the resulting distribution of $\rws$. The RMS of this distribution ($\pm 0.0041$) is taken as the systematic error associated with the efficiency correction. To estimate the contribution associated with event selection criteria, we vary each selection criterion over a suitable range and remeasure $\rws$ for each variation. The identification likelihood ratio $\mathcal{L}^{}_{K}$ is varied over the range 0.5–0.9 for kaon candidates and 0.1–0.5 for pion candidates. The momentum requirement for $D^{*}$ candidates is varied over the range 2.3–2.7 . For each selection criterion, the largest positive and negative deviation of $\rws$ from the nominal value is taken as the systematic error. The error due to multiple candidates is obtained by removing all events containing multiple candidates (8.6% of events) and refitting for $\rws$; the deviation observed is taken as the error. To determine the uncertainty associated with background PDF shapes (which are taken from MC and differ for RS and WS events), we vary the parameters of each background PDF by $\pm 1\sigma$, where $\sigma$ corresponds to the statistical error from the fit to MC. For each variation, the data is refit and the deviation of $\rws$ from the nominal value is recorded. The uncertainty due to a given background PDF is taken as the sum in quadrature of all deviations observed when varying the individual parameters. The systematic error due to uncertainty in the signal PDF is negligibly small. To check for possible bias in our fit results, we repeat the fit for Monte Carlo samples (each corresponding to the size of the data set) having different values of $\rws$. Comparing the fit results for $\rws$ with the true values shows no visible fit bias. The total systematic error is taken to be the sum in quadrature of all individual contributions. Our final result is $$\begin{aligned}
\rws & = & (0.324 \pm 0.008 \pm 0.007)\%.\end{aligned}$$ Multiplying this value by the well-measured RS branching fraction ${\cal B}(D^0\rightarrow K^-\pi^+\pi^+\pi^-)= (8.07\,^{+0.21}_{-0.19})$% [@pdg_2012] gives a WS branching fraction $$\begin{aligned}
{\cal B}(D^0\!\rightarrow\! K^+\pi^-\pi^+\pi^-) & = &
(2.61 \pm 0.06\,^{+0.09}_{-0.08}) \!\times \! 10^{-4}.~~~~~\end{aligned}$$ By combining our measurement of $\rws$ with world average values [@HFAG_avg] for $x$ and $y$, and recent measurements [@Lowrey] of $\alpha$ and $\delta$, we extract $R_{D}$ from Eq. \[eq:rws\]. We use a MC method to propagate the errors for the parameters and obtain $R_{D} = (0.327^{+0.019}_{-0.016})\%$.
[0.47]{} [@ l | c c ]{} Source & $+\Delta R$ (%) & $-\Delta R$ (%)\
Kaon ID & 0.0008 & 0.0006\
Pion ID & 0.0003 & 0.0024\
$D^*$ Momentum & 0.0029 & 0.0037\
Multiple Candidates & 0.0024 & 0.0024\
$uds$ & 0.0012 & 0.0002\
Combinatoric & 0.0034 & 0.0025\
Slow $\pi$ & 0.0009 & 0.0003\
Broken & 0.0010 & 0.0008\
Efficiency Correction & 0.0041 & 0.0041\
Sum & 0.0069 & 0.0070\
\[tab:systematics\]
In summary, we have measured the wrong-sign ratio $\rws=\Gamma(\dcs)/\Gamma(\cf)$ using $e^{+}e^{-}$ data collected at or near the $\Upsilon(4S)$ resonance. After correcting for differences in reconstruction efficiencies between RS and WS events, we obtain $\rws = (0.324 \pm 0.008 \pm 0.007)$%, where the first uncertainty is statistical and the second is systematic. This is the most precise measurement of $\rws$ to date. Using a MC method to extract $R_{D}$ from Eq. \[eq:rws\], we obtain $R_{D} = (0.327^{+0.019}_{-0.016})\%$. Multiplying $\rws$ by the branching fraction for $D^0\rightarrow K^-\pi^+\pi^+\pi^-$ gives ${\cal B}(D^0\!\rightarrow\! K^+\pi^-\pi^+\pi^-) =
(2.61 \pm 0.06\,^{+0.09}_{-0.08}) \!\times \! 10^{-4}$. This result is substantially more precise than the current PDG value of $(2.61\,^{+0.21}_{-0.19}) \! \times \! 10^{-4}$ [@pdg_2012].
We thank the KEKB group for the excellent operation of the accelerator; the KEK cryogenics group for the efficient operation of the solenoid; and the KEK computer group, the National Institute of Informatics, and the PNNL/EMSL computing group for valuable computing and SINET4 network support. We acknowledge support from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan, the Japan Society for the Promotion of Science (JSPS), and the Tau-Lepton Physics Research Center of Nagoya University; the Australian Research Council and the Australian Department of Industry, Innovation, Science and Research; Austrian Science Fund under Grant No. P 22742-N16; the National Natural Science Foundation of China under contract No. 10575109, 10775142, 10875115 and 10825524; the Ministry of Education, Youth and Sports of the Czech Republic under contract No. MSM0021620859; the Carl Zeiss Foundation, the Deutsche Forschungsgemeinschaft and the VolkswagenStiftung; the Department of Science and Technology of India; the Istituto Nazionale di Fisica Nucleare of Italy; The BK21 and WCU program of the Ministry Education Science and Technology, National Research Foundation of Korea Grant No. 2010-0021174, 2011-0029457, 2012-0008143, 2012R1A1A2008330, BRL program under NRF Grant No. KRF-2011-0020333, and GSDC of the Korea Institute of Science and Technology Information; the Polish Ministry of Science and Higher Education and the National Science Center; the Ministry of Education and Science of the Russian Federation and the Russian Federal Agency for Atomic Energy; the Slovenian Research Agency; the Basque Foundation for Science (IKERBASQUE) and the UPV/EHU under program UFI 11/55; the Swiss National Science Foundation; the National Science Council and the Ministry of Education of Taiwan; and the U.S.Department of Energy and the National Science Foundation. This work is supported by a Grant-in-Aid from MEXT for Science Research in a Priority Area (“New Development of Flavor Physics”), and from JSPS for Creative Scientific Research (“Evolution of Tau-lepton Physics”).
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---
abstract: '[The dynamic dipole polarizabilities for the Li atom and the Be$^+$ ion in the $2\,^2\!S$ and $2\,^2\!P$ states are calculated using the variational method with a Hylleraas basis. The present polarizabilities represent the definitive values in the non-relativistic limit. Corrections due to relativistic effects are also estimated. Analytic representations of the polarizabilities for frequency ranges encompassing the $n=3$ excitations are presented. The recommended polarizabilities for $^7$Li and $^9$Be$^+$ were $164.11\pm 0.03$ $a_0^3$ and $24.489 \pm 0.004$ $a_0^3$. ]{}'
author:
- 'Li-Yan Tang$^{1,2}$, Zong-Chao Yan$^{3,4}$, Ting-Yun Shi$^{1}$, and J. Mitroy$^{5}$'
title: '**The dynamic dipole polarizabilities of the Li atom and the Be$^{+}$ ion**'
---
Introduction
============
The advent of cold atom physics has lead to increased importance being given to the precise determination atomic polarizabilities and related quantities. One very important source of systematic error in the new generation of atomic frequency standards is the blackbody radiation (BBR) shift [@Rosenband; @Margolis; @Chwalla]. The differential Stark shifts caused by the ambient electromagnetic field leads to a temperature dependent shift in the transition frequency of the two states involved in the clock transition. The dynamic polarizability is also useful in the determination of the magic wavelength in optical lattices [@barber; @katori; @stan; @inouye]. Another area where polarization phenomena is important is in the determination of global potential surfaces for diatomic molecules [@leroy].
When consideration is given to all the atoms and ions commonly used in cold atom physics, the Li atom and Be$^+$ ion have the advantage that they have only three electrons. This makes them accessible to calculations using correlated basis sets with the consequence that many properties of these systems can be computed to a high degree of precision. The results of these first principle calculations can serve as atomic based standards for quantities that are not amenable to precision measurement. For example, cold-atom interferometry has been used to measure the ground state polarizabilities of the Li and Na atoms [@miffre; @ekstrom]. However the polarizability ratio, $\alpha_d(X)/\alpha_d(\text{Li})$ can be measured to a higher degree of precision than the individual polarizabilities [@cronin]. So measurements of this ratio, in conjunction with a high precision [*ab-initio*]{} calculation could lead to a new level of accuracy in polarizability measurements for the atomic species most commonly used in cold-atom physics.
Calculations and measurements of Stark shifts are particularly important in atomic clock research since the BBR shift is predominantly determined by the Stark shift of the two levels involved in the clock transition. The best experimental measurements of the Stark shift have been carried out for the alkali atoms and accuracies better than 0.1$\%$ have been reported [@miller; @hunter]. Experimental work at this level of accuracy relies on a very precise determination of the electric field strength in the interaction region [@hunter; @hunter92; @wijngaarden]. High precision Hylleraas calculations of the type presented here provide an invaluable test of the experimental reliability since they provide an independent means for the calibration of electric fields [@stevens].
The dynamic Stark shift in oscillating electromagnetic fields is also of interest. The so-called magic wavelength, i.e. the precise wavelength at which the Stark shifts for upper and lower levels of the clock transition are the same, is an important parameter for optical lattices. The present calculation is used to estimate the magic wavelength for the Li $2\,^2\!S$ $\to$ $2\,^2\!P$ transition. The present calculations of the AC Stark shift potentially provides an atomic based standard of electromagnetic (EM) field intensity for finite frequency radiation.
There have been many calculations of the static polarizabilities of the ground and excited states of the Li atom and the Be$^+$ ion [@tang; @yan; @yan1; @cohen; @zhang; @johnson; @tang2]. The most precise calculations on Li and Be$^+$ are the Hylleraas calculations by Tang and collaborators [@tang; @tang2]. The Hylleraas calculations were non-relativistic and also included finite mass effects for Li. Large scale calculations using fully correlated Hylleraas basis sets can attain a degree of precision not possible for calculations based on orbital basis sets [@McKDra91; @yan1; @yan-drake97]. There have been many calculations of the dynamic polarizability for Li [@merawa94; @merawa98; @pipin; @cohen; @chernov; @kobayashi; @muszynska; @safronova], but fewer for Be$^+$ [@merawa98; @muszynska]. The present calculation is by far the most precise calculation of the dynamic polarizability that is based upon a solution of the non-relativistic Schrödinger equation. One particularly noteworthy treatment is the relativistic single-double all-order many body perturbation theory calculation (MBPT-SD) by Safronova *et al.* [@safronova]. This calculation is fully relativistic and treats correlation effects to a high level of accuracy, although it does not achieve the same level of precision as the present Hylleraas calculation.
The present work computes the dynamic dipole polarizabilities of the Li atom and the Be$^+$ ion in the $2\,^2\!S$, and $2\,^2\!P$ levels using a large variational calculation with a Hylleraas basis set. This methodology allows for the determination of the computational uncertainty related to the convergence of the basis set. Analytic representations of the dynamic polarizabilities are made so they can subsequently be computed at any frequency. Finally, the difference between the calculated and experimental binding energies is used to estimate the size of the relativistic correction to the polarizability. The final polarizabilities should be regarded as the recommended polarizabilities for comparison with experiment. All quantities given in this work are reported in atomic units except where indicated otherwise.
------------ ------------------------------ ------------------------------ -----------
$2\,^2\!S$ $-0.19814691\underline{124}$ $-0.19813041\underline{084}$ –0.198142
$2\,^2\!P$ $-0.13024311\underline{963}$ $-0.13023623\underline{876}$ –0.130236
$3\,^2\!S$ $-0.07418\underline{381350}$ $-0.07417\underline{777025}$ –0.074182
$3\,^2\!P$ $-0.05723\underline{769823}$ $-0.05723\underline{424577}$ –0.057236
$3\,^2\!D$ $-0.05561012\underline{974}$ $-0.05560578\underline{543}$ –0.055606
$4\,^2\!S$ $-0.037\underline{52870957}$ $-0.037\underline{52445073}$ –0.038615
$4\,^2\!P$ $-0.031\underline{39073613}$ $-0.031\underline{38814390}$ –0.031975
$4\,^2\!D$ $-0.03127\underline{588444}$ $-0.03127\underline{343938}$ –0.031274
$4\,^2\!F$ $-0.031253555\underline{31}$ $-0.031251112\underline{02}$ –0.031243
$2\,^2\!S$ $-0.66919693\underline{847}$ $-0.66915422\underline{599}$ –0.669247
$2\,^2\!P$ $-0.52376705\underline{352}$ $-0.52375065\underline{365}$ –0.523769
$3\,^2\!S$ $-0.2672\underline{0549176}$ $-0.2671\underline{8867334}$ –0.267233
$3\,^2\!P$ $-0.2295\underline{6788615}$ $-0.2295\underline{5822005}$ –0.229582
$3\,^2\!D$ $-0.22248781\underline{972}$ $-0.22247429\underline{085}$ –0.222478
$4\,^2\!S$ $-0.13\underline{629487843}$ $-0.13\underline{628082370}$ –0.143152
$4\,^2\!P$ $-0.12\underline{222924451}$ $-0.12\underline{221999823}$ –0.128134
$4\,^2\!D$ $-0.12512\underline{688879}$ $-0.12511\underline{926908}$ –0.125124
$4\,^2\!F$ $-0.1250154671\underline{1}$ $-0.12500785\underline{769}$ –0.125008
------------ ------------------------------ ------------------------------ -----------
: Comparisons of the binding energies (in a.u.) of Li and Be$^+$ in their low-lying states. The experimental valence binding energies are taken from the National Institute of Standards database [@ralchenko]. The $J$-weighted average is used for states with $L \ge 1$. The ground-state energies for the $^{\infty}$Li$^+$ and $^{\infty}$Be$^{2+}$ ions are $-7.2799134126693059$ and $-13.6555662384235867$ a.u. respectively [@drake]. The ground-state energies for $^7$Li$^+$ and $^9$Be$^{2+}$ are $-7.2793215198156744$ and $-13.6547092682827917$ a.u. respectively [@drake]. Underlining is used to indicate digits that have not converged with respect to basis set enlargement.[]{data-label="tab:1"}
The structure calculations
==========================
Hamiltonian and Hylleraas coordinates
-------------------------------------
The Li atom and Be$^+$ ion are four-body Coulomb systems. After separating the center of mass coordinates, the nonrelativistic Hamiltonian can be written in the form [@zhang_yan] $$\begin{aligned}
H_0 &=& -\sum_{i=1}^3 \frac{1}{2\mu}\nabla_i^2
-\frac{1}{m_0}\sum_{i> j\ge 1}^3\nabla_i\cdot\nabla_j
-\sum_{i=1}^3\frac{Z}{r_i} \nonumber \\
& + &\sum_{i> j\ge 1}^3\frac{1}{r_{ij}}\,,
\label{eq:t1}\end{aligned}$$ where $r_{ij}=|\textbf{r}_{i}- \textbf{r}_{j}|$ is the distance between electrons $i$ and $j$, $\mu=m_0m_{\rm e}/(m_0+m_{\rm e})$ is the reduced mass between the electron and the nucleus, and $Z$ is the nuclear charge. In our calculation the wave functions are expanded in terms of the explicitly correlated basis set in Hylleraas coordinates: $$\begin{aligned}
\phi({\bf{r}}_{1},{\bf{r}}_{2},{\bf{r}}_{3})&=&r_{1}^{j_{1}}r_{2}^{j_{2}}r_{3}^{j_{3}}r_{12}^{j_{12}}r_{23}^{j_{23}}r_{31}^{j_{31}}e^{-\alpha
r_{1}-\beta r_{2}-\gamma r_{3}} \nonumber \\
& \times &
\mathcal{Y}_{(\ell_{1}\ell_{2})\ell_{12},\ell_{3}}^{LM_L}(\hat{{\bf{r}}}_{1},\hat{{\bf{r}}}_{2},
\hat{{\bf{r}}}_{3})\chi(1,2,3)\,, \label{eq:t2}\end{aligned}$$ where $\mathcal{Y}_{(\ell_{1}\ell_{2})\ell_{12},\ell_{3}}^{LM_L}$ is the vector-coupled product of spherical harmonics to form an eigenstate of total angular momentum $L$ and component $M_L$ $$\begin{aligned}
&&\mathcal{Y}_{(\ell_{1}\ell_{2})\ell_{12},\ell_{3}}^{LM_L}(\hat{{\bf{r}}}_{1},
\hat{{\bf{r}}}_{2},\hat{{\bf{r}}}_{3}) =\! \sum_{{\rm all\,}
m_{i}}\langle \ell_{1}m_{1}\ell_{2}m_{2}|\ell_{12}m_{12}\rangle
\nonumber\\
&&\times \langle\ell_{12}m_{12}\ell_{3}m_{3}|LM_L\rangle
Y_{\ell_{1}m_{1}}(\hat{{\bf{r}}}_{1})
Y_{\ell_{2}m_{2}}(\hat{{\bf{r}}}_{2})
Y_{\ell_{3}m_{3}}(\hat{{\bf{r}}}_{3})\,,\nonumber\\
\label{eq:t3}\end{aligned}$$ and $\chi(1,2,3)$ is the three-electron spin $1/2$ wave function. The variational wave function is a linear combination of anti-symmetrized basis functions $\phi$. With some truncations to avoid potential numerical linear dependence, all terms in Eq. (\[eq:t2\]) are included such that $$\begin{aligned}
j_1+j_2+j_3+j_{12}+j_{23}+j_{31} \le \Omega\,, \label{eq:t4}\end{aligned}$$ where $\Omega$ is an integer. The computational details in evaluating the necessary matrix elements of the Hamiltonian may be found in [@yan-drake97]. The nonlinear parameters $\alpha$, $\beta$, and $\gamma$ in Eq. (\[eq:t2\]) are optimized using Newton’s method.
The convergence for the energies and other expectation values is studied by increasing $\Omega$ progressively. The basis sets are essentially the same as two earlier Hylleraas calculations of the static polarizabilities [@tang; @tang2]. The maximum $\Omega$ used in the present calculations is 12. The uncertainty in the final value of any quantity is usually estimated to be equal to the size of the extrapolation from the largest explicit calculation.
Fig. \[f1\] is a schematic diagram showing the nonrelativistic energy levels of the most important states of the Li atom. The energy level diagram for the low lying states of Be$^+$ is similar.
The energies of the ground states for $^\infty$Li and $^7$Li were $-7.47806032391(5)$ and $-7.47745193065(5)$ a.u. respectively. The respective energies for the $^\infty$Be$^+$ and $^9$Be$^+$ ground states were $-14.3247631769(3)$ and $-14.3238634942(3)$ a.u.. Table \[tab:1\] gives the binding energies of the Li atom and Be$^+$ ion systems with respect to the two-electron Li$^+$ and Be$^{2+}$ cores. The Hylleraas basis was optimized to compute the $2\,^2\!S$ and $2\,^2\!P$ state polarizabilities, so some of the $n = 4$ state energies have significant deviations from the experimental $n = 4$ state energies. The states with significant energy differences can be regarded as pseudo-states. The uncertainties listed in Table \[tab:1\] represent the uncertainties in the energy with respect to an infinite basis calculation. The actual computational uncertainty is very small and there is no computational error in any of the calculated digits listed in Table \[tab:1\].
With one exception, all the finite mass binding energies are less tightly bound than experiment. The differences from experiment are most likely due to relativistic effects. The exception where experiment is less tightly bound than the finite mass calculation is the $4\,^2\!F$ state of Li. This exception was not investigated since the properties of this state do not enter into any of the polarizability calculations.
![Low lying energy levels of the Li atom. The energy level diagram for Be$^+$ is similar. []{data-label="f1"}](1.eps){width="49.00000%"}
Polarizability definitions
--------------------------
The dynamic polarizability provides a measure of the reaction of an atom to an external electromagnetic field. The dynamic polarizability at real frequencies can be expressed in terms of a sum over all intermediate states, including the continuum. The dynamic dipole polarizability is expressed in terms of the dynamic scalar and tensor dipole polarizabilities, $\alpha_1(\omega)$ and $\alpha_1^{T}(\omega)$, which can be expressed in terms of the reduced matrix elements of the dipole transition operator: $$\begin{aligned}
\alpha_1(\omega) &=& \sum_{L_a} \alpha_1(L_a,\omega)\,, \label{eq:t5}\\
\alpha_1^{T}(\omega) &=& \sum_{L_a}
W(L,L_a)\alpha_1(L_a,\omega)\,, \label{eq:t6}\end{aligned}$$ where $$\begin{aligned}
\alpha_1(L_a,\omega)=\frac{8\pi}{9(2L+1)}\sum_n\frac{\Delta
E_{0n}\big|\langle n_0L\|T_1\|nL_a\rangle\big|^2}{\Delta
E_{0n}^2-\omega^2} \,,\label{eq:t7}\end{aligned}$$ with $T_1=\sum_{i=0}^3 q_iR_i Y_{10}({\bf {\hat{R}}}_i)$ being the dipole transition operator, and $$\begin{aligned}
W(L,L_a) &=& (-1)^{L+L_a}\sqrt{\frac{30(2L+1)L(2L-1)}{(2L+3)(L+1)}} \nonumber \\
& \times& \left\{ \begin{matrix}
1 & 1 & 2 \\
L & L & L_a \\
\end{matrix} \right\}\,.
\label{eq:t8}\end{aligned}$$ In the above, $|n_0L\rangle$ is the initial state with principal quantum number $n_0$, angular momentum quantum number $L$, and energy $E_0$. The $n$th intermediate eigenfunction $|nL_a\rangle$, with principal quantum number $n$ and angular momentum quantum number $L_a$, has an energy $E_n$. The transition energy is $\Delta
E_{0n}=E_n-E_0$. The $q_i$ are the charges of the respective particles and ${\bf R}_i$ are defined in Ref. [@zhang_yan]. In particular, for the case of $L=0$, $$\begin{aligned}
\alpha_1(\omega) &=& \alpha_1(P,\omega)\,, \label{eq:t9}\\
\alpha_1^{T}(\omega)&=& 0\,;
\label{eq:t10}\end{aligned}$$ for $L=1$, $$\begin{aligned}
\alpha_1(\omega) \! &=& \! \alpha_1(S,\omega)+\alpha_1(P,\omega)+\alpha_1(D,\omega)\,, \label{eq:t11}\\
\alpha_1^{T}(\omega) \! &=& \! -\alpha_1(S,\omega)
+ \frac{1}{2}\alpha_1(P,\omega)-\frac{1}{10}\alpha_1(D,\omega).
\label{eq:t12}\end{aligned}$$ In Eqs. (\[eq:t11\]) and (\[eq:t12\]), $\alpha_1(P,\omega)$ is the contribution from the even-parity configuration $(pp')P$. The scalar and tensor polarizabilities can be easily related to the polarizabilities of the magnetic sub-levels, $\alpha_{1,M}(\omega)$, $$\begin{aligned}
\alpha_{1,0}(\omega) &=& \alpha_1(\omega) - 2\alpha_1^{T}(\omega) \nonumber \\
\alpha_{1,\pm 1}(\omega) &=& \alpha_1(\omega) + \alpha_1^{T}(\omega) \ .\end{aligned}$$
The dynamic polarizability for the $^{\infty}$Li atom and the $^{\infty}$Be$^+$ ion
===================================================================================
Ground state dynamic polarizabilities
-------------------------------------
![Dynamic dipole polarizability, $\alpha_1(\omega)$, of the Li atom in the ground state. The singularities in the polarizability at the $2\,^2\!S \rightarrow n\,^2\!P$ frequencies are marked.[]{data-label="f2"}](2.eps){width="49.00000%"}
Fig. \[f2\] shows the dynamic dipole polarizability of the lithium ground state as a function of photon energy. The chief errors in the dynamic polarizability are related to the convergence of the $n\,^2\!P$ excited state energies. The largest calculation used a basis with dimensions $(N_s,N_p)=(6412,5761)$. The difference between the $\alpha_1(\omega)$ and polarizability computed with a $(N_s,N_p)=(4172,3543)$ basis would be barely discernible in Fig. \[f2\]. The convergence of $\alpha_1(\omega)$ is best at photon energies far from the discrete excitation energies of the $n\,^2\!P$ excitations. The polarizability is very susceptible to small changes in the physical energies at photon energies close to the $n\,^2\!P$ excitation energies.
--------- -------------- -------------- -------------- ------------- --------- ------- --------
0.00000 164.112(1) 164.161(1) 164.11(3) 163.6 164.1 164.14
0.00500 164.996(1) 165.045(1) 165.00(3) 164.5 165.0 165.03
0.01000 167.707(1) 167.758(1) 167.71(3) 167.2 167.7 167.74
0.02000 179.517(1) 179.574(1) 179.52(3) 178.9 179.5 179.55
0.02931 201.242(2) 201.313(1) 201.24(3) 201.0(7)
0.03000 203.438(1) 203.512(1) 203.44(3) 202.6 203.4 203.47
0.03420 219.221(1) 219.307(1) 219.22(4) 219.0(8)
0.04000 250.265(1) 250.376(1) 250.26(4) 248.8 250.3 250.29
0.04624 304.278(1) 304.441(1) 304.26(5) 304.0(8)
0.05000 356.077(1) 356.300(1) 356.05(6) 355.2 356.1 356.60
0.05699 550.259(1) 550.790(1) 550.18(9) 549.7(1.1)
0.06000 741.165(2) 742.126(1) 741.00(12) 729.2 740.73
0.06507 1984.577(1) 1991.488(1) 1983.11(31) 1983(3)
0.07000 –2581.603(2) –2569.994(2) –2584.54(40) –2895.3
0.07592 –645.478(2) –644.749(1) –645.70(10) –645.9(1.3)
0.08000 –415.067(1) –414.763(1) –415.17(7) –427.1
0.09000 –211.518(2) –211.439(1) –211.55(3) –216.5
0.09110 –199.941(1) –199.868(1) –199.97(3) –200.1(0.8)
0.10000 –135.872(2) –135.838(1) –135.89(3) –0.819
0.11388 –86.266(1) –86.249(1) –86.27(2) –86.4(0.8)
0.15183 –38.210(9) –38.204(9) –38.22(1) –38.4(1.1)
0.16000 –31.08(5) –31.06(5) –31.08(6)
--------- -------------- -------------- -------------- ------------- --------- ------- --------
The uncertainties in the dynamic dipole polarizabilities of the Li ground state as well as the polarizabilities themselves are listed in Table \[tab:2\]. All of the values listed are accurate to about $\pm 1$ in the fifth digit for $\omega \leq 0.11388$ a.u.. Some of the alternate calculations of the $\alpha_1(\omega)$ polarizabilities [@safronova; @merawa94; @merawa98; @pipin; @cohen] are listed in Table \[tab:2\]. Dynamic polarizabilities from some less accurate calculations [@chernov; @kobayashi; @muszynska] have not been tabulated.
One feature of Table \[tab:2\] is the excellent agreement with the MBPT-SD calculation of Safronova [*et al.*]{} [@safronova]. The MBPT-SD calculation and the present Hylleraas calculation are in perfect agreement when the MBPT-SD theoretical uncertainty is taken into consideration. While the MBPT-SD calculation is fully relativistic, its treatment of electron correlation is less exact than the present calculation. The MBPT-SD calculation also gives no consideration of finite mass effects. Relativistic effects would tend to decrease $\alpha_1(\omega)$ at low $\omega$, and the MBPT-SD calculation gives slightly smaller $\alpha_1(\omega)$ at low $\omega$.
![The dynamic dipole polarizability, $\alpha_1(\omega)$ for the ground state of the Be$^+$ ion. The singularities in the polarizability at the $2\,^2\!S \rightarrow n\,^2\!P$ frequencies are marked.[]{data-label="f3"}](3.eps){width="49.00000%"}
The older CI-Hylleraas calculation of values of Pipin and Bishop [@pipin] compares excellently with the present more modern calculation. All digits in $\alpha_1(\omega)$ from the CI-Hylleraas calculation are in perfect agreement with the present Hylleraas calculation. The model potential polarizabilities of Cohen and Themelis [@cohen] are also very close to the present dynamic polarizability. The Cohen-Themelis potential was constructed using a Rydberg-Klein-Rees (RKR) inversion method. The Time-Dependent-Gauge-Invariant (TDGI) polarizabilities of Mérawa [*et al*]{} [@merawa94; @merawa98] are only accurate to 0.5$\%$ or larger. The moderate accuracy of TDGI calculations has also been noted in calculations of the static polarizabilities [@tang].
The static polarizabilities for Be$^+$ in the infinite mass approximation have been presented recently [@tang2]. The present calculation represents an extension of this earlier calculation since finite mass effects are now included. The dynamic dipole polarizabilities listed in Table \[tab:3\] includes transition frequencies that extend well into the ultraviolet region. The most accurate of the few alternate calculations should be the CI-Hylleraas calculation of Muszynska [*et al*]{} [@muszynska]. However, it gives an $\alpha_1(\omega)$ that is about 1$\%$ smaller than the present polarizability. Space limitations precluded tabulation of the TDGI polarizability [@merawa98]. The TDGI polarizability was of only moderate accuracy with errors of about $2\%$ for $\omega \leq 0.6$ a.u.. The uncertainty in the present polarizability is about 10$^{-4}$ a.u. for photon energies lower than 0.40 a.u., but has increased to 10$^{-2}$ a.u. at $\omega =
0.50$ a.u..
------ ---------------- ---------------- ---------------- ------
0.00 24.4966(1) 24.5064(1) 24.489(4) 24.3
0.01 24.6088(1) 24.6187(1) 24.601(4) 24.4
0.02 24.9518(1) 24.9620(1) 24.943(4) 24.7
0.04 26.4291(1) 26.4404(1) 26.419(4) 26.2
0.06 29.3390(1) 29.3528(1) 29.325(5) 29.1
0.08 34.7358(1) 34.7550(1) 34.715(6) 34.3
0.10 45.6509(1) 45.6836(1) 45.609(7) 44.9
0.12 74.7857(1) 74.8724(1) 74.656(12)
0.15 $-$367.8708(2) $-$365.8030(3) $-$371.860(60)
0.18 $-$43.2038(1) $-$43.1745(1) $-43.273(7)$
0.20 $-$25.3195(1) $-$25.3090(1) $-25.348(4)$
0.30 $-$5.7967(1) $-$5.7951(1) $-5.801(2)$
0.40 $-$0.2912(1) $-$0.2873(1) $-0.2961(2)$
0.50 $-$2.149(7) $-$2.161(7) $-2.164(8)$
------ ---------------- ---------------- ---------------- ------
: Dynamic dipole polarizabilities, $\alpha_1(\omega)$ (in a.u.), for the ground state of the Be$^+$ ion. The results of the fourth column incorporate relativistic effects. The numbers in brackets are the uncertainties in the last digits arising from incomplete convergence of the basis set. The recommended (Rec.) polarizabilities in the fourth column reflect uncertainties other than purely computational. []{data-label="tab:3"}
The dynamic polarizabilities for the Li and Be$^+$ ground states are depicted in Figures \[f2\] and \[f3\]. There are obvious similarities in shapes of the two $\alpha_1(\omega)$ curves but with the Li polarizability being about 5-10 times larger in magnitude at comparable values of $\omega/\omega_{2s \to 2p}$. One difference between the two curves is that Be$^+$ has zeroes in $\alpha_1({\omega})$ at a discernible frequency difference before the $3\,^2\!P$ and $4\,^2\!P$ excitations while the $\alpha_1(\omega)$ negative to positive crossovers for Li occur much closer to the transition frequencies.
Excited state dynamic polarizabilities
--------------------------------------
![The dynamic polarizabilities, $\alpha_1(\omega)$ and $\alpha_1^T(\omega)$ (in a.u.) of the Li $2\,^2\!P$ state for photon frequencies below 0.10 a.u.. The scalar polarizability is given by the solid line while the tensor polarizability is given by the chain curve.[]{data-label="f4"}](4.eps){width="49.00000%"}
The scalar and tensor dipole polarizabilities for the excited $2\,^2\!P$ state of the Li atom are listed in Table \[tab:4\]. As far as we know the present calculations are the only dynamic polarizabilities presented for this state. The structure of the dynamic polarizability is complicated since both downward and upward transitions leads to singularities. This is seen most clearly in Fig. \[f4\] which plots the polarizabilities for photon energies up to 0.10 a.u.. The tensor polarizability is generally small except in the vicinity of the $2\,^2\!S$, $3\,^2\!S$ and $3\,^2\!D$ transitions. The tensor polarizability can become large when a single transition tends to dominate Eq. (\[eq:t12\]). The scalar and tensor polarizabilities tend to be opposite in sign. The main contribution to the polarizabilities comes from transitions to the $S$ and $D$ states. The coefficients in the sum-rules, Eqs. (\[eq:t11\]) and (\[eq:t12\]), for these terms are opposite in sign.
-------- --------------- --------------- --------------- --------------- ---------------- ---------------
(a.u.)
0.00 126.9458(3) 1.6214(3) 126.9472(5) 1.6351(2) 126.970(4) 1.612(4)
0.01 129.2491(5) 1.4035(2) 129.2501(5) 1.4178(2) 129.273(4) 1.393(4)
0.02 136.8371(5) 0.5302(3) 136.8372(5) 0.5463(5) 136.864(4) 0.518(4)
0.03 152.469(1) $-$2.091(1) 152.468(2) $-$2.070(1) 152.503(4) $-2.106(4)$
0.04 185.542(5) $-$11.722(5) 185.535(5) $-$11.691(5) 185.593(6) $-11.747(6)$
0.05 301.24(8) $-$82.33(9) 301.23(9) $-$82.27(8) 301.33(10) $-82.38(10)$
0.06 $-$119.1(5) 446.7(5) $-$119.3(5) 446.9(5) $-119.0(6)$ 446.6(6)
0.07 1804.5(1) $-$904.2(2) 1801.2(1) $-$900.34(5) 1806.2(2) $-905.2(2)$
0.08 $-$593.1(2) $-$43.5(4) $-$592.7(3) $-$43.6(5) $-592.6(5)$ $-43.4(5)$
0.00 2.02476(1) 5.856012(1) 2.02319(1) 5.858938(1) 2.0285(10) 5.8528(10)
0.01 1.99755(1) 5.890887(1) 1.99595(1) 5.893842(1) 2.0013(10) 5.8876(10)
0.02 1.91389(1) 5.997630(1) 1.91221(1) 6.000672(1) 1.9178(12) 5.9942(12)
0.05 1.23144(1) 6.845876(1) 1.22905(1) 6.849643(1) 1.2363(13) 6.8415(13)
0.10 $-$3.88178(1) 12.61790 $-$3.89080(1) 12.62842 $-3.8666(15)$ 12.6033(13)
0.14 $-$94.71454 104.48595 $-$95.2476 105.02073 $-93.7873(16)$ 103.5592(16)
0.20 26.1505(1) $-$12.9481(2) 26.1499(1) $-$12.9451(1) 26.161(17) $-12.958(12)$
0.25 48.59(5) $-$26.10(1) 48.61(5) $-$26.11(1) 48.62(6) $-26.17(1)$
0.28 52.09(1) $-$3.43(1) 52.10(1) $-$3.44(1) 52.07(1) $-3.45(1)$
0.32 $-$49.87(1) 4.413(1) $-$49.85(1) 4.412(1) $-49.89(1)$ 4.40(1)
-------- --------------- --------------- --------------- --------------- ---------------- ---------------
![The dynamic polarizabilities, $\alpha_1(\omega)$ and $\alpha_1^T(\omega)$ (in a.u.), of the Be$^+$ $2\,^2\!P$ state for photon frequencies below 0.40 a.u.. The scalar polarizability is given by the solid line while the tensor polarizability is given by the chain curve.[]{data-label="f5"}](5.eps){width="49.00000%"}
![The polarizability difference between the $2\,^2\!S$ and $2\,^2\!P$ states of Li. Polarizability differences are shown for $M = 0$ and $M = 1$.[]{data-label="f6"}](6.eps){width="49.00000%"}
The dynamic polarizabilities for the Be$^+$ $2\,^2\!P$ state are also tabulated in Table \[tab:4\] and depicted in Figure \[f5\] for the photon frequencies below 0.40 a.u.. There are three resonances in this frequency range. The scalar and tensor dynamic polarizabilities are similar in shape but with the opposite sign. As far as we know, there has been no previous calculation of the $2\,^2\!P$ state dynamic polarizability.
![The polarizability difference between the $2\,^2\!S$ and $2\,^2\!P$ states of Be$^+$. Polarizability differences are shown for $M = 0$ and $M = 1$. []{data-label="f7"}](7.eps){width="49.00000%"}
The static $2\,^2\!S$ $\to$ $2\,^2\!P$ Stark shift
--------------------------------------------------
The static Stark shift for the $2\,^2\!S$ $\to$ $2\,^2\!P$ energy interval in an electric field of strength $F$ is written as $$\begin{aligned}
\Delta E_{\rm 2s-2p, M} &=& -\frac{1}{2} F^2 \left(
\alpha_{\rm 2s} - \alpha_{\rm 2p, M} \right) \nonumber \\
& - & \frac{1}{24} F^4
\left( \gamma_{\rm 2s} - \gamma_{\rm 2p, M} \right)
+ \ldots\,,\label{eq:t19}\end{aligned}$$ where $\gamma$ is the hyper-polarizability. The Stark shift depends on the magnetic quantum number $M$ of the $2\,^2\!P$ state. The relative size of $\Delta \alpha$ and $\Delta \gamma$ determines the extent to which the Stark shift is influenced by the hyper-polarizability at high field strengths. The relative importance of $\Delta \alpha$ and $\Delta \gamma$ is given by the ratio $$X=\frac{F^2(\gamma_{2s}-\gamma_{2p, M})}{12(\alpha_{2s}-\alpha_{2p,
M})}=\frac{F^2\Delta \gamma}{12\Delta \alpha}\,.\label{eq:t20}$$ Using the static polarizability and static hyper-polarizability for the Li atom results in $\Delta \alpha = 37.1$ and $\Delta \gamma =
9.99 \times 10^6$ giving $X = 0.0001$ at $F = 6.67 \times 10^{-5}$ a.u. (344 kV/cm) and $X=0.001$ at $F=2.11\times 10^{-4}$ a.u. (1087 kV/cm). These estimates of the critical field strength where the quadratic Stark shift is valid depend slightly on the magnetic quantum number and exact values can be determined by using $M$-dependent polarizabilities. Stark shifts of higher order than the hyper-polarizability can be comfortably ignored at the 0.01$\%$ level provided the field strength is less than 1100 kV/cm. The static Stark shift for Be$^+$ is not interesting since it is difficult to measure as a Be$^+$ ion immersed in a finite electric field is accelerated away from the finite field region.
----------------- ---------------------------- ----------------------------
$^\infty$Li $0.046317680\underline{6}$ $0.084763957\underline{1}$
$0.081021795\underline{5}$
$0.093664330\underline{5}$
$^7$Li $0.046335687\underline{8}$ $0.084766087\underline{0}$
$0.081024478\underline{9}$
$0.093661899\underline{1}$
Rec. $^7$Li $0.046297(4)$ $0.084756(2)$
$0.081014(2)$
$0.0936613(2)$
$^\infty$Be$^+$ $0.262920267\underline{8}$ $0.378457000\underline{4}$
$0.370371502\underline{7}$
$0.390752146\underline{3}$
$^9$Be$^+$ $0.262917360\underline{3}$ $0.378451843\underline{7}$
$0.370279952\underline{2}$
$0.390455356\underline{8}$
Rec. $^9$Be$^+$ $0.2628956(7)$ $0.378443(2)$
$0.370274(1)$
$0.3904546(2)$
----------------- ---------------------------- ----------------------------
: The photon energies for which there is no Stark shift for the $2\,^2\!S$ $\to$ $2\,^2\!P$ transition. Underlined digits indicate uncertain digits arising from lack of basis set convergence. Digits in brackets indicate possible uncertainties associated with relativistic corrections in the recommended (Rec.) values.[]{data-label="tab:5"}
The dynamic $2\,^2\!S$ $\to$ $2\,^2\!P$ Stark shift
----------------------------------------------------
The Li Stark shifts, $\alpha(2s)- \alpha(2p_M)$, are plotted as a function of frequency in Figure \[f6\]. It is seen that there are magic wavelengths for $M = 0$ just below the $2\,^2\!P$ $\to$ $3\,^2\!S$ threshold and between the $2\,^2\!S$ $\to$ $2\,^2\!P$ and $2\,^2\!P$ $\to$ $3\,^2\!D$ thresholds. The actual energies for which the polarizability difference is zero are given in Table \[tab:5\]. The Stark shifts get very large for frequencies between 0.058 and 0.070 a.u..
The Be$^+$ Stark shifts, $\alpha(2s)- \alpha(2p_M)$, are plotted as a function of frequency in Figure \[f7\]. The Stark shifts are much smaller in magnitude than the Li atom shifts. One difference from Li is that the Be$^+$ shift has no zero for energies below the $2\,^2\!S$ $\to$ $2\,^2\!P$ threshold. The first zero in the Stark shift (excepting those related to a singularity) is at 0.263 a.u..
------------------------ ------------------------------ ------------------------------ ------------------------------ ------------------------------ ------------------------------- ------------------------------
$f_{2s\rightarrow 2p}$ 0.746956$\underline{855381}$ 0.746961$\underline{871867}$ 0.747011$\underline{776131}$ 0.4980674$\underline{22721}$ 0.4980833$\underline{82699}$ 0.4982270$\underline{10322}$
$\Delta E_{2s2p}$ 0.06790379$\underline{1567}$ 0.06789417$\underline{2078}$ 0.06790605 0.14542988$\underline{4364}$ 0.14540357$\underline{2344}$ 0.14547806
$f_{2s\rightarrow 3p}$ 0.00473$\underline{1019443}$ 0.00473$\underline{7600312}$ 0.00472$\underline{8028090}$ 0.0832$\underline{43986131}$ 0.0832$\underline{889414647}$ 0.0832$\underline{09271939}$
$\Delta E_{2s3p}$ 0.14090$\underline{9212964}$ 0.14089$\underline{6165068}$ 0.14090640 0.4396$\underline{29051730}$ 0.4395$\underline{96005937}$ 0.43966521
$f_{2s\rightarrow 4p}$ 0.00496$\underline{0028680}$ 0.00496$\underline{4714658}$ 0.04$\underline{0874056901}$ 0.04$\underline{0896543934}$
$\Delta E_{2s4p}$ 0.166$\underline{756175058}$ 0.166$\underline{742266938}$ 0.54$\underline{6967693380}$ 0.54$\underline{6934227760}$
$S(-2)$ 1.697$\underline{71}$ 1.698$\underline{63}$ 0.37$\underline{9627}$ 0.37$\underline{9781}$
$S(-4)$ 21.$\underline{8714}$ 21.$\underline{8888}$ 0.5$\underline{13938}$ 0.5$\underline{14207}$
$S(-6)$ 4$ \underline{28.809}$ 4$\underline{29.237}$ 0.9$\underline{79476}$ 0.9$\underline{80088}$
$S(-8)$ 9.$\underline{69364}[3]$ 9.$\underline{70502}[3]$ 2.$\underline{05322}$ 2.$\underline{05469}$
$S(-10)$ 2.$\underline{37008}[5]$ 2.$\underline{37325}[5]$ 4.$\underline{52145}$ 4.$\underline{52508}$
$S(-12)$ 6.$\underline{08098}[6]$ 6.$\underline{09006}[6]$ 1$\underline{0.2473}$ 1$\underline{0.2564}$
$S(-14)$ 1.$\underline{61032}[8]$ 1.$\underline{61296}[8]$ 2$\underline{3.6368}$ 2$\underline{3.6596}$
$S(-16)$ 4.$\underline{35760}[9]$ 4.$\underline{36537}[9]$ 5$\underline{5.1280}$ 5$\underline{5.1852}$
$\eta_1$ 2$\underline{7.0605}$ 2$\underline{7.0643}$ 2.$\underline{33229}$ 2.$\underline{33246}$
------------------------ ------------------------------ ------------------------------ ------------------------------ ------------------------------ ------------------------------- ------------------------------
Analytic representation
-----------------------
The utility of the present calculations can be increased by constructing a closed form expression for the dynamic polarizability. This is done by retaining the first $3$ terms in Eq. (\[eq:t7\]) explicitly and then expanding the energy denominator in the remainder. The expressions explicitly include oscillator strengths up to the $n = 4$ principal quantum numbers. The closed form expression is $$\begin{aligned}
\alpha_1(\omega)& = & \left( \sum_{n=2}^{4}
\frac{f_{2s \rightarrow np}}{\Delta E_{2snp}^2-\omega^2} \right)
+ S(-2) + \omega^2 S(-4) \nonumber \\
&+& \omega^4 S(-6) + \ldots +\omega^{14} S(-16) + C(\omega)
\label{eq:t13}\end{aligned}$$ where $$S(-m)=\sum_{n=5} \frac{f_{2s \rightarrow np}}{(\Delta E_{2snp})^{m}}
\,,\label{eq:t14}$$ $$C(\omega)=\frac{\eta_1 \omega^{16}S(-16)}{1-\eta_1\omega^2} \,.$$ Here $f_{2s \rightarrow np}$ are the dipole oscillator strengths for the $2\,^2\!S \rightarrow n\,^2\!P$ transitions with transition energies $\Delta E_{2snp}$. The $S(-n)$ are the Cauchy moments of the remainder of the oscillator strength distribution and are independent of $\omega$. The $C(\omega)$ is an approximate term to represent the summation from the term $S(-18)$ to $S(\infty)$. The ratio, $\eta_1=S(-n-2)/S(-n)$, is assumed to be constant and its value is set at $S(-16)/S(-14)$. Numerical values of the various constants in Eq. (\[eq:t13\]) can be found in Table \[tab:6\]. Inclusion of the remainder term has greatly increased the precision of the analytic fit to the exact dynamic polarizability.
The analytic representation for the Li $2\,^2\!S$ state is accurate to 0.01 a.u. for $\omega \leqslant 0.1612$ a.u. and to an accuracy of 0.1 a.u. for $\omega \leqslant 0.1728$ a.u.. The dynamic polarizability for the Be$^+$ $2\,^2\!S$ state maintains its accuracy over a larger $\omega$ range. It is accurate to 0.001 a.u. for $\omega \leqslant 0.543$ a.u., to 0.01 a.u. for $\omega \leqslant
0.58605$ a.u. and 0.1 a.u. for $\omega \leqslant 0.6$ a.u..
The presence of zeroes in the dynamic polarizability near the singularities means that the relative error in the analytic representation can get very large in a frequency range very close to the zeroes. Neglecting these localized regions with anomalously high relative uncertainties, the relative difference between the analytic representation and actual dynamic polarizability for the Li $2\,^2\!S$ state was less than 0.001% for $\omega \leqslant 0.1399$ a.u., 0.01% for $\omega \leqslant 0.1551$ a.u., and 0.1% for $\omega \leqslant 0.1651$ a.u.. The relative difference for the Be$^+$ $2\,^2\!S$ state obtained by the variational Hylleraas method was less than 0.001% for $\omega \leqslant 0.4737$ a.u., 0.01% for $\omega \leqslant 0.5067$ a.u. and 0.1% for $\omega \leqslant
0.52575$ a.u.. The inclusion of the remainder term, $C(\omega)$ improved the accuracy of the analytic representation by one or two order of magnitude within the frequency range listed above.
The dynamic dipole polarizabilities of the $2\,^2\!P$ states of Li and Be$^+$ have both scalar, $\alpha_1(\omega)$, and tensor, $\alpha_1^{T}(\omega)$, parts. The scalar part can be written $$\begin{aligned}
\alpha_1(\omega) &=& \sum_{n=2}^{4}
\frac{f_{2p \rightarrow ns}}{\Delta E_{2pns}^2-\omega^2}
+ \sum_{n=3}^{4} \frac{f_{2p \rightarrow nd}}{\Delta E_{2pnd}^2-\omega^2} \nonumber \\
&+& S(-2) + \omega^2 S(-4) + \omega^4 S(-6) + \ldots \nonumber \\
&+& \omega^{14}S(-16)+C(\omega)\,,
\label{eq:t15}\end{aligned}$$ where $$\begin{aligned}
S(-m)&=&\sum_{n=5} \frac{f_{2p \rightarrow ns}}{(\Delta E_{2pns})^{m}}
+ \sum_{n'} \frac{f_{2p \rightarrow n'P}}{(\Delta E_{2pn'P})^{m}} \nonumber \\
&+& \sum_{n=5} \frac{f_{2p \rightarrow nd}}{(\Delta E_{2pnd})^{m}} \,.\label{eq:t16}\end{aligned}$$
The $2p \rightarrow n'P$ excitation involves a core excitation and the intermediate state is an unnatural parity $^2\!P^e$ state. The tensor part is $$\begin{aligned}
\alpha^T_1(\omega) &= & -\sum_{n=2}^{4}
\frac{f_{2p \rightarrow ns}}{\Delta E_{2pns}^2-\omega^2}
- \frac{1}{10} \sum_{n=3}^{4}
\frac{f_{2p \rightarrow nd}}{\Delta E_{2pnd}^2-\omega^2} \nonumber \\
&+& S^T(-2) + \omega^2 S^T(-4) + \omega^4 S^T(-6) + \ldots
\nonumber\\
&+& \omega^{14}S^T(-16)+C^T(\omega)\,,
\label{eq:t17}\end{aligned}$$ where $$\begin{aligned}
S^T(-m) & = & -\sum_{n=5} \frac{f_{2p \rightarrow ns}}{(\Delta E_{2pns})^{m}}
+ \frac{1}{2} \sum_{n'} \frac{f_{2p \rightarrow n'P}}{(\Delta E_{2pn'P})^{m}} \nonumber \\
&-& \frac{1}{10} \sum_{n=5} \frac{f_{2p \rightarrow nd}}{(\Delta E_{2pnd})^{m}} \,,\label{eq:t18}\end{aligned}$$ $$\begin{aligned}
C^T(\omega)=\frac{\eta^T_1\omega^{16} S^T(-16)}{1-\eta^T_1\omega^2} \,,\end{aligned}$$ where $f_{2p\rightarrow mL_1}$ means the oscillator strength from $2p$ state to $mL_1$ state transition. $S^T(-2)$, $S^T(-4)$, $S^T(-6)$ $\cdots$ are the coefficients corresponding to $\omega^0$, $\omega^2$, $\omega^4$ $\cdots$ terms of the tensor part. The remainder term, $C^T(\omega)$ is an approximate expression to take into account the $S^T(-18)$ $\to$ $S^T(\infty)$ summations. The factor $\eta^T_1$ is set to be $\eta^T_1 = S(-16)/S(-14)$. All parameters in the analytic representation are given in Table \[tab:7\].
The first two terms of Eqs. (\[eq:t15\]) and (\[eq:t17\]) include five resonances, which make the major contribution to the polarizability with the second term involving excitations to $D$ states being the most important. This is clearly seen in the Li $\alpha_1(\omega)$ of 185.542(5) a.u. at $\omega=0.04$ a.u.. The contribution of the first summation of Eq. (\[eq:t15\]) was $-8.8717$ a.u., while the second summation contributed 175.8241 a.u.. The value given by Eq. (\[eq:t15\]) was 185.5378 a.u., which agrees with the exact value at the level of 0.0004$\%$.
The analytic representation for the scalar polarizability $\alpha_1(\omega)$ of the Li $2\,^2\!P$ state is accurate to 0.01 a.u. for $\omega \leqslant 0.0855$ a.u. and to 0.1 a.u. for $\omega
\leqslant 0.0937$ a.u.. The analytic representation for the tensor polarizability, $\alpha_1^T(\omega)$, is accurate to 0.01 a.u. for $\omega \leqslant 0.0926$ a.u. and to 0.10 a.u. for $\omega
\leqslant 0.1$ a.u..
------------------------ ------------------------------- ------------------------------- ------------------------------ ------------------------------- ------------------------------- ------------------------------- --
$f_{2p\rightarrow 2s}$ $-0.2489856\underline{18454}$ $-0.2489872\underline{90622}$ $-0.2490039\underline{2538}$ $-0.1660224\underline{74240}$ $-0.1660277\underline{94233}$ $-0.1660756\underline{70107}$
$\Delta E_{2p2s}$ $-0.06790379\underline{1567}$ $-0.06789417\underline{2078}$ $-0.067906050$ $-0.14542988\underline{4364}$ $-0.14540357\underline{2344}$ $-0.145478060000$
$f_{2p\rightarrow 3s}$ 0.1105$\underline{78835460}$ 0.1105$\underline{75403831}$ 0.1105$\underline{89306872}$ 0.0643$\underline{85095804}$ 0.0643$\underline{85347515}$ 0.0644$\underline{07168594}$
$\Delta E_{2p3s}$ 0.05605$\underline{9306121}$ 0.05605$\underline{8468507}$ 0.056054150 0.25656$\underline{1561765}$ 0.25656$\underline{1980308}$ 0.256536125000
$f_{2p\rightarrow 4s}$ 0.014$\underline{979087136}$ 0.014$\underline{981586569}$ 0.012$\underline{876689561}$ 0.012$\underline{879003186}$
$\Delta E_{2p4s}$ 0.0927$\underline{14410055}$ 0.0927$\underline{11788029}$ 0.387$\underline{472175091}$ 0.387$\underline{469829948}$
$f_{2p\rightarrow 3d}$ 0.638568$\underline{044661}$ 0.638583$\underline{007678}$ 0.638583$\underline{083728}$ 0.63198$\underline{1700709}$ 0.63205$\underline{9480294}$ 0.63204$\underline{7210702}$
$\Delta E_{2p3d}$ 0.07463298$\underline{9884}$ 0.07463045$\underline{3329}$ 0.074630150 0.3012792$\underline{33806}$ 0.3012763$\underline{62793}$ 0.301291440000
$f_{2p\rightarrow 4d}$ 0.1227$\underline{46501135}$ 0.1227$\underline{56411337}$ 0.12271$\underline{1398708}$ 0.12273$\underline{4598764}$
$\Delta E_{2p4d}$ 0.098967$\underline{235189}$ 0.098962$\underline{799378}$ 0.39864$\underline{0164736}$ 0.39863$\underline{1384564}$
$S(-2)$ 16.8$\underline{408}$ 16.8$\underline{447}$ 1.075$\underline{95}$ 1.076$\underline{28}$
$S(-4)$ 97$\underline{4.832}$ 97$\underline{5.135}$ 3.69$\underline{533}$ 3.69$\underline{658}$
$S(-6)$ 6.4$\underline{0829}[4]$ 6.4$\underline{1083}[4]$ 14.9$\underline{422}$ 14.9$\underline{479}$
$S(-8)$ 4.4$\underline{8404}[6]$ 4.4$\underline{8620}[6]$ 64.3$\underline{838}$ 64.4$\underline{113}$
$S(-10)$ 3.2$\underline{6108}[8]$ 3.2$\underline{6295}[8]$ 288.$\underline{547}$ 288.$\underline{684}$
$S(-12)$ 2.4$\underline{3429}[10]$ 2.4$\underline{3592}[10]$ 132$\underline{8.18}$ 132$\underline{8.87}$
$S(-14)$ 1.8$\underline{5141}[12]$ 1.8$\underline{5282}[12]$ 62$\underline{32.21}$ 62$\underline{35.79}$
$S(-16)$ 1.4$\underline{2796}[14]$ 1.4$\underline{2918}[14]$ 296$\underline{67.5}$ 296$\underline{85.9}$
$\eta_1$ 77.$\underline{1282}$ 77.$\underline{1354}$ 4.$\underline{76034}$ 4.$\underline{76058}$
$S^T(-2)$ $-2.7\underline{3075}$ $-2.7\underline{3118}$ $-0.15\underline{6423}$ $-0.15\underline{6454}$
$S^T(-4)$ $-15\underline{4.670}$ $-15\underline{4.702}$ $-0.54\underline{4987}$ $-0.54\underline{5110}$
$S^T(-6)$ $-9.8\underline{8199}[3]$ $-9.8\underline{8458}[3]$ $-2.10\underline{990}$ $-2.11\underline{044}$
$S^T(-8)$ $-6.\underline{71603}[5]$ $-6.\underline{71818}[5]$ $-8.\underline{71572}$ $-8.\underline{71824}$
$S^T(-10)$ $-4.\underline{73561}[7]$ $-4.\underline{73742}[7]$ $-37.\underline{5083}$ $-37.\underline{5206}$
$S^T(-12)$ $-3.\underline{42263}[9]$ $-3.\underline{42417}[9]$ $-16\underline{6.164}$ $-16\underline{6.226}$
$S^T(-14)$ $-2.\underline{51902}[11]$ $-2.\underline{52034}[11]$ $-7\underline{52.681}$ $-7\underline{53.003}$
$S^T(-16)$ $-1.\underline{88083}[13]$ $-1.\underline{88197}[13]$ $-34\underline{71.37}$ $-34\underline{73.06}$
$\eta^T_1$ 74.$\underline{6651}$ 74.$\underline{6713}$ 4.$\underline{61201}$ 4.$\underline{61228}$
------------------------ ------------------------------- ------------------------------- ------------------------------ ------------------------------- ------------------------------- ------------------------------- --
The relative error in the analytic representation of $\alpha_1(\omega)$ for the $2\,^2\!P$ state of the Li atom is less than 0.001% for $\omega \leqslant 0.082$ a.u., 0.01% for $\omega
\leqslant 0.0871$ a.u., and 0.1% for $\omega \leqslant 0.0906$ a.u.. The relative error of the analytic representation for $\alpha_1^T(\omega)$ is less than 0.001% for $\omega \leqslant
0.0815$ a.u., 0.01% for $\omega \leqslant 0.0932$ a.u., and 0.1% for $\omega \leqslant 0.0995$ a.u..
The dynamic polarizability of the Be$^+$ $2\,^2\!P$ state maintains its accuracy over a larger range of $\omega$. It is accurate to 0.001 a.u. for $\omega\leqslant 0.3513$ a.u., to 0.01 a.u. for $\omega \leqslant 0.3837$ a.u. and 0.1 a.u. for $\omega \leqslant
0.4111$ a.u.. The absolute error for $\alpha_1^T(\omega)$ is 0.001 a.u. for $\omega\leqslant 0.3788$ a.u., 0.01 a.u. for $\omega\leqslant 0.4077$ a.u., and 0.1 a.u. for $\omega\leqslant
0.4291$ a.u..
The relative error between the analytic representation and Hylleraas values of $\alpha_1(\omega)$ for the Be$^+$ $2\,^2\!P$ is less than 0.001$\%$ for $\omega \leqslant 0.332$ a.u., 0.01% for $\omega
\leqslant 0.3536$ a.u. and $0.1\%$ for $\omega \leqslant 0.3708$ a.u.. The relative error for $\alpha_1^T(\omega)$ of the Be$^+$ $2\,^2\!P$ state is less than 0.001% for $\omega \leqslant 0.3265$ a.u., 0.01% for $\omega \leqslant 0.3434$ a.u. and 0.1% for $\omega \leqslant 0.3534$ a.u..
Finite mass corrections
=======================
The effect of the finite mass was to decrease the Li atom and Be$^+$ ion binding energies listed in the Table \[tab:1\]. Therefore, it is not surprisingly that the $\omega = 0$ polarizabilities of the Li and Be$^+$ ground states are increased in Tables \[tab:2\] and \[tab:3\]. The overall changes of the $\omega = 0$ polarizabilities are $0.03\%$ and $0.04\%$ for Li and Be$^+$ respectively. The finite mass polarizabilities are larger than the infinite mass values at $\omega = 0$ a.u.. These differences can be taken as indicative of the overall change in the polarizabilities at finite frequencies below the first excitation threshold. The differences are naturally larger near thresholds.
The finite mass effect for the Li $2\,^2\!P$ state increased its polarizability by 0.001$\%$ (Table \[tab:4\]) while decreasing the polarizability for the Be$^+$ $2\,^2\!P$ state (Table \[tab:4\]) by 0.08$\%$. This behavior for Be$^+$ is due to the $2\,^2\!P$ $\to$ $2\,^2\!S$ downward transition. The increased negative contribution from this transition is enough to outweigh the increased positive contributions from transitions to more highly excited states.
As a general rule, the magnitude of the polarizabilities for both upward and downward transitions increase for the finite mass calculations. The residual Cauchy moments $S(-n)$ in Table \[tab:6\] and Table \[tab:7\] all increase for the finite mass calculations since these are computed exclusively from upward transitions.
Other effects and Uncertainties
===============================
Estimate of Relativistic effects
--------------------------------
The major omission from the present calculation is the inclusion of relativistic effects. The larger part of the energy difference between the present finite mass calculations and the experimental binding energies in Table \[tab:1\] is due to the omission of relativistic effects. Relativistic effects will alter the polarizability calculation in two ways. First, the energy differences will be changed. Generally, the binding energies of all states can be expected to be slightly larger. Secondly, there will be some changes in the reduced matrix elements. The wave functions for the $n\,^2\!S$ and $n\,^2\!P$ states can be expected to be slightly more compact since they are more tightly bound.
Correcting for the relativistic energy is simply a matter of replacing the theoretical energies in the sum rules by the experimental values. The spin-orbit weighted averages were used for states with $L\geq 1$. The corrections to the transition matrix elements are made by recourse to calculations using a semi-empirical model potential that supplements the potential field of a frozen Hartree-Fock (HF) core with a tunable polarization potential [@zhang; @tang2; @mitroy]. Polarizabilities for Li and Be$^+$ computed with this approach reproduce Hylleraas calculation at the 0.1$\%$ accuracy level [@zhang; @tang; @tang2].
The method used to estimate the relativistic effect upon matrix elements relies on comparing two very similar calculations. One calculation has its polarization potentials tuned to reproduce the finite mass energies of Table \[tab:1\]. The other calculation is tuned to give the experimental energies. The matrix elements for the low lying transitions that dominate the dynamic polarizabilities are then compared. The differences between the “finite-mass” calculation and the “experimental” calculations are then determined. These changes in the matrix elements are then applied as corrections to the set of Hylleraas matrix elements. The only matrix elements that are changed are those involving transitions inside the $2\,^2\!L$ and $3\,^2\!L$ level space. Transitions to these states dominate the $2\,^2\!S$ and $2\,^2\!P$ polarizabilities. The actual change in the Li $2\,^2\!S$ $\to$ $2\,^2\!P$ matrix element was a reduction of 0.0054$\%$. The reduction in the Be$^+$ $2\,^2\!S$ $\to$ $2\,^2\!P$ matrix element was 0.011$\%$.
Using the new set of corrected matrix elements gives a ground state polarizability of 164.114 a.u. (Table \[tab:1\]). This represents a reduction of the polarizability by 0.047 a.u.. A coupled cluster calculation of the lithium ground state estimated that relativistic effects reduced its polarizability by 0.06 a.u. [@lim]. The static polarizability of the $2\,^2\!P$ state of $^7$Li, namely 126.947 a.u., was increased to 126.970 a.u. (Table \[tab:4\]). This gives a Stark shift of $-$37.144 a.u., which is in agreement with the experiment of Hunter [*et al*]{} [@hunter] which gave $-$37.14(2) for the $^7$Li $2\,^2\!S -2\,^2\!P_{1/2}$ Stark shift. Another calculation was made to check the $2\,^2\!P_{1/2}$:$2\,^2\!P_{3/2}$ polarizability difference. The MBPT-SD calculation gave a difference of 0.015 a.u. [@johnson]. Doing two calculations tuned to give a $2\,^2\!P$ spin-orbit splitting of $1.77 \times 10^{-6}$ a.u.. (the energy splitting in the MBPT-SD calculation [@johnson]) gave a polarizability difference of 0.0145 a.u.. A further test was made by examination of the line strengths of the Si$^{3+}$ $3\,^2\!S - 3\,^2\!P$ spin-orbit doublet. A MBPT-SD calculation gave a line strength ratio of 1.000524 (once angular momentum factors were removed) [@mitroy09a]. Turning the core potential in a semi-empirical model based on the Schrodinger equation [@mitroy09a] gave a value of 1.000618. The available evidence supports the conjecture that it is possible to use the energy differences between the Hylleraas and experimental energies to get an initial estimate of relativistic corrections for other properties such as the polarizability. The uncertainty in the correction would seem to be about 20$\%$. To a certain extent the cancelations involved in adding the finite mass and relativistic corrections together leads to polarizabilities that are close to the infinite mass polarizabilities.
The static polarizability of the Be$^+$ ground state was reduced from 24.506 to 24.489 a.u.. This represents a reduction of 0.4$\%$. However, the static scalar polarizability of the $2\,^2\!P$ state increased from 2.0231 to 2.0285 a.u., an increase of 0.24$\%$. The static tensor polarizability changed from 5.8589 to 5.8528 a.u.. The heavier mass and larger nuclear charge means relativistic effects are substantially larger than finite mass corrections.
Dynamic polarizabilities and their analytic representations from the set of matrix elements with the estimate of the relativistic effect are listed in Tables \[tab:2\] - \[tab:7\] as the recommended values. The changes to analytic representation only involved changes in the oscillator strength and energy differences for a few states.
Source Value
---------------------------------------------------------- --------------
Li$_2$ spectrum [@mcalexander], $C_6$ fixed from [@yan1] 11.0022(24)
Li$_2$ spectrum [@leroy], $C_6$ fixed from [@yan1] 11.00241(23)
Li$_2$ spectrum [@leroy], $C_6$ fixed from [@zhang] 11.00240(23)
Hylleraas $^{\infty}$Li 11.000221
Hylleraas $^{7}$Li 11.001853
Hylleraas $^{7}$Li: Recommended 11.0007
: Experimental $C_3$ values from analysis of the Li$_2$ spectrum and $C_3$ values from the Hylleraas calculations. []{data-label="tab:8"}
The $2\,^2\!S$ $\to$ $2\,^2\!P$ matrix element and uncertainties
----------------------------------------------------------------
Recently Le Roy [*et al*]{} [@leroy] analysed the ro-vibrational spectrum of the lithium dimer obtaining an estimate for the $C_3$ parameter describing the long range $C_3/R^3$ potential of the A-state that dissociates to the $2\,^2\!S$ and $2\,^2\!P$ states. The $C_3$ parameter can be related to the $2\,^2\!S - 2\,^2\!P$ multiplet strength. The determination of Le Roy [*et al*]{} represented an order of magnitude improvement in precision over any previous determination of $C_3$.
The current value of $C_3 = 11.0007$ a.u. computed with relativistic corrections is about 0.0155$\%$ smaller than the experimental value of Le Roy [*et al*]{}. The finite mass calculation with the relativistic correction is closer to experiment than the infinite mass $C_3$, but there is a remaining discrepancy of 0.0017 a.u.. It is not likely that QED effects can explain the discrepancy as Pachucki [*et al*]{} found that these were 2.5 times smaller than relativistic effects in the polarizability of helium [@pachucki]. It must be recalled that the Le Roy [*et al*]{} experiment is reporting an order of magnitude improvement in experimental precision. Going to such extreme levels of precision means there might be small corrections that need to be applied to the analysis of the data that have not received consideration. For example, the value of $C_3$ will be different for states asymptotic to the $2\,^2\!P_{1/2}$ and $2\,^2\!P_{3/2}$ levels. The analysis of Le Roy [*et al*]{} uses a common $C_3$ value for both members of the spin-orbit doublet. Irrespective of this, it should be noted that experiment and theory are incompatible at precisions better than 0.01$\%$.
The difference between the present and Le Roy $C_3$ is used to assign an error to the present polarizability calculation. Changes of 0.008$\%$ were made to the $2(3)\,^2\!S - 2(3)\,^2\!P$ matrix elements, the polarizabilities were recomputed, and the differences assigned as the uncertainty in the recommended values. This difference is actually larger than the estimated relativistic change in the matrix element. Therefore, the recommended static polarizability of the Li ground state is 164.11(3) a.u.. Uncertainties in the $2(3)\,^2\!P - 3\,^2\!D$ matrix elements are smaller (relativistic effects have a smaller impact on these matrix elements) and have not been included in the uncertainty analysis. The final value for the static scalar polarizability of the $2\,^2\!P$ state was 126.970(4) a.u., while the tensor polarizability was 1.612(4) a.u..
The same uncertainty analysis was applied to the Be$^+$ ion polarizabilities. The recommended value for the ground state is 24.489(4) a.u.. The static $2\,^2\!P$ scalar polarizability was set as 2.0285(10) a.u. while the tensor polarizability was set to 5.8528(10) a.u..
Uncertainties in the recommended dynamic polarizabilities in Tables \[tab:2\], \[tab:3\], \[tab:4\] and \[tab:5\] were computed by making corrections to the matrix elements, recomputing and then observing the change. These uncertainties are best interpreted as indicative as opposed to rigorous estimates.
Summary
=======
Definitive non-relativistic values for the dynamic dipole polarizabilities of Li and Be$^+$ in their low-lying $2\,^2\!S$ and $2\,^2\!P$ states have been established using the variational method with Hylleraas basis sets. Calculation for both finite and infinite nuclear mass systems have been performed. Analytic representations for the dynamic polarizabilities of the Li atom and the Be$^+$ ion have also been developed. These results can serve as a standard against which any other calculation can be judged.
Subsidiary calculations have been used to estimate the impact of relativistic effects that are not explicitly included in the Hylleraas calculation. It is recommended that the value of 164.11(3) a.u. be adopted as the static polarizability of $^7$Li. The uncertainty of 0.03 a.u. is based on the difference between the present $C_3$ and that of Le Roy [*et al*]{} [@leroy]. This accuracy level is also supported by the $^7$Li $2\,^2\!S-2\,^2\!P_{1/2}$ Stark shift of $-$37.14 a.u. which is in perfect agreement with the value of $-$37.14(2) given by the high precision experiment of Hunter [*et al*]{} [@hunter]. The recommended static polarizability for Be$^+$ is 24.489(4) a.u..
The dynamic polarizabilities that have been obtained can be used as an atom based standard for electromagnetic field intensity. These polarizabilities can be regarded as an initial attempt to develop atom based standards for polarizability and Stark shift measurements. The primary virtue of the method with which the relativistic corrections were evaluated was simplicity of computation. For present purposes, the estimate of the relativistic corrections only has to be accurate to 10-20$\%$ for the recommended polarizabilities to be valid. The comparisons that have been done with fully relativistic calculations suggest that the estimates of the relativistic corrections are indeed accurate at this level. However, a more rigorous estimate using the Briet-Pauli Hamiltonian and perturbation theory would be desirable [@pachucki; @cencek; @lach].
This work was supported by NNSF of China under Grant No. 10974224 and by the National Basic Research Program of China under Grant No. 2010CB832803. Z.-C.Y. was supported by NSERC of Canada and by the computing facilities of ACEnet, SHARCnet, WestGrid, and in Part by the CAS/SAFEA International Partnership Program for Creative Research Teams. J.M. would like to thank the Wuhan Institute of Physics and Mathematics for its hospitality during his visit. We would like thank M.S.Safronova and J.F.Babb for useful communications during this work.
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abstract: 'We contribute towards the classification programme for Conway groupoids associated to a $2-(n,4,\lambda)$ design. Our main results improve the known bounds for a hole stabilizer to be primitive, or to contain the alternating group, ${\operatorname{Alt}}(n-1)$. We exploit these improved bounds to give a partial classification for Conway groupoids when $\lambda=3$.'
address:
- 'Department of Mathematics, University of South Wales, Treforest, CF37 1DL '
- 'Escuela de Matemáticas, Universidad Nacional, Heredia, Costa Rica. '
author:
- Nick Gill
- Jeremías Ramírez
title: 'Groups Obtained from $2-(n,4,3)$ Supersimple Designs'
---
[^1]
Introduction
============
In his famous paper [@Co1], John Conway used a “game” played on the projective plane ${\mathbb P}_3$ of order $3$ to construct the sporadic Mathieu group $M_{12}$, as well as a special subset of ${\operatorname{Sym}}(13)$ which he called $M_{13}$, and which could be endowed with the structure of a groupoid.
In recent work ([@GGNS; @GGS]), Conway’s construction has been generalized to geometries other than ${\mathbb P}_3$, namely to supersimple $2-(n,4,\lambda)$ designs. In this more general context, the analogue of the group $M_{12}$ is a subgroup of ${\operatorname{Sym}}(n-1)$ which is known as the [*hole stabilizer*]{} of the design. In this paper we prove a number of results concerning hole stabilizers. Our first main result is the following. It is a strengthening of [@GGS Theorem E].
Suppose that ${\mathcal{D}}$ is a supersimple $2-(n,4,\lambda)$ design, and that $\infty$ is a point in ${\mathcal{D}}$. Let $G:=\pi_\infty({\mathcal{D}})$ be the hole stabilizer of $\infty$, considered as a permutation group via its natural embedding in ${\operatorname{Sym}}(n-1)$.
1. If $n>\frac{24}{7}\lambda+1$, then $G$ is transitive;
2. if $n> 9\lambda-6$, then $G$ is primitive;
3. if $n{\geqslant}10\lambda-5$ then $G$ is generously transitive;
4. if $n>9\lambda^2-12\lambda+5$, then one of the following holds:
1. $G$ contains ${\operatorname{Alt}}(n-1)$;
2. $\lambda=1$, ${\mathcal{D}}=\mathbb{P}_3$ (the projective plane of order $3$), and $G = M_{12}$.
The value $\frac{24}{7}\lambda+1$ in the first bound in Theorem A is an improvement on the $4\lambda+1$ which appears in [@GGS Theorem E]. In fact Lemmas \[l: intransitive\] and \[l: 2 orbs\], which are stated and proved in §\[s: trans\], give even stronger bounds, and the first bound in Theorem A follows directly from these results. The second bound in Theorem A is proved in §\[s: bound prim\].
The third bound in Theorem A is of a different flavour to results in [@GGS]. A definition is required: a permutation group $G{\leqslant}{\operatorname{Sym}}(n)$ is called *generously transitive* if for each $i,j \in \{1,2,\ldots , n\}$, $i\neq j$ exists an element $g\in G$ that interchanges this elements.
The fourth bound in Theorem A is already known and appears in [@GGS Theorem E]. We keep it as part of our Theorem A as it will be useful later. However we can also give another result which has a similar flavour.
Let $G=\pi_{\infty}({\mathcal{D}})$. If $n>18\lambda-17$, then one of the following holds:
1. ${\operatorname{Alt}}(\Omega\setminus \{\infty\}) {\leqslant}G$;
2. ${\mathcal{D}}$ is the projective plane of order $3$ and $G\cong M_{12}$;
3. $G\cong {\operatorname{Sym}}(m)$ with $m{\leqslant}3\lambda-1$, and the action of $G$ on $\Omega\setminus\{\infty\}$ is permutation isomorphic to the action on the set of $k$-subsets of $\{1,\dots, m\}$ for some $k\in\{2,\dots, \lfloor\frac{m}{2}\rfloor\}$;
4. $G\cong {\operatorname{Alt}}(m)$ with $m{\leqslant}2\lambda$, and the action of $G$ on $\Omega\setminus\{\infty\}$ is permutation isomorphic to the action on the set of $k$-subsets of $\{1,\dots, m\}$ for some $k\in\{2,\dots, \lfloor\frac{m}{2}\rfloor\}$.
The strength of Theorem B is that it yields conclusions (1) and (2), conditional only a [**linear**]{} lower bound in $\lambda$ for $n$, as opposed to the quadratic lower bound in Theorem A (4).
On the other hand, the weakness of Theorem B is in conclusions (3) and (4): these parts have the advantage that they explicitly describe the permutation group $G$, however there is an associated loss of control on the size of $n$ in terms of $\lambda$. Nonetheless, for many values of $m$ and $k$ these actions violate the original quadratic bound that is the third bound in Theorem A and so, in principle, one could use this to obtain further restrictions on $m$ and $k$ in terms of $\lambda$. We have elected not to do this as it would introduce “clutter” to the statement, but practical applications of this theorem will probably require such an analysis. We prove Theorem B in §\[s: b\].
Our final theorem extends the classification of hole stabilizers for small values of $\lambda$. When $\lambda=1$ or $2$, [@GGNS Theorem C] gives a full classification. We now partially deal with the case of $\lambda=3$.
Let ${\mathcal{D}}=(\Omega, {\mathcal{B} })$ be a $2-(n,4,3)$ supersimple design. Let $\infty\in\Omega$ and set $G:=\pi_\infty({\mathcal{D}})$. Then either $G$ contains ${\operatorname{Alt}}(\Omega\setminus\{\infty\})$ or one of the following holds:
1. $G$ is [**intransitive**]{}: $(n,G)=\doubleunderline{(8, \{1\})}$;
2. $G$ is [**imprimitive**]{}: $(n,G)$ is in $$\{ \doubleunderline{(9, {\operatorname{Alt}}(4) \wr C_2)}, (13, ?), (16, ?), (17, ?), (21, ?)\}$$
3. $G$ is [**primitive**]{} $(n,G)$ is in $$\left\{ \begin{array}{l} (13, M_{12}), (13, M_{11}), (16, {\mathrm{SL}}_4(2)), \underline{(16, {\operatorname{Sym}}(6))}, \\ (16, {\operatorname{Alt}}(7)), (16, {\operatorname{Alt}}(6)), (17, ?)\end{array} \right\}.$$
Furthermore, for those entries that are double underlined, all such examples are known and classified; for those entries that are single underlined, an example is known; for those entries that are not underlined, no such example is known.
Note that Theorem C asserts that the classification of hole stabilizers is complete for $\lambda=3$ except when $n\in \{13,16,17, 21\}$. In §\[s: l3\] we prove Theorem C, and give a full description of all relevant examples.
One final remark: as we have said, our three main results fit into the programme of classification for Conway groupoids. In fact, though, we never directly study the groupoid of a design itself – all of our results are stated in terms of “the hole stabilizer”, and our proofs are also couched in these terms. For a definition of the Conway groupoid associated with a supersimple $2-(n,4,\lambda)$ design we refer to [@GGS] where, in addition, the connection between the hole stabilizer and the Conway groupoid is made clear.
Acknowledgments
---------------
It is a pleasure to thank B. McKay and M. Meringer who very kindly did a number of computer calculations at our request.
Background
==========
Block designs {#s: block design}
-------------
Let $t,n,k,\lambda$ positive integers. A *balanced incomplete block design* $(\Omega, \mathcal{B})$, also known as a *$t-(n,k,\lambda)$ design*, is a finite set $\Omega$ of size $n$, together with a finite multiset $\mathcal{B}$ each of size $k$ (called *lines*) such that any subset of $\Omega$ of size $t$ is contained in exactly $\lambda$ lines.
In this paper we are mostly interested in $2-(n,4,\lambda)$ designs. Such a design is called *simple* if there are no repeated lines, and *supersimple* if any two lines intersect in at most two points. In what follows we will be interested exclusively in supersimple $2-(n,4,\lambda)$ designs and so we can assume that the multiset $\mathcal{B}$ is in fact a set.
For some values of $t,n,k,\lambda$ the set of $t-(n,k,\lambda)$ designs have been completely enumerated. We will use this information for computer calculation purposes. We refer the reader to [@Col] for more information.
Let us note a particularly important example: the *Boolean quadruple system of order $2^k$* is the design $(\Omega, \mathcal{B})$, where $\Omega$ is identified with the set of vectors of $\mathbb{F}_2^{k}$, and $$\mathcal{B}:=\{ \{ v_1,v_2,v_3,v_4 \} :v_i \in \Omega \quad \mathrm{and} \quad v_1+v_2+v_3+v_4=0 \}.$$
It is easy to see that ${\mathcal{D}}$ is a $2-(2^k,4,2^{k-1}-1)$ design (in particular, when $k=2$, it is a $2-(8,4,3)$ design).
Permutation Groups {#s: permutation groups}
------------------
In this subsection, we collect some related notions about permutation groups that will be used at long of this paper. For more details we refer the reader to [@Mor].
Suppose that $G$ is a group acting on a non-empty set $\Omega$. The action is called *transitive* if for all $x,y \in \Omega$ there is an element $g\in G$ such that $x^g=y$. Suppose that the action of $G$ on $\Omega$ is transitive. A *system of imprimitivity* is a partition of $\Omega$ into $l$ subsets $\Delta_1, \Delta_2, \ldots , \Delta_l$ each of size $k$ such that $1<k,l<n$, and so that for all $i\in \{1,2,\ldots , l\}$ and all $g\in G$ there exists $j\in \{1,2,\ldots l\}$ such that
$$\Delta_i^g=\Delta_j.$$
The sets $\Delta_i$ are called *blocks*. We say that G acts *imprimitively* if there exists a system of imprimitivity. If no such set exists then G acts *primitively* on $\Omega$.
The *support* of an element $g\in G$, denoted ${\operatorname{supp}}(g)$ is the set of points in $\Omega$ not fixed by $g$.
Hole Stabilizers
----------------
Suppose that ${\mathcal{D}}=(\Omega, {\mathcal{B} })$ is a supersimple $2-(n,4,\lambda)$ design. Two points $x,y \in \Omega$ are *collinear* if there is some line in $\mathcal{B}$ that contains $x$ and $y$.
Suppose that a pair of distinct elements $x,y\in \Omega$ are collinear. We define the *elementary move* $[x,y]$ to be the permutation
$$[x,y]:=(x,y)\prod_{i=1}^{\lambda}(x_i,y_i),$$
where $\{x,y,x_i,y_i\}$ is a line in $\mathcal{B}$ for every $1{\leqslant}i{\leqslant}\lambda$. This product is well defined because ${\mathcal{D}}$ is supersimple. We also define $[x,x]=\mathrm{Id}_{\Omega}$.
Let $a$ and $b$ be distinct points in $\Omega$. We define
$$\overline{a,b}:=\{x\in\Omega\,|\,\textnormal{there exists $\ell\in{\mathcal{B} }$ such that $x,a,b,\in\ell$}\}.$$
In particular, note that $a,b\in\overline{a,b}$. Clearly the set of points in $\Omega$ moved by the permutation $[a,b]$ (also called the *support* of $[a,b]$) is precisely the set $\overline{a,b}$.
A *move sequence* is
$$[a_0,a_1,a_2, \ldots , a_n]=[a_0,a_1][a_1,a_2][a_2,a_3]\ldots[a_{n-1},a_n]$$
where $a_{i},a_{i+1}$ are collinear for all $0{\leqslant}i {\leqslant}n-1$. A move sequence is called *closed* if $a_0=a_n$. For each $x\in \Omega$ we define the *hole stabilizer*, $\pi_x({\mathcal{D}})$, to be set of all closed move sequences such that $a_0=a_n=x$, that is
$$\pi_{x}(\mathcal{D}):=\{[a_0,a_1,\ldots ,a_n]:a_0=a_n=x\}.$$
It is easy to check that $\pi_{x}(\mathcal{D})$ is a subgroup of ${\operatorname{Sym}}(\Omega\setminus\{x\})={\operatorname{Sym}}(n-1)$. In what follows we will need two easy facts [@GGNS Lemma 3.1 and Theorem A].
\[Properties of D\] Suppose that ${\mathcal{D}}=(\Omega, {\mathcal{B} })$ is a supersimple $2-(n,4,\lambda)$ design and that $x,y\in \Omega$.
1. $\pi_{x}(\mathcal{D})=\langle [x,a,b,x]:a,b\in \Omega\setminus \{x\} \rangle$.
2. $\pi_x(\mathcal{D})$ and $\pi_y(\mathcal{D})$ are conjugate subgroups of ${\operatorname{Sym}}(\Omega)$.
The second statement above implies that all hole stabilizers for a supersimple design ${\mathcal{D}}$ are permutation isomorphic groups. This allows us to talk of “the” hole stabilizer ${\mathrm{\mathbf{D}}}$ (defined up to permutation isomorphism), and in the rest of this paper we denote this group as $\pi_{\infty}(\mathcal{D})$.
A proof of Theorem A
====================
In this section we prove Theorem A. Throughout this section ${\mathcal{D}}$ is a supersimple $2-(n,4,\lambda)$ design with point set $\Omega$ and $\infty$ one such point. We write $G=\pi_\infty({\mathcal{D}})$.
A bound for transitivity {#s: trans}
------------------------
The lemmas in this section immediately yield statement (1) in Theorem A.
\[l: intransitive\] Suppose that $G=\pi_\infty({\mathcal{D}})$ has $t$ orbits on $\Omega\setminus\{\infty\}$ with $t>1$. Then $$n{\leqslant}\frac{2t\lambda}{t-1}+1.$$ In particular, if $n>4\lambda+1$, then $G=\pi_\infty({\mathcal{D}})$ is transitive.
Suppose that $x\in \Omega\setminus\{\infty\}$, and write $\Delta_x$ for the orbit of $x$ under $G$. Observe that $$\label{e: b}
\Delta_x \supseteq (\overline{\infty, x})^c.$$ Choose $x$ so that $\Delta_x$ is as small as possible. Then $|\Delta_x|{\leqslant}\frac{n-1}{t}$. Now, by the observation above, $$\Omega\setminus\{\infty\} = \left(\Delta_x \cup \overline{\infty, x}\right)\setminus\{\infty\}.$$ Noting that $x\in \Delta_x\cap \overline{\infty, x}$, we obtain that $$n-1 {\leqslant}\frac{n-1}{t} + 2\lambda + 1 - 1.$$ Rearranging this inequality gives the result. The “in particular” part of the lemma follows by taking $t=2$.
\[l: 2 orbs\] Suppose that $G=\pi_\infty({\mathcal{D}})$ has $2$ orbits on $\Omega\setminus\{\infty\}$. Then $n{\leqslant}\frac{24}{7}\lambda+1$.
Note that if $\lambda{\leqslant}2$, then [@GGNS Theorem C] implies that $G=\pi_\infty({\mathcal{D}})$ is always transitive. Thus we assume that $\lambda {\geqslant}3$.
Suppose that $x\in \Omega\setminus\{\infty\}$. Write $\Delta_x$ for the orbit of $x$ under $G$, and note that still holds. Suppose that $y\in \Omega\setminus(\Delta_x \cup\{\infty\})$. Now, taking complements of both sides of we observe that $$\Delta_y=\Delta_x^c \subseteq \overline{\infty, x}.$$ Note, too, that implies that $\Delta_x\cup \overline{\infty, x}=\Omega$. We wish to give a lower bound for $\Delta_x\cap\overline{\infty, x}$.
Observe that there are $\frac{\lambda(n-1)}{3}$ lines through $\infty$. All of these lines have either at least two elements of $\Delta_x$ or at least two of $\Delta_y$. Choose $x$ so that at least half of them (i.e. at least $\frac{\lambda(n-1)}{6}$ of them) contain at least two elements of $\Delta_x$.
Define $$\Lambda=\{(x_1,y) \mid x_1,y\in \Delta_x, \, x_1\neq y, \, y\in \overline{x_1,\infty} \}.$$ Counting this in two different ways, we obtain that $$\Delta_x \cdot (\textrm{Average number of points in $\Delta_x \cap (\overline{x_1,\infty}\setminus\{x_1\})$}) {\geqslant}\frac{\lambda(n-1)}{6} \cdot 2$$ and so we conclude that there exists an element $x$ such that $$| \Delta_x \cap \overline{x,\infty} | {\geqslant}\frac{\lambda(n-1)}{3\Delta}+1,$$ where $\Delta = |\Delta_x|$. This means that $$n = |\Delta_x \cup \overline{\infty, x}| {\leqslant}\Delta+ 2\lambda+2 - \left(\frac{\lambda(n-1)}{3\Delta}+1\right).$$ Rearranging we obtain that $$n{\leqslant}\Delta+1 + \frac{5\Delta\lambda}{3\Delta+\lambda}.$$ Now, for fixed $\Lambda$, the function $\frac{5\Delta\lambda}{3\Delta+\lambda}$ is an increasing function in the variable $\Delta$. Since $\Delta_x\subset \overline{\infty, y}$, we know that $\Delta{\leqslant}2\lambda$ and we obtain that then $$\frac{5\Delta\lambda}{3\Delta+\lambda}{\leqslant}\frac{10}{7}\lambda,$$ and we obtain that $n{\leqslant}\Delta+1+\frac{10}{7}\lambda {\leqslant}\frac{24}{7}\lambda+1$.
A bound for primitivity {#s: bound prim}
-----------------------
In this section we prove statement (2) of Theorem A. Throughout this section we suppose that $G$ is transitive and preserves a system of imprimitivity with $\ell$ blocks each of size $k$ (so that $n-1=k\ell$). Let us start with the following lemma which is [@GGNS Lemma 6.2].
\[l: prim\] If $n > 4\lambda+1$, then at least one of the following holds:
1. if $a_1,a_2\in \Omega$ lie in the same block of imprimitivity, then $\infty\in\overline{a_1,a_2}$;
2. $n{\leqslant}\frac{6\ell}{\ell-1}\lambda+1$.
Let us label blocks in the system of imprimitivity by $A,B,C,\dots$. Now we label points in $A$ by $a_1,a_2,a_3,\dots$, points in $B$ by $b_1,b_2,b_3,\dots,$ and so on.
\[l: prim 2\] Suppose that there exists a line $\{a_1,a_2,b,\infty\}$. Then $n{\leqslant}\frac{\ell}{\ell-1}(6\lambda-7-\frac{1}{\ell})$.
Choose $x$, a point in $\Omega$ such that $x\not\in \overline{\infty, a_1}\cup \overline{a_1, b} \cup \overline{\infty, b}$. Let $g_x=[\infty, a_1, x, \infty]$ and observe, first, that $a_2^{g_x}=b$. Thus $A^{g_x}=B$. Observe, second, that $a_1^{g_x}=x$ and so $x\in B$. We conclude that $$B\supseteq \Omega\setminus \left(\overline{\infty, a_1}\cup \overline{a_1, b} \cup \overline{\infty, b}\right) \cup\{b\}.$$ In particular, $|B|{\geqslant}n-(6\lambda-6)+1$. Now use the fact that $|B|=\frac{n-1}{\ell}$, and the result follows.
\[l: ell 2\] Suppose that $G$ preserves a system of imprimitivity with $\ell=2$ blocks of size $\frac{n-1}2$. Then $n{\leqslant}6\lambda+3$.
This implies that $G$ contains an element of support of size $2k=n-1$ in any generating set. Now the result follows from the fact that $G$ is generated by elements with support of size at most $6\lambda+2$ ([@GGNS Lemma 7.3] – or see item (4) of Lemma \[Properties of D\]).
The following lemma is stated for $\ell=3$; it is possible that similar statements may hold more generally.
\[l: ell 3\] Suppose that $\ell=3$ and that any line containing $\infty$ contains points from all blocks of imprimitivity ($A$, $B$ and $C$). Then $n{\leqslant}9\lambda-8$.
Let $L=\{\infty, a, b,c\}$ be a line. Then observe that $$\begin{aligned}
[\infty, a]=(b,c)(b_1, c_1)(b_2, c_2)\cdots (b_{\lambda-1}, c_{\lambda-1}); \\
[\infty, b]=(a,c)(a_1, c'_1)(a_2, c'_2)\cdots (a_{\lambda-1}, c'_{\lambda-1}); \\\end{aligned}$$ Now consider the element $g=[\infty, a, b,\infty]$. If $g$ is to fix $B$ set-wise, then $[a,b]$ must move $c, c_1,\dots, c_{\lambda-1}$ to elements in $B$. If this is the case, then $g$ must interchange $A$ and $C$. The same argument works if we consider what happen when we fix $A$ or $C$ set-wise.
We conclude that in any case $g$, which is an element of support at most $6\lambda-6$, must move at least two blocks, and so $$\frac23(n-1){\leqslant}6\lambda-6.$$
\[l: ell 4\] Suppose that $L$ is any line containing $\infty$, then $L$ intersects a block of imprimitivity in at most $1$ point. Then a block of imprimitivity has size at most $2\lambda-1$.
Let $L=\{\infty, a, b,c\}$ and suppose that $x\in B$, $x\not\in\overline{a,\infty}$ and $x\not\in\overline{a,b}$. Observe that the supposition implies that $x\not\in \overline{b,\infty}$. Now let $g_x=[\infty, a,x,\infty]$ and observe that $a^{g_x}=x\in B$ and $c^{g_x}=b\in B$. This is a contradiction.
Thus either $x\in\overline{a,\infty}$ or $x\in\overline{a,b}$. The supposition ensures that $|B\cap \overline{a,\infty}|{\leqslant}\lambda$. Suppose, then that $x\in\overline{a,b}$ and $x\not\in\overline{a,\infty}$. Then there is a line $\{a,b,x, y\}$ and, defining $g_x$ as before, observe that $a^{g_x}=x\in B$ and $c^{g_x}=y^{[x,\infty]}$. If $y\in B$, then the supposition guarantees that $c^{g_x}=y^{[x,\infty]}=y\in B$, which is a contradiction. We conclude that $y\not\in B$. Thus $\overline{a,b}$ can contain at most $\lambda-1$ points of $B$ apart from $b$. The result follows.
Let us sum up the work of this section with the next lemma which is statement (2) of Theorem A.
\[l: prim final\] If $G$ preserves a non-trivial system of imprimitivity, then $n{\leqslant}9\lambda-6$.
If $\lambda {\leqslant}2$, then the result follows immediately from [@GGNS Theorem C]. Assume from here on that $\lambda {\geqslant}3$. If $\ell=2$, then Lemma \[l: ell 2\] implies that $n{\leqslant}6\lambda+3$ and the result follows.
Suppose from here on that $\ell{\geqslant}3$. If there exists a line $\{a_1,a_2,b,\infty\}$, then Lemma \[l: prim 2\] implies that $$n{\leqslant}\frac{\ell}{\ell-1}\left(6\lambda-7-\frac{1}{\ell}\right){\leqslant}9\lambda-11,$$ and the result follows.
Suppose from here on that if $L$ is any line containing $\infty$, then $L$ intersects a block of imprimitivity in at most $1$ point. If $\ell=3$, then Lemma \[l: ell 3\] implies that $n{\leqslant}9\lambda-8$, and the result follows. If $\ell=4$, then Lemma \[l: ell 4\] implies that $n{\leqslant}8\lambda-3$, and the result follows.
Suppose from here on that $\ell{\geqslant}5$. Then Lemma \[l: prim\] implies that $$\label{ee}
n{\leqslant}\frac{6\ell}{\ell-1}\lambda+1 {\leqslant}7.5\lambda+1,$$ and the result follows for $\lambda {\geqslant}4$. For $\lambda=3$, implies that $n{\leqslant}23$. Since $2-(n,4,3)$ designs only occur for $n\equiv 0,1\pmod 4$, we conclude that $n{\leqslant}21$, and the result follows.
A bound for generous transitivity
---------------------------------
In this section we prove the third bound in Theorem A.
\[gt:bound\] Let $G=\pi_{\infty}(\mathcal{D})$. If $n{\geqslant}10\lambda-5$ then $G$ is generously transitive.
Let $a,b\in \Omega\setminus\{\infty\}$. We must find $g\in G$ such that $a^g=b$. If $\infty \not \in \overline{a,b}$ then we can use $g:=\left[\infty,a,b,\infty\right]$.
Suppose that $\infty \in \overline{a,b}$. This means that there exists a line $\{\infty, a, b, c\}$ for some $c$. Choose $x$ such that $x$ is not in $\overline{a,b}$, $\overline{\infty,a}$, $\overline{\infty,b}$, $\overline{a,c}$, $\overline{b,c}$. Then we can take $g$ to be $\left[\infty,c,x,\infty\right]$. Finally, observe that the sets listed above together contain at most $10\lambda-6$ elements, so, assuming $n{\geqslant}10\lambda-5$ we obtain the result.
A proof of Theorem B {#s: b}
====================
Our aim in this section is to prove Theorem B. We need two background results. The first is [@GGS Theorem D].
\[Trivial E\] Suppose that ${\mathcal{D}}$ is a supersimple $2-(n,4,\lambda)$ design, and that $[\infty, a, b, \infty]=(\,)$ whenever $\infty$ is collinear with $a,b$. Then one of the following is true:
1. ${\mathcal{D}}$ is a Boolean quadruple system and $\pi_\infty({\mathcal{D}})$ is trivial;
2. ${\mathcal{D}}$ is the projective plane of order $3$ and $\pi_\infty({\mathcal{D}}) \cong M_{12}$; or
3. $\pi_\infty({\mathcal{D}})\supseteq {\operatorname{Alt}}(\Omega\setminus\{\infty\})$.
We also need a result of Liebeck and Saxl [@LS Theorem 2].
\[t: ls\] Let $G$ be a primitive group of degree $d$. Then either
1. all non-trivial elements have support at least $\frac13 d$ points, or
2. $G$ is a subgroup of ${\operatorname{Sym}}(m)\wr {\operatorname{Sym}}(r)$ containing $({\operatorname{Alt}}(m))^r$, with $m{\geqslant}5$, where the action of ${\operatorname{Sym}}(m)$ is on $k$-element subsets of $\{1,\dots, \ell\}$ and the wreath product has the product action of degree $d=\binom{\ell}{k}^r$.
We note that there is an improvement on Liebeck and Saxl’s result due to Guralnick and Magaard [@GM] – we have elected not to use their result as it includes a longer list of exceptions.
Recall that the product action of ${\operatorname{Sym}}(\ell)\wr {\operatorname{Sym}}(r)$ can be thought of as an action on the set of functions $\Delta\to{\varGamma}$, where $\Delta$ is a set of size $r$ and ${\varGamma}$ is a set of size $\ell$. Let $bg=(b_1,\dots, b_r)g$ be an element of ${\operatorname{Sym}}(\ell)\wr {\operatorname{Sym}}(r)$ (so $b_1,\dots, b_r\in{\operatorname{Sym}}(\ell)$ and $g\in {\operatorname{Sym}}(r)$), then for $\alpha:\Delta\to{\varGamma}$, we have $$\alpha^{(b,g)}: \Delta \to {\varGamma}, \,\, i \mapsto ( i^{g^{-1}} \alpha )^{b_{i^{g^{-1}}}}.$$ Note that there are $d=\ell^r$ functions $\Delta\to{\varGamma}$. Using the notation just established, we have the following lemma.
\[l: product fixed\]
1. Let $G={\operatorname{Sym}}(\ell)\wr {\operatorname{Sym}}(r)$, considered as a permutation group via the product action on $d=\ell^r$ points. Suppose that $g=b h \in G$, with $b \in {\operatorname{Sym}}(\ell)^r$, $h \in {\operatorname{Sym}}(r)$ and $h\neq 1$. Then the number of fixed points of $b h$ is maximal when $h$ is a transposition, and $b=1$. In this case $b h$ fixes $\ell^{r-1}=d/\ell$ points.
2. Let $G={\operatorname{Sym}}(m)$ acting on the set $\Lambda$ of $k$-subsets of $\{1,\dots, m\}$ for some $k\in\{2,\dots, \lfloor\frac{m}{2}\rfloor\}$.
1. If $g\in {\operatorname{Sym}}(m)\setminus\{1\}$, then $g$ moves at least $2m-4$ points of $\Lambda$.
2. If $g\in {\operatorname{Alt}}(m)\setminus\{1\}$, then $g$ moves at least $3m-6$ points of $\Lambda$.
For (1), let $h$ be non-trivial, and label elements so that $1^h=2$. If $g=b h$ fixes a function $\alpha$, then we require that $$(1 \alpha) = (2 \alpha) ^{b_2}.$$ Thus the image of $1$ under $\alpha$ is prescribed by the image of $2$, and we obtain immediately that there are at most $\ell^{r-1}$ possibilities for $\alpha$.
For (2), let $g$ be non-trivial, and label elements so that $1^g=2$. The number of $k$-sets that contain $1$ but don’t contain $2$ is $\binom{m-2}{k-1}$; likewise the number of $k$-sets that contain $2$ but don’t contain $2^g$ is $\binom{m-2}{k-1}$. These two families of sets are disjoint, and all sets contained therein are moved by $g$, hence $2\binom{m-2}{k-1}$ is a lower bound on the number of points moved by $g$.
If $k>2$ and $m>5$, then this immediately yields the lower bound $3m-6$ (recall that we may assume that $k{\leqslant}m/2$). Thus we must consider the cases $m{\leqslant}5$ or $k=2$; note, though, that if $m{\leqslant}5$, then we automatically have that $k{\leqslant}2$.
Suppose, then, that $k=2$. If in the cycle decomposition of $g$, we have $(1,2,\dots, t)$, then the number of $2$-sets containing $1$ but not $2$, then $2$ but not $3$ (and so on ) is at least $t(m-2)$. Thus if $g$ contains a cycle of length $3$ or more, then the result follows; the bound for (a) also follows. Suppose, then that $g$ is in ${\operatorname{Alt}}(m)$ and $g$ is a product of $k$ distinct transpositions with $k{\geqslant}2$; write $g=(1,2)(3,4)\cdots$. Then the same argument yields a lower bound of $4 m-8$, and the result follows.
We are ready to prove Theorem B.
The result is true for $\lambda{\leqslant}2$ by classification theorems in [@GGNS]. Note that $n>18\lambda-17>9\lambda+1$ for $\lambda {\geqslant}3$ and so $G=\pi_\infty({\mathcal{D}})$ is a primitive subgroup of ${\operatorname{Sym}}(n-1)$.
Now, by Theorem \[Trivial E\], we can assume that $[\infty,a,b,\infty]\neq 1$ for some $a,b$ collinear with $\infty$. Such an element has support at most $6\lambda-6$. Now we consider the possibilities given in Theorem \[t: ls\]. If possibility (1) occurs, then we conclude that $n=d-1$ with $$\frac13d {\leqslant}6\lambda-6$$ which is a contradiction.
Thus, possibility (2) occurs: $G$ is a subgroup of ${\operatorname{Sym}}(\ell)\wr{\operatorname{Sym}}(r)$ in the product action on $\ell^r$ points. If $r\neq 1$, then any set of generators for $G$ must include an element $bg$ with $g\neq 1$ (using the notation established before Lemma \[l: product fixed\]). However we know that the set of elements of the form $[\infty, a,b,\infty]$ generate $\pi_\infty({\mathcal{D}})$ and these elements have support at most $6\lambda+2$. Referring to Lemma \[l: product fixed\], we conclude that $n=d-1$ with $$d-d/\ell{\leqslant}6\lambda+2$$ and so $d<12\lambda+4$ which is a contradiction for $\lambda {\geqslant}4$. For $\lambda=3$, we have a contradiction when $\ell\neq 2$. When $\ell=2$ we must rule out $n\in \{37, 38, 39,40\}$ but, since none of these are powers of $2$, this is immediate.
Thus we are left with the possibility that $r=1$, $d=\binom{m}{k}$, and $G=\pi_\infty({\mathcal{D}})$ is either ${\operatorname{Sym}}(m)$ or ${\operatorname{Alt}}(m)$ with the action on $\Omega\setminus\{\infty\}$ isomorphic to the action on the set $\Lambda$ of $k$-subsets of $\{1,\dots, m\}$.
If $G\cong{\operatorname{Sym}}(m)$, then Lemma \[l: product fixed\] implies that a non-trivial element of $G$ must move at least $2m-4$ points of $\Lambda$. We know that there exist non-trivial elements that move at most $6\lambda-6$ elements, and so we conclude that $m{\leqslant}3\lambda-1$.
If $G\cong {\operatorname{Alt}}(m)$ with $m>5$, then Lemma \[l: product fixed\] implies that a non-trivial element of $G$ must move at least $3m-6$ points of $\Lambda$, and the same argument implies that $m{\leqslant}2\lambda$.
If $G\cong {\operatorname{Alt}}(5)$, then Lemma \[l: product fixed\] implies that a non-trivial element of $G$ must fix at least $8$ points of $\Lambda$. We conclude that $6\lambda-6{\geqslant}8$ and so $\lambda{\geqslant}3$, and we are done.
Theorem C {#s: l3}
=========
Our aim in this section is to classify puzzle groups arising from supersimple $2-(n,4,3)$ designs. Note, first, that such designs only occur for $n\equiv 0,1\pmod 4$ and $n{\geqslant}8$.
Throughout this section we let ${\mathcal{D}}$ be a supersimple $2-(n,4,3)$ design and set $G=\pi_{\infty}({\mathcal{D}})$. Since $\lambda=3$, we observe that all elementary moves are even permutations and so, by Lemma \[Properties of D\], $G$ is a subgroup of ${\operatorname{Alt}}(n-1)$. We start by applying Theorem A to this situation in which case we obtain the following lemma.
\[l: start 3\]
1. if $n>11$, then $G$ is transitive;
2. if $n>21$, then $G$ is primitive;
3. if $n>50$, then $G={\operatorname{Alt}}(n-1)$.
Small n
-------
The $2-(8,4,3)$ and $2-(9,4,3)$ designs are listed explicitly in [@Col]. Direct calculation then yields the following result.
\[l: small n\] The following statements holds:
1. There is a unique supersimple $2-(8,4,3)$ design, and its hole stabilizer is trivial.
2. There is a unique supersimple $2-(9,4,3)$ design, and its hole stabilizer, $G$, is transitive and imprimitive, with $G\cong {\operatorname{Alt}}(4)\wr C_2$.
Using the list in [@Col], for $n=8$, we can see that there exists exactly one $2-(8,4,3)$ supersimple design. This designs is (isomorphic to) the Boolean quadruple system of order $8$, and so $\pi_{\infty}(\mathcal{D})$ is trivial.
For $n=9$ we can also check that exists exactly one $2-(9,4,3)$ supersimple design. A calculation using [@GAP] shows that $\pi_{\infty}(\mathcal{D})\cong {\operatorname{Alt}}(4)\wr C_2$.
Let us be explicit for the case $n=9$: it turns out that the only supersimple $2-(9,4,3)$ design is $$\begin{array}{rcl}
\mathcal{D} & = & \{(1,2,3,4),(1,2,5,6),(1,2,7,8),(1,3,5,9),(1,3,6,7),(1,4,5,8),(1,4,7,9),\\[1.3ex]
& & (1,6,8,9),(2,3,5,7),(2,3,8,9),(2,4,5,9),(2,4,6,8),(2,6,7,9),(3,4,6,9),\\[1.3ex]
& & (3,4,7,8),(3,5,6,8),(4,5,6,7),(5,7,8,9)\}.\\[1.3ex]
\end{array}$$
Next, a computer calculation of Professor Brendan McKay confirms that there are 28,893 supersimple $2-(12,4,3)$ designs; more computer calculations with [@GAP] confirm that all of these designs have hole stabilizer isomorphic to ${\operatorname{Alt}}(11)$, thus we assume that $13{\leqslant}n {\leqslant}29$ from here on.
From here on we assume that $n{\geqslant}13$. Lemma \[l: start 3\] implies, then, that $G$ is transitive.
The imprimitive case
--------------------
Suppose that $G$ is transitive and preserves a system of imprimitivity with $\ell$ blocks of size $k$ (so $n-1=k\ell$). Lemma \[l: start 3\] implies that $n{\leqslant}21$. We know that $n-1$ cannot be prime, so this implies that $n\in\{13,16,17,21\}$.
The primitive case
------------------
In this section we assume that $G$ is primitive and not isomorphic to ${\operatorname{Alt}}(n-1)$. We know already, thanks to Lemma \[l: start 3\], that $n{\leqslant}50$ and, thanks to Lemma \[l: small n\], that $n{\geqslant}13$. We start by improving this.
\[L=3\] Suppose that $G$ is primitive. Then either $G\cong {\operatorname{Alt}}(n-1)$ or one of the following statements holds:
- $n=13$ and $G\in \{M_{12},M_{11}\}$;
- $n=16$ and $G\in \{\mathrm{SL}_4(2), {\operatorname{Sym}}(6), {\operatorname{Alt}}(7), {\operatorname{Alt}}(6)\}$;
- $n=17$ and $G$ is isomorphic to one of $18$ primitive groups in $2^4.\mathrm{SL}_4(2)$;
- $n=28$ and $G\in \{\mathrm{PSp}_4(3)\rtimes C_2\}$;
- $n=29$ and $G\in \{\mathrm{Sp}_6(2),{\operatorname{Sym}}(8)\}$.
We know, by Theorem \[Trivial E\], that there exist points $a,b \in \Omega$ such that $g=[\infty, a,b,\infty]$ is non-trivial and $a,b$ are collinear with $\infty$. Then $g$ is an element with support of size at most $6\lambda-6=12$.
Now the list above contains all but one of the primitive groups on $n-1$ points which
1. satisfy $13{\leqslant}n {\leqslant}50$ with $n\equiv 0,1\pmod 4$;
2. contain a non-trivial element with support at most $12$;
3. are subgroups of ${\operatorname{Alt}}(n-1)$.
Let us consider the missing entry which occurs when $n=13$ and $G\cong {\operatorname{PSL}}_2(11)$. It is easy to check that there are no non-trivial elements that fix more than 2 points. But now, by Theorem \[Trivial E\], we can assume that there exists $g:=[\infty, a,b,\infty]$ which is not trivial and for which there exists $c$ such that $\{\infty, a,b,c\}\in{\mathcal{B} }$. But now observe that $g$ fixes $a$, $b$ and $c$, and so we have a contradiction.
\[lem:5.4\] $n{\leqslant}17$.
Suppose that $n>17$. Then $n\in \{28,29\}$. The three possible permutation groups given in Theorem \[L=3\] have precisely one non-trivial conjugacy class of elements of support at most $12$. In every case it is a conjugacy class of involutions with support exactly $12$.
Using [@GAP] one can verify that if $G$ is one of these three permutation groups, $g, h\in G$ are two involutions of support $12$ and $\tau$ is one of the six disjoint transpositions whose product is $g$, then $\tau$ is not one of the six disjoint transpositions whose product is $h$.
Let $a\in \Omega\setminus\{\infty\}$ and consider the three lines connecting $\infty$ to $a$: $$\{\infty, a, b_1, c_1\} \,\, \{\infty, a, b_2, c_2\} \,\, \{\infty, a, b_3, c_3\}.$$ Consider the permutation $[\infty, a, b_1, \infty]$. This is either trivial, or has support $12$. Note that in the latter case, this implies that the sets $\overline{\infty, a}$, $\overline{a,b_1}$ and $\overline{b_1,\infty}$ must overlap only in the set $\{\infty, a, b_1, c_1\}.$
On the other hand if $[\infty, a,b_1,\infty]$ is trivial, then one can check that one must have $b_2\in\overline{a, b_1}$ and one obtains that $[\infty, a,b_2,\infty]$ is trivial, likewise $[\infty, a,b_3,\infty]$.
By running the same argument starting with $b_2$ and $b_3$ in place of $b_1$, one sees that $[\infty, a,b_1,\infty]$ is trivial if and only if $[\infty, a,b_2,\infty]$ is trivial if and only if $[\infty, a, b_3,\infty]$ is trivial.
Now Theorem \[Trivial E\] implies that we can choose $a$ so that $[\infty, a,b_1,\infty]$ is not trivial. Thus the same is true of $[\infty, a,b_2,\infty]$. But now, note that both $[\infty, a, b_1,\infty]$ and $[\infty, a,b_2,\infty]$ include the transposition $(b_3,c_3)$. On the other hand $[\infty, a,b_1,\infty]$ includes the transposition $(b_2,c_2)$ which $[\infty, a, b_2, \infty]$ does not. This is a contradiction and we are done.
Lemma \[lem:5.4\] completes the proof of Theorem C. The remaining couple of results rule out some of the open possibilities from Theorem C. The first of these results generalizes the idea of Lemma \[lem:5.4\].
\[lema 6(lambda - 1)\] Let $G:=\pi_{\infty}(\mathcal{D})$ a puzzle group, where $\mathcal{D}:=(\Omega, \mathcal{B})$ is a $2-(n, 4, \lambda)$ supersimple design. Then, one of the next statements is true:
1. there exists a non trivial element of support strictly less than $6(\lambda - 1)$.
2. there exist two different elements $g,h$, both with cycle type $2^{3(\lambda - 1 )}$ and so that in their cycle decomposition they have a common transposition.
Suppose that (1) is not true, i.e. suppose that the unique element of $G$ with support less than $6 (\lambda-1)$ is the identity. Let $a \in \Omega \smallsetminus \{\infty\}$, and consider the $\lambda$ lines connecting $\infty$ to $a$:
$$\ell_1 := \{\infty, a, b_1, c_1 \}, \quad \ell_2 := \{\infty, a, b_2, c_2 \}, \quad \ell_3 := \{\infty, a, b_3, c_3 \}, \quad \ldots ,
\quad \ell_{\lambda} := \{\infty, a, b_{\lambda}, c_{\lambda} \}.$$
Consider the permutations $g_r := [\infty, a , b_1 , \infty]$ for $r=1,\dots, \lambda$. Suppose first that $g_1$ is trivial. Observe that $(b_r,c_r)$ is a transposition in $[\infty, a]$, and, according to supersimplicity $b_r \in \ell_{r}$, and only in $\ell_{r}$. So, $b_r \in \overline{a,b_1}$ or $b_r \in \overline{b_1, \infty}$. Then, $g_r := [\infty, a, b_r , \infty]$ is trivial, also. Changing $b_1$ with $b_r$ and $b_r$ with $b_1$ we obtain that $g_r$ is trivial if and only if $g_1$ is trivial.
Now, according to Theorem \[Trivial E\] we can choose $a \in \Omega \smallsetminus \{\infty\}$ so that $g_1$ is not trivial. Thus the same is true for $g_r$, for $r=2, 3, \ldots , \lambda$. Note, moreover, that the support of $g_i$ has at most $6(\lambda-1)$ elements, for $r=1,\dots, \lambda$. Then, since (1) is not true, we conclude that $|{\operatorname{supp}}(g_r)|= 6 (\lambda - 1)$ for $r = 2, \ldots , \lambda$. It follows that the elements $g_r$ have cycle type $2^{3(\lambda - 1 )}$. Now, note that $g_1$ and $g_2$ include the transposition $(b_3,c_3)$, and the elements $b_3, c_3$ can’t appear in any other transposition of $g_1$ and $g_2$, so $g_1 \neq g_2$, and this proves $(2)$.
\[primit:lambda=3\] Let $G:=\pi_{\infty}(\mathcal{D})$ a puzzle group, where $\mathcal{D}:=(\Omega, \mathcal{B})$ is a $2-(17, 4, 3)$ supersimple design. If $G$ is primitive, then $G$ is not isomorphic to any of the groups $ 2^4:D(2*5)$, $({\operatorname{Alt}}(4) \times {\operatorname{Alt}}(4)):2$, $(2^4:5).4$, ${\mathrm{AGL}}_1(16):2$, ${\mathrm{A\Upgamma L}}_1(16)$.
A calculation with [@GAP] shows that all non-trivial elements in the listed groups have support at least $6(\lambda - 1 ) = 12$. Furthermore, these elements have cycle structure $2^{6}$. Also, for every pair of different elements $g,h$ of this cycle type, we have that every transposition in $g$ is different to every transposition in $h$.
Now, applying Lemma \[lema 6(lambda - 1)\] we get a contradiction.
Combining this result with the earlier restrictions given in Lemma \[L=3\], we find that there are precisely 14 possible hole-stabilizers when $n=17$. In the [GAP]{} library, they are [PrimitiveGroup(16,i)]{} for $i\in \{7,8,10,11,12,13,14,15,16,17,18,19,20,21\}$. The case $i=21$ corresponds to ${\operatorname{Alt}}(16)$.
More generally, for $n{\leqslant}17$, the only known example when $G$ is primitive and not equal to ${\operatorname{Alt}}(n-1)$ occurs when $n=16$, and $G\cong {\operatorname{Sym}}(6)={\rm Sp}_4(2)$. This example was first noted in [@GGNS], and then generalized to an infinite family in [@GGS].
The case when Lambda=4
----------------------
It would be interesting to see if it might be possible to extend the classification to include puzzle groups arising from supersimple $2-(n,4,4)$ designs. Note, first, that such designs only occur for $n\equiv 1\pmod 3$ and $n{\geqslant}10$.
If $n=10$, an easy counting argument confirms that a supersimple $2-(10,4,4)$ design is also a $3-(10,4,1)$ design. Now a computer calculation by Professor Brendan McKay confirms that there is only one such design, and its hole stabilizer is ${\operatorname{Sym}}(9)$. For the record, this design has 30 lines as follows: $$\begin{array}{l}
{[}5, 7 ,8, 9], [4, 6 ,8 ,9],[4 ,5 ,6, 7,], [3 ,6 ,7, 9],[3, 4, 5, 8],[2, 6, 7, 8], [2, 4, 5, 9], [2 ,3 ,8, 9], \\ {[}2, 3, 5, 6], [2, 3, 4, 7], [1, 5, 6, 8], [1, 4, 7, 9], [1, 3, 7, 8], [1, 3, 5, 9], [1, 3 ,4 ,6], [1, 2, 6, 9], \\ {[}1, 2, 5, 7],[1, 2, 4, 8], [10, 5, 6, 9], [10, 4, 7, 8], [10, 3, 6, 8], [10, 3, 5, 7], [10, 3, 4, 9],\\ {[}10, 2, 7, 9],[10, 2, 5, 8],[10, 2, 4, 6],[10, 1, 8, 9],[10, 1, 6, 7],[10, 1, 4, 5], [10, 1, 2, 3].
\end{array}$$
[GGNS16]{}
Charles J. [Colbourn]{} and Jeffrey H. [Dinitz]{}, editors. 2nd ed. edition, 2007.
J. H. Conway. . In [*Surveys in combinatorics, 1997 ([L]{}ondon)*]{}, volume 241 of [*London Math. Soc. Lecture Note Ser.*]{}, pages 1–11. Cambridge Univ. Press, Cambridge, 1997.
John D. [Dixon]{} and Brian [Mortimer]{}. New York, NY: Springer-Verlag, 1996.
The GAP Group. , 2019.
Nick [Gill]{}, Neil I. [Gillespie]{}, Anthony [Nixon]{}, and Jason [Semeraro]{}. , 67(1):29–52, 2016.
Nick [Gill]{}, Neil I. [Gillespie]{}, and Jason [Semeraro]{}. , 38(2):399–442, 2018.
Robert [Guralnick]{} and Kay [Magaard]{}. , 207(1):127–145, 1998.
Martin W. [Liebeck]{} and Jan [Saxl]{}. , 63(2):266–314, 1991.
[^1]:
|
---
address: |
CNRS, DMA\
École normale supérieure\
45 rue d’Ulm\
75230 Paris Cedex 05
author:
- Olivier Benoist
date: Janvier 2019
subtitle: 'd’après Kollár, Hacon–Xu...'
title: Réduction stable en dimension supérieure
---
Introduction {#introduction .unnumbered}
============
L’espace de modules $M_g$ des courbes lisses de genre $g\geqslant 2$ construit par Mumford [@GIT] est une variété algébrique dont les points complexes sont naturellement en bijection avec les classes d’isomorphisme de courbes projectives lisses complexes de genre $g$ (nous renvoyons à [@AJP] et à [@KoMum] pour un aperçu de l’histoire de ce sujet).
Que ce soit pour étudier les dégénérescences de familles de courbes lisses ou la géométrie de la variété $M_g$ elle-même, il est utile de disposer d’une compactification projective de $M_g$ qui soit *modulaire*, c’est-à-dire qui paramètre encore des courbes algébriques, éventuellement singuli\` eres. Une telle compactification a été construite par Deligne et Mumford [@DM] : c’est l’espace de modules des courbes stables $\overline{M}_g$.
La recherche d’espaces de modules analogues paramétrant des variétés de dimension supérieure a suscité de nombreux travaux. Pour obtenir une théorie similaire, on se restreint aux variétés dont le fibré canonique est ample [^1]. Le cas des surfaces a alors été résolu par Kollár, Shepherd-Barron et Alexeev [@KSB; @Kocomplete; @Ale], et Viehweg [@Viehweg] a traité le cas des variétés lisses en dimension arbitraire.
Le cas général a fait l’objet d’avancées récentes, décrites dans ce rapport. Ces progrès sont dus au développement du programme des modèles minimaux par Birkar, Cascini, Hacon, McKernan et Xu [@BCHM; @HX; @HMX], à de nombreux travaux de Kollár [@Kocomplete; @KoHH; @Kosing; @Kobook], ainsi qu’à Fujino, Kovács et Patakfalvi [@Fujino; @KoPa].
Nous expliquons tout d’abord une motivation pour ces travaux : obtenir des théorèmes de réduction stable en dimension supérieure (théorèmes \[redstable\] et \[redstabledimsup\]). Nous définissons ensuite les variétés stables qui jouent dans ce cadre le rôle des courbes stables de Deligne et Mumford, et énonçons le théorème d’existence des espaces de modules de variétés stables (théorème \[thedm\]). Dans les troisième et quatrième sections, nous esquissons enfin la preuve du théorème de réduction stable et la construction de ces espaces de modules.
[*Conventions.*]{} Tous les schémas sont des $\operatorname{\mathbb{Q}}$-schémas noethériens. Une variété est un schéma séparé de type fini sur un corps $k$ de caractéristique nulle, par exemple le corps $\operatorname{\mathbb{C}}$ des nombres complexes.
Réduction semi-stable et réduction stable
=========================================
Fixons dans cette section un morphisme propre et surjectif $f:\mathcal{X}\to B$ entre variétés réduites. Supposons $B$ intègre, et notons $\eta$ le point générique de $B$ et $\mathcal{X}_{\eta}$ la fibre générique de $f$. On voit $f$ comme une famille de variétés algébriques paramétrée par les points de $B$. Cette famille peut avoir de mauvaises propriétés : les fibres de $f$ peuvent ne pas toutes avoir la même dimension, être très singulières... On est ainsi amené à rechercher des modèles birationnels $f':\mathcal{X}'\to B'$ de $f$ dont la géométrie et les singularités sont contrôlées. Plus précisément, on recherche un diagramme commutatif: $$\label{diagred}
\begin{aligned}
\xymatrix{
\mathcal{X}'\ar_{f'}[dr]\ar@{-->}^{\phi}[r]
& \mathcal{X}_{B'}\ar[r]\ar[d] & \mathcal{X} \ar^f[d] \\
&B'\ar^{\pi}[r]&B .
}
\end{aligned}$$ dans lequel $B'$ est une variété intègre de point générique $\eta'$, le morphisme $\pi:B'\to B$ est propre, génériquement fini et surjectif, le carré est cartésien, $\phi_{\eta'}$ est birationnelle et $f'$ est propre. Quelles propriétés peut-on alors imposer au morphisme $f'$ ?
Réduction semi-stable
---------------------
Une première réponse est apportée par le théorème de réduction semi-stable de Kempf, Knudsen, Mumford et Saint-Donat [@KKMS p. 53].
\[KKMS\] Si $\dim(B)=1$, on peut choisir $f':\mathcal{X}'\to B'$ comme dans de sorte que $B'$ et $\mathcal{X}'$ soient lisses et les fibres de $f'$ soient des diviseurs réduits à croisements normaux stricts dans $\mathcal{X}'$.
L’assertion que les fibres sont réduites (c’est-à-dire sans multiplicités) est ici essentielle. Quand la base a dimension arbitraire, on dispose encore d’un théorème de réduction semi-stable, démontré dans une variante faible par Abramovich et Karu [@AbraKaru] et en toute généralité par Adiprasito, Liu et Temkin [@ALT].
On peut choisir $f':\mathcal{X}'\to B'$ comme dans de sorte que $B'$ et $\operatorname{\mathcal{X}}'$ soient lisses, et $f'$ soit plat à fibres réduites.
Les énoncés de [@AbraKaru; @ALT] sont plus précis: on peut garantir que $f'$ soit munie d’une structure toroïdale. On en déduit par exemple que les fibres de $f'$ sont Gorenstein [@AbraKaru Proposition 6.5]. Les théorèmes de réduction semi-stable ci-dessus ont l’avantage de donner lieu à des familles $f':\mathcal{X}'\to B'$ dont l’espace total $\mathcal{X}'$ est lisse. Ils ont cependant plusieurs inconvénients. Ils sont fortement non uniques. Par exemple, dans le cadre du théorème \[KKMS\], on peut sans dommage éclater un point de $\mathcal{X}'$ en lequel $f'$ est lisse. De cette manière, même si le morphisme $f$ est lisse (si $\mathcal{X}_{\eta}$ a *bonne réduction*), il se peut que $f'$ ne le soit pas. Ainsi, si les singularités des fibres sont très contrôlées, leur géométrie ne l’est pas du tout. Les théorèmes de réduction stable apportent une solution à ce problème.
Réduction stable pour les familles de courbes
---------------------------------------------
Le premier tel énoncé, pour les familles à un paramètre de courbes, est dû à Deligne et Mumford [@DM] (d’autres preuves ont été données, par exemple dans [@AW; @Temkin]).
\[defcourbestable\] Une **courbe stable** est une variété projective connexe $C$ de dimension $1$ dont les singularités sont au plus nodales et dont le faisceau dualisant $\omega_C$ est ample. Le genre de $C$ est l’entier $g(C)=h^0(C,\omega_C)$.
\[redstcourbes\] Si $\dim(B)=1$ et si $\mathcal{X}_{\eta}$ est une courbe stable, on peut choisir $f':\mathcal{X}'\to B'$ comme dans de sorte que $f'$ soit plat à fibres des courbes stables, et $\phi_{\eta'}$ soit un isomorphisme.
De plus, si $B'$ est fixée, un tel $f':\operatorname{\mathcal{X}}'\to B'$ est unique.
Le théorème \[redstcourbes\] s’applique en particulier quand $\operatorname{\mathcal{X}}_{\eta}$ est une courbe lisse de genre $\geqslant 2$. À la différence du théorème \[KKMS\], il ne restreint pas les singularités de $\operatorname{\mathcal{X}}'$. La géométrie des fibres de $f'$ est en revanche très contrainte.
Les énoncés d’unicité et d’existence dans le théorème \[redstcourbes\] reflètent la séparation et la propreté de l’espace de modules des courbes stables $\overline{M}_g$ (et même, plus précisément, du champ de modules $\overline{\operatorname{\mathcal{M}}}_g$ des courbes stables). La propreté de $\overline{\operatorname{\mathcal{M}}}_g$ implique à son tour un théorème de réduction stable sur des bases de dimension arbitraire.
\[redstdJ\] Si $\mathcal{X}_{\eta}$ est une courbe stable, on peut choisir $f':\mathcal{X}'\to B'$ comme dans de sorte que $f'$ soit plat à fibres des courbes stables, et $\phi_{\eta'}$ soit un isomorphisme.
Soit $g$ le genre de $\operatorname{\mathcal{X}}_{\eta}$. Il n’existe pas de famille universelle de courbes stables sur l’espace de modules $\overline{M}_g$. Il résulte en revanche du lemme de Chow pour les champs de Deligne-Mumford [@LMB Théorème 16.6], appliqué au champ de modules $\overline{\mathcal{M}}_g$ des courbes stables, qu’il existe une famille plate $p:\operatorname{\mathcal{C}}\to Z$ de courbes stables de genre $g$ telle que le morphisme induit $Z\to\overline{\operatorname{\mathcal{M}}}_g$ soit fini et surjectif. Remarquons que $Z$ est propre par propreté de $\overline{\operatorname{\mathcal{M}}}_g$. La courbe stable $\operatorname{\mathcal{X}}_{\eta}$ induit un morphisme $\eta\to \overline{\operatorname{\mathcal{M}}}_g$. Notons $\eta'$ une composante irréductible du produit fibré $\eta\times_{\overline{\operatorname{\mathcal{M}}}_g} Z$. Soient $\widetilde{B}$ la normalisation de $B$ dans $\eta'$ et $B'\to \widetilde{B}$ une modification résolvant les indéterminées de l’application rationnelle naturelle $\widetilde{B}\dashrightarrow Z$. Le morphisme $f':\operatorname{\mathcal{X}}' \to B'$ construit en changeant de base $p:\operatorname{\mathcal{C}}\to Z$ par le morphisme $B'\to Z$ a les propriétés requises.
Le théorème \[redstdJ\], appliqué à une famille de courbes balayant une variété arbitraire, est un outil crucial dans la preuve du théorème d’altération des singularités de de Jong [@dJ] (voir plus précisément [@dJ §2.24, §5.13] ou [@Berthelot §3.2.3]).
Réduction stable en dimension supérieure
----------------------------------------
Nous définirons plus loin une notion de variété stable (définition \[defstable\]) et de famille de variétés stables ou famille stable (définition \[deffam\]) en dimension supérieure, permettant de généraliser les théorèmes \[redstcourbes\] et \[redstdJ\].
Pour l’instant, disons seulement qu’une variété propre et lisse est stable si et seulement si son fibré canonique est ample. C’est une condition bien plus restrictive pour les variétés de dimension $\geqslant 2$ que pour les courbes. Par exemple, les théorèmes ci-dessous ne s’appliquent pas aux familles de variétés de Fano, de variétés abéliennes ou de surfaces $K3$.
\[redstable\] Si $\dim(B)=1$ et si $\mathcal{X}_{\eta}$ est une variété stable, il existe $f':\mathcal{X}'\to B'$ comme dans tel que $f'$ soit une famille stable, et $\phi_{\eta'}$ soit un isomorphisme.
De plus, si $B'$ est fixée, un tel $f':\operatorname{\mathcal{X}}'\to B'$ est unique.
Ce théorème est dû à Hacon et Xu [@HX] quand $\operatorname{\mathcal{X}}_{\eta}$ est normale et à Kollár en général [@Kosing; @Kobook] (voir §\[secredst\] pour plus de détails).
Comme dans le cas des courbes, une conséquence géométrique du théorème \[redstable\] est la propreté des espaces de modules de variétés stables (voir le théorème \[thedm\]). Une fois de tels espaces de modules construits (ce qui est significativement plus dur que pour les espaces de modules de courbes, comme on le verra au § \[consedm\]), l’argument expliqué dans la preuve du théorème \[redstdJ\] permet d’obtenir un théorème de réduction stable sur une base de dimension supérieure.
\[redstabledimsup\] Si $\mathcal{X}_{\eta}$ est une variété stable, il existe $f':\mathcal{X}'\to B'$ comme dans tel que $f'$ soit une famille stable, et $\phi_{\eta'}$ soit un isomorphisme.
Stabilité
=========
Dans cette section, nous définissons et étudions les analogues en dimension supérieure des courbes stables de Deligne et Mumford.
Variétés stables
----------------
On peut penser aux courbes lisses de genre $g\geqslant 2$ qui ne sont pas hyperelliptiques comme plongées, à l’aide de leur fibré canonique, dans l’espace projectif $\operatorname{\mathbb{P}}_k^{g-1}$. Si l’on veut aussi prendre en compte les courbes hyperelliptiques, il faut plutôt considérer leur plongement tricanonique dans $\operatorname{\mathbb{P}}_k^{5g-6}$. On voudra aussi penser aux variétés stables de dimension supérieure comme étant pluricanoniquement plongées. Ce point de vue va imprégner toute la suite de ce texte. Il explique le rôle prépondérant que vont jouer le faisceau canonique et ses puissances dans la définition des variétés stables.
### Singularités
Introduisons tout d’abord la classe des singularités que ces variétés stables pourront porter.
\[defslc\] Une variété $X$ est dite à **singularités semi-log canoniques (slc)** si elle satisfait les conditions (i)-(v) suivantes.
(i) $X$ est réduite et purement de dimension $d$,
(ii) $X$ est à croisements normaux doubles en codimension 1,
(iii) $X$ satisfait la condition $S_2$ de Serre,
(iv) il existe $m>0$ tel que $\omega_X^{[m]}$ soit inversible,
(v) les discrépances des diviseurs au-dessus de $X$ sont $\geqslant -1$.
Si $X$ est de plus normale ou de manière équivalente par le critère de Serre, si $X$ vérifie :
1. X est régulière en codimension $1$,
on dit que $X$ est à **singularités log canoniques (lc)**.
Expliquons ces conditions. Que $X$ soit à croisements normaux doubles en codimension 1 signifie qu’il existe un ouvert $U\subset X$ dont le complémentaire a codimension $\geqslant 2$, le long duquel $X$ est soit régulière, soit localement isomorphe (pour la topologie étale ou, si $k=\operatorname{\mathbb{C}}$, pour la topologie analytique) à la singularité $\{xy=0\}\subset \mathbb{A}_k^{d+1}$. Qu’il soit nécessaire d’autoriser de telles singularités est déjà apparent dans le cas des courbes stables.
La condition $S_2$ de Serre est la propriété de Hartogs: elle stipule que les fonctions régulières sur $X$ s’étendent au travers des fermés $Z\subset X$ de codimension $\geqslant 2$. Plus précisément, si $Z$ est un tel fermé et si $j:X\setminus Z\hookrightarrow X$ est l’inclusion, le morphisme naturel $\operatorname{\mathcal{O}}_X\to j_*\operatorname{\mathcal{O}}_{X\setminus Z}$ est un isomorphisme. C’est un substitut de la normalité de $X$.
Les variétés stables doivent être pensées comme (pluri)canoniquement plongées et il est donc important de contrôler les formes différentielles de degré maximal sur $X$. C’est le rôle des conditions (iv) et (v). Notons $j:U\hookrightarrow X$ le plus gros ouvert le long duquel les singularités de $X$ sont à croisements normaux doubles. Comme les croisements normaux doubles sont des singularités localement d’intersection complète, donc Gorenstein, le faisceau dualisant $\omega_U$ de $U$ est un faisceau inversible[^2]. On définit le **faisceau canonique**[^3] de $X$ par $\omega_X:=j_*\omega_U$ et on introduit, pour tout $n\in\operatorname{\mathbb{Z}}$, ses puissances réflexives $\omega_X^{[n]}:=j_*(\omega_U^{\otimes n})$ : les **faisceaux pluricanoniques** de $X$. Qu’il existe un entier $m>0$ tel que $\omega_X^{[m]}$ soit un faisceau inversible, donc associé à un fibré en droites, est bien sûr une condition nécessaire à toute tentative de voir $X$ comme plongée à l’aide de formes pluricanoniques !
La condition (v) donne un contrôle birationnel sur les formes pluricanoniques sur $X$. Soit $\pi:Y\to X$ une modification normale[^4] de $X$ (par exemple la normalisation de $X$ ou une résolution des singularités de $X$), et soient $(E_i)_{i\in I}$ les diviseurs exceptionnels de $\pi$. Soit $m>0$ un entier tel que $\omega_X^{[m]}$ soit inversible. Au-dessus du lieu $Y\setminus \bigcup_i E_i$ où $\pi$ est un isomorphisme, on dispose d’un isomorphisme évident $\rho:\omega^{[m]}_{Y\setminus \cup_i E_i}{{\ext@arrow 0359\myrightarrowfill@{}{\,\sim\,}}}(\pi^*\omega_X^{[m]})|_{Y\setminus \cup_i E_i}$. Comme $\omega^{[m]}_{Y}$ et $\pi^*\omega_X^{[m]}$ sont inversibles au point générique de chacun des $E_i$, le morphisme $\rho$ a des zéros ou des pôles d’une certaine multiplicité le long de ces diviseurs, de sorte qu’il existe des $a_{E_i}(X)\in\frac{1}{m}\operatorname{\mathbb{Z}}$ tels que $\rho$ se prolonge en un isomorphisme $$\label{discrepeq}
\rho:\omega^{[m]}_{Y}{{\ext@arrow 0359\myrightarrowfill@{}{\,\sim\,}}}\pi^*\omega_X^{[m]}(\sum_i m \cdot a_{E_i}(X)E_i).$$ Les nombres rationnels $a_{E_i}(X)$ ont été choisis pour ne pas dépendre du choix de l’entier $m$ : ce sont les **discrépances** des diviseurs $E_i$.
La condition (v) selon laquelle ces discrépances sont toujours $\geqslant -1$ signifie en substance que les formes canoniques sur $X$ s’étendent en des formes à pôles au plus logarithmiques sur les modifications de $X$. Il suffit de la vérifier pour les diviseurs apparaissant sur une résolution arbitraire des singularités de $X$ dont le diviseur exceptionnel est à croisements normaux stricts (combiner [@Kosing Lemma 5.10 et Corollary 2.13]). C’est la condition la plus subtile de la définition \[defslc\]. La preuve transparente de l’unicité dans le théorème de réduction stable au § \[unicite\] permet de se convaincre de sa pertinence.
On définit d’autres classes de singularités en conservant les conditions (i)-(iv), mais en demandant à ce que les discrépances des diviseurs au-dessus de $X$ soient $>-1$ (resp. $\geqslant 0$, resp. $>0$) : ce sont les singularités **kawamata log terminales** ou **klt** (resp. **canoniques**, resp. **terminales**). Ces singularités sont normales. Nous nous en servirons peu.
Ces définitions s’étendent sans difficultés à des schémas plus généraux que des variétés. Nous les utiliserons par exemple pour des schémas de type fini sur le spectre d’un anneau de valuation discrète au §\[secredst\] et au §\[ouvertslc\].
### Définition
La notion de stabilité combine les propriétés locales discutées ci-dessus et une condition globale d’amplitude du faisceau canonique.
\[defstable\] Une **variété stable** est une variété projective $X$ à singularités slc dont le faisceau canonique $\omega_X$ est ample.
Le faisceau $\omega_X$ n’est pas inversible en général. La définition \[defstable\] requiert seulement qu’il soit ample comme $\operatorname{\mathbb{Q}}$-fibré en droites, c’est-à-dire que $\omega_X^{[m]}$ soit ample pour un $m>0$ (de manière équivalente, pour tout $m>0$) tel que $\omega_X^{[m]}$ soit inversible.
Les courbes stables sont traditionnellement supposées connexes, comme dans la définition \[defcourbestable\]. Il est plus naturel de ne pas faire cette hypothèse (voir par exemple le théorème \[bij\]). La définition des variétés stables dans le cas des surfaces avait été dégagée par Kollár et Shepherd-Barron [@KSB §5.4] et la définition en dimension arbitraire en est une extension immédiate. En revanche, l’étude de ces variétés est bien plus difficile en dimension $\geqslant 3$ qu’en dimension $2$.
### Cas des paires {#paires}
Nous utiliserons la variante suivante des définitions \[defslc\] et \[defstable\]. Une **paire** $(X,\Delta)$ est constituée d’une variété $X$ et d’un $\operatorname{\mathbb{Q}}$-diviseur de Weil $\Delta=\sum c_i\Delta_i$, où les $\Delta_i$ sont des sous-variétés intègres de codimension $1$ de $X$ non incluses dans le lieu singulier de $X$ et où $c_i\in\operatorname{\mathbb{Q}}$ (dans la pratique, on aura même $c_i\in\operatorname{\mathbb{Q}}\cap[0,1]$). On étend les définitions à ce cadre en remplaçant partout $\omega_X$ par le faisceau canonique $\omega_X(\Delta)$ de la paire.
\[slcpaires\] La paire $(X,\Delta)$ est à **singularités slc** (resp. **lc**) si $c_i\in\operatorname{\mathbb{Q}}\cap[0,1]$, si $X$ est réduite, purement de dimension $d$, à croisements normaux doubles (resp. régulière) en codimension $1$ et $S_2$, s’il existe un entier $m>0$ tel que $\omega_X^{[m]}(m\Delta)$ est inversible et si les discrépances $a_E(X,\Delta)$ des diviseurs $E$ au-dessus de $X$ sont $\geqslant -1$.
Elle est **stable** si elle est à singularités slc si $X$ est projective et si $\omega_X(\Delta)$ est ample.
Dans cette définition, les hypothèses faites sur $X$ assurent l’existence d’un ouvert $j:U\hookrightarrow X$ dont le complémentaire a codimension $\geqslant 2$ le long duquel $X$ est Gorenstein et les $\Delta_i$ sont Cartier. Si $n\in \operatorname{\mathbb{Z}}$ est tel que les $nc_i$ sont des entiers, cela permet de définir le faisceau $n$-canonique $\omega_X^{[n]}(n\Delta):=j_*(\omega_U^{\otimes n}(n\Delta|_U))$ de $(X,\Delta)$. Les discrépances $a_{E}(X,\Delta)$ sont calculées par rapport au faisceau canonique $\omega_X(\Delta)$ de la paire. Si $\pi:Y\to X$ est une modification normale de $X$ avec diviseurs exceptionnels $E_i$, si $(\pi^{-1})_*\Delta$ est la transformée stricte de $\Delta$ dans $Y$, et si $m>0$ est tel que $\omega_X^{[m]}(m\Delta)$ est inversible, elles sont définies par l’isomorphisme naturel généralisant : $$\label{logdiscrepeq}\omega^{[m]}_{Y}(m(\pi^{-1})_*\Delta){{\ext@arrow 0359\myrightarrowfill@{}{\,\sim\,}}}\pi^*\omega_X^{[m]}(m\Delta)\Big(\sum_i m \cdot a_{E_i}(X,\Delta)E_i\Big).$$
Nous aurons à considérer des paires pour plusieurs raisons ; la principale est la suivante. Soit $X$ une variété satisfaisant aux conditions (i)-(iv) de la définition \[defslc\]. Soit $m>0$ tel que $\omega^{[m]}_X$ soit inversible et soit $\nu:\widetilde{X}\to X$ la normalisation de $X$. Notons $\Gamma\subset\widetilde{X}$ le lieu exceptionnel de $\nu$, muni de sa structure réduite. C’est un diviseur qui est l’adhérence de l’image inverse par $\nu$ du lieu où $X$ est à croisements normaux doubles. On appelle $\Gamma$ le **conducteur** de $X$. L’isomorphisme évident $(\nu^*\omega_X^{[m]})|_{\widetilde{X}\setminus\Gamma}{{\ext@arrow 0359\myrightarrowfill@{}{\,\sim\,}}}\omega_{\widetilde{X}}^{[m]}|_{\widetilde{X}\setminus\Gamma}$ se prolonge en un isomorphisme $$\label{conducteur}
\nu^*\omega_X^{[m]}{{\ext@arrow 0359\myrightarrowfill@{}{\,\sim\,}}}\omega_{\widetilde{X}}^{[m]}(m\Gamma),$$ comme le montre un calcul local sur le lieu où $X$ est à croisements normaux doubles [@Kosing (5.7.4)]. On déduit immédiatement de l’isomorphisme l’équivalence [@Kosing Lemma 5.10] : $$\label{slclc}
X\textrm{ est \`a singularit\'es slc }\iff (\widetilde{X},\Gamma)\textrm{ est \`a singularit\'es lc.}$$ Ce procédé de normalisation permettra de ramener l’étude des variétés à singularités slc au cas normal. Comprendre dans quelle mesure on peut reconstruire $X$ à partir de $(\widetilde{X},\Gamma)$ est une question difficile (voir le théorème \[bij\] pour un énoncé précis).
### Exemples {#exslc}
Les seules singularités slc de dimension $1$ sont les nœuds.
En dimension $2$, les singularités slc ont été classifiées par Kawamata [@Kawsing Theorem 2] dans le cas normal et par Kollár et Shepherd-Barron [@KSB Theorem 4.24] en général (voir aussi [@Kosing §3.3] ou [@Kobook]). Sans rappeler cette classification en détail, donnons quelques exemples représentatifs. Les singularités obtenues comme quotient de $\operatorname{\mathbb{A}}_k^2$ par un sous-groupe fini de $\operatorname{GL}_2(k)$ sont lc. Cela inclut toutes les singularités Du Val (ou points doubles rationnels). D’autres singularités lc de surface sont les singularités elliptiques obtenues comme cônes sur une courbe elliptique.
Des exemples de surfaces slc non normales sont les points à croisement normaux triples $\{xyz=0\}\subset \operatorname{\mathbb{A}}_k^3$, le parapluie de Whitney ou pinch point $\{x^2=yz^2\}\subset \operatorname{\mathbb{A}}_k^3$, ou un cône sur une courbe elliptique nodale $\{y^2=x^3+x^2\}\subset \operatorname{\mathbb{A}}_k^3$.
On ne dispose pas de classification en dimension supérieure. Les cônes $$\label{cone}
C(X,L):=\operatorname{Spec}\bigoplus_{l\geqslant 0} H^0(X,L^{\otimes l})$$ où $X$ est une variété projective munie d’un fibré ample $L$, fournissent une instructive source d’exemples. On calcule que $C(X,L)$ a des singularités slc (resp. lc) si et seulement si $X$ a des singularités slc (resp. lc) et s’il existe des entiers $m<0$ et $l\geqslant 0$ tels que $\omega_X^{[m]}\simeq L^{\otimes l}$ [@Kosing §3.1]. En particulier, le cône anticanonique sur une variété de Fano, ou un cône associé à un fibré ample arbitraire sur une variété de Calabi-Yau, ont des singularités lc.
D’autres exemples élémentaires sont les singularités quotient, c’est-à-dire les quotients de variétés lisses[^5] par l’action d’un groupe fini [@Kosing 3.18]. Des exemples plus riches, à la topologie plus compliquée, ont été construits par Kollár [@Konew].
Familles stables
----------------
### Définition {#pardeffamst}
Comme on le verra au §\[exfam\], les familles plates à fibres slc (resp. stables) ne donnent pas lieu à une bonne notion de famille de variétés slc (resp. stables). La raison pour cela est que, si l’on souhaite penser aux variétés stables comme étant pluricanoniquement plongées, il est important que les faisceaux (pluri)canoniques des fibres varient convenablement en famille ; c’est une condition que l’on doit imposer.
\[deffam\] Une **famille localement stable** est un morphisme plat à fibres slc $f:\mathcal{X}\to B$ tel que pour tout $n\in\operatorname{\mathbb{Z}}$, le faisceau $\omega_{\mathcal{X}/B}^{[n]}$ soit $f$-plat de formation commutant à tout changement de base. C’est une **famille de variétés stables** ou **famille stable** si $f$ est de plus propre à fibres stables.
Dans cette définition, les faisceaux pluricanoniques relatifs $\omega_{\mathcal{X}/B}^{[n]}$ sont construits comme dans le cas absolu. Plus précisément, on note $j:\mathcal{U}\hookrightarrow\mathcal{X}$ le plus gros ouvert le long duquel les singularités des fibres géométriques de $f$ sont à croisements normaux doubles. Le morphisme $f|_{\mathcal{U}}$ est plat à fibres Gorenstein, de sorte que le faisceau dualisant relatif $\omega_{\mathcal{U}/B}$ est inversible [@Conrad Theorem 3.5.1]. On pose $\omega_{\mathcal{X}/B}:=j_*\omega_{\mathcal{U}/B}$ et $\omega_{\mathcal{X}/B}^{[n]}:=j_*(\omega_{\mathcal{U}/B}^{\otimes n})$.
La définition \[deffam\] requiert tout d’abord que les faisceaux pluricanoniques relatifs $\omega_{\mathcal{X}/B}^{[n]}$ soient plats sur $B$. Cette hypothèse ne suffit pas à assurer que les fibres $\omega_{\mathcal{X}/B}^{[n]}|_{\operatorname{\mathcal{X}}_b}$ de ces faisceaux au-dessus d’un point $b\in B$ co" incident avec les faisceaux pluricanoniques $\omega_{\mathcal{X}_b}^{[n]}$ de la fibre. C’est le rôle de la condition de changement de base dans la définition \[deffam\] : elle revient à imposer que le morphisme naturel $\omega_{\mathcal{X}/B}^{[n]}|_{\operatorname{\mathcal{X}}_b}\to \omega_{\mathcal{X}_b}^{[n]}$ soit un isomorphisme, pour tout $b\in B$ et tout $n\in\operatorname{\mathbb{Z}}$. Ceci implique[^6] en effet la propriété, a priori plus forte, de commutation à tout changement de base : pour tout morphisme $g:B'\to B$, si l’on note $g_{\operatorname{\mathcal{X}}}:\operatorname{\mathcal{X}}'\to\operatorname{\mathcal{X}}$ le changement de base, le morphisme naturel $g_{\operatorname{\mathcal{X}}}^*\omega_{\mathcal{X}/B}^{[n]}\to \omega_{\mathcal{X}'/B'}^{[n]}$ est un isomorphisme.
Il suit de la définition \[deffam\] que si $f$ est localement stable, il existe $m>0$ tel que le faisceau $\omega^{[m]}_{\operatorname{\mathcal{X}}/B}$ soit inversible (et $f$-ample si $f$ est stable). En effet, par récurrence noethérienne sur la base $B$, on peut choisir $m$ de sorte que $\omega_{\mathcal{X}_b}^{[m]}$ soit inversible pour tout $b\in B$. Il résulte de sa platitude et de sa commutation au changement de base que $\omega^{[m]}_{\operatorname{\mathcal{X}}/B}$ est inversible (et $f$-ample si les fibres sont stables). Une famille stable est donc bien canoniquement polarisée, comme désiré.
Les conditions de platitude et de commutation au changement de base pour $\omega_{\mathcal{X}/B}^{[n]}$ sont subtiles. Elles sont automatiques pour $n=1$ par [@lcdB Theorem 7.9.3] et [@Kosing Corollary 6.32]. Elles sont toujours vérifiées si les fibres de $f$ sont à singularités canoniques[^7] (voir [@Kosurvey Aside 30]). Enfin, quand la base $B$ est réduite, on dispose d’un critère numérique : il est équivalent de demander que le degré de la polarisation canonique des fibres soit localement constant sur la base [Kobook]{}.
Dans la définition \[deffam\], la condition de commutation de tous les $\omega_{\mathcal{X}/B}^{[n]}$ aux changements de base est connue sous le nom de *condition de Kollár*. Une variante, dite *condition de Viehweg* [@Viehweg Assumptions 8.30], consiste à demander que $\omega_{\mathcal{X}/B}^{[m]}$ soit inversible (et par conséquent commute aux changements de base) seulement pour un $m>0$. Elle permet également de construire des espaces de modules projectifs de variétés stables ; ils diffèrent par leur structure schématique de ceux obtenus à l’aide de la condition de Kollár (voir [@Kodef]).
### Exemples {#exfam}
Illustrons, en suivant [@YPG 7.A] et [@Kosurvey Example 26], l’importance de la condition de changement de base dans la Définition \[deffam\].
Considérons d’une part le plongement de Veronese $\Sigma_1=\operatorname{\mathbb{P}}^2_{k}\hookrightarrow\operatorname{\mathbb{P}}^{5}_{k}$ et d’autre part le plongement $\Sigma_2=\operatorname{\mathbb{P}}^1_{k}\times\operatorname{\mathbb{P}}^1_{k}\hookrightarrow\operatorname{\mathbb{P}}^5_{k}$ induit par $\operatorname{\mathcal{O}}(1,2)$. Ces deux surfaces projectives ont pour sections hyperplanes lisses des courbes rationnelles normales quartiques $\Gamma$. Pour $i\in\{1,2\}$, soit $f_i:\operatorname{\mathcal{X}}_i\to\operatorname{\mathbb{P}}^1_{k}$ un pinceau général de sections hyperplanes du cône $C(\Sigma_i,\operatorname{\mathcal{O}}(1))$ sur $\Sigma_i$. Toutes les fibres de $f_i$ sont isomorphes à $\Sigma_i$, sauf la section hyperplane passant par le sommet du cône, qui est isomorphe au cône $C$ sur la courbe rationnelle normale quartique[^8]. On voit ainsi que les fibres de $f_i$ ont des singularités lc (voir § \[exslc\]).
On remarque cependant que les nombres d’intersection $\omega_{\Sigma_1}\cdot\omega_{\Sigma_1}=9$ et $\omega_{\Sigma_2}\cdot\omega_{\Sigma_2}=8$ des surfaces $\Sigma_1$ et $\Sigma_2$ diffèrent. Comme $f_1$ et $f_2$ ont une fibre spéciale isomorphe à $C$ en commun, cette remarque n’est pas compatible avec le fait que les faisceaux dualisants relatifs de ces familles forment un $\operatorname{\mathbb{Q}}$-fibré en droites. Cela s’explique par le fait que, si $f_1$ est bien localement stable (en particulier, le faisceau $\omega^{[2]}_{\operatorname{\mathcal{X}}_1/B}$ est inversible[^9]), la famille $f_2$ ne l’est pas ($\omega^{[n]}_{\operatorname{\mathcal{X}}_2/B}$ n’est inversible pour aucun $n>0$^(\[note1\])^).
En remplaçant les fibres des $f_i$ par des revêtements ramifiés appropriés, on obtient des exemples analogues pour lesquels $f_1$ est stable (et pas seulement localement stable).
On construit un exemple un peu différent en suivant [@KSB Example 5.12]. Effectuons la même construction à l’aide des deux surfaces $\Sigma'_1=\operatorname{\mathbb{P}}^1_{k}\times\operatorname{\mathbb{P}}^1_{k}\hookrightarrow\operatorname{\mathbb{P}}^8_{k}$ et $\Sigma'_2=\operatorname{\mathbb{P}}_{\operatorname{\mathbb{P}}^1_{k}}(\operatorname{\mathcal{O}}\oplus\operatorname{\mathcal{O}}(1))\hookrightarrow\operatorname{\mathbb{P}}^8_{k}$, plongées par leur fibré anticanonique, dont les sections hyperplanes lisses sont des courbes elliptiques octiques. Prenant, pour $i\in\{1,2\}$, un pinceau de sections hyperplanes du cône $C(\Sigma'_i,\operatorname{\mathcal{O}}(1))$ sur $\Sigma'_i$, on peut obtenir deux familles $f'_i:\operatorname{\mathcal{X}}'_i\to\operatorname{\mathbb{P}}^1_{k}$ dont les fibres générales sont toutes isomorphes à $\Sigma'_i$, sauf une qui est un cône sur une courbe elliptique octique fixée. À la différence de l’exemple précédent, les deux familles sont localement stables : on vérifie même que $\omega_{\operatorname{\mathcal{X}}^i/B}$ est inversible^(\[note1\])^ pour $i\in\{1,2\}$.
Comme ci-dessus, en remplaçant les $f_i$ par des revêtements ramifiés bien choisis, on peut obtenir deux familles stables qui ont une fibre singulière en commun et dont les fibres générales, lisses, ne peuvent être membres d’une même famille lisse de base irréductible. Il s’agit donc d’un exemple où deux composantes irréductibles distinctes de l’espace des modules des variétés stables s’intersectent. Ce phénomène n’apparaît pas en dimension $1$. Signalons que Horikawa [@Horikawa Theorem 3] a construit de tels exemples pour lesquels la fibre spéciale commune aux deux familles est de plus lisse.
Espaces de modules de variétés stables
--------------------------------------
### Existence
Nous pouvons à présent donner l’énoncé précis d’existence de l’espace de modules des variétés stables. La construction de l’espace de modules des courbes stables demandait de fixer le genre de ces courbes. En dimension supérieure, on doit aussi fixer un invariant discret : la fonction de Hilbert.
La **fonction de Hilbert** $F:\operatorname{\mathbb{Z}}\to\operatorname{\mathbb{Z}}$ d’une variété stable $X$ est $$F(n):=\chi(X,\omega_X^{[n]}).$$
Comme $\omega_X$ n’est pas inversible en général, la fonction de Hilbert de $X$ peut ne pas être un polynôme en $n$[^10]. L’hypothèse de platitude dans la définition \[deffam\] montre que cet invariant est localement constant sur la base d’une famille stable.
\[thedm\] Soit $F:\operatorname{\mathbb{Z}}\to\operatorname{\mathbb{Z}}$ une fonction. La catégorie fibrée en groupoïdes $$\label{fonctpoints}
B\mapsto\{\hspace{.1em}\textrm{familles stables }f:\operatorname{\mathcal{X}}\to B\textrm{ dont les fibres ont fonction de Hilbert $F$}\}$$ sur la catégorie des $k$-schémas est un champ de Deligne-Mumford $\overline{\mathcal{M}}_F$ propre sur $k$ admettant un espace de modules grossier projectif $\overline{M}_F$.
La preuve de ce théorème, due à de nombreux auteurs, sera esquissée au §\[consedm\]. Le lecteur qui ne serait pas familier avec les champs [@LMB; @Olsson] peut ne retenir que la seconde partie de son énoncé. Elle signifie qu’il existe une variété projective $\overline{M}_F$ sur $k$ et une manière d’associer à toute famille stable $f:\operatorname{\mathcal{X}}\to B$ de même fonction de Hilbert $F$ un morphisme $\psi(f):B\to \overline{M}_F$, qui soit fonctorielle en $B$, de sorte que $(\overline{M}_F,\psi)$ soit universel pour cette propriété, et induise une bijection $$\left\{
\begin{array}{c}
\mbox{classes d'isomorphisme de vari\'et\'es}\\
\mbox{stables sur $K$ de fonction de Hilbert $F$}
\end{array}
\right\}
{{\ext@arrow 0359\myrightarrowfill@{}{\,\sim\,}}}\overline{M}_F(K).$$ pour toute extension algébriquement close $K$ de $k$. Par exemple, le théorème \[thedm\] munit l’ensemble des classes d’isomorphisme de variétés stables complexes de fonction de Hilbert $F$ d’une structure naturelle de variété projective complexe.
Insistons sur l’importance de la définition des singularités slc pour la validité de cet énoncé. Admettre une classe plus large de singularités aurait nui au caractère séparé de $\overline{M}_F$ ; restreindre les singularités autorisées aurait empêché sa propreté.
Dans le cas des surfaces, le théorème \[thedm\] est connu depuis longtemps, par des travaux de Kollár, Shepherd-Barron et Alexeev [@KSB; @Kocomplete; @Ale], à deux subtilités près. D’une part, une structure schématique sur $\overline{M}_F$ prenant en compte les fonctions nilpotentes, n’a été construite rigoureusement que plus tard (voir [@HaKo; @KoHH; @AbraHa] et §\[ssHH\]). D’autre part, la propreté des composantes irréductibles de $\overline{M}_F$ paramétrant génériquement des variétés non normales n’a pu être établie que grâce aux techniques de recollement de Kollár (voir [@Kosing; @Kobook] et §\[nonnormale\]).
### Géométrie
La fonction de Hilbert $F(n)=(g-1)(2n-1)$ donne lieu au champ de modules $\overline{\mathcal{M}}_g$ des courbes stables de Deligne et Mumford [@DM] et à son espace de modules grossier $\overline{M}_g$. Le champ $\overline{\operatorname{\mathcal{M}}}_g$ est lisse et irréductible, de sorte que $\overline{M}_g$ est normal et irréductible. On a vu à la fin du §\[exfam\] que ces propriétés tombaient en défaut en dimension supérieure. Vakil [@Murphy Main Theorem M2] a même démontré que les singularités des variétés $\overline{M}_F$ peuvent être arbitrairement mauvaises.
La géométrie de $\overline{M}_g$ est aujourd’hui bien comprise et fait l’objet d’une abondante littérature. A contrario, on dispose de très peu de descriptions concrètes d’espaces de modules non triviaux de variétés stables en dimension supérieure (à l’exception notable de l’espace de modules des produits de courbes stables [@vOp]). On ne sait par exemple pas décrire l’adhérence de l’ouvert paramétrant des surfaces quintiques dans $\operatorname{\mathbb{P}}^3_k$ [@quintic1; @quintic2]. On trouvera dans [@Godeaux] l’état de l’art dans le cas des surfaces de Godeaux.
Comme la fonction de Hilbert d’une variété stable lisse est polynomiale, les variétés stables dont la fonction de Hilbert n’est pas polynomiale, comme dans la note de bas de page , donnent lieu à des composantes connexes de l’espace de modules qui ne paramètrent que des variétés singulières. Soit enfin $M$ une composante connexe de $\overline{M}_F$. On sait que si l’une des variétés que $M$ paramètre vérifie la condition $S_k$ de Serre, alors toutes ont cette propriété [@lcdB Corollary 1.3]. Par conséquent, si l’une d’entre elles est Cohen-Macaulay (par exemple : lisse), toutes sont Cohen-Macaulay. Il est malgré tout utile de considérer aussi des variétés stables qui ne sont pas Cohen-Macaulay ; on en verra une raison au §\[stableslc\].
### Variantes
De nombreuses variantes des espaces de modules de variétés stables sont utiles et ont été étudiées. Tout d’abord, il est naturel de considérer plut\^ ot des espaces de modules de paires stables, qui généralisent en dimension supérieure les espaces de modules de courbes stables pointées. Ce sujet a été développé dans [@Hassett; @Hacking; @Alelim; @KoPa] et le livre [@Kobook] en fait une étude approfondie.
Il est également intéressant de construire des espaces de modules de morphismes stables à valeurs dans une variété fixée. Quand la source du morphisme est une courbe, ces espaces ont été introduits par Kontsevich (voir [@FuPa]), et on pourra consulter [@Alemod; @DR] en dimension supérieure.
Les résultats en caractéristique positive sont limités. L’article [@Pata] contient le meilleur énoncé connu : sur un corps de caractéristique $p\geqslant 7$, l’espace de modules des surfaces stables existe comme espace algébrique séparé et ses sous-espaces propres sont projectifs.
Outils pour l’étude des singularités slc
----------------------------------------
Pour obtenir des compactifications modulaires $\overline{M}_F$ des espaces de modules de variétés projectives lisses canoniquement polarisées, nous avons dû autoriser des variétés à singularités slc. Que ce soit pour construire ces compactifications ou pour d’éventuelles applications de leur existence, il est important d’étudier cette classe de singularités. Il s’avère qu’elles ont des propriétés remarquables ; nous en décrivons ici quelques-unes.
### Adjonction {#paradj}
Soit $(X,\Delta+B)$ une paire dans laquelle le diviseur de Weil $B$ est affecté d’un coefficient $1$. On suppose que $X$ est normale, purement de dimension $d$, et qu’il existe un entier $m>0$ tel que $\omega_X^{[m]}(m\Delta+mB)$ est inversible. Considérons la normalisation $\nu:\widetilde{B}\to B$ de $B$ et soit $U\subset X$ le plus gros ouvert disjoint de $\Delta$ le long duquel $X$ et $B$ sont tous deux réguliers. L’isomorphisme canonique $\omega_X(B)|_{B\cap U}{{\ext@arrow 0359\myrightarrowfill@{}{\,\sim\,}}}\omega_{B\cap U}$ donné par le résidu des formes différentielles induit un isomorphisme $$\label{differente}
\omega_X^{[m]}(m\Delta+mB)|_{\widetilde{B}}{{\ext@arrow 0359\myrightarrowfill@{}{\,\sim\,}}}\omega_{\widetilde{B}}^{[m]}(m\operatorname{Diff}_{\widetilde{B}}(\Delta)),$$ où $\operatorname{Diff}_{\widetilde{B}}(\Delta)$ est un $\operatorname{\mathbb{Q}}$-diviseur de Weil sur $\widetilde{B}$ uniquement déterminé et indépendant de $m$ : c’est la **différente** de $\Delta$ sur $\widetilde{B}$ (voir [@Kosing Definition 4.2]).
Dans de nombreuses situations, par exemple dans le cadre d’une récurrence sur la dimension, il est utile de ramener l’étude de $(X,\Delta+B)$ à celle de $(\widetilde{B},\operatorname{Diff}_{\widetilde{B}}(\Delta))$. Le théorème \[invadj\], dû à Kawakita [@Kawakita], et qui fait suite à des travaux de Shokurov [@Shokurov] et de Kollár [@Koflips §17], est un outil précieux pour ce type d’arguments.
\[invadj\] Les conditions suivantes sont équivalentes:
(i) La paire $(X,\Delta+B)$ est lc dans un voisinage de $B$.
(ii) La paire $(\widetilde{B}, \operatorname{Diff}_{\widetilde{B}}(\Delta))$ est lc.
L’implication (i)$\implies$(ii), dite *adjonction*, est facile. C’est l’implication réciproque (ii)$\implies $(i), dite *inversion de l’adjonction*, qui est délicate. Sa preuve repose de manière essentielle sur le théorème d’annulation de Kawamata-Viehweg.
Par le biais de l’équivalence , on peut déduire du théorème \[invadj\] des énoncés portant sur les singularités slc (voir [@Patakfibre Lemma 2.10, Corollary 2.11]).
### Propriétés cohomologiques
La premi\` ere indication que les classes de singularités que nous considérons ont de bonnes propriétés cohomologiques a été le théorème d’Elkik [@Elkik] selon lequel les singularités canoniques sont rationnelles. Ce résultat reste valide plus généralement pour les singularités klt [@KM Theorem 5.22].
\[rationnelles\] Les singularités klt sont rationnelles.
On en déduit que les singularités klt sont Cohen-Macaulay [@KM Theorem 5.10].
Malheureusement, les singularités lc ne sont pas toujours rationnelles, ni même Cohen-Macaulay. Par exemple, un cône sur une surface abélienne est lc mais pas $S_3$ [@Kosing Example 3.6]. Il est donc nécessaire de trouver un substitut à la rationalité, qui s’applique aux variétés lc (ou plus généralement slc). Kollár et Kovács ont montré que les singularités Du Bois [@Kosing §6] remplissent ce rôle (voir [@lcdB], [@Kosing §6.2]).
Les singularités slc sont Du Bois.
Une conséquence concrète de cet énoncé est le fait que si $f:\operatorname{\mathcal{X}}\to B$ est une famille stable, les fonctions $b\mapsto h^i(\operatorname{\mathcal{X}}_b,\operatorname{\mathcal{O}}_{\operatorname{\mathcal{X}}_b})$ sont localement constantes sur $B$ [@DuBois Théorème 4.6]. Nous n’utiliserons pas les singularités Du Bois dans la suite de ce texte.
En revanche, nous devrons savoir contrôler précisément le défaut de la propriété $S_3$ des singularités slc. Nous utiliserons à cet effet un résultat d’Alexeev [@Alelim Lemma 3.2], étendu dans [@Kosing Theorem 7.20]. On dit qu’une sous-variété intègre d’une variété $X$ à singularités slc est un **centre log canonique** de $X$ si c’est l’image d’un diviseur au-dessus de $X$ dont la discrépance est égale à $-1$.
\[S3\] Soit $X$ une variété slc. Si $x\in X$ n’est pas le point générique d’un centre log canonique de $X$, on a $\operatorname{prof}(\omega^{[n]}_{X,x})\geqslant \min(3, \dim(\operatorname{\mathcal{O}}_{X,x}))$ pour tout $n\in\operatorname{\mathbb{Z}}$.
Les cas $n=0$ et $n=1$ sont explicités dans [@Kosing Corollaries 7.21 and 7.22], et le cas général se prouve de la même manière[^11].
Le théorème de réduction stable {#secredst}
===============================
Nous expliquons dans cette section la preuve du théorème \[redstable\]. On en considère plutôt une variante locale sur un anneau de valuation discrète $R$ de corps de fonctions $K$. On note $T=\operatorname{Spec}(R)$ son spectre, de point fermé $t$ et de point générique $\eta$.
\[redstable2\] Soit $X$ une variété stable sur $K$. Il existe une extension finie d’anneaux de valuations discrètes $R\subset R'$ de corps de fonctions $K\subset K'$ et une famille stable $f:\mathcal{X}\to \operatorname{Spec}(R')$ telle que $\mathcal{X}_{K'}\simeq X_{K'}$. Si $R'$ est fixé, cette famille est unique.
Que le théorème \[redstable2\] implique le théorème de réduction stable sous sa forme globale énoncée au théorème \[redstable\] est standard.
Soit $f:\operatorname{\mathcal{X}}\to B$ un morphisme propre de base une courbe intègre de corps de fonctions $K$. Si $\operatorname{\mathcal{X}}_{\eta}$ est stable, la famille $f$ est stable au-dessus d’un ouvert dense $U\subset B$ (par exemple, par les arguments des §§\[deminormal\]–\[ouvertslc\]). Pour tout $b\in B\setminus U$, le théorème \[redstable2\] appliqué à l’anneau de valuation discrète $R_b:=\operatorname{\mathcal{O}}_{B,b}$ fournit une extension finie $R'_b$ de $R_b$ de corps de fonctions $K'_b$ telle que $\operatorname{\mathcal{X}}_{K'_b}$ ait un modèle stable sur $R'_b$. Soit $K'$ une extension galoisienne de $K$ dans laquelle tous les $K'_b$ se plongent et soit $\pi:B'\to B$ la normalisation de $B$ dans $K'$. Par construction, la variété $\operatorname{\mathcal{X}}_{K'}$ possède un modèle stable au voisinage de tout point de $B'$. Ces modèles locaux se recollent par unicité.
Le théorème \[redstable2\] est dû à Hacon et Xu [@HX] et Kollár [@Kosing; @Kobook]. C’est ce théorème qui nous permettra de vérifier la propreté du champ de modules des variétés stables (voir §\[champ\]). Les résultats antérieurs de Birkar, Cascini, Hacon et McKernan [@BCHM] auraient cependant suffi à démontrer la propreté des composantes irréductibles de $\overline{\operatorname{\mathcal{M}}}_F$ qui paramètrent génériquement des variétés lisses.
La preuve du théorème \[redstable2\] repose sur le point de vue selon lequel les variétés stables doivent être considérées comme pluricanoniquement plongées. Plus précisément, si $X$ est une variété stable et si $m>0$ est tel que $\omega_X^{[m]}$ est inversible, on peut reconstruire $X$ à partir de son algèbre $m$-canonique par la formule $X\simeq\operatorname{Proj}\bigoplus_{l \geqslant 0}H^0(X,\omega_X^{[lm]})$. L’existence comme l’unicité des familles stables dans le théorème \[redstable2\] seront obtenues par le biais de ces algèbres $m$-canoniques.
Unicité {#unicite}
-------
Montrons la propriété d’unicité dans le théorème \[redstable2\]. La preuve donnée dans [@Kosurvey Proposition 6] quand les $f_i$ sont lisses s’étend au cas général [@Kobook]. Commençons par démontrer un lemme que nous utiliserons à plusieurs reprises.
\[XXt\] Soit $f:\operatorname{\mathcal{X}}\to T$ un morphisme propre et plat dont les fibres satisfont les conditions (i)-(iv) de la définition \[defslc\]. Soit $m>0$ un entier tel que $\omega_{\operatorname{\mathcal{X}}/T}^{[m]}$ soit inversible. Si la variété $\operatorname{\mathcal{X}}_t$ est à singularités slc, la paire $(\operatorname{\mathcal{X}},\operatorname{\mathcal{X}}_t)$ est à singularités slc.
On considère le diagramme commutatif $$\begin{aligned}
\xymatrix @R=0.4cm {
\widetilde{\operatorname{\mathcal{X}}_t}\ar^{\tilde{\iota}}[r]\ar^{\nu_t}[d] & \widetilde{\operatorname{\mathcal{X}}} \ar^{\nu}[d] \\
\operatorname{\mathcal{X}}_t\ar^{\iota}[r]&\operatorname{\mathcal{X}}, }
\end{aligned}$$ où $\nu$ et $\nu_t$ sont les normalisations de $\operatorname{\mathcal{X}}$ et $\operatorname{\mathcal{X}}_t$, et où $\iota$ et $\tilde{\iota}$ sont les morphismes naturels. On note $\Gamma\subset \widetilde{\operatorname{\mathcal{X}}}$ et $\Delta\subset \widetilde{\operatorname{\mathcal{X}}_t}$ les conducteurs de $\operatorname{\mathcal{X}}$ et $\operatorname{\mathcal{X}}_t$ (voir §\[paires\]). Par définition des conducteurs et de la différente (voir et ), on dispose d’isomorphismes naturels $$\omega_{\widetilde{\operatorname{\mathcal{X}}_t}}^{[m]}(m\Delta)\simeq \nu_t^*\omega_{\operatorname{\mathcal{X}}_t}^{[m]}\simeq \nu_t^*\iota^*\omega_{\operatorname{\mathcal{X}}}^{[m]}(m\operatorname{\mathcal{X}}_t)\simeq\tilde{\iota}^*\omega_{\widetilde{\operatorname{\mathcal{X}}}}^{[m]}(m(\widetilde{\operatorname{\mathcal{X}}})_t+m\Gamma)\simeq \omega_{\widetilde{\operatorname{\mathcal{X}}_t}}^{[m]}(m\operatorname{Diff}_{\widetilde{\operatorname{\mathcal{X}}_t}}(\Gamma)).$$ La composée de ces isomorphismes étant l’identité de $\omega_{\widetilde{\operatorname{\mathcal{X}}_t}}^{[m]}$ aux points génériques de $\widetilde{\operatorname{\mathcal{X}}_t}$, il suit que $\Delta=\operatorname{Diff}_{\widetilde{\operatorname{\mathcal{X}}_t}}(\Gamma)$. Comme $\operatorname{\mathcal{X}}_t$ est slc, la paire $(\widetilde{\operatorname{\mathcal{X}}_t},\Delta)$ est lc par , donc la paire $(\widetilde{\operatorname{\mathcal{X}}},\Gamma+(\widetilde{\operatorname{\mathcal{X}}})_t)$ est lc par inversion de l’adjonction (théorème \[invadj\]). On déduit que $(\operatorname{\mathcal{X}},\operatorname{\mathcal{X}}_t)$ est slc par , qui s’adapte immédiatement au cas des paires.
Nous pouvons à présent démontrer la propriété d’unicité dans le théorème \[redstable2\]. Soient $f_1:\mathcal{X}_1\to T$ et $f_2:\mathcal{X}_2\to T$ des familles stables et $\phi_{\eta}:\operatorname{\mathcal{X}}_{1,\eta}{{\ext@arrow 0359\myrightarrowfill@{}{\,\sim\,}}}\operatorname{\mathcal{X}}_{2,\eta}$ un isomorphisme. On souhaite démontrer que $\phi_{\eta}$ s’étend en un isomorphisme $\phi:\operatorname{\mathcal{X}}_1{{\ext@arrow 0359\myrightarrowfill@{}{\,\sim\,}}}\operatorname{\mathcal{X}}_2$.
Pour cela, notons $\operatorname{\mathcal{X}}\subset\operatorname{\mathcal{X}}_1\times_T\operatorname{\mathcal{X}}_2$ l’adhérence du graphe de $\phi_{\eta}$. Soit $\operatorname{\mathcal{Y}}\to\operatorname{\mathcal{X}}$ une modification $S_2$ de $\operatorname{\mathcal{X}}$ qui est un isomorphisme au-dessus de $\operatorname{\mathcal{X}}_\eta$, telle que les composantes irréductibles du lieu non normal de $\operatorname{\mathcal{Y}}$ dominent toutes $T$[^12]. Notons $g_i:\operatorname{\mathcal{Y}}\to\operatorname{\mathcal{X}}_i$ les projections naturelles, et choisissons un entier $m>0$ tel que les $\omega^{[m]}_{\operatorname{\mathcal{X}}_i/T}$ soient inversibles.
Si $i\in\{1,2\}$, la paire $(\operatorname{\mathcal{X}}_i,(\operatorname{\mathcal{X}}_i)_t)$ est slc par le lemme \[XXt\]. Pour $l\geqslant 0$, on dispose par d’un isomorphisme $\omega^{[lm]}_{\operatorname{\mathcal{Y}}}{{\ext@arrow 0359\myrightarrowfill@{}{\,\sim\,}}}g_i^*\omega_{\operatorname{\mathcal{X}}_i}^{[lm]}(\sum_E lm \cdot a_{E}(\operatorname{\mathcal{X}}_i)E)$, où la somme porte sur les diviseurs $g_i$-exceptionnels $E$ de $\operatorname{\mathcal{Y}}_t$, et où les $a_{E}(\operatorname{\mathcal{X}}_i)$ sont les discrépances de $\operatorname{\mathcal{X}}_i$. On en déduit un isomorphisme $\omega^{[lm]}_{\operatorname{\mathcal{Y}}}(lm\operatorname{\mathcal{Y}}_t){{\ext@arrow 0359\myrightarrowfill@{}{\,\sim\,}}}(g_i^*\omega_{\operatorname{\mathcal{X}}_i}^{[lm]}(lm(\operatorname{\mathcal{X}}_i)_t))(\sum_E lm \cdot a_{E}(\operatorname{\mathcal{X}}_i)E)$. En le comparant à l’isomorphisme définissant les discrépances de $(\operatorname{\mathcal{X}}_i,(\operatorname{\mathcal{X}}_i)_t)$, on voit que $a_{E}(\operatorname{\mathcal{X}}_i)=a_{E}(\operatorname{\mathcal{X}}_i,(\operatorname{\mathcal{X}}_i)_t)+b_E$ où $b_E$ est la multiplicité de $E$ dans $\operatorname{\mathcal{Y}}_t$. On a donc $a_{E}(\operatorname{\mathcal{X}}_i)\geqslant -1+1=0$ car $(\operatorname{\mathcal{X}}_i,(\operatorname{\mathcal{X}}_i)_t)$ est slc. On en déduit les égalités $$\label{discreptrick}
H^0(\operatorname{\mathcal{X}}_i, \omega_{\operatorname{\mathcal{X}}_i}^{[lm]})=H^0(\operatorname{\mathcal{Y}}, g_i^*\omega_{\operatorname{\mathcal{X}}_i}^{[lm]}(\sum_E lm \cdot a_{E}(\operatorname{\mathcal{X}}_i)E))=H^0(\operatorname{\mathcal{Y}}, \omega_{\operatorname{\mathcal{Y}}}^{[lm]}),$$ où seule la première égalité est à justifier. Que le membre de gauche soit inclus dans celui de droite est une conséquence de la positivité des $a_{E}(\operatorname{\mathcal{X}}_i)$. Pour voir l’autre inclusion, notons $U_i\subset\operatorname{\mathcal{X}}_i$ l’ouvert au-dessus duquel $g_i$ est un isomorphisme. Si $\sigma\in H^0(\operatorname{\mathcal{Y}}, g_i^*\omega_{\operatorname{\mathcal{X}}_i}^{[lm]}(\sum_E lm \cdot a_{E}(\operatorname{\mathcal{X}}_i)E))$, la restriction $\sigma|_{U_i}\in H^0(U_i,\omega_{U_i}^{[lm]})$ se relève à $H^0(\operatorname{\mathcal{X}}_i,\omega_{\operatorname{\mathcal{X}}_i}^{[lm]})$ par propriété $S_2$ de $\operatorname{\mathcal{X}}_i$, car $\operatorname{\mathcal{X}}_i\setminus U_i$ a codimension $\geqslant 2$ dans $\operatorname{\mathcal{X}}_i$.
On conclut en définissant $\phi$ par la chaîne d’isomorphismes naturels suivante, où nous utilisons l’amplitude de $\omega_{\operatorname{\mathcal{X}}_1}^{[m]}$ et $\omega_{\operatorname{\mathcal{X}}_2}^{[m]}$ : $$\operatorname{\mathcal{X}}_1\hspace{-.1em}\simeq \hspace{-.1em}\operatorname{Proj}_T\bigoplus _{l\geqslant 0}\hspace{-.1em}H^0(\operatorname{\mathcal{X}}_1, \omega_{\operatorname{\mathcal{X}}_1}^{[lm]})\hspace{-.1em}\simeq\hspace{-.1em} \operatorname{Proj}_T\bigoplus _{l\geqslant 0} \hspace{-.1em}H^0(\operatorname{\mathcal{Y}}, \omega_{\operatorname{\mathcal{Y}}}^{[lm]})\hspace{-.1em}\simeq \hspace{-.1em}\operatorname{Proj}_T\bigoplus _{l\geqslant 0} \hspace{-.1em}H^0(\operatorname{\mathcal{X}}_2, \omega_{\operatorname{\mathcal{X}}_2}^{[lm]})\hspace{-.1em}\simeq \hspace{-.1em}\operatorname{\mathcal{X}}_2.$$
La preuve ci-dessus fait clairement apparaître le rôle de la condition (v) sur les discrépances dans la définition \[defslc\] : c’est elle qui permet d’identifier les algèbres $m$-canoniques de $\operatorname{\mathcal{X}}_1$ et $\operatorname{\mathcal{X}}_2$, donc $\operatorname{\mathcal{X}}_1$ et $\operatorname{\mathcal{X}}_2$.
Fibre générique normale {#pargennormale}
-----------------------
Passons à l’assertion d’existence dans le théorème \[redstable2\]. On suppose dans ce paragraphe que la variété stable $X$ sur $K$ est normale, donc lc. Ce cas particulier crucial est dû à Hacon et Xu [@HX Corollary 1.5]. Il repose sur le programme des modèles minimaux par le biais de [@HX Theorem 1.1] que nous discuterons plus au §\[finitude\].
### Construction du modèle stable {#modelestable}
Soit $\overline{f}:\overline{\operatorname{\mathcal{X}}}\to T$ un morphisme projectif et plat tel que $\overline{\operatorname{\mathcal{X}}}_\eta\simeq X$. Par le théorème de réduction semi-stable de Kempf, Knudsen, Mumford et Saint-Donat [@KKMS p. 198] (voir le théorème \[KKMS\]) sous la forme plus précise énoncée dans [@KM Theorem 7.17], on peut supposer, quitte à remplacer $R$ par une extension finie et $\overline{\operatorname{\mathcal{X}}}$ par le changement de base normalisé, qu’il existe une modification $\mu:\operatorname{\mathcal{Y}}\to \overline{\operatorname{\mathcal{X}}}$ telle que $\operatorname{\mathcal{Y}}$ soit régulier et $\operatorname{\mathcal{Y}}_t$ soit réduit, et telle que si l’on note $\Delta\subset\operatorname{\mathcal{Y}}$ l’adhérence du lieu exceptionnel de $\mu_{\eta}$, le diviseur $\Delta+\operatorname{\mathcal{Y}}_t$ est à croisements normaux stricts dans $\operatorname{\mathcal{Y}}$. Cette application du théorème de réduction semi-stable est le seul moment où, dans la preuve du théorème \[redstable2\], on doit modifier l’anneau de valuation discrète de base.
Soit $m>0$ un entier tel que $\omega_{X}^{[m]}$ soit inversible. Comme $X$ est à singularités lc, on a $a_{\Delta_i}(X)\geqslant -1$ pour toute composante irréductible $\Delta_i$ de $\Delta$. On déduit, par un argument analogue à celui qui a démontré , que $H^0(X, \omega_X^{[lm]})=H^0(\operatorname{\mathcal{Y}}_{\eta},\omega_{\operatorname{\mathcal{Y}}_{\eta}}^{[lm]}(lm\Delta_{\eta}))$ pour tout $l\geqslant 0$. Comme $\omega_X^{[m]}$ est ample, l’algèbre $m$-canonique $$A_K:=\bigoplus_{l\geqslant 0} H^0(\operatorname{\mathcal{Y}}_\eta,\omega_{\operatorname{\mathcal{Y}}_\eta}^{[lm]}(lm\Delta_\eta))$$ est de type fini sur $K$ et $\operatorname{Proj}(A_K)\simeq X$. Par [@HX Theorem 1.1] (voir le théorème \[thHX\]) appliqué au morphisme $\overline{f}\circ\mu:\operatorname{\mathcal{Y}}\to T$, l’algèbre $m$-canonique $$A:=\bigoplus_{l\geqslant 0} H^0(\operatorname{\mathcal{Y}},\omega_{\operatorname{\mathcal{Y}}}^{[lm]}(lm\Delta+lm\operatorname{\mathcal{Y}}_t))$$ de $(\operatorname{\mathcal{Y}},\Delta+\operatorname{\mathcal{Y}}_t)$ est de type fini sur $R$. On peut donc former le modèle canonique relatif $f:\operatorname{\mathcal{X}}:=\operatorname{Proj}_T A\to T$ de $(\operatorname{\mathcal{Y}},\Delta+\operatorname{\mathcal{Y}}_t)$ au-dessus de $T$. Notons $\phi:\operatorname{\mathcal{Y}}\dashrightarrow\operatorname{\mathcal{X}}$ l’application rationnelle naturelle. On affirme que $f:\operatorname{\mathcal{X}}\to T$ est la famille stable recherchée.
Comme $\operatorname{\mathcal{X}}_\eta\simeq X$, il reste à démontrer que $f$ est stable. C’est le but des §§ \[total\]–\[famille\].
### Étude de l’espace total {#total}
On commence par étudier l’espace total $\operatorname{\mathcal{X}}$ du morphisme $f:\operatorname{\mathcal{X}}\to T$. Pour ce faire, on s’appuie sur des propriétés élémentaires des modèles canoniques relatifs, rassemblées dans [@Kosing Theorem 1.26].
On montre ainsi que $\operatorname{\mathcal{X}}$ est normal, que l’application birationnelle $\phi:\operatorname{\mathcal{Y}}\dashrightarrow\operatorname{\mathcal{X}}$ est une contraction rationnelle[^13] et que, quitte à remplacer $m$ par un multiple, le faisceau $\omega^{[m]}_{\operatorname{\mathcal{X}}}(m\phi_*(\Delta+\operatorname{\mathcal{Y}}_t))$ est inversible et $f$-ample. Comme $X\simeq \operatorname{\mathcal{X}}_{\eta}$ et comme $\Delta_\eta$ est contracté par $\mu_\eta$, on a $\phi_*\Delta=0$. La multiplication par une uniformisante de $R$ induit un isomorphisme $\operatorname{\mathcal{O}}_{\operatorname{\mathcal{X}}}{{\ext@arrow 0359\myrightarrowfill@{}{\,\sim\,}}}\operatorname{\mathcal{O}}_{\operatorname{\mathcal{X}}}(\operatorname{\mathcal{X}}_t)=\operatorname{\mathcal{O}}_{\operatorname{\mathcal{X}}}(\phi_*\operatorname{\mathcal{Y}}_t)$ ; on voit donc que $\omega^{[m]}_{\operatorname{\mathcal{X}}}$ est inversible et $f$-ample. Enfin, une dernière assertion de [@Kosing Theorem 1.26] est que comme la paire $(\operatorname{\mathcal{Y}},\Delta+\operatorname{\mathcal{Y}}_t)$ est à singularités lc, il en va de même pour $(\operatorname{\mathcal{X}},\phi_*(\Delta+\operatorname{\mathcal{Y}}_t))=(\operatorname{\mathcal{X}},\operatorname{\mathcal{X}}_t)$.
### Étude de la fibre spéciale {#speciale}
Il est temps de démontrer que la fibre spéciale $\operatorname{\mathcal{X}}_t$ de $f$ est stable. Il ne reste plus qu’à voir que ses singularités sont slc.
Comme la fibre spéciale $\operatorname{\mathcal{Y}}_t$ de $\operatorname{\mathcal{Y}}$ est réduite et que $\phi$ est une contraction birationnelle, on voit que $\operatorname{\mathcal{X}}_t$ est génériquement réduite. De plus, $\operatorname{\mathcal{X}}_t$ est $S_1$ comme diviseur de Cartier dans $\operatorname{\mathcal{X}}$ qui est normal donc $S_2$. Ces deux faits combinés montrent exactement que $\operatorname{\mathcal{X}}_t$ est réduite. Nous avons vérifié la condition (i) de la définition \[defslc\].
La condition (ii) selon laquelle $\operatorname{\mathcal{X}}_t$ est au plus nodale en codimension $1$ résulte du fait que $(\operatorname{\mathcal{X}},\operatorname{\mathcal{X}}_t)$ est à singularités lc et d’une étude fine des paires à singularités lc en un point de codimension $2$ se situant sur une composante affectée d’un coefficient $1$ du bord [@Kosing Corollary 2.32]. Qu’une telle étude soit possible est à rapprocher du fait que l’on sache classifier les singularités lc des surfaces [@KM §4.1].
Comme $(\operatorname{\mathcal{X}},\operatorname{\mathcal{X}}_t)$ est à singularités lc, il en va a fortiori de même pour $\operatorname{\mathcal{X}}$. Soit $E$ un diviseur au-dessus de $\operatorname{\mathcal{X}}$ dont l’image dans $\operatorname{\mathcal{X}}$ se situe sur la fibre spéciale $\operatorname{\mathcal{X}}_t$. L’inégalité $a_E(\operatorname{\mathcal{X}})\geqslant a_E(\operatorname{\mathcal{X}},\operatorname{\mathcal{X}}_t)+1\geqslant 0$ montre que $\operatorname{\mathcal{X}}_t$ ne contient aucun centre log canonique de $\operatorname{\mathcal{X}}$. Il suit du théorème \[S3\] que $\operatorname{prof}(\operatorname{\mathcal{O}}_{\operatorname{\mathcal{X}},x})\geqslant \min(3, \dim(\operatorname{\mathcal{O}}_{\operatorname{\mathcal{X}},x}))$ pour tout $x\in\operatorname{\mathcal{X}}_t$. Comme $\operatorname{\mathcal{X}}_t$ est un diviseur de Cartier dans $\operatorname{\mathcal{X}}$, on déduit que $\operatorname{prof}(\operatorname{\mathcal{O}}_{\operatorname{\mathcal{X}}_t,x})\geqslant \min(2, \dim(\operatorname{\mathcal{O}}_{\operatorname{\mathcal{X}}_t,x}))$ pour tout $x\in\operatorname{\mathcal{X}}_t$, ce qui est la condition (iii).
Le faisceau $\omega_{\operatorname{\mathcal{X}}}^{[m]}|_{\operatorname{\mathcal{X}}_t}$ est inversible car $\omega_{\operatorname{\mathcal{X}}}^{[m]}$ l’est. Il coïncide donc avec $\omega_{\operatorname{\mathcal{X}}_t}^{[m]}$ car ces deux faisceaux sont $S_2$ et isomorphes en codimension $1$. Ceci démontre que $\omega_{\operatorname{\mathcal{X}}_t}^{[m]}$ est inversible, donc que la condition (iv) est satisfaite.
Enfin, la paire $(\operatorname{\mathcal{X}},\operatorname{\mathcal{X}}_t)$ étant à singularités lc, il en va de même pour $(\widetilde{\operatorname{\mathcal{X}}_t},\operatorname{Diff}_{\widetilde{\operatorname{\mathcal{X}}_t}}(0))$ par adjonction (théorème \[invadj\]). Nous avons vu dans la preuve du lemme \[XXt\] que $\operatorname{Diff}_{\widetilde{\operatorname{\mathcal{X}}_t}}(0)$ est le conducteur de $\operatorname{\mathcal{X}}_t$. On déduit donc de que $\operatorname{\mathcal{X}}_t$ est slc.
### Stabilité de la famille {#famille}
Il reste enfin à démontrer que la famille $f:\operatorname{\mathcal{X}}\to T$ est stable au sens de la définition \[deffam\]. Le morphisme $f$ est plat puisque $\operatorname{\mathcal{X}}$ est réduit et que toutes ses composantes irréductibles dominent $T$. Nous avons déjà montré que ses fibres sont à singularités slc.
Soit $n\in\operatorname{\mathbb{Z}}$. Le faisceau $\omega_{\operatorname{\mathcal{X}}/T}^{[n]}$ est plat car il est $S_1$ par construction et car les composantes irréductibles de son support dominent $T$. Nous avons déjà vu au §\[speciale\] que $\operatorname{\mathcal{X}}_t$ ne contient aucun centre log canonique de $\operatorname{\mathcal{X}}$. Le théorème \[S3\] montre donc que $\operatorname{prof}(\omega^{[n]}_{\operatorname{\mathcal{X}}/T,x})\geqslant \min(3, \dim(\operatorname{\mathcal{O}}_{\operatorname{\mathcal{X}},x}))$ pour tout $x\in\operatorname{\mathcal{X}}_t$. Comme $\operatorname{\mathcal{X}}_t$ est un diviseur de Cartier dans $\operatorname{\mathcal{X}}$, on déduit que $\omega^{[n]}_{\operatorname{\mathcal{X}}/T}|_{\operatorname{\mathcal{X}}_t}$ est $S_2$. Les deux faisceaux $\omega^{[n]}_{\operatorname{\mathcal{X}}/T}|_{\operatorname{\mathcal{X}}_t}$ et $\omega^{[n]}_{\operatorname{\mathcal{X}}_t}$ sont $S_2$ et isomorphes en codimension $1$ ; ils coïncident donc. Cela entraîne la stabilité de la famille $f$ et achève la preuve du théorème \[redstable2\] quand $X$ est normale.
Fibre générique non normale {#nonnormale}
---------------------------
Expliquons maintenant l’énoncé d’existence du théorème \[redstable2\] dans le cas général.
### Normalisation {#techglue}
La preuve, due à Kollár [@Kobook], procède par réduction au cas normal. On applique le théorème de réduction stable à la normalisation de $X$ qui est justiciable des arguments du §\[pargennormale\], et on construit $f:\operatorname{\mathcal{X}}\to T$ en *recollant* le modèle stable obtenu le long de lui-même pour faire apparaître les singularités non normales requises. L’étape de recollement est surprenamment délicate à mettre en œuvre et constitue le cœur du livre [@Kosing]. Expliquons son principe.
Soit $X$ une variété à singularités slc. Notons $\pi:\widetilde{X}\to X$ sa normalisation et $\Gamma$ son conducteur. On sait par que la paire $(\widetilde{X},\Gamma)$ est lc. Le morphisme $\pi|_{\Gamma}:\Gamma\to\pi(\Gamma)$ est de degré deux au-dessus de l’ouvert dense de $\pi(\Gamma)$ le long duquel $X$ est à croisements normaux doubles. On en déduit une involution rationnelle $\tau:\Gamma\dashrightarrow\Gamma$, qui s’étend en une involution régulière $\tau:\widetilde{\Gamma}\to\widetilde{\Gamma}$ génériquement sans point fixe de la normalisation $\nu:\widetilde{\Gamma}\to\Gamma$ de $\Gamma$. Géométriquement, $\tau$ échange les deux branches des singularités à croisements normaux doubles de $X$. En comparant l’équation définissant la différente et son pull-back par $\tau$, on voit que le $\operatorname{\mathbb{Q}}$-diviseur $\operatorname{Diff}_{\widetilde{\Gamma}}(0)$ de $\widetilde{\Gamma}$ est $\tau$-invariant.
Donnons deux exemples de ces constructions. Si $X=\{x^2=yz^2\}\subset \operatorname{\mathbb{A}}_k^3$ est le parapluie de Whitney, on a $\widetilde{X}=\operatorname{\mathbb{A}}^2_k$, la normalisation $\pi:\widetilde{X}\to X$ est donnée par $(u,v)\mapsto(uv,v^2,u)$, on a $\Gamma=\widetilde{\Gamma}=\{u=0\}\subset \operatorname{\mathbb{A}}_k^2$ et $\tau:v\mapsto -v$, et $\operatorname{Diff}_{\widetilde{\Gamma}}(0)=0$.
Si $X=\{xyz=0\}\subset \operatorname{\mathbb{A}}_k^3$ est le point à croisements normaux triples, la normalisation $\widetilde{X}$ est une union disjointe de trois espaces affines de dimension $2$, le conducteur $\Gamma$ est l’union de leurs axes de coordonnées, de sorte que $\widetilde{\Gamma}$ est une union de six droites, et on calcule que $\operatorname{Diff}_{\widetilde{\Gamma}}(0)\subset\widetilde{\Gamma}$ est la réunion de leurs origines. L’involution $\tau:\widetilde{\Gamma}\to\widetilde{\Gamma}$ échange ces droites deux par deux. Dans cet exemple, l’involution rationnelle $\tau:\Gamma\dashrightarrow\Gamma$ n’est pas régulière en les trois points singuliers de $\Gamma$.
Kollár a remarqué que, sous des hypothèses appropriées, on peut construire la variété $X$ à partir des données $(\widetilde{X},\Gamma,\tau)$. Un exemple prototypique (qui n’est pas l’énoncé précis dont on aura besoin pour la preuve du théorème \[redstable2\]) est [@Kosing Theorem 5.13].
\[bij\] Les constructions ci-dessus induisent une bijection $$\left\{\hspace{-.5em}
\begin{array}{c}
\mbox{Classes d'isomorphisme}\\
\mbox{de vari\'et\'es stables $X$}
\end{array}
\hspace{-.3em}\right\}\hspace{-.2em}{{\ext@arrow 0359\myrightarrowfill@{}{\,\sim\,}}}\hspace{-.2em}\left\{\hspace{-.6em}
\begin{array}{c}
\mbox{Classes d'isomorphisme de paires lc stables}\\
\mbox{ $(\widetilde{X},\Gamma)$ munies d'une involution g\'en\'eriquement }\\
\mbox{ sans point fixe $\tau$ de $(\widetilde{\Gamma},\operatorname{Diff}_{\widetilde{\Gamma}}(0))$}
\end{array}
\hspace{-.5em}\right\}.$$
### Étapes du recollement {#recoller}
Il est facile de voir que l’application du théorème \[bij\] est injective [@Kosing Proposition 5.3]. C’est sa surjectivité qui est difficile. Le triplet $(\widetilde{X}, \Gamma,\tau)$ étant donné, il s’agit de construire $X$ en recollant $\widetilde{X}$ sur elle-même le long de $\Gamma$ de la manière indiquée par $\tau$. Plutôt que d’expliquer la démonstration, dont la structure inductive est complexe, décrivons les difficultés qu’il faut surmonter, qui correspondent aussi aux étapes de la preuve du théorème \[bij\].
\(i) On souhaite construire $X$ comme quotient de $\widetilde{X}$ par la relation d’équivalence qui identifie $\nu(x)$ et $\nu(\tau(x))$ pour tout point géométrique $x\in\widetilde{\Gamma}$. Comme le morphisme $\pi:\widetilde{X}\to X$ à construire est fini, il faut que la relation d’équivalence engendrée par ces relations ait des classes d’équivalence finies. Ce n’est pas du tout une évidence !
Par exemple, prenons $\widetilde{X}=\operatorname{\mathbb{A}}^3_k$ et $\Gamma=\{xy=0\}$ de sorte que $\widetilde{\Gamma}$ est l’union de deux plans affines de coordonnées respectives $(y_1,z_1)$ et $(x_2,z_2)$ et que $\operatorname{Diff}_{\widetilde{\Gamma}}(0)$ est l’union des deux droites d’équations $\{y_1=0\}$ et $\{x_2=0\}$. Définissons une involution $\tau$ échangeant ces deux plans par l’équation $\tau(y_1,z_1)=(x_2,z_2+1)$. Pour ces choix de $(\widetilde{X}, \Gamma,\tau)$, on voit que les points $(0,0,n)\in\widetilde{X}$ pour $n\in\operatorname{\mathbb{Z}}$ sont tous équivalents.
Dans le cadre du théorème \[bij\], ce sont les hypothèses globales de projectivité de $\widetilde{X}$ et d’amplitude de $\omega_{\widetilde{X}}(\Gamma)$ qui assureront la finitude de ces classes d’équivalences [@Kosing Corollary 5.37]. La preuve de ce fait repose en dernier lieu sur des résultats de finitude pour des groupes d’automorphismes birationnels de paires dont le fibré canonique a des propriétés de positivité [@Kosing Corollary 10.69].
\(ii) Supposons le problème décrit en (i) résolu. On dispose alors d’une relation d’équivalence finie sur $\widetilde{X}$ dont on souhaite construire le quotient comme variété algébrique. Ce serait la variété $X$ recherchée. Il n’est malheureusement pas du tout évident que ce soit possible.
Donnons un exemple en suivant [@Kosing Example 9.7]. Soient $\widetilde{X}$ l’union de deux espaces affines de dimension $3$, de coordonnées respectives $(x_1,y_1,z_1)$ et $(x_2,y_2,z_2)$, et $\Gamma\subset\widetilde{X}$ défini par les équations $\{y_i^3=z_i^2\}$ pour $i\in\{1,2\}$. La normalisation $\widetilde{\Gamma}$ de $\Gamma$ est une union de deux plans affines, de coordonnées respectives $(u_1,v_1)$ et $(u_2,v_2)$, et le morphisme $\nu:\widetilde{\Gamma}\to\Gamma$ est donné par $(u_i,v_i)\mapsto (u_i,v_i^2,v_i^3)$. Choisissons pour $\tau$ l’involution échangeant ces deux plans, définie par la formule $\tau(u_1,v_1)=(u_1+v_1,v_1)$.
On vérifie aisément que la relation d’équivalence engendrée par $\nu(x)\sim\nu(\tau(x))$ est finie, de sorte que le problème soulevé en (i) n’apparaît pas. De plus, cette relation d’équivalence admet bien un quotient catégorique dans la catégorie des $k$-schémas : le spectre de la sous-$k$-algèbre de $k[x_1,y_1,z_1]\times k[x_2,y_2,z_2]$ engendrée par les idéaux $\langle y_1,z_1\rangle$ et $\langle y_2,z_2\rangle$. Cette algèbre n’est pas de type fini sur $k$ (ni même noethérienne). Le quotient de $\widetilde{X}$ par la relation d’équivalence considérée n’est donc pas une variété.
Le problème avec cet exemple est que la paire $(\widetilde{X},\Gamma)$ n’est pas lc. C’est seulement sous l’hypothèse que les singularités de $(\widetilde{X},\Gamma)$ sont lc que Kollár montre l’existence du quotient $X$ recherché [@Kosing Theorem 5.32]. Cette hypothèse est utilisée de la manière suivante. La variété $X$ est obtenue par un procédé inductif qui consiste, en simplifiant, à d’abord construire les quotients des centres log canoniques de $(\widetilde{X},\Gamma)$, en commençant par ceux qui ont dimension minimale. Pour ce faire, on utilise de manière essentielle des propriétés de seminormalité des centres log canoniques [@Kosing §4.20], qui permettent en un sens de les manipuler topologiquement. Dans l’exemple ci-dessus, c’est le défaut de seminormalité de $\Gamma$ qui pose véritablement problème.
\(iii) Maintenant que la variété $X$ est construite, il faut vérifier qu’elle a les propriétés requises. Si la plupart sont faciles à vérifier, l’existence d’un entier $m>0$ tel que $\omega_X^{[m]}$ soit inversible est hautement non triviale. À nouveau, illustrons-le sur un exemple.
Soient $\widetilde{X}$ l’union disjointe de trois plans affines de coordonnées $(x_1,y_1)$, $(x_2,y_2)$ et $(x_3,y_3)$, et $\Gamma\subset\widetilde{X}$ le diviseur défini par les équations $\{y_1=0\}$, $\{x_2y_2=0\}$ et $\{x_3=0\}$. La normalisation $\widetilde{\Gamma}$ de $\Gamma$ est une union disjointe de quatre droites affines de coordonnées respectives $x_1$, $x_2$, $y_2$ et $y_3$. Choisissons pour $\tau$ l’involution de $\widetilde{\Gamma}$ échangeant les deux premières droites par la formule $\tau(x_1)=x_2$, et les deux dernières par $\tau(y_2)=y_3$. Aucun des problèmes décrits en (i) et (ii) ne se pose et l’on peut donc considérer la variété $X$ quotient de $\widetilde{X}$ par la relation d’équivalence engendrée par $\nu(x)\sim\nu(\tau(x))$. Avec les notations de , on a $X=C(Y,L)$, où $Y$ est une chaîne de trois droites projectives et où le fibré en droites ample $L$ sur $Y$ a degré $1$ sur chacune de ces trois composantes. On vérifie alors en adaptant [@Kosing Proposition 3.14 (4)] que $\omega_X^{[m]}$ n’est inversible pour aucun $m>0$.
Pour expliquer cela, remarquons que la différente $\operatorname{Diff}_{\widetilde{\Gamma}}(0)\subset \widetilde{\Gamma}$ est l’union des origines de la deuxième et de la troisième composante de $\widetilde{\Gamma}$. On voit donc que $\tau$ ne préserve pas $\operatorname{Diff}_{\widetilde{\Gamma}}(0)$. Dans la preuve du théorème \[bij\], c’est l’hypothèse que $\operatorname{Diff}_{\widetilde{\Gamma}}(0)$ soit $\tau$-invariant qui assure l’existence d’un $m>0$ tel que $\omega_X^{[m]}$ est inversible [@Kosing Theorem 5.38]. Cette hypothèse est utilisée comme suit. Soit $m>0$ tel que $\omega^{[m]}_{\widetilde{X}}(m\Gamma)$ est inversible. On souhaite descendre $\omega^{[m]}_{\widetilde{X}}(m\Gamma)$ (ou une de ses puissances) en un faisceau inversible sur $X$, isomorphe à $\omega_X^{[m]}$ (ou à une de ses puissances). Pour ce faire, on raisonne géométriquement en considérant l’espace total $\widetilde{p}:\widetilde{L}\to\widetilde{X}$ du fibré en droites associé à $\omega^{[m]}_{\widetilde{X}}(m\Gamma)$ sur $\widetilde{X}$. Notons $\Delta:=\widetilde{p}^{-1}(\Gamma)$. Le fait que la différente $\operatorname{Diff}_{\widetilde{\Gamma}}(0)$ soit $\tau$-invariante implique que $\tau$ se relève naturellement en une involution $\sigma$ de la normalisation $\widetilde{\Delta}$ de $\Delta$. On peut alors appliquer les étapes (i) et (ii) de la technique de recollement au triplet $(\widetilde{L},\Delta,\sigma)$. On construit de la sorte une variété $p:L\to X$ qu’on vérifie être (quitte à remplacer $m$ par un multiple) le fibré en droites associé à $\omega^{[m]}_X$. Cela implique en particulier que $\omega^{[m]}_X$ est inversible, ce qu’on désirait montrer.
### Réduction stable {#stableslc}
Expliquons maintenant, en suivant [@Kobook], comment la méthode de recollement est utilisée pour démontrer l’assertion d’existence dans le théorème \[redstable2\].
Soit $X$ une variété stable sur $K$. Notons $\widetilde{X}$ sa normalisation, $\Gamma$ son conducteur et $\tau:\widetilde{\Gamma}\to \widetilde{\Gamma}$ l’involution naturelle, qui préserve $\operatorname{Diff}_{\widetilde{\Gamma}}(0)$. Le théorème de réduction stable pour les variétés normales (voir §\[pargennormale\]), convenablement étendu au cas des paires, montre que $(\widetilde{X},\Gamma)$ admet un modèle stable $\widetilde{f}:(\widetilde{\operatorname{\mathcal{X}}},\mathit{\Gamma})\to T$ sur $T$. Le lemme \[XXt\], adapté au cas des paires, montre que la paire $(\widetilde{\operatorname{\mathcal{X}}},\mathit{\Gamma}+\widetilde{\operatorname{\mathcal{X}}}_t)$ est lc, et on déduit donc de l’adjonction (théorème \[invadj\]) que $(\widetilde{\mathit{\Gamma}},\operatorname{Diff}_{\widetilde{\mathit{\Gamma}}}(\widetilde{\operatorname{\mathcal{X}}}_t))=(\widetilde{\mathit{\Gamma}},\operatorname{Diff}_{\widetilde{\mathit{\Gamma}}}(0)+\widetilde{\mathit{\Gamma}}_t)$ est lc. De plus, pour $m>0$ bien choisi, $\omega^{[m]}_{\widetilde{\mathit{\Gamma}}}(m\operatorname{Diff}_{\widetilde{\mathit{\Gamma}}}(0))=\omega^{[m]}_{\widetilde{\operatorname{\mathcal{X}}}}(m\mathit{\Gamma})|_{\widetilde{\mathit{\Gamma}}}$ est un faisceau inversible ample relativement à $T$. Argumentant comme aux §§\[speciale\]–\[famille\], on voit que $(\widetilde{\mathit{\Gamma}},\operatorname{Diff}_{\widetilde{\mathit{\Gamma}}}(0))\to T$ est une famille stable. Par l’énoncé d’unicité dans le théorème de réduction stable (voir §\[unicite\]), convenablement étendu au cas des paires, l’involution $\tau$ sur la fibre générique s’étend en une involution encore notée $\tau:(\widetilde{\mathit{\Gamma}},\operatorname{Diff}_{\widetilde{\mathit{\Gamma}}}(0))\to (\widetilde{\mathit{\Gamma}},\operatorname{Diff}_{\widetilde{\mathit{\Gamma}}}(0))$. On applique alors la technique de recollement décrite au §\[recoller\][^14] au triplet $(\widetilde{\operatorname{\mathcal{X}}},\mathit{\Gamma},\tau)$, ce qui donne lieu à un morphisme $f:\operatorname{\mathcal{X}}\to T$, dont on vérifie qu’elle est la famille stable recherchée.
La preuve que nous venons de décrire ne permet pas de se limiter aux variétés stables qui sont Cohen-Macaulay. En effet, la normalisation $\widetilde{X}$ d’une variété stable Cohen-Macaulay $X$ peut ne pas être elle-même Cohen-Macaulay [@Kosurvey Example 23].
Finitude de l’algèbre canonique {#finitude}
-------------------------------
Revenons sur le théorème de Hacon et Xu que nous avons utilisé au §\[pargennormale\], et qui est un ingrédient décisif de la preuve du théorème \[redstable\].
Soit $(X,\Delta)$ une paire à singularités slc. Définissons l’**algèbre canonique** de $(X,\Delta)$ comme étant $A(X,\Delta):=\bigoplus_{\l\geqslant 0} H^0(X,\omega_X^{[l]}(\left \lfloor{l\Delta}\right \rfloor))$, où $\left \lfloor{l\Delta}\right \rfloor$ est le diviseur de Weil sur $X$ obtenu en arrondissant les coefficients de $l\Delta$ à l’entier inférieur. On conjecture (voir par exemple [@FG Conjecture A]) la propriété suivante.
\[conjfin\] L’algèbre canonique d’une paire projective lc est de type fini.
Les premiers résultats concernant la conjecture \[conjfin\] en dimension arbitraire ont été obtenus par Birkar, Cascini, Hacon et McKernan [@BCHM]. Ils la résolvent en particulier pour les paires de type général[^15] à singularités klt. Ce travail remarquable a déjà fait l’objet d’un exposé dans ce séminaire [@Druel] et est à la base des développements ultérieurs.
De manière surprenante, la conjecture \[conjfin\] tombe en défaut pour les variétés slc : Kollár a donné un exemple de surface projective slc qui est de type général mais dont l’algèbre canonique n’est pas de type fini [@Kotwo Proposition 1]. Les singularités slc se comportent donc moins bien vis-à-vis du programme des modèles minimaux que leurs homologues normales que sont les singularités klt, lc... C’est pour cette raison que nous avons dû traiter séparément, dans la preuve du théorème de réduction stable, les variétés normales au §\[pargennormale\] et les variétés non normales au §\[nonnormale\]. Le théorème de Hacon et Xu constitue un progrès sur ces questions dans le cas lc, dans un contexte adapté à la preuve du théorème \[redstable2\] : ils travaillent dans une situation relative, et supposent connue l’existence du modèle canonique de la fibre générique. Énonçons le cas particulier[^16] de [@HX Theorem 1.1] qui nous a été utile.
\[thHX\] Soit $f:\operatorname{\mathcal{X}}\to T$ un morphisme projectif, avec $\operatorname{\mathcal{X}}$ régulier et $\Delta\subset \operatorname{\mathcal{X}}$ un diviseur à croisements normaux stricts. Supposons que $(\operatorname{\mathcal{X}}_\eta,\Delta_\eta)$ est de type général. Si $A(\operatorname{\mathcal{X}}_\eta,\Delta_\eta)$ est de type fini sur $K$, alors $A(\operatorname{\mathcal{X}},\Delta)$ est de type fini sur $R$.
La preuve utilise de manière cruciale les techniques de [@BCHM]. Nous nous contentons d’en décrire la structure.
Après avoir peut-être remplacé $(\operatorname{\mathcal{X}},\Delta)$ par un modèle birationnel (un modèle minimal, construit en adaptant les techniques de [@BCHM]), on souhaite démontrer qu’il existe un entier $m>0$ tel que $\omega^{[m]}_{\operatorname{\mathcal{X}}}(m\Delta)$ est inversible et sans point base : ceci entraîne en effet la finitude de l’algèbre canonique. Il est bien sûr nécessaire de savoir démontrer que la restriction $\omega^{[m]}_{\operatorname{\mathcal{X}}}(m\Delta)|_{\Delta}$ est elle-même sans point base, et un théorème d’extension dû à Fujino [@Fubpf Theorem 1.1] montre que cela serait en fait suffisant.
On voudrait obtenir cette information dans le cadre d’une récurrence sur la dimension. Malheureusement, $\Delta$ n’est en général pas normal : il a seulement des singularités slc et on ne peut lui appliquer l’hypothèse de récurrence. L’idée de Hacon et Xu est de plutôt appliquer l’hypothèse de récurrence à la normalisation de $\Delta$, puis de redescendre l’information obtenue à $\Delta$ à l’aide de la technique de recollement de Kollár que nous avons décrite aux §§\[techglue\]–\[recoller\].
Construction de l’espace de modules {#consedm}
===================================
Cette section est consacrée à la preuve du théorème \[thedm\]. Si la stratégie est connue depuis longtemps [@KSB; @Kocomplete; @Viehweg], beaucoup de détails cruciaux n’ont été mis au point que très récemment [@KoHH; @Kosing; @HX; @HMX; @Fujino]. Le lecteur pourra consulter avec profit les textes de survol [@YPG; @Kosurvey], ainsi que le livre [@Kobook] pour une présentation détaillée.
La démonstration exploite à nouveau les plongements pluricanoniques des variétés stables. Ils permettent de paramétrer les variétés stables de fonction de Hilbert fixée par une union de sous-schémas localement fermés d’un schéma de Hilbert (§§\[borne\]–\[repr\]) qu’on quotiente ensuite par le groupe des transformations projectives pour construire l’espace de modules recherché (§§\[champ\]–\[grossier\]).
Caractère borné {#borne}
---------------
Fixons une fonction de Hilbert $F:\operatorname{\mathbb{Z}}\to\operatorname{\mathbb{Z}}$. La première étape de la construction de $\overline{M}_F$ est la recherche d’une famille de variétés projectives, dans laquelle toutes les variétés stables de fonction de Hilbert $F$ apparaissent et qui soit *bornée* au sens où la base de la famille est elle-même une variété, donc de type fini sur le corps de base $k$. Pour cela, il suffit de démontrer l’énoncé suivant.
\[thborne\] Il existe un entier $m$ tel que pour toute variété stable $X$ de fonction de Hilbert $F$, le faisceau $\omega_X^{[m]}$ est inversible, très ample et sans cohomologie supérieure.
En effet, la famille universelle $g:\operatorname{\mathcal{Y}}\to H$ au-dessus du schéma de Hilbert $H$ paramétrant les sous-schémas fermés de $\operatorname{\mathbb{P}}_{k}^{F(m)-1}$ de fonction de Hilbert $n\mapsto F(nm)$ a alors les propriétés requises.
En restriction aux variétés lisses, le théorème \[thborne\] est un cas particulier du grand théorème de Matsusaka [@Matsusaka]. Le cas des courbes est facile (on peut prendre $m=3$) et c’est Alexeev qui a résolu le cas des surfaces [@Ale]. En général, le théorème \[thborne\] a été démontré par Hacon, McKernan et Xu [@HMX]. On ne donnera ici aucune indication sur sa preuve, qui repose sur le programme des modèles minimaux : on renvoie le lecteur au texte de survol [@HMX2].
Représentabilité de la stabilité {#repr}
--------------------------------
La seconde étape de la preuve consiste à isoler, dans le schéma de Hilbert $H$ construit au §\[borne\], le lieu paramétrant des variétés stables $m$-canoniquement plongées de fonction de Hilbert $F$.
\[threpr\] Il existe un morphisme $\iota:H'\to H$ tel que le changement de base $g':\operatorname{\mathcal{Y}}'\to H'$ de $f$ par $\iota$ soit une famille de variétés stables $m$-canoniquement plongées de fonction de Hilbert $F$, et qui soit universel pour cette propriété.
On procède par étapes, en effectuant plusieurs changements de base successifs, chacun améliorant les propriétés du morphisme $g:\operatorname{\mathcal{Y}}\to H$. Remarquons que $g$ est déjà plat par définition du schéma de Hilbert.
### {#deminormal}
On commence par remplacer $H$ par l’ouvert $H_1\subset H$ paramétrant des variétés réduites, $S_2$ et équidimensionnelles [@EGA43 Théorème 12.2.1], et dont les singularités en codimension $1$ sont au plus des croisements normaux doubles (pour ce dernier point, on remarque que cette condition est équivalente à avoir des singularités au plus nodales aux points de codimension $1$ et on utilise le fait que les déformations des singularités nodales sont au plus nodales ; voir [@Kosing §1.41.2] pour un énoncé précis).
Notant $g_1:\operatorname{\mathcal{Y}}_1\to H_1$ le changement de base, on peut alors définir le faisceau canonique relatif $\omega_{\operatorname{\mathcal{Y}}_1/H_1}$ et ses puissances réflexives $\omega_{\operatorname{\mathcal{Y}}_1/H_1}^{[n]}$ pour $n\in\operatorname{\mathbb{Z}}$, comme au §\[pardeffamst\].
### {#ssHH}
La seconde étape consiste à assurer que les faisceaux $\omega_{\operatorname{\mathcal{Y}}_1/H_1}^{[n]}$ soient plats de formation commutant à tout changement de base, pour tout $1\leqslant n\leqslant m$. Cette étape est cruciale si l’on souhaite munir $\overline{M}_F$ d’une structure schématique raisonnable. Elle a été entièrement clarifiée par Kollár [@KoHH] ; d’autres approches avaient été proposées par Hassett et Kovács [@HaKo][^17] et par Abramovich et Hassett [@AbraHa].
Kollár construit, pour $1\leqslant n\leqslant m$, des décompositions $\operatorname{Hull}(\omega_{\operatorname{\mathcal{Y}}_1/H_1}^{[n]})\to H_1$ de $H_1$ en sous-schémas localement fermés au-dessus desquels les faisceaux cohérents $\omega_{\operatorname{\mathcal{Y}}_{1,s}}^{[n]}$ pour $s\in H_1$ s’organisent en une famille plate, et qui sont universelles pour cette propriété. Il ne reste plus qu’à définir $H_2:=\operatorname{Hull}(\omega_{\operatorname{\mathcal{Y}}_1/H_1}^{[1]})\times_{H_1}\dots\times_{H_1}\operatorname{Hull}(\omega_{\operatorname{\mathcal{Y}}_1/H_1}^{[m]})$ comme étant la décomposition de $H_1$ en sous-schémas localement fermés qui les raffine toutes, et à considérer le morphisme $g_2:\operatorname{\mathcal{Y}}_2\to H_2$ obtenu par changement de base.
Un des attraits du point de vue de Kollár est sa généralité : il n’utilise pas de propriétés particulières des variétés stables, ni des faisceaux pluricanoniques. Il considère plutôt un morphisme projectif $p:X\to S$ arbitraire et un faisceau cohérent $\operatorname{\mathcal{F}}$ sur $X$. Dans cette situation, il construit une décomposition $\operatorname{Hull}(\operatorname{\mathcal{F}})\to S$ de $S$ en sous-schémas localement fermés au-dessus de laquelle les hulls $\operatorname{\mathcal{F}}_s^{[**]}$ des $\operatorname{\mathcal{F}}_s$ pour $s\in S$ s’organisent en une famille plate, et qui est universelle pour cette propriété [@KoHH Theorem 21]. Si $X_s$ est $S_2$ et $\operatorname{Supp}(\operatorname{\mathcal{F}}_s)=X_s$, le hull $\operatorname{\mathcal{F}}_s^{[**]}$ n’est autre que le double dual $\operatorname{\mathcal{F}}_s^{**}$ de $\operatorname{\mathcal{F}}_s$ (voir [@KoHH Definition 14] ou ci-dessous pour une définition générale des hulls). Ceci s’applique dans la situation que nous avons considérée. Avec les notations ci-dessus, on a donc $(\omega_{\operatorname{\mathcal{Y}}_1/H_1, s}^{[n]})^{[**]}=(\omega_{\operatorname{\mathcal{Y}}_1/H_1, s}^{[n]})^{**}=\omega_{\operatorname{\mathcal{Y}}_1,s}^{[n]}$ et la décomposition $\operatorname{Hull}(\omega_{\operatorname{\mathcal{Y}}_1/H_1}^{[n]})$ a bien les propriétés voulues.
Pour construire $\operatorname{Hull}(\operatorname{\mathcal{F}})$, Kollár en identifie une compactification naturelle : l’espace de modules $\operatorname{QHusk}(\operatorname{\mathcal{F}})$ des husks quotients cohérents de $\operatorname{\mathcal{F}}$. Un **husk quotient** cohérent de $\operatorname{\mathcal{F}}$ relativement à $S$ est un morphisme de faisceaux cohérents $q:\operatorname{\mathcal{F}}\to\operatorname{\mathcal{G}}$ sur $X$ où $\operatorname{\mathcal{G}}$ est $f$-plat, tel que pour tout $s\in S$, le faisceau $\operatorname{\mathcal{G}}_s$ sur $X_s$ est pur et $q$ est surjectif aux points génériques du support de $\operatorname{\mathcal{G}}_s$ [@KoHH Definition 9]. Ce husk quotient est un **hull** si pour tout $s\in S$, le morphisme $q_s$ est un isomorphisme aux points génériques du support de $\operatorname{\mathcal{G}}_s$, est surjectif aux points de codimension $1$ du support de $\operatorname{\mathcal{G}}_s$ et est maximal pour ces propriétés [@KoHH Definition 17]. Par des techniques inspirées de la construction des schémas $\operatorname{Quot}$ de Grothendieck, on démontre [@KoHH Theorem 10] que le foncteur qui à un $S$-schéma $T$ associe l’ensemble des hulls quotients cohérents $q:\operatorname{\mathcal{F}}_T\to \operatorname{\mathcal{G}}$ de $\operatorname{\mathcal{F}}_T$ relativement à $T$ est représentable par une union dénombrable d’espaces algébriques propres sur $S$, qu’on note $\operatorname{QHusk}(\operatorname{\mathcal{F}})$ : c’est l’espace de modules $\operatorname{QHusk}(\operatorname{\mathcal{F}})$ des husks quotients cohérents de $\operatorname{\mathcal{F}}$. On vérifie enfin que le sous-foncteur des hulls est représentable par un ouvert $\operatorname{Hull}(\operatorname{\mathcal{F}})$ de $\operatorname{QHusk}(\operatorname{\mathcal{F}})$ et que $\operatorname{Hull}(\operatorname{\mathcal{F}})$ est une décomposition de $S$ en sous-schémas localement fermés [@KoHH Theorem 21].
### {#inversible}
On remplace ensuite $H_2$ par l’ouvert $H_3\subset H_2$ au-dessus duquel $\omega_{\operatorname{\mathcal{Y}}_2/H_2}^{[m]}$ est un faisceau inversible. Pour construire un tel ouvert, on procède comme suit.
Par le lemme de Nakayama, l’ensemble $\{y\in\operatorname{\mathcal{Y}}_2| \dim_{\kappa(y)}(\omega_{\operatorname{\mathcal{Y}}_2/H_2}^{[m]}\otimes \kappa(y))>1\}$ est fermé (voir [@Hartshorne III Example 12.7.2]). On note $H_3\subset H_2$ le complémentaire de son image dans $H_2$ et $g_3:\operatorname{\mathcal{Y}}_3\to H_3$ le changement de base. Montrons que $\operatorname{\mathcal{F}}:=\omega_{\operatorname{\mathcal{Y}}_3/H_3}^{[m]}$ est inversible. Soit $y\in\operatorname{\mathcal{Y}}_3$ et soit $s\in \operatorname{\mathcal{F}}_{y}$ engendrant $\operatorname{\mathcal{F}}_{y}\otimes \kappa(y)$. Par le lemme de Nakayama, le morphisme $\operatorname{\mathcal{O}}_{\operatorname{\mathcal{Y}}_3,y}\xrightarrow{s}\operatorname{\mathcal{F}}_y$ est surjectif ; on note $\operatorname{\mathcal{N}}$ son noyau. La suite courte $$0\to \operatorname{\mathcal{N}}\otimes\kappa(g_3(y))\to\operatorname{\mathcal{O}}_{\operatorname{\mathcal{Y}}_3,y}\otimes\kappa(g_3(y))\to\operatorname{\mathcal{F}}_y\otimes\kappa(g_3(y))\to 0$$ est exacte car $\operatorname{\mathcal{F}}_y$ est $\operatorname{\mathcal{O}}_{H_3,g_3(y)}$-plat par §\[ssHH\]. Comme le support ensembliste de $\operatorname{\mathcal{F}}$ est $\operatorname{\mathcal{Y}}_3$ et que $\operatorname{\mathcal{O}}_{\operatorname{\mathcal{Y}}_3,y}\otimes\kappa(g_3(y))$ est réduit, on déduit que $\operatorname{\mathcal{N}}\otimes\kappa(g_3(y))=0$. A fortiori, $\operatorname{\mathcal{N}}\otimes\kappa(y)=0$, donc $\operatorname{\mathcal{N}}=0$ par le lemme de Nakayama. On a bien montré que $\operatorname{\mathcal{F}}$ est inversible en $y$.
On voit maintenant en écrivant $n=am+b$ avec $1\leqslant b\leqslant m$ que $\omega^{[n]}_{\operatorname{\mathcal{Y}}_3/H_3}$ est plat de formation commutant à tout changement de base, pour tout $n\in \operatorname{\mathbb{Z}}$.
### {#ouvertslc}
On se restreint alors à l’ouvert $H_4\subset H_3$ le long duquel les fibres de $g_3$ sont à singularités slc (et on note $g_4:\operatorname{\mathcal{Y}}_4\to H_4$ le changement de base). L’existence d’un tel ouvert est démontrée dans [@AbraHa Proposition A.1.1], où la preuve est attribuée à Alexeev. L’argument repose crucialement sur l’inversion de l’adjonction (voir § \[paradj\]).
Voir que le lieu $\{x\in H_3|\operatorname{\mathcal{Y}}_{3,x}\textrm{\hspace{.2em}a des singularit\'es slc}\}$ est un sous-ensemble constructible de $H_3$ est aisé. En effet, si $\eta$ est le point générique d’une composante irréductible de $H_3$, une log résolution de $\operatorname{\mathcal{Y}}_{3,\eta}$ s’étend en une log résolution des fibres de $g_3$ au-dessus d’un voisinage $U$ de $\eta$ dans la variété réduite $H_3^{\operatorname{red}}$. En calculant les discrépances des fibres de $g_3$ au-dessus de $U$ sur ces log résolutions, on montre que l’une est slc si et seulement si les autres le sont. On conclut par récurrence noethérienne.
Il reste à démontrer que ce lieu est stable par générisation. On se ramène à une situation relative sur le spectre $T$ d’un anneau de valuation discrète, de point fermé $t$ et de point générique $\eta$. On dispose d’un morphisme propre et plat $f:\operatorname{\mathcal{X}}\to T$ dont les fibres satisfont les conditions (i)-(iv) de la définition \[defslc\], tel que $\operatorname{\mathcal{X}}_t$ est slc et $\omega_{\operatorname{\mathcal{X}}/T}^{[m]}$ inversible. Le lemme \[XXt\], dont la preuve reposait sur l’inversion de l’adjonction, assure que $(\operatorname{\mathcal{X}},\operatorname{\mathcal{X}}_t)$ est slc, donc que $\operatorname{\mathcal{X}}_\eta$ est slc.
### {#section}
On dispose de deux fibrés en droites naturels sur $\operatorname{\mathcal{Y}}_4$ : le faisceau $m$-canonique $\omega_{\operatorname{\mathcal{Y}}_4/H_4}^{[m]}$, qui est inversible par le §\[inversible\], et le fibré tautologique $\operatorname{\mathcal{O}}_{\operatorname{\mathcal{Y}}_4/H_4}(1)$ induit par $\operatorname{\mathcal{O}}_{\operatorname{\mathbb{P}}^{F(m)-1}_k}(1)$. On souhaite maintenant se restreindre au sous-schéma localement fermé $H_5\subset H_4$ au-dessus duquel ces deux fibrés en droites coïncident, Zariski-localement sur $H_5$. L’existence d’un tel sous-schéma n’est pas évidente, car comme les fibres de $g_4$ peuvent ne pas être irréductibles, le foncteur de Picard $\operatorname{Pic}_{\operatorname{\mathcal{Y}}_4/H_4}$ pourrait ne pas être séparé. Par conséquent, $H_5$ pourrait ne pas être fermé dans $H_4$.
Par [@EGA32 Corollaire 7.8.7], le faisceau $g_{4,*}\operatorname{\mathcal{O}}_{\operatorname{\mathcal{Y}}_4}$ est localement libre de formation commutant au changement de base. Les fibres de $g_4$ étant réduites par le §\[deminormal\], le morphisme naturel $H'_4:=\operatorname{Spec}_{H_4}(g_{4,*}\operatorname{\mathcal{O}}_{\operatorname{\mathcal{Y}}_4})\to H_4$ est fini étale. En considérant la factorisation de Stein $g'_4:\operatorname{\mathcal{Y}}_4\to H'_4$ de $g_4$, on se ramène aisément au cas où les fibres de $g_4$ sont connexes, ce qu’on suppose désormais. Dans ce cas, l’argument qui suit est donné dans [@Viehweg Lemma 1.19].
On note $\operatorname{\mathcal{L}}:=\omega_{\operatorname{\mathcal{Y}}_4/H_4}^{[m]}\otimes\operatorname{\mathcal{O}}_{\operatorname{\mathcal{Y}}_4/H_4}(-1)$. Si $0\to\operatorname{\mathcal{L}}\to\operatorname{\mathcal{E}}^0\to\operatorname{\mathcal{E}}^1\to\dots$ est une résolution de $\operatorname{\mathcal{L}}$ par des sommes de fibrés en droites assez amples et si l’on pose $\operatorname{\mathcal{F}}^i:=g_{4,*}\operatorname{\mathcal{E}}^i$, la cohomologie du complexe de fibrés vectoriels $0\to \operatorname{\mathcal{F}}^0\to \operatorname{\mathcal{F}}^1\to\dots$ sur $H_4$ calcule les $R^ig_{4,*}\operatorname{\mathcal{L}}$, et ce après tout changement de base (par cohomologie et changement de base). Soit $\operatorname{\mathcal{Q}}:=\operatorname{Coker}(\operatorname{\mathcal{F}}^0\to\operatorname{\mathcal{F}}^1)$. On commence par se restreindre à l’ouvert où $\operatorname{\mathcal{Q}}$ a rang $\leqslant\operatorname{rg}(\operatorname{\mathcal{F}}^1)-\operatorname{rg}(\operatorname{\mathcal{F}}^0)+1$, puis au fermé défini par l’annulation des mineurs de taille $\operatorname{rg}(\operatorname{\mathcal{F}}^0)$ de $\operatorname{\mathcal{F}}^0\to\operatorname{\mathcal{F}}^1$. Le faisceau $\operatorname{\mathcal{Q}}$ est maintenant localement libre de rang $\operatorname{rg}(\operatorname{\mathcal{F}}^1)-\operatorname{rg}(\operatorname{\mathcal{F}}^0)+1$ par [@Eisenbud Proposition 20.8]. Il suit que le noyau $\operatorname{\mathcal{K}}:=\operatorname{Ker}(\operatorname{\mathcal{F}}^0\to\operatorname{\mathcal{F}}^1)$ est localement libre de rang $1$, de formation commutant à tout changement de base. On déduit que $g_{4,*}\operatorname{\mathcal{L}}=\operatorname{\mathcal{K}}$ est inversible et de formation commutant à tout changement de base. Il suffit pour conclure de se restreindre à l’ouvert $H_5$ au-dessus duquel le morphisme d’adjonction $g_4^*g_{4,*}\operatorname{\mathcal{L}}\to\operatorname{\mathcal{L}}$ est un isomorphisme. On note bien sûr $g_5:\operatorname{\mathcal{Y}}_5\to H_5$ le changement de base.
### {#section-1}
Considérons l’ouvert $H_6\subset H_5$ au-dessus duquel $\operatorname{\mathcal{O}}_{\operatorname{\mathcal{Y}}_5/H_5}(1)$ n’a pas de cohomologie supérieure [@Hartshorne Theorem 12.8]. Notant $g_6:\operatorname{\mathcal{Y}}_6\to H_6$ le changement de base, le faisceau $g_{6,*}\operatorname{\mathcal{O}}_{\operatorname{\mathcal{Y}}_6/H_6}(1)$ est un fibré vectoriel par [@Hartshorne Theorem 12.11]. On se restreint finalement à l’ouvert $H'\subset H_6$ où le morphisme $H^0(\operatorname{\mathbb{P}}_{k}^{F(m)-1}, \operatorname{\mathcal{O}}(1))\to g_{6,*}\operatorname{\mathcal{O}}_{\operatorname{\mathcal{Y}}_6/H_6}(1)$ de fibrés vectoriels sur $H_6$ est un isomorphisme. Ce dernier point assure que la famille $g':\operatorname{\mathcal{Y}}'\to H'$ obtenue par changement de base est $m$-canoniquement plongée et achève la preuve du théorème \[threpr\].
Le champ de modules {#champ}
-------------------
On peut maintenant construire le champ de modules $\overline{\operatorname{\mathcal{M}}}_F$. Le groupe $\operatorname{PGL}_{F(m)}$ agit sur $\operatorname{\mathbb{P}}^{F(m)-1}_k$ par changement de coordonnées, donc aussi sur son schéma de Hilbert $H$. Comme le morphisme $\iota:H'\to H$ est défini par une propriété universelle, cette action se relève naturellement en une action sur $H'$. J’affirme que la catégorie fibrée en groupoïdes $\overline{\operatorname{\mathcal{M}}}_F$ définie en s’identifie canoniquement au champ quotient $[H'/\operatorname{PGL}_{F(m)}]$[^18] et est donc un champ algébrique [@LMB (4.6.1)].
Intuitivement, cette assertion est claire : $H'$ paramètre les variétés stables de fonction de Hilbert $F$ qui sont $m$-canoniquement plongées dans $\operatorname{\mathbb{P}}^{F(m)-1}_k$, et deux telles sous-variétés de $\operatorname{\mathbb{P}}^{F(m)-1}_k$ sont isomorphes comme variétés abstraites si et seulement si elles diffèrent par un changement de coordonnées projectives.
Justifions-le plus formellement. On se contente ici de construire un $1$-morphisme $\overline{\operatorname{\mathcal{M}}}_F\to [H'/\operatorname{PGL}_{F(m)}]$ ; il est aisé de vérifier que c’est un isomorphisme. Considérons un $B$-point de $\overline{\operatorname{\mathcal{M}}}_F$, qui correspond à une famille stable $f:\operatorname{\mathcal{X}}\to B$ dont les fibres ont fonction de Hilbert $F$ comme en . Le faisceau $\omega_{\operatorname{\mathcal{X}}/B}^{[m]}$ est plat et inversible sur les fibres de $f$ par le théorème \[thborne\]. Il est donc inversible par l’argument du §\[inversible\]. Par cohomologie et changement de base [@Hartshorne III Theorem 12.11], qui s’applique par le théorème \[thborne\] et comme $\omega_{\operatorname{\mathcal{X}}/B}^{[m]}$ est $f$-plat, le faisceau $f_*\omega_{\operatorname{\mathcal{X}}/B}^{[m]}$ est localement libre de rang $F(m)$. Le faisceau inversible $\omega_{\operatorname{\mathcal{X}}/B}^{[m]}$ est très ample relativement à $f$, et plonge $\operatorname{\mathcal{X}}$ dans le fibré projectif $\operatorname{\mathbb{P}}_B(f_*\omega_{\operatorname{\mathcal{X}}/B}^{[m]})$ sur $B$. Soit $I$ le schéma des isomorphismes $\operatorname{Isom}_B (\operatorname{\mathbb{P}}^{F(m)-1}_B,\operatorname{\mathbb{P}}_B(f_*\omega_{\operatorname{\mathcal{X}}/B}^{[m]}))$ : c’est un $\operatorname{PGL}_{F(m)}$-torseur sur $B$. En tirant $f:\operatorname{\mathcal{X}}\to B$ sur $I$ et en utilisant l’isomorphisme universel au-dessus de $I$, on obtient un morphisme $\lambda:I\to H$, qui factorise à travers un morphisme $\mu:I\to H'$ par la propriété universelle de $H'$. Le couple $(I,\mu)$ est le $B$-point de $[H'/\operatorname{PGL}_{F(m)}]$ que nous cherchions à construire.
Remarquons que l’action de $\operatorname{PGL}_{F(m)}$ se relève naturellement à l’espace total $\operatorname{\mathcal{Y}}'$ de la famille $g':\operatorname{\mathcal{Y}}'\to H'$. Notant $\overline{\operatorname{\mathcal{U}}}_F:=[\operatorname{\mathcal{Y}}'/\operatorname{PGL}_{F(m)}]$, on voit que le champ $\overline{\operatorname{\mathcal{M}}}_F$ porte une famille stable universelle $u:\overline{\operatorname{\mathcal{U}}}_F\to \overline{\operatorname{\mathcal{M}}}_F$.
Le champ $\overline{\operatorname{\mathcal{M}}}_F$ est propre. En effet, le critère valuatif de propreté [@Olsson Theorem 11.5.1] est vérifié par le théorème de réduction stable, sous sa forme donnée au théorème \[redstable2\] (la séparation de $\overline{\operatorname{\mathcal{M}}}_F$ correspondant à la propriété d’unicité dans le théor\` eme de réduction stable).
On en déduit la finitude des groupes d’automorphismes des variétés stables. En effet, le critère valuatif de propreté pour ces groupes d’automorphismes est un cas particulier du critère valuatif de séparation pour $\overline{\operatorname{\mathcal{M}}}_F$ et est donc vérifié. Ces groupes d’automorphismes sont donc propres. Comme ils sont de plus affines (ce sont des sous-groupes de $\operatorname{PGL}_{F(m)}$), ils sont finis.
Le schéma en groupes des automorphismes d’une variété stable est de plus réduit, comme tout schéma en groupes en caractéristique nulle. Appliquant [@Olsson Theorem 8.3.3, Remark 8.3.4], on voit que $\overline{\operatorname{\mathcal{M}}}_F$ est un champ de Deligne-Mumford.
L’espace de modules grossier {#grossier}
----------------------------
C’est un théorème de Keel et Mori [@KeMo] que tout champ séparé de type fini sur un corps admet un espace de modules grossier, qui est un espace algébrique séparé de type fini sur ce corps. Notons $\overline{M}_F$ l’espace de modules grossier de $\overline{\operatorname{\mathcal{M}}}_F$. La propreté de $\overline{\operatorname{\mathcal{M}}}_F$ entraîne celle de $\overline{M}_F$.
Il reste à démontrer la projectivité de $\overline{M}_F$. Les premières approches à ce problème ont reposé sur une autre stratégie de construction de $\overline{M}_F$ que celle présentée ici : la théorie géométrique des invariants de Mumford [@GIT], dont l’avantage est de produire des variétés qui sont, par construction, munies d’un fibré ample. C’est ainsi que Knudsen [@Knudsen] et Gieseker et Mumford [@Mumford §5] ont démontré la projectivité de $\overline{M}_g$. Cette technique a également permis à Gieseker de démontrer la quasi-projectivité des espaces de modules de surfaces lisses canoniquement polarisées [@Gieseker], et Viehweg a réussi, dans un véritable tour de force, à faire de même en dimension arbitraire [@Viehweg]. Cette méthode se heurte cependant à des difficultés sérieuses, déjà soulevées par Mumford [@Mumford §3] et Shah [@Shah], dans le cas des variétés singulières.
Kollár a proposé dans [@Kocomplete] une autre méthode pour démontrer la projectivité d’un espace de modules qu’on sait être propre. Expliquons comment elle s’applique à $\overline{M}_F$.
L’espace algébrique $\overline{M}_F$ porte de nombreux fibrés en droites naturels. Rappelons que $u:\overline{\operatorname{\mathcal{U}}}_F\to\overline{\operatorname{\mathcal{M}}}_F$ est la famille universelle construite au §\[champ\]. Si $n>0$ est un entier assez divisible, le faisceau $\omega^{[n]}_{\overline{\operatorname{\mathcal{U}}}_F/\overline{\operatorname{\mathcal{M}}}_F}$ est un fibré en droites $u$-ample dont les restrictions aux fibres de $u$ n’ont pas de cohomologie supérieure. Pour un tel $n$, le faisceau $u_*\omega^{[n]}_{\overline{\operatorname{\mathcal{U}}}_F/\overline{\operatorname{\mathcal{M}}}_F}$ est un fibré vectoriel sur $\overline{\operatorname{\mathcal{M}}}_F$. Pour $N>0$ assez divisible, le fibré en droites $\det(u_*\omega^{[n]}_{\overline{\operatorname{\mathcal{U}}}_F/\overline{\operatorname{\mathcal{M}}}_F})^{\otimes N}$ sur $\overline{\operatorname{\mathcal{M}}}_F$ descend en un unique fibré en droites $\operatorname{\mathcal{L}}_{n, N}$ sur $\overline{M}_F$ par [@Rydh] (voir aussi [@Viehweg Lemma 9.26]).
\[amplitude\] Si $n$ est assez divisible, $\operatorname{\mathcal{L}}_{n, N}$ est un fibré ample sur $\overline{M}_F$.
Ce théor\` eme a été démontré par Kollár dans [@Kocomplete Corollary 5.6] pour les espaces de modules de surfaces stables et par Fujino [@Fujino] en général. Kollár a remarqué qu’il suffisait de vérifier que pour toute courbe projective lisse $C$ et pour tout morphisme $\psi:C\to\overline{\operatorname{\mathcal{M}}}_F$, le fibré vectoriel $\psi^*(u_*\omega^{[n]}_{\overline{\operatorname{\mathcal{U}}}_F/\overline{\operatorname{\mathcal{M}}}_F})$ est nef [@Kocomplete Theorem 2.6]. Qu’on puisse déduire le théorème \[amplitude\], qui est un énoncé de positivité *stricte*, d’un tel résultat de positivité *au sens large* est remarquable. L’argument repose sur le critère d’amplitude de Nakai-Moishezon. Le gain de positivité est fourni par la variation, dans une famille de variétés stables, des équations de ces variétés dans leurs plongements pluricanoniques (voir [@Kocomplete §2.9]).
La vérification du fait que $\psi^*(u_*\omega^{[n]}_{\overline{\operatorname{\mathcal{U}}}_F/\overline{\operatorname{\mathcal{M}}}_F})$ est nef est due à Kollár [@Kocomplete Theorem 4.12] pour les familles de surfaces et à Fujino [@Fujino Theorem 1.7] en général. Elle s’appuie sur des théorèmes de semipositivité en théorie de Hodge, qui remontent à Fujita [@Fujita] et qui sont démontrés dans la généralité requise dans [@FF Corollary 5.23].
Patakfalvi et Xu [@PatakXu] ont montré l’amplitude d’un autre fibré en droites, défini seulement sur la normalisation de $\overline{M}_F$ : le *fibré en droites CM*. Comme $\overline{M}_F$ est propre, cela fournit une autre preuve de sa projectivité (voir [@EGA31 Proposition 2.6.2]).
[**Remerciements.**]{} Merci à Olivier Debarre, Stéphane Druel, Javier Fresán, Christopher Hacon, János Kollár et Bertrand Rémy pour leurs utiles commentaires.
[666]{}
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[^1]: Par le biais de leurs modèles canoniques, cela prend en compte toutes les variétés de type général. Des compactifications modulaires ont aussi été construites, par d’autres méthodes que celles expliquées ici, pour d’autres espaces de modules : variétés abéliennes [@Aleva], certaines variétés de Fano [@LWX].
[^2]: Sur l’ouvert de lissité de $X$, il s’agit du faisceau canonique des formes différentielles de degré maximal. On peut décrire très concrètement $\omega_U$ en général: une section locale est une $d$-forme différentielle sur la normalisation, à pôles au plus logarithmiques le long de l’image inverse du lieu double, et dont les résidus le long des deux branches du lieu double sont opposés [@Kosing Proposition 5.8].\[fn:repeat\]
[^3]: On prendra garde que, $X$ n’étant pas Cohen-Macaulay en général, ce faisceau peut ne pas coïncider avec le complexe dualisant de $X$ : il n’en est qu’un des faisceaux de cohomologie.
[^4]: Une **modification** est un morphisme propre birationnel. On n’a pas vraiment besoin de supposer $Y$ normale : il suffit que $Y$ soit $S_2$ et régulière aux points génériques des diviseurs exceptionnels de $\pi$.
[^5]: Il est faux en général que le quotient d’une variété lc par un groupe fini est encore lc. Soit $\pi:S\to T$ une surface $K3$ obtenue comme revêtement double de $T=\operatorname{\mathbb{P}}^1\times\operatorname{\mathbb{P}}^1$ ramifié au-dessus d’un diviseur lisse de bidegré $(4,4)$, et notons $L:=\operatorname{\mathcal{O}}_T(1,2)$. Le morphisme de cônes $C(S,\pi^*L)\to C(T,L)$ est le quotient par une action de $\operatorname{\mathbb{Z}}/2\operatorname{\mathbb{Z}}$, mais $C(S,\pi^*L)$ est lc alors que $C(T,L)$ ne l’est pas.
[^6]: Pour le voir, on peut combiner [@EGA42 Théorème 5.10.5 et Proposition 6.3.1].
[^7]: Justifions-le. Par classification des singularités canoniques de surfaces [@KM Theorem 4.20], les fibres de $f$ sont Gorenstein en codimension $2$. Par [@Conrad Theorem 3.5.1], il existe un ouvert $j:\mathcal{U}\hookrightarrow\mathcal{X}$ tel que $\operatorname{\mathcal{X}}\setminus\operatorname{\mathcal{U}}$ a codimension $\geqslant 3$ dans les fibres de $f$ et tel que $f|_{\operatorname{\mathcal{U}}}$ est Gorenstein, de sorte que $\omega_{\operatorname{\mathcal{U}}/B}$ est inversible. Comme les fibres de $f$ sont de plus $S_3$ par [@Elkik] (voir le théorème \[rationnelles\]), on peut conclure à l’aide de [@Koflat Theorem 12].
[^8]: Cette section hyperplane pourrait a priori avoir un point immergé au sommet. On vérifie que ce n’est pas le cas en remarquant que $\Sigma_i$ et $C$ ont même polynôme de Hilbert.
[^9]: Dans ces exemples, l’unique singularité de l’espace total est celle d’un cône. On vérifie alors ces assertions à l’aide du calcul du groupe des classes d’un cône [@Kosing Proposition 3.14 (4)].\[note1\]
[^10]: \[nonpol\]Soit $L$ un fibré en droites très ample sur une surface d’Enriques $S$ et soit $Z=C(S,L)$ le cône sur $S$ dans le plongement induit par $L$. Soit $X\to S$ un revêtement double ramifié le long d’une section lisse de $L^{\otimes 4}$. Nous laissons au lecteur le soin de vérifier que $X$ est stable et que sa fonction de Hilbert n’est pas un polynôme.
[^11]: On travaille localement au voisinage de $x$ et on applique [@Kosing Theorem 7.20] avec $\Delta=\Delta'=0$ à un diviseur de Weil $D\subset X$ tel que $\operatorname{\mathcal{O}}_X(-D)\simeq\omega^{[n]}_X$.
[^12]: Pour construire $\operatorname{\mathcal{Y}}$, on note $Z\subset\operatorname{\mathcal{X}}$ l’union des composantes irréductibles du lieu non normal de $\operatorname{\mathcal{Y}}$ qui dominent $T$, on remarque que les faisceaux $\operatorname{\mathcal{O}}_{\operatorname{\mathcal{X}}_\eta}$ sur $\operatorname{\mathcal{X}}_{\eta}$ et $\widetilde{\operatorname{\mathcal{O}}_{\operatorname{\mathcal{X}}\setminus Z}}$ sur $\operatorname{\mathcal{X}}\setminus Z$ se recollent en un faisceau d’algèbres cohérent $\operatorname{\mathcal{A}}$ sur $\operatorname{\mathcal{X}}\setminus Z_t$ et on définit $\operatorname{\mathcal{Y}}:=\operatorname{Spec}_{\operatorname{\mathcal{O}}_{\operatorname{\mathcal{X}}}}(j_*\operatorname{\mathcal{A}})$, où $j:\operatorname{\mathcal{X}}\setminus Z_t\hookrightarrow\operatorname{\mathcal{X}}$ est l’inclusion et où $j_*\operatorname{\mathcal{A}}$ est un faisceau d’algèbres cohérent $S_2$ par [@EGA42 Propositions 5.11.1 et 5.10.10].
[^13]: Cela signifie que son inverse ne contracte pas de diviseurs.
[^14]: L’étape (i) du procédé de recollement est plus facile à mettre en œuvre ici que dans le cadre du théorème \[bij\]. En effet, on peut exploiter l’existence du recollement $X$ de $(\widetilde{X},\Gamma,\tau)=(\widetilde{\operatorname{\mathcal{X}}}_\eta,\mathit{\Gamma}_\eta,\tau_\eta)$, et le fait (déjà expliqué au §\[speciale\]) qu’aucun centre log canonique de $(\widetilde{\operatorname{\mathcal{X}}},\mathit{\Gamma})$ n’est inclus dans la fibre spéciale $\widetilde{\operatorname{\mathcal{X}}}_t$, et appliquer [@Kosing Lemma 9.55]. Les étapes (ii) et (iii) sont en revanche inchangées.
[^15]: La paire $(X,\Delta)$ est **de type général** s’il existe un entier $m>0$ tel que $\omega_X^{[m]}(m\Delta)$ soit inversible et induise une application rationnelle $X\dashrightarrow\operatorname{\mathbb{P}}^N_k$ qui est birationnelle sur son image.
[^16]: La finitude des algèbres canoniques dans l’énoncé du théorème \[thHX\] est équivalente à l’existence des bons modèles minimaux dans l’énoncé de [@HX Theorem 1.1], par [@HMX Lemma 2.9.1].
[^17]: L’article [@HaKo] utilise la condition de Viehweg plutôt que celle de Kollár (voir §\[pardeffamst\]).
[^18]: Rappelons qu’un B-point du champ quotient $[H'/\operatorname{PGL}_{F(m)}]$ est la donnée d’un $\operatorname{PGL}_{F(m)}$-torseur $I$ sur $B$, et d’un morphisme $\operatorname{PGL}_{F(m)}$-équivariant $\mu:I\to H'$.
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abstract: 'In this work we investigate how the details of the quark-gluon interaction vertex affect the quantitative description of chiral symmetry breaking and dynamical mass generation through the gap equation. We employ the Maris-Tandy (MT) [@Maris:1999nt] and Qin-Chang (QC) [@Qin:2011dd] models for the gluon propagator and the effective strong running coupling. The gap equation is solved by employing several vertex Ans${\rm \ddot{a}}$tze which have been constructed in order to implement some of the key aspects of a gauge field theory such as gauge invariance and multiplicative renormalizability. We find that within a small variation of MT and QC model parameters, all truncations point towards the same quantitative pattern of chiral symmetry breaking, the running quark mass function, ensuring the robustness of this approach.'
author:
- 'M. Atif Sultan'
- Faisal Akram
- Bilal Masud
- Khépani Raya
date:
-
-
title: 'Effect of the Quark-Gluon Vertex on Dynamical Chiral Symmetry Breaking'
---
[^1]
[^2]
[^3]
[^4]
\[sec:intro\]Introduction
=========================
If quantum chromodynamics (QCD) is the underlying theory of strong interactions, we expect all hadronic observables to be calculable from the complete knowledge of its Green functions. There is an infinite set of integral field theoretic equations which describe these $n$-point functions in a coupled and highly non-linear manner. These are the well-known Schwinger-Dyson equations (SDEs), [@Schwinger:1951ex; @Schwinger:1951hq; @Dyson:1949ha]. Their structure is such that the two point one-particle irreducible (1PI) Green functions (propagators) are related to three point such functions (vertices) which in turn are entangled with the four-point functions (scattering kernels), [*ad infinitum*]{}. In a general formalism, not limited to the perturbative domain, this infinite set must be truncated by introducing mathematical model(s) of some suitable set of Green functions before a solution becomes tractable. The most favorite choice, which lies on the borderline of a daunting computational complexity while still maintaining predictable exploration of non-perturbative QCD and hadronic physics is to model the 3-point interactions. It is natural to demand any truncation of SDEs to resemble the true dynamics of quarks and gluons to as much extent as possible, while successfully describing the observable degrees of freedom, namely, mesons and baryons [@Binosi:2014aea; @Binosi:2016wcx]. Ideally, we can impose the following restrictions on the quark-gluon vertex which enters the gap equation directly and also constrains the kernel of the Bethe-Salpeter equation [@Chang:2009zb; @Binosi:2016rxz]:
- It must satisfy the Slavnov-Taylor identity (STI), [@Slavnov:1972fg; @Taylor:1971ff]. This implies that the requirement of gauge invariance fixes the longitudinal part of the quark-gluon vertex.
- The transverse part is constrained by the requirement of the generalized Landau-Khalatnikov-Fradkin transformations (LKFTs) [@DeMeerleer:2018txc; @Aslam:2015nia; @Bashir:2005wt] and the transverse Takahashi identities (TTIs) [@Takahashi:1985yz; @Kondo:1996xn; @He:2000we] for QCD.
- It should reduce to its perturbation theory Feynman expansion in the limit of weak coupling expansion. Note that a truncation of the complete set of SDEs, which maintains gauge invariance and MR of a gauge theory at every level of approximation, is perturbation theory. Therefore, physically meaningful solutions of the SDEs must agree with perturbative results in the weak coupling regime [@Davydychev:2000rt; @Bashir:1999bd; @Bashir:2000rv; @Bashir:2011vg; @Bermudez:2017bpx].
- It should transform correctly under the discrete symmetries of charge, conjugation, parity and time reversal ($C$,$P$ and $T$).
- It should be free of any kinematic singularities.
- Physical observables should be strictly gauge-independent [@Bashir:1994az; @Bashir:1995qr; @Bashir:2011dp].
However, the fact remains that any truncation of the SDEs can only be considered sensible only if it is consistently able to reproduce the experimental observations pertaining QCD and hadron physics. The well-know Maris-Tandy (MT) model [@Maris:1999nt] achieves that goal quite successfully, especially to study low lying mesons such as pions and $\rho$. In conjunction with the bare vertex and gluon propagator in the Landau gauge, it uses an ans${\rm \ddot{a}}$tze for the effective coupling which is constrained by perturbation theory in the ultraviolet region. The chiral quark condensate dictates the strength of its kernel in the infrared. Note that the chiral symmetry breaking pattern of the pseudoscalar meson sector is governed by the relationship between the gap equation and the Bethe-Salpeter kernel, supplied by the axial vector Ward-Green-Takahashi identity [@Qin:2014vya].
Notice that The MT model was put forward before the SDEs prediction for the massive gluon solution, [@Aguilar:2004sw], which was later confirmed in modern lattice studies, [@Cucchieri:2007md; @Bogolubsky:2007ud; @Cucchieri:2010xr; @Bogolubsky:2009dc; @Oliveira:2009eh], supporting a finite but infrared enhanced scalar form factor of the gluon propagator, the so called decoupling solution. It is also in agreement with subsequent SDE results, [@Aguilar:2008xm; @Boucaud:2008ky; @Aguilar:2009nf; @Pennington:2011xs], exact renormalization group (RG) equations, [@Fischer:2008uz], refined Gribov-Zwanziger formalism, [@Dudal:2007cw; @Dudal:2008sp; @Dudal:2010tf], the earlier suggestion of Cornwall, [@Cornwall:1981zr] and the contemporary process independent effective charge described in [@Binosi:2016nme; @Rodriguez-Quintero:2018wma], which combines the best from modern lattice and continuum studies. Even if one includes the effect of dynamical quarks, [@Bowman:2007du; @Ayala:2012pb; @Aguilar:2012rz], the qualitative behavior of the gluon propagator remains the same and feeds expected physics back into the gap equation, [@Bashir:2013zha]. This feature of the gluon propagator is conveniently captured in a later kernel-construction for the gap equation which goes by the name of Qin-Chang (QC) model [@Qin:2011dd]. However, connection with one-loop perturbation theory is still maintained along with the use of the bare vertex.
In this article, we employ the effective coupling of both the MT and the QC models in association with a set of refined ans${\rm \ddot{a}}$tze for the quark-gluon vertex. We extend the decomposition of the QED vertex suggested by Ball and Chiu [@Ball:1980ay] to the QCD case, as also adopted in [@Davydychev:2000rt; @Bermudez:2017bpx].
We then carry out our numerical analysis of the quark gap equation for the following vertices: bare, Ball-Chiu [@Ball:1980ay], Curtis-Pennington [@Curtis:1990zs], Kizilersu-Pennington [@Kizilersu:2009kg] and Bashir [*et al*]{}. [@Bashir:2011dp].
The article is organized as follows: in \[sec:models\] we discuss the preliminaries of the gap equation, introducing the MT and the QC models. In \[sec:vertices\], we explicitly discuss each of the vertex Ans${\rm \ddot{a}}$tze we employ and discuss their merits. \[sec:gapequation\] details the algebraic expressions that stem from the choice of each vertex. \[sec:results\] contains numerical results for each truncation as well as a comparative analysis. Finally, in \[sec:conclusions\], we summarize our conclusions.
Gap equation: preliminaries and the gluon propagator {#sec:models}
====================================================
The starting point to investigate how dynamical chiral symmetry breaking (DCSB) is realized and how it is affected by the choice of the quark-gluon vertex, is the renormalized Schwinger-Dyson equation for the quark propagator. This equation, which is depicted in fig. \[fig:SDE\], can be written in the following form: $$\label{eq:1}
S^{-1}(p)= Z_{2}(i\slashed{p}+m_{0}) + \Sigma(p),$$ where $\Sigma(p)$ is the quark self energy $$\label{eq:2}
\Sigma(p) = Z_{1}\int^{\Lambda_{UV}}\frac{d^{4}k}{(2\pi)^{4}} \; C_{F} \hspace{0.5mm} g^2 \;\gamma_{\mu} \hspace{0.5mm} S(k) \; D_{\mu\nu}(q) \; \Gamma_{\nu}(k,p).$$ Here $q=k-p$, $C_{F}=4/3$, and $\Lambda_{UV}$ is the ultraviolet mass regulator. $Z_1$ and $Z_2$ are the quark and quark-gluon vertex renormalization constants, respectively, which depend on the UV regulator and the renormalization point $\mu$. This equation, also known as *gap equation*, involves not only the quark propagator $S(p)$, but also the gluon propagator $D_{\mu\nu}(q)$ and the fully dressed quark-gluon vertex $\Gamma_{\nu}(k,P)$. Each of those green functions, which also depend on the renormalization point, obey their own Schwinger-Dyson equations. This yields an infinite tower of coupled equations, which must be systematically truncated in order to extract the encoded physics.
Regardless of the truncation scheme, the full quark propagator can be defined in terms of two scalar functions, namely mass function $M(p^{2})$ and quark wavefunction renormalization function $F(p^{2};\mu^2)$, such that $$\label{eq:quarkprop1}
S(p;\mu) = \frac{F(p^{2};\mu^2)}{i\slashed{p}+M(p^{2})},$$ in analogy to its bare counterpart $$S_{0}(p) = \frac{1}{i\slashed{p}+m_{0}},$$ where $m_0$ is the bare mass of the quark. An alternative representation of $S(p;\mu)$ is given by $$\label{eq:4}
S^{-1}(p;\mu) = A(p^{2};\mu^2) \hspace{0.5mm} i\slashed{p}+B(p^{2};\mu^2),$$ where we can readily identify $$M(p^{2}) = \frac{B(p^{2};\mu^2)}{A(p^{2};\mu^2)}, \hspace{.3in} F(p^{2};\mu^2) = \frac{1}{A(p^{2};\mu^2)}.$$ Notably, multiplicative renormalizability ensures that the mass function does not depend on the renormalization point. Not that for the simplicity of notation, we will omit displaying the $\mu^2$ dependence altogether.\
![Schwinger Dyson equation for full quark propagator. \[fig:SDE\]](SDE.pdf){width="8cm"}
The general form of the gluon propagator is [@Aguilar:2009nf]: $$D_{\mu\nu}(q) = \frac{D(q^{2})}{q^{2}}\left[\delta_{\mu\nu}-\frac{q_{\mu} \hspace{0.5mm} q_{\nu}}{q^{2}}\right]+\xi \frac{q_{\mu} \hspace{0.5mm} q_{\nu}}{q^{4}},$$ where $D(q^2)$ is the gluon dressing function and $\xi$ is the covariant gauge. A typical choice is the Landau gauge [@Bashir:2008fk; @Bashir:2009fv; @Raya:2013ina], which corresponds to $\xi = 0$. It occupies a special place in field theories since, among other things, model dependent differences between ans${\rm \ddot{a}}$tze for the gluon-quark vertex are least noticeable in this gauge. Moreover, it is a covariant gauge which is readily implemented in lattice QCD simulations [@Cucchieri:2009kk; @Boucaud:2018xup].
The first model of gluon propagator employed herein is the well known Maris-Tandy model [@Maris:1999nt]. In this model, the effective coupling $\alpha_{s}(q^2)\equiv g^2 D(q^2)/4\pi$ is given by: $$\begin{aligned}
\label{eq:MTMODEL}
\frac{\alpha_s(q^2)}{q^2} = \frac{\pi D}{\omega^{6}} q^{2} e^{-q^{2}/\omega^{2}}
+\frac{ \gamma_{m} \; \pi \textit{F}(q^{2})}{\frac{1}{2}\ln[\tau+(1+q^{2}/\Lambda_{QCD}^{2})^{2}]},\end{aligned}$$ with $\textit{F}(q^{2}) = \{1-e^{-q^{2}/[4 m_{t}^{2}]}\}/q^{2}$, $\tau = e^{2}-1$, $\gamma_{m} = 12/(33-2 N_{f})$, $N_{f} = 4$, $m_t = 0.5$ GeV and $\Lambda_{QCD} = 0.234$ GeV. The first term provides an infrared enhancement, controlled by the parameters $\omega$ and $D$, while the second term reproduces the one-loop renormalization group equation of QCD. Another model choice is the Qin-Chang interaction: $$\begin{aligned}
\label{eq:QCMODEL}
\frac{\alpha_s(q^2)}{q^2} = \frac{2\pi D}{\omega^{4}} e^{-q^{2}/\omega^{2}}
+\frac{ \gamma_{m} \; \pi \textit{F}(q^{2})}{\frac{1}{2}\ln[\tau+(1+q^{2}/\Lambda_{QCD}^{2})^{2}]},\end{aligned}$$ which differs from the MT model in the infrared enhancement term, such that the behavior of the effective gluon is in agreement with our modern understanding of QCD’s gauge sector; viz., the gluon propagator is a bounded, regular function of spacelike momenta, which achieves its maximum value on this domain at $q^2=0$. Typically, $\omega \in (0.3,0.6)$ GeV and the product $(\omega D) = m_G^3$ is associated with a gluon mass scale $m_G \sim 400 - 800$ MeV [@Qin:2011dd]. Both MT and QC models have been widely employed in the context of DSE, in order to successfully obtain a wide range of hadron observables: light meson spectrum [@Maris:1999nt; @Chang:2011ei], parton distribution amplitudes of light and heavy mesons [@Chang:2013pq; @Ding:2015rkn], and the associated elastic and transition form factors [@Maris:2002mz; @Chang:2013nia; @Raya:2015gva; @Raya:2016yuj; @Chen:2018rwz].
Quark-gluon vertex Ans${\rm \ddot{A}}$tze {#sec:vertices}
=========================================
For the 1PI quark-gluon vertex the simplest choice is to replace the fermion-boson vertex by free (tree-level) vertex, which is known as the rainbow truncation. This (bare) vertex satisfies WGTI only in massless quenched approximation in the Landau gauge [@Kizilersu:2013hea]. The simplicity of that choice brings many deficiencies, such as the lack of information about DCSB on its structure (no explicit appearance of the quark mass function in the vertex dressing). Moreover, the associated dressed quark anomalous chromomagnetic moment is strictly zero. These drawbacks can be compensated by a proper choice of parameters for the effective gluon propagator, obtaining an excellent description of pseudoscalars and light vector mesons, see [@Maris:1999nt; @Chang:2011ei], for example. However, a proper description of axial vector mesons can not be achieved, since the bare vertex lacks a proper enhancement of spin-orbit splitting in the meson spectrum, provided by the DCSB [@Chang:2009zb; @Lu:2017cln].
In constructing a fully consistent fermion-boson vertex ansätz, many efforts have been carried out over the last few decades [@Ball:1980ay; @Curtis:1990zs; @Kizilersu:2009kg; @Bashir:2011dp; @Aguilar:2014lha; @Binosi:2016wcx; @Bermudez:2017bpx; @Aguilar:2018epe]; we explore those mentioned in the introduction [@Ball:1980ay; @Curtis:1990zs; @Kizilersu:2009kg; @Bashir:2011dp]. The general structure of the full vertex consists of 12 independent vectors which can be obtained from three vectors $k_{\mu}$, $p_{\mu}$, $\gamma_{\mu}$ and four spin scalars 1, $\slashed{k}$, $\slashed{p}$, $\slashed{k}\slashed{p}$ [@Kizilersu:1995iz]. Longitudinal WGTI entails [@Qin:2014vya]: $$iq_{\mu} \hspace{0.5mm} \Gamma_{\mu} = S^{-1}(k)-S^{-1}(p).$$ Then, the full quark-gluon vertex can be split into 4 longitudinal and 8 transverse parts $$\Gamma_{\mu}(k,p) = \Gamma^{L}_{\mu}(k,p)+\Gamma^{T}_{\mu}(k,p),$$ such that $q_{\mu}\Gamma^{T}_{\mu} = 0. $ The longitudinal part is fixed by the above WGTI, while the remaining transverse part must be written in a covariant basis, free of kinematical singularities [@Ball:1980ay]. The longitudinal part, in the so-called Ball-Chiu (BC) basis, is given by $$\begin{aligned}
\nonumber
\Gamma_{\mu}^{(BC)}(k,p) &=& \lambda_{1}(k^{2},p^{2}) \hspace{0.5mm} \gamma_{\mu}+\lambda_{2}(k^{2},p^{2})(k+p)_{\mu}
\\
\nonumber
&+& \lambda_{3}(k^{2},p^{2})(k+p)_{\mu}(\slashed{k} + \slashed{p}) \\
\label{eq:BCV}
&+& \lambda_{4}(k^{2},p^{2})(k+p)_{\nu}\; \sigma_{\mu\nu},\end{aligned}$$ where the dressing functions are $$\begin{aligned}
\nonumber
\lambda_{1}(k^{2},p^{2}) &=& \frac{1}{2}(A(k^{2})+A(p^{2}))\;, \\
\nonumber
\lambda_{2}(k^{2},p^{2}) &=& -i \Delta_B(k^2,p^2) \;, \\
\nonumber
\lambda_{3}(k^{2},p^{2}) &=& \frac{1}{2} \Delta_A(k^2,p^2) \;, \\
\lambda_{4}(k^{2},p^{2}) &=& 0\;,\end{aligned}$$ where $(k^2-p^2) \Delta_{\varphi}(k^2,p^2) = \varphi(k^2) - \varphi(p^2)$. The transverse part is written as a linear combination of the 8 basis vectors. That is, $$\Gamma^{T}_{\mu} = \sum_{i=1}^{8} \tau_{i}(k^{2},p^{2},q^{2}) \hspace{0.5mm} T_{i\mu}(k,p),$$ where $\tau_{i}$ are unknown scalar functions and the tensor structures $T_{i\mu}$ are written as $$\begin{aligned}
\nonumber
T_{1\mu}(k,p) &=& p_{\mu}(k\cdot q)-k_{\mu}(p\cdot q)\;, \\
\nonumber
T_{2\mu}(k,p) &=& [p_{\mu}(k\cdot q)-k_{\mu}(p\cdot q)](\slashed{k}+\slashed{p})\;, \\
\nonumber
T_{3\mu}(k,p) &=& q^{2}\gamma_{\mu}-q_{\mu} \; \slashed{q}\;, \\
\nonumber
T_{4\mu}(k,p) &=& q^{2}[\gamma_{\mu} \; (\slashed{k}+\slashed{p})-(p+k)_{\mu}]\\
\nonumber
&+&2(p-k)_{\mu} \; k_{\nu} \; p_{\lambda} \; \sigma_{\nu\lambda}\;, \\
\nonumber
T_{5\mu}(k,p) &=& q_{\nu} \; \sigma_{\nu\mu}\;, \\
\nonumber
T_{6\mu}(k,p) &=& \gamma_{\mu} \; (p^{2}-k^{2})+(p+k)_{\mu} \; \slashed{q}\;, \\
\nonumber
T_{7\mu}(k,p) &=& \frac{1}{2} \; (p^{2}-k^{2})[\gamma_{\mu} \; (\slashed{k}+\slashed{p})-(p+k)_{\mu}]
\\
\nonumber
&+&(k+p)_{\mu} \; k_{\nu} \; p_{\lambda} \; \sigma_{\nu\lambda}\;, \\
T_{8\mu}(k,p) &=& -\gamma_{\mu} \; k_{\nu} \; p_{\lambda} \; \sigma_{\nu\lambda}+k_{\mu} \; \slashed{p}-p_{\mu} \; \slashed{k}\;.\end{aligned}$$ This is the most general vector structure that can be constructed. In 1990, Curtis and Pennington [@Curtis:1990zs] realized that a simple choice of only of the transverse coefficients is enough to ensure multiplicative renormalizability of the massless quark propagator in the quenched approximation of QED. The full form of the vertex, the CP vertex, is $$\Gamma_{\mu} = \Gamma^{(CP)}_{\mu} = \Gamma^{(BC)}_{\mu}+\Gamma^{T(CP)}_{\mu},$$ where the correction term is given by $$\begin{aligned}
\Gamma^{T(CP)}_{\mu} = \frac{\gamma_{\mu} \; (k^{2}-p^{2})-t_{\mu} i \gamma \cdot t}{2 \; d(k,p)} [A(k^{2})-A(p^{2})]\;,\end{aligned}$$ where $t=k+p$ and $$\begin{aligned}
d(k,p) &=& \frac{(k^{2}-p^{2})^{2}+\left[M^2(k^2)+ M^2(p^2)\right]^{2}}{k^{2}+p^{2}}\,.\end{aligned}$$ An unquenched version, in the chiral limit, was introduced in 2009 by Kizilersu and Pennington (KP) [@Kizilersu:2009kg]. They proposed two vertex constructions that satisfy all necessary constraints but differ only beyond the leading logarithmic order. Notably, both constructions give similar results in the Landau gauge [@Kizilersu:2009kg; @Kizilersu:2013hea]. So one could use either. We have used the following: $$\Gamma^{T(KP)}_{\mu} = \tau_{2}T_{2\mu}+\tau_{3}T_{3\mu}+\tau_{6}T_{6\mu}+\tau_{8}T_{8\mu},$$ where $$\begin{aligned}
\nonumber
\tau_{2}(k^{2},p^{2},q^{2}) &=& -\frac{4}{3} \; \frac{1}{k^{4}-p^{4}} \; (A(k^{2})-A(p^{2}))
\\
\nonumber
&-&\frac{1}{3} \; \frac{A(k^{2})+A(p^{2})}{(k^{2}+p^{2})^{2}} \; \; \ln\left[\frac{A(k^{2}) \; A(p^{2})}{A(q^{2})^{2}}\right]\;, \\
\nonumber
\tau_{3}(k^{2},p^{2},q^{2}) &=& -\frac{5}{12}\frac{1}{k^{2}-p^{2}} \; (A(k^{2})-A(p^{2}))\;, \\
\nonumber
&-&\frac{1}{6}\frac{A(k^{2})+A(p^{2})}{(k^{2}+p^{2})^{2}} \; \; \ln\left[\frac{A(k^{2}) \; A(p^{2})}{A(q^{2})^{2}}\right]\;, \\ \nonumber
\tau_{6}(k^{2},p^{2},q^{2}) &=& \frac{1}{4}\frac{1}{k^{2}+p^{2}} \; (A(k^{2})-A(p^{2}))\;, \\
\tau_{8}(k^{2},p^{2},q^{2}) &=& 0\;.\end{aligned}$$ In 2012, Bashir [*et al.*]{}(BB) [@Bashir:2011dp] presented a fermion-boson vertex expressed solely in terms of the vector and scalar functions appearing in the fermion propagator, which is always consistent with one-loop perturbation theory and is independent of the angle between the relative momenta. Strinkingly, it has also no explicit dependance on the covariant-gauge parameter although the gauge-independence of the critical coupling in QED, above which chiral symmetry is broken, is achieved. The transverse structure is chosen as $$\begin{aligned}
\label{eq:2.69}
\tau_{1}(k^{2},p^{2}) &= \frac{a_{1} \Delta_{B}(k^{2},p^{2})}{(k^{2} + p^{2})}\;, \nonumber \\
\tau_{2}(k^{2},p^{2}) &= \frac{a_{2} \Delta_{A}(k^{2},p^{2})}{(k^{2} + p^{2})}\;, \nonumber \\
\tau_{3}(k^{2},p^{2}) &= a_{3} \Delta_{A}(k^{2},p^{2})\;, \nonumber \\
\tau_{4}(k^{2},p^{2}) &= \frac{a_{4} \Delta_{B}(k^{2},p^{2}) }{[k^2 + M^2(k^2)][p^2 + M^2(p^2)]}\;, \nonumber \\
\tau_{5}(k^{2},p^{2}) &= a_{5} \Delta_{B}(k^{2},p^{2})\;, \nonumber \\
\tau_{6}(k^{2},p^{2}) &= \frac{a_{6} (k^2 + p^2) \Delta_{A}(k^{2},p^{2}) }{[(k^2 - p^2)^{2} + (M^2(k^2) + M^2(p^2))^{2}]}\;, \nonumber \\
\tau_{7}(k^{2},p^{2}) &= \frac{a_{7} \Delta_{B}(k^{2},p^{2})}{(k^{2} + p^{2})}\;, \nonumber \\
\tau_{8}(k^{2},p^{2}) &= a_{8} \Delta_{A}(k^{2},p^{2})\;.\end{aligned}$$ where. The values of the momentum independent constants, $\{ a_{i}, i = 1,2, ...,8 \}$ are shown in Table \[table:tab21\]. The fixing procedure of these constants can be found in Ref. [@Bashir:2011dp]. In the next section we will discuss eq. (\[eq:2\]) along with (\[eq:1\]) for all these vertices.
---------- --------- --------- --------- --------- --------- --------- --------- ---------
Constant $a_{1}$ $a_{2}$ $a_{3}$ $a_{4}$ $a_{5}$ $a_{6}$ $a_{7}$ $a_{8}$
Value 0 3.4 1 6 -4/3 -1/2 -1/3 -3.7
---------- --------- --------- --------- --------- --------- --------- --------- ---------
: \[table:tab21\]Values of the momentum independent constants.
Gap equation {#sec:gapequation}
============
Equations for $B(k^{2})$ and $A(k^{2})$ can be obtained from proper projections of eq. \[eq:1\], that is, multiplying eq. \[eq:1\] by $\textbf{1},\;$ $\slashed{p},$ respectively, and then taking Dirac traces. In the minimal approximation of the fully dressed quark-gluon vertex, the bare vertex $ \Gamma_{\mu} = \gamma_{\mu},$ quark self-energy given by equation (\[eq:2\]) acquires the following simple form $$\label{eq:bare}
\Sigma(p) = Z_{1}\int^{\Lambda_{UV}}\frac{d^{4}k}{(2\pi)^{4}} \; C_{F} \; g^2 \; \gamma_{\mu} \; S(k) \; D_{\mu\nu}(q) \; \gamma_{\nu}.$$ Thus, one arrives at the following expressions: $$\begin{aligned}
B(p^{2}) &=& m_{0} \; Z_2+16 \; \pi Z_1\int^{\Lambda_{UV}}\frac{d^{4}k}{(2\pi)^{4}}\frac{\alpha_{s}(q^{2})}{q^{2}}
\\
\label{intbare1}
&\times& \frac{B(k^{2})}{A^{2}(k^{2}) \; k^{2}+B^{2}(k^{2})}\;,\\
\nonumber
A(p^{2}) &=& Z_2+\frac{16 \; \pi}{3 \; p^{2}}Z_1\int^{\Lambda_{UV}}\frac{d^{4}k}{(2 \; \pi)^{4}}\frac{\alpha_{s}(q^{2})}{q^{2}}
\\
\label{intbare2}
&\times& \frac{A(k^{2})}{A^{2}(k^{2}) \; k^{2}+B^{2}(k^{2})}\left[k\cdot p+\frac{2 \; k\cdot q \; p\cdot q}{q^{2}}\right]. \qquad\end{aligned}$$ Notably, all of the QCD’s gauge sector contributions are effectively contained in the models for $\alpha_s(q^2)$, thus arriving at an abelianized version of the theory, such that $Z_{2}(\mu^{2},\Lambda^{2}_{UV})=Z_{1}(\mu^{2},\Lambda^{2}_{UV})$, which are fixed by applying the renormalization boundary condition $$\begin{aligned}
S^{-1}(p)|_{p^2=\mu^2}=i\slashed{p}+m_\mu\;,\end{aligned}$$ where $m_\mu$ is a scale dependent renormalization mass. It implies $A(p^2)_{p^2=\mu^2} =1,\; B(p^2)_{p^2=\mu^2} = m_\mu$. This procedure applies for all the rest of the truncations explored herein, since it is a feature of the effective gluon models employed. In a similar way, substituting BC vertex from eq. \[eq:BCV\] in eq. (\[eq:2\]), one arrives at the following expresion for $B(p^{2})$ $$\begin{gathered}
\label{eq:23}
B(p^{2}) = m_{0}\; Z_1+\frac{16 \; \pi}{3}Z_1\int^{\Lambda_{UV}}\frac{d^{4}k}{(2 \; \pi)^{4}}\frac{\alpha_{s}(q^{2})}{q^{2}}
\\
\times \frac{1}{A^{2}(k^{2})\; k^{2}+B^{2}(k^{2})} \{ I_{B1}^{BC}+I_{B2}^{BC}-I_{B3}^{BC} \},\end{gathered}$$ where $I_{B1}^{BC}$, $I_{B2}^{BC}$ and $I_{B3}^{BC}$ are the integrands related to the BC vertex, such that $$\begin{aligned}
I_{B1}^{BC} &=& 3 B(k^{2})\frac{A(k^{2})+A(p^{2})}{2}, \\
I_{B2}^{BC} &=& B(k^{2}) \Delta_A(k^2,p^2)
\left\{ \frac{t^{2} \; q^{2}- ( t\cdot q )^{2}}{ 2q^{2}}\right\}, \\
I_{B3}^{BC} &=& A(k^{2}) \Delta_B(k^2,p^2)
\left\{ \frac{q^{2}t \cdot k-t \cdot q \; q\cdot k}{q^{2}}\right\}.\end{aligned}$$ Analogously, the corresponding equation for $A(p^2)$, for the BC vertex, is $$\begin{aligned}
\nonumber
A(p^{2}) &=& Z_1+\frac{16 \; \pi}{3}Z_1\int^{\Lambda_{UV}}\frac{d^{4}k}{(2 \; \pi)^{4}}
\\
\label{eq:24}
&\times& \frac{\alpha_{s}(q^{2})}{q^{2}}\frac{ I_{A1}^{BC}-I_{A2}^{BC}+I_{A3}^{BC}}{A^{2}(k^{2}) \; k^{2}+B^{2}(k^{2})},\end{aligned}$$ where the integrands are written as $$\begin{aligned}
I_{A1}^{BC} &=& A(k^{2})\frac{A(k^{2})+A(p^{2})}{2}\frac{1}{p^{2}}
\\
&\times& \left\{\frac{k\cdot p\; q^{2}+2 \; [(k^{2}+p^{2}) \; k\cdot p-k^{2} \; p^{2}-k\cdot p^{2}]}{q^{2}} \right\}\;, \\
\nonumber
I_{A2}^{BC} &=& \frac{A(k^{2})}{2p^{2}} \Delta_A(k^2,p^2)
\\
&\times&
\bigg\{[p^2 k + k^2 p] \cdot t
- \frac{p^{2} t\cdot q \; k\cdot q - k^{2} t \cdot q \; p\cdot q}{q^{2}} \bigg\}\;,\\
\nonumber
I_{A3}^{BC} &=& B(k^{2}) \Delta_B(k^2,p^2) \frac{1}{p^{2}} \left\{\frac{t\cdot q \; p\cdot q-t\cdot p \; q^{2}}{q^{2}} \right\}\;.\end{aligned}$$ By taking into account the transverse correction term put forward by Curtis-Pennington [@Curtis:1990zs], one arrives at $$\begin{aligned}
\nonumber
B(p^{2}) &=& m_{0}Z_1+\frac{16\pi}{3}Z_1\int^{\Lambda_{UV}}\frac{d^{4}k}{(2\pi)^{4}}\frac{\alpha_{s}(q^{2})}{q^{2}} \\
\nonumber
&\times& \frac{1}{A^{2}(k^{2})k^{2}+B^{2}(k^{2})} \{ I_{B1}^{BC}+I_{B2}^{BC}-I_{B3}^{BC} \\
&+& \frac{3}{2}B(k^{2})(k^{2}+p^{2})\;L(k^{2},p^{2}) \},\end{aligned}$$ where $L(k^{2},p^{2})$ is defined as $$\begin{aligned}
\nonumber
L =
\frac{[A^{2}(k^{2})A^{2}(p^{2})]^{2}
\Delta_A(k^2,p^2) }{[A^{2}(k^{2})A^{2}(p^{2})]^{2}+[A^{2}(p^{2})B^{2}(k^{2})+A^{2}(k^{2})B^{2}(p^{2})]^{2}}. \quad
\nonumber\end{aligned}$$ On the other hand, the corresponding expression for $A(p^2)$ reads as $$\begin{aligned}
\nonumber
A(p^{2}) &=& Z_1+\frac{16 \hspace{0.5mm} \pi}{3}Z_1\int^{\Lambda_{UV}}\frac{d^{4}k}{(2 \hspace{0.5mm} \pi)^{4}}\frac{\alpha_{s}(q^{2})}{q^{2}}
\\
\nonumber
&\times& \frac{1}{A^{2}(k^{2}) \; k^{2}+B^{2}(k^{2})}\{ I_{A1}^{BC}-I_{A2}^{BC}+I_{A3}^{BC}
\\
&+& 2 \; A(k^{2})\frac{k^{2}+p^{2}}{k^2-p^2}(I_{A1}^{CP}+I_{A2}^{CP}) \; L(k^{2},p^{2}) \}. \quad\end{aligned}$$ The integrands $I_{A1}^{CP}$ and $I_{A2}^{CP}$, which are related to the CP term, are written as $$\begin{aligned}
I_{A1}^{CP} &=& (k^{2}-p^{2})\left\{\frac{3(k^{2}+p^{2})k\cdot p-2 k^{2} p^{2}-4k\cdot p^{2}}{q^{2}}\right\}, \\
I_{A2}^{CP} &=& k^{2} t \cdot p -p^{2} t\cdot k + \frac{p^{2} t \cdot q k\cdot q - k^{2} t \cdot q p\cdot q}{q^{2}}. \nonumber\end{aligned}$$ When employing KP vertex ansätz [@Kizilersu:2009kg], one arrives at the following equation for $B(p^{2})$ $$\begin{aligned}
\nonumber
B(p^{2}) &=& m_{0} \; Z_1+\frac{16 \; \pi}{3}Z_1\int^{\Lambda_{UV}}\frac{d^{4}k}{(2\pi)^{4}}\frac{\alpha_{s}(q^{2})}{q^{2}}
\\
\nonumber
&\times& \frac{1}{A^{2}(k^{2}) \; k^{2}+B^{2}(k^{2})} \{ I_{B1}^{BC}+I_{B2}^{BC}-I_{B3}^{BC}
\\
&+&B(k^{2})(I_{B1}^{KP}-I_{B2}^{KP}-I_{B3}^{KP}) \}\;.\end{aligned}$$ The integrands related specifically to the KP vertex, $I_{B1}^{KP}$, $I_{B2}^{KP}$ and $I_{B3}^{KP}$, are expressed as $$\begin{aligned}
I_{B1}^{KP} &=& 2 \; (k\cdot p^{2}-k^{2}p^{2})\bigg\{\frac{4}{3}\frac{A(k^{2})-A(p^{2})}{k^{4}-p^{4}}
\\
&+&\frac{1}{3}\frac{A(k^{2})+A(p^{2})}{(k^{2}+p^{2})^{2}} \; \ln\left[\frac{A(k^{2}) \; A(p^{2})}{A^{2}(q^{2})}\right]\bigg\}\;, \\
I_{B2}^{KP} &=& q^{2}\bigg\{\frac{5}{4} \Delta_A(k^2,p^2) \;,
\\
&+& \frac{1}{2}\frac{A(k^{2})+A(p^{2})}{k^{2}+p^{2}}\ln\left[\frac{A(k^{2}) \; A(p^{2})}{A^{2}(q^{2})}\right]\bigg\}\;, \\
I_{B3}^{KP} &=& \frac{3}{4}(k^{2}-p^{2})\frac{A(k^{2})-A(p^{2})}{k^{2}+p^{2}}\;.\end{aligned}$$ The analogous equation for $A(p^2)$, for KP vertex, is $$\begin{aligned}
\nonumber
A(p^{2}) &=& Z_1+\frac{16 \; \pi}{3}\int^{\Lambda_{UV}}\frac{d^{4}k}{(2\pi)^{4}}\frac{\alpha_{s}(q^{2})}{q^{2}}
\\
\nonumber
&\times& \frac{1}{A^{2}(k^{2}) \; k^{2}+B^{2}(k^{2})}\{ I_{A1}^{BC}-I_{A2}^{BC}+I_{A3}^{BC}
\\
&+&A(k^{2})(I_{A1}^{KP}+I_{A2}^{KP}-I_{A3}^{KP}) \}\;,\end{aligned}$$ where the related integrands are $$\begin{aligned}
I_{A1}^{KP} &=& \frac{(k^{2}+p^{2})(k^{2} \; p^{2}-k.p^{2})}{p^{2}}\bigg\{\frac{4}{3}\frac{A(k^{2})-A(p^{2})}{k^{4}-p^{4}}
\\
\nonumber
&+&\frac{1}{3}\frac{A(k^{2})+A(p^{2})}{(k^{2}+p^{2})^{2}} \; \ln\left[\frac{A(k^{2}) \; A(p^{2})}{A^{2}(q^{2})}\right]\bigg\}, \\
\nonumber
I_{A2}^{KP} &=& \frac{k\cdot p \; (4 \; k\cdot p-3 \; p^{2})+k^{2} \; (2 \; p^{2}-3 \; k\cdot p)}{p^{2}} \\
\nonumber
&\times& \bigg\{\frac{5}{12} \Delta_A(k^2,p^2) \\
&+&\frac{1}{6}\frac{A(k^{2})+A(p^{2})}{k^{2}+p^{2}} \; \ln\left[\frac{A(k^{2}) \; A(p^{2})}{A^{2}(q^{2})}\right]\bigg\},
\\
I_{A3}^{KP} &=& \frac{3}{4} \; A(k^{2}) \; [A(k^{2})-A(p^{2})]\frac{(k^{2}-p^{2}) \; }{(k^{2}+p^{2})}\frac{k\cdot p}{p^{2} }\;.\end{aligned}$$ Unlike the other vertex ans${\rm \ddot{a}}$tze, the transverse part of the KP vertex introduces a non-trivial angular dependence, related to the logarithmic terms which contain $A(q^2)$. Thus, the numerical evaluation of such integrals is much more complicated. In particular, this non-trivial angular dependence becomes more problematic when the quark propagator needs to be computed for complex values of momenta.\
Finally, the integral equations for the $B(p^2)$ and $A(p^2)$ using the BB vertex [@Bashir:2011dp] are $$\begin{aligned}
B(p^2) &= \mbox{r.h.s. of eq.} (\ref{eq:23}) \nonumber \\
&+ \frac{16\pi}{3} Z_1 \int \frac{d^4 k}{(2\pi)^4} \frac{\alpha_{s}(q^2)}{q^2} \frac{1}{k^2 A^2(k^2)+ B^2(k^2)} \nonumber \\
& \times \bigg\{ a_{1} A(k^2) \frac{k \cdot p^2-k^2 p^2}{k^2+p^2} \Delta_B(k^2,p^2) + 2 a_{2} B(k^2) \nonumber \\
& + 2 a_{2} B(k^2)\frac{k \cdot p^2-k^2 p^2}{k^2+p^2} \Delta_A(k^2,p^2) \nonumber\\
& - 3 a_{3} B(k^2) (k^2 + p^2 - 2 k \cdot p) \Delta_A(k^2,p^2) \nonumber \\
& - a_{4} A(k^2) \frac{k \cdot p \;(k^2 p^2-k \cdot p^2)}{ (k^2+ M^2(k^2)) (p^2+M^2(p^2))} \Delta_B(k^2,p^2) \nonumber \\
& - 3 a_{6} B(k^2) \frac{(k^2-p^2)(k^4-p^4) \Delta_A(k^2,p^2)}{(k^2-p^2 )^2 + ( M^2(k^2)+M^2(p^2) )^2 } \nonumber \\
& + 2 a_{7} A(k^2)\frac{k \cdot p\; (k \cdot p^2-k^2 p^2)}{(k^2+p^2)(k^2+p^2-2 k \cdot p)} \Delta_B(k^2,p^2) \nonumber \\
&+ 3 a_{8} B(k^2) k \cdot p \;\Delta_A(k^2,p^2) \bigg\} ,\end{aligned}$$ $$\begin{aligned}
A(p^2) &= \mbox{r.h.s. of eq.} (\ref{eq:24}) \nonumber \\
& + \frac{16\pi}{3} Z_1 \int \frac{d^4 k}{(2\pi)^4} \frac{\alpha_{s}(q^2)}{q^2} \frac{A(k^2)}{k^2 A^2(k^2)+ B^2(k^2)} \nonumber \\
&\times \bigg\{ a_{1} B(k^2) \frac{k \cdot p^2-k^2 p^2}{k^2+p^2} \Delta_B(k^2,p^2) \nonumber \\
& + a_{2} A(k^2) (k^2 p^2-k \cdot p^2) \Delta_A(k^2,p^2) \nonumber \\
& - a_{3} A(k^2) [k^2(3 k \cdot p-2 p^2) \nonumber \\
& \quad \quad \quad + k \cdot p \;(3 p^2-4 k \cdot p)]\Delta_A(k^2,p^2) \nonumber \\
& - a_{4} B(k^2) \frac{k \cdot p\; (k^2 p^2-k \cdot p^2)\Delta B(k^2,p^2)}{(k^2+ M^2(k^2))
(p^2+ M(p^2))} \nonumber \\
& - 3 a_{6} A(k^2)\Delta_A(k^2,p^2) \nonumber \\
& \quad \quad \quad \times \frac{(k^2-p^2) \; k \cdot p \; (k^4-p^4)}{(k^2-p^2 )^2 + ( M^2(k^2) + M^2(p^2))^2} \nonumber \\
& + a_{7} B(k^2) \Delta_B(k^2,p^2) \nonumber \\
& \quad \quad \quad \times \frac{k^4 p^2 - k^2 (k \cdot p+p^2)^2 + k \cdot p^2 (2 k \cdot p+p^2)}{(k^2+p^2)(k^2+p^2-2 k \cdot p)^2} \nonumber \\
&+ a_{8} A(k^2) \Delta A(k^2,p^2) \bigg[ \frac{4 k \cdot p^2 (p^2-k \cdot p)}{k^2+p^2-2 k \cdot p} \nonumber \\
&\quad \quad \quad + \frac{k^4 p^2 + k^2(2 k \cdot p^2 - 2 p^2 k \cdot p - p^4)}{k^2+p^2-2 k \cdot p} \bigg] \bigg\} .\end{aligned}$$ In the next section, we show and discuss results of the SDE for the quark propagator, employing the different quark-gluon vertex ans${\rm \ddot{a}}$tze described herein.
Results {#sec:results}
=======
We solved the SDE for the quark propagator for many values of the current quark mass: $ m_{u/d}=0.00374 \;$GeV, $ m_{s}=0.1 \;$GeV, $ m_{c}=1.0 \;$GeV, $ m_{b}=4.1 \;$GeV and in the chiral limit $m_q = 0$; the renormalization point is chosen as $\mu=2.85$ GeV. The results using the bare, BC, CP, KP and BB vertices, with the MT model of gluon propagator are shown in figures (2)-(6), respectively. In figures (7)-(11) we show the analogous resuls for the QC interaction. The parameters $\omega$ and $D$ are chosen such that the obtained chiral condensates are in agreement with modern estimates [@Williams:2007ey; @Brodsky:2010xf; @Chang:2011mu]. In table \[tab:condensates\], we show the computed condensates for different current quark masses, according to eq. (15) in ref. [@Williams:2007ey]. Obtained values for the different constituent quark masses, defined as $M(p^2=0)$, are shown in table \[tab:constitutemass\]. In the infrared region, the mass function saturates at a finite value and monotonically decreases as $p^2$ increases. The saturation point, in comparison with the current quark mass, is much larger in the case of the light quarks. In fact, the mass function decreases sharply with $p^2$ for light quarks whereas it is approximately constant for the heavy ones. Notably, even in the chiral limit, it is possible to generate mass and a non-zero value of the condensate (which is a primary order parameter for DCSB, [@Chang:2011mu]). This important facet of QCD cannot be achieved in perturbation theory. Dynamical mass generation via strong-interaction processes (DCSB) is the dominant mass generating mechanism in the light sector, while the heavy sector is more affected by its larger coupling to the Higgs field. A corollary of the last fact, is that the extracted condensate tends to a constant as the current quark mass increases. The same trend is observed in [@Brodsky:2010xf]. Finally, it is observed that the effects of the dressing of the vertices diminish as the current quark mass increases, producing almost the same mass function and the same quark condensate. Notwithstanding that, to produce the same quantitative results in the light sector, the minimal approximation, the bare quark-gluon vertex, requires the largest compensation from the effective gluon, as can be read from the $m_G = (\omega D)^{1/3}$ parameters shown in tables \[tab:condensates\] and \[tab:constitutemass\]. The above statements are clear signs of the intimate connection between DCSB and the quark-gluon vertex.
![Quark mass function for different current quark masses ($ m_{u/d}=0.00374 \;$GeV, $ m_{s}=0.1 \;$GeV, $ m_{c}=1.0 \;$GeV, $ m_{b}=4.1 \;$GeV and the chiral limit $m_q = 0$), obtained with the **bare** quark-gluon vertex and **MT** effective gluon. \[fig:MTBare\]](MTBare.pdf){width="8cm"}
![Quark mass function for different current quark masses ($ m_{u/d}=0.00374 \;$GeV, $ m_{s}=0.1 \;$GeV, $ m_{c}=1.0 \;$GeV, $ m_{b}=4.1 \;$GeV and the chiral limit $m_q = 0$), obtained with the **BC** quark-gluon vertex and **MT** effective gluon. \[fig:MTBC\]](MTBC.pdf){width="8cm"}
![Quark mass function for different current quark masses ($ m_{u/d}=0.00374 \;$GeV, $ m_{s}=0.1 \;$GeV, $ m_{c}=1.0 \;$GeV, $ m_{b}=4.1 \;$GeV and the chiral limit $m_q = 0$), obtained with the **CP** quark-gluon vertex and **MT** effective gluon. \[fig:MTCP\]](MTCP.pdf){width="8cm"}
![Quark mass function for different current quark masses ($ m_{u/d}=0.00374 \;$GeV, $ m_{s}=0.1 \;$GeV, $ m_{c}=1.0 \;$GeV, $ m_{b}=4.1 \;$GeV and the chiral limit $m_q = 0$), obtained with the **KP** quark-gluon vertex and **MT** effective gluon. \[fig:MTKP\]](MTKP.pdf){width="8cm"}
![Quark mass function for different current quark masses ($ m_{u/d}=0.00374 \;$GeV, $ m_{s}=0.1 \;$GeV, $ m_{c}=1.0 \;$GeV, $ m_{b}=4.1 \;$GeV and the chiral limit $m_q = 0$), obtained with the **BB** quark-gluon vertex and **MT** effective gluon. \[fig:MTBB\]](MTBB.pdf){width="8cm"}
![Quark mass function for different current quark masses ($ m_{u/d}=0.00374 \;$GeV, $ m_{s}=0.1 \;$GeV, $ m_{c}=1.0 \;$GeV, $ m_{b}=4.1 \;$GeV and the chiral limit $m_q = 0$), obtained with the **bare** quark-gluon vertex and **QC** effective gluon. \[fig:QCBare\]](QCBare.pdf){width="8cm"}
![Quark mass function for different current quark masses ($ m_{u/d}=0.00374 \;$GeV, $ m_{s}=0.1 \;$GeV, $ m_{c}=1.0 \;$GeV, $ m_{b}=4.1 \;$GeV and the chiral limit $m_q = 0$), obtained with the **BC** quark-gluon vertex and **QC** effective gluon. \[fig:QCBC\]](QCBC.pdf){width="8cm"}
![Quark mass function for different current quark masses ($ m_{u/d}=0.00374 \;$GeV, $ m_{s}=0.1 \;$GeV, $ m_{c}=1.0 \;$GeV, $ m_{b}=4.1 \;$GeV and the chiral limit $m_q = 0$), obtained with the **CP** quark-gluon vertex and **QC** effective gluon. \[fig:QCCP\]](QCCP.pdf){width="8cm"}
![Quark mass function for different current quark masses ($ m_{u/d}=0.00374 \;$GeV, $ m_{s}=0.1 \;$GeV, $ m_{c}=1.0 \;$GeV, $ m_{b}=4.1 \;$GeV and the chiral limit $m_q = 0$), obtained with the **KP** quark-gluon vertex and **QC** effective gluon. \[fig:QCKP\]](QCKP.pdf){width="8cm"}
![Quark mass function for different current quark masses ($ m_{u/d}=0.00374 \;$GeV, $ m_{s}=0.1 \;$GeV, $ m_{c}=1.0 \;$GeV, $ m_{b}=4.1 \;$GeV and the chiral limit $m_q = 0$), obtained with the **BB** quark-gluon vertex and **QC** effective gluon. \[fig:QCBB\]](QCBB.pdf){width="8cm"}
Vertex ans$\ddot{a}$tz $ \omega \;$ $ m_G \;$ $ m_{q}=0 \;$ $ m_{u/d}=0.00374 \;$ $ m_{s}=0.1 \;$ $ m_{c}=1.0 \;$ $ m_{b}=4.1 \;$
------------------------------------------ -------------- ----------- --------------- ----------------------- ----------------- ----------------- -----------------
MT Model
Bare 0.40 0.663 0.246 0.260 0.436 0.763 0.772
Ball-Chiu [@Ball:1980ay] 0.30 0.485 0.250 0.262 0.425 0.761 0.773
Curtis-Pennington [@Curtis:1990zs] 0.30 0.462 0.248 0.261 0.428 0.763 0.773
Kizilersu-Pennington [@Kizilersu:2009kg] 0.32 0.458 0.243 0.257 0.430 0.763 0.773
Bashir-Bermudez [@Bashir:2011dp] 0.30 0.371 0.249 0.260 0.431 0.770 0.773
QC Model
Bare 0.50 0.737 0.249 0.263 0.438 0.763 0.772
Ball-Chiu [@Ball:1980ay] 0.30 0.513 0.243 0.255 0.423 0.759 0.773
Curtis-Pennington [@Curtis:1990zs] 0.30 0.481 0.247 0.259 0.426 0.761 0.773
Kizilersu-Pennington [@Kizilersu:2009kg] 0.33 0.470 0.249 0.262 0.432 0.763 0.773
Bashir-Bermudez [@Bashir:2011dp] 0.30 0.428 0.245 0.257 0.420 0.765 0.773
Vertex ans$\ddot{a}$tz $ \omega \;$ $ m_G \;$ $ m_{q}=0 \;$ $ m_{u/d}=0.00374 \;$ $ m_{s}=0.1 \;$ $ m_{c}=1.0 \;$ $ m_{b}=4.1 \;$
------------------------------------------ -------------- ----------- --------------- ----------------------- ----------------- ----------------- -----------------
MT Model
Bare 0.40 0.663 0.436 0.446 0.612 1.434 4.221
Ball-Chiu [@Ball:1980ay] 0.30 0.485 0.401 0.405 0.502 1.290 4.188
Curtis-Pennington [@Curtis:1990zs] 0.30 0.462 0.388 0.393 0.489 1.273 4.185
Kizilersu-Pennington [@Kizilersu:2009kg] 0.32 0.458 0.331 0.337 0.455 1.272 4.183
Bashir-Bermudez [@Bashir:2011dp] 0.30 0.371 0.337 0.334 0.474 1.378 4.177
QC Model
Bare 0.50 0.737 0.459 0.469 0.635 1.457 4.225
Ball-Chiu [@Ball:1980ay] 0.30 0.513 0.375 0.372 0.495 1.355 4.191
Curtis-Pennington [@Curtis:1990zs] 0.30 0.481 0.408 0.411 0.495 1.281 4.187
Kizilersu-Pennington [@Kizilersu:2009kg] 0.33 0.470 0.338 0.344 0.463 1.276 4.184
Bashir-Bermudez [@Bashir:2011dp] 0.30 0.428 0.449 0.453 0.551 1.319 4.191
Conclusions and Scope {#sec:conclusions}
=====================
Features of the dressed-quark-gluon vertex and their role in the gap equation have been explored. In particular, we solve the SDE for the quark propagator, using the following vertex ans$\ddot{a}$tze: bare, BC, CP, KP, BB. Light quarks, which are weakly coupled to the Higgs field, owe their mass primarily to the strong QCD dynamics. This is qualitatively a robust feature of the SDE studies, independent of the choice of the quark-gluon vertex. The quantitative details, to which several hadron observables, such as elastic and transition form factors, may be sensitive to, can distinguish between the choice of the three point vertex.
In fact there is a natural interplay between the role of the gluon propagator and the quark-gluon vertex. For example, the bare vertex requires larger infrared enhancement in the gluon propagator, in order to generate a phenomenologically acceptable amount of mass (and chiral quark condensate). It also works the other way around: a realistic and currently converging understanding of the gluon propagator can generate an acceptable running quark mass via QCD’s gap equation only so long as the quark-gluon vertex exhibits material infrared enhancement itself. Thus an intimate connection between the quark-gluon vertex and DCSB, via the gap equation, is established. In fact, half of the structures which define this vertex only appear if chiral symmetry is dunamically broken.
In our current study, within a small variation of MT and QC model parameters, all truncations employed herein point towards the same qualitative pattern of chiral symmetry breaking with fairly good quantitative comparisons. Thus the running quark mass function can be viewed as a robust feature of the strong interactions of QCD and not a model dependent artefact. Next immediate step is to investigate if the vertices ans$\ddot{a}$tze studied in this work are suitable for use in non-perturbative studies of sophisticated hadron physics phenomenology (such as ground-state masses and decay constants). Therefore, a first task is to write a consistent Bethe-Salpeter kernel for all those vertices. For the bare vertex, it is well-known that a ladder-like kernel is sufficient for many needs, providing an accurate description of pseudo-scalars and light-vector mesons (see for example, [@Chang:2011ei; @Chang:2013pq; @Ding:2015rkn; @Raya:2015gva; @Raya:2016yuj]); it corresponds to the popular rainbow-ladder truncation. Nevertheless, for a fully-dressed quark-gluon vertex, the construction of a consistent Bethe-Salpeter kernel could be an outstanding challenge [@Chang:2009zb; @Binosi:2016rxz; @Qin:2016fbu]. Moreover, DCSB generates a momentum-dependent dressed-quark anomalous chromomagnetic moment, which is large at infrared momenta and has a non negligible impact on a good description of axial-vector mesons [@Chang:2010hb; @Bashir:2011dp]. All of those aspects are currently being investigated.
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---
abstract: 'It is a classical result that any finite tree with positively weighted edges, and without vertices of degree 2, is uniquely determined by the weighted path distance between each pair of leaves. Moreover, it is possible for a (small) strict subset ${{\mathcal L}}$ of leaf pairs to suffice for reconstructing the tree and its edge weights, given just the distances between the leaf pairs in ${{\mathcal L}}$. It is known that any set ${{\mathcal L}}$ with this property for a tree in which all interior vertices have degree 3 must form a [*cover*]{} for $T$ – that is, for each interior vertex $v$ of $T$, ${{\mathcal L}}$ must contain a pair of leaves from each pair of the three components of $T-v$. Here we provide a partial converse of this result by showing that if a set ${{\mathcal L}}$ of leaf pairs forms a cover of a certain type for such a tree $T$ then $T$ and its edge weights can be uniquely determined from the distances between the pairs of leaves in ${{\mathcal L}}$. Moreover, there is a polynomial-time algorithm for achieving this reconstruction. The result establishes a special case of a recent question concerning ‘triplet covers’, and is relevant to a problem arising in evolutionary genomics.'
address: |
(K.T.H.): School of Computing Sciences, University of East Anglia, UK; phone: +44 (0) 1603 593211; FAX: +44 (0) 1603 593345.\
(M.S.): Department of Mathematics and Statistics, University of Canterbury, New Zealand.
author:
- 'Katharina T. Huber and Mike Steel'
title: 'Reconstructing fully-resolved trees from triplet cover distances'
---
[**Keywords:**]{} phylogenetic tree, tree metric, tree reconstruction, triplet cover.
[**Email:**]{} [email protected] (Corresponding author)
Introduction {#intro}
============
Any tree $T$ with positively weighted edges, induces a metric $d$ on the set of leaves by considering the weighted path distance in $T$ between each pair of leaves. Moreover, provided $T$ has no vertices of degree 2, and that we ignore the labeling of interior vertices, both $T$ and its edge weights are uniquely determined by the metric $d$. This uniqueness result has been known since the $1960$s and fast algorithms exist for reconstructing both the tree and its edge weights from $d$ (for further background the interested reader may consult [@bar] and [@sem] and the references therein).
The uniqueness result and the algorithms are important in evolutionary biology for reconstructing an evolutionary tree of species from genetic data [@fel]. However in this setting one frequently may not have $d$-values available for all pairs of species, due to the patchy nature of genomic coverage [@san].
This raises a fundamental mathematical question – for which subsets of pairs of leaves of a tree do we need to know the $d$-values in order to uniquely recover the tree and its edge weights? In general this appears a difficult question (indeed determining whether such a partial $d$-metric is realized by [*any*]{} tree is NP-hard [@far]). However, some sufficient conditions (as well as some necessary conditions) for uniqueness to hold have been found, in [@cha; @yus], and more recently in [@dre3], and [@gue]. In this paper we consider the uniqueness question for trees that are ‘fully-resolved’ (i.e. all the interior vertices have degree 3) as these trees are of particular importance in evolutionary biology, and because the uniqueness question is easier to study for this class of trees.
The structure of this paper is as follows. First we introduce some background terminology and concepts, and then we define the particular type of subsets of leaf pairs (called ‘stable triplet covers’) which we show suffice to uniquely determine a fully-resolved tree. Moreover, we show how this comes about by establishing two combinatorial properties of stable triplet covers - a ‘shellability’ property and a graph-theoretic property related to tree-width, which we show is quite different to shellability. We conclude by providing a proof that a polynomial-time algorithm will reconstruct a tree and its edge weights for any set of leaf pairs that contains a stable triplet cover (or more generally a shellable subset). Our result answers a special case of the question posed at the end of [@dre3] of whether every ‘triplet cover’ of a fully-resolved tree determines the tree and its edge weights.
Preliminaries
=============
We now introduce some precise definitions required to state and prove our main results. We mostly follow the notation and terminology of [@sem] and [@dre3].
$X-$trees, edge-weightings and distances
----------------------------------------
For the rest of the paper, assume that $|X|\geq 3$. An $X-$tree $T=(V,E)$ is a graph theoretical tree whose leaf set is $X$ and which does not have any vertices of degree 2. We call an $X-$tree [*fully-resolved*]{} if every [*interior vertex*]{} of $T$, that is, every non-leaf vertex of $T$, has degree three. Moreover, we call two distinct leaves $x$ and $y$ of $T$ a [*cherry*]{} of $T$, denoted by $x,y$, if the parent of $x$ is simultaneously the parent of $y$. For any subset $Y\subseteq X$, we denote by $T|Y$ the $Y$-tree obtained by restricting $T$ to $Y$ (suppressing resulting degree two vertices).
An example of a fully-resolved $X-$tree for $X=\{a,b,c,d,e,f,g\}$, and having two cherries, is shown in Fig. \[figure1\](i).
In case $|Y|=4$, say $Y=\{a,b,c,d\}$, and the path from $a$ to $b$ does not share a vertex with the path from $c$ to $d$ in $T|Y$, we refer to $T|Y$ as a [*quartet tree*]{} and denote it by $ab||cd$. Note that by deleting any edge $e\in E$ from $T$ the leaf sets $A_e$ and $B_e:=X-A_e$ of the resulting two trees induce a bipartition of $X$. We refer to such a bipartition as [*$X-$split*]{} and denote it by $A|B$ where $A:=A_e$ and $B:=B_e$ and $A_e$ and $B_e$ are as above. We say that two $X-$trees $T=(V,E)$ and $T'=(V',E')$ are [*equivalent*]{} if there exists a bijection $\phi:V\to V'$ that is the identity on $X$ and extends to a graph isomorphism from $T$ to $T'$.
Suppose for the following that $T=(V,E)$ is an $X-$tree. Then we call a map $w:E\to \mathbb R_{\geq 0}$ that assigns a [*weight*]{}, that is, a non-negative real number, to every edge of $T$ an [*edge-weighting*]{} for $T$. Note that this definition implies that some of the edges of $T$ might have weight zero. We denote an $X-$tree $T$ together with an edge-weighting $w$ by the pair $(T,w)$ and call an edge-weighting that assign non-zero weight to every edge of $T$ that is not incident with a leaf of $T$ [*proper*]{}. Note that for any edge-weighting $w$ of $T$, taking the sum of the weights of the edges on the shortest path from some $x\in X$ to some $y\in X$ induces a distance between $x$ and $y$ and thus a distance $d=d_{(T,w)}$ on $X$.
For example, in the tree in Fig. \[figure1\](i), if each edge has weight 1, then $d(a,b)=2, d(c,e)=4$, and $d(c,f)=5$.
Lassos
------
We call a subset of $X$ of size two a [*cord*]{} of $X$ and, for $a,b\in X$ distinct write $ab$ rather than $\{a,b\}$ for the cord containing $a$ and $b$. Also, for any non-empty set ${{\mathcal L}}\subseteq {X\choose 2}$ of cords of $X$, we denote the edges of the graph $(X,{{\mathcal L}})$ whose vertex set is $X$ and whose edge set is the set $\{\{a,b\}: ab\in{{\mathcal L}}\}$ by $ab$ rather than $\{a,b\}$, $ab\in {{\mathcal L}}$.
Suppose for the following that ${{\mathcal L}}\subseteq {X\choose 2}$ is a non-empty set of cords of $X$. If $T'=(V',E')$ is a further $X-$tree and $w$ and $w'$ are edge-weightings for $T$ and $T'$, respectively, such that $d_{(T,w)}(x,y)=d_{(T',w')}(x,y)$ holds for all $xy\in {{\mathcal L}}$ then we say that $(T,w)$ and $(T',w')$ are [*${{\mathcal L}}$-isometric*]{}. Moreover we say that ${{\mathcal L}}$ is
1. an [*edge-weight lasso*]{} for $T$ if for any two proper edge-weightings $w$ and $w'$ for $T$ such that $(T,w)$ and $(T,w')$ are ${{\mathcal L}}$-isometric we have that $w=w'$.
2. a [*topological lasso*]{} for $T$ if for any other $X-$tree $T'$ and any two proper edge-weightings $w$ and $w'$ for $T$ and $T'$, respectively, such that $(T,w)$ and $(T',w')$ are ${{\mathcal L}}$-isometric we have that $T$ and $T'$ are equivalent.
3. a [*strong lasso*]{} for $T$ if ${{\mathcal L}}$ is simultaneously an edge-weight and a topological lasso for $T$.
If ${{\mathcal L}}$ is a strong lasso for an $X-$tree then the graph $(X, {{\mathcal L}})$ must be connected, and each component of this graph must be non-bipartite [@dre3]. An example of a strong lasso ${{\mathcal L}}$ of the tree in Fig. \[figure1\](i) is the set of cords corresponding to the edges of the graph in Fig. \[figure1\](ii).
Shellability
------------
Given a subset ${{\mathcal L}}$ of ${{X \choose 2}}$ with $X=\bigcup {{\mathcal L}}$, and an $X-$tree $T$, we say that ${{X \choose 2}}-{{\mathcal L}}$ is [*$T$–shellable*]{} if there exists an ordering of the cords in ${{X \choose 2}}-{{\mathcal L}}$ as, say, $a_1b_1, a_2b_2, \dots, a_mb_m$ such that, for every $\mu\in \{1,2,\dots,m\}$, there exists a pair $x_\mu,y_\mu$ of ‘pivots’ for $a_\mu b_\mu$, i.e., two distinct elements $x_\mu,y_\mu\in X-\{a_\mu,b_\mu\}$, for which the tree $T|{Y_\mu}$ obtained from $T$ by restriction to $Y_\mu:=\{a_\mu,b_\mu,x_\mu,y_\mu\}$, is the quartet tree $a_\mu x_\mu||y_\mu b_\mu$, and all cords in $\binom{Y_\mu}{2}$ except $a_\mu b_\mu$ are contained in ${{\mathcal L}}_\mu:={{\mathcal L}}\cup \big\{a_{\mu'}b_{\mu'}: \mu'\in
\{1,2,\dots,\mu-1\}\big\}$. Any such ordering of ${{X \choose 2}}-{{\mathcal L}}$ will also be called a [*shellable ordering*]{} of ${{X \choose 2}}-{{\mathcal L}}$, and any subset ${{\mathcal L}}$ of ${{X \choose 2}}$ for which a shellable ordering of ${{X \choose 2}}-{{\mathcal L}}$ exists will also be called an [*shellable lasso for $T$*]{}. In [@dre3 Theorem 6], it was established that every shellable lasso for an $X-$tree is in particular a strong lasso for that tree.
Example 1
---------
Consider the seven-taxon tree, shown in Fig. \[figure1\](i), and the lasso ${{\mathcal L}}= \{ab, bd, ad, bc, bf, ag, dg, eb, ef, fg, gc\}$ (the edges of the graph in Fig. \[figure1\](ii)). The remaining ten chords in $\binom{X}{2}-{{\mathcal L}}$ have a shellable ordering, described as follows: $$bg, cd, ac, cf, ce, af, df, ae, eg, ed,$$ where the corresponding cord pivots are: $$(a,d), (b,g), (b,d), (b,g), (b,f), (b,g), (b,g), (b,f), (a,f), (b,f),$$ and so ${{\mathcal L}}$ is a shellable (and hence strong) lasso for $T$. $\Box$
Covers, triplet covers
----------------------
A necessary condition for ${{\mathcal L}}\subseteq \binom{X}{2}$ to be a edge-weight lasso or a topological lasso for a fully-resolved $X-$tree is that ${{\mathcal L}}$ forms a [*cover*]{} for $T$ – that is, for each interior vertex $v$ of $T$, ${{\mathcal L}}$ contains a pair of leaves from each pair of the three components of $T-v$. However this condition is not sufficient for ${{\mathcal L}}$ to be either an edge-weight lasso or a topological lasso (examples are given in [@dre3]).
A particular type of cover for a fully-resolved $X-$tree is a [*triplet cover*]{} which is defined as any subset ${{\mathcal L}}$ of $\binom{X}{2}$ with the property that for each interior vertex $v$ of $T$ we can select leaves $a,b,c$ from each of the three components of $T-v$ so that $ab, ac, bc \in {{\mathcal L}}$. It can be shown that if ${{\mathcal L}}$ is a triplet cover for a fully-resolved $X-$tree $T$ then ${{\mathcal L}}$ is an edge-weight lasso. However it is not known whether or not every triplet cover of every such $T$ is also a topological (and thereby a strong) lasso for $T$.
A special class of triplet covers
=================================
Suppose that $T=(V,E)$ is a fully-resolved $X-$tree, and let $${\rm clus}(T) := \bigcup_{e \in E} \{A_e, X-A_e\},$$ where $A_e|(X-A_e)$ denotes the $X-$split associated with edge $e\in E$. We call the elements in ${\rm clus}(T)$ ‘clusters’ (in biology, they are also sometimes referred to as ‘clans’ [@wil]). Thus a cluster is a subset of $X$ that corresponds to the leaf labels on one side of some edge of $T$.
Given a collection ${{\mathcal C}}$ of non-empty subsets of $X$ we say that any function $f: {{\mathcal C}}\rightarrow X$ is a [*stable transversal*]{} for ${{\mathcal C}}$ if it satisfies the two properties:
- (transversality) $f(A) \in A$, for all $A \in {{\mathcal C}}$;
- (stability) $f(A) \in B \subseteq A \Longrightarrow f(A) = f(B)$ for all $A, B \in {{\mathcal C}}$.
Mostly we will be concerned with stable transversals for ${\rm clus}(T)$, which were introduced in [@boc], though for a different purpose.
Example 2
---------
An example of a stable transversals for ${\rm clus}(T)$ is as follows: Consider [*any*]{} stable transversal $g$ for $2^X$ (equivalently, the function $g(A) = \min A$ under some total ordering of $X$), and consider [*any*]{} proper edge weighting $w$ of $T$. For a cluster $A \in {\rm clus}(T)$, consider the subset $A_w$ of leaves of $T$ in $A$ that are a closest to the edge $e$ whose deletion induces the split $A|X-A$. Here ‘closest’ refers to the path distance in $T$ from each leaf in $A$ to $e$ under the edge weighting $w$. If we let $f(A) = g(A_w)$, for each $A \in {\rm clus}(T)$ then $f$ is a stable transversal for ${\rm clus}(T)$. Notice that this holds also for the corresponding function in which ‘closest’ is replaced by ‘furthest’ throughout. $\Box$
Example 3
---------
Consider the fully-resolved $X-$tree shown in Fig. \[figure2\](i), and the function $f$ defined as follows: $f(\{x\}) = x$ for all $x$ in $X$, and $$f(\{a,a'\})=a, f(\{b,b'\})=b, f(\{c,c'\})=c$$ and $$f(X-\{a,a'\})=b, f(X-\{b,b'\})=c, f(X-\{c,c'\})=a.$$ Then $f$ is a stable transversal for $T$. Note that the choices of $b,c,a$ in the last line could be replaced by, for example, $c,a,b$ or $c,c,a$ and we would still have a stable transveral. $\Box$
Stable triplet covers are minimal strong lassos for $T$
=======================================================
Given a fully-resolved $X-$tree $T$, a stable transversal $f$ of ${\rm clus}(T)$ defines a triplet cover for $T$ as follows: For each interior vertex $v$ of $T$, consider the three components of the graph $T-v$, and let $A^1_v, A^2_v, A^3_v$ denote their leaf sets. Then let $${{\mathcal L}}_{(T, f)} := \bigcup_{v \in V_{\rm int}} \{ f(A^1_v)f(A^2_v), f(A^2_v)f(A^3_v), f(A^3_v)f(A^1_v)\}$$ where $V_{\rm int}$ denotes the set of interior vertices of $T$. We say that ${{\mathcal L}}$ is a [*stable triplet cover*]{} (generated by $f$) if ${{\mathcal L}}= {{\mathcal L}}_{(T,f)}$ for some stable transversal $f$ of ${\rm clus}(T)$. For example, for the pair $(T, f)$ described in Example 3, we have: $${{\mathcal L}}_{(T, f)} = \{ab, ac, bc, aa', a'b, bb' ,b'c, cc', c'a\},$$ and the graph $(X, {{\mathcal L}})$ for ${{\mathcal L}}={{\mathcal L}}_{(T, f)}$ is shown in Fig. \[figure2\](ii). Notice that not all triplet covers are stable; indeed the set of triplet covers of a fully-resolved $X-$tree $T$ is precisely the set of subsets of $\binom{X}{2}$ of the form ${{\mathcal L}}_{(T, f)}$ where $f$ is required to satisfy only the transversality property above for some $f: {\rm clus}(T) \rightarrow X$.
2d-trees
--------
Interestingly, Fig. \[figure2\](ii) shows that for the set ${{\mathcal L}}={{\mathcal L}}_{(T, f)}$ with $T$ and $f$ from Example 3, the graph $(X,{{\mathcal L}})$ is a $2d$-tree, where a graph $G=(V,E)$ is called a [*$2d$-tree*]{} if there exists an ordering $x_1,x_2,\ldots, x_n$ of $V$ such that $\{x_1,x_2\}\in E$ and, for $i=3,\ldots, n$ the vertex $x_i$ has degree 2 in the subgraph of $G$ induced by $\{x_1,x_2,\ldots, x_i\}$. $2d$-trees are examples of $kd$-trees which were characterized in [@todd] and also studied in e.g.[@gue]. In Fig. \[figure2\](ii) an acceptable vertex ordering is $a,b,c, a',b',c'$. The graph in Fig. \[figure1\](ii) is also a 2d-tree, as can be seen by considering the vertex ordering $a,b,d, g,c, f,e$.
Main result
-----------
We can now state our first main result which relates stable triplet covers with 2d-trees and shellable lassos.
\[maintheorem\] If ${{\mathcal L}}$ is a stable triplet cover of a fully-resolved $X-$tree $T$ with $n:=|X|\geq 3$, then
- $(X,{{\mathcal L}})$ is a 2d-tree.
- ${{\mathcal L}}$ is an shellable lasso for $T$, and so ${{\mathcal L}}$ is a strong lasso for $T$.
- $|{{\mathcal L}}|= 2n-3$, and so ${{\mathcal L}}$ is also a minimal strong lasso for $T$.
We prove parts (i)–(iii) simultaneously by induction on $n=|X|$. Shellability holds trivially for $n=3$ (since then $\binom{X}{2}-{{\mathcal L}}= \emptyset$), so suppose that it holds when $n=k \geq 3$, and that $T$ is a fully-resolved tree with $k+1$ leaves, and that ${{\mathcal L}}$ is a triplet cover for $T$ generated by a stable transversal $f$ of ${\rm clus}(T)$. Select any cherry $x,y$ of $T$. Without loss of generality, we may suppose that $f(\{x,y\})=x$. Let $$z := f(X-\{x,y\}), X': = X-\{y\}, T':=T|X', {{\mathcal L}}':= {{\mathcal L}}-\{xy, yz\},$$ and define $f': {\rm clus}(T) \rightarrow X$ by setting $$f'(A) =
\begin{cases}
& f(A), \mbox{ if } x \not\in A; \\
& f(A\cup \{y\}), \mbox{ if } x \in A.
\end{cases}$$ Note that, since $f$ is a stable transversal for ${\rm clus}(T)$, it follows that $y$ is not an element of any cord of ${{\mathcal L}}'$, and so ${{\mathcal L}}' \subseteq \binom{X'}{2}$. Moreover, $y\not=f'(A)$ for any $A \in {\rm clus}(T')$, and so $f': {\rm clus}(T') \rightarrow X'$. It can now be checked that $f'$ is a stable transversal for ${\rm clus}(T')$ and so ${{\mathcal L}}'$ is a stable triplet cover of $T'$, generated by $f'$. By the inductive hypothesis (applied to $T'$ and $ {{\mathcal L}}'$) it follows with regards to (i) that $(X',{{\mathcal L}}')$ is a 2d-tree. Clearly adding $y$ to the vertex set of that graph and $xy$ and $zy$ to its edge set preserves the 2d-tree property. By the definition of ${{\mathcal L}}'$ it is easy to see that the resulting graph is $(X,{{\mathcal L}})$.
Note that regarding (ii) and (iii) the induction hypothesis implies that $|{{\mathcal L}}'| = 2k-3$, and so $|{{\mathcal L}}| = 2(k+1)-3$ and that $\binom{X'}{2} - {{\mathcal L}}'$ is shellable. So let us fix an ordering of $\binom{X'}{2} - {{\mathcal L}}'$ that provides such a shelling. This will form the initial segment of a shellable ordering of $\binom{X}{2}-{{\mathcal L}}$.
To describe this extended ordering, let $v$ be the interior vertex of $T$ adjacent to leaves $x$ and $y$, and let $u$ be the interior vertex of $T$ adjacent to $v$. Consider the three components of the graph $T-u$. One component contains $x,y$, and we will denote the leaf sets of the other two components by $X_2$ and $X_3$, where, without loss of generality, $z \in X_3$. Notice that $\binom{X}{2}-{{\mathcal L}}$ is the disjoint union of the three sets: $$\binom{X'}{2}-{{\mathcal L}}', \{ty: t \in X_2\} \mbox{ and } \{ty: t \in X_3-\{z\}\}.$$ We order $\binom{X}{2}-{{\mathcal L}}$ as follows: the elements of $\binom{X'}{2}-{{\mathcal L}}'$ come first, ordered by their shellable ordering, followed by the elements $ty$ with $t \in X_2$ (in any order), followed by the elements $ty$ with $t \in X_3-\{z\}$ (in any order).
We claim that any such ordering provides a shellable ordering of $\binom{X}{2}-{{\mathcal L}}$. To see this, observe first that, for any leaf $t \in X_2$, the elements $x,z$ provide ‘pivots’ for the pair $t,y$, since $T|\{x,y,z,t\} = xy||zt$ and all cords in $\binom{\{x,y,z,t\}}{2}$ except $ty$ are contained in ${{\mathcal L}}\cup (\binom{X'}{2}- {{\mathcal L}}')$. Also, for any leaf $t \in X_3$, if we select any leaf $z' \in X_2$ then the pair $x,z'$ provides a ‘pivot’ for $t,y$, since $T|\{x,y,z',t\} = xy||z't$, and all cords in $\binom{\{x,y,z',t\}}{2}$ except $ty$ are contained in ${{\mathcal L}}\cup (\binom{X'}{2}- {{\mathcal L}}') \cup \{t'y: t' \in X_2\}$. In all cases, the cords required for pivoting come earlier in the ordering.
Thus, we have established that ${{\mathcal L}}'$ is an shellable lasso for $T$, and so, by Theorem 6 of [@dre3], ${{\mathcal L}}$ is also a strong lasso for $T$. Moreover, we showed that $|{{\mathcal L}}| = 2|X|-3$, and since this equals the number of edges in any fully-resolved $X$–tree, linear algebra ensures that no strict subset of ${{\mathcal L}}'$ could be an edge weight-lasso for $T$. Hence, ${{\mathcal L}}$ is a minimal strong lasso for $T$, which completes the proof of the induction step, and thereby of the theorem.
Remarks
-------
- Just because a graph $(X, {{\mathcal L}})$ is a 2d-tree, it does not follow that ${{\mathcal L}}$ forms a strong (let alone a shellable) lasso for every given fully-resolved $X-$tree $T$. A simple example is furnished by $X=\{a,b,c,d\}$ and ${{\mathcal L}}= \{ab, ac, bc, ad, bd\}$, for which $(X, {{\mathcal L}})$ is a 2d-tree, and yet ${{\mathcal L}}$ fails to be a strong lasso for $T= ab||cd$.
However, if $(X, {{\mathcal L}})$ forms a 2d-tree, or more generally if ${{\mathcal L}}$ contains a subset ${{\mathcal L}}'$ such that $(X, {{\mathcal L}}')$ is a 2d-tree, then ${{\mathcal L}}$ is a strong lasso for at least one fully-resolved $X-$tree. The proof is constructive based on the ordering $x_1, x_2, \ldots, x_n$ in the definition of a 2d-tree: Start with the tree consisting of leaves $x_1$ and $x_2$, and construct a fully-resolved tree as follows: for each $i>2$, if $x_i$ is adjacent to $x_j$ and $x_k$ in $(X,{{\mathcal L}}')$ (where $j,k<i$) then let $x_i$ be the leaf that is attached by a new edge to a [*new*]{} subdivision vertex on the path connecting $x_j$ and $x_k$ in the tree so-far constructed.
It may be of interest to explore further the connection between shellability and 2d-trees, and in particular, the question of when the former property for some set ${{\mathcal L}}\subseteq \binom{X}{2}$ entails the latter property for ${{\mathcal L}}$, or for some subset of ${{\mathcal L}}$.
- Suppose that $T$ is a fully-resolved $X-$tree, and ${{\mathcal L}}\subseteq \binom{X}{2}$ contains a stable triplet cover. A natural setting in which this situation arises is the following. Suppose $(T,w)$ is a properly edge-weighted fully-resolved $X-$tree, and ${{\mathcal L}}\subseteq \binom{X}{2}$ has the property that, for any interior vertex, $v$, ${{\mathcal L}}$ contains every chord $xy$ for which $x$ is a closest leaf to $v$ in one subtree of $T-v$ and $y$ is a closest leaf to $v$ in another subtree of $T-v$. Then, as noted in Example 2 above, ${{\mathcal L}}$ contains a stable triplet cover.
Now, when ${{\mathcal L}}$ contains a stable triplet cover for $T$, it follows by Theorem \[maintheorem\] that ${{\mathcal L}}$ is a shellable, and thereby also a strong lasso for $T$ (since any superset of a strong lasso for a tree is also a strong lasso for that tree). However, it is perhaps not clear how one might efficiently construct $(T,w)$ from the distances induced by ${{\mathcal L}}$, particularly when the subset of ${{\mathcal L}}$ corresponding to the stable triplet cover is not also given explicitly. Thus, in the next section we describe a polynomial-time algorithm for reconstructing $(T, w)$ whenever ${{\mathcal L}}$ contains some (unknown) shellable lasso for $T$.
An algorithm for reconstructing $(T, w)$ from $d_{(T,w)}|{{\mathcal L}}$ when ${{\mathcal L}}$ contains an shellable lasso for $T$.
===================================================================================================================================
Suppose that ${{\mathcal L}}\subseteq {X\choose 2}$ and that $T$ is a fully-resolved $X-$tree, $w$ is a proper edge-weighting of $T$ and $d=d_{(T,w)}$. Starting with ${{\mathcal L}}^*= {{\mathcal L}}$ add cords to ${{\mathcal L}}^*$ and extend the domain of $d$ to those cords, by repeated application of the following extension rule (${{\mathcal R}}$), described in [@gue] (Section 6.2, page 246):
- Whenever $x,y,z,u \in X$ and $$\binom{\{x,y,u,z\}}{2} - \{xz\} \subseteq {{\mathcal L}}^*, xz \not\in {{\mathcal L}}^*,
\mbox{ and }$$ $$d(x,y)+d(u,z) < d(x,u)+d(y,z)$$ add $xz$ to ${{\mathcal L}}^*$, and let $d(x,z) := d(x,u)+d(y,z)-d(y,u).$
Let ${\rm cl}_{{\mathcal R}}({{\mathcal L}})$ be the set of resulting set of cords obtained from the initial set ${{\mathcal L}}$ when this extension rule no longer yields any new cords.
Note that ${\rm cl}_{{\mathcal R}}({{\mathcal L}})$ can be computed in polynomial time, and that $d-$values are assigned for all cords in ${\rm cl}_{{\mathcal R}}({{\mathcal L}})$. Moreover, if ${\rm cl}_{{\mathcal R}}({{\mathcal L}}) = \binom{X}{2}$, then ${\rm cl}_{{\mathcal R}}({{\mathcal L}})$ is a strong lasso for $T$, however the converse does not hold (Example 6.2 of [@dre3] provides a counterexample).
If ${{\mathcal L}}\subseteq {X\choose 2}$ contains an shellable lasso for a fully-resolved $X-$tree $T$, and $d= d_{(T,w)}$, for some proper edge weighting $w$, then ${\rm cl}_{{\mathcal R}}({{\mathcal L}}) = \binom{X}{2}$. Consequently, $T$ and $w$ can be reconstructed in polynomial time from the restriction of $d$ to ${{\mathcal L}}$.
Suppose that ${{\mathcal L}}' \subseteq {{\mathcal L}}$ is a shellable lasso for $T$; we will show that ${\rm cl}_{{\mathcal R}}({{\mathcal L}}') = \binom{X}{2}$ and so ${\rm cl}_{{\mathcal R}}({{\mathcal L}}) = \binom{X}{2}$. Suppose to the contrary that ${\rm cl}_{{\mathcal R}}({{\mathcal L}}')$ is a strict subset of $\binom{X}{2}$, and consider any shelling $a_1b_1, \ldots, a_mb_m$ of the cords in $\binom{X}{2}-{{\mathcal L}}'$ (such a shelling exists by the assumption that ${{\mathcal L}}'$ is an shellable lasso for $T$). Let $j \in \{1, \ldots, m\}$ be the smallest index for which $a_jb_j \not\in {\rm cl}_{{\mathcal R}}({{\mathcal L}}')$. Then the condition on the shelling ensures that there exists pivots $x_j, y_j \in X-\{a_j,b_j\}$ so that for $Y=\{a_j,b_j,x_j,y_j\} $ we have $T|Y$ is the quartet tree $a_jx_j||b_jy_j$ and that each cord in $\binom{Y}{2} -\{a_jb_j\}$ either is an element of ${{\mathcal L}}'$ or it occurs earlier in the ordering for the shelling than $a_jb_j$, and so, by the minimality assumption concerning $j$, all these cords lie in ${\rm cl}_{{\mathcal R}}({{\mathcal L}}')$. Consequently, $a_jb_j \in {\rm cl}_{{\mathcal R}}({\rm cl}_{{\mathcal R}}({{\mathcal L}}')) = {\rm cl}_{{\mathcal R}}({{\mathcal L}}')$, a contradiction. Thus, our assumption that ${\rm cl}_{{\mathcal R}}({{\mathcal L}}')$ is a strict subset of $\binom{X}{2}$ is not possible, as required.
Finally, to efficiently recover $(T, w)$, once $d$ has been defined on all of $\binom{X}{2}$, one can apply standard distance-based reconstruction methods for fully-resolved trees, such as the Neighbor-Joining method [@fel].
[Acknowledgments]{} {#acknowledgments .unnumbered}
===================
[*M.S. thanks the Royal Society of NZ under its James Cook Fellowship scheme. Both authors thank the organizers of the “Structure Discovery in Biology: Motifs, Networks $\&$ Phylogenies”, Schloss Dagstuhl, Germany, meeting where this paper was conceived.* ]{}
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---
address: 'Dipartimento di Astronomia, Via Ranzani 1, 40127 Bologna, Italy'
author:
- 'G. Sorrentino, B. Kelm, P. Focardi'
title: 'Active Star-forming galaxies in pairs in the 2dF'
---
Introduction
============
The environment of a galaxy may have a significant effect on its spectral characteristics (Balogh et al.2003astro-ph/0311379). In a cluster the role of singular encounters has a negligible impact, but it is important to understand whether close approaches in low density environments are important, as the majority of galaxies are found in a group-like environment. Close encounters are predicted to trigger star-formation and eventually AGN activity. We investigate whether any excess of active star-forming galaxies is found in small systems identified in the 2dF (Colless et al. 2003). The 2dF sub-sample we examine includes 10695 galaxies with m$_B$ between 17 and 17.5. For each galaxy close neighbours (with 17$\leq$m$_B$$\leq$19.5) have been identified lying within a circular area of 0.25$h^{-1}$Mpc projected radius and $\pm$1000$km/s$ from the galaxy. To each galaxy a surface density parameter $\sigma$$_{0.25}$ has been assigned, which is the ratio between the number of neighbours and the area of the region enclosing its most distant neighbour.
Active Star Forming galaxies in pairs
=====================================
In Fig.1 we show the average value of $\sigma$$_{0.25}$ vs. spectral type for galaxies in our sample. Spectral type is assigned as in Madgwick et al. (2002, MNRAS333,133). Galaxies in pairs, triplets and quartets are shown. It can be seen that the average value of $\sigma$$_{0.25}$ of ASF galaxies (T4) is significantly higher than those of non active galaxies (T1 and T2) in the pair sample, while no significant differences are seen in triplets and quartets. ASF galaxies appear therefore much more likely than non-active galaxies to inhabit extremely close pairs located in low density regions. High $\sigma$$_{0.25}$ values derived for ASF galaxies in pairs confirm that it is the distance to the nearest neighbour that is relevant (Lambas et al.,2002,astro-ph/0212222), and additionally, that the triggering efficiency rises in more isolated pairs. In Fig.2 we show $\sigma$$_{0.25}$ as a function of absolute magnitude for all galaxies in our sample and for ASF and T1 galaxies separately. The left panel shows $\sigma$$_{0.25}$ mean values derived for all galaxies displaying 1 or more neighbours (Neigh$\geq$1), the right panel shows values obtained when excluding pairs (Neigh$\geq$2). High $\sigma$$_{0.25}$ values associated to ASF galaxies are seen in the sample including pairs, while ASF galaxies in denser systems do show a less significant excess. Figure 2 also reveals that ASF galaxies with high values of $\sigma$$_{0.25}$ concentrate in a narrow magnitude range \[$-20$$\leq$M$_B$$\leq$$-19$\]. In this range ASF galaxies constitute between 7% and 15% of the whole galaxy population. Below M$_B$$\simeq$$-20$ the values of $\sigma$$_{0.25}$ are broadly similar between T1 and T4 galaxies indicating that, unlike the number of large scale neighbours (Kelm et al.2003astro-ph/0309268), the number of close neighbours does not discriminate between bright T1 and bright T4 galaxies.
|
---
abstract: |
We present pre-perihelion infrared 8 to 31 spectrophotometric and imaging observations of comet C/2012 K1 (Pan-STARRS), a dynamically new Oort Cloud comet, conducted with NASA’s Stratospheric Observatory for Infrared Astronomy (SOFIA) facility (+FORCAST) in 2014 June. As a “new” comet (first inner solar system passage), the coma grain population may be extremely pristine, unencumbered by a rime and insufficiently irradiated by the Sun to carbonize its surface organics. The comet exhibited a weak 10 silicate feature $\simeq 1.18
\pm 0.03$ above the underlying best-fit $215.32 \pm 0.95$ K continuum blackbody. Thermal modeling of the observed spectral energy distribution indicates that the coma grains are fractally solid with a porosity factor $D = 3$ and the peak in the grain size distribution, $a_{peak} = 0.6$ , large. The sub-micron coma grains are dominated by amorphous carbon, with a silicate-to-carbon ratio of $0.80^{ +0.25}_{- 0.20}$. The silicate crystalline mass fraction is $0.20^{ +0.30}_{ -0.10}$, similar to with other dynamically new comets exhibiting weak 10 silicate features. The bolometric dust albedo of the coma dust is $0.14 \pm 0.01$ at a phase angle of 34.76, and the average dust production rate, corrected to zero phase, at the epoch of our observations was $Af\rho \simeq 5340$ cm.\
author:
- |
Charles E. Woodward, Michael S. P. Kelley, David E. Harker, Erin L. Ryan,\
Diane H. Wooden, Michael L. Sitko, Ray W. Russell,\
William T. Reach, Imke de Pater, Ludmilla Kolokolova, Robert D. Gehrz
title: |
SOFIA Infrared Spectrophotometry of\
Comet C/2012 K1 (Pan-STARRS)
---
INTRODUCTION
============
Solar System formation was an engine that simultaneously preserves and transforms interstellar medium (ISM) ices, organics, and dust grains into cometesimals, planetesimals and, ultimately, planets. Observing and modeling the properties of small, primitive bodies in the solar system whose origins lie beyond the water frost line ($>5$ AU) provides critical insight into the formation of Solar System solids and establishes observation constraints for planetary system formation invoking migration – the ‘Grand Tack’ epoch [@walsh2011], followed by the ‘Nice Model’ events [@levison2009; @gomes2005]. The characteristics of comet dust can provide evidence to validate the new, emerging picture of small body populations – including comet families – resulting from planetary migration in the early Solar System.
Inside cometary nuclei, the bulk of the dust likely has been preserved since formation of the nucleus. Comet grains (and ices) also trace the pre-accretion history of comet materials extant in the outer disk. Comet dust composition can be studied via *Stardust* samples, selected collections of Interplanetary Dust Particles (IDPs), and in situ analysis in comet flyby and/or rendezvous missions. Dust species that are best explained as products of aqueous alteration (e.g., magnitite, cubanite, possibly pentlandite) are rare [@stodolna2012; @berger2011; @zolensky2008] and corresponding altered silicates (e.g., phyllosilicates, smectite) are missing suggesting that aqueous alteration in cometary nuclei is limited, is not well represented in the *Stardust* samples, or that these minerals have exogenous origins [@brownlee2014]. Thus, the bulk of comet grain properties including dust size, porosity, and composition relate to grain formation, radial mixing, and particle agglomeration in the proto-solar disk [for an extensive review see @brownlee2014]. However, opportunities to study actual samples of cometary dust are rare, motivating the need for telescopic remote sensing observations of dust whenever apparitions are accessible from terrestrial observatories.
In this paper we report our pre-perihelion (TP = 2014 Aug 27.65 UT) infrared 8 to 31 micron spectrophotometric observations of comet C/2012 K1 (Pan-STARRS), a dynamically new [see @oort1950 for a definition based on orbital elements] Oort Cloud comet – $(1/a_{org}) = 42.9 \times 10^{-6}$ AU$^{-1}$ [@williams2015] – conducted with NASA’s Stratospheric Observatory for Infrared Astronomy (SOFIA) facility during a series of four flights over the period from 2014 June 04 to 13 UT. Contemporaneous optical imaging observations are also presented.
OBSERVATIONS
============
Ground-based Optical Imaging {#obs-bok}
----------------------------
Comet C/2012 K1 (Pan-STARRS) was observed on 2012 June 01.22 UT and again on June 04.24 UT with the 2.3-m Bok Telescope at the Kitt Peak National Observatory. The comet was at heliocentric distance ($r_{h}$) of 1.74 AU and 1.71 AU, a geocentric distance ($\Delta$) of 1.66 AU and 1.69 AU, a phase angle of $34.62^{\circ}$ and $34.76^{\circ}$, for each date respectively.
The images were obtained with the 90Prime camera [@williams04], a prime focus imager built for the Bok Telescope. At the time of observation, the 90Prime camera utilized a thinned back-illuminated CCD detector with 4064 $\times$ 4064 pixels with a pixel size of $15.0~\micron$. At prime focus the camera pixel scale is $0.45^{\prime\prime}$ which yields a field of view of 30.5 $\times$ 30.5 square-arcmin. The instrument was equipped with Cousins/Bessel system broadband $V$ and $R$ filters. Multiple exposures (23 images in $R$ band and 9 images in $V$ band of 30 seconds each) were obtained of the nucleus and coma of the comet with the telescope tracking at the non-sidereal rate corresponding to the predicted motion of the comet provided by JPL Horizons[^1] in an airmass range of 1.40 to 1.74.
All images were corrected for overscan, bias and flat-fielding with standard IRAF[^2] routines. The data was photometrically calibrated using eight field stars of various spectral types with known $V$ and $R$ magnitudes selected from the Naval Observatory Merged Astrometric Dataset (NOMAD) catalog [@zacharias2004] on the same CCD amplifier as the comet. The standard deviation of the photometric $V$ and $R$ zero points derived from the average of the field stars is of order 1% and no color corrections for spectral type were applied. The average nightly seeing was $\sim 2.2^{\prime\prime}$ in both bands. A single 30 sec exposure in the $R$ band obtained on 2014 June 04.24 UT is shown in Fig. \[fig:bokVandR\].
SOFIA {#obs-sofia}
-----
Mid-infrared (mid-IR) spectrophotometric observations of comet C/2012 K1 (Pan-STARRS) were obtained using the Faint Object InfraRed CAmera for the SOFIA Telescope [FORCAST; @herter2012] mounted at the Nasmyth focus of the 2.5-m telescope of the SOFIA Observatory [@young2012; @gehrz2009]. The data were acquired over a series of four flights, originating from Palmdale, CA at altitudes of $\simeq 11.89$ km in 2014 June, that were conducted as part of our SOFIA Cycle 2 programs to observe comets (P.I. Woodward, AOR\_IDs 01\_001 and 02\_0002). Details of all SOFIA observations and the orbital parameters of comet C/2012 K1 (Pan-STARRS) at those epochs are summarized in Table \[tab:sobstab\_tab\].
FORCAST is a dual-channel mid-IR imager and grism spectrometer operating from 5 to 40 . Light is fed to two $256 \times 256$ pixel blocked-impurity-band (BIB) arrays, each with a plate scale of $0.768^{\prime\prime}$ per pixel and a distortion corrected field of view of $3.2^{\prime} \times 3.4^{\prime}$. The Short Wavelength Camera (SWC) covers the spectral region from 5 to 25 , while the Long Wavelength Camera (LWC) operates at wavelengths from 25 to 40 . Imaging data can be acquired in either dual channel mode (with some loss of throughput due to the dichroic) or single channel mode.
Imaging observations of C/2012 K1 (Pan-STARRS) in three filters were conducted on the first flight series, prior to three flights dedicated to spectroscopy. Spectroscopic observations of the comet used two grisms, one in the SWC (G111) and one in the LWC (G227), and the instrument was configured using a long-slit ($4.7^{\prime\prime} \times
191^{\prime\prime}$) which yields a spectral resolution $R = \lambda/\Delta\lambda \sim$ 140-300. The comet was imaged in the SWC using the F197 filter to position the target in the slit. Both imaging and spectroscopic data were obtained using a 2-point chop/nod in the Nod-Match-Chop (C2N) mode with 45$^{\prime\prime}$ chop and 90$^{\prime\prime}$ nod amplitudes at angles of 30$^{\circ}$/210$^{\circ}$ in the equatorial reference frame.
All FORCAST raw image data products were processed using the FORCAST\_REDUX Data Pipeline, v1.0.1beta [cf., @clarke2014], which employed the reduction packages FORCAST\_FSPEXTOOL, v1.1.0, and FORCAST\_DRIP, v1.1.0. Processing of the raw spectroscopic data was performed using the same packages, with the exception of FORCAST\_DRIP, which utilized v1.0.4. Details of the FORCAST\_REDUX Data Pipeline can be found in the Guest Investigator Handbook for FORCAST Data Products, Rev. B[^3]
SOFIA Imagery and Photometry
----------------------------
Aperture photometry of the SOFIA image data of comet C/2012 K1 (Pan-STARRS) was performed on the Level 3 pipeline coadded (\*.COA) data products using the Aperture Photometry Tool [APT v2.4.7; @laher2012]. At all FORCAST filter wavelengths, the comet exhibited extended emission beyond the PSF of point sources observed with FORCAST under optimal telescope jitter performance.[^4] The photometry was therefore conducted using a circular aperture centroided on the photocenter of the comet nucleus. We used an aperture of radius 13 pixels, corresponding to 9.984$^{\prime\prime}$, with a background aperture annulus of inner radius 30 pixels (23.58$^{\prime\prime}$) and outer radius of 60 pixels (47.16$^{\prime\prime}$). This aperture, which is $\simeq 3 \times$ the nominal point-source FWHM, encompassed the majority of the emission of the comet and coma. Sky-annulus median subtraction [ATP Model B as described in @laher2012] was used in the computation of the source intensity. The systematic source intensity uncertainty was computed using a depth of coverage value equivalent to the number of coadded image frames. The dominant source of overall uncertainty in the image photometry were image gradients due to imperfect atmospheric background subtraction. The calibration factors (and associated uncertainties) applied to the resultant aperture sums were included in the Level 3 data distribution and were derived from the weighted average of 3 calibrator observations of $\beta$ And (2 each) and $\alpha$ Boo (1 each). The resultant SOFIA photometry is presented in Table \[tab:simage\_phot\_tab\].
Due to turbulence, telescope jitter, and differing chop-nod patterns, i.e., the chopping difference between beams and the nodding of the entire telescope field-of-view [for a for a discussion and illustration of this standard infrared observing technique with SOFIA – see @temi2014; @young2012] executed in flight, the multi-filter imagery data could not be used to generate color temperature maps due to the unstable PSF.
During flights primarily devoted to obtaining grism data (§\[sec:ssp\]), images of the comet where obtained through a single filter at 19.7 micron. Figure \[fig:s195\_image\] shows the 19.7 surface brightness distribution of comet C/2012 K1 (Pan-STARRS) observed on 2014 June 13.17 UT. The nucleus is unresolved and azimuthally symmetric with a radial profile FWHM of $\sim 1.01^{\prime\prime}$ and the coma is extended and diffuse. Low surface brightness emission extends in a vector direction commensurate with that expected for a dust tail.
SOFIA Spectra {#sec:ssp}
-------------
Three temporally distinct spectra of comet C/2012 K1 (Pan-STARRS) were obtained in both grism over a series of flight sequences spanning 6 days (Table \[tab:sobstab\_tab\]). Many comets exhibit temporal variability in the infrared over periods of hours [e.g., @wooden2004] to days at relatively similar heliocentric distances due to coma jets related to nucleus activity and/or nucleus rotation period [e.g., @keller2007; @gehrz1995] that produces observable changes in the observed SEDs. Inter-comparison of each SED over this period showed no substantial changes in overall continuum flux densities nor spectral features to within the uncertainty per spectral resolution element. In addition, the 19.7 aperture photometry suggests also that the level of coma emission did not markedly change (cf., Table \[tab:simage\_phot\_tab\]). Apparently, comet C/2012 K1 (Pan-STARRS) was fairly quiescent in its infrared behavior during this epoch given our signal-to-noise ratio and aperture size (12,300 km radius). Thus, the three independent spectra were summed together in pipeline processing to produce an average spectral energy distribution (SED). A 3-point unweighted rectangular smoothing function was applied to this average SED to increase the point-to-point signal-to-noise ratio of the data product used in our thermal model spectral decomposition analysis. The calibrated data products do exhibit a few artifacts near the edges of the 17–27 spectral order where a few data points deviate upwards (near 17 ) or downwards (near 27 ) from the apparent spectral trend. The spectra of comet C/2012 K1 (Pan-STARRS) are presented in Fig. \[fig:sofia\_grating\].
RESULTS
=======
Taxonomically comet C/2012 K1 (Pan-STARRS) is a member of the dynamical comet family denoted as nearly isotropic comets (NICs), also commonly referred to as Oort Cloud comets [cf., @dones2004]. The interior composition of the ecliptic comets (ECs) and the NICs likely are preserved during their residence in the Scattered Disk and the Oort Cloud, but their surfaces are subject to various processing effects. Modeling the coma dust properties provides insight into the origin and evolution of dynamic comet families.
Thermal Modeling of the Coma SED {#sec:sofia_ans}
--------------------------------
Thermal modeling of the observed thermal infrared SED of comets obtained using remote sensing techniques enables derivation of coma dust grain properties. In particular, SOFIA (+FORCAST) provides spectroscopic coverage with the G111 grism to the region 9–12 which contains features from amorphous and crystalline silicates (e.g., 11.2 ) and organic species (e.g., PAHs). The G227 grism spans 17.6–27.7 , encompassing discrete resonances from crystalline silicates as well as spectral signatures from carbonates and phyllosilicates, putatively argued to be extant in comets [@lisse2006]. The SED slope at long thermal (15 ) wavelengths provides constraints on the abundance of the larger grain population in the coma. Observations in these spectral regimes are key to ascertaining the origins of silicates within the solar protoplanetary disk, and placing early solar disk evolution within the context of other circumstellar disks observed today through comparison to model and laboratory data [cf., @lindsay2013; @koike2010].
Modeling the mid-IR SED of C/2012 K1 (Pan-STARRS) yields estimates of the coma grain properties. We constrain the grain parameters by chi-squared fitting thermal emission models to the observed spectrum. The grain parameters included in the modeling are size distributions (n(a) da), porosity, the crystalline mass fraction (i.e., the fraction of the coma silicate grains that are crystalline), and relative material abundances. The dust temperature is calculated assuming thermal equilibrium of the grains; wherein the composition (mineralogy), size, and heliocentric distance determine the temperature of the grains.
Comet grains are dominated in composition by a handful of silicate-type materials [@hannerzol2010; @wooden2008]: Mg-Fe olivine- and pyroxene-types in amorphous (glassy) forms and their crystalline Mg-end-members forsterite (Mg$_{2}$SiO$_{4}$) and enstatite (MgSiO$_{3}$). Cometary aggregates also contain organics [@sandford2006] or amorphous-carbon-like materials [@matrajt2008; @formenkova1999] that may be the glue that holds the amorphous and crystalline materials together [@flynn2013; @cieslasanford2012]. Our model [@harker2002 and references therein] uses five materials: amorphous olivine and amorphous pyroxene with broad 10, 18, and 20 emission features, amorphous carbon with featureless emission, and crystalline olivine (Mg-rich) and orthopryoxene with narrow peaks. Broad and narrow resonances near 10 and 20 are modeled by warm chondritic (50% Fe; 50% Mg) amorphous silicates (i.e., glasses) and strong 11.25, 19.5, and 24 narrow features from cooler Mg-rich crystalline silicate materials.
The amorphous carbon component in our dust model is representative of several key dust species – e.g., elemental carbon dust [@formenkova1994], an organic component with C=C bonds, identified by XANES spectra near 285 eV that can include amorphous carbon [@flynn2013; @flynn2003; @wirick2009], and possibly other carbonaceous grains; however, overall model results do not depend on this degeneracy. We do not specifically include Fe-Ni sulfides (such as pyrrhotite or troilite) in our models nor carbonates or phyllosilicate-rich materials. The latter materials have not been detected in track analysis of *Stardust* samples [@nakamura2011; @wooden2008; @zolensky2008] nor are they unequivocally evident in remote sensing data [@bursentova2012; @cew2007]. Phyllosilicates, specifically smectites including montmorillonite, chlorite, and serpentine, have 18-23 resonances that worsen spectral fitting of comet C/1995 O1 (Hale-Bopp) [@wooden1999]. Hybrid IDPs may contain up to 10% smectite [@nakamura2011]. Smectitie is spectrally distinguishable from amorphous anhydrous olivine-type and amorphous pyroxene materials [@nakamura2011; @wooden1999], yet it is not required for spectral decomposition.
While FeS-type grains are present in IDPs [@bradleydai2000], meteoritics samples, and comets grains, such as Wild 2 [@heck2012; @zolensky2008; @velbel2007], our SOFIA spectra (Fig. \[fig:sofia\_grating\]) do not exhibit the broad 23 spectral features often associated with fine-grained FeS [@bursentova2012; @min2005; @hony2002; @keller2002]. Larger FeS particles would be spectrally indistinguishable from larger amorphous carbon particles at mid- to far-IR wavelengths, yet robust optical constants spanning visible through the far-IR are lacking for FeS due to measurement challenges of an inherently extremely absorbing material (L. Keller, private comm.). Thus, thermal modeling of FeS gains is uncertain. *Stardust* samples appear to be richer in FeS and poorer in carbonaceous matter [@joswiak2012], so there is no basis as yet to make an assumption about the relative abundance of FeS and amorphous carbon-like materials in comet comae. Hence we presume that the majority of absorbing materials in cometary dust re-radiating the observed infrared SED is dominated by olivine, pyroxene, and carbonaceous (amorphous carbon-like) materials [@brownlee2014; @wooden2008; @zolensky2008]. This presumption provides a foundation for comparing compositional similarities and diversities of comet dust composition derived from thermal models.
The best-fit chi-square model results are summarized in Table \[tab:bfmods\_tab\]. The model fit to the observed grism spectra with the corresponding spectral decomposition of grain components is presented in Fig. \[fig:sofia\_model\]. Mineralogically, the grains in the coma of C/2012 K1 (Pan-STARRS) are dominated by amorphous materials, especially carbon. Our models produce a Hanner (modified-power law) differential grain size distribution (HGSD)[^5] peaking with grains of radii $a_{peak} = 0.6$ , indicating relative moderately larger grains are present, and the grain power-law slope $N = 3.4$. In a HGSD the small radii grains at the peak of the grain size distribution dominate the surface area and the flux density.
Grains in comets are likely fractal porous aggregates [@schulz2015]. The grain porosity ($P$ versus the dust radius $a$), parameterized by $D$, is defined as $P = (a/0.1~\mu\rm{m})^{(D-3)}$ with $ D = 3$ for solid and $D = 2.5$ for highly porous grains [@cew2011]. Grains in the coma of C/2012 K1 (Pan-STARRS) also are solid (the fractal porosity parameter $D = 3.0$). Solid grains are not unusual, 65% of the *Stardust* tracks are carrot-shaped from solid terminal particles [@horz2006]. The sub-micron sized silicate-to-carbon ratio derived from our models is $0.80^{+ 0.25}_{- 0.20}$. The uncertainty in the parameters derived from the thermal models are at the 95% confidence level.
The 10 Silicate Emission Feature {#sec:10sef-disc}
---------------------------------
The 10 silicate feature in comet C/2012 K1 (Pan-STARRS) is quite weak compared to comets like C/1995 O1 (Hale-Bopp) or 17P/Holmes [e.g., @wooden1999; @watanabe2009]. Following [@sitko2004], at 10.5 we find the silicate emission (defined as $[F_{10}/F^{BB}_{continuum}]$) is $1.18 \pm 0.03$ above a blackbody curve fit to the observed grism spectra continua longwards of 12.5 . The best-fit blackbody is $T_{bb} = 215.32 \pm 0.95$ K (using Gaussian weighted errors) and color excess, defined as T$_{bb}$(fit)/(278K $r_{h}^{-0.5}$) is $= 0.992 \pm 0.004$. The normalized ($F_{\lambda}/F_{\lambda,T}$) SED in the region near the silicate feature at 10 is presented in Fig. \[fig:fbyfctbb\]. Typically data near 8 are used to establish the blue-continua ($\lambda\lambda 7.7-8.4$ ). Our estimate of the local 10 continua may yield slightly lower temperatures than an estimate that included 8 photometry.
The large value of $a_{peak}$ inferred from the thermal modeling of the observed SED of comet C/2012 K1 (Pan-STARRS) in 2014 June is commensurate with the weak 10 silicate feature. Smaller grains ($a_{peak} {\raisebox{-0.6ex}{$\,\stackrel
{\raisebox{-.2ex}{$\textstyle <$}}{\sim}\,$}}0.3$ ) produce higher contrast silicate features. Grains of greater porosity also produce higher contrast silicate features in the 10 band. Long period NICs have ‘typical’ HGSD slopes of $3.4 {\raisebox{-0.6ex}{$\,\stackrel
{\raisebox{-.2ex}{$\textstyle <$}}{\sim}\,$}}N {\raisebox{-0.6ex}{$\,\stackrel
{\raisebox{-.2ex}{$\textstyle <$}}{\sim}\,$}}3.7$ and silicate-to-amorphous carbon ratios $\gg 1$. The grain size distribution slope of C/2012 K1 (Pan-STARRS), $N = 3.4$, is not atypical. The preponderance of larger sub-micron grains ($a_{peak} = 0.6$ ) in the coma of comet C/2012 K1 (Pan-STARRS) results in cooler radiating dust that contributes to the ‘continuum’ under the 10 silicate feature and to the far-infrared flux density (see Fig. \[fig:sofia\_model\]). The sub-micron mass fraction is dominated by amorphous carbon grains. Amorphous carbon has a featureless emission spectrum that extends through the 10 region, so low amorphous silicate-to-carbon ratio also can weaken the silicate feature strength [@wooden2008; @wooden2004].
The Silicate Crystalline Mass Fraction {#sec:fcryst-disc}
--------------------------------------
The mass fraction of silicate sub-micron grains that are crystalline in comet comae is a keystone for models of early planet-forming processes [@bv2002; @ciesla2007; @ha2010]. This fraction is defined as
$$f_{cryst}^{silicates} \equiv \sum_{x=1}^{n} \frac{m_{cryst,x}}{(m_{cryst,x} +
m_{amorphous,x}^{silicates})}
\label{eqn:fcseqn}$$
where m$_{x}$ is the mass of species $x$. Crystalline species in comet grains provide a record of the high temperature process that formed dust in the inner disk of the solar system and the large scale mixing that transported these hot nebular products to the cold comet forming zones. Crystals, their composition [e.g., @wooden2008] and shape [@lindsay2013] trace inner-solar disk conditions [e.g., @ogliore2011] and offer a view into the earliest planet-forming processes that occurred in our early Solar System.
Crystals from the inner disk were transported out to the comet-forming regime and mixed with “amorphous” silicates [cf. @ciesla2011]. The “amorphous” silicates are thought to be outer disk materials that probably were inherited from the ISM [@brownlee2014; @watson2009; @kemper2004; @kemper2005; @lidraine2001] in the infall phase of the disk. They have non-stoichiometric compositions [GEMS-like, @matsuno2012; @bradleydai2004; @bradley1999 and references therein] that include the compositional ranges of olivine \[(Mg$_{y}\,$,Fe$_{(1-y)}$)$_{2}$ SiO$_{4}$\], with y $\approx$ 0.5 for amorphous olivine, and \[(Mg$_{x}\,$,Fe$_{(1-x)}$) SiO$_{3}$\] with x $\approx$ 0.5 for amorphous pyroxene-type materials. Crystals are identified by narrow IR emission features (e.g., 11.2, 19, 23.5, 27.5, 33 ) superposed on an underlying thermal continuum in remote sensing spectra. Crystalline silicates have been detected using remote sensing techniques in the dust comae of all comet classes including C/1995 O1 (Hale-Bopp) [@wooden1999; @harker2002; @harker2004a] the Deep Impact coma of 9P/Tempel 1 [@harker2005; @lisse2006; @harker2007], the fragmentation outburst of 17P/Holmes [@reach2010], and several other comets [@mskdw2009; @cew2011]. Amorphous silicates are also detected in these comets as well. Crystalline silicates are found in abundance in the *Stardust* samples of 81P/Wild 2; however, the amorphous grains are difficult to identify [@ishii2008] and may be limited to the smallest dust grains [@brownlee2014].
Crystalline silicates are rare in the ISM; however, they account for 2.2% of the total silicate component in the direction of the Galactic Center [@kemper2004; @kemper2005] and 5% along other lines-of-sight [@lidraine2001]. Solar System crystalline silicates detected in comets must be formed in the early stages of our disk’s evolution [@brownlee2014]. Crystalline silicates require T 1000K to form through either gas phase condensation or annealing of amorphous (glassy) silicate grains [@wooden2008; @wooden2005; @davoisne2006; @henning2003; @fabian2000] implying that the crystalline silicates must have been processed in the disk near the young Sun or in shocks out to a maximum distance of 3 to 5 AU [@dehsjd2002; @wg2008]. Post-formation, they were transported radially outward into the comet-formation zones [@cm2007] – a process that is apparently ubiquitous in observations of external protoplanetary disks [@olofsson2010]. Glassy silicate spherules (GEMS) and crystals are seen in aggregates in cometary IDPs. Large ‘terminal particle’ crystals and sub-micron crystals (crystallites) are components of aggregate grains captured in *Stardust* samples [@brownlee2012; @brownlee2006; @zolensky2008; @zolensky2006].
Thus to first order, the diversity of comet dust properties reflects the temporal and radial gradients in our Solar System’s early history and similarities and differences in dust characteristics, including $f_{cryst}$, may provide observational tests of of planetary migration models within the early solar system during the epoch of planet formation that resulted in a variety of small body dynamical populations. We find a that the silicate crystalline mass fraction in comet C/2012 K1 (Pan-STARRS) is $f_{cryst} = 0.20^{+ 0.30}_{- 0.10}$. This range is similar to that found for comet C/20007 N3 (Lulin), $f_{cryst} = 0.48 \pm 0.06$ [derived from the mass fractions presented in Table 3 of @cew2011 and Eqn. \[eqn:fcseqn\] of §\[sec:fcryst-disc\]] which also exhibited a weak 10 silicate emission feature.
EC and NIC Dust Characteristics {#sec:ecnic-disc}
-------------------------------
As a result of giant planet migration, some comet nuclei were dynamically scattered into the Oort cloud to be exposed to the Galactic environment, whereas those bodies comprising the bulk of the ECs population have nuclei exposed and processed (at depths ranging from mm to few cm) by solar insolation, space weathering, and heliocentric activity variations (sublimation of CO, CO$_{2}$; crystallization of water and other ices) which affects materials lofted into the comae. Although the interior compositions of ECs and NICs likely are preserved, their surfaces have differing processing histories.
Typically, active comets (arising from a population of NICs dynamically derived from the Oort Cloud and moving on long-period orbits) exhibit high contrast 10 $\micron$ silicate features. In contrast, short-period ECs (i.e., Jupiter-family comets) have, on average, lower 10 silicate features strengths [@sitko2004], and are thought to have lower activities [cf., the active area and active fraction measurements of @ahearn1995]. For decades, the low-activity of ECs has been attributed to the accumulation of a rime of insulating larger grains that were launched on non-escape orbits [@jewitt2007]. Thermal models that fit observed infrared spectra of comets reveal that high contrast silicate features arise from comae having a preponderance of sub-micron grains [@harker2002; @hanner1994]. Comae without these sub-micron grains have weaker silicate features. In individual comets, variations in the silicate feature strength have been seen on short time scales corresponding to the aperture-crossing times of jets or coma features [@wooden2004; @harker2005; @harker2007; @gicquel2012]. These variations are best explained by changes in the differential grain size (n(a) da) or fluctuations in the silicate-to-carbon grain ratio.
Differences between EC and NIC coma grain populations may arise from the surface layers EC nuclei being “processed” or weathered [e.g., @lijy2015]. Processing of ECs surfaces may result from their frequent perihelion passages that decreases surface volatiles and small grains and leads to the creation of rimes and dust mantles. Evidence suggesting such processing occurs over millennia may be found in the analysis of material excavated from comet 9P/Tempel 1 by *Deep Impact*: the dust grains in the ejecta were smaller than those in the ambient coma [@harker2007] and the immediate comet surface contained a layer of carbon rich grains [@sugita2005] and a dust mantle comprised of compact 20 -sized dust aggregates [@kobayashi2013]. However, this conjecture is not definitive as it unknown whether or not the impact location reflects the global surface dust properties of the nucleus. In ECs, the coma 10 silicate feature strengths are low [@mskdw2009] and the dust production rates are modest. However, when EC nuclei have either fragmented [i.e., 73P/SW3, @harker2011; @sitko2011], explosively released materials from subsurface cavities [i.e., 17P/Holmes, @reach2010], or have had subsurface materials excavated from depth [i.e., the 9P/Tempel 1 *Deep Impact* encounter, @harker2005] the IR SEDs exhibit 10 relatively strong silicate feature emission (${\raisebox{-0.6ex}{$\,\stackrel
{\raisebox{-.2ex}{$\textstyle >$}}{\sim}\,$}}1.2$) arising from a population of sub-micron size silicate grain species. Whether or not the strong 10 silicate features arise from the release of sub-micron sized grains or the disruption of loose aggregates of fine particles (e.g., through gas-pressure disruption or impact fragmentation) is not known. Indeed the silicate feature in 9P/Tempel 1 changed from a EC-like spectra to NIC-like spectra immediately after *Deep Impact* event, returning to an EC-like state several tens of hours later [cf., @harker2005].
NICs are canonically considered to be more pristine with higher surface volatile abundance [cf., @wooden2008] – the effects of dwell time in the Galactic environment being more benign. Also, NICs are often considered a homologous population lacking significant nucleus evolution. Inner solar system apparitions of these comets frequently result in brilliant comae, with large dust production rates and pronounced silicate feature emission at IR wavelengths. It is not entirely clear whether the highly active nuclear regions of NICs can spawn small sub-micron grains responsible for the silicate feature emission, either by heritage or by fragmentation induced within the gas acceleration zone. However, whether comet evolution, such as processing in the Galactic environment, can be ignored when comparing the Oort cloud comet dust composition (including that expressed in $f_{cryst}$) is an open question.
Table \[tab:fcryst\_tab\] present estimates of $f_{cryst}$ and select characteristics of the dust derived from thermal modeling of the mid-IR SEDs for a set of well-studied Oort cloud and “disrupted” Jupiter-family comets. The crystalline silicate fraction ranges appreciably, from $\sim10$% to $\sim80$%. The compositional similarity suggests that Oort cloud and Jupiter-family comets have common origin sites within the early solar system [an argument that parallels that derived from volatile composition studies, @ahearn2012], but the range of $f_{cryst}$ values in Oort cloud comets suggest this class may be sampling a particular region that is not represented in the Jupiter-family members. This inference is intriguing; however, limited in robustness as any tentative conclusions are based on a limited sample size. Large sample sizes are required to substantiate or vitiate these trends.
Comet C/2012 K1 (Pan-STARRS) and C/2007 N3 (Lulin) have modest mean values for $f_{cryst}$ (${\raisebox{-0.6ex}{$\,\stackrel
{\raisebox{-.2ex}{$\textstyle <$}}{\sim}\,$}}48\%$), low silicate-to-carbon ratios, and grain size distributions that peak at large radii, ${\raisebox{-0.6ex}{$\,\stackrel
{\raisebox{-.2ex}{$\textstyle >$}}{\sim}\,$}}0.6~\micron$. The 10 silicate feature is weak and/or absent in these NICs. Perhaps these bodies represent a population of more carbon dominated bodies, similar to the dark organic KBOs, whose surfaces are devoid of small grains. Indeed the low albedo (see §\[sec:c\_albedo\]) of C/2012 K1 (Pan-STARRS) and the dominance of amorphous carbon grain materials maybe providing clues.
Dust Production Rates {#sec:opt_ans}
---------------------
The radial profile of comet C/2012 K1 (Pan-STARRS) was plotted to assess the quality of the data for calculating a dust production rate near the epoch of our SOFIA observations. The radial profile of C/2012 K1 (Pan-STARRS) in the $V$ band shows a deviation from the $1/\rho$ profile [@gehrzney92], suggesting contamination from gas such as $C_{2}$ ($\Delta = 0$) band(s) near 5141 Å. Strong $C_{2}$ emission is present in spectra [@mckay2014 and also A. McKay, priv. comm.] contemporaneous with our optical imagery. We therefore only calculate the dust production in $R$ band. The $R$ band radial profile of C/2012 K1 (Pan-STARRS) is shown in Fig. \[fig:rradial\_profile\].
To estimate the rate of dust production in comet C/2012 K1 (Pan-STARRS), we utilize the $Af\rho$ quantity introduced by @ahearn84. This quantity serves as a proxy for dust production and when the cometary coma is in steady state, the value for $A(\Theta)f \rho$ is an aperture independent parameter,
$$A(\Theta)f \rho = \frac{4 \; r_{h}^{2} \; \Delta^{2} \; 10^{-0.4(m_{comet} - m_{\odot})}}{\rho} \; (cm)$$
where $A(\Theta)$ is four times the *geometric* albedo at a phase angle $\Theta$, $f$ is the filling factor of the coma, $m_{comet}$ is the measured cometary magnitude, $m_{\odot}$ is the apparent solar magnitude, $\rho$ is the linear radius of the aperture at the comet’s position (cm) and $r_{h}$ and $\Delta$ are the heliocentric and geocentric distances measured in AU and cm, respectively. To correct our comet measurements for phase angle effects we applied the Halley-Marcus (HM) [@marcus2007a; @marcus2007b; @schleicher1998] phase angle correction.[^6] We adopt an interpolated value of 0.3864 (appropriate for the 2014 June 04.24 UT dataset) to normalize $A(\Theta)f \rho$ to $0\degr$ phase angle. Table \[tab:afr\_tab\] reports values of $Af \rho = [(A({\Theta})f \rho/$HM\] at a selection of distances from the comet photocenter in the $R$-band.
In addition to $Af\rho$, we also compute the $\epsilon f \rho$ parameter of comet C/2012 K1 (Pan-STARRS) based on our FORCAST broadband photometry (Table \[tab:simage\_phot\_tab\]). The $\epsilon f \rho$ parameter [defined by @kelley2013 Appendix A] can be considered to be the thermal emission corollary to the scattered-light-based $Af\rho$:
$$\epsilon f \rho = \frac{\Delta^2}{\pi \rho} \frac{F_\nu}{B_\nu}\,\rm{(cm)},
\label{eqn:efrhoeqn}$$
where $\epsilon$ is the effective dust emissivity, $F_\nu$ is the flux density (Jy) of the comet within the aperture $\rho$, $B_\nu$ is the Planck function (Jy/sr) evaluated at the temperature $T = T_{scale}\, (278~\rm{K})\, r_{h}^{-0.5}$, where the scaling factor $T_{scale} = 0.99$ based on the 215 K measured continuum temperature discussed in §\[sec:10sef-disc\]. Derived values of $\epsilon\, f\rho$ for comet C/2012 K1 (Pan-STARRS) are presented in Table \[tab:simage\_phot\_tab\].
Coma Averaged Dust Albedo {#sec:c_albedo}
-------------------------
Dust albedo is a basic parameter characterizing the size distribution and physical properties of comet dust that is, surprisingly, infrequently measured. Following the convention of @gehrzney92 the *bolometric* albedo, ($A_{bolometric} \equiv (Energy_{\, scattered} /Energy_{\, incident}$) is
$$A_{bolometric} \simeq \frac{f(\Theta)}{1 + f(\Theta)},
\label{eqn:abgn}$$
where for comet dust the incident energy is the sum of the energy scattered by the coma plus the total energy of the coma’s thermal emission at an observed phase angle $\Theta$ (Sun-comet-observer angle). The term $f({\Theta})$ can be determined from fitting the observed spectral energy distribution of the comet with appropriate Planck blackbody functions in the infrared (thermal dust emission) and reflected solar spectra at optical (scattering) wavelengths
$$f({\Theta}) = \frac{[\lambda{\rm F}_\lambda]_{max, scattering}}{[\lambda{\rm F}_\lambda]_{max, IR}}
\label{eqn:ftheta}$$
where the $[\lambda{\rm F}_{\lambda}]_{max}$ is the peak of the SEDs in the respective wavelength ranges. Lab experiments and theoretical calculations of the scattered light from particles indicate that the total brightness, color, polarization, and polarization color depend on the optical constants, particle size distribution, structure, and porosity of the dust as well as the solar phase angle [@lindsay2013; @hadamcik2007; @kolok2004]. The spectral shape of the IR thermal emission provides a direct link with the mineralogy and grain size. Both of these processes provide information on the size and composition of the dust. The scattered light and thermal emission are also connected to one another through the grain albedo, the ratio of the scattered light to the total incident radiation. Because light is not isotropically scattered by comet dust the measured albedo will depend not only on the composition and structure of the dust grains, but also on the phase angle (Sun-comet-observer angle) of the observations.
The coma SED of comet C/2012 K1 (Pan-STARRS) was measured on 2014 June 04 UT using filter photometry at both mid-IR as well as the optical (scattered sunlight) wavelengths. These data enable computation of the coma averaged *bolometric* albedo [@gehrzney92]. Using the integrated flux densities in a circular aperture of radius $9.989^{\prime\prime}$, $[\lambda{\rm F}_\lambda]_{max, IR}$ was derived from the SOFIA photometry by $\chi$-square fitting a blackbody to the mid-IR data using Gaussian weighted errors, resulting in T$_{bb} = 214.04 \pm 14.94$ K with a peak flux of $ 2.02^{+ 0.12}_{- 0.15} \times 10^{-16}$ W cm$^{-2}.$ The \[V\] band photometry is contaminated by gas emission (§\[sec:opt\_ans\]). However, the $C_{2}$ bands fall outside the bandpass of the \[R\] filter and a 5800 K blackbody (the Sun) emission peaks near the \[R\] filter central wavelength ($\lambda_{c} = 0.64$ ) in $\lambda\rm{F}_{\lambda}$ (W cm$^{-2}$) space. Hence, $[\lambda{\rm F}_\lambda]_{max, scattering} = 3.33 \pm 0.03 \times 10^{-17}$ W cm$^{-2}$ derived the \[R\] band photometry measured in a circular aperture of radius $9.989^{\prime\prime}$ (Table \[tab:afr\_tab\]). The dust bolometric albedo of comet C/2012 K1 (Pan-STARRS) is $ 0.14 \pm 0.01$ at phase angle of $34.76^{\circ}$ (from Eqns. \[eqn:abgn\] and \[eqn:ftheta\]).
@kolok2004 reviewed published visual albedos of comets and found only eight 8 comets have measured albedos [excluding comets Kohoutek and Crommelin discussed in @gehrzney92], and all were from the NIC dynamical class. @mskdw2009 found only one EC with visual albedo, 21P/Giacobini-Zinner [@pittchov2008]. Recently, the albedos of 73P/Schwassmann-Wachmann 3, 103P/Hartley 2, and C/2009 P1 (Garradd) also have been measured [@meech2011; @sitko2011; @sitko2013]. Fig. \[fig:all\_comet\_albedo\] is a compilation of the the bolometric albedo data that exists on comets, including our determination for comet C/2012 K1 (Pan-STARRS). There is considerable scatter for multi-epoch observations of individual comets. Such scatter arises from variations in activity of a comet at different epochs of observation. For example, comet C/1995 O1 (Hale-Bopp), whose data are most scattered, had numerous and fast changing morphological structures [jets, shells, envelopes, e.g., @harker1997; @woodward1998]. All of these features were characterized by differing size and particle composition [e.g., @rodriguez1997; @schleicher1997]. Thus, the difference in the dust albedo for the same comet indicates variations in comet activity, specifically development of jets and other morphological features.
The ensemble albedos compiled by @kolok2004 also shows a broad distribution of values for each phase angle. The causes for these latter albedo ranges and the scatter in multi-epoch observations of comets are unclear, but must reside in the physical properties of the comet particles, including the grain size distribution, porosity, grain structure (i.e., prolate spheroids, crystalline needles, etc.), and composition [e.g., @lindsay2013]. However, observations have not yet demonstrated to what extent grain structure or grain compositions are important. To assess these latter aspects, thermal emission models and albedo observations of a additional comets are needed.
SUMMARY
=======
We discuss the pre-perihelion mid-infrared spectrophotometry and narrow band filter imagery obtained in 2014 June with FORCAST on the NASA SOFIA airborne platform of the dynamically new comet C/2012 K1 (Pan-STARRS) at a heliocentric distance of $\simeq 1.70$ AU. The spectral energy distribution of the comet at this epoch exhibits a 10 silicate feature, $[F_{10}/F^{BB}_{continuum}] = 1.18 \pm 0.03$ above a blackbody curve ($T_{bb} = 215.32 \pm 0.95$ K) fit to the spectra continua longwards of 12.5 which is quite weak compared to comets such as C/1999 O1 (Hale-Bopp) or 17P/Holmes. The coma dust bolometric albedo, $0.14 \pm 0.01$, derived using contemporaneous optical imagery is similar to other comets at the observed phase angle ($\sim 35$), while the dust production rate ($Af\rho$) from scattered light observations is $\simeq 5340$ cm.
From the observed infrared spectral energy distribution, thermal modeling analysis is used to determine the physical characteristics of the coma dust population and deduce the silicate crystalline mass fraction ($0.20^{ +0.30}_{ -0.10}$) and silicate-to-carbon dust ratio ($0.80^{ +0.25}_{- 0.20}$). We find that grains in the coma of C/2012 K1 (Pan-STARRS) are dominated by amorphous materials, especially carbon, and the differential grain size distribution peaks at radii of 0.6 , the slope of the distribution $N = 3.4$, and the grains are solid, having a fractal porosity parameter $D = 3.0$. The bulk grain properties of comet C/2012 K1 (Pan-STARRS) are comparable to other Nearly Isotropic comets (NICs) with weak 10 silicate features and similar in respect to coma grains seen in the small-set of Ecliptic family comets (ECs) that have fragmented, explosively released subsurface materials, or have had materials excavated from depth.
SOFIA observations of comet C/2012 K1 (Pan-STARRS) and other future comets enables characterization grain properties in the NIC and EC dynamical families. These properties, including dust size, porosity, and composition, relate to grain formation, radial mixing, and particle agglomeration in the proto-solar disk and provide insight to the evolution of the early solar system. As the number of well-studied comets increases at infrared wavelengths (from whence dust properties can be characterized), the fundamental differences between comets originating from different regions and times in the solar system may be eventually discerned.
Acknowledgments
===============
CEW and his team acknowledge support from Universities Space Research Association (USRA)/NASA contract NAS2-97001. CEW, MSK, DEH also acknowledge support from NASA Planetary Astronomy Program grant 12-PAST12-0016, while CEW and ELR also note support from NASA Planetary Astronomy Program grant NNX13AJ11G. The authors would also like to acknowledge the support and insight of Drs. J. DeBuzier and L. A. Helton of the SOFIA Science Ctr. for their assistance with flight planning and data reduction activities. This work is supported at The Aerospace Corporation by the Independent Research and Development program. We also thank the comments and suggestions of an anonymous referee that improved the clarity of our work.
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[^1]: http://ssd.jpl.nasa.gov/horizons.cgi
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|
---
abstract: 'Each distributor between categories enriched over a small quantaloid $\mathcal{Q}$ gives rise to two adjunctions between the categories of contravariant and covariant presheaves, and hence to two monads. These two adjunctions are respectively generalizations of Isbell adjunctions and Kan extensions in category theory. It is proved that these two processes are functorial with infomorphisms playing as morphisms between distributors; and that the free cocompletion functor of $\mathcal{Q}$-categories factors through both of these functors.'
address:
- |
Lili Shen\
School of Mathematics, Sichuan University, Chengdu 610064, China
- |
Dexue Zhang\
School of Mathematics, Sichuan University, Chengdu 610064, China
author:
- Lili Shen
- Dexue Zhang
title: 'Categories enriched over a quantaloid: Isbell adjunctions and Kan adjunctions'
---
[^1]
Introduction
============
A quantaloid [@Rosenthal1996; @Stubbe_2005] is a category enriched over the symmetric monoidal closed category consisting of complete lattices and join-preserving functions. Since a quantaloid ${\mathcal{Q}}$ is a closed and locally complete bicategory, one can develop a theory of categories enriched over ${\mathcal{Q}}$ [@Benabou1967]. It should be stressed, that for such categories, coherence issues will not be a concern in most cases. For an overview of this theory the reader is referred to [@Heymans:2010:SQG:2049377; @heymans2011symmetry; @Stubbe_2005; @Stubbe_2006].
This paper is concerned with an extension of Isbell adjunctions and Kan extensions for ${\mathcal{Q}}$-categories. In order to state the question clearly, we recall here Isbell adjunctions and Kan extensions in category theory.
Let ${{\mathbb{A}}}$ be a small category. The Isbell adjunction (or Isbell conjugacy) refers to the adjunction between ${\bf Set}^{{{\mathbb{A}}}^{\rm op}}$ and $({\bf Set}^{{{\mathbb{A}}}})^{\rm op}$ arising from the Yoneda embedding ${{\sf Y}}:{{\mathbb{A}}}{\longrightarrow}{\bf Set}^{{{\mathbb{A}}}^{\rm op}}$ and the co-Yoneda embedding ${{{\sf Y}}^{\dag}}:{{\mathbb{A}}}{\longrightarrow}({\bf Set}^{{{\mathbb{A}}}})^{\rm op}$. Given a functor $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ between small categories, composition with $F$ induces a functor $-\circ F:{\bf Set}^{{{\mathbb{B}}}^{\rm op}}{\longrightarrow}{\bf Set}^{{{\mathbb{A}}}^{\rm op}}$. The functor $-\circ F$ has both a left and a right adjoint, called respectively the left and the right Kan extension of $F$. Isbell adjunctions and Kan extensions have also been considered for categories enriched over a symmetric monoidal closed category [@Borceux1994; @Day2007651; @kelly1982basic; @kelly2005notes; @Lawvere1973; @lawvere1986taking].
In this paper, it is shown that for a small quantaloid ${\mathcal{Q}}$, each ${\mathcal{Q}}$-distributor $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$ between ${\mathcal{Q}}$-categories induces two adjunctions: $$\label{Isbell_intro}
{\phi_{{\uparrow}}}{\dashv}{\phi^{{\downarrow}}}:{{\mathcal{P}}{{\mathbb{A}}}}{\rightharpoonup}{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}$$ and $$\label{Kan_intro}
\phi^*{\dashv}\phi_*:{{\mathcal{P}}{{\mathbb{B}}}}{\rightharpoonup}{{\mathcal{P}}{{\mathbb{A}}}},$$ where ${{\mathcal{P}}{{\mathbb{A}}}}$ and ${{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}$ are the counterparts of ${\bf Set}^{{{\mathbb{A}}}^{\rm op}}$ and $({\bf Set}^{{\mathbb{A}}})^{\rm op}$, respectively.
If $\phi$ is the identity distributor on ${{\mathbb{A}}}$, then the adjunction ${\phi_{{\uparrow}}}{\dashv}{\phi^{{\downarrow}}}$ reduces to the Isbell adjunction in [@Stubbe_2005]. Given a ${\mathcal{Q}}$-functor $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$, consider the graph $F_\natural :{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$ and the cograph $F^\natural :{{\mathbb{B}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{A}}}$. Then it holds that (Theorem \[why\_kan\]) $$(F^{\natural})^*{\dashv}(F^{\natural})_*= F^{{\leftarrow}}=(F_{\natural})^*{\dashv}(F_{\natural})_*,$$ where $F^{\leftarrow}:{{\mathcal{P}}{{\mathbb{B}}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{A}}}}$ is the counterpart of the functor $-\circ F$ for ${\mathcal{Q}}$-categories.
Therefore, the adjunctions (\[Isbell\_intro\]) and (\[Kan\_intro\]) extend the fundamental construction of Isbell adjunctions and Kan extensions, so, they will be called Isbell adjunctions and Kan adjunctions by abuse of language. This paper is mainly concerned with the functoriality of these constructions.
For each ${\mathcal{Q}}$-distributor $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$, the related Isbell adjunction and Kan adjunction give rise to a monad ${\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}$ on ${{\mathcal{P}}{{\mathbb{A}}}}$ (called a closure operator on ${{\mathbb{A}}}$ in this paper) and a monad $\phi_*\circ\phi^*$ on ${{\mathcal{P}}{{\mathbb{B}}}}$, respectively. The correspondence $$(\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}})\ \mapsto\ ({{\mathbb{A}}},{\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}})$$ is functorial from the category of ${\mathcal{Q}}$-distributors and infomorphisms (defined below) to that of ${\mathcal{Q}}$-closure spaces (a ${\mathcal{Q}}$-category together with a closure operator) and continuous functors; and the correspondence $$(\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}})\ \mapsto\ ({{\mathbb{B}}},\phi_*\circ\phi^*)$$ defines a contravariant functor from the category of ${\mathcal{Q}}$-distributors and infomorphisms to that of ${\mathcal{Q}}$-closure spaces. Furthermore, the fixed points of the closure operator ${\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}:{{\mathcal{P}}{{\mathbb{A}}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{A}}}}$ (or equivalently, all the algebras if we consider ${\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}$ as a monad) constitute a complete ${\mathcal{Q}}$-category ${\mathcal{M}}(\phi)$; the fixed points of the closure operator $\phi_*\circ\phi^*:{{\mathcal{P}}{{\mathbb{B}}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{B}}}}$ also constitute a complete ${\mathcal{Q}}$-category ${\mathcal{K}}(\phi)$. Thus, each distributor $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$ generates two complete ${\mathcal{Q}}$-categories: ${\mathcal{M}}(\phi)$ and ${\mathcal{K}}(\phi)$. It will be shown that ${\mathcal{M}}$ is functorial and ${\mathcal{K}}$ contravariant functorial from the category of ${\mathcal{Q}}$-distributors and infomorphisms to that of complete ${\mathcal{Q}}$-categories and left adjoints. Moreover, the free cocompletion functor ${\mathcal{P}}$ of ${\mathcal{Q}}$-categories factors through both ${\mathcal{M}}$ and ${\mathcal{K}}$.
It should be pointed out that some conclusions in this paper have been proved, in the circumstance of concept lattices, in [@Shen2013166] for discrete ${\mathcal{Q}}$-categories in the case that ${\mathcal{Q}}$ is an one-object quantaloid, i.e., a unital quantale. The situation dealt with here is much more involved, and the method developed here allows for a wide range of applicability.
Categories enriched over a quantaloid {#quantaloid}
=====================================
The theory of categories enriched over a quantaloid has been studied systematically in [@Stubbe_2005; @Stubbe_2006]. In this section, we recall some basic concepts and fix some notations that will be used in the sequel.
Complete lattices and join-preserving functions constitute a symmetric monoidal closed category [**Sup**]{}. A [*quantaloid*]{} ${\mathcal{Q}}$ is a [**Sup**]{}-enriched category [@Rosenthal1996; @Stubbe_2005]. Explicitly, a quantaloid ${\mathcal{Q}}$ is a category with a class of objects ${\mathcal{Q}}_0$ such that ${\mathcal{Q}}(A,B)$ is a complete lattice for all $A,B\in{\mathcal{Q}}_0$, and the composition $\circ$ of morphisms preserves joins in both variables, i.e., $$g\circ\Big({\bigvee}_{i}f_i\Big)={\bigvee}_{i}(g\circ f_i)\quad\text{and}\quad\Big({\bigvee}_{j}g_j\Big)\circ f={\bigvee}_{j}(g_j\circ f)$$ for all $f,f_i\in{\mathcal{Q}}(A,B)$ and $g,g_j\in{\mathcal{Q}}(B,C)$. The complete lattice ${\mathcal{Q}}(A,B)$ has a top element $\top_{A,B}$ and a bottom element $\bot_{A,B}$.
In this paper, ${\mathcal{Q}}$ is always assumed to be a small quantaloid, i.e., ${\mathcal{Q}}_0$ is a set.
For each $X\in{\mathcal{Q}}_0$ and $f\in{\mathcal{Q}}(A,B)$, both functions $$-\circ f:{\mathcal{Q}}(B,X){\longrightarrow}{\mathcal{Q}}(A,X):g\mapsto g\circ f,$$ $$f\circ -:{\mathcal{Q}}(X,A){\longrightarrow}{\mathcal{Q}}(X,B):g\mapsto f\circ g$$ have respective right adjoints: $$-{\swarrow}f:{\mathcal{Q}}(A,X){\longrightarrow}{\mathcal{Q}}(B,X):g\mapsto g{\swarrow}f,$$ $$f{\searrow}-:{\mathcal{Q}}(X,B){\longrightarrow}{\mathcal{Q}}(X,A):g\mapsto f{\searrow}g.$$ The operators ${\swarrow}$ and ${\searrow}$ are respectively the [*left*]{} and [*right implications*]{}.
A [*${\mathcal{Q}}$-category*]{} [@Stubbe_2005] ${{\mathbb{A}}}$ consists of a set ${{\mathbb{A}}}_0$ equipped with a map $t:{{\mathbb{A}}}_0{\longrightarrow}{\mathcal{Q}}_0:x\mapsto tx$ ($tx$ is called the [*type*]{} of $x$ and ${{\mathbb{A}}}_0$ is called a [*${\mathcal{Q}}$-typed set*]{}) and hom-arrows ${{\mathbb{A}}}(x,y)\in{\mathcal{Q}}(tx,ty)$ such that
- $1_{tx}\leq{{\mathbb{A}}}(x,x)$ for all $x\in{{\mathbb{A}}}_0$;
- ${{\mathbb{A}}}(y,z)\circ{{\mathbb{A}}}(x,y)\leq{{\mathbb{A}}}(x,z)$ for all $x,y,z\in{{\mathbb{A}}}_0$.
A [*${\mathcal{Q}}$-functor*]{} [@Stubbe_2005] $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ between ${\mathcal{Q}}$-categories is a map $F:{{\mathbb{A}}}_0{\longrightarrow}{{\mathbb{B}}}_0$ such that
- $F$ is [*type-preserving*]{} in the sense that $\forall x\in{{\mathbb{A}}}_0$, $tx=t(Fx)$;
- $\forall x,x'\in{{\mathbb{A}}}_0$, ${{\mathbb{A}}}(x,x')\leq{{\mathbb{B}}}(Fx,Fx')$.
A ${\mathcal{Q}}$-functor $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ is [*fully faithful*]{} if ${{\mathbb{A}}}(x,x')={{\mathbb{B}}}(Fx,Fx')$ for all $x,x'\in{{\mathbb{A}}}_0$. Bijective fully faithful ${\mathcal{Q}}$-functors are exactly the isomorphisms in the category ${\mathcal{Q}}$-[**Cat**]{} of ${\mathcal{Q}}$-categories and ${\mathcal{Q}}$-functors.
A ${\mathcal{Q}}$-category ${{\mathbb{B}}}$ is a (full) ${\mathcal{Q}}$-subcategory of ${{\mathbb{A}}}$ if ${{\mathbb{B}}}_0$ is a subset of ${{\mathbb{A}}}_0$ and ${{\mathbb{B}}}(x,y)={{\mathbb{A}}}(x,y)$ for all $x,y\in{{\mathbb{B}}}_0$.
Given a ${\mathcal{Q}}$-category ${{\mathbb{A}}}$, there is a natural underlying preorder $\leq$ on ${{\mathbb{A}}}_0$. For $x,y\in{{\mathbb{A}}}_0$, $x\leq y$ if and only if they are of the same type $tx=ty=A$ and $1_A\leq{{\mathbb{A}}}(x,y)$. Two objects $x,y$ in ${{\mathbb{A}}}$ are [*isomorphic*]{} if $x\leq y$ and $y\leq x$, written $x\cong y$. ${{\mathbb{A}}}$ is [*skeletal*]{} if no two different objects in ${{\mathbb{A}}}$ are isomorphic.
The underlying preorders on ${\mathcal{Q}}$-categories induce an order between ${\mathcal{Q}}$-functors: $$F\leq G:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}\iff\forall x\in{{\mathbb{A}}}_0,Fx\leq Gx\ \text{in}\ {{\mathbb{B}}}_0.$$ We denote $F\cong G:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ if $F\leq G$ and $G\leq F$.
A pair of ${\mathcal{Q}}$-functors $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ and $G:{{\mathbb{B}}}{\longrightarrow}{{\mathbb{A}}}$ form an [*adjunction*]{} [@Stubbe_2005], written $F{\dashv}G:{{\mathbb{A}}}{\rightharpoonup}{{\mathbb{B}}}$, if $1_{{{\mathbb{A}}}}\leq G\circ F$ and $F\circ G\leq 1_{{{\mathbb{B}}}}$, where $1_{{\mathbb{A}}}$ and $1_{{\mathbb{B}}}$ are respectively the identity ${\mathcal{Q}}$-functors on ${{\mathbb{A}}}$ and ${{\mathbb{B}}}$. In this case, $F$ is called a [*left adjoint*]{} of $G$ and $G$ a [*right adjoint*]{} of $F$.
A [*${\mathcal{Q}}$-distributor*]{} [@Stubbe_2005] $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$ between ${\mathcal{Q}}$-categories is a map that assigns to each pair $(x,y)\in{{\mathbb{A}}}_0\times{{\mathbb{B}}}_0$ a morphism $\phi(x,y)\in{\mathcal{Q}}(tx,ty)$ in ${\mathcal{Q}}$, such that
- $\forall x\in{{\mathbb{A}}}_0$, $\forall y,y'\in{{\mathbb{B}}}_0$, ${{\mathbb{B}}}(y',y)\circ\phi(x,y')\leq\phi(x,y)$;
- $\forall x,x'\in{{\mathbb{A}}}_0$, $\forall y\in{{\mathbb{B}}}_0$, $\phi(x',y)\circ{{\mathbb{A}}}(x,x')\leq\phi(x,y)$.
${\mathcal{Q}}$-categories and ${\mathcal{Q}}$-distributors constitute a quantaloid ${\mathcal{Q}}$-[**Dist**]{} [@Stubbe_2005] in which
- the local order is defined pointwise, i.e., for ${\mathcal{Q}}$-distributors $\phi,\psi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$, $$\phi\leq\psi\iff\forall x\in{{\mathbb{A}}}_0,\forall y\in{{\mathbb{B}}}_0,\phi(x,y)\leq\psi(x,y);$$
- the composition $\psi\circ\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{C}}}$ of ${\mathcal{Q}}$-distributors $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$ and $\psi:{{\mathbb{B}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{C}}}$ is given by $$\forall x\in{{\mathbb{A}}}_0,\forall z\in{{\mathbb{C}}}_0,(\psi\circ\phi)(x,z)={\bigvee}_{y\in{{\mathbb{B}}}_0}\psi(y,z)\circ\phi(x,y);$$
- the identity ${\mathcal{Q}}$-distributor on a ${\mathcal{Q}}$-category ${{\mathbb{A}}}$ is the hom-arrows of ${{\mathbb{A}}}$ and will be denoted by ${{\mathbb{A}}}:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{A}}}$;
- for ${\mathcal{Q}}$-distributors $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$, $\psi:{{\mathbb{B}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{C}}}$ and $\eta:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{C}}}$, the left implication $\eta{\swarrow}\phi:{{\mathbb{B}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{C}}}$ and the right implication $\psi{\searrow}\eta:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$ are given by $$\forall y\in{{\mathbb{B}}}_0,\forall z\in{{\mathbb{C}}}_0,(\eta{\swarrow}\phi)(y,z)={\bigwedge}_{x\in{{\mathbb{A}}}_0}\eta(x,z){\swarrow}\phi(x,y)$$ and $$\forall x\in{{\mathbb{A}}}_0,\forall y\in{{\mathbb{B}}}_0,(\psi{\searrow}\eta)(x,y)={\bigwedge}_{z\in{{\mathbb{C}}}_0}\psi(y,z){\searrow}\eta(x,z).$$
An [*adjunction*]{} [@Stubbe_2005] in a quantaloid ${\mathcal{Q}}$, $f{\dashv}g:A{\rightharpoonup}B$ in symbols, is a pair of morphisms $f:A{\longrightarrow}B$ and $g:B{\longrightarrow}A$ in ${\mathcal{Q}}$ such that $1_A\leq g\circ f$ and $f\circ g\leq 1_B$. In this case, $f$ is a left adjoint of $g$ and $g$ a right adjoint of $f$. In particular, a pair of ${\mathcal{Q}}$-distributors $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$ and $\psi:{{\mathbb{B}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{A}}}$ form an adjunction $\phi{\dashv}\psi:{{\mathbb{A}}}{\rightharpoonup}{{\mathbb{B}}}$ in the quantaloid ${\mathcal{Q}}$-[**Dist**]{} if ${{\mathbb{A}}}\leq\psi\circ\phi$ and $\phi\circ\psi\leq{{\mathbb{B}}}$.
Every ${\mathcal{Q}}$-functor $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ induces an adjunction $F_{\natural}{\dashv}F^{\natural}:{{\mathbb{A}}}{\rightharpoonup}{{\mathbb{B}}}$ in ${\mathcal{Q}}$-[**Dist**]{} with $F_{\natural}(x,y)={{\mathbb{B}}}(Fx,y)$ and $F^{\natural}(y,x)={{\mathbb{B}}}(y,Fx)$ for all $x\in{{\mathbb{A}}}_0$ and $y\in{{\mathbb{B}}}_0$. The ${\mathcal{Q}}$-distributors $F_{\natural}:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$ and $F^{\natural}:{{\mathbb{B}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{A}}}$ are called the [*graph*]{} and [*cograph*]{} of $F$, respectively.
[[@Heymans:2010:SQG:2049377]]{} \[graph\_cograph\_implication\] If $f{\dashv}g:A{\rightharpoonup}B$ in a quantaloid ${\mathcal{Q}}$, then the following identities hold for all ${\mathcal{Q}}$-arrows $h,h'$ whenever the compositions and implications make sense:
- $h\circ f=h{\swarrow}g$, $g\circ h=f{\searrow}h$.
- $(f\circ h){\searrow}h'=h{\searrow}(g\circ h')$, $(h'\circ f){\swarrow}h=h'{\swarrow}(h\circ g)$.
- $(h{\searrow}h')\circ f=h{\searrow}(h'\circ f)$, $g\circ(h'{\swarrow}h)=(g\circ h'){\swarrow}h$.
- $g\circ(h{\searrow}h')=(h\circ f){\searrow}h'$, $(h'{\swarrow}h)\circ f=h'{\swarrow}(g\circ h)$.
The identities in Proposition \[graph\_cograph\_implication\] will be frequently applied in the next sections to the adjunction $F_{\natural}{\dashv}F^{\natural}:{{\mathbb{A}}}{\rightharpoonup}{{\mathbb{B}}}$ induced by a ${\mathcal{Q}}$-functor $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$.
[[@Stubbe_2005]]{} Let $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ and $G:{{\mathbb{B}}}{\longrightarrow}{{\mathbb{A}}}$ be a pair of ${\mathcal{Q}}$-functors. The following conditions are equivalent:
- $F{\dashv}G:{{\mathbb{A}}}{\rightharpoonup}{{\mathbb{B}}}$.
- $F_{\natural}={{\mathbb{B}}}(F-,-)={{\mathbb{A}}}(-,G-)=G^{\natural}$.
- $G_{\natural}{\dashv}F_{\natural}:{{\mathbb{B}}}{\rightharpoonup}{{\mathbb{A}}}$ in ${\mathcal{Q}}$-[**Dist**]{}.
- $G^{\natural}{\dashv}F^{\natural}:{{\mathbb{A}}}{\rightharpoonup}{{\mathbb{B}}}$ in ${\mathcal{Q}}$-[**Dist**]{}.
\[fully\_faithful\_graph\_cograph\] Let $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ be a ${\mathcal{Q}}$-functor.
- If $F$ is fully faithful, then $F^{\natural}\circ F_{\natural}={{\mathbb{A}}}$.
- If $F$ is essentially surjective in the sense that there is some $x\in{{\mathbb{A}}}_0$ such that $Fx\cong y$ in ${{\mathbb{B}}}$ for all $y\in{{\mathbb{B}}}_0$, then $F_{\natural}\circ F^{\natural}={{\mathbb{B}}}$.
\(1) If $F$ is fully faithful, then for all $x,x'\in{{\mathbb{A}}}_0$, $$(F^{\natural}\circ F_{\natural})(x,x')={\bigvee}_{y\in{{\mathbb{B}}}_0}{{\mathbb{B}}}(y,Fx')\circ{{\mathbb{B}}}(Fx,y)={{\mathbb{B}}}(Fx,Fx')={{\mathbb{A}}}(x,x').$$
\(2) If $F$ is essentially surjective, then for all $y,y'\in{{\mathbb{B}}}_0$, there is some $x\in{{\mathbb{A}}}_0$ such that $Fx\cong y$. Thus $$\begin{aligned}
(F_{\natural}\circ F^{\natural})(y,y')&={\bigvee}_{a\in{{\mathbb{A}}}_0}{{\mathbb{B}}}(Fa,y')\circ{{\mathbb{B}}}(y,Fa)\\
&\geq{{\mathbb{B}}}(Fx,y')\circ{{\mathbb{B}}}(y,Fx)\\
&={{\mathbb{B}}}(y,y')\circ{{\mathbb{B}}}(y,y)\\
&\geq{{\mathbb{B}}}(y,y').\end{aligned}$$ Since $F_{\natural}\circ F^{\natural}\leq{{\mathbb{B}}}$ holds trivially, it follows that $F_{\natural}\circ F^{\natural}={{\mathbb{B}}}$.
Following [@Stubbe_2005], for each $X\in{\mathcal{Q}}_0$, write $*_X$ for the ${\mathcal{Q}}$-category with only one object $*$ of type $t*=X$ and hom-arrow $1_X$.
A [*contravariant presheaf*]{} [@Stubbe_2005] on a ${\mathcal{Q}}$-category ${{\mathbb{A}}}$ is a ${\mathcal{Q}}$-distributor $\mu:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}*_X$ with $X\in{\mathcal{Q}}_0$. Contravariant presheaves on a ${\mathcal{Q}}$-category ${{\mathbb{A}}}$ constitute a ${\mathcal{Q}}$-category ${{\mathcal{P}}{{\mathbb{A}}}}$ in which $$t\mu=X\quad\text{and}\quad{{\mathcal{P}}{{\mathbb{A}}}}(\mu,{\lambda})={\lambda}{\swarrow}\mu$$ for all $\mu:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}*_X$ and ${\lambda}:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}*_{Y}$ in $({{\mathcal{P}}{{\mathbb{A}}}})_0$.
Dually, a [*covariant presheaf*]{} on a ${\mathcal{Q}}$-category ${{\mathbb{A}}}$ is a ${\mathcal{Q}}$-distributor $\mu:*_X{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{A}}}$. Covariant presheaves on ${{\mathbb{A}}}$ constitute a ${\mathcal{Q}}$-category ${{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}$ in which $$t\mu=X\quad\text{and}\quad{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}(\mu,{\lambda})={\lambda}{\searrow}\mu$$ for all $\mu:*_X{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{A}}}$ and ${\lambda}:*_{Y}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{A}}}$.
In particular, we denote ${\mathcal{P}}(*_X)={\mathcal{P}}X$ and ${{\mathcal{P}}^{\dag}}(*_X)={{\mathcal{P}}^{\dag}}X$ for each $X\in{\mathcal{Q}}_0$.
\[PdA\_QDist\_order\] For each ${\mathcal{Q}}$-category ${{\mathbb{A}}}$, it follows from the definition that the underlying preorder in ${{\mathcal{P}}{{\mathbb{A}}}}$ coincides with the local order in ${\mathcal{Q}}$-[**Dist**]{}, while the underlying preorder in ${{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}$ is the [*reverse*]{} local order in ${\mathcal{Q}}$-[**Dist**]{}. That is to say, for all $\mu,{\lambda}\in{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}$, we have $$\mu\leq{\lambda}\ \text{in}\ ({{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}})_0\iff{\lambda}\leq\mu\ \text{in}\ {\mathcal{Q}}\text{-}{\bf Dist}.$$ In order to get rid of the confusion about the symbol $\leq$, from now on we make the convention that the symbol $\leq$ between ${\mathcal{Q}}$-distributors always denotes the local order in ${\mathcal{Q}}$-[**Dist**]{} if not otherwise specified.
Given a ${\mathcal{Q}}$-category ${{\mathbb{A}}}$ and $a\in{{\mathbb{A}}}_0$, write ${{\sf Y}}a$ for the ${\mathcal{Q}}$-distributor $${{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}*_{ta},\quad x\mapsto{{\mathbb{A}}}(x,a);$$ write ${{{\sf Y}}^{\dag}}a$ for the ${\mathcal{Q}}$-distributor $$*_{ta}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{A}}},\quad x\mapsto{{\mathbb{A}}}(a,x).$$
The following lemma implies that both ${{\sf Y}}:{{\mathbb{A}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{A}}}}, a\mapsto{{\sf Y}}a$ and ${{{\sf Y}}^{\dag}}:{{\mathbb{A}}}{\longrightarrow}{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}, a\mapsto{{{\sf Y}}^{\dag}}a$ are fully faithful ${\mathcal{Q}}$-functors (hence embeddings if ${{\mathbb{A}}}$ is skeletal). Thus, ${{\sf Y}}$ and ${{{\sf Y}}^{\dag}}$ are called respectively the [*Yoneda embedding*]{} and the [*co-Yoneda embedding*]{}.
\[Yoneda\_lemma\] [[@Stubbe_2005]]{} ${{\mathcal{P}}{{\mathbb{A}}}}({{\sf Y}}a,\mu)=\mu(a)$ and ${{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}({\lambda},{{{\sf Y}}^{\dag}}a)={\lambda}(a)$ for all $a\in{{\mathbb{A}}}_0$, $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$ and ${\lambda}\in{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}$.
For each ${\mathcal{Q}}$-distributor $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$ and $x\in{{\mathbb{A}}}_0,y\in{{\mathbb{B}}}_0$, write $\phi(x,-)$ for the ${\mathcal{Q}}$-distributor $\phi\circ{{{\sf Y}}^{\dag}}_{{{\mathbb{A}}}}x: *_{tx}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$; and write $\phi(-,y)$ for the ${\mathcal{Q}}$-distributor ${\sf Y}_{{{\mathbb{B}}}}y\circ\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}*_{ty}$. Then the Yoneda lemma can be phrased as the commutativity of the following diagrams: $$\bfig
\ptriangle/->`<-`<-/[{{\mathcal{P}}{{\mathbb{A}}}}`*_{t\mu}`{{\mathbb{A}}};{{\mathcal{P}}{{\mathbb{A}}}}(-,\mu)`{{\sf Y}}_{\natural}`\mu] \place(250,500)[\circ]\place(0,250)[\circ]\place(250,250)[\circ]
\ptriangle(1200,0)/<-`->`->/[{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}`*_{t{\lambda}}`{{\mathbb{A}}};{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}({\lambda},-)`({{{\sf Y}}^{\dag}})^{\natural}`{\lambda}] \place(1450,500)[\circ]\place(1200,250)[\circ]\place(1450,250)[\circ]
\efig$$ That is, $\mu={{\mathcal{P}}{{\mathbb{A}}}}({{\sf Y}}-,\mu)$ and ${\lambda}={{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}({\lambda},{{{\sf Y}}^{\dag}}-)$.
\[distributor\_notion\] Given ${\mathcal{Q}}$-distributors $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$, $\psi:{{\mathbb{B}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{C}}}$ and $\eta:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{C}}}$, one can form ${\mathcal{Q}}$-distributors such as $\phi(x,-),\eta{\swarrow}\phi,\eta{\searrow}\psi(y,-)$, etc. We list here some basic formulas related to these ${\mathcal{Q}}$-distributors that will be used in the sequel. $$\forall x\in{{\mathbb{A}}}_0,\forall z\in{{\mathbb{C}}}_0,(\psi\circ\phi)(x,z)=\psi(-,z)\circ\phi(x,-);$$ $$\forall x\in{{\mathbb{A}}}_0,(\psi\circ\phi)(x,-)=\psi\circ\phi(x,-);$$ $$\forall y\in{{\mathbb{B}}}_0, (\eta{\swarrow}\phi)(y,-)=\eta{\swarrow}\phi(-,y);$$ $$\forall x\in{{\mathbb{A}}}_0,\forall y\in{{\mathbb{B}}}_0,(\psi{\searrow}\eta)(x,y)=\psi(y,-){\searrow}\eta(x,-).$$
For a ${\mathcal{Q}}$-functor $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ between ${\mathcal{Q}}$-categories, define ${\mathcal{Q}}$-functors $F^{{\rightarrow}}:{{\mathcal{P}}{{\mathbb{A}}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{B}}}}$ and $F^{{\leftarrow}}:{{\mathcal{P}}{{\mathbb{B}}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{A}}}}$ by $F^{{\rightarrow}}(\mu)=\mu\circ F^{\natural}$ and $F^{{\leftarrow}}({\lambda})={\lambda}\circ F_{\natural}$. Then $$F^{{\rightarrow}}{\dashv}F^{{\leftarrow}}:{{\mathcal{P}}{{\mathbb{A}}}}{\rightharpoonup}{{\mathcal{P}}{{\mathbb{B}}}}$$ in ${\mathcal{Q}}$-[**Cat**]{}. For all ${\lambda}\in{{\mathcal{P}}{{\mathbb{B}}}}$ and $x\in{{\mathbb{A}}}_0$, it can be verified that $$\label{F_la_lam_Fx}
F^{{\leftarrow}}({\lambda})(x)={\lambda}(Fx)\in {\mathcal{Q}}(tx,t{\lambda}).$$
Dually, we may also define ${\mathcal{Q}}$-functors $F^{{\rightarrow}}:{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}{\longrightarrow}{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}$ and $F^{{\leftarrow}}:{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}{\longrightarrow}{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}$ by $F^{{\rightarrow}}(\mu)=F_{\natural}\circ\mu$ and $F^{{\leftarrow}}({\lambda})=F^{\natural}\circ{\lambda}$. Then $$F^{{\leftarrow}}{\dashv}F^{{\rightarrow}}:{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}{\rightharpoonup}{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}$$ in ${\mathcal{Q}}$-[**Cat**]{}. For all ${\lambda}\in{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}$ and $x\in{{\mathbb{A}}}_0$, it can be verified that $$\label{F_La_lam_Fx}
F^{{\leftarrow}}({\lambda})(x)={\lambda}(Fx)\in{\mathcal{Q}}(t{\lambda}, tx).$$
Note that the symbol $F^{\rightarrow}$ is used for both of the ${\mathcal{Q}}$-functors ${{\mathcal{P}}{{\mathbb{A}}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{B}}}}$ and ${{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}{\longrightarrow}{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}$. This should cause no confusion since it can be easily detected from the context which one it stands for. So is the symbol $F^{\leftarrow}$.
We would like to stress that $$\label{Fla_Fra_adjuntion}
\mu\leq F^{{\leftarrow}}\circ F^{{\rightarrow}}(\mu)\quad\text{and}\quad F^{{\rightarrow}}\circ F^{{\leftarrow}}({\lambda})\leq{\lambda}$$ for all $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$ and ${\lambda}\in{{\mathcal{P}}{{\mathbb{B}}}}$; whereas $$\label{FLa_FRa_adjuntion}
\nu\leq F^{{\leftarrow}}\circ F^{{\rightarrow}}(\nu)\quad\text{and}\quad F^{{\rightarrow}}\circ F^{{\leftarrow}}({\gamma})\leq{\gamma}$$ for all $\nu\in{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}$ and $\gamma\in{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}$ by Remark \[PdA\_QDist\_order\].
For a ${\mathcal{Q}}$-functor $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ and a contravariant presheaf $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$, the [*colimit of $F$ weighted by $\mu$*]{} [@Stubbe_2005] is an object ${{\rm colim}}_{\mu}F\in{{\mathbb{B}}}_0$ (necessarily of type $t\mu$) such that $${{\mathbb{B}}}({{{\rm colim}}}_{\mu}F,-)=F_{\natural}{\swarrow}\mu.$$
Dually, for a covariant presheaf ${\lambda}\in{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}$, the [*limit of $F$ weighted by ${\lambda}$*]{} is an object $\lim_{{\lambda}}F\in{{\mathbb{B}}}_0$ (necessarily of type $t{\lambda}$) such that $${{\mathbb{B}}}(-,{\lim}_{{\lambda}}F)={\lambda}{\searrow}F^{\natural}.$$
A ${\mathcal{Q}}$-category ${{\mathbb{B}}}$ is [*cocomplete*]{} (resp. [*complete*]{}) if ${{\rm colim}}_{\mu}F$ (resp. $\lim_{{\lambda}}F$) exists for each ${\mathcal{Q}}$-functor $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ and $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$ (resp. ${\lambda}\in{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}$).
In particular, for a ${\mathcal{Q}}$-category ${{\mathbb{A}}}$ and $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$ (resp. ${\lambda}\in{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}$), the colimit ${{\rm colim}}_{\mu}1_{{{\mathbb{A}}}}$ (the limit ${\lim}_{{\lambda}}1_{{{\mathbb{A}}}}$, resp.) exists if there is some $a\in{{\mathbb{A}}}_0$ such that $${{\mathbb{A}}}(a,-)={{\mathbb{A}}}{\swarrow}\mu\quad(\text{resp.}\ {{\mathbb{A}}}(-,a)={\lambda}{\searrow}{{\mathbb{A}}}).$$ In this case, we say that $a$ is a [*supremum*]{} of $\mu$ (resp. an [*infimum*]{} of ${\lambda}$), and denote it by $\sup\mu$ (resp. $\inf{\lambda}$). Note that for any ${\mathcal{Q}}$-functor $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ and $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$ (resp. ${\lambda}\in{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}$), $${{\rm colim}}_{\mu}F={\sup}_{{{\mathbb{B}}}}F^{{\rightarrow}}(\mu)\quad(\text{resp.}\ {\lim}_{{\lambda}}F={\inf}_{{{\mathbb{B}}}}F^{{\rightarrow}}({\lambda}))$$ when it exists.
Let ${{\mathbb{A}}}$ be a ${\mathcal{Q}}$-category. For $x\in{{\mathbb{A}}}_0$ and $f\in{\mathcal{P}}(tx)$ (resp. $f\in{{\mathcal{P}}^{\dag}}(tx)$), the [*tensor*]{} (resp. [*cotensor*]{}) [@Stubbe_2006] of $f$ and $x$, denoted by $f\otimes x$ (resp. $f{\!\rightarrowtail\!}x$), is an object in ${{\mathbb{A}}}_0$ of type $t(f\otimes x)=tf$ (resp. $t(f{\!\rightarrowtail\!}x)=tf$) such that $${{\mathbb{A}}}(f\otimes x,-)={{\mathbb{A}}}(x,-){\swarrow}f\quad(\text{resp}.\ {{\mathbb{A}}}(-,f{\!\rightarrowtail\!}x)=f{\searrow}{{\mathbb{A}}}(-,x)).$$
For $x\in{{\mathbb{A}}}_0$ and $f\in{\mathcal{P}}(tx)$, it is easily seen that the tensor $f\otimes x$ is exactly the supremum of $f\circ{{\sf Y}}x\in{{\mathcal{P}}{{\mathbb{A}}}}$ if it exists. Dually, for $y\in{{\mathbb{A}}}_0$ and $g\in{{\mathcal{P}}^{\dag}}(ty)$, the cotensor $g{\!\rightarrowtail\!}y$ is the infimum of ${{{\sf Y}}^{\dag}}y\circ g\in{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}$ if it exists.
A ${\mathcal{Q}}$-category ${{\mathbb{A}}}$ is said to be [*tensored*]{} (resp. [*cotensored*]{}) if the tensor $f\otimes x$ (resp. the cotensor $f{\!\rightarrowtail\!}x$) exists for all choices of $x$ and $f$.
\[PA\_tensor\] Let ${{\mathbb{A}}}$ be a ${\mathcal{Q}}$-category.
- ${{\mathcal{P}}{{\mathbb{A}}}}$ is a tensored and cotensored ${\mathcal{Q}}$-category in which $$f\otimes\mu=f\circ\mu,\quad g{\!\rightarrowtail\!}\mu=g{\searrow}\mu$$ for all $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$ and $f\in{\mathcal{P}}(t\mu)$, $g\in{{\mathcal{P}}^{\dag}}(t\mu)$.
- ${{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}$ is a tensored and cotensored ${\mathcal{Q}}$-category in which $$f\otimes{\lambda}={\lambda}{\swarrow}f,\quad g{\!\rightarrowtail\!}{\lambda}={\lambda}\circ g$$ for all ${\lambda}\in{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}$ and $f\in{\mathcal{P}}(t{\lambda})$, $g\in{{\mathcal{P}}^{\dag}}(t{\lambda})$.
Let ${{\mathbb{A}}}$ be a ${\mathcal{Q}}$-category and $X\in{\mathcal{Q}}_0$. The objects in ${{\mathbb{A}}}$ with type $X$ constitute a subset of the underlying preordered set ${{\mathbb{A}}}_0$ and we denote it by ${{\mathbb{A}}}_X$. A ${\mathcal{Q}}$-category ${{\mathbb{A}}}$ is said to be [*order-complete*]{} [@Stubbe_2006] if each ${{\mathbb{A}}}_X$ admits all joins in the underlying preorder.
For each subset $\{x_i\}\subseteq{{\mathbb{A}}}_X$, if the join (resp. meet) of $\{x_i\}$ in ${{\mathbb{A}}}_X$ exists, then $${\bigvee}_i x_i=\sup{\bigvee}_i{{\sf Y}}x_i\quad\Big(\text{resp.}\quad{\bigwedge}_i x_i=\inf{\bigwedge}_i{{{\sf Y}}^{\dag}}x_i\Big),$$ where $\displaystyle{\bigvee}_i x_i$ and $\displaystyle{\bigwedge}_i x_i$ denote respectively the join and the meet in ${{\mathbb{A}}}_X$; $\displaystyle{\bigvee}_i{{\sf Y}}x_i$ denotes the join in $({{\mathcal{P}}{{\mathbb{A}}}})_X$ and $\displaystyle{\bigwedge}_i{{{\sf Y}}^{\dag}}x_i$ the meet in $({{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}})_X$.
[[@Stubbe_2005; @Stubbe_2006]]{} \[complete\_cocomplete\_equivalent\] For a ${\mathcal{Q}}$-category ${{\mathbb{A}}}$, the following conditions are equivalent:
- ${{\mathbb{A}}}$ is complete.
- ${{\mathbb{A}}}$ is cocomplete.
- ${{\mathbb{A}}}$ is tensored, cotensored, and order-complete.
- Each $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$ has a supremum.
- ${{\sf Y}}$ has a left adjoint in ${\mathcal{Q}}$-[**Cat**]{}, given by $\sup:{{\mathcal{P}}{{\mathbb{A}}}}{\longrightarrow}{{\mathbb{A}}}$.
- Each ${\lambda}\in{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}$ has an infimum.
- ${{{\sf Y}}^{\dag}}$ has a right adjoint in ${\mathcal{Q}}$-[**Cat**]{}, given by $\inf:{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}{\longrightarrow}{{\mathbb{A}}}$.
In this case, for each $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$ and ${\lambda}\in{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}$, $$\sup\mu={\bigvee}_{a\in{{\mathbb{A}}}_0}(\mu(a)\otimes a),\quad\inf{\lambda}={\bigwedge}_{a\in{{\mathbb{A}}}_0}({\lambda}(a){\!\rightarrowtail\!}a),$$ where $\displaystyle{\bigvee}$ and $\displaystyle{\bigwedge}$ denote respectively the join in ${{\mathbb{A}}}_{t\mu}$ and the meet in ${{\mathbb{A}}}_{t{\lambda}}$.
\[PX\_PA\_complete\] Let ${{\mathbb{A}}}$ be a ${\mathcal{Q}}$-category.
- ${{\mathcal{P}}{{\mathbb{A}}}}$ is a complete ${\mathcal{Q}}$-category in which $$\sup\Phi={\bigvee}_{\mu\in{{\mathcal{P}}{{\mathbb{A}}}}}\Phi(\mu)\circ\mu=\Phi\circ({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}$$ $$\bfig
\ptriangle/->`<-`<-/[{{\mathcal{P}}{{\mathbb{A}}}}`*_{t\Phi}`{{\mathbb{A}}}; \Phi`({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}`\sup\Phi] \place(250,500)[\circ] \place(0,250)[\circ]\place(250,250)[\circ]
\efig$$ for all $\Phi\in{\mathcal{P}}({{\mathcal{P}}{{\mathbb{A}}}})$ [@Stubbe_2005] and $$\inf\Psi={\bigwedge}_{\mu\in{{\mathcal{P}}{{\mathbb{A}}}}}\Psi(\mu){\searrow}\mu=\Psi{\searrow}({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}$$ $$\bfig
\btriangle[{{\mathbb{A}}}`*_{t\Psi}`{{\mathcal{P}}{{\mathbb{A}}}}; \inf\Psi`({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}`\Psi]
\place(250,0)[\circ]\place(0,250)[\circ] \place(250,250)[\circ] \place(125,125)[\twoar(1,1)]
\efig$$ for all $\Psi\in{{\mathcal{P}}^{\dag}}({{\mathcal{P}}{{\mathbb{A}}}})$, i.e., $\inf\Psi$ is the largest ${\mathcal{Q}}$-distributor $\mu:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}*_{t\Psi}$ such that $\Psi\circ\mu\leq ({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}$.
- ${{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}$ is a complete ${\mathcal{Q}}$-category in which $$\sup\Phi=({{{\sf Y}}^{\dag}}_{{{\mathbb{A}}}})^{\natural}{\swarrow}\Phi\quad\text{and} \quad\inf\Psi=({{{\sf Y}}^{\dag}}_{{{\mathbb{A}}}})^{\natural}\circ\Psi$$ for all $\Phi\in{\mathcal{P}}({{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}})$ and $\Psi\in{{\mathcal{P}}^{\dag}}({{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}})$.
In particular, ${\mathcal{P}}X$ and ${{\mathcal{P}}^{\dag}}X$ are both complete ${\mathcal{Q}}$-categories for all $X\in{\mathcal{Q}}_0$.
\[F\_la\_ra\_condition\] [[@Stubbe_2006]]{} Let $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ be a ${\mathcal{Q}}$-functor between ${\mathcal{Q}}$-categories, with ${{\mathbb{A}}}$ complete, then $F$ is a left (resp. right) adjoint in ${\mathcal{Q}}$-[**Cat**]{} if and only if
1. $F$ preserves tensors (resp. cotensors) in the sense that $F(f\otimes_{{{\mathbb{A}}}}x)= f\otimes_{{{\mathbb{B}}}}Fx$ (resp. $F(f{\!\rightarrowtail\!}_{{{\mathbb{A}}}}x)=f{\!\rightarrowtail\!}_{{{\mathbb{B}}}}Fx$) for all $x\in{{\mathbb{A}}}_0$ and $f\in{\mathcal{P}}(tx)$ (resp. $f\in{{\mathcal{P}}^{\dag}}(tx)$).
2. For all $X\in{\mathcal{Q}}_0$, $F:{{\mathbb{A}}}_X{\longrightarrow}{{\mathbb{B}}}_X$ preserves arbitrary joins (resp. meets).
[[@Stubbe_2006]]{} \[left\_adjoint\_preserves\_sup\] Let $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ be a ${\mathcal{Q}}$-functor between ${\mathcal{Q}}$-categories, with ${{\mathbb{A}}}$ complete, then $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ is a left (resp. right) adjoint if and only if $F$ preserves supremum (resp. infimum) in the sense that $F(\sup_{{{\mathbb{A}}}}\mu)=\sup_{{{\mathbb{B}}}}F^{{\rightarrow}}(\mu)$ for all $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$ (resp. $F(\inf_{{{\mathbb{A}}}}\mu)=\inf_{{{\mathbb{B}}}}F^{{\rightarrow}}(\mu)$ for all $\mu\in{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}$).
Thus, left (resp. right) adjoint ${\mathcal{Q}}$-functors between complete ${\mathcal{Q}}$-categories are exactly suprema-preserving (resp. infima-preserving) ${\mathcal{Q}}$-functors. Complete ${\mathcal{Q}}$-categories and left adjoint ${\mathcal{Q}}$-functors constitute a subcategory of ${\mathcal{Q}}$-[**Cat**]{} which will be denoted by ${\mathcal{Q}}$-[**CCat**]{}.
The forgetful functor ${\mathcal{Q}}\text{-}{\bf CCat}{\longrightarrow}{\mathcal{Q}}\text{-}{\bf Cat}$ has a left adjoint ${\mathcal{P}}:{\mathcal{Q}}\text{-}{\bf Cat}{\longrightarrow}{\mathcal{Q}}\text{-}{\bf CCat}$ that sends a ${\mathcal{Q}}$-functor $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ to the left adjoint ${\mathcal{Q}}$-functor $F^{{\rightarrow}}:{{\mathcal{P}}{{\mathbb{A}}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{B}}}}$. This implies that ${\mathcal{P}}{{\mathbb{A}}}$ is the free cocompletion of ${{\mathbb{A}}}$ [@Stubbe_2005].
Now, we introduce the crucial notion in this paper, that of infomorphisms between ${\mathcal{Q}}$-distributors. An infomorphism between ${\mathcal{Q}}$-distributors is what a Chu transform between Chu spaces [@Barr1991; @Pratt1995]. The terminology “infomorphism” is from [@Barwise1997; @Ganter2007].
Given ${\mathcal{Q}}$-distributors $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$ and $\psi:{{\mathbb{A}}}'{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}'$, an infomorphism $(F,G):\phi{\longrightarrow}\psi$ is a pair of ${\mathcal{Q}}$-functors $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{A}}}'$ and $G:{{\mathbb{B}}}'{\longrightarrow}{{\mathbb{B}}}$ such that $G^\natural \circ\phi=\psi\circ F_\natural$, or equivalently, $\phi(-,G-)=\psi(F-,-)$. $$\bfig
\square[{{\mathbb{A}}}`{{\mathbb{B}}}`{{\mathbb{A}}}'`{{\mathbb{B}}}';\phi`F_\natural `G^\natural`\psi]
\place(250,0)[\circ]
\place(250,500)[\circ]
\place(0,250)[\circ]
\place(500,250)[\circ]
\efig$$
An adjunction $F{\dashv}G:{{\mathbb{A}}}{\rightharpoonup}{{\mathbb{B}}}$ in ${\mathcal{Q}}$-[**Cat**]{} is exactly an infomorphism from the identity ${\mathcal{Q}}$-distributor on ${{\mathbb{A}}}$ to the identity ${\mathcal{Q}}$-distributor on ${{\mathbb{B}}}$. Thus, infomorphisms are an extension of adjoint ${\mathcal{Q}}$-functors.
${\mathcal{Q}}$-distributors and infomorphisms constitute a category ${\mathcal{Q}}$-[**Info**]{}. The primary aim of this paper is to show that the constructions of Isbell adjunctions and Kan adjunctions are functors defined on ${\mathcal{Q}}$-[**Info**]{}.
\[Y\_functor\_Cat\_Info\] Let $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ be a ${\mathcal{Q}}$-functor, then $$(F,F^{{\leftarrow}}):(({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathcal{P}}{{\mathbb{A}}}}){\longrightarrow}(({{\sf Y}}_{{{\mathbb{B}}}})_{\natural}:{{\mathbb{B}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathcal{P}}{{\mathbb{B}}}})$$ is an infomorphism.
For all $x\in{{\mathbb{A}}}_0$ and ${\lambda}\in{{\mathcal{P}}{{\mathbb{B}}}}$, $$\begin{aligned}
({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}(x,F^{{\leftarrow}}({\lambda}))&= {{\mathcal{P}}{{\mathbb{A}}}}({{\sf Y}}_{{\mathbb{A}}}(x),F^{{\leftarrow}}({\lambda}))\\
&=F^{{\leftarrow}}({\lambda})(x)&\text{(by Yoneda lemma)}\\
&={\lambda}(Fx)&\text{(by Equation (\ref{F_la_lam_Fx}))}\\
&= {{\mathcal{P}}{{\mathbb{B}}}}({{\sf Y}}_{{\mathbb{B}}}(Fx),{\lambda})&\text{(by Yoneda lemma)}\\
&=({{\sf Y}}_{{{\mathbb{B}}}})_{\natural}(Fx,{\lambda}).\end{aligned}$$ Hence the conclusion holds.
The above proposition gives rise to a fully faithful functor ${{\bf Y}}:{\mathcal{Q}}\text{-}{\bf Cat}{\longrightarrow}{\mathcal{Q}}\text{-}{\bf Info}$ that sends each ${\mathcal{Q}}$-category ${{\mathbb{A}}}$ to the graph $({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}$ of the Yoneda embedding.
\[Y\_U\_adjunction\] ${{\bf Y}}:{\mathcal{Q}}\text{-}{\bf Cat}{\longrightarrow}{\mathcal{Q}}\text{-}{\bf Info}$ is a left adjoint of the forgetful functor ${{\bf U}}:{\mathcal{Q}}\text{-}{\bf Info}{\longrightarrow}{\mathcal{Q}}\text{-}{\bf Cat}$ that sends an infomorphism $$(F,G):(\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}){\longrightarrow}(\psi:{{\mathbb{A}}}'{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}')$$ to the ${\mathcal{Q}}$-functor $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{A}}}'$.
It is clear that ${{\bf U}}\circ{{\bf Y}}={\bf id}_{{\mathcal{Q}}\text{-}{\bf Cat}}$, the identity functor on ${\mathcal{Q}}$-[**Cat**]{}. Thus $\{1_{{{\mathbb{A}}}}\}$ is a natural transformation from ${\bf id}_{{\mathcal{Q}}\text{-}{\bf Cat}}$ to ${{\bf U}}\circ{{\bf Y}}$. It remains to show that for each ${\mathcal{Q}}$-category ${{\mathbb{A}}}$, ${\mathcal{Q}}$-distributor $\psi:{{\mathbb{A}}}'{\longrightarrow}{{\mathbb{B}}}'$ and ${\mathcal{Q}}$-functor $H:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{A}}}'$, there is a unique infomorphism $$(F,G):{{\bf Y}}({{\mathbb{A}}}){\longrightarrow}(\psi:{{\mathbb{A}}}'{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}')$$ such that the diagram $$\bfig
\qtriangle<700,500>[{{\mathbb{A}}}`{{\bf U}}\circ{{\bf Y}}({{\mathbb{A}}})`{{\mathbb{A}}}';1_{{{\mathbb{A}}}}`H`{{\bf U}}(F,G)]
\efig$$ is commutative. By definition, ${{\bf Y}}({{\mathbb{A}}})$ is the graph $({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathcal{P}}{{\mathbb{A}}}}$ and ${{\bf U}}(F,G)=F$. Thus, we only need to show that there is a unique ${\mathcal{Q}}$-functor $G:{{\mathbb{B}}}'{\longrightarrow}{{\mathcal{P}}{{\mathbb{A}}}}$ such that $$(H,G):(({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathcal{P}}{{\mathbb{A}}}}){\longrightarrow}(\psi:{{\mathbb{A}}}'{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}')$$ is an infomorphism.
Let $G=H^{{\leftarrow}}\circ{\overline{\psi}}:{{\mathbb{B}}}'{\longrightarrow}{{\mathcal{P}}{{\mathbb{A}}}}$, where ${\overline{\psi}}:{{\mathbb{B}}}'{\longrightarrow}{{\mathcal{P}}{{\mathbb{A}}}}'$ is the ${\mathcal{Q}}$-functor assigning each $y'\in{{\mathbb{B}}}'_0$ to $\psi(-,y')$ in ${{\mathcal{P}}{{\mathbb{A}}}}$. Then $$(H,G):(({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathcal{P}}{{\mathbb{A}}}}) {\longrightarrow}(\psi:{{\mathbb{A}}}'{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}')$$ is an infomorphism since $$({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}(x,Gy')=(Gy')(x)=H^{{\leftarrow}}\circ{\overline{\psi}}(y')(x)={\overline{\psi}}(y')(Hx)=\psi(Hx,y')$$ for all $x\in{{\mathbb{A}}}_0$ and $y'\in{{\mathbb{B}}}'_0$. This proves the existence of $G$.
To see the uniqueness of $G$, suppose that $G':{{\mathbb{B}}}'{\longrightarrow}{{\mathcal{P}}{{\mathbb{A}}}}$ is another ${\mathcal{Q}}$-functor such that $$(H,G'):(({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathcal{P}}{{\mathbb{A}}}}){\longrightarrow}(\psi:{{\mathbb{A}}}'{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}')$$ is an infomorphism. Then for all $x\in{{\mathbb{A}}}_0$ and $y'\in{{\mathbb{B}}}'_0$, $$(G'y')(x)=({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}(x,G'y')=\psi(Hx,y')={\overline{\psi}}(y')(Hx)=H^{{\leftarrow}}\circ{\overline{\psi}}(y')(x)=(Gy')(x),$$ hence $G'=G$.
Similar to Proposition \[Y\_functor\_Cat\_Info\], one can check that sending a ${\mathcal{Q}}$-functor $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ to the infomorphism $$(F^{{\leftarrow}},F):(({{{\sf Y}}^{\dag}}_{{{\mathbb{B}}}})^{\natural}:{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}){\longrightarrow}(({{{\sf Y}}^{\dag}}_{{{\mathbb{A}}}})^{\natural}:{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{A}}})$$ induces a fully faithful functor ${{\bf Y}}^{\dag}:{\mathcal{Q}}\text{-}{\bf Cat}{\longrightarrow}({\mathcal{Q}}\text{-}{\bf Info})^{\rm op}$.
\[Y\_U\_adjunction\_contravariant\] ${{\bf Y}}^{\dag}:{\mathcal{Q}}\text{-}{\bf Cat}{\longrightarrow}({\mathcal{Q}}\text{-}{\bf Info})^{\rm op}$ is a left adjoint of the contravariant forgetful functor $({\mathcal{Q}}\text{-}{\bf Info})^{\rm op}{\longrightarrow}{\mathcal{Q}}\text{-}{\bf Cat}$ that sends each infomorphism $$(F,G):(\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}){\longrightarrow}(\psi:{{\mathbb{A}}}'{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}')$$ to the ${\mathcal{Q}}$-functor $G:{{\mathbb{B}}}'{\longrightarrow}{{\mathbb{B}}}$.
Similar to Proposition \[Y\_U\_adjunction\].
${\mathcal{Q}}$-closure spaces {#closure_space}
==============================
Let ${{\mathbb{A}}}$ be a ${\mathcal{Q}}$-category.
- An isomorphism-closed ${\mathcal{Q}}$-subcategory ${{\mathbb{B}}}$ of ${{\mathbb{A}}}$ is a [*${\mathcal{Q}}$-closure system*]{} (resp. [*${\mathcal{Q}}$-interior system*]{}) of ${{\mathbb{A}}}$ if the inclusion ${\mathcal{Q}}$-functor $I:{{\mathbb{B}}}{\longrightarrow}{{\mathbb{A}}}$ is a right (resp. left) adjoint.
- A ${\mathcal{Q}}$-functor $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{A}}}$ is a [*${\mathcal{Q}}$-closure operator*]{} (resp. [*${\mathcal{Q}}$-interior operator*]{}) on ${{\mathbb{A}}}$ if $1_{{{\mathbb{A}}}}\leq F$ (resp. $F\leq 1_{{{\mathbb{A}}}}$) and $F^2\cong F$.
\[adjunction\_closure\_interior\] Let $F{\dashv}G:{{\mathbb{A}}}{\rightharpoonup}{{\mathbb{B}}}$ be an adjunction in ${\mathcal{Q}}$-[**Cat**]{}. Then $G\circ F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{A}}}$ is a ${\mathcal{Q}}$-closure operator and $F\circ G:{{\mathbb{B}}}{\longrightarrow}{{\mathbb{B}}}$ is a ${\mathcal{Q}}$-interior operator.
\[closure\_interior\_system\_operator\] Let ${{\mathbb{A}}}$ be a ${\mathcal{Q}}$-category, ${{\mathbb{B}}}$ an isomorphism-closed ${\mathcal{Q}}$-subcategory of ${{\mathbb{A}}}$. The following conditions are equivalent:
- ${{\mathbb{B}}}$ is a ${\mathcal{Q}}$-closure system (resp. ${\mathcal{Q}}$-interior system) of ${{\mathbb{A}}}$.
- There is a ${\mathcal{Q}}$-closure operator (resp. ${\mathcal{Q}}$-interior operator) $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{A}}}$ such that ${{\mathbb{B}}}_0=\{x\in{{\mathbb{A}}}_0:Fx\cong x\}$.
$(1)\Rightarrow(2)$: If the inclusion ${\mathcal{Q}}$-functor $I:{{\mathbb{B}}}{\longrightarrow}{{\mathbb{A}}}$ has a left adjoint $G:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$, let $F=I\circ G$, then $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{A}}}$ is a ${\mathcal{Q}}$-closure operator. Since $Fx=Gx\in{{\mathbb{B}}}_0$ for all $x\in{{\mathbb{A}}}_0$ and ${{\mathbb{B}}}$ is isomorphism-closed, it is clear that $\{x\in{{\mathbb{A}}}_0:Fx\cong x\}\subseteq{{\mathbb{B}}}_0$. Conversely, for all $x\in{{\mathbb{B}}}_0$, $${{\mathbb{B}}}(Fx,x)={{\mathbb{B}}}(Gx,x)={{\mathbb{A}}}(x,Ix)={{\mathbb{A}}}(x,x)\geq 1_{tx},$$ and ${{\mathbb{B}}}(x,Fx)\geq 1_{tx}$ holds trivially, hence $x\cong Fx$, as required.
$(2)\Rightarrow(1)$: We show that the inclusion ${\mathcal{Q}}$-functor $I:{{\mathbb{B}}}{\longrightarrow}{{\mathbb{A}}}$ is a right adjoint. View $F$ as a ${\mathcal{Q}}$-functor from ${{\mathbb{A}}}$ to ${{\mathbb{B}}}$, then $1_{{{\mathbb{A}}}}\leq I\circ F$. Since $F^2\cong F$, it follows that $F\circ I\cong 1_{{{\mathbb{B}}}}$. Thus $F{\dashv}I:{{\mathbb{A}}}{\rightharpoonup}{{\mathbb{B}}}$, as required.
For a ${\mathcal{Q}}$-category ${{\mathbb{A}}}$, a ${\mathcal{Q}}$-closure operator (resp. ${\mathcal{Q}}$-interior operator) $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{A}}}$ is exactly a monad (resp. comonad) [@Lane1998] on ${{\mathbb{A}}}$. The above proposition states that a ${\mathcal{Q}}$-closure system (resp. ${\mathcal{Q}}$-interior system) of ${{\mathbb{A}}}$ is exactly the category of algebras (resp. coalgebras) for a monad (resp. comonad) on ${{\mathbb{A}}}$. The terminology “${\mathcal{Q}}$-closure operator” (resp. “${\mathcal{Q}}$-interior operator”) comes from its similarity to closure (resp. interior) operators in topology.
\[closure\_system\_complete\] Each ${\mathcal{Q}}$-closure system (resp. ${\mathcal{Q}}$-interior system) of a complete ${\mathcal{Q}}$-category is itself a complete ${\mathcal{Q}}$-category.
Let ${{\mathbb{B}}}$ be a ${\mathcal{Q}}$-closure system of a complete ${\mathcal{Q}}$-category ${{\mathbb{A}}}$. By Proposition \[closure\_interior\_system\_operator\], there is a ${\mathcal{Q}}$-closure operator $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{A}}}$ such that ${{\mathbb{B}}}_0=\{x\in{{\mathbb{A}}}_0:Fx\cong x\}$. View $F$ as a ${\mathcal{Q}}$-functor from ${{\mathbb{A}}}$ to ${{\mathbb{B}}}$, then $F$ is essentially surjective and $F{\dashv}I:{{\mathbb{A}}}{\rightharpoonup}{{\mathbb{B}}}$, where $I$ is the inclusion ${\mathcal{Q}}$-functor. For all $\mu\in{{\mathcal{P}}{{\mathbb{B}}}}$, $$\begin{aligned}
F({\sup}_{{{\mathbb{A}}}}I^{{\rightarrow}}(\mu))&={\sup}_{{{\mathbb{B}}}}F^{{\rightarrow}}\circ I^{{\rightarrow}}(\mu)&(\text{by Corollary \ref{left_adjoint_preserves_sup}})\\
&={\sup}_{{{\mathbb{B}}}}\mu\circ I^{\natural}\circ F^{\natural}&(\text{by the definition\ of}\ F^{{\rightarrow}}\ \text{and}\ I^{{\rightarrow}})\\
&={\sup}_{{{\mathbb{B}}}}\mu\circ F_{\natural}\circ F^{\natural}\circ I^{\natural}\circ F^{\natural}&(\text{by Proposition \ref{fully_faithful_graph_cograph}(2)})\\
&={\sup}_{{{\mathbb{B}}}}\mu\circ F_{\natural}\circ (F\circ I\circ F)^{\natural}\\
&={\sup}_{{{\mathbb{B}}}}\mu\circ F_{\natural}\circ F^{\natural}&(\text{since}\ F{\dashv}I:{{\mathbb{A}}}{\rightharpoonup}{{\mathbb{B}}})\\
&={\sup}_{{{\mathbb{B}}}}\mu.&(\text{by Proposition \ref{fully_faithful_graph_cograph}(2)})\end{aligned}$$ Then it follows from Proposition \[complete\_cocomplete\_equivalent\] that $F({{\mathbb{A}}})$ is a complete ${\mathcal{Q}}$-category.
\[closure\_system\_cotensor\_meet\] Let ${{\mathbb{A}}}$ be a complete ${\mathcal{Q}}$-category with tensor $\otimes$ and cotensor $\ {\!\rightarrowtail\!}\ $, ${{\mathbb{B}}}$ an isomorphism-closed ${\mathcal{Q}}$-subcategory of ${{\mathbb{A}}}$. Then ${{\mathbb{B}}}$ is a ${\mathcal{Q}}$-closure system (resp. ${\mathcal{Q}}$-interior system) of ${{\mathbb{A}}}$ if and only if
- for every subset $\{x_i\}\subseteq{{\mathbb{B}}}_0$ of the same type $X$, the meet $\displaystyle{\bigwedge}\limits_i x_i$ (resp. the join $\displaystyle{\bigvee}\limits_i x_i$) in ${{\mathbb{A}}}_X$ belongs to ${{\mathbb{B}}}_0$.
- for each $x\in{{\mathbb{B}}}_0$ and $f\in{{\mathcal{P}}^{\dag}}(tx)$ (resp. $f\in{\mathcal{P}}(tx)$), the cotensor $f{\!\rightarrowtail\!}x$ (resp. the tensor $f\otimes x$) in ${{\mathbb{A}}}$ belongs to ${{\mathbb{B}}}_0$.
Follows immediately from Proposition \[F\_la\_ra\_condition\].
An immediate consequence of Proposition \[closure\_system\_cotensor\_meet\] is that the infimum (resp. supremum) in a ${\mathcal{Q}}$-closure system (resp. ${\mathcal{Q}}$-interior system) ${{\mathbb{B}}}$ of a complete ${\mathcal{Q}}$-category ${{\mathbb{A}}}$ can be calculated as $$\label{closure_system_infimum}
{\inf}_{{{\mathbb{B}}}}{\lambda}={\bigwedge}_{b\in{{\mathbb{B}}}_0}({\lambda}(b){\!\rightarrowtail\!}b),\quad\Big(\text{resp.}\ {\sup}_{{{\mathbb{B}}}}\mu={\bigvee}_{b\in{{\mathbb{B}}}_0}(\mu(b)\otimes b)\Big)$$ for ${\lambda}\in{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}$ (resp. $\mu\in{{\mathcal{P}}{{\mathbb{B}}}}$), where the cotensors and meets (resp. tensors and joins) are calculated in ${{\mathbb{A}}}$.
A [*${\mathcal{Q}}$-closure space*]{} is a pair $({{\mathbb{A}}},C)$ that consists of a ${\mathcal{Q}}$-category ${{\mathbb{A}}}$ and a ${\mathcal{Q}}$-closure operator $C:{{\mathcal{P}}{{\mathbb{A}}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{A}}}}$. A [*continuous ${\mathcal{Q}}$-functor*]{} $F:({{\mathbb{A}}},C){\longrightarrow}({{\mathbb{B}}},D)$ between ${\mathcal{Q}}$-closure spaces is a ${\mathcal{Q}}$-functor $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ such that $F^{{\rightarrow}}\circ C\leq D\circ F^{{\rightarrow}}$. The category of ${\mathcal{Q}}$-closure spaces and continuous ${\mathcal{Q}}$-functors is denoted by ${\mathcal{Q}}$-[**Cls**]{}.
If $C, D$ are viewed as monads on ${\mathcal{P}}{{\mathbb{A}}}, {\mathcal{P}}{{\mathbb{B}}}$ respectively, then a ${\mathcal{Q}}$-functor $F:({{\mathbb{A}}},C){\longrightarrow}({{\mathbb{B}}},D)$ between ${\mathcal{Q}}$-closure spaces is continuous if and only if $F^{\rightarrow}:{\mathcal{P}}{{\mathbb{A}}}{\longrightarrow}{\mathcal{P}}{{\mathbb{B}}}$ is a lax map of monads from $C$ to $D$ in the sense of [@Leinster2004].
Note that for a ${\mathcal{Q}}$-closure space $({{\mathbb{A}}},C)$, the ${\mathcal{Q}}$-closure operator $C$ is idempotent since ${{\mathcal{P}}{{\mathbb{A}}}}$ is skeletal. Let $C({{\mathcal{P}}{{\mathbb{A}}}})$ denote the ${\mathcal{Q}}$-subcategory of ${{\mathcal{P}}{{\mathbb{A}}}}$ consisting of the fixed points of $C$. Since ${{\mathcal{P}}{{\mathbb{A}}}}$ is a complete ${\mathcal{Q}}$-category, $C({{\mathcal{P}}{{\mathbb{A}}}})$ is also a complete ${\mathcal{Q}}$-category. A contravariant presheaf ${{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}*_X$ is said to be [*closed*]{} in the ${\mathcal{Q}}$-closure space $({{\mathbb{A}}},C)$ if it belongs to $C({{\mathcal{P}}{{\mathbb{A}}}})$. The following lemma states that continuous ${\mathcal{Q}}$-functors behave in a manner similar to the continuous maps between topological spaces: the inverse image of a closed contravariant presheaf is closed.
A ${\mathcal{Q}}$-functor $F:({{\mathbb{A}}},C){\longrightarrow}({{\mathbb{B}}},D)$ between ${\mathcal{Q}}$-closure spaces is continuous if and only if $F^{{\leftarrow}}({\lambda})\in C({{\mathcal{P}}{{\mathbb{A}}}})$ whenever ${\lambda}\in D({{\mathcal{P}}{{\mathbb{B}}}})$.
It suffices to show that $F^{{\rightarrow}}\circ C\leq D\circ F^{{\rightarrow}}$ if and only if $C\circ F^{{\leftarrow}}\circ D\leq F^{{\leftarrow}}\circ D$.
Suppose $F^{{\rightarrow}}\circ C\leq D\circ F^{{\rightarrow}}$, then $$F^{{\rightarrow}}\circ C\circ F^{{\leftarrow}}\circ D\leq D\circ F^{{\rightarrow}}\circ F^{{\leftarrow}}\circ D\leq D\circ D=D,$$ and consequently $C\circ F^{{\leftarrow}}\circ D\leq F^{{\leftarrow}}\circ D$.
Conversely, suppose $C\circ F^{{\leftarrow}}\circ D\leq F^{{\leftarrow}}\circ D$, then $$C\leq C\circ F^{{\leftarrow}}\circ F^{{\rightarrow}}\leq C\circ F^{{\leftarrow}}\circ D\circ F^{{\rightarrow}}\leq F^{{\leftarrow}}\circ D\circ F^{{\rightarrow}},$$ and consequently $F^{{\rightarrow}}\circ C\leq D\circ F^{{\rightarrow}}$.
Thus a continuous ${\mathcal{Q}}$-functor $F:({{\mathbb{A}}},C){\longrightarrow}({{\mathbb{B}}},D)$ between ${\mathcal{Q}}$-closure spaces induces a pair of ${\mathcal{Q}}$-functors $$F^{\triangleright}=D\circ F^{{\rightarrow}}:C({{\mathcal{P}}{{\mathbb{A}}}}){\longrightarrow}D({{\mathcal{P}}{{\mathbb{B}}}})\quad\text{and}\quad F^{\triangleleft}=F^{{\leftarrow}}:D({{\mathcal{P}}{{\mathbb{B}}}}){\longrightarrow}C({{\mathcal{P}}{{\mathbb{A}}}}).$$
If $F:({{\mathbb{A}}},C){\longrightarrow}({{\mathbb{B}}},D)$ is a continuous ${\mathcal{Q}}$-functor between ${\mathcal{Q}}$-closure spaces, then $F^{\triangleright}{\dashv}F^{\triangleleft}:C({{\mathcal{P}}{{\mathbb{A}}}}){\rightharpoonup}D({{\mathcal{P}}{{\mathbb{B}}}})$.
It is sufficient to check that $${{\mathcal{P}}{{\mathbb{B}}}}(D\circ F^{{\rightarrow}}(\mu),{\lambda})={{\mathcal{P}}{{\mathbb{B}}}}(F^{{\rightarrow}}(\mu),{\lambda})$$ for all $\mu\in C({{\mathcal{P}}{{\mathbb{A}}}})$ and ${\lambda}\in D({{\mathcal{P}}{{\mathbb{B}}}})$ since it holds that ${{\mathcal{P}}{{\mathbb{A}}}}(\mu,F^{\leftarrow}(\lambda))={{\mathcal{P}}{{\mathbb{B}}}}(F^{\rightarrow}(\mu),\lambda)$. Indeed, since $D$ is a ${\mathcal{Q}}$-closure operator, $$\begin{aligned}
{{\mathcal{P}}{{\mathbb{B}}}}(F^{{\rightarrow}}(\mu),{\lambda})&\leq{{\mathcal{P}}{{\mathbb{B}}}}(D\circ F^{{\rightarrow}}(\mu),D({\lambda}))\\
&={{\mathcal{P}}{{\mathbb{B}}}}(D\circ F^{{\rightarrow}}(\mu),{\lambda})\\
&={\lambda}{\swarrow}(D\circ F^{{\rightarrow}}(\mu))\\
&\leq{\lambda}{\swarrow}F^{{\rightarrow}}(\mu)\\
&={{\mathcal{P}}{{\mathbb{B}}}}(F^{{\rightarrow}}(\mu),{\lambda}),\end{aligned}$$ hence ${{\mathcal{P}}{{\mathbb{B}}}}(D\circ F^{{\rightarrow}}(\mu),{\lambda})={{\mathcal{P}}{{\mathbb{B}}}}(F^{{\rightarrow}}(\mu),{\lambda})$.
Skeletal complete ${\mathcal{Q}}$-categories constitute a full subcategory of ${\mathcal{Q}}$-[**CCat**]{} and we denote it by $({\mathcal{Q}}\text{-}{\bf CCat})_{{{\sf skel}}}$. The above proposition gives rise to a functor $${\mathcal{T}}:{\mathcal{Q}}{\text -}{\bf Cls}{\longrightarrow}({\mathcal{Q}}\text{-}{\bf CCat})_{{{\sf skel}}}$$ that maps a continuous ${\mathcal{Q}}$-functor $F:({{\mathbb{A}}},C){\longrightarrow}({{\mathbb{B}}},D)$ to a left adjoint ${\mathcal{Q}}$-functor $F^{\triangleright}:C({{\mathcal{P}}{{\mathbb{A}}}}){\longrightarrow}D({{\mathcal{P}}{{\mathbb{B}}}})$ between skeletal complete ${\mathcal{Q}}$-categories.
For each complete ${\mathcal{Q}}$-category ${{\mathbb{A}}}$, it follows from Theorem \[complete\_cocomplete\_equivalent\] and Example \[adjunction\_closure\_interior\] that $C_{{{\mathbb{A}}}}={{\sf Y}}\circ\sup:{{\mathcal{P}}{{\mathbb{A}}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{A}}}}$ is a ${\mathcal{Q}}$-closure operator, hence $({{\mathbb{A}}},C_{{{\mathbb{A}}}})$ is a ${\mathcal{Q}}$-closure space.
If $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ is a left adjoint ${\mathcal{Q}}$-functor between complete ${\mathcal{Q}}$-categories, then $F:({{\mathbb{A}}},C_{{{\mathbb{A}}}}){\longrightarrow}({{\mathbb{B}}},C_{{{\mathbb{B}}}})$ is a continuous ${\mathcal{Q}}$-functor.
For all $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$, $$\begin{aligned}
F^{{\rightarrow}}\circ C_{{{\mathbb{A}}}}(\mu)&=C_{{{\mathbb{A}}}}(\mu)\circ F^{\natural}\\
&={{\mathbb{A}}}(-,{\sup}_{{{\mathbb{A}}}}\mu)\circ F^{\natural}\\
&\leq{{\mathbb{B}}}(F-,F({\sup}_{{{\mathbb{A}}}}\mu))\circ F^{\natural}\\
&=F_{\natural}(-,F({\sup}_{{{\mathbb{A}}}}\mu))\circ F^{\natural}\\
&\leq{{\mathbb{B}}}(-,F({\sup}_{{{\mathbb{A}}}}\mu))&(\text{since}\ F_{\natural}{\dashv}F^{\natural}:{{\mathbb{A}}}{\rightharpoonup}{{\mathbb{B}}}\ \text{in}\ {\mathcal{Q}}\text{-}{\bf Dist})\\
&={{\mathbb{B}}}(-,{\sup}_{{{\mathbb{B}}}}F^{{\rightarrow}}(\mu))&(\text{by Corollary \ref{left_adjoint_preserves_sup}})\\
&=C_{{{\mathbb{B}}}}\circ F^{{\rightarrow}}(\mu).\end{aligned}$$ Hence $F:({{\mathbb{A}}},C_{{{\mathbb{A}}}}){\longrightarrow}({{\mathbb{B}}},C_{{{\mathbb{B}}}})$ is continuous.
The above proposition gives a functor ${\mathcal{D}}:({\mathcal{Q}}\text{-}{\bf CCat})_{{{\sf skel}}}{\longrightarrow}{\mathcal{Q}}\text{-}{\bf Cls}$.
For each ${\mathcal{Q}}$-category ${{\mathbb{A}}}$, it is clear that $C_{{{\mathbb{A}}}}({{\mathcal{P}}{{\mathbb{A}}}})=\{{{\sf Y}}_{{{\mathbb{A}}}} a\mid a\in{{\mathbb{A}}}_0\}$. So, for a skeletal ${\mathcal{Q}}$-category ${{\mathbb{A}}}$, if we identify ${{\mathbb{A}}}$ with the ${\mathcal{Q}}$-subcategory $C_{{{\mathbb{A}}}}({{\mathcal{P}}{{\mathbb{A}}}})$ of ${{\mathcal{P}}{{\mathbb{A}}}}$, then the functor ${\mathcal{T}}:{\mathcal{Q}}\text{-}{\bf Cls}{\longrightarrow}({\mathcal{Q}}\text{-}{\bf CCat})_{{{\sf skel}}}$ is a left inverse of ${\mathcal{D}}:({\mathcal{Q}}\text{-}{\bf CCat})_{{{\sf skel}}}{\longrightarrow}{\mathcal{Q}}\text{-}{\bf Cls}$ since ${\mathcal{T}}\circ{\mathcal{D}}({{\mathbb{A}}})=C_{{{\mathbb{A}}}}({{\mathcal{P}}{{\mathbb{A}}}})$.
\[T\_D\_adjunction\] ${\mathcal{T}}:{\mathcal{Q}}\text{-}{\bf Cls}{\longrightarrow}({\mathcal{Q}}\text{-}{\bf CCat})_{{{\sf skel}}}$ is a left inverse and left adjoint of ${\mathcal{D}}:({\mathcal{Q}}\text{-}{\bf CCat})_{{{\sf skel}}} {\longrightarrow}{\mathcal{Q}}\text{-}{\bf Cls}$.
It remains to show that ${\mathcal{T}}$ is a left adjoint of ${\mathcal{D}}$ . Given a ${\mathcal{Q}}$-closure space $({{\mathbb{A}}},C)$, denote $C({{\mathcal{P}}{{\mathbb{A}}}})$ by ${{\mathbb{X}}}$, then ${\mathcal{D}}\circ{\mathcal{T}}({{\mathbb{A}}},C)=({{\mathbb{X}}},C_{{{\mathbb{X}}}})$. Let $\eta_{({{\mathbb{A}}},C)}=C\circ{{\sf Y}}_{{{\mathbb{A}}}}:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{X}}}$. We show that $\eta=\{\eta_{({{\mathbb{A}}},C)}\}$ is a natural transformation from the identity functor to ${\mathcal{D}}\circ{\mathcal{T}}$ and it is the unit of the desired adjunction.
[**Step 1**]{}. $\eta_{({{\mathbb{A}}},C)}:({{\mathbb{A}}},C){\longrightarrow}({{\mathbb{X}}},C_{{{\mathbb{X}}}})$ is a continuous ${\mathcal{Q}}$-functor, i.e. $\eta_{({{\mathbb{A}}},C)}^{{\rightarrow}}\circ C\leq C_{{{\mathbb{X}}}}\circ\eta_{({{\mathbb{A}}},C)}^{{\rightarrow}}$.
Firstly, we show that $C(\mu)=\sup_{{{\mathbb{X}}}}\circ\eta_{({{\mathbb{A}}},C)}^{{\rightarrow}}(\mu)$ for all $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$. Consider the diagram: $$\bfig
\Square[{\mathcal{P}}({{\mathcal{P}}{{\mathbb{A}}}})`{{\mathcal{P}}{{\mathbb{A}}}}`{{\mathcal{P}}{{\mathbb{X}}}}`{{\mathbb{X}}};\sup_{{{\mathcal{P}}{{\mathbb{A}}}}}`C^{\rightarrow}`C`\sup_{{{\mathbb{X}}}}]
\morphism(-600,500)<600,0>[{{\mathcal{P}}{{\mathbb{A}}}}`{\mathcal{P}}({{\mathcal{P}}{{\mathbb{A}}}});{{\sf Y}}_{{{\mathbb{A}}}}^{{\rightarrow}}]
\morphism(-600,500)|l|<600,-500>[{{\mathcal{P}}{{\mathbb{A}}}}`{\mathcal{P}}{{\mathbb{X}}};\eta_{({{\mathbb{A}}},C)}^{\rightarrow}]
\efig$$ The commutativity of the left triangle follows from $\eta_{({{\mathbb{A}}},C)}=C\circ {{\sf Y}}_{{{\mathbb{A}}}}$. Since $C:{{\mathcal{P}}{{\mathbb{A}}}}{\longrightarrow}{{\mathbb{X}}}$ is a left adjoint in ${\mathcal{Q}}$-[**Cat**]{} (obtained in the proof of Proposition \[closure\_interior\_system\_operator\]), it preserves supremum (Corollary \[left\_adjoint\_preserves\_sup\]), thus the right square commutes. The whole diagram is then commutative. For each $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$, we have that $$\label{mu_sup_ymu}
\mu=\mu\circ{{\mathbb{A}}}=\mu\circ{{\sf Y}}_{{{\mathbb{A}}}}^{\natural}\circ({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}={{\sf Y}}_{{{\mathbb{A}}}}^{\rightarrow}(\mu)\circ({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}={\sup}_{{{\mathcal{P}}{{\mathbb{A}}}}}\circ{{\sf Y}}_{{{\mathbb{A}}}}^{\rightarrow}(\mu),$$ where the second equality comes from the fact that the Yoneda embedding ${{\sf Y}}_{{{\mathbb{A}}}}$ is fully faithful and Proposition \[fully\_faithful\_graph\_cograph\](1), while the last equality comes from Example \[PX\_PA\_complete\]. Consequently, $$C(\mu)=C\circ{\sup}_{{{\mathcal{P}}{{\mathbb{A}}}}}\circ{{\sf Y}}_{{{\mathbb{A}}}}^{\rightarrow}(\mu)={\sup}_{{{\mathbb{X}}}}\circ\eta_{({{\mathbb{A}}},C)}^{{\rightarrow}}(\mu)$$ for all $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$.
Secondly, we show that $\eta_{({{\mathbb{A}}},C)}^{{\rightarrow}}(\mu)\leq{{\sf Y}}_{{{\mathbb{X}}}}(\mu)={{\mathbb{X}}}(-,\mu)$ for each $\mu\in{{\mathbb{X}}}$. Indeed, $$\begin{aligned}
\eta_{({{\mathbb{A}}},C)}^{{\rightarrow}}(\mu)&=\mu\circ\eta_{({{\mathbb{A}}},C)}^{\natural}\\
&=\mu\circ(C\circ{{\sf Y}}_{{{\mathbb{A}}}})^{\natural}\\
&={{\mathcal{P}}{{\mathbb{A}}}}({{\sf Y}}_{{\mathbb{A}}}-,\mu)\circ{{\sf Y}}_{{{\mathbb{A}}}}^{\natural}\circ C^{\natural}&(\text{by Yoneda lemma})\\
&=({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}(-,\mu)\circ{{\sf Y}}_{{{\mathbb{A}}}}^{\natural}\circ C^{\natural}& \\
&\leq{{\mathcal{P}}{{\mathbb{A}}}}(-,\mu)\circ C^{\natural}&(\text{since}\ ({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}{\dashv}{{\sf Y}}_{{{\mathbb{A}}}}^{\natural}:{{\mathbb{A}}}{\rightharpoonup}{{\mathcal{P}}{{\mathbb{A}}}}\ \text{in}\ {\mathcal{Q}}\text{-}{\bf Dist})\\
&\leq{{\mathbb{X}}}(C-,\mu)\circ C^{\natural}&(\text{since}\ C\ \text{is a}\ {\mathcal{Q}}\text{-functor and}\ C(\mu)=\mu)\\
&=C_{\natural}(-,\mu)\circ C^{\natural}\\
&\leq{{\mathbb{X}}}(-,\mu).&(\text{since}\ C_{\natural}{\dashv}C^{\natural}:{{\mathcal{P}}{{\mathbb{A}}}}{\rightharpoonup}{{\mathbb{X}}}\ \text{in}\ {\mathcal{Q}}\text{-}{\bf Dist})\end{aligned}$$
Therefore, for all $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$, $$\eta_{({{\mathbb{A}}},C)}^{{\rightarrow}}\circ C(\mu)\leq{{\sf Y}}_{{{\mathbb{X}}}}\circ{\sup}_{{{\mathbb{X}}}}\circ\eta_{({{\mathbb{A}}},C)}^{{\rightarrow}}(\mu)=C_{{{\mathbb{X}}}}\circ\eta_{({{\mathbb{A}}},C)}^{{\rightarrow}}(\mu),$$ as desired.
[**Step 2**]{}. $\eta=\{\eta_{({{\mathbb{A}}},C)}\}$ is a natural transformation. Let $F:({{\mathbb{A}}},C){\longrightarrow}({{\mathbb{B}}},D)$ be a continuous ${\mathcal{Q}}$-functor, we must show that $$D\circ{{\sf Y}}_{{{\mathbb{B}}}}\circ F=\eta_{({{\mathbb{B}}},D)}\circ F={\mathcal{D}}\circ{\mathcal{T}}\circ F\circ\eta_{({{\mathbb{A}}},C)}=D\circ F^{{\rightarrow}}\circ C\circ{{\sf Y}}_{{{\mathbb{A}}}}.$$
Firstly, we show that $$\label{Yoneda_natural}
{{\sf Y}}_{{{\mathbb{B}}}}\circ F=F^{{\rightarrow}}\circ{{\sf Y}}_{{{\mathbb{A}}}}$$ for each ${\mathcal{Q}}$-functor $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$. Indeed, for all $x\in{{\mathbb{A}}}_0$, $$\begin{aligned}
{{\sf Y}}_{{{\mathbb{B}}}}\circ Fx&=F^{\natural}(-,x)&(\text{by the definition of}\ F^{\natural})\\
&=({{\mathbb{A}}}\circ F^{\natural})(-,x)\\
&={{\mathbb{A}}}(-,x)\circ F^{\natural}&(\text{by Remark \ref{distributor_notion}})\\
&=F^{{\rightarrow}}\circ{{\sf Y}}_{{{\mathbb{A}}}}x.&(\text{by the definition of}\ F^{{\rightarrow}})\end{aligned}$$
Secondly, since $C$ is a ${\mathcal{Q}}$-closure operator, $${{\sf Y}}_{{{\mathbb{B}}}}\circ F=F^{{\rightarrow}}\circ{{\sf Y}}_{{{\mathbb{A}}}}\leq F^{{\rightarrow}}\circ C\circ{{\sf Y}}_{{{\mathbb{A}}}},$$ and consequently $D\circ{{\sf Y}}_{{{\mathbb{B}}}}\circ F\leq D\circ F^{{\rightarrow}}\circ C\circ{{\sf Y}}_{{{\mathbb{A}}}}$.
Thirdly, the continuity of $F$ leads to $$F^{{\rightarrow}}\circ C\circ{{\sf Y}}_{{{\mathbb{A}}}}\leq D\circ F^{{\rightarrow}}\circ{{\sf Y}}_{{{\mathbb{A}}}}=D\circ{{\sf Y}}_{{{\mathbb{B}}}}\circ F,$$ hence $D\circ F^{{\rightarrow}}\circ C\circ{{\sf Y}}_{{{\mathbb{A}}}}\leq D\circ{{\sf Y}}_{{{\mathbb{B}}}}\circ F$.
[**Step 3**]{}. $\eta_{({{\mathbb{A}}},C)}:({{\mathbb{A}}},C){\longrightarrow}({{\mathbb{X}}},C_{{{\mathbb{X}}}})$ is universal in the sense that for any skeletal complete ${\mathcal{Q}}$-category ${{\mathbb{B}}}$ and continuous ${\mathcal{Q}}$-functor $F:({{\mathbb{A}}},C){\longrightarrow}({{\mathbb{B}}},C_{{{\mathbb{B}}}})$, there exists a unique left adjoint ${\mathcal{Q}}$-functor $\overline{F}:{{\mathbb{X}}}{\longrightarrow}{{\mathbb{B}}}$ that makes the following diagram commute: $$\label{eta_universal}
\bfig
\qtriangle<600,500>[({{\mathbb{A}}},C)`({{\mathbb{X}}},C_{{{\mathbb{X}}}})`({{\mathbb{B}}},C_{{{\mathbb{B}}}});\eta_{({{\mathbb{A}}},C)}`F`\overline{F}]
\efig$$
[**Existence.**]{} Let $\overline{F}=\sup_{{{\mathbb{B}}}}\circ F^{\rightarrow}:{{\mathbb{X}}}{\longrightarrow}{{\mathbb{B}}}$ be the following composition of ${\mathcal{Q}}$-functors $${{\mathbb{X}}}\hookrightarrow{{\mathcal{P}}{{\mathbb{A}}}}\to^{F^{\rightarrow}}{{\mathcal{P}}{{\mathbb{B}}}}\to^{\sup_{{{\mathbb{B}}}}}{{\mathbb{B}}}.$$
First, $\overline{F}:{{\mathbb{X}}}{\longrightarrow}{{\mathbb{B}}}$ is a left adjoint in ${\mathcal{Q}}$-[**Cat**]{}. Indeed, $\overline{F}$ has a right adjoint $G:{{\mathbb{B}}}{\longrightarrow}{{\mathbb{X}}}$ given by $G=F^{\triangleleft}\circ{{\sf Y}}_{{{\mathbb{B}}}}$. $G$ is well-defined since ${{\sf Y}}_{{{\mathbb{B}}}}b$ is a closed in $({{\mathbb{B}}},C_{{{\mathbb{B}}}})$ for each $b\in{{\mathbb{B}}}_0$. For all $\mu\in{{\mathbb{X}}}_0$ and $y\in{{\mathbb{B}}}_0$, it holds that $$\begin{aligned}
{{\mathbb{B}}}(\overline{F}(\mu),y)&={{\mathbb{B}}}(-,y){\swarrow}F^{{\rightarrow}}(\mu)\\
&={{\mathbb{B}}}(-,y){\swarrow}(\mu\circ F^{\natural})\\
&=({{\mathbb{B}}}(-,y)\circ F_\natural){\swarrow}\mu&(\text{by Proposition \ref{graph_cograph_implication}(2)})\\
&=F_\natural (-,y){\swarrow}\mu&(\text{by Remark \ref{distributor_notion}})\\
&={{\mathcal{P}}{{\mathbb{A}}}}(\mu,F^{\triangleleft}\circ{{\sf Y}}_{{{\mathbb{B}}}}y)&(\text{by the definition of}\ F_{\natural}\ \text{and}\ F^{\triangleleft})\\
&={{\mathbb{X}}}(\mu,Gy),\end{aligned}$$ hence $\overline{F}$ is a left adjoint of $G$.
Second, $F=\overline{F}\circ\eta_{({{\mathbb{A}}},C)}$. Note that for all $x\in{{\mathbb{A}}}_0$, $$\begin{aligned}
{{\mathbb{B}}}(Fx,-)&=F_{\natural}(x,-)\\
&=({{\mathbb{B}}}{\swarrow}F^{\natural})(x,-)& (\text{by Proposition \ref{graph_cograph_implication}(1)})\\
&={{\mathbb{B}}}{\swarrow}F^{\natural}(-,x)\\
&={{\mathbb{B}}}{\swarrow}({{\sf Y}}_{{{\mathbb{B}}}}\circ Fx)\\
&={{\mathbb{B}}}{\swarrow}(F^{{\rightarrow}}\circ{{\sf Y}}_{{{\mathbb{A}}}}x),&(\text{by Equation (\ref{Yoneda_natural})})\end{aligned}$$ thus $F=\sup_{{{\mathbb{B}}}}\circ F^{{\rightarrow}}\circ{{\sf Y}}_{{{\mathbb{A}}}}$. Consequently $$\begin{aligned}
\overline{F}\circ\eta_{({{\mathbb{A}}},C)}&={\sup}_{{{\mathbb{B}}}}\circ F^{{\rightarrow}}\circ C\circ{{\sf Y}}_{{{\mathbb{A}}}}\\
&\leq{\sup}_{{{\mathbb{B}}}}\circ C_{{{\mathbb{B}}}}\circ F^{{\rightarrow}}\circ{{\sf Y}}_{{{\mathbb{A}}}}&(\text{since}\ F\ \text{is continuous})\\
&={\sup}_{{{\mathbb{B}}}}\circ {{\sf Y}}_{{{\mathbb{B}}}}\circ{\sup}_{{{\mathbb{B}}}}\circ F^{{\rightarrow}}\circ{{\sf Y}}_{{{\mathbb{A}}}}\\
&={\sup}_{{{\mathbb{B}}}}\circ F^{{\rightarrow}}\circ{{\sf Y}}_{{{\mathbb{A}}}}&(\text{since}\ {\sup}_{{{\mathbb{B}}}}{\dashv}{{\sf Y}}_{{{\mathbb{B}}}}:{{\mathcal{P}}{{\mathbb{B}}}}{\rightharpoonup}{{\mathbb{B}}})\\
&=F.\end{aligned}$$ Conversely, since $C$ is a ${\mathcal{Q}}$-closure operator, it is clear that $$F={\sup}_{{{\mathbb{B}}}}\circ F^{{\rightarrow}}\circ{{\sf Y}}_{{{\mathbb{A}}}}\leq{\sup}_{{{\mathbb{B}}}}\circ F^{{\rightarrow}}\circ C\circ{{\sf Y}}_{{{\mathbb{A}}}}=\overline{F}\circ\eta_{({{\mathbb{A}}},C)},$$ hence $F\cong\overline{F}\circ\eta_{({{\mathbb{A}}},C)}$, and consequently $F=\overline{F}\circ\eta_{({{\mathbb{A}}},C)}$ since ${{\mathbb{B}}}$ is skeletal.
[**Uniqueness.**]{} Suppose $H:{{\mathbb{X}}}{\longrightarrow}{{\mathbb{B}}}$ is another left adjoint ${\mathcal{Q}}$-functor that makes Diagram (\[eta\_universal\]) commute. For each $\mu\in{{\mathbb{X}}}$, since $C:{{\mathcal{P}}{{\mathbb{A}}}}{\longrightarrow}{{\mathbb{X}}}$ is a left adjoint in ${\mathcal{Q}}$-[**Cat**]{}, we have $$\mu=C(\mu)=C(\mu\circ{{\mathbb{A}}})=C\Big({\bigvee}_{x\in{{\mathbb{A}}}_0}\mu(x)\circ{{\sf Y}}_{{{\mathbb{A}}}}x\Big)={\bigvee}_{x\in{{\mathbb{A}}}_0}\mu(x)\otimes_{{{\mathbb{X}}}}C({{\sf Y}}_{{{\mathbb{A}}}}x),$$ where the last equality follows from Example \[PA\_tensor\] and Proposition \[F\_la\_ra\_condition\]. It follows that $$\begin{aligned}
H(\mu)&=H\Big({\bigvee}_{x\in{{\mathbb{A}}}_0}\mu(x)\otimes_{{{\mathbb{X}}}}C({{\sf Y}}_{{{\mathbb{A}}}}x)\Big) \\
&={\bigvee}_{x\in{{\mathbb{A}}}_0}\mu(x)\otimes_{{{\mathbb{B}}}}(H\circ\eta_{({{\mathbb{A}}},C)}(x))&\text{(by Proposition \ref{F_la_ra_condition})}\\
&={\bigvee}_{x\in{{\mathbb{A}}}_0}\mu(x)\otimes_{{{\mathbb{B}}}} Fx.\end{aligned}$$ Consequently, $$\begin{aligned}
{{\mathbb{B}}}(H(\mu),-)&={{\mathbb{B}}}\Big({\bigvee}_{x\in{{\mathbb{A}}}_0}\mu(x)\otimes_{{{\mathbb{B}}}} Fx,-\Big)\\
&={\bigwedge}_{x\in{{\mathbb{A}}}_0}\Big({{\mathbb{B}}}(Fx,-){\swarrow}\mu(x)\Big)\\
&=F_\natural {\swarrow}\mu\\
&={{\mathbb{B}}}{\swarrow}(\mu\circ F^{\natural})&(\text{by Proposition \ref{graph_cograph_implication}(2)})\\
&={{\mathbb{B}}}{\swarrow}F^{{\rightarrow}}(\mu).\end{aligned}$$ Since ${{\mathbb{B}}}$ is skeletal, it follows that $H(\mu)=\sup_{{{\mathbb{B}}}}\circ F^{{\rightarrow}}(\mu)$. Therefore, $H=\sup_{{{\mathbb{B}}}}\circ F^{{\rightarrow}}=\overline{F}$.
Isbell adjunctions {#Isbell_adjunction}
==================
Given a ${\mathcal{Q}}$-distributor $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$, define a pair of ${\mathcal{Q}}$-functors $${\phi_{{\uparrow}}}:{{\mathcal{P}}{{\mathbb{A}}}}{\longrightarrow}{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}\quad\text{and}\quad{\phi^{{\downarrow}}}:{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{A}}}}$$ by $${\phi_{{\uparrow}}}(\mu)=\phi{\swarrow}\mu\quad\text{and}\quad{\phi^{{\downarrow}}}({\lambda})={\lambda}{\searrow}\phi.$$ It should be warned that ${\phi_{{\uparrow}}}$ and ${\phi^{{\downarrow}}}$ are both contravariant with respect to local orders in ${\mathcal{Q}}$-[**Dist**]{} by Remark \[PdA\_QDist\_order\], i.e., $$\label{uphi_contravariant}
\forall\mu_1,\mu_2\in{{\mathcal{P}}{{\mathbb{A}}}},\mu_1\leq\mu_2\Longrightarrow{\phi_{{\uparrow}}}(\mu_2)\leq{\phi_{{\uparrow}}}(\mu_1)$$ and $$\label{dphi_contravariant}
\forall{\lambda}_1,{\lambda}_2\in{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}},{\lambda}_1\leq{\lambda}_2\Longrightarrow{\phi^{{\downarrow}}}({\lambda}_2)\leq{\phi^{{\downarrow}}}({\lambda}_1).$$
\[uphi-dphi-adjunction\] ${\phi_{{\uparrow}}}{\dashv}{\phi^{{\downarrow}}}:{{\mathcal{P}}{{\mathbb{A}}}}{\rightharpoonup}{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}$ in ${\mathcal{Q}}$-[**Cat**]{}.
For all $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$ and ${\lambda}\in{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}$, $$\begin{aligned}
{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}({\phi_{{\uparrow}}}(\mu),{\lambda})&={\lambda}{\searrow}{\phi_{{\uparrow}}}(\mu)\\
&={\lambda}{\searrow}(\phi{\swarrow}\mu)\\
&=({\lambda}{\searrow}\phi){\swarrow}\mu\\
&={\phi^{{\downarrow}}}({\lambda}){\swarrow}\mu\\
&={{\mathcal{P}}{{\mathbb{A}}}}(\mu,{\phi^{{\downarrow}}}({\lambda})).\end{aligned}$$ Hence the conclusion holds.
Letting ${{\mathbb{B}}}={{\mathbb{A}}}$ and $\phi={{\mathbb{A}}}$ in Proposition \[uphi-dphi-adjunction\] gives the following
[[@Stubbe_2005]]{} \[ub\_lb\_adjunction\] ${{\mathbb{A}}}\swarrow(-){\dashv}(-)\searrow{{\mathbb{A}}}:{{\mathcal{P}}{{\mathbb{A}}}}{\rightharpoonup}{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}$.
The adjunction in Corollary \[ub\_lb\_adjunction\] is known as the Isbell adjunction in category theory. So, the adjunction ${\phi_{{\uparrow}}}{\dashv}{\phi^{{\downarrow}}}:{{\mathcal{P}}{{\mathbb{A}}}}{\rightharpoonup}{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}$ is a generalization of the Isbell adjunction. As we shall see, all adjunctions between ${{\mathcal{P}}{{\mathbb{A}}}}$ and ${{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}$ are of this form, and will be called Isbell adjunctions by abuse of language.
Each ${\mathcal{Q}}$-functor $F:{{\mathbb{A}}}{\longrightarrow}{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}$ corresponds to a ${\mathcal{Q}}$-distributor ${{\ulcorner}F {\urcorner}}:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$ given by ${{\ulcorner}F {\urcorner}}(x,y)=F(x)(y)$ for all $x\in{{\mathbb{A}}}_0$ and $y\in{{\mathbb{B}}}_0$, and each ${\mathcal{Q}}$-functor $G:{{\mathbb{B}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{A}}}}$ corresponds to a ${\mathcal{Q}}$-distributor ${{\ulcorner}G {\urcorner}}:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$ given by ${{\ulcorner}G {\urcorner}}(x,y)=G(y)(x)$ for all $x\in{{\mathbb{A}}}_0$ and $y\in{{\mathbb{B}}}_0$.
\[uphi\_dphi\_Yoneda\] Let $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$ be a ${\mathcal{Q}}$-distributor, then ${\ulcorner}{\phi_{{\uparrow}}}\circ{{\sf Y}}_{{{\mathbb{A}}}}{\urcorner}=\phi={\ulcorner}{\phi^{{\downarrow}}}\circ{{{\sf Y}}^{\dag}}_{{{\mathbb{B}}}}{\urcorner}$.
For all $x\in{{\mathbb{A}}}_0$ and $y\in{{\mathbb{B}}}_0$, $$\begin{aligned}
{\ulcorner}{\phi_{{\uparrow}}}\circ{{\sf Y}}_{{{\mathbb{A}}}}{\urcorner}(x,y)&=({\phi_{{\uparrow}}}\circ{{\sf Y}}_{{{\mathbb{A}}}}x)(y)\\
&=(\phi{\swarrow}({{\sf Y}}_{{{\mathbb{A}}}}x))(y)\\
&=\phi(-,y){\swarrow}{{\mathbb{A}}}(-,x)\\
&=\phi(x,y)\\
&={{\mathbb{B}}}(y,-){\searrow}\phi(x,-)\\
&=(({{{\sf Y}}^{\dag}}_{{{\mathbb{B}}}}y){\searrow}\phi)(x)\\
&=({\phi^{{\downarrow}}}\circ{{{\sf Y}}^{\dag}}_{{{\mathbb{B}}}}y)(x)\\
&={\ulcorner}{\phi^{{\downarrow}}}\circ{{{\sf Y}}^{\dag}}_{{{\mathbb{B}}}}{\urcorner}(x,y),\end{aligned}$$ showing that the conclusion holds.
\[Isbell\_distributor\_bijection\] The correspondence $\phi\mapsto{\phi_{{\uparrow}}}$ is an isomorphism of posets $${\mathcal{Q}}\text{-}{\bf Dist}({{\mathbb{A}}},{{\mathbb{B}}})\cong{\mathcal{Q}}\text{-}{\bf CCat}^{\rm co}({{\mathcal{P}}{{\mathbb{A}}}},{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}),$$ where the “${\rm co}$” means reversing order in the hom-sets.
Let $F:{{\mathcal{P}}{{\mathbb{A}}}}{\longrightarrow}{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}$ be a left adjoint ${\mathcal{Q}}$-functor. We show that the correspondence $F\mapsto{\ulcorner}F\circ{{\sf Y}}_{{{\mathbb{A}}}}{\urcorner}$ is an inverse of the correspondence $\phi\mapsto{\phi_{{\uparrow}}}$, and thus they are both isomorphisms of posets between ${\mathcal{Q}}\text{-}{\bf Dist}({{\mathbb{A}}},{{\mathbb{B}}})$ and ${\mathcal{Q}}\text{-}{\bf CCat}^{\rm co}({{\mathcal{P}}{{\mathbb{A}}}},{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}})$.
Firstly, we show that both of the correspondences are order-preserving. Indeed, $$\begin{aligned}
&\phi\leq\psi\ \text{in}\ {\mathcal{Q}}\text{-}{\bf Dist}({{\mathbb{A}}},{{\mathbb{B}}})\\
\iff&\forall\mu\in{{\mathcal{P}}{{\mathbb{A}}}},{\phi_{{\uparrow}}}(\mu)=\phi{\swarrow}\mu\leq\psi{\swarrow}\mu={\psi_{{\uparrow}}}(\mu)\ \text{in}\ {\mathcal{Q}}\text{-}{\bf Dist}\\
\iff&\forall\mu\in{{\mathcal{P}}{{\mathbb{A}}}},{\phi_{{\uparrow}}}(\mu)\geq{\psi_{{\uparrow}}}(\mu)\ \text{in}\ ({{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}})_0\\
\iff&{\phi_{{\uparrow}}}\leq{\psi_{{\uparrow}}}\ \text{in}\ {\mathcal{Q}}\text{-}{\bf CCat}^{\rm co}({{\mathcal{P}}{{\mathbb{A}}}},{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}})\end{aligned}$$ and $$\begin{aligned}
&F\leq G\ \text{in}\ {\mathcal{Q}}\text{-}{\bf CCat}^{\rm co}({{\mathcal{P}}{{\mathbb{A}}}},{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}})\\
\iff&\forall\mu\in{{\mathcal{P}}{{\mathbb{A}}}},F(\mu)\geq G(\mu)\ \text{in}\ ({{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}})_0\\
\iff&\forall\mu\in{{\mathcal{P}}{{\mathbb{A}}}},F(\mu)\leq G(\mu)\ \text{in}\ {\mathcal{Q}}\text{-}{\bf Dist}({{\mathbb{A}}},{{\mathbb{B}}})\\
\Longrightarrow{}&\forall x\in{{\mathbb{A}}}_0,{\ulcorner}F\circ{{\sf Y}}_{{{\mathbb{A}}}}{\urcorner}(x,-)=F\circ{{\sf Y}}_{{{\mathbb{A}}}}x\leq G\circ{{\sf Y}}_{{{\mathbb{A}}}}x={\ulcorner}G\circ{{\sf Y}}_{{{\mathbb{A}}}}{\urcorner}(x,-)\ \text{in}\ {\mathcal{Q}}\text{-}{\bf Dist}({{\mathbb{A}}},{{\mathbb{B}}})\\
\iff&{\ulcorner}F\circ{{\sf Y}}_{{{\mathbb{A}}}}{\urcorner}\leq{\ulcorner}G\circ{{\sf Y}}_{{{\mathbb{A}}}}{\urcorner}\ \text{in}\ {\mathcal{Q}}\text{-}{\bf Dist}({{\mathbb{A}}},{{\mathbb{B}}}).\end{aligned}$$
Secondly, $F=({\ulcorner}F\circ{{\sf Y}}_{{{\mathbb{A}}}}{\urcorner})_{{\uparrow}}$. For all $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$, since $F$ is a left adjoint in ${\mathcal{Q}}$-[**Cat**]{}, by Example \[PA\_tensor\] and Proposition \[F\_la\_ra\_condition\] we have $$\begin{aligned}
F(\mu)&=F(\mu\circ{{\mathbb{A}}})\\
&=F\Big({\bigvee}_{x\in{{\mathbb{A}}}_0}\mu(x)\circ{{\sf Y}}_{{{\mathbb{A}}}}x\Big)\\
&={\bigwedge}_{x\in{{\mathbb{A}}}_0}(F\circ{{\sf Y}}_{{{\mathbb{A}}}}x){\swarrow}\mu(x)\\
&={\ulcorner}F\circ{{\sf Y}}_{{{\mathbb{A}}}}{\urcorner}{\swarrow}\mu\\
&=({\ulcorner}F\circ{{\sf Y}}_{{{\mathbb{A}}}}{\urcorner})_{{\uparrow}}(\mu).\end{aligned}$$
Finally, $\phi={\ulcorner}{\phi_{{\uparrow}}}\circ{{\sf Y}}_{{{\mathbb{A}}}}{\urcorner}$. This is obtained in Proposition \[uphi\_dphi\_Yoneda\].
For a ${\mathcal{Q}}$-distributor $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$, one obtains two ${\mathcal{Q}}$-functors ${\underline{\phi}}:{{\mathbb{A}}}{\longrightarrow}{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}$ and ${\overline{\phi}}:{{\mathbb{B}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{A}}}}$ by letting ${\underline{\phi}}x=\phi(x,-)$ for all $x\in{{\mathbb{A}}}_0$ and ${\overline{\phi}}y=\phi(-,y)$ for all $y\in{{\mathbb{B}}}_0$. Stubbe [@Stubbe_2005] shows that the maps $\phi\mapsto{\underline{\phi}}$ and $F\mapsto{{\ulcorner}F {\urcorner}}$ establish an isomorphism of posets between ${\mathcal{Q}}\text{-}{\bf Dist}({{\mathbb{A}}},{{\mathbb{B}}})$ and ${\mathcal{Q}}\text{-}{\bf Cat}^{\rm co}({{\mathbb{A}}},{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}})$, while the maps $\phi\mapsto{\overline{\phi}}$ and $F\mapsto{{\ulcorner}F {\urcorner}}$ establish an isomorphism of posets between ${\mathcal{Q}}\text{-}{\bf Dist}({{\mathbb{A}}},{{\mathbb{B}}})$ and ${\mathcal{Q}}\text{-}{\bf Cat}({{\mathbb{B}}},{{\mathcal{P}}{{\mathbb{A}}}})$. Together with Theorem \[Isbell\_distributor\_bijection\], we have the following isomorphisms of posets $$\label{Isbell_isomorphism}
{\mathcal{Q}}\text{-}{\bf Dist}({{\mathbb{A}}},{{\mathbb{B}}})\cong{\mathcal{Q}}\text{-}{\bf Cat}^{\rm co}({{\mathbb{A}}},{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}})\cong{\mathcal{Q}}\text{-}{\bf Cat}({{\mathbb{B}}},{{\mathcal{P}}{{\mathbb{A}}}})\cong{\mathcal{Q}}\text{-}{\bf CCat}^{\rm co}({{\mathcal{P}}{{\mathbb{A}}}},{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}).$$
Given a ${\mathcal{Q}}$-distributor $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$, it follows from Example \[adjunction\_closure\_interior\] that ${\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}:{{\mathcal{P}}{{\mathbb{A}}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{A}}}}$ is a ${\mathcal{Q}}$-closure operator and ${\phi_{{\uparrow}}}\circ{\phi^{{\downarrow}}}:{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}{\longrightarrow}{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}$ is a ${\mathcal{Q}}$-interior operator. For each $y\in{{\mathbb{B}}}_0$, since $$\label{phiF_fixed}
{\overline{\phi}}y=\phi(-,y)={\phi^{{\downarrow}}}\circ{{{\sf Y}}^{\dag}}_{{{\mathbb{B}}}}y= {\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}\circ{\phi^{{\downarrow}}}\circ{{{\sf Y}}^{\dag}}_{{{\mathbb{B}}}}y,$$ it follows that ${\overline{\phi}}y=\phi(-,y)$ is closed in the ${\mathcal{Q}}$-closure space $({{\mathbb{A}}}, {\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}})$. Dually, for all $x\in{{\mathbb{A}}}_0$, $$\label{Fphi_fixed}
{\underline{\phi}}x=\phi(x,-)={\phi_{{\uparrow}}}\circ{{\sf Y}}_{{{\mathbb{A}}}}x={\phi_{{\uparrow}}}\circ{\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}\circ{{\sf Y}}_{{{\mathbb{A}}}}x$$ is a fixed point of the ${\mathcal{Q}}$-interior operator ${\phi_{{\uparrow}}}\circ{\phi^{{\downarrow}}}$. These facts will be used in the proofs of Theorem \[F\_U\_adjunction\] and Theorem \[complte\_category\_sup\_dense\].
\[F\_G\_info\_F\_uphi\_dphi\_continuous\] Let $(F,G):\phi{\longrightarrow}\psi$ be an infomorphism between ${\mathcal{Q}}$-distributors $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$ and $\psi:{{\mathbb{A}}}'{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}'$. Then $F:({{\mathbb{A}}},{\phi_{{\uparrow}}}\circ{\phi^{{\downarrow}}}){\longrightarrow}({{\mathbb{A}}}',{\psi_{{\uparrow}}}\circ{\psi^{{\downarrow}}})$ is a continuous ${\mathcal{Q}}$-functor.
Consider the following diagram: $$\bfig
\square|alrb|[{{\mathcal{P}}{{\mathbb{A}}}}`{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}`{{\mathcal{P}}{{\mathbb{A}}}}'`{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}';{\phi_{{\uparrow}}}`F^{{\rightarrow}}`G^{{\leftarrow}}`{\psi_{{\uparrow}}}]
\square(500,0)/>``>`>/[{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}`{{\mathcal{P}}{{\mathbb{A}}}}`{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}'`{{\mathcal{P}}{{\mathbb{A}}}}';{\phi^{{\downarrow}}}``F^{{\rightarrow}}`{\psi^{{\downarrow}}}]
\efig$$ We must prove $F^{{\rightarrow}}\circ{\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}\leq{\psi^{{\downarrow}}}\circ{\psi_{{\uparrow}}}\circ F^{{\rightarrow}}$. To this end, it suffices to check that
\(a) the left square commutes if and only if $(F,G):\phi{\longrightarrow}\psi$ is an infomorphism; and
\(b) $F^{{\rightarrow}}\circ{\phi^{{\downarrow}}}\leq{\psi^{{\downarrow}}}\circ G^{{\leftarrow}}$ if and only if $G^{\natural}\circ\phi\leq\psi\circ F_{\natural}$.
For (a), suppose $G^{{\leftarrow}}\circ{\phi_{{\uparrow}}}={\psi_{{\uparrow}}}\circ F^{{\rightarrow}}$, then for all $x\in{{\mathbb{A}}}_0$, $$\begin{aligned}
G^{\natural}\circ\phi(x,-)&=G^{{\leftarrow}}(\phi(x,-))&(\text{by the definition of}\ G^{{\leftarrow}})\\
&=G^{{\leftarrow}}({\phi_{{\uparrow}}}\circ{{\sf Y}}_{{{\mathbb{A}}}}x)&\text{(by Proposition \ref{uphi_dphi_Yoneda})}\\
&={\psi_{{\uparrow}}}(F^{{\rightarrow}}\circ{{\sf Y}}_{{{\mathbb{A}}}}x)\\
&={\psi_{{\uparrow}}}({{\sf Y}}_{{{\mathbb{A}}}}x\circ F^{\natural})&(\text{by the definition of}\ F^{{\rightarrow}})\\
&=\psi{\swarrow}({{\sf Y}}_{{{\mathbb{A}}}}x\circ F^{\natural})&(\text{by the definition of}\ {\psi_{{\uparrow}}})\\
&=(\psi\circ F_{\natural}){\swarrow}{{\mathbb{A}}}(-,x)&(\text{by Proposition \ref{graph_cograph_implication}(2)})\\
&=\psi\circ F_{\natural}(x,-).\end{aligned}$$
Conversely, if $(F,G):\phi{\longrightarrow}\psi$ is an infomorphism, then for all $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$, $$\begin{aligned}
G^{{\leftarrow}}\circ{\phi_{{\uparrow}}}(\mu)&=G^{\natural}\circ(\phi{\swarrow}\mu)&(\text{by the definition of}\ G^{{\leftarrow}}\ \text{and}\ {\phi_{{\uparrow}}})\\
&=(G^{\natural}\circ\phi){\swarrow}\mu&(\text{by Proposition \ref{graph_cograph_implication}(3)})\\
&=(\psi\circ F_{\natural}){\swarrow}\mu\\
&=\psi{\swarrow}(\mu\circ F^{\natural})&(\text{by Proposition \ref{graph_cograph_implication}(2)})\\
&={\psi_{{\uparrow}}}\circ F^{{\rightarrow}}(\mu).\end{aligned}$$
For (b), suppose $F^{{\rightarrow}}\circ{\phi^{{\downarrow}}}\leq{\psi^{{\downarrow}}}\circ G^{{\leftarrow}}$, then for all $y'\in{{\mathbb{B}}}'_0$, $$\begin{aligned}
G^{\natural}(-,y')\circ\phi&=G_{\natural}(y',-){\searrow}\phi&(\text{by Proposition \ref{graph_cograph_implication}(1)})\\
&={\phi^{{\downarrow}}}(G_{\natural}(y',-))&(\text{by the definition of}\ {\phi^{{\downarrow}}})\\
&\leq F^{{\leftarrow}}\circ F^{{\rightarrow}}\circ{\phi^{{\downarrow}}}(G_{\natural}(y',-))&(\text{since}\ F^{{\rightarrow}}{\dashv}F^{{\leftarrow}}:{{\mathcal{P}}{{\mathbb{A}}}}{\rightharpoonup}{{\mathcal{P}}{{\mathbb{A}}}}')\\
&\leq F^{{\leftarrow}}\circ{\psi^{{\downarrow}}}\circ G^{{\leftarrow}}(G_{\natural}(y',-))\\
&=F^{{\leftarrow}}\circ{\psi^{{\downarrow}}}\circ G^{{\leftarrow}}\circ G^{{\rightarrow}}\circ{{{\sf Y}}^{\dag}}_{{{\mathbb{B}}}'}y'&(\text{by the definition of}\ G^{{\rightarrow}})\\
&\leq F^{{\leftarrow}}\circ{\psi^{{\downarrow}}}\circ{{{\sf Y}}^{\dag}}_{{{\mathbb{B}}}'}y'&\text{(by Inequality (\ref{FLa_FRa_adjuntion}) and (\ref{dphi_contravariant}))}\\
&=F^{{\leftarrow}}(\psi(-,y'))&\text{(by Proposition \ref{uphi_dphi_Yoneda})}\\
&=\psi(-,y')\circ F_{\natural}.&(\text{by the definition of}\ F^{{\leftarrow}})\end{aligned}$$
Conversely, if $G^{\natural}\circ\phi\leq\psi\circ F_{\natural}$, then for all ${\lambda}\in{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}$, $$\begin{aligned}
F^{{\rightarrow}}\circ{\phi^{{\downarrow}}}({\lambda})&=({\lambda}{\searrow}\phi)\circ F^{\natural}&(\text{by the definition of}\ F^{{\rightarrow}}\ \text{and}\ {\phi^{{\downarrow}}})\\
&\leq((G^{\natural}\circ{\lambda}){\searrow}(G^{\natural}\circ\phi))\circ F^{\natural}\\
&\leq((G^{\natural}\circ{\lambda}){\searrow}(\psi\circ F_{\natural}))\circ F^{\natural}\\
&\leq(G^{\natural}\circ{\lambda}){\searrow}(\psi\circ F_{\natural}\circ F^{\natural})\\
&\leq(G^{\natural}\circ{\lambda}){\searrow}\psi&(\text{since}\ F_{\natural}{\dashv}F^{\natural}:{{\mathbb{A}}}{\rightharpoonup}{{\mathbb{B}}}\ \text{in}\ {\mathcal{Q}}\text{-}{\bf Dist})\\
&={\psi^{{\downarrow}}}\circ G^{{\leftarrow}}({\lambda}).&(\text{by the definition of}\ {\psi^{{\downarrow}}}\ \text{and}\ G^{{\leftarrow}})\end{aligned}$$ This completes the proof.
By virtue of Proposition \[F\_G\_info\_F\_uphi\_dphi\_continuous\] we obtain a functor ${\mathcal{U}}:{\mathcal{Q}}\text{-}{\bf Info}{\longrightarrow}{\mathcal{Q}}\text{-}{\bf Cls}$ that sends an infomorphism $$(F,G):(\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}){\longrightarrow}(\psi:{{\mathbb{A}}}'{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}')$$ to a continuous ${\mathcal{Q}}$-functor $$F:({{\mathbb{A}}},{\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}){\longrightarrow}({{\mathbb{A}}}',{\psi^{{\downarrow}}}\circ{\psi_{{\uparrow}}}).$$
Given a ${\mathcal{Q}}$-closure space $({{\mathbb{A}}},C)$, define a ${\mathcal{Q}}$-distributor $\zeta_C:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}C({{\mathcal{P}}{{\mathbb{A}}}})$ by $$\zeta_C(x,\mu)=\mu(x)$$ for all $x\in{{\mathbb{A}}}_0$ and $\mu\in C({{\mathcal{P}}{{\mathbb{A}}}})$. It is clear that $\zeta_C$ is obtained by restricting the domain and the codomain of the ${\mathcal{Q}}$-distributor $${{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathcal{P}}{{\mathbb{A}}}}, \quad ({\lambda},\mu)\mapsto \mu\circ{\lambda}.$$
Given a continuous ${\mathcal{Q}}$-functor $F:({{\mathbb{A}}},C){\longrightarrow}({{\mathbb{B}}},D)$ between ${\mathcal{Q}}$-closure spaces, consider the ${\mathcal{Q}}$-functor $F^{\triangleleft}:D({{\mathcal{P}}{{\mathbb{B}}}}){\longrightarrow}C({{\mathcal{P}}{{\mathbb{A}}}})$ that sends each closed contravariant presheaf ${\lambda}$ to $F^{\triangleleft}({\lambda})=F^{{\leftarrow}}({\lambda})$. Then similar to Proposition \[Y\_functor\_Cat\_Info\] one can check that $$(F,F^{\triangleleft}):(\zeta_C:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}C({{\mathcal{P}}{{\mathbb{A}}}})){\longrightarrow}(\zeta_D:{{\mathbb{B}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}D({{\mathcal{P}}{{\mathbb{B}}}}))$$ is an infomorphism. Thus, we obtain a functor ${\mathcal{F}}:{\mathcal{Q}}\text{-}{\bf Cls}{\longrightarrow}{\mathcal{Q}}\text{-}{\bf Info}$.
\[F\_U\_adjunction\] ${\mathcal{F}}:{\mathcal{Q}}\text{-}{\bf Cls}{\longrightarrow}{\mathcal{Q}}\text{-}{\bf Info}$ is a left adjoint and right inverse of ${\mathcal{U}}:{\mathcal{Q}}\text{-}{\bf Info}{\longrightarrow}{\mathcal{Q}}\text{-}{\bf Cls}$.
[**Step 1.**]{} ${\mathcal{F}}$ is a right inverse of ${\mathcal{U}}$.
For each ${\mathcal{Q}}$-closure space $({{\mathbb{A}}},C)$, by the definition of the functor ${\mathcal{F}}$, ${\mathcal{F}}({{\mathbb{A}}},C)$ is the ${\mathcal{Q}}$-distributor $\zeta_C:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}C({{\mathcal{P}}{{\mathbb{A}}}})$, where $\zeta_C(x,\mu)=\mu(x)$ for all $x\in{{\mathbb{A}}}_0$ and $\mu\in C({{\mathcal{P}}{{\mathbb{A}}}})$. In order to prove ${\mathcal{U}}\circ{\mathcal{F}}({{\mathbb{A}}},C)=({{\mathbb{A}}},C)$, we show that $C=\zeta_C^{\downarrow}\circ(\zeta_C)_{\uparrow}$.
For all $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$ and ${\lambda}\in C({{\mathcal{P}}{{\mathbb{A}}}})$, since $C$ is a ${\mathcal{Q}}$-functor, $${\lambda}{\swarrow}\mu={{\mathcal{P}}{{\mathbb{A}}}}(\mu,{\lambda})\leq{{\mathcal{P}}{{\mathbb{A}}}}(C(\mu),{\lambda})={\lambda}{\swarrow}C(\mu),$$ and consequently $C(\mu)\leq({\lambda}{\swarrow}\mu){\searrow}{\lambda}$. Since $C$ is a ${\mathcal{Q}}$-closure operator, we have $$(C(\mu){\swarrow}\mu){\searrow}C(\mu)\leq 1_{t\mu}{\searrow}C(\mu)=C(\mu),$$ hence $$\begin{aligned}
C(\mu)&={\bigwedge}_{{\lambda}\in C({{\mathcal{P}}{{\mathbb{A}}}})}({\lambda}{\swarrow}\mu){\searrow}{\lambda}\\
&={\bigwedge}_{{\lambda}\in C({{\mathcal{P}}{{\mathbb{A}}}})}(\zeta_C(-,{\lambda}){\swarrow}\mu){\searrow}\zeta_C(-,{\lambda})\\
&={\bigwedge}_{{\lambda}\in C({{\mathcal{P}}{{\mathbb{A}}}})}(\zeta_C)_{\uparrow}(\mu)({\lambda}){\searrow}\zeta_C(-,{\lambda})\\
&=\zeta_C^{\downarrow}\circ(\zeta_C)_{\uparrow}(\mu),\end{aligned}$$ as required.
[**Step 2.**]{} ${\mathcal{F}}$ is a left adjoint of ${\mathcal{U}}$.
For each ${\mathcal{Q}}$-closure space $({{\mathbb{A}}},C)$, ${\rm id}_{({{\mathbb{A}}},C)}:({{\mathbb{A}}},C){\longrightarrow}{\mathcal{U}}\circ{\mathcal{F}}({{\mathbb{A}}},C)$ is clearly a continuous ${\mathcal{Q}}$-functor and $\{{\rm id}_{({{\mathbb{A}}},C)}\}$ is a natural transformation from the identity functor on ${\mathcal{Q}}$-[**Cls**]{} to ${\mathcal{U}}\circ{\mathcal{F}}$. Thus, it remains to show that for each ${\mathcal{Q}}$-distributor $\psi:{{\mathbb{A}}}'{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}'$ and each continuous ${\mathcal{Q}}$-functor $H:({{\mathbb{A}}},C){\longrightarrow}({{\mathbb{A}}}',{\psi^{{\downarrow}}}\circ{\psi_{{\uparrow}}})$, there is a unique infomorphism $$(F,G):{\mathcal{F}}({{\mathbb{A}}},C){\longrightarrow}(\psi:{{\mathbb{A}}}'{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}')$$ such that the diagram $$\bfig
\qtriangle<700,500>[({{\mathbb{A}}},C)`{\mathcal{U}}\circ{\mathcal{F}}({{\mathbb{A}}},C)`({{\mathbb{A}}}', {\psi^{{\downarrow}}}\circ{\psi_{{\uparrow}}});{\rm id}_{({{\mathbb{A}}},C)}`H`{\mathcal{U}}(F,G)]
\efig$$ is commutative.
By definition, ${\mathcal{F}}({{\mathbb{A}}},C)=\zeta_C:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}C({{\mathcal{P}}{{\mathbb{A}}}})$ and ${\mathcal{U}}(F,G)=F$, where $\zeta_C(x,\mu)=\mu(x)$. Thus, we only need to show that there is a unique ${\mathcal{Q}}$-functor $G:{{\mathbb{B}}}'{\longrightarrow}C({{\mathcal{P}}{{\mathbb{A}}}})$ such that $$(H,G):(\zeta_C:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}C({{\mathcal{P}}{{\mathbb{A}}}})){\longrightarrow}(\psi:{{\mathbb{A}}}'{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}')$$ is an infomorphism.
Let $G=H^{\triangleleft}\circ{\overline{\psi}}:{{\mathbb{B}}}'{\longrightarrow}C({{\mathcal{P}}{{\mathbb{A}}}})$. That $G$ is well-defined follows from the fact that ${\overline{\psi}}y'\in{\psi^{{\downarrow}}}\circ{\psi_{{\uparrow}}}({{\mathcal{P}}{{\mathbb{A}}}}')$ for all $y'\in{{\mathbb{B}}}'_0$ by Equation (\[phiF\_fixed\]) and that $H:({{\mathbb{A}}},C){\longrightarrow}({{\mathbb{A}}}',{\psi^{{\downarrow}}}\circ{\psi_{{\uparrow}}})$ is continuous. Now we check that $$(H,G):(\zeta_C:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}C({{\mathcal{P}}{{\mathbb{A}}}})){\longrightarrow}(\psi:{{\mathbb{A}}}'{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}')$$ is an infomorphism. This is easy since $$\zeta_C(x,Gy')=(Gy')(x)=H^{\triangleleft}\circ{\overline{\psi}}(y')(x) ={\overline{\psi}}(y')(Hx)=\psi(Hx,y')$$ for all $x\in{{\mathbb{A}}}_0$ and $y'\in{{\mathbb{B}}}'_0$. This proves the existence of $G$.
To see the uniqueness of $G$, suppose that $G':{{\mathbb{B}}}'{\longrightarrow}C({{\mathcal{P}}{{\mathbb{A}}}})$ is another ${\mathcal{Q}}$-functor such that $$(H,G'):(\zeta_C:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}C({{\mathcal{P}}{{\mathbb{A}}}})){\longrightarrow}(\psi:{{\mathbb{A}}}'{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}')$$ is an infomorphism. Then for all $x\in{{\mathbb{A}}}_0$ and $y'\in{{\mathbb{B}}}'_0$, $$(G'y')(x)=\zeta_C(x,G'y')=\psi(Hx,y')={\overline{\psi}}(y')(Hx) =H^{\triangleleft}\circ{\overline{\psi}}(y')(x)=(Gy')(x),$$ hence $G'=G$.
\[Cls\_coreflective\_Info\] The category ${\mathcal{Q}}$-[**Cls**]{} is a coreflective subcategory of ${\mathcal{Q}}$-[**Info**]{}.
The composition $${\mathcal{M}}={\mathcal{T}}\circ{\mathcal{U}}:{\mathcal{Q}}\text{-}{\bf Info}{\longrightarrow}({\mathcal{Q}}\text{-}{\bf CCat})_{{{\sf skel}}}$$ sends a ${\mathcal{Q}}$-distributor $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$ to a complete ${\mathcal{Q}}$-category ${\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}({{\mathcal{P}}{{\mathbb{A}}}})$. Conversely, since ${\mathcal{F}}$ is a right inverse of ${\mathcal{U}}$ (Theorem \[F\_U\_adjunction\]) and ${\mathcal{T}}$ is a left inverse of ${\mathcal{D}}$ (up to isomorphism, Theorem \[T\_D\_adjunction\]), we have the following
\[complete\_category\_fca\] Every skeletal complete ${\mathcal{Q}}$-category is isomorphic to ${\mathcal{M}}(\phi)$ for some ${\mathcal{Q}}$-distributor $\phi$.
The following proposition shows that the free cocompletion functor of ${\mathcal{Q}}$-categories factors through the functor ${\mathcal{M}}$.
\[M\_Yoneda\_PA\] The diagram $$\bfig
\qtriangle<700,500>[{\mathcal{Q}}\text{-}{\bf Cat}`{\mathcal{Q}}\text{-}{\bf Info}`{\mathcal{Q}}\text{-}{\bf CCat};{{\bf Y}}`{\mathcal{P}}`{\mathcal{M}}]
\efig$$ commutes.
First, ${\mathcal{M}}(({{\sf Y}}_{{{\mathbb{A}}}})_{\natural})= (({{\sf Y}}_{{{\mathbb{A}}}})_{\natural})^{{\downarrow}}\circ (({{\sf Y}}_{{{\mathbb{A}}}})_{\natural})_{{\uparrow}}({{\mathcal{P}}{{\mathbb{A}}}})={{\mathcal{P}}{{\mathbb{A}}}}$ for each ${\mathcal{Q}}$-category ${{\mathbb{A}}}$. To see this, it suffices to check that $$\mu= (({{\sf Y}}_{{{\mathbb{A}}}})_{\natural})^{{\downarrow}}\circ (({{\sf Y}}_{{{\mathbb{A}}}})_{\natural})_{{\uparrow}}(\mu)= (({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}{\swarrow}\mu){\searrow}({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}$$ for all $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$. On one hand, by Yoneda lemma we have $$({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}{\swarrow}\mu=({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}{\swarrow}({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}(-,\mu)\geq{{\mathcal{P}}{{\mathbb{A}}}}(\mu,-),$$ thus $$(({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}{\swarrow}\mu){\searrow}({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}\leq{{\mathcal{P}}{{\mathbb{A}}}}(\mu,-){\searrow}({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}=({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}(-,\mu)=\mu.$$ On the other hand, $\mu\leq(({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}{\swarrow}\mu){\searrow}({{\sf Y}}_{{{\mathbb{A}}}})_{\natural}$ holds trivially.
Second, it is trivial that for each ${\mathcal{Q}}$-functor $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$, $${\mathcal{M}}\circ{{\bf Y}}(F)=F^{\rightarrow}={\mathcal{P}}(F).$$
Therefore, the conclusion holds.
Corollary \[Cls\_coreflective\_Info\] says that the category ${\mathcal{Q}}$-[**Cls**]{} is a coreflective subcategory of ${\mathcal{Q}}$-[**Info**]{}. In the following we show that ${\mathcal{Q}}$-[**Cls**]{} is equivalent to a subcategory of ${\mathcal{Q}}$-[**Info**]{}. This equivalence is a generalization of that between closure spaces and state property systems in [@Aerts1999].
A ${\mathcal{Q}}$-state property system is a triple $({{\mathbb{A}}},{{\mathbb{B}}},\phi)$, where ${{\mathbb{A}}}$ is a ${\mathcal{Q}}$-category, ${{\mathbb{B}}}$ is a skeletal complete ${\mathcal{Q}}$-category and $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$ is a ${\mathcal{Q}}$-distributor, such that
- $\phi(-,{\inf}_{{{\mathbb{B}}}}{\lambda})={\lambda}{\searrow}\phi$ for all ${\lambda}\in{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}$,
- ${{\mathbb{B}}}(y,y')=\phi(-,y'){\swarrow}\phi(-,y)$ for all $y,y'\in{{\mathbb{B}}}_0$.
${\mathcal{Q}}$-state property systems and infomorphisms constitute a category ${\mathcal{Q}}$-[**Sp**]{}, which is a subcategory of ${\mathcal{Q}}$-[**Info**]{}.
For each ${\mathcal{Q}}$-closure space $({{\mathbb{A}}},C)$, $({{\mathbb{A}}},C({{\mathcal{P}}{{\mathbb{A}}}}),\zeta_C)$ is a ${\mathcal{Q}}$-state property system. First, for all $\Psi\in{{\mathcal{P}}^{\dag}}(C({{\mathcal{P}}{{\mathbb{A}}}}))$, it follows from Example \[PX\_PA\_complete\] and Equation (\[closure\_system\_infimum\]) that $$\begin{aligned}
\zeta_C(-,{\inf}_{C({{\mathcal{P}}{{\mathbb{A}}}})}\Psi)&= {\inf}_{C({{\mathcal{P}}{{\mathbb{A}}}})}\Psi\\
&= {\bigwedge}_{\mu\in C({{\mathcal{P}}{{\mathbb{A}}}})}\Psi(\mu){\searrow}\mu\\
&={\bigwedge}_{\mu\in C({{\mathcal{P}}{{\mathbb{A}}}})}\Psi(\mu){\searrow}\zeta_C(-,\mu)\\
&=\Psi{\searrow}\zeta_C.\end{aligned}$$ Second, it is trivial that $$C({{\mathcal{P}}{{\mathbb{A}}}})(\mu,{\lambda})={\lambda}{\swarrow}\mu=\zeta_C(-,{\lambda}) {\swarrow}\zeta_C(-,\mu)$$ for all $\mu,{\lambda}\in C({{\mathcal{P}}{{\mathbb{A}}}})$.
Therefore, the codomain of the functor ${\mathcal{F}}:{\mathcal{Q}}\text{-}{\bf Cls}{\longrightarrow}{\mathcal{Q}}\text{-}{\bf Info}$ can be restricted to the subcategory ${\mathcal{Q}}$-[**Sp**]{}.
\[F\_U\_equivalence\] The functors ${\mathcal{F}}:{\mathcal{Q}}\text{-}{\bf Cls}{\longrightarrow}{\mathcal{Q}}\text{-}{\bf Sp}$ and ${\mathcal{U}}:{\mathcal{Q}}\text{-}{\bf Sp}{\longrightarrow}{\mathcal{Q}}\text{-}{\bf Cls}$ establish an equivalence of categories.
It is shown in Theorem \[F\_U\_adjunction\] that ${\mathcal{U}}\circ{\mathcal{F}}={\bf id}_{{\mathcal{Q}}\text{-}{\bf Cls}}$, so, it suffices to prove that ${\mathcal{F}}\circ{\mathcal{U}}\cong {\bf id}_{{\mathcal{Q}}\text{-}{\bf Sp}}$.
Given a ${\mathcal{Q}}$-state property system $({{\mathbb{A}}},{{\mathbb{B}}},\phi)$, we have by definition $${\mathcal{F}}\circ{\mathcal{U}}({{\mathbb{A}}},{{\mathbb{B}}},\phi)=({{\mathbb{A}}},{\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}({{\mathcal{P}}{{\mathbb{A}}}}), \zeta_{{\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}}).$$ By virtue of Equation (\[phiF\_fixed\]), the images of the ${\mathcal{Q}}$-functor ${\overline{\phi}}:{{\mathbb{B}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{A}}}}$ are contained in ${\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}({{\mathcal{P}}{{\mathbb{A}}}})$, so, it can be viewed as a ${\mathcal{Q}}$-functor ${\overline{\phi}}:{{\mathbb{B}}}{\longrightarrow}{\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}({{\mathcal{P}}{{\mathbb{A}}}})$. Since for any $x\in{{\mathbb{A}}}_0$ and $y\in{{\mathbb{B}}}_0$, $$\phi(x,y)=({\overline{\phi}}y)(x)=\zeta_{{\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}}(x,{\overline{\phi}}y),$$ it follows that $\eta_{\phi}=(1_{{{\mathbb{A}}}},{\overline{\phi}})$ is an infomorphism from $\zeta_{{\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}}: {{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}({{\mathcal{P}}{{\mathbb{A}}}})$ to $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$. Hence $\eta_{\phi}$ is a morphism from ${\mathcal{F}}\circ{\mathcal{U}}({{\mathbb{A}}},{{\mathbb{B}}},\phi)$ to $({{\mathbb{A}}},{{\mathbb{B}}},\phi)$ in ${\mathcal{Q}}$-[**Sp**]{}. We claim that $\eta$ is a natural isomorphism from ${\mathcal{F}}\circ{\mathcal{U}}$ to the identity functor ${\bf id}_{{\mathcal{Q}}\text{-}{\bf Sp}}$.
Firstly, $\eta_{\phi}$ is an isomorphism. It suffices to show that $${\overline{\phi}}:{{\mathbb{B}}}{\longrightarrow}{\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}({{\mathcal{P}}{{\mathbb{A}}}})$$ is an isomorphism between ${\mathcal{Q}}$-categories.
Since $${{\mathbb{B}}}(y,y')=\phi(-,y'){\swarrow}\phi(-,y)={{\mathcal{P}}{{\mathbb{A}}}}({\overline{\phi}}y,{\overline{\phi}}y')$$ for all $y,y'\in{{\mathbb{B}}}_0$, it follows that ${\overline{\phi}}$ is fully faithful. For each $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$, let $y=\inf_{{{\mathbb{B}}}}{\phi_{{\uparrow}}}(\mu)$, then $${\overline{\phi}}y=\phi(-,y)=\phi(-,{\inf}_{{{\mathbb{B}}}}{\phi_{{\uparrow}}}(\mu))={\phi_{{\uparrow}}}(\mu){\searrow}\phi={\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}(\mu),$$ hence ${\overline{\phi}}$ is surjective. Since ${{\mathbb{B}}}$ is skeletal, we deduce that ${\overline{\phi}}:{{\mathbb{B}}}{\longrightarrow}{\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}({{\mathcal{P}}{{\mathbb{A}}}})$ is an isomorphism.
Secondly, $\eta$ is natural. For this, we check the commutativity of the following diagram for any infomorphism $(F,G):({{\mathbb{A}}},{{\mathbb{B}}},\phi){\longrightarrow}({{\mathbb{A}}}',{{\mathbb{B}}}',\psi)$ between ${\mathcal{Q}}$-state property systems: $$\bfig
\Square[{\mathcal{F}}\circ{\mathcal{U}}({{\mathbb{A}}},{{\mathbb{B}}},\phi)`({{\mathbb{A}}},{{\mathbb{B}}},\phi)`{\mathcal{F}}\circ{\mathcal{U}}({{\mathbb{A}}}',{{\mathbb{B}}}',\psi)`({{\mathbb{A}}}',{{\mathbb{B}}}',\psi);
(1_{{{\mathbb{A}}}},{\overline{\phi}})`(F,F^{\triangleleft})`(F,G)`(1_{{{\mathbb{A}}}'},{\overline{\psi}})] \efig$$ In fact, the equality $F\circ 1_{{{\mathbb{A}}}}=1_{{{\mathbb{A}}}'}\circ F$ is clear; and for all $x\in{{\mathbb{A}}}_0$ and $y'\in{{\mathbb{B}}}'_0$, $${\overline{\phi}}\circ G(y')(x)=\phi(x,Gy')=\psi(Fx,y')={\overline{\psi}}(y')(Fx)=F^{\triangleleft}\circ{\overline{\psi}}(y')(x),$$ thus the conclusion follows.
Together with Theorem \[T\_D\_adjunction\] we have
The composition $${\mathcal{T}}\circ{\mathcal{U}}:{\mathcal{Q}}\text{-}{\bf Sp}{\longrightarrow}({\mathcal{Q}}\text{-}{\bf CCat})_{{{\sf skel}}}$$ is a left adjoint of $${\mathcal{F}}\circ{\mathcal{D}}:({\mathcal{Q}}\text{-}{\bf CCat})_{{{\sf skel}}}{\longrightarrow}{\mathcal{Q}}\text{-}{\bf Sp}.$$
We end this section with a characterization of the complete ${\mathcal{Q}}$-category ${\mathcal{M}}(\phi)$ for a ${\mathcal{Q}}$-distributor $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$.
Given a ${\mathcal{Q}}$-distributor $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$, let ${\mathcal{M}}_{\phi}({{\mathbb{A}}},{{\mathbb{B}}})$ denote the set of pairs $(\mu,{\lambda})\in{{\mathcal{P}}{{\mathbb{A}}}}\times{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}$ such that ${\lambda}={\phi_{{\uparrow}}}(\mu)$ and $\mu={\phi^{{\downarrow}}}({\lambda})$. ${\mathcal{M}}_{\phi}({{\mathbb{A}}},{{\mathbb{B}}})$ becomes a ${\mathcal{Q}}$-typed set if we assign $t(\mu,{\lambda})=t\mu=t{\lambda}$. For $(\mu_1,{\lambda}_1),(\mu_2,{\lambda}_2)\in{\mathcal{M}}_{\phi}({{\mathbb{A}}},{{\mathbb{B}}})$, let $$\label{B_A_B_phi_order}
{\mathcal{M}}_{\phi}({{\mathbb{A}}},{{\mathbb{B}}})((\mu_1,{\lambda}_1),(\mu_2,{\lambda}_2))={{\mathcal{P}}{{\mathbb{A}}}}(\mu_1,\mu_2)={{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}({\lambda}_1,{\lambda}_2),$$ Then ${\mathcal{M}}_{\phi}({{\mathbb{A}}},{{\mathbb{B}}})$ becomes a ${\mathcal{Q}}$-category.
The projection $$\pi_1:{\mathcal{M}}_{\phi}({{\mathbb{A}}},{{\mathbb{B}}}){\longrightarrow}{{\mathcal{P}}{{\mathbb{A}}}},\quad (\mu,{\lambda})\mapsto\mu$$ is clearly a fully faithful ${\mathcal{Q}}$-functor. Since the image of $\pi_1$ is exactly the set of fixed points of the ${\mathcal{Q}}$-closure operator ${\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}:{{\mathcal{P}}{{\mathbb{A}}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{A}}}}$, we obtain that ${\mathcal{M}}_{\phi}({{\mathbb{A}}},{{\mathbb{B}}})$ is isomorphic to the complete ${\mathcal{Q}}$-category ${\mathcal{M}}(\phi)={\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}({{\mathcal{P}}{{\mathbb{A}}}})$.
Similarly, the projection $$\pi_2:{\mathcal{M}}_{\phi}({{\mathbb{A}}},{{\mathbb{B}}}){\longrightarrow}{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}},\quad (\mu,{\lambda})\mapsto{\lambda}$$ is also a fully faithful ${\mathcal{Q}}$-functor and the image of $\pi_2$ is exactly the set of fixed points of the ${\mathcal{Q}}$-interior operator ${\phi_{{\uparrow}}}\circ{\phi^{{\downarrow}}}:{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}{\longrightarrow}{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}$. Hence ${\mathcal{M}}_{\phi}({{\mathbb{A}}},{{\mathbb{B}}})$ is also isomorphic to the complete ${\mathcal{Q}}$-category ${\phi_{{\uparrow}}}\circ{\phi^{{\downarrow}}}({{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}})$, which is a ${\mathcal{Q}}$-interior system of the skeletal complete ${\mathcal{Q}}$-category ${{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}$.
Equation (\[B\_A\_B\_phi\_order\]) shows that $${\phi_{{\uparrow}}}:{\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}({{\mathcal{P}}{{\mathbb{A}}}}){\longrightarrow}{\phi_{{\uparrow}}}\circ{\phi^{{\downarrow}}}({{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}})$$ and $${\phi^{{\downarrow}}}:{\phi_{{\uparrow}}}\circ{\phi^{{\downarrow}}}({{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}){\longrightarrow}{\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}({{\mathcal{P}}{{\mathbb{A}}}})$$ are inverse to each other. Therefore, ${\mathcal{M}}(\phi)(={\phi^{{\downarrow}}}\circ{\phi_{{\uparrow}}}({{\mathcal{P}}{{\mathbb{A}}}}))$, ${\phi_{{\uparrow}}}\circ{\phi^{{\downarrow}}}({{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}})$ and ${\mathcal{M}}_{\phi}({{\mathbb{A}}},{{\mathbb{B}}})$ are isomorphic to each other.
\[sup\_dense\] A ${\mathcal{Q}}$-functor $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ is $\sup$-dense (resp. $\inf$-dense) if for any $y\in{{\mathbb{B}}}_0$, there is some $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$ (resp. ${\lambda}\in{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}$) such that $y=\sup_{{{\mathbb{B}}}}F^{{\rightarrow}}(\mu)$ (resp. $y=\inf_{{{\mathbb{B}}}}F^{{\rightarrow}}({\lambda})$).
For each ${\mathcal{Q}}$-category ${{\mathbb{A}}}$, the Yoneda embedding ${{\sf Y}}:{{\mathbb{A}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{A}}}}$ is $\sup$-dense in ${{\mathcal{P}}{{\mathbb{A}}}}$. Indeed, we have that $\mu=\sup_{{{\mathcal{P}}{{\mathbb{A}}}}}\circ{{\sf Y}}^{\rightarrow}(\mu)$ for all $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$ (see Equation (\[mu\_sup\_ymu\]) in the proof of Theorem \[T\_D\_adjunction\]). Dually, the co-Yoneda embedding ${{{\sf Y}}^{\dag}}:{{\mathbb{A}}}{\longrightarrow}{{{\mathcal{P}}^{\dag}}{{\mathbb{A}}}}$ is $\inf$-dense.
The following characterization of ${\mathcal{M}}_{\phi}({{\mathbb{A}}},{{\mathbb{B}}})$ (hence ${\mathcal{M}}(\phi)$) extends Theorem 4.8 in [@Lai2009695] to the general setting.
\[complte\_category\_sup\_dense\] Given a ${\mathcal{Q}}$-distributor $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$, a skeletal complete ${\mathcal{Q}}$-category ${{\mathbb{X}}}$ is isomorphic to ${\mathcal{M}}_{\phi}({{\mathbb{A}}},{{\mathbb{B}}})$ if and only if there exist a $\sup$-dense ${\mathcal{Q}}$-functor $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{X}}}$ and an $\inf$-dense ${\mathcal{Q}}$-functor $G:{{\mathbb{B}}}{\longrightarrow}{{\mathbb{X}}}$ such that $\phi=G^{\natural}\circ F_{\natural}={{\mathbb{X}}}(F-,G-)$.
[**Necessity.**]{} It suffices to prove the case ${{\mathbb{X}}}={\mathcal{M}}_{\phi}({{\mathbb{A}}},{{\mathbb{B}}})$. Define ${\mathcal{Q}}$-functors $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{X}}}$ and $G:{{\mathbb{B}}}{\longrightarrow}{{\mathbb{X}}}$ by $$Fa=({\phi^{{\downarrow}}}\circ{\underline{\phi}}a,{\underline{\phi}}a),\quad Gb=({\overline{\phi}}b,{\phi_{{\uparrow}}}\circ{\overline{\phi}}b),$$ then $F,G$ are well defined by equations (\[phiF\_fixed\]) and (\[Fphi\_fixed\]). It follows that $$\begin{aligned}
{{\mathbb{X}}}(F-,G-)&={{\mathcal{P}}{{\mathbb{A}}}}({\phi^{{\downarrow}}}\circ{\underline{\phi}}-,{\overline{\phi}}-)\\
&={{\mathcal{P}}{{\mathbb{A}}}}({\phi^{{\downarrow}}}\circ{\underline{\phi}}-,{\phi^{{\downarrow}}}\circ{{{\sf Y}}^{\dag}}_{{{\mathbb{B}}}}-)&(\text{by Equation (\ref{phiF_fixed})})\\
&={{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}({\phi_{{\uparrow}}}\circ{\phi^{{\downarrow}}}\circ{\underline{\phi}}-,{{{\sf Y}}^{\dag}}_{{{\mathbb{B}}}}-)&(\text{by Proposition \ref{uphi-dphi-adjunction}})\\
&={{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}({\underline{\phi}}-,{{{\sf Y}}^{\dag}}_{{{\mathbb{B}}}}-)&(\text{by Equation (\ref{Fphi_fixed})})\\
&=({\underline{\phi}}-)(-)&(\text{by Yoneda lemma})\\
&=\phi.\end{aligned}$$ Now we show that $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{X}}}$ is $\sup$-dense. For all $(\mu,{\lambda}),(\mu',{\lambda}')\in{{\mathbb{X}}}_0$, $$\begin{aligned}
{{\mathbb{X}}}((\mu,{\lambda}),(\mu',{\lambda}'))&={\lambda}'{\searrow}{\lambda}\\
&={\lambda}'{\searrow}{\phi_{{\uparrow}}}(\mu)\\
&={\lambda}'{\searrow}(\phi{\swarrow}\mu)\\
&=({\lambda}'{\searrow}\phi){\swarrow}\mu\\
&={{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}}({\underline{\phi}}-,{\lambda}'){\swarrow}\mu\\
&={{\mathbb{X}}}(F-,(\mu',{\lambda}')){\swarrow}\mu&(\text{by Equation (\ref{B_A_B_phi_order})})\\
&=({{\mathbb{X}}}(-,(\mu',{\lambda}'))\circ F_{\natural}){\swarrow}\mu\\
&={{\mathbb{X}}}(-,(\mu',{\lambda}')){\swarrow}(\mu\circ F^{\natural})&(\text{by Proposition \ref{graph_cograph_implication}(2)})\\
&={{\mathbb{X}}}(-,(\mu',{\lambda}')){\swarrow}F^{{\rightarrow}}(\mu),\end{aligned}$$ thus $(\mu,{\lambda})={\sup}_{{{\mathbb{X}}}}\circ F^{{\rightarrow}}(\mu)$, as desired.
That $G:{{\mathbb{B}}}{\longrightarrow}{{\mathbb{X}}}$ is $\inf$-dense can be proved similarly.
[**Sufficiency.**]{} We show that the type-preserving function $$H:{{\mathbb{X}}}{\longrightarrow}{\mathcal{M}}_{\phi}({{\mathbb{A}}},{{\mathbb{B}}}),\quad Hx=(F_{\natural}(-,x),G^{\natural}(x,-))$$ is an isomorphism of ${\mathcal{Q}}$-categories.
[**Step 1.**]{} ${{\mathbb{X}}}=F_{\natural}{\swarrow}F_{\natural}=G^{\natural}{\searrow}G^{\natural}$.
For all $x\in{{\mathbb{X}}}_0$, since $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{X}}}$ is $\sup$-dense, there is some $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$ such that $x=\sup_{{{\mathbb{X}}}}F^{{\rightarrow}}(\mu)$, thus $$\label{x_sup_Fmu}
{{\mathbb{X}}}(x,-)={{\mathbb{X}}}{\swarrow}F^{{\rightarrow}}(\mu)={{\mathbb{X}}}{\swarrow}(\mu\circ F^{\natural})=({{\mathbb{X}}}\circ F_{\natural}){\swarrow}\mu=F_{\natural}{\swarrow}\mu,$$ where the third equality follows from Proposition \[graph\_cograph\_implication\](2). Consequently $$\begin{aligned}
{{\mathbb{X}}}(x,-)&\leq F_{\natural}{\swarrow}F_{\natural}(-,x)\\
&\leq(F_{\natural}{\swarrow}F_{\natural}(-,x))\circ{{\mathbb{X}}}(x,x)\\
&=(F_{\natural}{\swarrow}F_{\natural}(-,x))\circ(F_{\natural}(-,x){\swarrow}\mu)&(\text{by Equation (\ref{x_sup_Fmu})})\\
&\leq F_{\natural}{\swarrow}\mu\\
&={{\mathbb{X}}}(x,-),&(\text{by Equation (\ref{x_sup_Fmu})})\end{aligned}$$ hence ${{\mathbb{X}}}(x,-)=F_{\natural}{\swarrow}F_{\natural}(-,x)=(F_{\natural}{\swarrow}F_{\natural})(x,-)$.
Since $G:{{\mathbb{B}}}{\longrightarrow}{{\mathbb{X}}}$ is $\inf$-dense, similar calculations lead to ${{\mathbb{X}}}=G^{\natural}{\searrow}G^{\natural}$.
[**Step 2.**]{} $Hx\in{\mathcal{M}}_{\phi}({{\mathbb{A}}},{{\mathbb{B}}})$ for all $x\in{{\mathbb{X}}}_0$, thus $H$ is well defined. Indeed, $$\begin{aligned}
{\phi_{{\uparrow}}}(F_{\natural}(-,x))&=\phi{\swarrow}F_{\natural}(-,x)\\
&=(G^{\natural}\circ F_{\natural}){\swarrow}F_{\natural}(-,x)&(\text{since}\ \phi=G^{\natural}\circ F_{\natural})\\
&=G^{\natural}\circ(F_{\natural}{\swarrow}F_{\natural}(-,x))&(\text{by Proposition \ref{graph_cograph_implication}(3)})\\
&=G^{\natural}\circ{{\mathbb{X}}}(x,-)&(\text{by Step 1})\\
&=G^{\natural}(x,-).\end{aligned}$$ Similar calculation shows that ${\phi^{{\downarrow}}}(G^{\natural}(x,-))=F_{\natural}(-,x)$. Hence, $Hx\in{\mathcal{M}}_{\phi}({{\mathbb{A}}},{{\mathbb{B}}})$.
[**Step 3.**]{} $H$ is a fully faithful ${\mathcal{Q}}$-functor. Indeed, for all $x,x'\in{{\mathbb{X}}}_0$, by Step 1, $${{\mathbb{X}}}(x,x')=F_{\natural}(-,x'){\swarrow}F_{\natural}(-,x)={{\mathcal{P}}{{\mathbb{A}}}}(F_{\natural}(-,x),F_{\natural}(-,x'))={\mathcal{M}}_{\phi}({{\mathbb{A}}},{{\mathbb{B}}})(Hx,Hx').$$
[**Step 4.**]{} $H$ is surjective. For each pair $(\mu,{\lambda})\in{\mathcal{M}}_{\phi}({{\mathbb{A}}},{{\mathbb{B}}})$, we must show that there is some $x\in{{\mathbb{X}}}_0$ such that $F_{\natural}(-,x)=\mu$ and $G^{\natural}(x,-)={\lambda}$. Indeed, let $x=\sup_{{{\mathbb{X}}}}F^{{\rightarrow}}(\mu)$, then $$\begin{aligned}
G^{\natural}(x,-)&=G^{\natural}\circ{{\mathbb{X}}}(x,-)\\
&=G^{\natural}\circ(F_{\natural}{\swarrow}\mu)&(\text{by Equation (\ref{x_sup_Fmu})})\\
&=(G^{\natural}\circ F_{\natural}){\swarrow}\mu&(\text{by Proposition \ref{graph_cograph_implication}(3)})\\
&=\phi{\swarrow}\mu&(\text{since}\ \phi=G^{\natural}\circ F_{\natural})\\
&={\phi_{{\uparrow}}}(\mu)\\
&={\lambda},\end{aligned}$$ and it follows that $F_{\natural}(-,x)={\phi^{{\downarrow}}}(G^{\natural}(x,-))={\phi^{{\downarrow}}}({\lambda})=\mu$.
\[M\_example\] (1) If the quantaloid ${\mathcal{Q}}$ has only one object, i.e., ${\mathcal{Q}}$ is a unital quantale (in particular the $2$-element Boolean algebra), then a ${\mathcal{Q}}$-distributor $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$ between discrete ${\mathcal{Q}}$-categories is exactly a ${\mathcal{Q}}$-valued relations between two sets.[^2] In this case an element $(\mu,{\lambda})$ in ${\mathcal{M}}_{\phi}({{\mathbb{A}}},{{\mathbb{B}}})$ is a formal concept of the formal context $({{\mathbb{A}}},{{\mathbb{B}}},\phi)$ in the sense of [@Radim2004; @Ganter:1997:FCA:550737; @Shen2013166] and ${\mathcal{M}}_{\phi}({{\mathbb{A}}},{{\mathbb{B}}})$ is the (fuzzy) formal concept lattice of $({{\mathbb{A}}},{{\mathbb{B}}},\phi)$. So, the construction of ${\mathcal{M}}({\phi})$ provides an extension of Formal Concept Analysis [@Radim2004; @Ganter:1997:FCA:550737].
\(2) If the quantaloid ${\mathcal{Q}}$ degenerates to a unital commutative quantale, then ${\mathcal{Q}}$-categories have been treated as quantitative (fuzzy) ordered sets, e.g. [@Radim2004; @Wagner94solvingrecursive]. In this case, for each ${\mathcal{Q}}$-category ${{\mathbb{A}}}$, ${\mathcal{M}}({{\mathbb{A}}})$ is the enriched MacNeille completion of ${{\mathbb{A}}}$ given in [@Radim2004; @Wagner94solvingrecursive]. Thus, the construction of ${\mathcal{M}}(\phi)$ also generalizes the MacNeille completion of (quantitative) ordered sets.
Kan adjunctions {#Kan_adjunction}
===============
Given a ${\mathcal{Q}}$-distributor $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$, composing with $\phi$ yields a ${\mathcal{Q}}$-functor $$\phi^*:{{\mathcal{P}}{{\mathbb{B}}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{A}}}}$$ defined by $$\phi^*({\lambda})={\lambda}\circ\phi.$$ for all ${\lambda}\in{{\mathcal{P}}{{\mathbb{B}}}}$. Define another ${\mathcal{Q}}$-functor $$\phi_*:{{\mathcal{P}}{{\mathbb{A}}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{B}}}}$$ by $$\phi_*(\mu)=\mu{\swarrow}\phi.$$
The following propositions \[phistar\_adjoint\] and \[phi\_star\_Yoneda\] can be verified in a way similar to that for propositions \[uphi-dphi-adjunction\] and \[uphi\_dphi\_Yoneda\].
\[phistar\_adjoint\] $\phi^*{\dashv}\phi_*:{{\mathcal{P}}{{\mathbb{B}}}}{\rightharpoonup}{{\mathcal{P}}{{\mathbb{A}}}}$ in ${\mathcal{Q}}$-[**Cat**]{}.
\[phi\_star\_Yoneda\] Let $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$ be a ${\mathcal{Q}}$-distributor, then ${\ulcorner}\phi^*\circ{{\sf Y}}_{{{\mathbb{B}}}}{\urcorner}=\phi$.
If $\phi:{{\mathbb{A}}}{\rightharpoonup}{{\mathbb{B}}}$ is itself a left adjoint ${\mathcal{Q}}$-distributor, then $\phi^*$ is not only a left adjoint ${\mathcal{Q}}$-functor, but also a right adjoint ${\mathcal{Q}}$-functor as asserted in the following
$\phi{\dashv}\psi:{{\mathbb{A}}}{\rightharpoonup}{{\mathbb{B}}}$ in ${\mathcal{Q}}$-[**Dist**]{} if and only if $\psi^*{\dashv}\phi^*:{{\mathcal{P}}{{\mathbb{A}}}}{\rightharpoonup}{{\mathcal{P}}{{\mathbb{B}}}}$ in ${\mathcal{Q}}$-[**Cat**]{}.
[**Necessity.**]{} By Proposition \[graph\_cograph\_implication\](2), for all $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$ and ${\lambda}\in{{\mathcal{P}}{{\mathbb{B}}}}$, $${{\mathcal{P}}{{\mathbb{B}}}}(\psi^*(\mu),{\lambda})={\lambda}{\swarrow}(\mu\circ\psi)=({\lambda}\circ\phi){\swarrow}\mu={{\mathcal{P}}{{\mathbb{A}}}}(\mu,\phi^*({\lambda})).$$
[**Sufficiency.**]{} We must show that ${{\mathbb{A}}}\leq\psi\circ\phi$ and $\phi\circ\psi\leq{{\mathbb{B}}}$. Indeed, for all $x\in{{\mathbb{A}}}_0$ and $y\in{{\mathbb{B}}}_0$, by Proposition \[phi\_star\_Yoneda\], $$\psi(-,x)\circ\phi=\phi^*(\psi(-,x))=\phi^*\circ\psi^*\circ{{\sf Y}}_{{{\mathbb{A}}}}x\geq 1_{{{\mathcal{P}}{{\mathbb{A}}}}}\circ{{\sf Y}}_{{{\mathbb{A}}}}x={{\mathbb{A}}}(-,x),$$ $$\phi(-,y)\circ\psi=\psi^*(\phi(-,y))=\psi^*\circ\phi^*\circ{{\sf Y}}_{{{\mathbb{B}}}}y\leq 1_{{{\mathcal{P}}{{\mathbb{B}}}}}\circ{{\sf Y}}_{{{\mathbb{B}}}}y={{\mathbb{B}}}(-,y).$$ This completes the proof.
Therefore, for a left adjoint ${\mathcal{Q}}$-distributor $\phi$, $\phi^*$ has both a right adjoint $\phi_*$ and a left adjoint $\psi^*$, where $\psi$ is the right adjoint of $\phi$ in ${\mathcal{Q}}$-[**Dist**]{}. In particular, given a ${\mathcal{Q}}$-functor $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$, since the cograph $F^{\natural}:{{\mathbb{B}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{A}}}$ of $F$ is a right adjoint of the graph $F_\natural :{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$ of $F$, it follows that both $(F^\natural )_*$ and $(F_\natural )^*$ are right adjoints of $(F^\natural )^*$, hence equal to each other. Since $F^{\leftarrow}:{{\mathcal{P}}{{\mathbb{B}}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{A}}}}$ is the counterpart of the functor $-\circ F$ for ${\mathcal{Q}}$-categories, we arrive at the following conclusion which asserts that the adjunction $\phi^*{\dashv}\phi_*$ generalizes Kan extensions in category theory.
\[why\_kan\] For each ${\mathcal{Q}}$-functor $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$, it holds that $$(F^{\natural})^*{\dashv}(F^{\natural})_*= F^{{\leftarrow}}=(F_{\natural})^*{\dashv}(F_{\natural})_*.$$
\(1) The left Kan extension $(F^{\natural})^*:{{\mathcal{P}}{{\mathbb{A}}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{B}}}}$ of a ${\mathcal{Q}}$-functor $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ given in Theorem \[why\_kan\] is exactly the pointwise left Kan extension of ${{\sf Y}}_{{{\mathbb{B}}}}\circ F:{{\mathbb{A}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{B}}}}$ along ${{\sf Y}}_{{{\mathbb{A}}}}:{{\mathbb{A}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{A}}}}$ in Stubbe [@Stubbe_2005]. Indeed, it can be verified that if the pointwise left Kan extension $\langle F,G\rangle$ of a ${\mathcal{Q}}$-functor $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ along $G:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{C}}}$ exists, then for each $c\in{{\mathbb{C}}}_0$, $$\langle F,G\rangle(c)={{\mathbb{B}}}{\swarrow}(F^{\natural})^*(G_{\natural}(-,c)).$$
\(2) Consider the Boolean algebra ${\bf 2}=\{0,1\}$ as an one-object quantaloid. Then every set can be regarded as a discrete ${\bf 2}$-category. Given sets $X$ and $Y$, a distributor $F:X{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}Y$ is essentially a relation from $X$ to $Y$, or a set-valued map $X{\longrightarrow}{\bf 2}^Y$. If we write $F^{\rm op}$ for the dual relation of $F$, then both $F_*$ and $(F^{\rm op})^*$ are maps from ${\bf 2}^Y$ to ${\bf 2}^X$. Explicitly, for each $V\subseteq Y$, $$F_*(V)=\{x\in X\mid F(x)\subseteq V\} \ {\rm and}\ (F^{\rm op})^*(V)=\{x\in X\mid F(x)\cap V\not=\emptyset\}.$$
If both $X$ and $Y$ are topological spaces, then the upper and lower semi-continuity of $F$ (as a set-valued map) [@Berge1963] can be phrased as follows: $F$ is upper (resp. lower) semi-continuous if $F_*(V)$ (resp. $(F^{\rm op})^*(V)$) is open in $X$ whenever $V$ is open in $Y$. In particular, if $F$ is the graph of some map $f:X{\longrightarrow}Y$, then $(F^{\rm op})^*(V)=F_*(V)=f^{-1}(V)$ for all $V\subseteq Y$, hence $f$ is continuous iff $F$ is lower semi-continuous iff $F$ is upper semi-continuous [@Berge1963].
The following corollary shows that for a fully faithful ${\mathcal{Q}}$-functor $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$, both $(F^{\natural})^*$ and $(F_{\natural})_*$ can be regarded as extensions of $F$ [@Lawvere1973].
\[left\_right\_Kan\_equivalent\] If $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ is a fully faithful ${\mathcal{Q}}$-functor, then for all $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$, it holds that $(F^{\natural})^*(\mu)\circ F_{\natural}=\mu$ and $(F_{\natural})_*(\mu)\circ F_{\natural}=\mu$.
The first equality is a reformulation of Proposition \[fully\_faithful\_graph\_cograph\](1). For the second equality, $$\begin{aligned}
(F_{\natural})_*(\mu)\circ F_{\natural}&=(\mu{\swarrow}F_{\natural})\circ F_{\natural}\\
&=\mu{\swarrow}(F^{\natural}\circ F_{\natural})&(\text{by Proposition \ref{graph_cograph_implication}(4)})\\
&=\mu{\swarrow}{{\mathbb{A}}}&(\text{by Proposition \ref{fully_faithful_graph_cograph}(1)})\\
&=\mu.\end{aligned}$$ This completes the proof.
Adjunctions of the form $\phi^*{\dashv}\phi_*:{{\mathcal{P}}{{\mathbb{B}}}}{\rightharpoonup}{{\mathcal{P}}{{\mathbb{A}}}}$ will be called Kan adjunctions by abuse of language. The following theorem states that all adjunctions between ${{\mathcal{P}}{{\mathbb{B}}}}$ and ${{\mathcal{P}}{{\mathbb{A}}}}$ are of this form.
\[Kan\_distributor\_bijection\] The correspondence $\phi\mapsto\phi^*$ is an isomorphism of posets $${\mathcal{Q}}\text{-}{\bf Dist}({{\mathbb{A}}},{{\mathbb{B}}})\cong{\mathcal{Q}}\text{-}{\bf CCat}({{\mathcal{P}}{{\mathbb{B}}}},{{\mathcal{P}}{{\mathbb{A}}}}).$$
The proof is similar to Theorem \[Isbell\_distributor\_bijection\]. The correspondence $G\mapsto {\ulcorner}G\circ{{\sf Y}}_{{{\mathbb{B}}}}{\urcorner}$ is an inverse of the correspondence $\phi\mapsto\phi^*$.
Theorem \[Kan\_distributor\_bijection\] adds one more isomorphism of posets to (\[Isbell\_isomorphism\]): $$\begin{aligned}
{\mathcal{Q}}\text{-}{\bf Dist}({{\mathbb{A}}},{{\mathbb{B}}})&\cong{\mathcal{Q}}\text{-}{\bf Cat}^{\rm co}({{\mathbb{A}}},{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}})\cong{\mathcal{Q}}\text{-}{\bf Cat}({{\mathbb{B}}},{{\mathcal{P}}{{\mathbb{A}}}})\\
&\cong{\mathcal{Q}}\text{-}{\bf CCat}^{\rm co}({{\mathcal{P}}{{\mathbb{A}}}},{{{\mathcal{P}}^{\dag}}{{\mathbb{B}}}})\cong{\mathcal{Q}}\text{-}{\bf CCat}({{\mathcal{P}}{{\mathbb{B}}}},{{\mathcal{P}}{{\mathbb{A}}}}).\end{aligned}$$
Since $\phi_*\circ\phi^*:{{\mathcal{P}}{{\mathbb{B}}}}{\longrightarrow}{{\mathcal{P}}{{\mathbb{B}}}}$ is a ${\mathcal{Q}}$-closure operator for each ${\mathcal{Q}}$-distributor $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$, it follows that $({{\mathbb{B}}},\phi_*\circ\phi^*)$ is a ${\mathcal{Q}}$-closure space.
\[F\_G\_infomorhpism\_F\_phistar\_continuous\] Let $(F,G):(\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}){\longrightarrow}(\psi:{{\mathbb{A}}}'{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}')$ be an infomorphism. Then $G:({{\mathbb{B}}}',\psi_*\circ\psi^*){\longrightarrow}({{\mathbb{B}}},\phi_*\circ\phi^*)$ is a continuous ${\mathcal{Q}}$-functor.
Consider the following diagram: $$\bfig
\square|alrb|[{{\mathcal{P}}{{\mathbb{B}}}}'`{{\mathcal{P}}{{\mathbb{A}}}}'`{{\mathcal{P}}{{\mathbb{B}}}}`{{\mathcal{P}}{{\mathbb{A}}}};\psi^*`G^{{\rightarrow}}`F^{{\leftarrow}}`\phi^*]
\square(500,0)/>``>`>/[{{\mathcal{P}}{{\mathbb{A}}}}'`{{\mathcal{P}}{{\mathbb{B}}}}'`{{\mathcal{P}}{{\mathbb{A}}}}`{{\mathcal{P}}{{\mathbb{B}}}};\psi_*``G^{{\rightarrow}}`\phi_*]
\efig$$ One must show that $G^{{\rightarrow}}\circ\psi_*\circ\psi^*\leq\phi_*\circ\phi^*\circ G^{{\rightarrow}}$. We leave it to the reader to check that the left square commutes if and only if $(F,G):\phi{\longrightarrow}\psi$ is an infomorphism and that if $G^{\natural}\circ\phi\leq\psi\circ F_{\natural}$ then $G^{{\rightarrow}}\circ\psi_*\leq\phi_*\circ F^{{\leftarrow}}$.
By virtue of Proposition \[F\_G\_infomorhpism\_F\_phistar\_continuous\] we obtain a functor ${\mathcal{V}}:({\mathcal{Q}}\text{-}{\bf Info})^{\rm op}{\longrightarrow}{\mathcal{Q}}\text{-}{\bf Cls}$ that sends an infomorphism $$(F,G):(\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}){\longrightarrow}(\psi:{{\mathbb{A}}}'{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}')$$ to a continuous ${\mathcal{Q}}$-functor $$G:({{\mathbb{B}}}',\psi_*\circ\psi^*){\longrightarrow}({{\mathbb{B}}},\phi_*\circ\phi^*).$$
The composition of $${\mathcal{V}}:({\mathcal{Q}}\text{-}{\bf Info})^{\rm op}{\longrightarrow}{\mathcal{Q}}\text{-}{\bf Cls}$$ and $${\mathcal{T}}:{\mathcal{Q}}{\text -}{\bf Cls}{\longrightarrow}({\mathcal{Q}}\text{-}{\bf CCat})_{{{\sf skel}}}$$ gives a functor $${\mathcal{K}}={\mathcal{T}}\circ{\mathcal{V}}:({\mathcal{Q}}\text{-}{\bf Info})^{\rm op}{\longrightarrow}({\mathcal{Q}}\text{-}{\bf CCat})_{{{\sf skel}}}$$ that sends each ${\mathcal{Q}}$-distributor $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$ to the complete ${\mathcal{Q}}$-category $\phi_*\circ\phi^*({{\mathcal{P}}{{\mathbb{B}}}})$.
The following conclusion asserts that the free cocompletion functor of ${\mathcal{Q}}$-categories factors through ${\mathcal{K}}$.
\[K\_Yoneda\_PA\] If $F:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$ is a fully faithful ${\mathcal{Q}}$-functor, then ${\mathcal{K}}(F^{\natural})={{\mathcal{P}}{{\mathbb{A}}}}$. In particular, the diagram $$\bfig
\qtriangle<700,500>[{\mathcal{Q}}\text{-}{\bf Cat}`({\mathcal{Q}}\text{-}{\bf Info})^{\rm op}`{\mathcal{Q}}\text{-}{\bf CCat};{{\bf Y}}^{\dag}`{\mathcal{P}}`{\mathcal{K}}]
\efig$$ commutes.
In order to see that ${\mathcal{K}}(F^{\natural})=(F^{\natural})_*\circ(F^{\natural})^*({{\mathcal{P}}{{\mathbb{A}}}})={{\mathcal{P}}{{\mathbb{A}}}}$, it suffices to check that $(F^{\natural})_*\circ(F^{\natural})^*(\mu)=\mu$ for all $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$. Indeed, $$\begin{aligned}
(F^{\natural})_*\circ(F^{\natural})^*(\mu)&=(F_{\natural})^*\circ(F^{\natural})^*(\mu)&(\text{by Theorem\ \ref{why_kan}})\\
&=(F^{\natural}\circ F_{\natural})^*(\mu)\\
&={{\mathbb{A}}}^*(\mu)&(\text{by Proposition \ref{fully_faithful_graph_cograph}(1)})\\
&=\mu.\end{aligned}$$
Furthermore, it is easy to verify that ${\mathcal{K}}\circ{{\bf Y}}^{\dag}(G)=G^{\rightarrow}={\mathcal{P}}(G)$ for each ${\mathcal{Q}}$-functor $G:{{\mathbb{A}}}{\longrightarrow}{{\mathbb{B}}}$. Thus, the conclusion follows.
Theorem \[complete\_category\_fca\] shows that every skeletal complete ${\mathcal{Q}}$-category is of the form ${\mathcal{M}}(\phi)$. It is natural to ask whether every complete ${\mathcal{Q}}$-category can be written of the form ${\mathcal{K}}(\phi)$ for some ${\mathcal{Q}}$-distributor $\phi$. A little surprisingly, this is not true in general. This fact was pointed out in [@Lai2009695] in the case that ${\mathcal{Q}}$ is a unital commutative quantale. However, the answer is positive for a special kind of quantaloids.
Let ${\frak{D}}=\{d_A:A{\longrightarrow}A\mid A\in{\mathcal{Q}}_0\}$ be a family of morphisms in a quantaloid ${\mathcal{Q}}$. ${\frak{D}}$ is called a [*cyclic family*]{} [@Rosenthal1996] if $d_A{\swarrow}f=f{\searrow}d_B$ for all $f\in{\mathcal{Q}}(A,B)$. ${\frak{D}}$ is called a [*dualizing family*]{} [@Rosenthal1996] if $(d_A{\swarrow}f){\searrow}d_A=f=d_B{\swarrow}(f{\searrow}d_B)$ for all $f\in{\mathcal{Q}}(A,B)$.
A [*Girard quantaloid*]{} [@Rosenthal1996] is a quantaloid with a cyclic dualizing family ${\frak{D}}$ of morphisms.
[[@Rosenthal1996]]{} \[Girard\_quantaloid\_properties\] Suppose ${\mathcal{Q}}$ has a dualizing family $${\frak{D}}=\{d_A:A{\longrightarrow}A\mid A\in{\mathcal{Q}}_0\}.$$ Then for all ${\mathcal{Q}}$-arrows $f,f_t:A{\longrightarrow}B$, $g:B{\longrightarrow}C$, $h:A{\longrightarrow}C$:
- $g\circ f=d_C{\swarrow}(f{\searrow}(g{\searrow}d_C))=((d_A{\swarrow}f){\swarrow}g){\searrow}d_A$.
- $(h{\swarrow}f){\searrow}d_C=f\circ(h{\searrow}d_C)$, $d_A{\swarrow}(g{\searrow}h)=(d_A{\swarrow}h)\circ g$.
- $(d_B{\swarrow}g){\searrow}f=g{\swarrow}(f{\searrow}d_B)$.
Let ${\mathcal{Q}}$ be a Girard quantaloid with a cyclic dualizing family $${\frak{D}}=\{d_A:A{\longrightarrow}A\mid A\in{\mathcal{Q}}_0\}.$$ For all $f\in{\mathcal{Q}}(A,B)$, let $$\neg f=d_A{\swarrow}f=f{\searrow}d_B:B{\longrightarrow}A.$$ Then $\neg\neg f=f$ since ${\frak{D}}$ is a dualizing family. For each ${\mathcal{Q}}$-category ${{\mathbb{A}}}$, set $$(\neg{{\mathbb{A}}})(y,x)=\neg{{\mathbb{A}}}(x,y)$$ for all $x,y\in{{\mathbb{A}}}_0$. It is easy to verify that $\neg{{\mathbb{A}}}:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{A}}}$ is a ${\mathcal{Q}}$-distributor and $${\frak{D}}'=\{\neg{{\mathbb{A}}}:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{A}}}\mid {{\mathbb{A}}}\in{\mathcal{Q}}\text{-}{\bf Dist}\}$$ is a cyclic dualizing family of ${\mathcal{Q}}$-[**Dist**]{}. Thus
[[@Rosenthal1996]]{} \[DistQ\_Girard\] If ${\mathcal{Q}}$ is a Girard quantaloid, then ${\mathcal{Q}}$-[**Dist**]{} is a Girard quantaloid.
Therefore, by assigning $\neg\phi=\neg{{\mathbb{A}}}{\swarrow}\phi=\phi{\searrow}\neg{{\mathbb{B}}}$ for each ${\mathcal{Q}}$-distributor $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$, we obtain a functor $\neg:{\mathcal{Q}}\text{-}{\bf Info}{\longrightarrow}({\mathcal{Q}}\text{-}{\bf Info})^{\rm op}$ that sends an infomorphism $$(F,G):(\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}){\longrightarrow}(\psi:{{\mathbb{A}}}'{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}')$$ to $$(G,F):(\neg\psi:{{\mathbb{B}}}'{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{A}}}'){\longrightarrow}(\neg\phi:{{\mathbb{B}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{A}}}).$$
It is clear that $\neg\circ\neg=1_{{\mathcal{Q}}\text{-}{\bf Info}}$. We leave it to the reader to check that $(\neg\phi)(y,x)=\neg\phi(x,y)$ for any distributor $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$ and $x\in{{\mathbb{A}}}_0,y\in{{\mathbb{B}}}_0$.
\[V\_G\_id\] Suppose ${\mathcal{Q}}$ is a Girard quantaloid. Then for any ${\mathcal{Q}}$-distributor $\phi:{{\mathbb{A}}}{{ \ext@arrow 0055{\orightarrowfill@}{}{\ \ \ \ }}}{{\mathbb{B}}}$, it holds that $\phi^*=\neg\circ(\neg\phi)_{{\uparrow}}$ and $\phi_*=(\neg\phi)^{{\downarrow}}\circ\neg$.
For all ${\lambda}\in{{\mathcal{P}}{{\mathbb{B}}}}$ and $\mu\in{{\mathcal{P}}{{\mathbb{A}}}}$, we have $$\begin{aligned}
\phi^*({\lambda})&={\lambda}\circ\phi\\
&={\lambda}\circ(\neg\phi{\searrow}\neg{{\mathbb{A}}})&(\text{by Proposition \ref{DistQ_Girard}})\\
&=(\neg\phi{\swarrow}{\lambda}){\searrow}\neg{{\mathbb{A}}}&(\text{by Proposition \ref{Girard_quantaloid_properties}(2)})\\
&=\neg\circ(\neg\phi)_{{\uparrow}}({\lambda})\end{aligned}$$ and $$\begin{aligned}
\phi_*(\mu)&=\mu{\swarrow}\phi\\
&=\mu{\swarrow}(\neg\phi{\searrow}\neg{{\mathbb{A}}})&(\text{by Proposition \ref{DistQ_Girard}})\\
&=(\neg{{\mathbb{A}}}{\swarrow}\mu){\searrow}\neg\phi&(\text{by Proposition \ref{Girard_quantaloid_properties}(3)})\\
&=\neg\mu{\searrow}\neg\phi&(\text{by Proposition \ref{DistQ_Girard}})\\
&=(\neg\phi)^{{\downarrow}}\circ\neg\mu.\end{aligned}$$ The conclusion thus follows.
\[G\_V\_adjunction\] Suppose ${\mathcal{Q}}$ is a Girard quantaloid. Then ${\mathcal{V}}={\mathcal{U}}\circ\neg$ and it has a left adjoint right inverse given by $${\mathcal{G}}=\neg\circ{\mathcal{F}}:{\mathcal{Q}}\text{-}{\bf Cls}{\longrightarrow}({\mathcal{Q}}\text{-}{\bf Info})^{\rm op}.$$ Therefore, every skeletal complete ${\mathcal{Q}}$-category is isomorphic to ${\mathcal{K}}(\phi)$ for some ${\mathcal{Q}}$-distributor $\phi$.
This is an immediate consequence of Theorem \[F\_U\_adjunction\] and Lemma \[V\_G\_id\].
Concluding remarks and questions
================================
Isbell adjunctions and Kan extensions are fundamental constructions in category theory, both of them can be viewed as adjunctions between categories of (contravariant) functors. This paper investigates the functoriality of these constructions in a special setting: categories enriched over a small quantaloid ${\mathcal{Q}}$. To this end, infomorphisms (an extension of adjunctions between categories) are introduced to play the role of morphisms between distributors. It is shown that each distributor between categories enriched over a small quantaloid gives rise to two adjunctions (which are respectively generalizations of Isbell adjunctions and Kan extensions), hence to two monads; and that these two processes are functorial from the category of distributors and infomorphisms to the category of complete ${\mathcal{Q}}$-categories and left adjoints.
This paper is a first step (in a very special setting) to the functoriality of the constructions of Isbell adjunctions and Kan extensions, many things remain to be discovered. We end this paper with two questions.
The definition of infomorphisms is meaningful for distributors between small categories. The first question is: Is it possible to establish similar results for distributors between small categories?
The infomorphisms between distributors introduced here can be composed vertically, but not horizontally. So, the second question is: Is it possible to find a certain kind of morphisms between distributors that can be composed in both directions and behave in a nice way with respect to the construction of Kan extension and Isbell adjunction?
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[^1]: This work is supported by Natural Science Foundation of China (11071174).
[^2]: A ${\mathcal{Q}}$-category ${{\mathbb{A}}}$ is discrete if ${{\mathbb{A}}}(x,x)= 1_{tx}$ for all $x\in{{\mathbb{A}}}_0$ and ${{\mathbb{A}}}(x,y)=\bot_{tx,ty}$ whenever $x\neq y$.
|
---
author:
- 'L.-M. Cheng'
- 'S. Borgani'
- 'P. Tozzi'
- 'L. Tornatore'
- 'A. Diaferio'
- 'K. Dolag'
- 'X.-T. He'
- 'L. Moscardini'
- 'G. Murante'
- 'G. Tormen'
title: 'Simulating the Soft X-ray excess in clusters of galaxies'
---
Introduction
============
Clusters of galaxies provide a reservoir of baryons in the form of a hot plasma with typical temperatures of $10^7$–$10^8$ K, which emits over a broad band from Extreme Ultraviolet (EUV) to $\sim 10$ keV X–rays. Most of the observed X-ray features of clusters can be well accounted for within the framework of thermal bremsstrahlung emission plus emission lines associated to the metal content of the intra–cluster medium (ICM). However, a number of observations with the Extreme Ultraviolet Explorer (EUVE; e.g. Lieu et al. 1996a,b; Mittaz et al. 1998; Maloney & Bland-Hawthorn 2001), ROSAT (e.g. Bonamente et al. 2001a,b) and XMM-Newton (e.g. Finoguenov et al. 2003; Kaastra et al. 2003) have claimed the detection of an excess of EUV and soft X-ray emission in the spectrum of several clusters, with respect to what expected from a one–temperature plasma model.
A number of suggestions have been proposed for the origin of this excess, the two most popular scenarios being the non–thermal origin from inverse Compton scattering of the cosmic microwave background photons by relativistic electrons in the intracluster gas (e.g. Hwang 1997; En$\beta$lin & Biermann 1998; De Paolis et al. 2003), and the thermal origin due to warm gas at $T\sim 10^6\,K$, either from inside clusters or from diffuse filaments outside clusters (Lieu et al. 1996b; Nevalainen et al. 2003). However, the existence of the excess has been disputed by Bowyer et al. (1999), who argued that the EUV excess is an artifact caused by improper subtraction of the instrumental background (see also Berghofer & Bowyer 2002; Durret et al. 2002), although they concluded that a relatively weak EUV excess in the Virgo and Coma clusters may be real. Bregman et al. (2004) also argued that the excess may be caused by an improper inclusion in the data analysis of the effect of fluctuations of the galactic hydrogen column density.
Even within the framework of the thermal models, it is a matter of debate whether the gas responsible for the excess is located within clusters or in large–scale filamentary structures. For instance, Ettori (2003) found evidence for as much as $17$ per cent of the baryons in clusters to be presumably in the form of warm ($10^5$–$10^7$ K) material, which must emit EUV or soft X-ray photons in excess to those expected from the hot phase of the ICM. On the other hand, Kaastra et al. (2003) and Finoguenov et al. (2003) claim that the soft excess may have originated in filaments in the gas distribution in the vicinity of clusters.
In this paper, we present an analysis of a set of clusters extracted from a large cosmological hydrodynamical simulation (Borgani et al. 2004, Paper I), which is aimed at investigating the presence and origin of a soft X–ray excess in their spectra. By taking advantage of the cosmological environment of our simulation, which includes radiative cooling, star formation and galactic winds triggered by supernova (SN), we “observe” clusters in projection and estimate their spectra also including the contribution from the background/foreground large–scale gas distribution. Since in our simulation we treat only thermal emissivity processes, the two main questions that we intend to address are the following: (a) does a realistic description of the evolution of cosmic baryons account for a soft X–ray excess of thermal origin as large as that observed in the spectra of clusters? (b) is the excess associated to warm gas residing within clusters or to large–scale filaments observed in projection?
The structure of the paper is as follows. In Section 2 we describe our set of simulated clusters. After describing the procedure to compute the synthetic spectra, we present in Section 3 our results on the excess, its origin and a comparison with observations. We draw our main conclusions in Section 4.
The simulated clusters
======================
We analyze a representative set of 20 simulated clusters, which are extracted from a cosmological box for a standard flat $\Lambda$CDM cosmological model, with $\Omega_m=0.3, \Omega_\Lambda=0.7$, Hubble constant $H_0=100h$ km s$^{-1}$Mpc$^{-1}$,$h=0.7$, baryon density $\Omega_b=0.04$ and $\sigma_8=0.8$ for the normalization of the power spectrum. The box has side-length of $192h^{-1}\rm Mpc$ and contains $480^3$ dark matter particles and an initially equal number of gas particles, thus resulting in $m_{\rm DM}=4.6\times 10^9 h^{-1}M_\odot$ and $m_{\rm gas}=6.9\times 10^8 h^{-1}M_\odot$ for the mass of the two particle species. The Plummer–equivalent gravitational force softening is set to $7.5\,h^{-1}$kpc in physical units from $z=2$ to $z=0$, while it is kept fixed in comoving units at higher redshift. The SPH smoothing scale is allowed to decrease at most to one–fourth of the gravitational softening (see Paper I, for a more detailed description of this simulation). The run has been evolved using [P-GADGET2]{}, a massively parallel Tree–SPH code (Springel et al. 2001) with fully adaptive time-step integration. The implementation of SPH adopted in the code follows the entropy–conserving formulation by Springel & Hernquist (2002). The simulation includes a treatment of star formation based on a sub–resolution model of the interstellar medium and the effect of galactic winds powered by SN-II explosions (Springel & Hernquist 2003, SH03). The code also includes a treatment of metal production from SN-II. The resulting metallicity value, which is assigned to each gas and star particle, has to be interpreted as a global value which is contributed by different heavy elements with solar relative abundances. We exclude those particles from the computation of X–ray emissivity having temperature below $3\times 10^4$ K and gas density above $500\bar\rho_{\rm bar}$, being $\bar\rho_{\rm bar}$ the mean baryon density. Furthermore, following SH03, each gas particle which lies above a limiting density threshold is assumed to be composed of a hot ionized phase and a cold neutral phase, whose relative amounts depend on the local conditions of density and temperature. Since such particles are aimed at describing the multi-phase nature of the inter-stellar medium, we decided to exclude also their contribution in the computation of the ICM X–ray emissivity. Although the number of such particles is always very small, their high density may cause a sizable, although spurious, contribution to the soft X–ray emission.
[cccccccc]{} Cluster& $M_{\rm vir}$ & $R_{\rm vir}$& $T_{\rm ew}$ & $Z_{\rm ew}$ & $\eta_x$ & $\eta_y$ & $\eta_z$\
Index & $[10^{14}h^{-1}M_{\odot}]$ & $[h^{-1}Mpc]$& $[keV]$ & $[Z_{\odot}]$& & &\
CL01 & 1.60& 1.11 & 2.43 & 0.16 &0.97 & 0.19 & 0.10\
CL02 & 2.46& 1.28 & 2.74 & 0.17 &0.12 & 0.80 & 0.53\
CL03 & 2.59& 1.30 & 3.05 & 0.17 &0.17 & 0.18 &0.21\
CL04 & 7.00& 1.82 & 5.12 & 0.13 &0.11 & 0.40 &0.10\
CL05 & 2.00& 1.20 & 2.42 & 0.19 &0.48 & 0.20 &0.21\
CL06 & 1.72& 1.14 & 2.40 & 0.18 &0.17 & 0.17 &0.19\
CL07 & 1.99& 1.20 & 2.59 & 0.14 &0.28 & 0.23 &0.24\
CL08 & 13.0& 2.23 & 6.50 & 0.11 &0.58 & 0.26 &0.36\
CL09 & 2.57& 1.30 & 3.02 & 0.12 &0.53 & 0.44 &0.11\
CL10 & 3.76& 1.48 & 3.74 & 0.11 &0.34 & 0.31 &0.17\
CL11 &1.46 & 1.08 & 2.02 &0.18&0.56 & 0.30 &0.68\
CL12 &1.57 & 1.21 & 1.56 &0.24&0.67 & 0.61 &0.26\
CL13 &1.08 & 0.97 & 1.87 &0.20&0.22 & 1.47 &1.34\
CL14 &1.45 & 1.07 & 2.17 &0.13&1.73 & 1.36 &1.68\
CL15 &2.07 & 1.21 & 2.60 &0.16&0.17 & 0.18 &0.12\
CL16 &1.76 & 1.15 & 1.79 &0.26&0.60 & 0.70 &0.26\
CL17 &6.04 & 1.81 & 5.14 &0.12&0.09 & 0.07 &0.34\
CL18 &1.06 & 1.04 & 1.21 &0.26&1.79 & 1.19 &1.22\
CL19 &3.34 & 1.41 & 3.50 &0.15&0.17 & 0.20 &0.13\
CL20 &2.90 & 1.42 & 2.58 &0.11&0.57 & 0.53 &0.50\
\
The selected clusters have virial masses spanning about a decade from $\sim 10^{14}h^{-1}M_\odot$ to $\sim 10^{15}h^{-1}M_\odot$ (see Table 1). We did not apply any particular criterion to select the clusters to be analysed and, therefore, our set is representative of the whole cluster population in our simulation within this mass range. For each cluster, we measure the virial radius, $R_{\rm vir}$, as the distance from the most–bound DM particle, which encompasses an average density equal to the virial density for our cosmological model (e.g., Eke et al. 1996). Accordingly, the virial mass, $M_{\rm vir}$, is defined as the mass contained within $R_{\rm vir}$. We refer to Paper I for a description of the cluster identification algorithm that we have applied. The emission–weighted temperature and metallicity in a given energy band used to model the hot ICM also are listed in Table 1. Around each cluster we extract a spherical region extending out to $6\,R_{\rm vir}$. This region is then observed in projection, by extracting a cylinder with axis on the center of each cluster. This allows us to account for the contribution of the surrounding large–scale structure to the X–ray emission of each cluster. As we will show below, taking the fore/back–ground structure out to $6R_{\rm vir}$ is sufficient to obtain converged estimates of this contribution.
Analysis and Results
====================
Measuring the Soft Excess
-------------------------
The X–ray luminosity contributed by the $i$–th gas particle in the simulation is computed according to $$L_{X,i}=(\mu m_p)^{-2}
\left(\frac{n_e}{n_H}\right)_i
m_i\rho_i\Lambda(T_i,Z_i;E_1,E_2)$$ where $m_i$ and $\rho_i$ are the mass and the density of the hot phase of that particle, respectively, $\mu$ is the mean molecular weight, $n_e$ and $n_H$ are the number densities of electrons and protons, respectively. The cooling function $\Lambda(T,Z;E_1,E_2)$ is calculated within the energy band \[$E_1$–$E_2$\] using the plasma emission model by Raymond & Smith (1977), where $Z$ is the gas metallicity. Using this cooling function, the spectra are computed by binning the emissivity within energy intervals, so as to have an energy resolution $\Delta \log
E=0.01$.
In their analysis of the soft X–ray excess of the Coma cluster from ROSAT–PSPC data, Bonamente et al. (2003) apply a MEKAL model to fit the high–energy (\[1–2\] keV) portion of the spectrum. Then they extrapolate the best–fitting model to a lower energy (\[0.2–1\] keV) band, where the predicted spectrum is compared with the actually observed one. In order to reproduce this same procedure, we would be required to simulate mock observations of our simulated clusters, and extract a spectrum with signal–to–noise appropriate for a realistic exposure time and using the appropriate response function. Since reproducing in detail the observational setup is beyond the scope of this paper, we adopt the approach of computing for each cluster the emission–weighted temperature, $$T_{\rm ew}=\frac{\sum_i m_i \rho_i \Lambda(T_i,Z_i;E_1,
E_2)T_i}{\sum_i m_i \rho_i \Lambda(T_i,Z_i;E_1, E_2)}
\label{eq:tew}$$ and the emission–weighted metallicity, $$Z_{\rm ew}=\frac{\sum_i m_i \rho_i \Lambda(T_i,Z_i;E_1,
E_2)Z_i}{\sum_i m_i \rho_i \Lambda(T_i,Z_i;E_1,E_2)}.
\label{eq:zew}$$ in the \[1–2\] keV band. Then, we compute the corresponding MEKAL spectrum for this temperature and metallicity, to be compared with the actual synthetic spectrum in the \[0.2–1\] keV band (we use in the following the solar photospheric abundance by Anders & Grevesse 1989, when expressing metallicity in solar units). Clearly, our approach is correct as long as emission–weighted measures coincide with the corresponding quantities obtained from spectral fitting. Mazzotta et al. (2004) have recently pointed out that the complex ICM temperature structure in simulated clusters causes the spectral–fitting temperatures to be about 20 per cent lower than $T_{\rm ew}$ (see also Mathiesen & Evrard 2001). They provide an expression for a spectroscopic–like temperature, which represents an effective recipe to compute the actual spectral temperature in simulations. However, the fitting function given by Mazzotta et al. is specific to the response function and energy coverage of the Chandra and XMM–Newton detectors, therefore not applicable to the ROSAT–PSPC. Furthermore, no such effective recipe has been calibrated so far to estimate the spectroscopic metallicity in simulated clusters. Finally, the spectroscopic–like temperature is well defined only for clusters hotter than 3 keV, while a significant fraction of the simulated clusters in our set has lower temperature (see Table 1). For these reasons, we prefer to use here eqs.(\[eq:tew\]) and (\[eq:zew\]) for the definition of temperature and metallicity. We defer to a forthcoming paper a more detailed analysis, based on synthetic XMM-Newton pn and MOS spectra of simulated clusters, to investigate the detectability of a soft excess under realistic observing conditions (Cheng et al., in preparation).
The values of $T_{\rm ew}$ and $Z_{\rm ew}$ in the \[1–2\] keV band are given in Table 1 for all our clusters. The resulting metallicity values are smaller than the typical observed Fe abundance, $\sim 0.3
Z_{\odot}$ (e.g., Arnaud et al. 2001; Ikebe et al. 2002; Baumgartner et al. 2003). A careful study of the ICM metallicity would require including in simulations the contribution of the SN-Ia and accounting for the different yields of different elements (e.g., Tornatore et al. 2004). In principle, this may represent a limitation of the present analysis, since metal lines are expected to give a significant contribution to the total emissivity in the soft part of the spectrum, where we are seeking for the excess. In order to test whether our final results are affected by the uncertain description of the ICM metal enrichment, we have verified by how much our results change when we assume either $Z=0$ or $Z=0.25 Z_{\odot}$ in the MEKAL spectrum of a plasma with temperature $T=2$ keV. We find that the contribution to the total emission from metal lines turns out to be $\sim 3$ per cent in the \[0.2–1\] keV band, this tiny difference being due to the lack of prominent lines in the above energy range. Therefore, we expect that our final results should be largely insensitive to the approximate treatment of the ICM metal enrichment.
Having estimated $T_{\rm ew}$ and $Z_{\rm ew}$ for each cluster by emission-weighting in the \[1–2\] keV band, we can now compute the corresponding one–temperature and one–metallicity spectrum. This spectrum is then compared with the synthetic one, obtained by summing the contributions from all gas particles within the “observational” cylinder of each cluster. The presence and amount of a soft excess is then established by comparing the two spectra in the \[0.2–1\] keV band.
The result of this comparison is shown in Figure \[fi:sp04\], where we plot the projected luminosity density as a function of the frequency, $F_\nu$, for cluster-centric distances $R\le 0.4\,R_{\rm
vir}$. Instead of showing spectra for the whole cluster set, we report here the results only for the first nine clusters of the list. The solid lines show the synthetic spectra, while the dotted lines are the prediction from the one–temperature and one–metallicity model. Quite apparently, the two spectra are always very similar, thus demonstrating that no appreciable soft excess is generated in the central regions of our simulated clusters. The only appreciable difference is related to the presence of soft metal lines in the synthetic spectra. A quite different result is found if we concentrate, instead, on more external cluster regions, $0.4\le R/R_{\rm vir}\le 0.7$ (see Figure \[fi:sp07\]), which corresponds to the typical scales where the soft excess is detected from observational data. In this case, a fair number of clusters in our set shows evidence for a significant excess at $h\nu{\raise
-2.truept\hbox{\rlap{\hbox{$\sim$}}\raise5.truept \hbox{$<$}\ }}1$ keV, thus witnessing the presence of a warm gas component, which contributes to the soft X–ray emission. This result is consistent with the observational evidence for a X–ray excess, that is more pronounced at larger cluster–centric radii (e.g., Bonamente et al. 2002; see the discussion here below).
In order to quantify the amount of soft excess, we define the relative excess $$\eta=\frac{p-q}{q}
\label{eq:relex}$$ (see Bonamente et al. 2002), where $p$ is the flux in a given soft band, as computed from the synthetic spectrum, and $q$ is the prediction from the hot ICM model. We provide in Table 1 the values of the relative excess measured in this region, after projecting each cluster along three orthogonal directions. In about 30 per cent of the cases, we find a relative excess $\eta >0.5$. In several cases, the $\eta$ value for the same cluster changes quite substantially with the projection direction. This already indicates that the excess is sensitive to the occurrence of a few significant structures along the line-of-sight, rather than to the overall large–scale distribution of the gas.
Having established that the soft excess phenomenon is rather common in the outskirt region of simulated clusters, when observed in projection, it is worth asking whether such an excess is generated by warm gas inside clusters or is associated to fore/back–ground large–scale filamentary structures, extending over scales of several Mpc. Large-scale hydrodynamic simulations (e.g. Cen & Ostriker 1999; Cen et al. 2001; Davé et al. 2001) indicate that about half of the baryons in the local Universe exist in the form of a warm-hot intergalactic medium (WHIM) with temperature $ T\sim 10^5$–$10^7
K$. The WHIM permeates the so-far elusive large–scale filamentary structures, from which galaxy clusters continuously accrete gas. Therefore, it is quite possible that the soft excess originates from large–scale diffuse filaments. On the other hand, an observational census of the baryons inside clusters (Ettori 2003) suggests that about 20 per cent of the ICM may be contributed by warm gas in the above temperature range. This gas would also provide a soft X–ray emissivity on top of that generated by the hot gas heated to the cluster virial temperature.
Tracing in simulations the gas distribution in the large–scale environment of galaxy clusters offers the possibility to distinguish between the internal and the external origin for the soft excess. To this purpose, we compute the synthetic spectrum of each cluster by excluding from the observational cylinder all the gas particles lying at a distance larger than $R_{\rm vir}$ along the line-of-sight from the cluster center. The corresponding spectra are shown with the thick lines in Figure \[fi:sp07\]. In the large majority of the cases we find that, after excluding the contribution from the gas outside $R_{\rm vir}$, the spectrum shows only a very modest change. This implies that the soft excess is mostly contributed by warm gas residing inside the virial cluster regions, while excluding a significant contribution from the large–scale filaments. The only exception, among the nine clusters shown in Fig. \[fi:sp07\], is represented by CL03, for which a significant part of the excess is generated outside the virial region. A visual inspection of the gas distribution along the projection direction reveals the presence of a small group just approaching the virial region of the cluster, which is responsible for this excess.
In order to investigate this point further, we show in Figure \[fi:exdis\] how the relative excess changes with the extension along the line-of-sight, $R_{\rm los}$, of the region where the spectra are computed. In most cases the values of $\eta$ do not increase beyond $R_{\rm los}=R_{\rm vir}$, thus confirming that the excess does not generally receive a significant contribution outside the cluster virial regions. In the few cases where the excess increases along the line-of-sight, this takes place always in a narrow interval of distance. This confirms that the excess is contributed by individual small–scale structures (i.e. fore/background groups), rather than by the integrated effect of large–scale filaments.
In general, the amount of soft X–ray emission from filaments may depend on the energy feedback from SN and AGN, that heats the diffuse baryons: an efficient feedback brings the gas on a high adiabat, thus preventing it from reaching the density contrast of dark matter within filaments, therefore suppressing its X–ray emissivity. As discussed in Paper I, the feedback used in this simulation may be somewhat too weak to prevent overcooling within clusters and to break to the observed level the self–similarity of X-ray scaling relations at the scale of galaxy groups. If more efficient feedback needs to be introduced, then we expect an even smaller contribution of filaments to the soft X–ray excess.
In order to verify whether the soft excess is generated by a diffuse warm gas phase or by high density gas within subhalos, we plot in the left panel of Figure \[fi:phase\] the entropy–density phase diagram of the gas lying at projected cluster–centric distance $0.4<R/R_{\rm
vir}<0.7$ for the CL02 cluster. This structure is the one displaying the largest excess among those shown in Fig.\[fi:sp07\]. The entropy of the $i$–th gas particle is defined as $S_i=T_i/n_{e,i}^{2/3}$, where $T_i$ is its temperature (in keV) and $n_{e,i}$ is the associated number density of electrons (in cm$^{-3}$). Quite apparently, gas particles occupy two distinct regions of the $S$–$\rho$ plane; (1) the high–entropy low–density region, which corresponds to the shocked phase formed by gas whose high entropy has been generated by diffuse accretion; (2) a tail of condensed gas at lower entropy and high density, which is formed by gas within merging subhalos that preserved its low entropy during the accretion phase. As shown in the right panel of Fig. \[fi:phase\], the soft excess disappears once we remove the condensed gas phase (i.e., particles with $S<400$ keV cm$^2$) from the computation of the spectrum. This demonstrates that the soft excess is associated with the presence of previously virialized clumps of high density gas at a temperature of a few tenths of keV, which are still surviving in the ICM, rather than to a diffuse phase of warm gas superimposed on the hot cluster atmosphere.
Comparison with observations
----------------------------
Using ROSAT–PSPC pointings of the Coma cluster, Bonamente et al. (2003) reported the detection of a soft excess in the \[0.2–1\] keV band at a high statistical significance. After performing the analysis at different angular distances from the cluster center, they concluded that the relative excess is an increasing function of this distance. In order to verify how typical is this result within our set of simulated clusters, we convert angular scales at the redshift of Coma to physical scales, and then rescale physical scales in units of the virial radius. To this purpose, we take the value, $R_{\rm vir}=1.9{\,h^{-1}{\rm Mpc}}$ obtained for the Coma cluster by Lokas & Mamon (2003) from their analysis of the kinematics of cluster galaxies. The result of this comparison is shown in Figure \[obs\_coma\]. Crosses are the observational values of the average excess within different annuli, as reported in Table 2 of Bonamente et al. (2003). For each of the 20 simulated clusters, we plot the excess obtained by projecting along three orthogonal directions. The solid line marks the upper 90 percentile in the distribution of the relative soft excesses in simulated clusters.
As a general result, both simulations and observational data show a relative excess that increases with the projected radius. The lower soft excess at small radii is due to the less complex thermal structure of the ICM in the inner cluster regions: small sub–clumps reaching such regions had time to be disrupted and their gas to get thermalized with the surrounding hot cluster atmosphere. Still, for $0.4{\raise
-2.truept\hbox{\rlap{\hbox{$\sim$}}\raise5.truept \hbox{$<$}\ }}R/R_{\rm vir}{\raise
-2.truept\hbox{\rlap{\hbox{$\sim$}}\raise5.truept \hbox{$<$}\ }}0.7$, the excess observed in Coma is somewhat higher than the level of the 90 percentile in the distribution of the simulated soft X–ray excess. This indicates that an excess as large as that observed in the Coma cluster represents a rather rare event in our set of simulated clusters. We refrain from drawing any strong conclusion from this comparison, since most of the simulated clusters in our set are substantially smaller than the Coma cluster.
From their analysis of an ensemble of galaxy clusters, observed with the ROSAT–PSPC, Bonamente et al. (2002) report that the soft excess in the narrower \[0.2–0.4\] keV band is in fact a rather widespread phenomenon, which is detected at high significance in a fair fraction, $\sim 20$ per cent, of the sources. We compare in Figure \[observ\] the relative excess in this band as a function of the cluster temperature, for both observed and simulated systems. As for data, the values of the excess are those reported in Figure 3 by Bonamente et al. (2002), while the temperature values have been taken from White (2000). The simulation results are obtained by computing the mean relative excess within $0.7\,R_{\rm vir}$, therefore including also the contribution from the innermost regions. This scale represents a good approximation to the typical size of the regions covered by observations. Within the temperature interval sampled by both observed and simulated clusters, the corresponding values of the relative excess are quite consistent with each other. We note that few observed systems have a significant negative excess. Bonamente et al. (2002) interpret this as due to a soft absorption associated to the presence of cold gas in central cluster regions. For $T{\raise
-2.truept\hbox{\rlap{\hbox{$\sim$}}\raise5.truept \hbox{$<$}\ }}2$ keV the scatter in the relative excess increases for simulations, with few clusters being characterized by a soft X–ray excess larger than 20 per cent. This indicates that the relative importance of the warm component of the ICM increases in a few cases for colder systems, thus potentially leading to an intrinsic difficulty of defining a single temperature for systems with temperature below 2 keV (see also Mazzotta et al. 2004).
As a concluding remark, it is worth mentioning that the approximate treatment of metal production in our simulation represents a potential source of uncertainty in the estimate of the soft excess. As already mentioned, the simulation includes energy feedback and the production of metals only from SN-II, using global metal yields (SH03). This is the main reason for the rather low metallicity measured in the simulated clusters (see Table 1). An improved treatment of the chemical enrichment of the gas in simulations requires accounting for the contribution from SN–Ia, hereby including detailed stellar yields and stellar evolution models (e.g., Tornatore et al. 2004, and references therein). As we have shown, the soft excess is associated to clumped gas at a temperature of a few tenths of keV. At such temperatures, a significant fraction of the emissivity is associated to metal lines. Therefore, an underestimate of the plasma metallicity would lead to an underestimate of the synthetic soft flux from the warm clumps, while leaving nearly unaffected the soft flux from the hot cluster atmosphere. In this respect, the values of the soft excess presented here may be somewhat underestimated.
Conclusions and future perspectives
===================================
We have studied a sample of 20 simulated clusters, extracted from a large SPH cosmological simulation of a concordance $\Lambda$CDM model, with the aim of exploring the presence and the origin of the soft X–ray excess in galaxy clusters. Besides the [*non–radiative*]{} gas dynamics, our simulation includes the treatment of star formation and energy feedback from galactic winds powered by supernovae. As such it provides a quite realistic description of the evolution of the diffuse gas in a cosmological environment.
Each cluster is observed in projection, by including the line-of-sight contribution from the surrounding large-scale gas distribution. This allows us to verify whether simulated clusters predict any soft excess of thermal origin and whether such an excess is generated within or outside the cluster virial region. Our analysis is aimed at mimicking the observational procedure followed by Bonamente et al. (2002, 2003), in their analysis of ROSAT–PSPC data. For each cluster we compute the corresponding emission–weighted temperature and metallicity in a relatively hard energy band. These values are then used to calculate the spectrum of a one-temperature and one-metallicity plasma model. The flux predicted by this spectrum in a softer band is then compared to that from the synthetic spectrum, computed by summing over the contribution of all the gas particles lying within the “observational cylinder” of each simulated cluster.
Our main results can be summarized as follows.
(a)
: A significant soft X–ray excess is detected for cluster-centric distances $0.4<R/R_{\rm vir}<0.7$, whose amount varies from cluster to cluster (see Table 1 and Fig.\[fi:sp07\]). In about 30 per cent of the cases we detect a relative excess at least as large as 50 per cent. However, the excess turns out to be always negligible in the central cluster regions, $R<0.4R_{\rm vir}$ (see Fig.\[fi:sp04\]).
(b)
: In most of the cases, the excess is found to originate inside the virial region of the simulated clusters (see Fig.\[fi:exdis\]). In the few cases in which a sizable contribution to the excess arises from the large–scale cluster environment, we find that it is contributed by fore/background groups lying along the cluster line-of-sight. Based on this result, we predict that the soft excess detected in observations receives only a minor contribution from the warm–hot medium permeating the large–scale cosmic web.
(c)
: Even within the virial cluster regions, the soft excess is contributed by high–density and low–entropy gas, rather than by a diffuse phase of warm gas (see Fig.\[fi:phase\]). This gas is associated to merging sub–groups, which still preserve their identity before being destroyed and thermalized in the hot ICM.
(d)
: We compare our results with those reported by Bonamente et al. (2003) for the Coma cluster. Although both observations and simulations show that the relative excess increases with the cluster-centric distance, its value in Coma is larger than for most of the simulated clusters. This implies that an excess as large as that observed for Coma is a rather rare event in our set of simulated systems.
A general conclusion of our analysis is that a soft X–ray excess of thermal origin is naturally predicted by hydrodynamical simulations of galaxy clusters in a cosmological environment. While this is an interesting result [*per se*]{}, we believe that the comparison between data and simulation does not yet have the required level of accuracy to exclude the presence of a non–thermal origin (i.e., inverse Compton scattering), at least for part of the observed excess. There is little doubt that solving this issue requires both a more refined analysis of simulations and higher quality data.
Recent XMM–Newton observations have confirmed the presence of a soft excess in the Coma cluster (Finoguenov et al. 2003) and in other four clusters (Kaastra et al. 2003), which are consistent with having a thermal origin. Finoguenov et al. (2003) claims that the soft excess arises from a fairly large filamentary structure containing warm gas. Although this result may be in contradiction with our results from the simulations, it is clear that a detailed comparison would require mimicking the observational procedure as close as possible on a cluster-by-cluster basis.
It is clear that establishing the exact amount, nature and origin of the soft X–ray excess requires an optimal control of observational systematics, such as the instrumental calibration, accounting for instrumental and cosmic background, as well as for the galactic hydrogen column density in the cluster direction. Indeed, since the soft emission occurs close to the lower boundary of the useful energy range of currently available detectors, it is of paramount importance to assess the accuracy of their calibration.
On the other hand, a detailed analysis of simulations requires reproducing as close as possible the observational setup. In the analysis presented in this paper, synthetic spectra of the simulated clusters have been generated under the assumptions of dealing with an ideal detector, with a flat response function across the whole energy range of interest, and with no, or perfectly controlled, background. It is clear that, with the steadily increasing quality of data and level of sophistication of numerical simulations, accounting for the instrumental features is mandatory for a self–consistent and meaningful comparison between theoretical predictions and observations, with both operating (e.g., Gardini et al. 2003) and planned satellites (e.g., Yoshikawa et al. 2004). We are currently undertaking a program of simulation analysis, based on mimicking XMM–Newton spectrum observations with both pn and MOS detectors. This will allow us to perform a more quantitative comparison with observations, so as to shed light on the implication of the soft excess on the physical properties of baryons within and around galaxy clusters.
The simulation has been realized using the IBM-SP4 machine at the “Consorzio Interuniversitario del Nord-Est per il Calcolo Elettronico” (CINECA, Bologna), with CPU time assigned thanks to an INAF-CINECA grant. We are greatly indebted to Volker Springel, who provided the GADGET code for the simulation and for a careful reading of the paper. We acknowledge useful discussions with Andrea Biviano, Pasquale Mazzotta, Silvano Molendi and Xiang-Ping Wu. L.-M.C. has been supported by a pre–doctoral fellowship of Regione Friuli Venezia–Giulia. This work has been partially supported by the PD51 INFN grant.
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abstract: 'This paper focuses on recovering an unknown vector $\beta$ from the noisy data $Y=X\beta +\sigma\xi$, where $X$ is a known $n\times p$-matrix, $\xi $ is a standard white Gaussian noise, and $\sigma$ is an unknown noise level. In order to estimate $\beta$, a spectral regularization method is used, and our goal is to choose its regularization parameter with the help of the data $Y$. In this paper, we deal solely with regularization methods based on the so-called ordered smoothers (see [@K]) and extend the oracle inequality from [@G1] to the case, where the noise level is unknown.'
author:
- |
Yuri Golubev [^1]\
`[email protected]`
title: 'Adaptive spectral regularizations of high dimensional linear models [^2]'
---
Introduction and main results {#s1}
=============================
This paper deals with recovering an unknown vector $\beta\in \mathbb{R}^n$ from the noisy data $$Y=X\beta +\sigma\xi,$$ where $X$ is a known $n\times p$-matrix with $n\ge p$, $\xi=\bigl(\xi(1),\ldots,\xi(n)\bigr)^\top
$ is a standard white Gaussian noise ($\mathbf{E}\xi(k)=0,\, \mathbf{E}\xi^2(k)=1, \, k=1,\ldots,n$ ), and $\sigma$ is an unknown noise level.
In spite of its simplicity, this mathematical model plays an important role in solving practical inverse problems like gravity problems (see, e.g. [@BMP]), tomography inverse problems [@H], and many others. As a rule, in inverse problems $n$ and $p$ are very large and therefore the main goal in this paper is to propose an approach suitable for $n=\infty,\, p=\infty$, severely ill-posed matrices $X^\top X$, and the unknown noise level.
We begin with the standard maximum likelihood estimate $$\hat{\beta}_0=\argmin_{\beta\in \mathbb{R}^p}\|Y-X\beta\|^2= (X^{\top}X)^{-1}X^{\top}Y ,$$ where $ \|z\|^2=\sum_{k=1}^nz^2(k).
$ It is well known and easy to check that $$\mathbf{E}(\hat\beta_0-\beta) (\hat\beta_0-\beta)^\top=\sigma^2 (X^\top X)^{-1}$$ and thus, the mean square risk of $\hat\beta_0$ is computed as follows: $$\label{eqn.1}
\mathbf{E} \|\hat\beta_0-\beta\|^2 =\sigma^2{\rm trace }\Bigl[(X^\top X)^{-1}\Bigr] =\sigma^2\sum_{k=1}^p
\lambda^{-1}(k),$$ where $\lambda(k)$ and $\psi_k\in \mathbb{R}^n$ here and below are the eigenvalues and the eigenvectors of $X^{\top}X$ $$X^{\top}X\psi_k =\lambda(k) \psi_k.$$
In this paper, it is assumed solely that $\lambda(1) \ge \lambda(2) \ge \cdots\ge \lambda(p)$. This assumption together with (\[eqn.1\]) reveals the main difficulty in $\hat{\beta}_0$: *its risk may be very large when $p$ is large or when $X^\top X$ has a large condition number.*
The natural idea to improve $\hat\beta_0$ is to suppress large $\lambda^{-1}(k)$ in (\[eqn.1\]) with the help of a linear smoother. Therefore we make use of the following family of linear estimates $$\label{eqn.2}
\hat{\beta}_\alpha=H_\alpha\hat\beta_0,\, \alpha\in(0,\alpha^\circ] ,$$ where $H_\alpha,\, \alpha\in(0,\alpha^\circ]$ is a family of $p\times p$-smoothing matrices.
In what follows, we deal with the smoothing matrices admitting the following representation $$H_\alpha=\sum_{k=1}^p \mathcal{H}_\alpha[\lambda(k)]\psi_k\psi_k^\top,$$ where $\mathcal{H}_\alpha(\lambda): \ \mathbb{R}^+\rightarrow [0,1]$ is such that $$\lim_{\alpha\rightarrow 0}\mathcal{H}_\alpha(\lambda)=1,\quad \lim_{\lambda\rightarrow
0}\mathcal{H}_\alpha(\lambda)=0.$$
In the literature (see, e.g., [@EHN]), this method is called [*spectral regularization*]{}. It covers widely used regularizations methods such as the Tikhonov-Phillips regularization [@TX] known in the statistical literature as *ridge regression*, Landweber’s iterations [@L], the $\mu$-method (see e.g. [@EHN]), and many others.
Summarizing, $\beta$ is estimated with the help of the family of linear estimates $\hat{\beta}_\alpha, \ \alpha\in(0,\alpha^\circ]$ defined by (\[eqn.2\]) and our goal is to find based on the data at hand the best estimator within this family. Notice that for given $\alpha$, the mean square risk of $\hat \beta_\alpha$ is computed as follows: $$\label{eqn.3}
\begin{split}
L_\alpha(\beta)\,\stackrel{\rm def}{=}\,&\mathbf{E}\|\hat
\beta_\alpha-\beta\|^2 =\sum_{k=1}^p\bigl[1-
h_\alpha(k)\bigr]^2\langle\beta,\psi_k\rangle^2+\sigma^2
\sum_{k=1}^p \lambda^{-1}(k) h_\alpha^2(k),
\end{split}$$ where $$h_\alpha(k)\stackrel{\rm def}{=}\mathcal{H}_\alpha[\lambda(k)]\quad \text{and} \quad
\langle\beta,\psi_k\rangle \stackrel{\rm def}{=}\sum_{l=1}^p \beta(l)\psi_k(l).$$ It is easily seen from (\[eqn.3\]) that the variance of $\hat \beta_\alpha$ is always smaller than that one of $\hat \beta_0$, but $\hat \beta_\alpha$ has a non-zero bias and therefore adjusting $\alpha$ we may improve the risk of $\hat\beta_0$. This improvement may be very significant when $\langle\beta,\psi_k\rangle^2$ are small for large $k$.
In practice, a good choice of the regularizing matrix family $H_\alpha$ is a delicate problem related to the computational complexity of $\hat\beta_\alpha$. For details, we refer interested readers to [@EHN].
As a rule, practical spectral regularization methods (the spectral cut–off, the Tikhonov-Phillips regularization, Landweber’s iterations) represent the so-called *ordered smoothers* [@K]. This means that the family of functions $\{\mathcal{H}_\alpha(\lambda),\, \alpha\in (0,\alpha^{\circ}]\}$ is ordered in the following sense:
\[d1\] The family of functions $\{{F}_\alpha(\lambda),\, \alpha\in A, \, \lambda \in\Lambda\subseteq\mathbb{R}^+\}$ is ordered if:
1. For any given $\alpha\in A$, $ F_\alpha(\lambda): \ \Lambda\rightarrow [0,1]$ is a monotone function of $\lambda$.
2. If for some $ \alpha_1,\alpha_2\in A$ and some $ \lambda'\in \Lambda$, $F_{\alpha_1}(\lambda') < F_{\alpha_2}(\lambda')$, then for all $ \lambda\in \Lambda$, $
F_{\alpha_1}(\lambda) \le F_{\alpha_2}(\lambda)$.
The next important question usually arising in practice is related to the data-driven choice of the regularization parameter $\alpha$. In statistical literature, one can find several general approaches to this problem. We cite here, for instance, the Lepski method which has been adopted to inverse problems in [@Ma], [@BH], [@BHMR], and the model selection technique which was used in [@LL].
The approach proposed in this paper is a slight modification of the unbiased risk estimation. To make the presentation simpler, we begin with the case, where the noise level $\sigma^2$ is known. Intuitively, a good data-driven regularization parameter should minimize in some sense the risk $L_\alpha(\beta)$ (see (\[eqn.3\])). Obviously, the best regularization parameter minimizing $L_\alpha(\beta)$ cannot be used since it depends on the unknown parameter of interest $\beta$. However, the idea of minimization of $L_\alpha(\beta)$ may be put into practice with the help of the empirical risk minimization principle defining the regularization parameter as follows: $$\label{eqn.4}
\hat\alpha=\argmin_\alpha {R}_{\alpha}^\sigma[Y,Pen],$$ where $${R}_\alpha^\sigma[Y,Pen]\stackrel{\rm def}{=}
\|\hat\beta_0-\hat\beta_\alpha\|^2+\sigma^2Pen(\alpha),$$ and $Pen(\alpha): (0,\alpha^{\circ}]\rightarrow\mathbb{R}^+$ is a given function called *penalty*. The main idea in this approach is to link $L_\alpha(\beta)$ and ${R}_\alpha^\sigma[Y,Pen]$. Heuristically, we want to find a minimal penalty $Pen(\alpha)$ that ensures the following inequality $$\label{eqn.5}
L_\alpha(\beta)\lesssim
{R}_{\alpha}^\sigma[Y,Pen]+\mathcal{C},$$ where $\mathcal{C}$ is a random variable that doesn’t depend on $\alpha$. It is convenient to define this constant as follows: $$\mathcal{C}=-\|\beta-\hat\beta_0\|^2=-\sigma^2\sum_{k=1}^p \lambda^{-1}(k)\xi^2(k).$$
The traditional approach to solving (\[eqn.5\]) is based on the minimization of the unbiased risk estimate. In this method, the penalty is computed as a root of the equation $$\label{eqn.6}
L_\alpha(\beta)=
\mathbf{E}\Bigl\{{R}_{\alpha}^\sigma[Y,Pen_u]+\mathcal{C}\Bigr\}.$$ One can check with a simple algebra that $$Pen_{u}(\alpha)=2 \sum_{k=1}^p
\lambda^{-1}(k) h_\alpha(k).$$ The idea of this penalty goes back to [@A] and [@CGPT] provides some oracle inequalities related to this approach.
Another well-known and widely used approach to the data-driven choice of $\alpha$ is related to the cross validation technique [@DMR]. In the framework of our statistical model, this method prompts a data-driven regularization parameter which is close to $$\hat{\alpha}_{CV}=\arg\min_{\alpha}\Bigl\{\|Y-X\hat{\beta}_\alpha\|^2+
\sigma^2Pen_{CV}(\alpha)\Bigr\},$$ with $$Pen_{CV}(\alpha)=2 \sum_{k=1}^p
h_\alpha(k).$$ It is well-known (see e.g. [@K]) that if the risk of $\hat{\beta}$ is measured by $\mathbf{E}\|X\hat{\beta}-X\beta\|^2$, then this penalty is nearly optimal and it works always well.
However, the question *Does $\hat{\alpha}_{CV}$ works well when the risk is measured by $\mathbf{E}\|\hat{\beta}-\beta\|^2$ ?* has a delicate answer depending on the spectrum of $X^\top X$. To the best of our knowledge there are no oracle inequalities controlling the risk of $\hat{\beta}{\hat{\alpha}_{CV}}$ uniformly in $\beta$. Notice, however, that one can show with the help of the method for computing minimal penalties in [@BM], that if $\lambda(k)\le \exp(-\kappa k)$, then the risk of this method blows up starting from some $\kappa>0$.
The similar effect takes place in the unbiased risk estimation. This happens because the standard deviation of ${R}_{\alpha}^\sigma[Y,Pen_{u}]+\mathcal{C}$ may be very large with respect to the mean $\mathbf{E}\bigl\{{R}_{\alpha}^\sigma[Y,Pen_{u}]+\mathcal{C}\big\}$ and therefore (\[eqn.5\]) may fail with a high probability.
To improve the above mentioned drawbacks of the unbiased risk estimation, we define, following [@G1], the penalty as a minimal root of the equation $$\label{eqn.7}
\begin{split}
&\mathbf{E}\sup_{ \alpha\le \alpha^{\circ}} \Bigl[L_\alpha(\beta)-
{R}_{\alpha}^\sigma[Y,Pen]-\mathcal{C}\Bigr]_+ \le C_1\mathbf{E} \Bigl[ L_{\alpha^{\circ}}(\beta)-
{R}_{\alpha^{\circ}}^\sigma[Y,Pen]-\mathcal{C}\Bigr]_+,
\end{split}$$ where $[x]_+=\max\{0,x\}$ and $C_1>1$ is a constant. Heuristic motivation behind this approach is rather transparent. We are looking for the minimal penalty that balances the excess risks corresponding to all possible $\alpha\in (0,\alpha^{\circ}]$. Recall that the excess risk is defined by the difference between the risk of the estimate and its penalized empirical risk. Note that in view of (\[eqn.5\]), we can deal solely with the positive part of the excess risk.
In order to explain heuristically how Equation (\[eqn.7\]) may be solved, we begin with the spectral representation of the underlying statistical problem. One can check easily that $$y(k)\stackrel{\rm def }{=} \langle X^\top Y,\psi_k\rangle/ \lambda(k) =
\langle \beta,\psi_k\rangle +\sigma \xi'(k)/\sqrt{\lambda(k)},$$ where $\xi'(k)$ are i.i.d. $\mathcal{N}(0,1)$. With these notations, $\hat\beta_\alpha$ admits the following representation $$\langle \hat\beta_\alpha,\psi_k\rangle =h_\alpha(k)y(k)=h_\alpha(k)\beta(k)+\sigma h_\alpha(k)\xi'(k)/\sqrt{\lambda(k)},$$ where $\beta(k)=\langle \beta,\psi_k\rangle$, and therefore $$\label{eqn.8}
\begin{split}
\|\hat\beta_0-\hat\beta_\alpha\|^2&=\sum_{k=1}^p\bigl[1-h_\alpha(k)\bigr]^2
y^2(k),\\
\|\beta-\hat\beta_\alpha\|^2&=\sum_{k=1}^p\bigl[\beta(k)-h_\alpha(k)y(k)\bigr]^2.
\end{split}$$
In what follows, it is assumed that the penalty has the following structure $$Pen(\alpha) =Pen_{u}(\alpha)+(1+\gamma)Q(\alpha),$$ where $\gamma$ is a small positive number and $Q(\alpha), \, \alpha>0$ is a positive function of $\alpha$ to be defined later on. Recall that the first term at the right-hand side is obtained from the unbiased risk estimation (see Equation (\[eqn.6\])). With $Pen(\alpha)$ we can rewrite the excess risk as follows: $$\begin{aligned}
\label{eqn.9}
\begin{split}
& L_\alpha(\beta)-{R}_{\alpha}^\sigma[Y,Pen]-\mathcal{C}\\
&\quad =\sigma^2 \sum_{k=1}^p \lambda^{-1}(k)\bigl[2h_\alpha(k)-h_\alpha^2(k)\bigr](\xi'^2(k)-1)-(1+\gamma)
\sigma^2Q(\alpha)\\
&\qquad +2\sigma \sum_{k=1}^p \lambda^{-1/2}(k)\bigl[2h_\alpha(k)-h_\alpha^2(k)\bigr]\xi'(k)\beta(k).
\end{split}
\end{aligned}$$
The first idea in solving (\[eqn.7\]) is based on the the fact that the cross term $$2\sigma \sum_{k=1}^p \lambda^{-1/2}(k)\bigl[2h_\alpha(k)-h_\alpha^2(k)\bigr]\xi'(k)\beta(k)$$ is typically small with respect to $\mathbf{E}\bigl\{R_\alpha^\sigma[Y,Pen]+\mathcal{C}\bigr\}$ (see for more details Lemma 9 in [@G1]). With this in mind, omitting the cross term, Equation (\[eqn.7\]) can be rewritten in the following nearly equivalent form $$\label{eqn.10}
\mathbf{E}\sup_{\alpha\le \alpha^{\circ}}[\eta_\alpha-(1+\gamma)Q(\alpha)]_+ \lesssim C_1\mathbf{E}[\eta_{\alpha^{\circ}}-(1+\gamma)Q(\alpha^{\circ})]_+ \asymp D(\alpha^{\circ}),$$ where $$\eta_\alpha\stackrel{\rm def }{=}\sum_{k=1}^p \lambda^{-1}(k)\bigl[2h_\alpha(k)-h_\alpha^2(k)\bigr][\xi'^2(k)-1]$$ and $$D(\alpha)\stackrel{\rm def }{=} \sqrt{\mathbf{E}\eta^2_\alpha}=\biggl\{2\sum_{k=1}^p \lambda^{-2}(k)\bigl[2h_\alpha(k)-h_\alpha^2(k)\bigr]^2\biggr\}^{1/2}.$$
Now we are in a position to compute an approximation of the minimal root for (\[eqn.10\]). It is clear that $Q(\alpha)\ge Q^+(\alpha) $, where $ Q^+(\alpha) $ is a root of $$\label{eqn.11}
\mathbf{E}[\eta_\alpha-Q^+(\alpha)]_+=D(\alpha^{\circ}).$$
To find a feasible solution to (\[eqn.11\]), we make use of the exponential Chebychev inequality resulting in $$\label{eqn.12}
\mathbf{E}[\eta-x]_+^p\le \Gamma(p+1)\lambda^{-p}\exp(-\lambda x)
\mathbf{E}\exp(\lambda\eta),$$ where $\eta$ is a random variable, $\Gamma(\cdot)$ is the gamma function, and $\lambda>0$.
Therefore we define $Q^+(\alpha)$ as a root of equation $$\inf_{\lambda}\exp[-\lambda Q^+(\alpha)]\mathbf{E}\exp(\lambda \eta_\alpha)=D(\alpha^{\circ}).$$
It is easy to check with a simple algebra that $$\label{eqn.13}
Q^+(\alpha)=2D(\alpha)\mu_\alpha\sum_{k=1}^p\frac{\rho^2_\alpha(k)}{1-2\mu_\alpha \rho_\alpha(k)},$$ where $\mu_\alpha$ is a root of the equation $$\label{eqn.14}
\sum_{k=1}^p F[\mu_\alpha \rho_\alpha(k)]=\log \frac{D(\alpha)}{D(\alpha^{\circ})},$$ and $$\label{eqn.15}
\begin{split}
F(x)&=\frac{1}{2}\log(1-2x)+x +\frac{2x^2}{1-2x},\\
\rho_\alpha(k)&=\sqrt{2}D^{-1}(\alpha)\lambda^{-1}(k)\bigl[2h_\alpha(k)
-h_\alpha^2(k)\bigr].
\end{split}$$
The next result (see also Theorem 1 in [@G1]) shows that $Q^+(\alpha)$ is a nearly optimal solution to (\[eqn.10\]).
\[proposition.1\] For any $\gamma>q\ge 0$ $$\mathbf{E}\sup_{\alpha\le \alpha^{\circ}}\Bigl[\eta_\alpha-(1+\gamma)Q^+(\alpha)\Bigr]_+^{1+q}\le \frac{CD^{1+q}(\alpha^{\circ})}{(\gamma-q)^3},$$ where here and throughout the paper $C$ denotes a generic constant.
Let us now turn to the case, where $\sigma$ is unknown. To compute the data-driven regularization parameter in this situation, we replace $\sigma^2$ in $R^\sigma_\alpha[Y,Pen]$ by the standard variance estimator $$\hat{\sigma}^2_\alpha=\frac{\|Y-X\hat\beta_{\alpha}\|^2}{\|1-H_{\alpha}\|^2}.$$ Thus we arrive at the following approximation of the empirical risk $$R_\alpha[Y,Pen]\stackrel{\rm def}{=}
\|\hat\beta_0-\hat\beta_\alpha\|^2+\frac{\|Y-X\hat\beta_\alpha\|^2}{\|1-H_\alpha\|^2}Pen(\alpha)$$ and the data-driven regularization parameter is computed now as follows: $$\hat\alpha =\argmin_{\alpha_\circ\le\alpha\le\alpha^\circ}R_\alpha[Y,Pen].$$
Notice that in contrast to the case of known $\sigma$, it assumed that $\alpha$ is bounded from below by $ \alpha_\circ$. This constraint ensures that with a hight probability $[\sigma^2-\hat\sigma_{\hat{\alpha}}^2]_+< \sigma^2/2$ uniformly in $\beta\in \mathbb{R}^p$. Unfortunately, when this inequality fails we cannot control correctly the risk of $\hat\beta_{\hat\alpha}$ since it may blow up (see [@BM] for similar phenomenon in the model selection). So, to avoid the blowup, we need a relatively good estimate of $\sigma$, or equivalently, large $\|1-H_\alpha\|^2$.
Stress also that since $\alpha_\circ$ cannot depend on $\sigma$, we would like to have $\alpha_\circ$ as small as possible to be sure that the methods works for small noise levels. From a mathematical viewpoint, this means that we need a relatively good upper bound for $\mathbf{E}| \sigma-\hat\sigma_{\hat{\alpha}}^2|Pen(\hat{\alpha})$. Roughly speaking, we have to check that with a hight probability $$| \sigma^2-\hat\sigma_{\hat{\alpha}}^2|Pen(\hat{\alpha})\ll\sigma^2 Pen(\hat{\alpha}).$$ The main difficulty in proving this equation is related to the fact that the random variables $\sigma^2-\hat\sigma_{\hat{\alpha}}^2$ and $ Pen(\hat{\alpha})$ are dependent. To overcome this difficulty we make use of the law of the iterated logarithm for $\sigma^2-\hat\sigma_{\hat{\alpha}}^2$ combined with a generalization of the Hölder inequality (see Lemmas \[lemma.4\] and \[lemma.3\] below). To carry out this approach, we need the following additional condition: *there exists a positive constant $C_2$ such that for all $\alpha\in (0,\alpha^\circ]$ $$\begin{aligned}
\label{eqn.16}
&\|h_\alpha\|^2_\lambda \ge C_2 \sum_{k=1}^p \lambda^{-1}(k)h_\alpha(k), \\[5pt]
\label{eqn.17}
&\frac{\|h_\alpha\|^2_\lambda}{\log[D(\alpha)/D(\alpha^\circ)]}+\max_k \frac{h_\alpha(k)}{\lambda(k)}\ge C_2 D(\alpha),\end{aligned}$$ where $$\|h_\alpha\|^2_\lambda =\sum_{k=1}^p \lambda^{-1}(k)h^2_\alpha(k).$$*
Denote for brevity $$\begin{split}
\Ps(\alpha_\circ,\alpha^\circ) \stackrel{\rm def}{=}
\frac{1}{\|1-h_{\alpha_\circ}\|}\biggl\{\biggl[\log\log\biggl(1+\frac{\|1-h_{\alpha_\circ}\|^2}{\|1-h_{\alpha^\circ}\|^2}\biggr)\biggr]^{1/2}
\\ {}+\log \biggl(1+\frac{Pen(\alpha_\circ)}{Pen(\alpha^\circ)}\biggr)\biggr\}
.
\end{split}$$
The following theorem controls the risk of $\hat\beta_{\hat\alpha}$ via the penalized oracle risk defined by $$r(\beta)\stackrel{\rm def}{=}\inf_{\alpha_\circ\le \alpha\le \alpha^{\circ}} \bar{R}_{\alpha}(\beta),$$ where $$\begin{split}
\bar{R}_{\alpha}(\beta)\,\stackrel{\rm def}{=}\,\mathbf{E}_\beta\bigl\{
R_\alpha[Y,Pen]+\mathcal{C}\bigr\} = L_\alpha(\beta) +(1+\gamma)\sigma^2Q^+(\alpha)\\ {}+ \frac{Pen(\alpha)}{\|1-h_\alpha\|^2}\sum_{k=1}^p \bigl[1-h_\alpha(k)\bigr]^2 \lambda(k) \beta^2(k).
\end{split}$$
\[theorem.1\] Let $ Pen(\alpha) =2 \sum_{k=1}^p
\lambda^{-1}(k) h_\alpha(k)+(1+\gamma)Q^+(\alpha)
$ with $Q^+(\alpha)$ defined by (\[eqn.13\]–\[eqn.15\]) and suppose (\[eqn.16\]–\[eqn.17\]) hold. Then, uniformly in $\beta\in \mathbb{R}^p$, $$\begin{aligned}
\label{eqn.18}
\begin{split}
\mathbf{E}_\beta\|\beta -\hat \beta_{\hat\alpha}\|^2\le
\biggl[1+C\Ps(\alpha_\circ,\alpha^\circ)+ C\log^{-1/2}\frac{ r(\beta)}{\sigma^2 D(\alpha^{\circ})}\biggr]
r(\beta)\\
{}+\frac{C\sigma^2D(\alpha^\circ)}{[1-C\Ps(\alpha_\circ,\alpha^\circ)/\gamma]_+\sqrt{\gamma} }\R\biggl[\frac{r(\beta)}{\sigma^2\gamma D(\alpha^{\circ})}+ \frac{1}{\gamma^4}\biggr]
,
\end{split}
\end{aligned}$$ where $
\R(x)={x}/{\log(x)}.
$
Notice that Equation \[eqn.18\] can be rewritten in the following concise form $$\label{eqn.19}
\mathbf{E}_\beta\|\beta -\hat \beta_{\hat\alpha}\|^2\le \biggl[1+C\Ps(\alpha_\circ,\alpha^\circ)+ \Psi_{\alpha_\circ,\gamma}\biggl(\frac{ r(\beta)}{\sigma^2 D(\alpha^{\circ})}\biggr)\biggr]r(\beta),$$ where $\Psi_{\alpha_\circ,\gamma}(\cdot)$ is a bounded function such that $$\label{eqn.20}
\lim_{x\rightarrow \infty}\Psi_{\alpha_\circ,\gamma}(x)=0.$$
The statistical sense of (\[eqn.19\]) is rather transparent: this equation shows that in typical nonparametric situations the method works like the ideal penalized oracle with the risk $r(\beta)$. *The typical nonparametric situation* means that
- $p$ is large, so, for properly chosen $\alpha_\circ$, $ \Ps(\alpha_\circ,\alpha^\circ)$ is small,
- the vector $(\langle\beta,\psi_1\rangle,\ldots \langle\beta,\psi_p\rangle)^\top$ contains many significant components, and thus $r(\beta)\gg \sigma^2D(\alpha^\circ)$.
These assumptions are typical in the minimax estimation, where it is assumed that $\beta$ belongs to an ellipsoid. Notice that with the help of (\[eqn.19\]–\[eqn.20\]) one can check relatively easily that for a proper chosen spectral regularization, $\hat\beta_{\hat\alpha}$ is the asymptotically minimax estimate up to a constant (see for details [@G1] and [@P]).
We finish this section with a short discussion of Conditions (\[eqn.16\]–\[eqn.17\]). Equation (\[eqn.16\]) means that $h_\alpha(k)$ vanishes rather rapidly for large $k$. This is always true for the spectral cut-off method ($h_\alpha(k)=\mathbf{1}\{\alpha \lambda(k)\ge 1\}$). Indeed, if $\lambda(k)\asymp k^{-p}$ with some $p\ge 0$, then $$\|h_\alpha\|^2_\lambda \asymp {\alpha}^{-p-1}, \quad
D(\alpha) \asymp {\alpha}^{-p-1/2}$$ and it is seen easily that (\[eqn.17\]) is fulfilled. Assume now that $X^\top X$ is severely ill-posed, i.e., $\lambda(k)\asymp
\exp(-\kappa k)$ with $\kappa>0$. Then $$\max_k \lambda(k)h_\alpha(k)\asymp \exp(\kappa/\alpha) \quad \text{and}\quad
D(\alpha) \asymp \kappa^{-1/2}\exp(\kappa/\alpha).$$ Therefore (\[eqn.17\]) holds with $C_2 = \kappa^{-1/2}$.
Proofs
======
Ordered processes and their basic properties
--------------------------------------------
The main results in this paper are based on a general fact which is similar to Dudley’s entropy bound (see, e.g., [@VW]). Let $\zeta_t$ be a separable zero mean random process on $ \mathbb{R}^+$. Denote for brevity $$\Delta^\zeta(t_1,t_2)=\zeta_{t_1}-\zeta_{t_2}.$$ The following fact (see Lemma 1 in [@G1]) plays a cornerstone role in the proof of Proposition \[proposition.1\] and Theorem \[theorem.1\].
\[proposition.2\] Let $v^2_u,\ u\in \mathbb{R}^+$, be a continuous strictly increasing function with $v^2_0=0$. Then for any $\lambda>0$, $$\begin{split}
\log \mathbf{E}\exp\biggl\{\lambda \max_{0\le s\le
t}\frac{\Delta^\zeta(s,t)}{\sigma_t}\biggr\}\le \frac{\log(2)\sqrt{2}}{\sqrt{2}-1}\qquad \qquad\\
\qquad + \max_{0< s'< s\le t}\max_{| z|\le \sqrt{2}/(\sqrt{2}-1)}\log
\mathbf{E}\exp\biggl\{z\lambda
\frac{\Delta^\zeta(s',s)}{\bar{\Delta}^v(s',s)}\biggr\},
\end{split}$$ where $
\bar{\Delta}^v(s',s)=\sqrt{|v^2_s-v^2_{s'}|}.
$
A zero mean process $\zeta_t,\ t\in \mathbb{R}^+$ is called ordered if there exists a continuous strictly monotone function $v^2_t,\ t\in \mathbb{R}^+$ and some $\Lambda > 0$ such that $$\sup_{s',s\in \mathbb{R}^+:\,s' \neq s}\mathbf{E}\exp\biggl[ \Lambda \frac{\Delta^\zeta(s',s)}{\bar{\Delta}^v(s',s)}\biggr]<\infty.$$
The next two propositions (see Lemmas 2 and 3 in [@G1]) show that the ordered process $\zeta_t$ can be controlled by the deterministic function $v_t$.
\[proposition.3\] Let $\zeta_t$ be an ordered process with $\zeta_0=0$. Then there exists a constant $C(q',q)$ such that for all $1< q',q\le 2$, uniformly in $\free>0$ $$\mathbf{E}\sup_{t\ge 0}\bigl[\zeta_t-\free v^q_t\bigr]_+^{q'} \le
\frac{C(q',q)}{\free^{q'/(q-1)}},$$ where $[x]_+=\max(0,x)$.
\[proposition.4\] Assume that there exists a monotone function $v_t, t\ge 0$ such that a random process $\zeta_t, \ t\in \mathbb{R}^+$, satisfies $$\mathbf{E}\sup_{t\ge 0}\bigl[\zeta_t-\free v^q_t\bigr]_+^{q'} \le
\frac{C}{\free^{q'/(q-1)}},$$ for any $\free>0$ and some $ q'\ge 1$, $q> 1 $. Then there exists a constant $C'$ such that for any random variable $\tau\in \mathbb{R}^+$ the following inequality holds $$\bigl[\mathbf{E}|\zeta_\tau|^{q'}\bigr]^{1/q'}\le C'\bigl[\mathbf{E}v^{q q'}_\tau\bigr]^{1/(q q')}.$$
In what follows, we focus on typical ordered processes related to the empirical risk. The following two propositions (see Lemmas 4 and 5 in [@G1]) are essential in controlling the cross term $$\sigma \sum_{k=1}^p \lambda^{-1/2}(k)[2h_\alpha(k)-h_\alpha^2(k)]\xi'(k)\beta(k)$$ in the case, where $\alpha$ is a random variable depending on $\xi'(k),\, k=1,\ldots,p$.
\[proposition.5\] For any given $\bar\alpha> 0$ and any $\free>0$, $$\begin{split}
&\mathbf{E}\sup_{0\le \alpha\le \alpha^\circ}\biggl\{\sum_{k=1}^p
\bigl[h_{\bar\alpha}(k)-h_\alpha(k)\bigr] b(k) \xi'(k)
\\&\qquad-\free\biggl[\sum_{k=1}^p
\bigl[h_{\bar\alpha}(k)-h_\alpha(k)\bigr]^2 b^2(k)\biggr]^{q'}\biggr\}_+^{q}
\le \frac{C}{\free^{q/(2q'-1)}}, \quad q'>1/2.
\end{split}$$
\[proposition.6\] Let $\bar \alpha$ be a given smoothing parameter. Then for any $p\in [1,2)$, there exists a constant $C(p)$ so that for any data-driven smoothing parameter $\hat\alpha$, $$\begin{split}
&\mathbf{E} \biggl| \sum_{k=1}^p
\bigl[h_{\hat\alpha}(k)-h_{\bar\alpha}(k)\bigr]\lambda^{-1/2}(k) \beta(k)
\xi'(k)\biggr|^p\\&\quad \le C(p)
\Bigl\{\mathbf{E}\max_{k} \lambda^{-1}(k)
h_{\hat\alpha}^2(k)\Bigr\}^{p/2}\biggl[ \sum_{k=1}^p
\bigr[1-h_{\bar\alpha}(k)\bigl]^2\beta^2(k)\biggr]^{p/2}
\\&\qquad + C(p)\Bigl\{\max_{k} \lambda^{-1}(k)
h_{\bar\alpha}^2(k)\Bigr\}^{p/2}\biggl[\mathbf{E} \sum_{k=1}^p
\bigr[1-h_{\hat\alpha}(k)\bigl]^2\beta^2(k)\biggr]^{p/2}.
\end{split}$$
In order to obtain oracle inequalities in the case, where the noise variance is unknown, we will need the following lemma generalizing Proposition \[proposition.3\].
\[lemma.1\] Let $$\zeta_\alpha(b) =\sum_{k=1}^p [1-h_\alpha(k)]\xi'(k)b(k),\
v^2_\alpha(b) = \sum_{k=1}^p [1-h_\alpha(k)]^2 b^2(k), \
K=\frac{2}{(\sqrt{2}-1)^2}.$$ Then uniformly in $b\in \mathbb{R}^p$ $$\mathbf{E}\exp\bigl\{ \sup_{\alpha\in \mathbb{R}^+}\bigl[\zeta_\alpha(b)-Kv_\alpha^2(b)\bigr] \bigr\} \le C.$$
Since $h_\alpha(\cdot), \alpha \ge 0$, is the family of ordered functions, it is not difficult to check that $$\label{eqn.21}
\mathbf{E}[\zeta_{\alpha'}(b)-\zeta_\alpha(b)]^2 \le |v^2_{\alpha'}(b)-v^2_{\alpha}(b)|.$$ Indeed, we can rewrite (\[eqn.21\]) in the following equivalent form $$\mathbf{E}\zeta_{\alpha'}(b)\zeta_\alpha(b)\ge \min\Bigl\{
v^2_{\alpha'}(b),v^2_{\alpha}(b) \Bigr\}.$$ Assume for definiteness that $h_\alpha(k)\ge h_{\alpha'}(k),\, k=1,2,\ldots,p$. Then $1-h_\alpha(k)\le 1-h_{\alpha'}(k),\, k=1,2,\ldots,p$, and we get $$\begin{split}
\mathbf{E}\zeta_{\alpha'}(b)\zeta_\alpha(b) = \sum_{k=1}^p [1-h_\alpha(k)] [1-h_{\alpha'}(k)] b^2(k)\\
\ge \sum_{k=1}^p [1-h_{\alpha}(k)]^2 b^2(k)
=v_{\alpha}^2(b),
\end{split}$$ thus proving (\[eqn.21\]).
Since $\zeta_\alpha(b)$ is a Gaussian process, we obtain by (\[eqn.21\]) $$\label{eqn.22}
\log \mathbf{E}\exp\biggl\{\lambda \frac{\zeta_{\alpha'}(b)-\zeta_\alpha(b)}{\sqrt{|v^2_{\alpha'}(b)-v^2_{\alpha}(b)|}}\biggr\}\le \frac{\lambda^2}{2}.$$
We may assume without loss of generality that $\sigma_\alpha$ is a continuous function in $\alpha\in \mathbb{R}^+$. Then let us fix some $\epsilon>0$ and define $\alpha_l\in \mathbb{R}^+$ as roots of equations $$v^2_{\alpha_l}(b)=(1+\epsilon)^{l}, \ l\ge 0.$$ Since $v_\alpha^2(b)\le \sum_{k=1}^p b^2(k)$, the set of $\alpha_l$ is always finite but it may be empty.
Let $\alpha^*$ be a root of the equation $v_{\alpha^*}^2(b)=1$. Then by Proposition \[proposition.2\] and (\[eqn.22\]) we obtain $$\begin{split}
&\mathbf{E}\exp\Bigl\{\max_{\alpha\in \mathbb{R}^+}\bigl[\zeta_\alpha(b) -Kv_\alpha^2(b)\bigr]\Bigr\}\\&\quad \le \mathbf{E}\exp\Bigl\{\max_{\alpha>\alpha^* }\bigl[\zeta_\alpha(b) -Kv_\alpha^2(b)\bigr]\Bigr\} +\mathbf{E}\exp\Bigl\{\max_{\alpha\le \alpha^*}\bigl[\zeta_\alpha(b) -Kv_\alpha^2(b)\bigr]\Bigr\}
\\ &\quad \le \mathbf{E}\exp\Bigl\{\max_{\alpha\le \alpha^*}\zeta_\alpha(b) \Bigr\}+\sum_{l\ge 0}
\mathbf{E}\exp\Bigl\{\max_{\alpha_{l}<\alpha\le \alpha_{l+1}}\bigl[\zeta_\alpha(b) -Kv^2_{\alpha_{l-1}}(b)\bigr]\Bigr\}\\
&\quad\le C+\sum_{l\ge 0}
\mathbf{E}\exp\biggl\{\max_{0<\alpha\le \alpha_{l}}\biggl[v_{\alpha_l}(b)\frac{\zeta_\alpha(b)} {v_{\alpha_l}(b)}
-Kv^2_{\alpha_{l-1}}(b)\biggr]\biggr\}\\
&\quad \le C+C \sum_{l\ge 0}
\exp\biggl\{\biggl[\frac{v_{\alpha_l}^2(b)} {(\sqrt{2}-1)^2}
-Kv^2_{\alpha_{l-1}}(b)\biggr]\biggr\}\\
&\quad \le C+C\sum_{l\ge 0}
\exp\biggl\{\biggl[(1+\epsilon)^{l}\biggl(\frac{1} {(\sqrt{2}-1)^2}
-\frac{K}{1+\epsilon}\biggr)\biggr]\biggr\}\\
&\quad = C+C\sum_{l\ge 0}
\exp\biggl\{-(1+\epsilon)^{l-1}\frac{1-\epsilon} {(\sqrt{2}-1)^2}
\biggr\}.
\end{split}$$ This equation with $\epsilon =0.5$ completes the proof.
Recovering the noise variance
-----------------------------
In this section, we focus on basic probabilistic properties of the variance estimator $$\hat\sigma^2_{\hat\alpha} =\frac{\|Y-X\hat\beta_{\hat\alpha}\|^2}{\|1-H_{\hat\alpha}\|^2},$$ in the case, where $\hat\alpha$ is a data-driven smoothing parameter. We begin with a simple auxiliary fact.
\[lemma.2\] Let $\eta'$ and $\eta$ be nonnegative random variables. Then the following inequality $$\begin{aligned}
\label{eqn.23}
\begin{split}
\mathbf{E}\eta'\eta^q \le\, &\frac{ 2^{q-1} \lambda^q }{(2-q)^q} \mathbf{E}\eta' \log^q
\biggl\{1+\frac{\eta'}{\mathbf{E}\eta'}\biggr\}\\
& {}+\frac{ 2^{q-1} \lambda^q}{(2-q)^q} \mathbf{E}\eta' \log^q \biggl\{1+\mathbf{E}
\biggl[\exp\biggl(\frac{\eta}{\lambda}\biggr)
-\frac{\eta}{\lambda}-1\biggr]\biggr\}
+q \lambda^q \mathbf{E}\eta'
\end{split}\end{aligned}$$ holds for any $\lambda>0$ and $ q\in (1,2)$.
Consider the following function $$F(z,y)\stackrel{\rm def}{=}\max_{x\ge 0}\Bigl\{x^qy-z[\exp(x)-1-x]\Bigr\}.$$ Differentiating $x^qy-z[\exp(x)-1-x]$ in $x$, it is easy to check that $$F(z,y)=x_*^qy-z[\exp(x_*)-1-x_*]\le x_*^qy,$$ where $x_*$ is a root of the equation $$x_*=\log\biggl(1+\frac{qy x_*^{q-1}}{z}\biggr).$$ Since $\log(x)$ is convex, it is clear $$\log\biggl(1+\frac{qy x_*^{q-1}}{z}\biggr)\le
\log \biggl(1+\frac{qy}{z}\biggr)+\biggl(1+\frac{qy}{z}\biggr)^{-1}
\frac{q(q-1)y}{z}(x^*-1).$$ Therefore $x_*\le x^*$, where $x^*$ is a root of the following linear equation $$x^*= \log \biggl(1+\frac{qy}{z}\biggr)+\biggl(1+\frac{qy}{z}\biggr)^{-1}
\frac{q(q-1)y}{z}(x^*-1).$$ Since $q>1$, with a little algebra we get $$x^*\le \biggl(1+\frac{qy}{z}\biggr)\biggl[1+\frac{q(2-q)y}{z}\biggr]^{-1}
\log \biggl(1+\frac{qy}{z}\biggr)\le \frac{1}{2-q} \log \biggl(1+\frac{qy}{z}\biggr),$$ thus arriving at the following upper bound $$F(z,y)\le \frac{y}{(2-q)^q} \log^q \biggl(1+\frac{qy}{z}\biggr).$$
Now we are in a position to finish the proof. Notice that for any $\lambda >0$ $$\begin{split}
&\eta' \biggl(\frac{\eta}{\lambda}\biggr)^q -z\biggl[\exp\biggl(\frac{\eta}{\lambda}\biggr)
-1-\frac{\eta}{\lambda}\biggr]\\ &\quad \le \max_{x\ge 0}\biggl\{\eta' \biggl(\frac{x}{\lambda}\biggr)^q -z\biggl[\exp\biggl(\frac{x}{\lambda}\biggr)-1-\frac{x}{\lambda}\biggr]\biggr\}=F(z,\eta'),
\end{split}$$ and therefore $$\begin{split}
\mathbf{E}\eta'\eta^q \le\,& \lambda^q \biggl\{\mathbf{E}F(z,\eta')
+z\mathbf{E}\biggl[\exp\biggl(\frac{\eta}{\lambda}\biggr)
-1-\frac{\eta}{\lambda}\biggr]\biggr\}
\\ \le \,&\lambda^q \biggl\{\frac{1}{(2-q)^q}\mathbf{E}\eta'
\log^q \biggl(1+\frac{q\eta'}{z}\biggr)+
z\mathbf{E}\biggl[\exp\biggl(\frac{\eta}{\lambda}\biggr)
-1-\frac{\eta}{\lambda}\biggr]\biggr\}.
\end{split}$$ Next, substituting in the above equation $$z=q \mathbf{E} \eta' \biggr\{ \mathbf{E}\biggl[\exp\biggl(\frac{\eta}{\lambda}\biggr)
-1-\frac{\eta}{\lambda}\biggr]\biggr\}^{-1},$$ we obtain $$\label{eqn.24}
\begin{split}
\mathbf{E}\eta'\eta^q \le \lambda^q \biggl\{\frac{1}{(2-q)^q}\mathbf{E}\eta'
\log^q \biggl(1+\frac{\eta'}{\mathbf{E}\eta'}\mathbf{E}\biggl[\exp\biggl(\frac{\eta}{\lambda}\biggr)
-1-\frac{\eta}{\lambda}\biggr]\biggr)+
q\mathbf{E}\eta'\biggr\}.
\end{split}$$ Finally, applying the following inequality $$\begin{split}
\log^q(1+xy)\le [\log(1+y)+\log(1+x)]^q \le 2^{q-1}\log^{q
}(1+x)\\{}+2^{q-1}\log^q(1+y),\ x,y>0,
\end{split}$$ we get $$\begin{split}
\mathbf{E}\eta'
\log^q \biggl\{1+\frac{\eta'}{\mathbf{E}\eta'}\mathbf{E}\biggl[\exp\biggl(\frac{\eta}{\lambda}\biggr)
-1-\frac{\eta}{\lambda}\biggr]\biggr\}
\quad \le 2^{q-1}\mathbf{E}\eta'
\log^q \biggl(1+\frac{\eta'}{\mathbf{E}\eta'}\biggr)\\{}+
2^{q-1}
\log^q \biggl\{1+\mathbf{E}\biggl[\exp\biggl(\frac{\eta}{\lambda}\biggr)
-1-\frac{\eta}{\lambda}\biggr]\biggr\},
\end{split}$$ and combining this equation with (\[eqn.24\]), we finish the proof of (\[eqn.23\]).
\[lemma.3\] Let $\eta$ be a nonnegative sub-Gaussian random variable, i.e., such that for all $\lambda >0$ and some $S>0$ $$\label{eqn.25}
\mathbf{E}\exp(\eta/\lambda)\le C\exp(S^2/\lambda^2).$$ Then for any $q\in [1,2)$ $$\label{eqn.26}
\Bigl[\mathbf{E}\eta'^q\eta^q\Bigr]^{1/q}\le \frac{CS}{2-q}\biggl[ \mathbf{E}\eta'^q\log^q\biggl(1+\frac{\eta'^q}
{\mathbf{E}\eta'^q}\biggr)\biggr]^{1/q}.$$
Replacing $\eta'$ in (\[eqn.23\]) by $ \eta'^q$ and substituting (\[eqn.25\]) in (\[eqn.23\]), we get with $\lambda=S$ $$\begin{split}
\mathbf{E}\eta'^q\eta^q
\le \frac{ 2^{q-1} S^q }{(2-q)^q} \mathbf{E}\eta'^q \log^q \biggl[1+\frac{\eta'^q}{\mathbf{E}\eta'^q}\biggr]
+\frac{ 2^{q-1} S^q}{(2-q)^q} \mathbf{E}\eta'^q
+q S^q \mathbf{E}\eta'^q.
\end{split}$$ Let $F(x)=x\log^q(1+x)$. It is clear that $$F'(x)=\log^{q}(1+x)+\frac{qx\log^{q-1}(1+x)}{1+x}$$ is increasing in $x$ and therefore $F(x)$ is convex. Therefore by Jensen’s inequality $$\mathbf{E}\eta'^q \log^q \biggl[1+\frac{\eta'^q}{\mathbf{E}\eta'^q}\biggr]
\ge \log(2)\mathbf{E}\eta'^q,$$ and thus, we arrive at (\[eqn.26\]).
\[lemma.4\] Let $$\zeta_\alpha =\sum_{k=1}^p
\bigl[1-h_\alpha(k)\bigr]^2[1-\xi'^2(k)]$$ and $$\Sigma_\alpha= 2 \|(1-h_\alpha)^2\|\sqrt{\log\log \frac{ \|(1-h_{\alpha^{\circ}})^2\|^2\exp(2)}{ \|(1-h_\alpha)^2\|^2}}.$$ Then for any $s\in (1,2]$, $$\mathbf{P}\biggl\{\sup_{\alpha\le \alpha^{\circ}}\frac{\zeta_\alpha-s\Sigma_\alpha}{\|(1-h_\alpha)^2\|}\ge x\biggr\}\le \frac{C}{(s-1)^{3}}\exp\biggl\{-\frac{(3-s)^2x^2}{16}\biggr\}.$$
For some $\epsilon>0$ define $\alpha_k,\, k\ge 0$, as roots of equations $$\|(1-h_{\alpha_k})^2\|^2=(1+\epsilon)^{-k}\|(1-h_{\alpha^\circ})^2\|^2.$$ Then, denoting for brevity $$G_{k+1}(x)= s\Sigma_{\alpha_{k+1}}+x \|(1-h_{\alpha_{k+1}})^2\|,$$ we obtain $$\label{eqn.27}
\begin{split}
\mathbf{P}\, \biggl\{ \sup_{\alpha\le \alpha^{\circ}}\frac{\zeta_\alpha-s\Sigma_\alpha}{\|(1-h_\alpha)^2\|}\ge x \biggr\}\le \sum_{k=0}^\infty \mathbf{P}\, \biggl\{ \sup_{\alpha\in [\alpha_{k+1},\alpha_k]}\frac{\zeta_\alpha-s\Sigma_\alpha}{\|(1-h_\alpha)^2\|}\ge x \biggr\}\\ \quad \le \sum_{k=0}^\infty \mathbf{P}\, \biggl\{ \sup_{\alpha\in [\alpha_{k+1},\alpha_k]}\zeta_\alpha\ge G_{k+1}(x) \biggr\}\\ \le \sum_{k=0}^\infty \mathbf{P}\, \biggl\{ \zeta_{\alpha_{k+1}}\ge [1-f(\epsilon)]G_{k+1}(x)
\biggr\}\\ \qquad +
\sum_{k=0}^\infty \mathbf{P}\, \biggl\{ \sup_{\alpha\in [\alpha_{k+1},\alpha_k]}[\zeta_\alpha-\zeta_{\alpha_{k+1}}]\ge f(\epsilon)G_{k+1}(x) \biggr\},
\end{split}$$ where $f(\epsilon)$ will be chosen later on.
Since $\log(1+x)\ge x -x^2/2,\ x\ge 0$, then for any $\lambda>0$ $$\label{eqn.28}
\mathbf{E}\exp(\lambda \zeta_\alpha)\le \exp\bigl[\lambda^2\|(1-h_\alpha)^2\|^2\bigr],$$ and by the exponential Tchebychev inequality we get $$\label{eqn.29}
\begin{split}
\mathbf{P}\, \biggl\{ \zeta_{\alpha_{k+1}}\ge [1-f(\epsilon)]G_{k+1}(x) \biggr\} \le
\exp\biggl\{-\frac{[1-f(\epsilon)]^2 G_{k+1}^2(x)}{4\|(1-h_{\alpha_{k+1}})^2\|^2}\biggr\}\\ \quad \le \exp\biggl\{-s^2[1-f(\epsilon)]^2\log[(k+1)\log(1+\epsilon)]-\frac{[1-f(\epsilon)]^2 x^2}{4}\biggr\}.
\end{split}$$
To bound from above the last term in Equation (\[eqn.27\]), we make use of that $2h_\alpha(k)-h^2_\alpha(k)$ is a family of ordered functions, and thus (see (\[eqn.21\])) $$\|(1-h_{\alpha_{k}})^2-(1-h_{\alpha_{k+1}})^2\|^2\le \|(1-h_{\alpha_{k}})^2\|^2-\|(1-h_{\alpha_{k+1}})^2\|^2.$$ Similarly to (\[eqn.28\]) $$\mathbf{E}\exp\bigl\{\lambda [\zeta_{\alpha_{k}}-\zeta_{\alpha_{k+1}}]\bigr\}\le \exp\bigl[\lambda^2\|(1-h_{\alpha_{k}})^2-(1-h_{\alpha_{k+1}})^2\|^2\bigr].$$ Therefore with the help of Proposition \[proposition.2\] and the exponential Tchebychev inequality we obtain $$\label{eqn.30}
\begin{split}
\mathbf{P}\, \biggl\{ \sup_{\alpha_{k+1}<\alpha\le \alpha_k}[\zeta_\alpha-\zeta_{\alpha_{k+1}}]
\ge f(\epsilon) G_{k+1}(x) \biggr\}
\le \min_{\lambda>0}\exp\biggl\{-\lambda f(\epsilon) G_{k+1}(x)\\[3pt]
\qquad {}+ \frac{\lambda^2(\sqrt{2}-1)^2\|(1-h_{\alpha_{k}})^2-(1-h_{\alpha_{k+1}})^2\|^2}
{4[\|(1-h_{\alpha_{k}})^2\|^2-\|(1-h_{\alpha_{k+1}})^2\|^2]}\biggr\}\\[3pt]
\quad \le
C\exp\biggl\{-\frac{(\sqrt{2}-1)^2f^2(\epsilon) G_{k+1}^2(x)}
{8[\|(1-h_{\alpha_{k}})^2\|^2-\|(1-h_{\alpha_{k+1}})^2\|^2]}\biggr\}\\[3pt]
\quad =
C\exp\biggl\{-\frac{(\sqrt{2}-1)^2s^2 f^2(\epsilon)}{4\epsilon}\log[(k+1)\log(1+\epsilon)]\\[3pt]
\qquad \quad {}- \frac{(\sqrt{2}-1)^2x^2 f^2(\epsilon)}{8\epsilon}\biggr\}.
\end{split}$$
Now we chose $f(\epsilon)$ to balance the exponents at the right-hand sides in (\[eqn.29\]) and (\[eqn.30\]), thus arriving at following equation for this function $$\frac{(\sqrt{2}-1)^2f^2(\epsilon)}{2\epsilon}=[1-f(\epsilon)]^2.$$ This yields $$f(\epsilon)=\frac{\sqrt{2\epsilon}}{\sqrt{2}-1+\sqrt{2\epsilon}}
.$$ With this $f(\epsilon)$ and with (\[eqn.27\]–\[eqn.30\]) we get $$\label{equ.21}
\begin{split}
\mathbf{P}\, \biggl\{ \sup_{\alpha\le \alpha^{\circ}}\frac{\zeta_\alpha-s\Sigma_\alpha}{\|(1-h_\alpha)^2\|} \ge x \biggr\}\le& \frac{C\exp\{-[1-f(\epsilon)]^2x^2/4\}}{\epsilon^{s^2[1-f(\epsilon)]^2}\bigl\{s^2[1-f(\epsilon)]^2-1\bigr\}_+}.
\end{split}$$
Finally, choosing $\epsilon $ as a root of $
f(\epsilon)=(s-1)/2
$, we finish the proof.
We summarize the main properties of the variance estimator in the following lemma.
\[lemma.5\] For any $q \in (1,2)$ $$\mathbf{E}\Bigl\{[\sigma^2-\hat\sigma_{\hat\alpha}^2]_+\sigma^{-2}
Pen(\hat\alpha)\Bigr\}^q \le C(2-q)^{-q}\Ps^q(\alpha_\circ,\alpha^\circ)
\mathbf{E}\bigl[Pen(\hat{\alpha})\bigr]^q .$$
By (\[eqn.8\]) we obtain $$\label{eqn.31}
\begin{split}
\sigma^2-\hat\sigma^2_{\hat\alpha}=\frac{\sigma^2}{\|1-h_{\hat\alpha}\|^2}\sum_{k=1}^p \bigl[1-h_{\hat\alpha}(k)\bigr]^2[1-\xi'^2(k)]\\
{}-\frac{2\sigma}{\|1-h_{\hat\alpha}\|^2}\sum_{k=1}^p \bigl[1-h_{\hat\alpha}(k)\bigr]^2\xi'(k)\beta(k)\sqrt{\lambda(k)}\\
{}-\frac{1}{\|1-h_{\hat\alpha}\|^2}\sum_{k=1}^p \bigl[1-h_{\hat\alpha}(k)\bigr]^2\beta^2(k)\lambda(k).
\end{split}$$
The first term at the right-hand side in (\[eqn.31\]) is controlled with the help of Lemmas \[lemma.3\] and \[lemma.4\] (with $s=2$) as follows: $$\label{eqn.32}
\begin{split}
\biggl\{\mathbf{E}\biggl|\frac{Pen(\hat{\alpha})}{\|1-h_{\hat\alpha}\|^2}\sum_{k=1}^p \bigl[1-h_{\hat\alpha}(k)\bigr]^2[1-\xi'^2(k)]\biggr|^q\biggr\}^{1/q}\\
\quad =\biggl\{\mathbf{E}\frac{Pen^q(\hat{\alpha})}{\|1-h_{\hat\alpha}\|^q}\biggl|\frac{\zeta_{\hat{\alpha}}-2\Sigma_{\hat{\alpha}}+2\Sigma_{\hat{\alpha}}}{\|1-h_{\hat\alpha}\|}\biggr|^q\biggr\}^{1/q}
\\ \quad \le \frac{C}{\|1-h_{\alpha_\circ}\|}\biggl\{\log\log\biggl(1+\frac{\|1-h_{\alpha_\circ}\|^2}{\|1-h_{\alpha^\circ}\|^2}\biggr)\biggr\}^{1/2}
\bigl[\mathbf{E}Pen^q(\hat{\alpha})\bigr]^{1/q}
\\ \qquad {}+\frac{C}{(2-p)\|1-h_{\alpha_\circ}\|}\log \biggl[1+\frac{Pen(\alpha_\circ)}{Pen(\alpha^\circ)}\biggr]
\bigl[\mathbf{E}Pen^q(\hat{\alpha})\bigr]^{1/q}.
\end{split}$$
To control the last two terms in (\[eqn.31\]), notice that $\tilde{h}_\alpha(k)=
2h_\alpha(k)- h_\alpha^2(k),\ \alpha>0,$ is a family of ordered functions. Hence, applying Lemma \[lemma.1\] with $$b(k)=\frac{2\beta(k)\sqrt{\lambda(k)}}{K\sigma},$$ we have $$\label{eqn.33}
\begin{split}
\mathbf{E}\exp\biggl\{\frac{2}{K\sigma^{2}} \biggl[2\sigma \sum_{k=1}^p \bigl[1-\tilde{h}_{\hat\alpha}(k)\bigr]\xi'(k)\beta(k)\sqrt{\lambda(k)}
\\ - \sum_{k=1}^p \bigl[1-\tilde{h}_{\hat\alpha}(k)\bigr]^2\beta^2(k)\lambda(k)
\biggr]\biggr\}\le C .
\end{split}$$ This inequality and Lemma \[lemma.2\] with $$\begin{split}
\eta'&=Pen^q(\alpha)\\
\eta&=\frac{2}{K\sigma^{2}} \biggl[2\sigma \sum_{k=1}^p \bigl[1-\tilde{h}_{\hat\alpha}(k)\bigr]\xi'(k)\beta(k)\sqrt{\lambda(k)}
- \sum_{k=1}^p \bigl[1-\tilde{h}_{\hat\alpha}(k)\bigr]^2\beta^2(k)\lambda(k)
\biggr],
\end{split}$$ yield $$\label{eqn.34}
\begin{split}
\mathbf{E}\biggl[\frac{2\sigma}{\|1-h_{\hat\alpha}\|^2}\sum_{k=1}^p \bigl[1-h_{\hat\alpha}(k)\bigr]^2\xi'(k)\beta(k)\sqrt{\lambda(k)}\\
\qquad {}-\frac{1}{\|1-h_{\hat\alpha}\|^2}\sum_{k=1}^p \bigl[1-h_{\hat\alpha}(k)\bigr]^2\beta^2(k)\lambda(k)\biggr]^q
Pen^q(\hat{\alpha})\\
\quad \le \frac{C\sigma^{2q}}{(2-q)^q\|1-h_{\alpha_\circ}\|^{2q}}\log^q \biggl[1+\frac{Pen(\alpha_\circ)}{Pen(\alpha^\circ)}\biggr]\mathbf{E}Pen^q(\hat{\alpha}).
\end{split}$$
Finally, combining (\[eqn.31\]), (\[eqn.32\]), and (\[eqn.34\]) and using Jensen’s inequality, we finish the proof.
Proof of Theorem \[theorem.1\]
------------------------------
The following proposition (see Lemma 7 in [@G1]) summarizes some basic properties of the penalty defined by (\[eqn.13\]–\[eqn.15\]).
\[proposition.7\] $$\begin{aligned}
\nonumber
{Q^+(\alpha)}\ge D(\alpha)\max\biggl\{\sqrt{\log \frac{D(\alpha)}{D(\alpha^{\circ})}},\frac{1}{\mu_\alpha} \log \frac{D(\alpha)}{D(\alpha^{\circ})}\biggr\},
\\
\nonumber
\mu_\alpha \ge \min\biggl\{\frac{1}{2}\sqrt{\log \frac{D(\alpha)}{D(\alpha^{\circ})}}, \frac14\biggr\},\end{aligned}$$ If ${D(\alpha)}\ge \exp(2)D(\alpha^{\circ})$, then $$\begin{aligned}
\nonumber
&{D(\alpha)}\ge {\mu_\alpha Q^+(\alpha)}\biggl[\log \frac{\mu_\alpha Q^+(\alpha)}{D(\alpha^{\circ})}\biggr]^{-1}.\end{aligned}$$ For any $\alpha_1 \le \alpha_2$ $$\begin{aligned}
\nonumber
&\frac{D(\alpha_1)}{D(\alpha_2)}\le \frac{Q^+(\alpha_1)}{Q^+(\alpha_2)}.\end{aligned}$$
We begin the proof of Theorem \[theorem.1\] with a simple generalization of Proposition \[proposition.3\]. Consider the following random process $$\eta_\alpha^\epsilon =\sum_{k=1}^p \lambda^{-1}(k)h_\alpha^\epsilon(k)[\xi'^2(k)-1],$$ where $h_\alpha^\epsilon(k)=[2(1+\epsilon) h_{\alpha}(k)-\epsilon h_{\alpha}^2(k)]/(2+\epsilon)$.
\[lemma.6\] Let $q\in (1,2]$. Then for any random variable $\hat\alpha\le \alpha^\circ$ $$\mathbf{E}\eta_{\hat \alpha}^\epsilon \le \frac{C \sigma_{\alpha^\circ}^\epsilon}{\sqrt{q-1}} \biggl[\mathbf{E}\biggl(\frac{\sigma_{\hat\alpha}^\epsilon}
{\sigma_{\alpha^\circ}^\epsilon}\biggr)^{q}
\biggr]^{1/q},$$ where $$\sigma_\alpha^\epsilon=\biggl\{2\sum_{k=1}^p \lambda^{-2}(k)[h_\alpha^\epsilon(k)]^2
\biggr\}^{1/2}.$$
It is based on the following fact. Let $S(x)=x^{1/(q-1)}, \ x \in \mathbb{R}^+$. Then $$\label{equ.26}
\begin{split}
\rho(z)\,\stackrel{\rm def}{=}\, \mathbf{E}\sup_{\alpha\le \alpha^\circ}\biggl\{\eta_\alpha^\epsilon-z\sigma_{\alpha}^\epsilon S^{-1}\biggl(\frac{\sigma_\alpha^\epsilon}
{\sigma_{\alpha^\circ}^\epsilon}\biggr)
\biggr\}_+ \\ \le C\sigma_{\alpha^\circ}^\epsilon \int_0^\infty xS\biggl(\frac{x}{z}\biggr){\rm e}^{-Cx^2}\, dx,
\end{split}$$ where $S^{-1}(x)=x^{q-1}$ denotes the inverse function to $S(x)$.
To prove this inequality, define $\alpha_k, \ k=0,1,2,\ldots$ as roots of the following equations $${\sigma_{\alpha_k}^\epsilon}=\sigma_{\alpha^\circ}^\epsilon S \bigl(1/z\bigr){\rm e}^k.$$ Then, noticing that $\eta_\alpha^\epsilon-\eta_{\alpha_k}^\epsilon$ is an ordered process, we obtain by (\[eqn.12\]) and Proposition \[proposition.2\] $$\begin{split}
\rho(z) \le \mathbf{E} \sup_{\alpha_0\le \alpha\le \alpha^\circ}|\eta_\alpha^\epsilon|
+ \sum_{k=2}^\infty \mathbf{E}\sup_{\alpha_{k}\le \alpha < \alpha_{k-1} }\biggl\{\eta_\alpha^\epsilon-z \sigma_{\alpha_{k-1}}^\epsilon S^{-1}\biggl(\frac{\sigma_{\alpha_{k-1}}^\epsilon}
{\sigma_{\alpha^\circ}^\epsilon}
\biggr)\biggr]\biggr\}_+\\ \le C\sigma_{\alpha^\circ}^\epsilon S \biggl(\frac1z\biggr) \sum_{k=0}^\infty
{\rm e}^k \exp\Bigl\{-C\bigl[zS^{-1}\bigl(S \bigl(z^{-1}\bigr){\rm e}^k\bigl)\bigr]^2\Bigr\} \\
\le C\sigma_{\alpha^\circ}^\epsilon\int_{0}^\infty\exp\Bigl\{-C\bigl[z S^{-1}(u)\bigr]^2\bigr\}
\, du= C\sigma_{\alpha^\circ}^\epsilon \int_{0}^\infty{\rm e}^{-Cz^2 v^2}\, d S(v)
\\ \le C\sigma_{\alpha^\circ}^\epsilon z^2\int_{0}^\infty S(v)v {\rm e}^{-Cz^2v^2}\, dv
=C\sigma_{\alpha^\circ}^\epsilon \int_0^\infty x S \biggl(\frac{x}{z}\biggr){\rm e}^{-Cx^2}\, dx.
\end{split}$$
Next we get by the Laplace method $$\label{eqn.35}
\begin{split}
\rho(z)= C^{q/(q-1)} S\biggl(\frac1z\biggr)
\int_0^\infty x^{q/(q-1)}{\rm e}^{-x^2/2}\, dx\\
\le C^{q/(q-1)} \biggl(\frac1z\biggr)^{1/(q-1)}\exp\biggl[\frac{q}{2(q-1)}\log \frac{q}{q-1}\biggr].
\end{split}$$
To finish the proof, denote for brevity $$E=\mathbf{E}\biggl(\frac{\sigma_{\hat \alpha}^\epsilon}{\sigma_{\alpha^\circ}^\epsilon}\biggr)^q.$$ Then by (\[eqn.35\]) we obtain with a simple algebra $$\begin{split}
\mathbf{E}\eta_\alpha^\epsilon
\le \min_{z}\biggl\{z \mathbf{E}\sigma_{\hat{\alpha}}^\epsilon \sigma_{\alpha^\circ}S^{-1}
\biggl(\frac{\sigma_{\hat\alpha}^\epsilon}
{\sigma_{\alpha^\circ}^\epsilon}\biggr)+ S\biggl(\frac{C}{z}\biggr)\exp\biggl[\frac{q}{2(q-1)}\log \frac{q}{q-1}\biggr]\biggr\}\\
\\
\le C\sigma_{\alpha^\circ}^\epsilon \min_{z}\biggl\{z E+ \biggl(\frac{C}{z}\biggr)^{1/(q-1)}\exp\biggl[\frac{q}{2(q-1)}\log \frac{q}{q-1}\biggr]\biggr\}\\ \le\frac{C}{\sqrt{q-1}}
\sigma_{\alpha^\circ}^\epsilon E^{1/q}.\\[-12pt] %\quad \blacksquare
\end{split}
%\vspace*{-12pt}$$
The following important lemma provides an upper bound for $L_{\hat\alpha}(\beta)+(1+\gamma)Q^+(\hat\alpha)$.
\[lemma.7\]For any data-driven $\hat\alpha$ and any given $\bar\alpha\in [\alpha_\circ,\alpha^\circ]$, the following inequality $$\begin{split}
&\Bigl\{\mathbf{E}\bigl[\sigma^{-2}L_{\hat\alpha}(\beta)+(1+\gamma)
Q^+(\hat\alpha)\bigr]^{1+\gamma/4}\Bigr\}^{1/(1+\gamma/4)}\\&\quad \le \frac{C}{[1-C \Ps(\alpha_\circ,\alpha^\circ)/\gamma]_+} \biggl[\frac{\bar{R}_{\bar\alpha}(\beta)}{\gamma \sigma^2}
+\frac{D(\alpha^\circ)}{\gamma^4}\biggr]
\end{split}$$ holds uniformly in $\beta\in \mathbb{R}^p$ and $\gamma\in (0,1/{4})$.
In view of the definition of $\hat\alpha$, for any given smoothing parameter $\bar\alpha$, $R_{\hat\alpha}[Y,Pen]\le R_{\bar\alpha}[Y,Pen]$. It is easy to check with the help of (\[eqn.8\]) that this inequality is equivalent to the following one $$\label{eqn.36}
\begin{split}
L_{\hat\alpha}(\beta)+(1+\gamma)
\sigma^2Q^+(\hat\alpha)
-\sigma^2 \sum_{k=1}^p \lambda^{-1}(k)\tilde{h}_{\hat\alpha}(k)[\xi'^2(k)-1]\\
\qquad {}+2\sigma \sum_{k=1}^p \lambda^{-1/2}(k)[1-h_{\hat\alpha}(k)]^2\xi'(k)\beta(k) +[\hat\sigma_{\hat\alpha}^2-\sigma^2]Pen(\hat\alpha)
\\ \quad \le
L_{\bar\alpha}(\beta)+(1+\gamma)
\sigma^2Q^+(\bar\alpha)
-\sigma^2 \sum_{k=1}^p \lambda^{-1}(k)\tilde{h}_{\bar\alpha}(k)[\xi'^2(k)-1]\\
\qquad {}+2\sigma \sum_{k=1}^p \lambda^{-1/2}(k)[1-h_{\bar\alpha}(k)]^2\xi'(k)\beta(k) +[\hat\sigma_{\bar\alpha}^2-\sigma^2]Pen(\bar\alpha),
\end{split}$$ where $
\tilde{h}_\alpha(k)=2h_\alpha(k)-h^2_\alpha(k)
$. We can rewrite (\[eqn.36\]) as follows: $$\label{eqn.37}
\begin{split}
\frac{\gamma}{2}\bigl[L_{\hat\alpha}(\beta)+(1+\gamma)
\sigma^2Q^+(\hat\alpha)\bigr]\le
L_{\bar\alpha}(\beta)+(1+\gamma)
\sigma^2Q^+(\bar\alpha)
\\\quad {}-\sigma^2 \sum_{k=1}^p \lambda^{-1}(k)\tilde{h}_{\bar\alpha}(k)[\xi'^2(k)-1]\\
\quad {}+\sigma^2 \sum_{k=1}^p \lambda^{-1}(k)\tilde{h}_{\hat\alpha}(k)[\xi'^2(k)-1]
-\bigg(1+\frac{\gamma}{2}-\frac{\gamma^2}{2}\biggr)\sigma^2Q^+(\hat\alpha)\\
\quad {}+2\sigma \sum_{k=1}^p \lambda^{-1/2}(k)[\tilde{h}_{\hat\alpha}(k)-\tilde{h}_{\bar\alpha}(k)]
\xi'(k)\beta(k)-\biggl(1-\frac\gamma2\biggr)L_{\hat\alpha}(\beta) \\ \quad+[\hat\sigma_{\bar\alpha}^2-\sigma^2]_+Pen(\bar\alpha)
+[\sigma^2-\hat\sigma_{\hat\alpha}^2]_+Pen(\hat\alpha).
\end{split}$$
Since $\bar\alpha$ is given, we get by Jensen’s inequality $$\label{eqn.38}
\begin{split}
\mathbf{E}\biggl|\sum_{k=1}^p \lambda^{-1}(k)\tilde{h}_{\bar\alpha}(k)[\xi'^2(k)-1]\biggr|^{1+\gamma/4}\le
C\biggl\{\sum_{k=1}^p \lambda^{-2}(k)\tilde{h}_{\bar\alpha}^2(k)\biggr\}^{1/2+\gamma/8}\\
\le C\bigl[D(\bar\alpha)\bigr]^{1+\gamma/4}
\le C\bigl[\sigma^{-2}\bar{R}_{\bar\alpha}(\beta)\bigr]^{1+\gamma/4}.
\end{split}$$
The third line in (\[eqn.37\]) is bounded by Proposition \[proposition.1\] as follows: $$\label{eqn.39}
\begin{split}
\mathbf{E}\biggl[\sum_{k=1}^p \lambda^{-1}(k)\tilde{h}_{\hat\alpha}(k)[\xi'^2(k)-1]
-\bigg(1+\frac{\gamma}{2}-\frac{\gamma^2}{2}\biggr)Q^+(\hat\alpha)\biggr]^{
1+\gamma/4}_+\\
\le\frac{CD^{1+\gamma/4}(\alpha^\circ)}{\gamma^3},
\end{split}$$ where $\gamma\le 1/\sqrt{2}$.
The upper bound for the fourth line in (\[eqn.37\]) is a little bit more tricky. Since $\bigl\{\tilde{h}_{\alpha}(\cdot), \alpha\in (0,\alpha^{\circ}]\bigr\}$ is a family of ordered functions, we obtain by Proposition \[proposition.5\] that for any $\epsilon>0$ and given $q'>1/2$, $$\label{eqn.40}
\begin{split}
&\mathbf{E}\biggl|2\sigma\sum_{k=1}^p \lambda^{-1/2}(k)\bigl[\tilde{h}_{\hat\alpha}(k)-\tilde{h}_{\bar\alpha}(k)
\bigr]\xi'(k)\beta(k)\\&\quad - \epsilon\biggl\{\sigma^2\sum_{k=1}^p \lambda^{-1}(k)\bigl[\tilde{h}_{\hat\alpha}(k)-\tilde{h}_{\bar\alpha}(k)
\bigr]^2\beta^2(k)\biggr\}^{q'}\biggr|^{q} \le \frac{1}{(C\epsilon)^{q/(2q'-1)}}.
\end{split}$$
To continue this inequality, notice that if $\hat\alpha\ge \bar \alpha$, then $$\frac{\tilde{h}_{\hat\alpha}(k)}{\tilde{h}_{\bar\alpha}(k)}\le 1, \quad \frac{\tilde{h}_{\hat\alpha}(k)}{\tilde{h}_{\bar\alpha}(k)}\ge \tilde{h}_{\hat\alpha}(k)$$ and therefore $$\label{eqn.41}
\begin{split}
\sum_{k=1}^\infty
\bigl[\tilde{h}_{\bar\alpha}(k)-\tilde{h}_{\hat\alpha}(k)\bigr]^2\frac{\beta^2(k)}{\lambda(k)}
=\sum_{k=1}^\infty
\tilde{h}^2_{\bar\alpha}(k)
\biggl[1-\frac{\tilde{h}_{\hat\alpha}(k)}{\tilde{h}_{\bar\alpha}(k)}\biggr]^2
\frac{\beta^2(k)}{\lambda(k)}
\\ \qquad
\le \max_s \frac{\tilde{h}_{\bar\alpha}^2(s)}{\lambda(s)} \sum_{k=1}^\infty
\bigl[1-\tilde{h}_{\hat\alpha}(k)\bigr]^2\beta^2(k)
\\ \qquad
\le \max_s \frac{4{h}_{\bar\alpha}^2(s)}{\lambda(s)} \sum_{k=1}^\infty
\bigl[1-{h}_{\hat\alpha}(k)\bigr]^2\beta^2(k).
\end{split}$$ Similarly, if $\hat\alpha < \bar\alpha$, then $$\label{eqn.42}
\begin{split}
\sum_{k=1}^\infty
\bigl[\tilde{h}_{\bar\alpha}(k)-\tilde{h}_{\hat\alpha}(k)\bigr]^2
\frac{\beta^2(k)}{\lambda(k)}
\le \max_s \frac{ \tilde{h}_{\hat\alpha}^2(s) }{\lambda(s)} \sum_{k=1}^\infty
\bigl[1-\tilde{h}_{\bar\alpha}(k)\bigr]^2\beta^2(k)\\
\quad\le \max_s \frac{4 {h}_{\hat \alpha}^2(s)}{\lambda(s)} \sum_{k=1}^\infty
\bigl[1-{h}_{\bar\alpha}(k)\bigr]^2\beta^2(k).
\end{split}$$
So, combining (\[eqn.41\]–\[eqn.42\]) with Young’s inequality $$yx^s-x \le (1-s)s^{s/(1-s)}y^{1/(1-s)}, \quad
x,y\ge 0,\ s<1,$$ gives $$\label{eqn.43}
\begin{split}
\epsilon\biggl[\sigma^2\sum_{k=1}^\infty
\bigl[\tilde{h}_{\bar\alpha}(k)-\tilde{h}_{\hat\alpha}(k)\bigr]^2\lambda^{-1}(k)\beta^2(k)\biggr]^{q'}-\biggl(1-\frac{\gamma}{2}\biggr)L_{\hat\alpha}(\beta)\\
\quad \le\biggl(1-\frac{\gamma}{2}\biggr)^{-1}\epsilon\biggl[\sigma^2\sum_{k=1}^\infty
\bigl[\tilde{h}_{\bar\alpha}(k)-\tilde{h}_{\hat\alpha}(k)\bigr]^2\lambda^{-1}(k)\beta^2(k)\biggr]^{q'}-L_{\hat\alpha}(\beta)\\
\quad \le C\epsilon^{1/(1-q')}\biggl[\sum_{k=1}^\infty
\bigl[1-{h}_{\bar\alpha}(k)\bigr]^2\beta^2(k)+\sigma^2\max_k \frac{ {h}_{\bar\alpha}^2(k)}{\lambda(k)}\biggr]^{q'/(1-q')}.
\end{split}$$
Thus, by (\[eqn.40\]) and (\[eqn.43\]) we obtain $$\begin{split}
\mathbf{E}\biggl|2\sigma \sum_{k=1}^p \lambda^{-1/2}(k)\bigl[\tilde{h}_{\hat\alpha}(k)-\tilde{h}_{\bar\alpha}(k)
\bigr]\xi'(k)\beta(k) - \biggl(1-\frac{\gamma}{2}\biggr)L_{\hat\alpha}(\beta)\biggr|^{q}\\
\le C\mathbf{E}\biggl|2\sigma \sum_{k=1}^p \lambda^{-1/2}(k)\bigl[\tilde{h}_{\hat\alpha}(k)-\tilde{h}_{\bar\alpha}(k)
\bigr]\xi'(k)\beta(k)\\
\qquad - \biggl[\sigma^2\sum_{k=1}^p \lambda^{-1}(k)\bigl[\tilde{h}_{\hat\alpha}(k)-\tilde{h}_{\bar\alpha}(k)
\bigr]^2\beta^2(k)\biggr]^{q'}\biggr|^{q}\\
\qquad {}+ C \mathbf{E}\biggl| \biggl[\sigma^2\sum_{k=1}^p \lambda^{-1}(k)\bigl[\tilde{h}_{\hat\alpha}(k)-\tilde{h}_{\bar\alpha}(k)
\Bigr]^2\beta^2(k)\biggr]^{q'}- \biggl(1-\frac\gamma2\biggr) L_{\hat\alpha}(\beta)\biggr|^{q}
\\ \le (C\epsilon)^{-\frac{q}{2q'-1}}
+C\epsilon^{\frac{q}{1-q'}}\biggl[\sum_{k=1}^\infty
\bigl[1-{h}_{\bar\alpha}(k)\bigr]^2\beta^2(k)+\sigma^2\max_k \frac{ {h}_{\bar\alpha}^2(k)}{\lambda(k)}\biggr]^{\frac{qq'}{1-q'}} .
\end{split}$$ Therefore, substituting in the above equation $q'=2/3$ and $$\epsilon =\biggl[\sum_{k=1}^\infty\bigl[1-{h}_{\bar\alpha}(k)\bigr]^2\beta^2(k)+\sigma^2\max_k \frac{ {h}_{\bar\alpha}^2(k)}{\lambda(k)}\biggr]^{-3q},$$ we get $$\label{eqn.44}
\begin{split}
\mathbf{E}\biggl|2\sigma \sum_{k=1}^p \lambda^{-1/2}(k)\bigl[\tilde{h}_{\hat\alpha}(k)-\tilde{h}_{\bar\alpha}(k)
\bigr]\xi'(k)\beta(k)-\biggl(1-\frac\gamma2\biggr)L_{\hat\alpha}(\beta)\biggr|^{q}\\
\quad \le C\biggl[\sum_{k=1}^\infty
\bigl[1-{h}_{\bar\alpha}(k)\bigr]^2\beta^2(k)+\sigma^2\max_k \frac{ {h}_{\bar\alpha}^2(k)}{\lambda(k)}\biggr]^{q}.
\end{split}$$
Now we proceed with the last line in Equation (\[eqn.37\]). Since $\bar\alpha$ is given, we have by (\[eqn.31\]) $$\begin{split}
\bigl\{\mathbf{E}[\hat\sigma_{\bar\alpha}^2-\sigma^2]^2\bigr\}^{1/2}
\le \frac{\sigma^2}{\|1-h_{\bar\alpha}\|^2}\biggl\{\sum_{k=1}^p \bigl[1-h_{\bar\alpha}(k)\bigr]^4\biggr\}^{1/2}\\ {}+\frac{2\sigma}{\|1-h_{\bar\alpha}\|^2}\biggl\{\sum_{k=1}^p \bigl[1-h_{\bar\alpha}(k)\bigr]^4{\beta^2(k)}{\lambda(k)}\biggr\}^{1/2}\\ {}+\frac{1}{\|1-h_{\bar\alpha}\|^2}\sum_{k=1}^p \bigl\{1-h_{\bar\alpha}(k)\bigr\}^2{\beta^2(k)}{\lambda(k)}\\
\le \frac{C\sigma^2}{\|1-h_{\bar\alpha}\|}+\frac{C}{\|1-h_{\bar\alpha}\|^2}\sum_{k=1}^p \bigl[1-h_{\bar\alpha}(k)\bigr]^2{\beta^2(k)}{\lambda(k)}
\end{split}$$ and therefore $$\label{eqn.45}
\begin{split}
&\mathbf{E}\bigl\{[\hat\sigma_{\bar\alpha}^2-\sigma^2]_+Pen(\bar\alpha)\bigr\}^{1+\gamma/4} \le
C\bigl[\sigma^{-2} \Ps(\alpha_\circ,\alpha^\circ)\bar{R}_{\bar\alpha}(\beta)\bigr]^{1+\gamma/4}.
\end{split}$$
The last term in (\[eqn.37\]) can be bounded by Lemma \[lemma.5\] and (\[eqn.16\]) as follows: $$\label{eqn.46}
\begin{split}
&\mathbf{E}\bigl\{[\sigma^2-\hat\sigma_{\hat\alpha}^2]_+Pen(\hat\alpha)\bigr\}^{1+\gamma/4}\\ &\quad \le C\Ps^{1+\gamma/4}(\alpha_\circ) \mathbf{E}\bigl\{\sigma^{-2}L_{\hat\alpha}(\beta)+(1+\gamma)
Q^+(\hat\alpha)\bigr\}^{1+\gamma/4} .
\end{split}$$
Finally combining Equations (\[eqn.37\]), (\[eqn.38\]), (\[eqn.39\]), (\[eqn.44\]), (\[eqn.45\]), (\[eqn.46\]), we finish the proof.
The next idea in the proof of Theorem \[theorem.1\] is that the data-driven parameter $\hat\alpha$ defined by (\[eqn.4\]) cannot be very small, or equivalently, that the ratio $D(\hat\alpha)/D(\alpha^{\circ})$ cannot be very large.
\[lemma.8\] For any data-driven $\hat\alpha$ and any given $\bar\alpha\in [\alpha_\circ,\alpha^\circ]$, the following upper bound holds $$\label{eqn.47}
\begin{split}
&\biggl\{\mathbf{E}\biggl[\frac {D(\hat\alpha)}{D(\alpha^{\circ})}\biggr]^{1+\gamma/4}\biggr\}^{1/(1+\gamma/4)}\\&\qquad\le \frac{C}{[1-C \Ps(\alpha_\circ,\alpha^\circ)/\gamma]_+}
\R \biggl[\frac{\bar{R}_{\bar\alpha}(\beta)}{\sigma^2\gamma D(\alpha^{\circ})}+ \frac{1}{\gamma^4}\biggr],
\end{split}$$ for any $ \gamma\in (0,1/4)$.
Representing $$(1+\gamma)Q^+(\hat\alpha)=\biggl(1+\frac{\gamma}{2}\biggr)Q^+(\hat\alpha)
+\frac{\gamma}{2} Q^+(\hat\alpha),$$ we obtain with a simple algebra from (\[eqn.37\]) $$\begin{split}
\frac {\gamma\sigma^2}{2}\biggl[\sum_{k=1}^p \frac{h_{\hat\alpha }^2(k)}{ \lambda(k)} +(1+\gamma)Q^+(\hat\alpha)\biggr]
\le \bar{R}_{\bar\alpha}(\beta) -\sigma^2 \sum_{k=1}^p \frac{\tilde{h}_{\bar\alpha}(k)}{\lambda(k)}[\xi'^2(k)-1]\\
\quad {}+\sigma^2\sup_{\alpha\le \alpha^{\circ}} \biggl[\sum_{k=1}^p \frac{\tilde{h}_{\alpha}(k)}{\lambda(k)}[\xi'^2(k)-1]
-\biggl(1+\frac{\gamma}{2}-\frac{\gamma^2}{2}\biggr)
Q^+(\alpha)\biggr]_+\\
\quad {}+2\sigma \sum_{k=1}^p \lambda^{-1/2}(k)\bigl[\tilde{h}_{\hat\alpha}(k)-\tilde{h}_{\bar\alpha}(k)
\bigr]\xi'(k)\beta(k) -\biggl(1-\frac{\gamma}{2}\biggr)L_{\hat\alpha}(\beta) \\
\quad {}+[\hat\sigma_{\bar\alpha}^2-\sigma^2]Pen(\bar\alpha)+
[\sigma^2-\hat\sigma_{\hat\alpha}^2]_+Pen(\hat\alpha).
\end{split}$$
Combining this with Equations (\[eqn.38\]), (\[eqn.39\]), (\[eqn.44\]), (\[eqn.45\]), (\[eqn.46\]), we obtain $$\label{eqn.48}
\begin{split}
&\biggl\{\mathbf{E} \biggl[\frac{\|h_{\hat\alpha}\|^2_\lambda +Q^+(\hat\alpha)}
{D(\alpha^\circ)}\biggr]^{1+\gamma/4}\biggr\}^{1/(1+\gamma/4)}
\\ & \qquad\le \frac{C}{[1-C \Ps(\alpha_\circ,\alpha^\circ)/\gamma]_+} \biggl[\frac{C\bar{R}_{\bar\alpha}(\beta)}{\sigma^2 \gamma D(\alpha^\circ)}
+\frac{1}{\gamma^4}\biggr].
\end{split}$$
To continue this inequality, we need a lower bound for $\|h_\alpha\|_\lambda^2 +Q^+(\alpha)$. Notice that $$f(x)\stackrel{\rm def}{=}F(x)-\frac{x^2}{1-2x}=\frac{1}{2}\log(1-2x)+x+\frac{x^2}{1-2x}$$ is a non-negative function for $x\ge 0$ since $$f'(x)=\frac{2x^2}{(1-2x)^2}\ge 0 \text{ and }f(0)=0.$$ Therefore the following inequality holds $$\label{eqn.49}
F(x)\ge \frac{x^2}{1-2x}.$$
Let $$\label{eqn.50}
k_\alpha =\arg\max _k \frac{h_\alpha(k)}{\lambda(k)},$$ then by (\[eqn.14\]) and (\[eqn.49\]) we obviously get $$\log\frac{D(\alpha)}{D(\alpha^\circ)}\ge F[\mu_\alpha \rho_\alpha(k_\alpha)]\ge \frac{[\mu_\alpha \rho_\alpha(k_\alpha)]^2}{1-2 \mu_\alpha \rho_\alpha(k_\alpha) }.$$ With this inequality we obtain $$\mu_\alpha \rho_\alpha(k_\alpha)\le \biggl\{1+\biggl[1+\log^{-1}\frac{D(\alpha)}{D(\alpha^\circ)}\biggr]^{1/2}\biggr\}^{-1},$$ thus arriving at $$\label{eqn.51}
\mu_\alpha^{-1} \ge 2\rho_\alpha(k_\alpha).$$
Now we are in a position to bound from below $ \|h_\alpha\|_\lambda^2 +Q^+(\alpha)$. By (\[eqn.13\]–\[eqn.15\]), (\[eqn.50\]–\[eqn.51\]), and (\[eqn.17\]) we obtain $$\label{eqn.52}
\begin{split}
\|h_\alpha\|_\lambda^2 +Q^+(\alpha)\ge \|h_\alpha\|_\lambda^2+\frac{2D(\alpha)}{\mu_\alpha}\sum_{k=1}^p \frac{[\mu_\alpha\rho_\alpha(k) ]^2}{1-2\mu_\alpha\rho_\alpha(k)}\\
\ge \|h_\alpha\|_\lambda^2+\frac{D(\alpha)}{\mu_\alpha}\sum_{k=1}^p F[\mu_\alpha\rho_\alpha(k)] =
\|h_\alpha\|_\lambda^2+\frac{D(\alpha)}{\mu_\alpha}\log\frac{D(\alpha)}{D(\alpha^\circ)}\\
\ge \|h_\alpha\|_\lambda^2+2\rho_\alpha(k_\alpha)D(\alpha)\log\frac{D(\alpha)}{D(\alpha^\circ)}\\
\ge \|h_\alpha\|_\lambda^2+ \frac{h_\alpha(k_\alpha)}{\lambda(k_\alpha)}\log\frac{D(\alpha)}{D(\alpha^\circ)}
\\
\ge \|h_\alpha\|_\lambda^2+ \log\frac{D(\alpha)}{D(\alpha^\circ)}\max _k \frac{h_\alpha(k)}{\lambda(k)}
%\\ &
\ge CD(\alpha)\log\frac{D(\alpha)}{D(\alpha^\circ)}.
\end{split}$$
With the help of (\[eqn.52\]) we continue (\[eqn.48\]) as follows: $$\label{eqn.53}
\begin{split}
\biggl\{\mathbf{E} \biggl[\frac {D(\hat\alpha)}{D(\alpha^{\circ})}\log\frac {D(\hat\alpha)}{D(\alpha^{\circ})}\biggr]^{1+\gamma/4}\biggr\}^{1/(1+\gamma/4)}\qquad\qquad
\\[6pt] \quad \le \frac{C}{[1-C \Ps(\alpha_\circ,\alpha^\circ)/\gamma]_+} \biggl[\frac{C\bar{R}_{\bar\alpha}(\beta)}{\sigma^2 \gamma D(\alpha^\circ)}
+\frac{1}{\gamma^4}\biggr].
\end{split}$$ To control from below the left-hand side in the above equation, notice that $$\label{eqn.54}
\begin{split}
\mathbf{E}\biggl[\frac {D(\hat\alpha)}{D(\bar{\alpha})}\log\frac {D(\hat\alpha)}{D(\bar{\alpha})}
\biggr]^{1+\gamma/4}=\frac{1}{(1+\gamma/4)^{1+\gamma/4}}
\mathbf{E}\biggl[\frac {D(\hat\alpha)}{D(\bar{\alpha})}\biggr]^{1+\gamma/4}%\times
\\[6pt]
\times \biggl\{\log\biggr[\frac {D(\hat\alpha)}{D(\bar{\alpha})}\biggr]^{1+\gamma/4}\biggr\}^{1+\gamma/4}.
\end{split}$$
To finish the proof, let us consider the function $f(x)=x\log^{1+\gamma/4}(x)$, $ x\ge 1$. Computing its second order derivative, one can easily check that $f(x)$ is convex for all $x\ge \exp(1)={\rm e}$. So, $f(x+{\rm e}-1)$ is convex for $x\ge 1$. It is easily seen there exists a constant $C>0$ such that for all $x\ge 1$ $$f(x)\ge \frac{1}{2} f(x+{\rm e}-1)-C.$$ Therefore by (\[eqn.54\]) and Jensen’s inequality, $$\label{eqn.55}
\begin{split}
\mathbf{E}\biggl[\frac {D(\hat\alpha)}{D(\bar{\alpha})}\log\frac {D(\hat\alpha)}{D(\bar{\alpha})}\biggr]^{1+\gamma/4}\ge C \biggl\{\mathbf{E}\biggl[\frac {D(\hat\alpha)}{D(\bar{\alpha})}\biggr]^{1+\gamma/4}+{\rm e}-1\biggr\}
\\[6pt]
\quad \times \log^{1+\gamma/4}\biggl\{\mathbf{E}\biggr[\frac {D(\hat\alpha)}{D(\bar{\alpha})}\biggr]^{1+\gamma/4}+{\rm e}-1\biggr\}-C.
\end{split}$$
Let $$\psi(x)=(x+{\rm e}-1)\log^{1+\gamma/4}(x+{\rm e}-1).$$ It is easy to check that the inverse function $\psi^{-1}(x)$ satisfies the following inequality $$\label{equ.46}
\psi^{-1}(x)\le (x+{\rm e}-1)\log^{-1-\gamma/4}(x+{\rm e}-1).$$ Therefore combining this equation and (\[eqn.55\]) with (\[eqn.53\]), we arrive at (\[eqn.47\]).
Now we are ready to proceed with the proof of Theorem \[theorem.1\]. Let $\epsilon>0$ be a small given number to be defined later on. By (\[eqn.8\]) and (\[eqn.9\]), the following equation for the skewed excess risk $$\label{eqn.56}
\begin{split}
\mathcal{E}(\epsilon)\stackrel{\rm def}{=} \sup_{\beta \in
\mathbb{R}^p}\mathbf{E}_\beta\Bigl\{\|\beta -\hat
\beta_{\hat\alpha}\|^2 -(1+\epsilon) \bigl\{{R}_{\hat \alpha}[Y]+\mathcal{C}\bigr\} \Bigr\}\\
\ =\sup_{\beta \in
\mathbb{R}^p}\mathbf{E}_\beta\biggl\{ -\epsilon
\sum_{k=1}^p\bigl[1-h_{\hat\alpha}(k)\bigr]^2 \beta^2(k)-\epsilon\sigma^2 \sum_{k=1}^p
\lambda^{-1}(k) h_{\hat\alpha}^2(k)\\ \quad- (1+\epsilon)(1+\gamma)\sigma^2 Q^+(\hat\alpha)
\\
\quad -2\sigma
\sum_{k=1}^p\bigl\{1+\epsilon-[(1+2\epsilon)h_{\hat\alpha}(k)-\epsilon
h_{\hat\alpha}^2(k)
]\bigr\}\beta(k)\lambda^{-1/2}(k)
\xi'(k) \\ \quad +\sigma^2\sum_{k=1}^p
\lambda^{-1}(k)\bigl[2(1+\epsilon) h_{\hat\alpha}(k)-\epsilon h_{\hat\alpha}^2(k)\bigr][\xi'^2(k)-1]
\\ \qquad
+[\sigma^2-\hat\sigma_{\hat\alpha}^2]Pen(\hat\alpha)
\biggr\}
\end{split}$$ holds.
We proceed with the second line from below at the right-hand side of this display. By Lemmas \[lemma.6\] and \[lemma.8\], we obtain $$\label{eqn.57}
\begin{split}
\sigma^2\mathbf{E}\sum_{k=1}^p
\lambda^{-1}(k) \bigl[(1+\epsilon) h_{\hat\alpha}(k)-\epsilon h_{\hat\alpha}^2(k)\bigr][\xi'^2(k)-1]\\
\le \frac{C}{[1-C \Ps(\alpha_\circ,\alpha^\circ)/\gamma]_+\sqrt{\gamma} }
\R \biggl[\frac{\bar{R}_{\bar\alpha}(\beta)}{\sigma^2\gamma D(\alpha^{\circ})}+ \frac{1}{\gamma^4}\biggr].
\end{split}$$
The next step is to bound the third line from below at the right-hand side of (\[eqn.56\]). It suffices to note that $\tilde{h}_\alpha^\epsilon (k)=\bigl[(1+2\epsilon) h_{\alpha}(k)-\epsilon h_{\alpha}^2(k)\bigr]/(1+\epsilon)$ is the family of ordered functions. Hence, by Proposition \[proposition.6\], we get with $\bar\alpha=\arg\min_{\alpha\in[\alpha_\circ,\alpha^\circ]}\bar{R}_\alpha(\beta)$ $$\label{eqn.58}
\begin{split}
2\sigma\mathbf{E}
\sum_{k=1}^p\bigl\{1+\epsilon-[(1+2\epsilon)h_{\hat\alpha}(k)-\epsilon
h_{\hat\alpha}^2(k)
]\bigr\}\beta(k)\lambda^{-1/2}(k)
\xi'(k)\\ \quad =2(1+\epsilon)\sigma\mathbf{E}
\sum_{k=1}^p\bigl[h_{\bar\alpha}^\epsilon(k)-h_{\hat\alpha}^\epsilon(k)\bigr]\beta(k)\lambda^{-1/2}(k)
\xi'(k)\\ \quad \le C \biggl[\sigma^2\mathbf{E}\max_k \lambda^{-1}(k) {h}_{\hat \alpha}^2(k) \sum_{k=1}^p
\bigl[1-{h}_{\bar\alpha}(k)\bigr]^2\beta^2(k)\bigg]^{1/2}\\
\qquad {}+C \biggl[\sigma^2\max_k \lambda^{-1}(k) {h}_{\bar \alpha}^2(k) \mathbf{E} \sum_{k=1}^p
\bigl[1-{h}_{\hat\alpha}(k)\bigr]^2\beta^2(k)\bigg]^{1/2}\\ \quad \le
C\sigma^2\epsilon^{-1}\mathbf{E}\max_k \lambda^{-1}(k) {h}_{\hat \alpha}^2(k)+\epsilon
\sum_{k=1}^p
\bigl[1-{h}_{\bar\alpha}(k)\bigr]^2\beta^2(k)
\\ \qquad {}+
C\sigma^2\epsilon^{-1}\max_k \lambda^{-1}(k) {h}_{\bar \alpha}^2(k)+\epsilon
\mathbf{E}\sum_{k=1}^p
\bigl[1-{h}_{\hat\alpha}(k)\bigr]^2\beta^2(k)
.
\end{split}$$
Therefore, substituting (\[eqn.45\]), (\[eqn.46\]), (\[eqn.57\]), (\[eqn.58\]) in (\[eqn.56\]), we obtain the following upper bound for the skewed excess risk $$\label{eqn.59}
\begin{split}
\mathcal{E}(\epsilon)\le C\sigma^2\epsilon^{-1}\mathbf{E}\max_k \lambda^{-1}(k) {h}_{\hat \alpha}^2(k)-\sigma^2\mathbf{E}Q^+(\hat\alpha) +C \Ps(\alpha_\circ,\alpha^\circ)\bar{R}_{\bar\alpha}(\beta)\\ +C\sigma^2\epsilon^{-1}\max_k \lambda^{-1}(k) {h}_{ \bar\alpha}^2(k)+\epsilon \sum_{k=1}^p
\bigl[1-{h}_{\bar\alpha}(k)\bigr]^2\beta^2(k)\\ {}+ \frac{C\sigma^2D(\alpha^\circ)}{[1-C \Ps(\alpha_\circ,\alpha^\circ)/\gamma]_+\sqrt{\gamma} }
\R \biggl[\frac{\bar{R}_{\bar\alpha}(\beta)}{\sigma^2\gamma D(\alpha^{\circ})}+ \frac{1}{\gamma^4}\biggr].
\end{split}$$
Let us consider the function $$U(\epsilon)=\max_{\alpha\le \alpha^{\circ}}\bigl\{C\epsilon^{-1}\max_k \lambda^{-1}(k) {h}_{ \alpha}^2(k)-Q^+(\alpha)\bigr\}.$$ Since $$\begin{split}
\max_k \frac{{h}_{ \alpha}^2(k)}{\lambda(k)} \le\max_k \frac{{h}_{ \alpha}(k)}{\lambda(k)}\le \biggl[\sum_{k=1}^p \frac{{h}_{ \alpha}^2(k)}{\lambda^2(k)}\biggr]^{1/2}\\
\le \biggl\{\sum_{k=1}^p \frac{{h}_{ \alpha}^2(k)}{\lambda^2(k)}[2-h_\alpha(k)]^2\biggr\}^{1/2}
\le \frac{D(\alpha)}{\sqrt{2}}
\end{split}$$ and by Proposition \[proposition.7\] $$Q^+(\alpha)\ge D(\alpha)\sqrt{\log \frac{D(\alpha)}{D(\alpha^{\circ})}},$$ we get $$\begin{split}
U(\epsilon)\le D(\alpha^{\circ}) \max_{\alpha\le \alpha^{\circ}}\biggl\{\frac{C}{\epsilon}\frac{D(\alpha)}{D(\alpha^{\circ})}-
\frac{D(\alpha)}{D(\alpha^{\circ})}\biggl[\log\frac{D(\alpha)}{D(\alpha^{\circ})}\biggr]^{1/2}\biggr\}\\
\le D(\alpha^{\circ}) \max_{x\ge 1}\biggl\{ \frac{Cx}{\epsilon}-x\sqrt{\log(x)}\biggr\}.
\end{split}$$ One can easily check with a simple algebra that $$\label{eqn.60}
\max_{x\ge 1}\biggl\{ \frac{Cx}{\epsilon}-x\sqrt{\log(x)}\biggr\}\le\frac{\epsilon}{C}
\exp\biggl[\frac{C^2}{\epsilon^2}\biggr].$$ Indeed, let $x^*=\arg\max_x\Bigl\{Cx/\epsilon-x\sqrt{\log(x)}\Bigr\}$. Then, differentiating $Cx/\epsilon-x\sqrt{\log(x)}$ in $x$, we obtain the following equation for $x^*$ $$\frac{C}{\epsilon}-\sqrt{\log(x^*)}-\frac{1}{2\sqrt{\log(x^*)}}=0.$$ Therefore $$x^*=\exp\biggl\{\biggl(\frac{C}{2\epsilon}+\sqrt{\frac{C^2}{4\epsilon^2}-1}\biggl)^2\biggr\}\le \exp\biggl(\frac{C^2}{\epsilon^2}\biggr).$$ This equation proves (\[eqn.60\]) since $$\max_{x\ge 1} \biggl\{\frac{Cx}{\epsilon }-x\sqrt{\log(x)}\biggr\}\le \frac{Cx^*}{\epsilon}.$$ With (\[eqn.60\]) we continue (\[eqn.59\]) as follows: $$\label{equ.61}
\begin{split}
\mathcal{E}(\epsilon)\le C\sigma^2D(\alpha^{\circ}){\epsilon}
\exp\frac{C^2}{\epsilon^2}+C\Ps(\alpha_\circ,\alpha^\circ)\bar{R}_{\bar\alpha}(\beta)\\ {}+C\sigma^2\epsilon\sum_{k=1}^p \lambda^{-1}(k) {h}_{ \bar\alpha}^2(k)+\epsilon \sum_{k=1}^p
\bigl[1-{h}_{\bar\alpha}(k)\bigr]^2\beta^2(k)\\ {}+
\frac{C \sigma^2D(\alpha^\circ)}{[1-C \Ps(\alpha_\circ,\alpha^\circ)/\gamma]_+\sqrt{\gamma} }
\R \biggl[\frac{\bar{R}_{\bar\alpha}(\beta)}{\sigma^2\gamma D(\alpha^{\circ})}+ \frac{1}{\gamma^4}\biggr]\\
\le C\sigma^2D(\alpha^{\circ}){\epsilon}
\exp\frac{C^2}{\epsilon^2}+C\epsilon\bar{R}_{\bar\alpha}(\beta) +C \Ps(\alpha_\circ,\alpha^\circ)\bar{R}_{\bar\alpha}(\beta)\\ {}+ \frac{C}{[1-C \Ps(\alpha_\circ,\alpha^\circ)/\gamma]_+\sqrt{\gamma} }
\R \biggl[\frac{\bar{R}_{\bar\alpha}(\beta)}{\sigma^2\gamma D(\alpha^{\circ})}+ \frac{1}{\gamma^4}\biggr] .
\end{split}$$
Therefore, substituting this equation in $$\mathbf{E}\|\beta-\hat\beta_{\hat\alpha}\|^2 \le (1+\epsilon)\bar{R}_{\bar\alpha}(\beta)+\mathcal{E}(\epsilon),$$ we get $$\label{eqn.61}
\begin{split}
\mathbf{E}\|\beta-\hat\beta_{\hat\alpha}\|^2 \le [1+C \Ps(\alpha_\circ,\alpha^\circ)]r(\beta)+
C\sigma^2D(\alpha^{\circ})\times \\ \times \inf_{\epsilon}\biggl[\epsilon\exp\frac{C^2}{\epsilon^2}
+\frac{\epsilon r(\beta)}{\sigma^2 D(\alpha^{\circ})}\biggr]\\ {}+
\frac{C \sigma^2D(\alpha^\circ) }{[1-C \Ps(\alpha_\circ,\alpha^\circ)/\gamma]_+\sqrt{\gamma} }
\R \biggl[\frac{r(\beta)}{\sigma^2\gamma D(\alpha^{\circ})}+ \frac{1}{\gamma^4}\biggr].
\end{split}$$
Hence, to finish the proof of the theorem, it remains to check that $$\label{eqn.62}
\inf_\epsilon F(\epsilon,u)=\inf_{\epsilon}\biggl[\epsilon\exp\frac{C^2}{\epsilon^2}
+\epsilon u\biggr]\le \frac{Cu}{\sqrt{\log(u)}}.$$ Let $\epsilon_*=\arg\min_{\epsilon}F(\epsilon,u)$. Then, differentiating $F(\epsilon,u)$ in $\epsilon$, we arrive at the following equation for $\epsilon_*$ $$\exp\biggl(\frac{C^2}{\epsilon_*^2}\biggr)-\frac{C^2}{\epsilon_*^2}\exp\biggl(\frac{C^2}{\epsilon_*^2}\biggr)+u=0.$$ Thus $$\frac{C^2}{\epsilon_*^2}+\log\biggl(\frac{C^2}{\epsilon_*^2}-1\biggr)=u$$ and it follows immediately from the above equation that $$\epsilon_*\le \frac{C}{\sqrt{\log(u)}}$$ and therefore $$F(\epsilon_*,u)\le 2u\epsilon_*\le \frac{2Cu}{\sqrt{\log(u)}},$$ thus proving (\[eqn.62\]).
Finally, substituting (\[eqn.62\]) with $u=r(\beta)/[\sigma^2 D(\alpha^\circ)]$ in (\[eqn.61\]), we complete the proof of the theorem.
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[^1]: Université de Provence, Marseille, France and Moscow Institute of Phisics and Technology, Russia.
[^2]: The author was partially supported by the ANR grant no. ANR-BLAN-0234-01 and by Laboratory of Structural Methods of Predictive Modeling and Optimization, MIPT, RF government grant, ag. 11.G34.31.0073.
|
---
abstract: 'The object of this paper is to discuss the physical interpretation of quantum behaviour of Schrodinger electron (SchEl) and bring to light on the cause for the Heisenberg convenient operator way of its describing, using the nonrelativistic quantum mechanics laws and its mathematical results. We describe the forced stochastically diverse circular harmonic oscillation motion, created by force of the electrical interaction of the SchEl’s elementary electric charge (ElmElcChrg) with the electric intensity (ElcInt) of the resultant quantum electromagnetic field (QntElcMgnFld) of the existing StchVrtPhtns, as a solution of Abraham-Lorentz equation. By dint of this equation we obtain that the smooth thin line of a classical macro particle is rapidly broken of many short and disorderly orientated lines, owing the continuous dispersion of the quantum micro particle (QntMicrPrt) on the StchVrtPhtns. Between two successive scattering the centers of diverse circular oscillations with stochastically various radii are moving along this short disordered line. These circular harmonic oscillations lie within the flats, perpendicular to same disordered short line, along which are moving its centers. In a result of same forced circular harmonic oscillation motion the smooth thin line of the LrEl is roughly spread and turned out into some cylindrically wide path of the SchEl. Hence the dispersions of different dynamical parameters, determining the state of the SchEl, which are results of its continuously interaction with the resultant QntElcMgnFld of the StchVrtPhtns. The absence of the smooth thin line trajectory at the circular harmonic oscilation moving of the QntMicrPrt forces us to use the matrix elements (Fourier components) of its roughly spread wide cylindrical path for its description.'
author:
- |
Josiph Mladenov Rangelov,\
Institute of Solid State Physics,Bulgarian Academy of Sciences,\
72Tsarigradsko chaussee,1 784 Sofia,Bulgaria.
title: |
**PHYSICAL MODEL OF SCHRODINGER ELECTRON.\
HEISENBERG CONVENIENT WAY FOR\
DESCRIPTION OF ITS QUANTUM BEHAVIOUR**
---
Introduction
=============
We assume that the vacuum fluctuations (VcmFlcs) through zero-point quantum electromagnetic field (QntElcMgnFld) perform an important role in a behaviour of the micro particles (MicrPrts). As thing turned out that if the Brownian stochastic motion (BrnStchMtn) of some classical micro particle (ClsMicrPrt) is a result of fluctuating deviations of averaged values of all having an effect forces on a ClsMcrPrt, coming from many molecule blows from a surround environment, then the quantized stochastic dualistic wave-particle behaviour of every QntMicrPrt is a result of the continuous uncontrolled electromagnetic interaction (ElcMgnIntAct) between its well spread (WllSpr) elementary electric charge (ElmElcChrg) of the charged one (a Schrodinger’s electron (SchEl)) or its magnetic dipole moment (MgnDplMmn) for uncharged one (as a neutron), and the averaged electric intensity for charged MicrPrts or the averaged magnetic intensity for uncharged one, of the resultant quantized electromagnetic field (QntElcMgnFld) of all stochastic virtual photons (StchVrtPhtns), excited within the FlcVcm and existing within its neighborhood, which exercises a very power influence on its state and behaviour. Consequently the continuously scattering of the well spread (WllSpr) elementary electric charge (ElmElcChrg) of the SchEl on the StchVrtPhtns at its creation powerfully broken the smooth thin line of the classical trajectory of many short and very disorderly orientated small lines and the powerfully its interaction (IntAct) with the electric intensity (ElcInt) or magnetic intensity (MgnInt) of the resultant QntElcMgnFld of the existing StchVrtPhtns forced it to make the circular harmonic oscillations with various radii and the centers, lying over the small disordered lines. In a result of this complicated motion the narrow smooth line of the classical trajectory is turned out into some wide rough cylindrically spread path of the QntMcrPrt. Although in further we will give the necessary calculations, we wish to repeated, that as a result of the continuously scattering of the QntMicrPrt on the StchVrtPhtns at their creations the smooth thin line of the classical trajectory is turned out into powerfully often broken of many small and very disorderly orientated short lines. The uninterrupted ElcMgnIntAct of the ElmElcChrg or of the MgnDplMmn of the QntMicrPrt with the ElcInt or the MgnInt of the resultant QntElcMgnFld of the StchVrtPhtns, existent within the fluctuating vacuum (FlcVcm) between two consecutive scatterings forced the QntMicrPrt to carry out the stochastic circular oscillation motion, which exercise an influence of its behavior within a neighborhood of the smooth classically line into the cylindrically spread by different radii wide path. It isn’t allowed us to forget that the broken of the smooth thin line of very short and very disorderly orientated small line is a result of its continuously scattering on the StchVrtPhtns, over which are found the centers of the forced stochastic circular oscillation motion of the QntMicrPrt, owing a result of the ElcMgnIntAct of its WllSpr ElmElcChrg and MgnDplMmn with the intensities of the resultant electric field (RslElcFld) and resultant magnetic field (RslMgnFld) of the stochastic virtual photons (StchVrtPhtns). The WllSpd ElmElcChrg of the SchEl is moving at its circular oscillations of different radii within the flats, which are perpendicular to the very short and very disorderly orientated small lines, obtained in a result of its continuously scattering on the StchVrtPhtns, at its Furtian quantized stochastic circular harmonic oscillation motion through the fluctuating vacuum (FlcVcm). Therefore in our transparent survey about the physical model (PhsMdl) of the nonrelativistic quantized SchrEl one will be regarded as some WllSpr ElmElcChrg), participating simultaneously in two different motions: A) The classical motion of a classical Lorentz’ electron (LrEl) along an well contoured smooth thin trajectory, realized in a consequence of some known interaction (IntAct) of its over spread (OvrSpr) ElmElcChrg, MgnDplMnt or bare mass with the intensity of some external classical fields (ClsFlds) as in the Newton nonrelativistic classical mechanics (NrlClsMch) and Maxwell nonrelativistic classical electrodynamics (ClsElcDnm). B) The isotropic three-dimensional nonrelativistic quantized (IstThrDmnNrlQnt) Furthian stochastic boson circular harmonic oscillations motion (FrthStchBsnCrcHrmOscMtn) of the SchEl as a natural result of the permanent ElcIntAct of its WllSpr ElmElcChrg with the ElcInt of the resultant QntElcMgnFld of a large number StchVrtPhtns. This ElcIntAct between the WllSpr ElmElcChrg and the FlcVcm (zero-point ElcMgnFld) is generated by dint of StchVrtPhtns exchanged between the fluctuating vacuum (FlcVcm) and the WllSpr ElmElcChrg during a time interval of their life. As soon as this Furthian quantized stochastic wave-particle behaviour of the SchEl is very similar to known Brownian classical stochastic behaviour of the ClsMacrPrt, therefore the QntMicrPrt cannot has the classical sharp contoured smooth and thin trajectory but has a cylindrical broad rough path, obtained as a sum of circular oscillations motions of different radii and centers, lying on accidental broken short lines, strongly disordered within a space. Hence the often broken trajectory of the moving QntMicrPrt present itself a sum of small parts from some circumferences with different radii and centers, lying within flats, which are perpendicular to accidental broken short lines, strongly disordered in space. Therefore in a principle the exact description of the resultant behaviour of the SchEl owing of its joint participation in both mentioned above motions could be done only by means of the NrlQntMch’s and nonrelativistic ClsElcDnm’s laws.
It is known of many scientists the existence of three different ways [@WH], [@MBPJ] and [@BHJ], [@EScha] and [@RPF], [@FH] for the description of the quantum behaviour [@LB] of the nonrelativistic SchEl. It is turned out that there is some possibility enough to show by means of the existence intrinsic analogy between the quadratic differential wave equation in partial derives (QdrDfrWvEqtPrtDrv) of Schrodinger and the quadratic differential particle equation in partial derivative (QdrDfrPrtEqtPrtDrv) of Hamilton-Jacoby that the addition of the kinetic energy of the Furthian stochastic boson circular harmonic oscillation of some QntMicrPrt to the kinetic energy of such ClsMacrPrt determines their dualistic wave-particle quantized behaviour. It turns out the stochastic motion over the powerfully break up the sharp contoured smooth thin classical line of the in many shortly and very disorderly (stochastically orientated) small lines. As in such a natural way we have ability enough to obtain the minimal value of the dispersion product, determined with the Heisenberg uncertainty relation. Science there exists an essential analogy between the registration forms of the quadratic differential diffusive equation (QdrDfrDfsEqt) of Focker-Plank for the distribution function $P(r,t)$ of a probability density (DstFncPrbDns) of the free Brownian ClsMacrPrt (BrnClsMacrPrt) in a motionless coordinate system in a respect to it and quadratic differential wave equation in partial derivative (QdrDfrWvEqtPrtDrv) of Schrodinger for the orbital wave function (OrbWvFnc) of some free Furthian QntMicrPrt (FrthQntMicrPrt) in a motionless coordinate system in a respect to it we come to an essential conclusion that there are also some possibility enough to describe the quantized stochastic behaviour of the SchEl by means of the analogy between the classical Wiener continual integral and the quantized Feynman continual integral. Feynman has used for the description of transition between two OrbWvFncs of some free FrthQntMicrPrt with different coordinates and times some formula, analogous of such the formula, which early had been used by Einstein [@AE], [@ES], Smoluchowski [@MS] and Wiener [@NW] for the description of same transition between two DstFncsPrbDns of the free BrnClsMacrPrts. In this way we understand why the behaviour of the QntMicrPrt must be described by the OrbWvFnc $\Psi$ , although the behaviour of the ClsMacrPrt may be described by a line.
Mathematical description of the physical cause ensuring the display of the QntMicrPrt behaviour.
=================================================================================================
The object of this paper is to discuss the fundamental problems of the physical interpretation of the nonrelativistic quantized behaviour of the SchEl and bring to light for understanding the cause,securing the existence of this uncommon state of each the QntMicrPrt. It is necessary to understand why the QntMicrPrt has no classical smooth thin trajectory and why its behaviour must be described by the Heisenberg matrix of the convenient operator way, using the laws of the NrlQntMch and its effective mathematical results. The PhsMdl of the SchEl is built by means of the equation of the forced motion of the dumping classical oscillator under the force action of electric interaction (ElcIntAct) between its WllSpr ElmElcChrg and the ElcInt of the RslQntElcMgnFld of the StchVrtPhtns, created in the FlcVcm. The unusual behaviour of the SchEl may be described by the following motion equation in Maxwell nonrelativistic classical electrodynamics (ClsElcDnm): $$\label{FF}
\ddot{r_j}\,+\,\omega_o^2\,r_j\,=\,-\,(\frac{e}{m})\,\{\,E_j\,+
\,E_j^i\,\}\,=\,\frac{e}{mC}\frac{\partial\,A_j\,}{\partial\,t\,}\,+
\,\frac{2e^2}{3mC^3}\,\stackrel{\cdots}{r_j},$$ where $E_j^i$ and $E_j$ denote the ElcInt of both the ElcFld $E_j^i$ of radiative friction, that is to say of LwEng unemitted longitudinal (Lng) VrtPhtn (VrtLngPht), and ElcInt $E_j$ of the LwEng-VrtPhtn in the FlcVcm. In accordance of the relation (\[FF\]) the ElcInt $E_j$ of an external QntElcMgnFld may be described by means of its $A_j$, having the following analytical presentation: $$\label{GG}
A_j\,=\,\frac{i}{L}\,\sum_q\,\sqrt{\frac{2\pi\hbar C}{L\,q}}\,I_{jq}\,
\left[\,a_{jq}^+\,e^{i(t\omega\,-\,qr)}\,-
\,a_{jq}\,e^{-i(t\omega\,-\,qr)}\,\right],$$
Indeed the ElcInt $E_j$ of StchVrtFtn could be obtained by taking of a particle derivative of the expression (\[GG\]) relatively for the $$\label{HH}
E_j\,=\,\frac{1}{L}\,\sum_q\,\sqrt{\frac{2\pi\hbar\omega}{\,L\,}}\,I_{jq}\,
\left[\,a_{jq}^+\,e^{i(t\omega\,-\,qr)}\,+
\,a_{jq}\,e^{-i(t\omega\,-\,qr)}\,\right],$$
There is a necessity to note here that we have exchanged the signs in eqs. (\[GG\]) and (\[HH\]). Indeed, in order to get the necessary correspondences between operator expressions of the $\hat{p}_j$ and $\hat{A}_j$, it is appropriate to use the sign (-) in the eq.(\[GG\]) and the sign (+) in an eq.(\[HH\]). The helpful of this exchange of signs will be letter seen in following expressions (\[KK\]) of $\hat{r_j}$ and (\[LL\]) of $\hat{p_j}$. Hence by substituting the eq.(\[GG\]) in the eq.(\[FF\]) and transposition of same term in its left-hand side one can obtain motion equation in Lorentz-Abrahams nonrelativistic presentation (LAP): $$\label{II}
\ddot{r_j}\,-\,\tau\,\stackrel{\cdots}{r_j}\,+\,\omega_o^2\,r_j\,=
\,-\,(\frac{e}{m})\,E_j,$$
The temporary dependence of $r_j$ contains two frequencies $\omega_o$ and $\omega$. In a spite of $\omega_o\,\ge\,\omega$, then the very greatest magnitude of the term $\tau\,\stackrel{\cdots}{r_j}$ is -$\tau\,\omega_o^2\,\dot{r_j}$. Although of that the term $\tau\,\stackrel{\cdots}{r_j}$ still presents itself by -$\tau\,\omega^2\,\dot{r_j}$. Indeed, the general solution of eq.(\[II\]) is given by sum of the general solution of the homogeneous equation and a particular solution to the inhomogeneous equation. At $\omega\,\tau\,=\,\frac{2e^2}{3mC^2}\frac{\omega}{C}\,=\,\frac{\pi}{3}\,
(\frac{2e^2}{C\hbar})\,\frac{\hbar}{mC}\,\frac{2}{\lambda}\,\le\,1,$ the general solution of the homogeneous equation has a form of a relaxing oscillation of a frequency $\omega$. The particular solution has a form of a forced oscillation of a frequency $\omega$. Therefore we may rewrite eq. (\[II\]) in the following form : $$\label{KK}
\ddot{r_j}\,+\,\tau\,\omega^2\,\dot{r_j}\,+\,\omega_o^2\,r_j\,=
\,-\,(\frac{e}{m})\,E_j(r,t),$$
From eq.(\[KK\]) it is easily seen that the motion dumping of the SchEl is caused by well-known Lorentz’ dumping force owing to radiation friction of its moving WllSpr ElmElcChrg. In a rough approximation of the Maxwell nonrelativistic ClsElcDnm the minimum time interval for an emission or absorption of a real photon (RlPhtn) by the WllSpr ElmElcChrg of the SchEl may be evidently determined by the parameter of Lorentz-Abrahams : $$\label{LL}
\,\tau\,=\,\frac{2e^2}{3mC^3},$$
The particular solution of the motion eq.(\[KK\]), describing the forced quantized stochastic circular harmonic motion of the QntMicrPrt, have been written by Welton [@ThW], Kalitchin [@NK] and Sokolov and Tumanov [@ASBT], cite[AAS]{} by the way of the operator division in the following analytical form : $$\label{MM}
\,\hat{r_j}\,=\,\sum_q\,\frac{e\,q}{m\,L}\,
\sqrt{\frac{2\pi \hbar\omega }{\,L\,q\,}}\,I_{jq}\,
\left[\,\frac{a_{jq}^{+}\,\exp{\{i\,t\omega \,-\,i\,qr\,\}}}
{\omega _o^2\,-\,\omega ^2\,+ \,i\tau \omega ^3\,}\,+
\,\frac{a_{jq}\,\exp{\{-\,i\,t\omega \,+\,i\,qr\,\}}}
{\omega _o^2\,-\,\omega ^2\,-\,i\tau \omega ^3\,}\,\right],$$
$$\label{NN}
\,\hat{P_j}\,=\,i\,\sum_q\,\frac{e\,\omega _o^2}{C\,L}\,
\sqrt{\frac{2\pi\hbar\omega }{\,L\,q\,}}\,I_{jq}\,
\left[\,\frac{\,a_{jq}^{+}\,\exp{\{\,i\,t\omega \,-\,i\,qr\,\}}}
{\omega _o^2\,-\,\omega ^2\,+\,i\tau \omega ^3\,}\,-
\,\frac{\,a_{jq}\,\exp{\{-\,i\,t\omega \,+\,i\,qr\,\}}}
{\omega _o^2\,-\,\omega ^2\,-\,i\tau \omega ^3}\,\right],$$
The analytical presentation (\[NN\]) of the SchEl’s momentum components have been calculated through using the relation known from Maxwell ClsElcDnm : $$\label{OO}
\hat{P}_j\,=\,m\,\dot{r}_j\,-\,(\frac{e}{C})\left[\,A_j\,+\,A_j^i\,\right],$$
Further they have calculated the well-known Heisenberg’s commutation relations (HsnCmtRlts) between the operators of the dynamic variables $\hat{r}_j$ (\[MM\]) and $\hat{P}_j$ (\[NN\]) by virtue of the following definition : $$\label{PP}
{\hat P}_j\,{\hat r}_k\,-\,{\hat r}_k\,{\hat P}_j\,\approx
\,-i\,\hbar\,\delta_{jk}$$
Since then it is easily to understand by means of the upper account that if the ClsMacrPrt’s motion is occurred along a clear definite smooth thin trajectory in the NrlClsMch, then the QntMicrPrt’s motion is performed in a form of the RndTrmMtn along a pete very small line, stochastically orientated in the space near the clear-cut smooth thin trajectory in the NrlQntMch. As a result of that we can suppose that the QntStchBhv of the QntMicrPrt can be described by means of the following physical quantities in the NrlQntMch : $$\label{QQ}
\quad r_j\,=\,{\bar r}_j\,+\,\delta{r}_j\quad;
\quad p_j\,=\,\bar{p}_j\,+\,\delta{p}_j\quad;$$
Mathematical description of the minimal dispersions of some dynamical parameters of a QntMicrPrt
=================================================================================================
Indeed,because of the eqs.(\[QQ\]) the values of the averaged physical parameters in the NrlQntMch $\langle p_j^2 \rangle$ is different from the values of the same physical parameters in the NrlClsMch $\bar{p}_j^2$ as it is seen : $$\label{RR}
\quad\langle{r}_j^2\rangle\,=\,{\bar r}_j^2\,+\,\langle\delta{r}_j^2\rangle\,;
\quad\langle{p}_j^2\rangle\,=\,\bar{p}_j^2\,+\,\langle\delta{p}_j^2\rangle\,;$$
In spite of that the averaged value of the orbital (angular) mechanical momentum of the QntMicrPrt has the following value : $$\label{SS}
\,\langle L^2\rangle\,=\,\sum_j\,(\bar{L}_j)^2\,+
\,\sum_j\,\langle(\delta{L}_j)^2\rangle\,=
\,(\bar{L}_x)^2\,+\,\langle(\delta{L}_x)^2\rangle\,+
\,(\bar{L}_y)^2\,+\,\langle(\delta{L}_y)^2\rangle\,+
\,(\bar{L}_z)^2\,+\,\langle(\delta{L}_z)^2\rangle\,;$$
or at the $(\bar{L}_x)^2\,=\,0$ and $(\bar{L}_y)^2\,=\,0$ we must obtain : $$\label{TT}
\,\langle L^2\rangle\,=
\,(\bar{L}_z)^2\,+\,\langle(\delta{L}_z)^2\rangle\,+
\,\langle(\delta{L}_y)^2\rangle\,+\,\langle(\delta{L}_x)^2\rangle\,$$
As both the value of the $\,\langle(\delta{L}_x)^2\rangle\,$ and $\,\langle(\delta{L}_y)^2\rangle\,$ are equal of the $\,\frac{\bar{L}_z\hbar^2}{2}\,$ and the value of the $\,\langle(\delta{L}_z)^2\rangle\,$ is equal of the $\,\frac{\hbar^2}{4}\,$. Therefore : $$\label{UU}
\quad\langle L^2\rangle\,=\,l^2\hbar^2\,+\,l\hbar^2\,+
\,\frac{\hbar^2}{4}\,=\,(\,l\,+\,\frac{1}{2}\,)^2\hbar^2\,;$$
The realized above investigation assists us to come to the conclusion that the dispersions of the dynamical parameters of the QntMicrPrt are natural results of their forced stochastic oscillation motions along the very small line stochastically orientated in space near to the classical clear-cut smooth thin line of the corresponding dynamical parameters values of the ClsMacrPrt, owing to ElcMgnIntAct of its OvrSpr ElmElcChrg or MgnDplMm with the intensities of the RslElcFld or RslMgnFld of the QntElcMgnFlds of the StchVrtPhtns at its motion through the FlcVcm. It is turned out that the kinetic energy of the IstThrDmnNrlQnt FrthStchBsnCrcHrmOscs, which the QntMicrPrt takes from the FlcVcm, called as its localized energy, one ensures the stability of the SchEl in its ground state in the H-like atom. We have the ability to obtain the minimal value of the dispersion product, determined by the Heisenberg uncertainty relation.
In a consequence of what was asserted above in order to obtain the QntQdrDfr WvEqn of Sch we must add to the kinetic energy $\,\frac{(\nabla_l\,S_1)^2}
{2m}\,$ of the NtnClsPrt in the following ClsQdrDifPrtEqt of Hml-Jcb : $$\label{f1}
-\frac{\partial S_1}{\partial t}\,=\,\frac{(\nabla_j\,S_1)^2}{2m}\,+\,U\,;$$
the kinetic energy $\,\frac{(\nabla_l\,S_2)^2}{2m}\,$ of the BrnClsPrt. In such the natural way we obtain the following analytic presentation of the QntQdrDfrWvEqt of Sch : $$\label{f2}
-\frac{\partial S_1}{\partial t}\,=\,\frac{(\nabla_j\,S_1)^2}{2m}\,+
\,\frac{(\nabla_j\,S_2)^2}{2m}\,+\,U\,;$$
The purpose of our investigation in henceforth is to obtain the eq. (\[f2\]) by means of physically obvious and mathematically correct proof. Therefore we could desire a voice of a supposition that all uncommon ways of the SchEl’s behaviour in the NrlQntMch or of other QntMicrPrts in the micro world are natural consequences of unconstrained stochastic joggles on account of continuously accidently exchanges of LwEnr-StchVrtPhtn between its WllSpr ElmElcChrg and the VcmFlc. In consequence of the absence of SchEl’s trajectory within the NrlQntMch as within the NrlClsMch and the stochastical character of its random trembling motion together with the probably interpretation of the SchEl’s OrbWvFnc module square are naturally consequences of the continuous ElcMgnIntAct between the SchEl’s WllSpr ElmElcChrg and EfcElcInt $E_j$ of existent LwEnr-VrtPhtns, stochastically generated by fluctuating energy within FlcVcm through continuous incident exchange of LwEnr-StchVrtPhtns, which are either emitted or adsorbable by either the VcmFlcs or the Schel’s Wllspr ElmElcChrg. Really, a deep understanding of the physics of the random trembling motion, in accordance with the description of the Brownian stochastic behaviour of BrnCslPrts we can determine both as the value $V^{-}$ of the SchEl’s velocity before the moment $t$ of the scattering time of some LwEnr-StchVrtPhtns from its WllSpr ElmElcChrg, so the value $V^{+}$ after the same moment $t$ of the scattering time by means of the following definitions : $$\label{q1}
V_j^{-}\,=\,lim_{\Delta t\to\,o} \left\{\,\frac{r(t)_j\,-\,r(t-\Delta t)_j\,}
{\Delta t}\,\right\}\,= \,(\,V_j\,-\,i\,U_j\,)\,;$$
$$\label{q2}
V_j^{+}\,=\,lim_{\Delta t\to\,o}\left\{\,\frac{r(t+\Delta t)_j\,-\,r(t)_j\,}
{\Delta t}\,\right\}\,=\,(\,V_j\,+\,i\,U_j\,)\,;$$
In addition we may determine two new velocities $V_j$ and $U_j$ by dint of the following equations : $$\label{r}
2\,V_j\,=\,V_j^+\,+\,V_j^- \qquad {\rm and} \qquad
2\,i\,U_j\,=\,V_j^+\,-\,V_j^-\,,$$
In conformity with the eq.(\[r\]) it is obviously followed that the current velocity $V$ describes the regular drift of the SchEl and the osmotic velocity $U$ describes its nonrelativistic quantized stochastic bozon oscillations. Afterwards by virtue of the well-known definition equations : $$\label{s1}
2\,m\,V_j\,=\,m\,(\,V_j^+\,+\,V_j^-\,)\,=\,2\,\nabla_j\,S_1$$
and $$\label{s2}
2\,i\,m\,U_j\,=\,m\,(\,V_j^+\,-\,V_j^-\,)\,=\,2\,i\,\nabla_j\,S_2$$
one can obtain the following presentation of the SchEl’s OrbWvFnc $\psi(r,t)$ : $$\label{t}
\psi(r,t)\,=\,\exp\{\,i\,\frac{S_1}{\hbar}\,-\,\frac{S_2}{\hbar}\,\}\,=
\,B\,\exp\{\,i\,(\frac{S_1}{\hbar}\}$$
It is easily to verify the results (\[r\]), (\[s1\]) (\[s2\]). In an effect ones may be obtained by means of the following natural equations : $$\label{ur1}
m\,V_j^+\,\psi(r,t)\,=
\,-\,i\,\hbar\,\nabla_j\,\exp\{\frac{i\,S_1}{\hbar}\,-\,\frac{S_2}{\hbar}\}\,
=\,(\,\nabla _j S_1\,+\,i\,\nabla _j S_2\,)\,\psi(r,t)$$
and $$\label{ur2}
m\,V_j^-\,\psi(r,t)^+\,=
\,+\,i\,\hbar\,\nabla _j \exp\{\frac{i\,S_1}{\hbar}\,-\,\frac{S_2}{\hbar}\}\,
=\,(\,\nabla _j S_1\,-\,i\,\nabla _j S_2\,)\,\psi(r,t)^+$$
Indeed, $$\begin{aligned}
\label{us1}
2\,m\,V_j\,=\,m\,(\,V_j^+\,+\,V_j^-\,)\,= \nonumber \\
\,\left\{\,(\,\nabla _j S_1\,+\,i\,\nabla _j S_2\,)\,+
\,(\,\nabla _j S_1\,-\,i\,\nabla _j S_2\,)\,\right\}\,
\quad {\rm or} \quad 2\,m\,V_j\,=\,2\,\nabla _j S_1\,\end{aligned}$$
and $$\begin{aligned}
\label{us2}
2\,i\,m\,U_j\,=\,m\,(\,V_j^+\,-\,V_j^-\,)\,= \nonumber \\
\,\left\{\,(\,\nabla _j S_1\,+\,i\,\nabla _j S_2\,)\,-
\,(\,\nabla _j S_1\,-\,i\,\nabla _j S_2\,)\,\right\}\,
\quad {\rm or} \quad 2\,i\,m\,U_j\,=\,2\,i\,\nabla _j S_2\,\end{aligned}$$
In consequence we could assume that the module square of the SchEl’s OrbWvFnc $\psi(r,t)$ describes the probability density of its location close by the space point $r$ at the time moment $t$ in the good light of our obvious interpretation. Further in order to obtain the partial differential equation of the continuity we are going to calculate one by virtue of its well-known definitions : $$\begin{aligned}
\label{v1}
\frac{\partial {\mid\psi\mid}^2}{\partial t}\,+
\,\nabla _j (V_j^{+}\,{\mid\psi\mid}^2\,)\,=
\,\frac{\partial (\exp{\{-\,2\,\frac {S_2}{\hbar }\}})}{\partial t}\,+
\,\nabla _j \left[\,(\nabla _j \frac{S_1}{m}\,+\,i\,\nabla _j \frac{S_2}{m})\,
\exp{\{-\,2\,\frac{S_2}{\hbar}\}}\,\right]\,= \nonumber \\
\,\left[\,-\frac{2}{\hbar}\,\frac{\partial {S_2}}{\partial t}\,+
\,\frac{1}{m}\,(\nabla _j)^2 {S_1}\,+
\,\frac{i}{m}\,(\nabla _j)^2 {S_2}\,-
\,\frac{2}{m\hbar}\,\,\nabla _j {S_1}\,\nabla _j {S_2}\,-
\,\frac{2i}{\hbar}\,\nabla _j {S_2}\,\nabla _j {S_2}\,\right]\,
\left [\,\exp{\{\,-\,2\,\frac{S_2}{\hbar}\}}\,\right ]\end{aligned}$$
$$\begin{aligned}
\label{v2}
\frac{\partial {\mid\psi\mid}^2}{\partial t}\,+
\,\nabla _j (V_j^{-}\,{\mid\psi\mid}^2\,)\,=
\,\frac{\partial (\exp{\{-\,2\,\frac{S_2}{\hbar}\}})}{\partial t}\,+
\,\nabla _j \left[\,(\nabla _j \frac{S_1}{m}\,-\,i\,\nabla _j \frac{S_2}{m})\,
\exp{\{-\,2\,\frac{S_2}{\hbar}\}}\,\right]\,= \nonumber \\
\,\left[\,-\frac{2}{\hbar}\,\frac{\partial {S_2}}{\partial t}\,-
\,\frac{1}{m}\,(\nabla _j)^2 {S_1}\,-
\,\frac{i}{m}\,(\nabla _j)^2 {S_2}\,-
\,\frac{2}{m\hbar}\,\nabla _j\,{S_1}\,\nabla _j\,{S_2}\,+
\,\frac{2i}{\hbar}\,\nabla _j\,{S_2}\,\nabla _j\,{S_2}\,\right]\
\,\left[\,\exp{\{-\,2\,\frac{S_2}{\hbar }\}}\,\right]\end{aligned}$$
With the purpose to calculate the last expressions of the continuity equations (\[v1\]) and (\[v2\]) we are going to turn the expression (\[t\]) of the SchEl’s OrbWvFnc $\psi(r,t)$ in the quadratic differential wave equation in partial derivatives of Schrodinger : $$\label{w}
\,i\,\hbar\,\frac{\partial \psi(r,t)}{\partial t}\,=
\,-\,\frac{\hbar^2}{2}\,\frac{(\nabla_j)^2}{m}\,\psi(r,t)\,+\,U(r,t)\,\psi(r,t)$$
Further we are able to obtain the following result : $$\begin{aligned}
\label{z}
\left(\,-\,\frac{\partial {S_1}}{\partial t}\,-
\,i\,\frac{\partial {S_2}}{\partial t}\,\right)\,\psi(r,t)\,= \nonumber \\
\,\left\{\,\frac{(\nabla _j {S_1})^2}{2m}\,-
\,\frac{(\nabla_j {S_2})^2}{2m}\,+
\,\frac{\hbar}{2m}\,(\nabla_j)^2 {S_2}\,-
\,i\,\frac{\hbar}{2m}\,(\nabla_j)^2 {S_1}\,+
\,\frac{i}{m}\,\nabla_j {S_1}\,\nabla_j {S_2}\,+
\,U(r,t)\,\right\}\,\psi(r,t)\,\end{aligned}$$
As there exist both the real and imaginary parts in the complex valued eq. (\[z\]) , it is obviously that from this one follows two quadratic differential equations in partial derivatives : $$\label{aa1}
\frac{\partial {S_2}}{\partial t}\,=
\,\frac{\hbar}{2m}\,(\nabla _j)^2\,{S_1}\,-
\,\frac{1}{m}\,(\nabla _j {S_1})\,(\nabla _j {S_2})$$
and $$\label{aa2}
-\,\frac{\partial {S_1}}{\partial t}\,=
\,\frac{1}{2\,m}(\nabla _j {S_1})^2\,-
\,\frac{1}{2\,m}(\nabla _j {S_2})^2\,+
\,\frac{\hbar}{2\,m}(\nabla _j)^2 {S_2}\,+\,U(r,t)$$
Inasmuch as it is well-known from the NrlQntMch the continuity partial differential equation can be obtained by means of the eqs.(\[aa1\]), (\[s1\] and (\[t\]) in the following form : $$\begin{aligned}
\label{ab}
\frac{\partial {\left|\psi\right|^2}}{\partial t}\,+
\,\nabla _j \left(V_j\,{\left|\psi\right|^2}\,\right)\,=
\,\frac{\partial \exp{\{-\,2\,\frac{S_2}{\hbar}\}}}{\partial t}\,+
\,\frac{1}{m}\,\nabla _j \left(\nabla _j {S_1}
\,\exp{\{-\,2\,\frac{S_2}{\hbar}\}}\,\right)\,=\,0 ;\end{aligned}$$
Thence the eq.(\[v1\]) and eq.(\[v2\]) can be simplified by means of the eqs.(\[ab\]) and (\[aa1\]). In a result of such substitutions the following continuity partial differential equations could be obtained : $$\label{ac1}
\frac{\partial {\left|\psi\right|^2}}{\partial t}\,+
\,\nabla _j \left(V_j^{+}\,{\left|\psi\right|^2}\right)\,=
\,\frac{i}{m}\,\nabla _j \left(\,\nabla _j {S_2}
\,\exp{\{-2\frac{S_2}{\hbar}\}}\,\right)$$
$$\label{ac2}
\frac{\partial {\left|\psi\right|^2}}{\partial t}\,+
\,\nabla _j \left(V_j^{-}\,{\left|\psi\right|^2}\right)\,=
\,-\,\frac{i}{m}\,\nabla _j \left(\nabla _j {S_2}
\,\exp{\{-2\frac{S_2}{\hbar}\}}\right)$$
In order to calculate the value of the expressions in the brackets in the right-hand side of the eqs.(\[ac1\]) and (\[ac2\]) we will determine the relation between the values of both integrals : $$\label{ad1}
\,\int\,\int_{V_R}\,\int\,{\nabla_j}^2\,{S_2}\,
\exp{\{-2\frac{S_2}{\hbar}\}}\;dV
\quad {\rm and} \quad
\,\int\,\int_{V_R}\,\int\,(\nabla _j {S_2}\,)^2\,
\exp{\{-2\frac{S_2}{\hbar}\}}\;dV$$
The first integral in (\[ad1\]) may be calculated through integration by parts. In this easily way we could obtain : $$\begin{aligned}
\label{ad2}
\,\int\,\int_{V_R}\,\int\,(\nabla _j)^2\,{S_2}
\,\exp{\{-2\frac{S_2}{\hbar}\}}\,dV\,=
\,\int\,\int_{S_R}\,\nabla _j {S_2}
\,\exp{\{-2\frac{S_2}{\hbar}\}}\,dS_j\,- \nonumber \\
\,\int\,\int_{S_o}\,\nabla_j {S_2}
\,\exp{\{-2\frac{S_2}{\hbar}\}}\,dS_j\,+
\,\frac{2}{\hbar}\,\int\,\int_{V_R}\,\int\,(\nabla _j {S_2})^2\,
\exp{\{-2\frac{S_2}{\hbar}\}}\,dV\end{aligned}$$
From above it is evidently that the second two-dimensional integral over the surface $S_o$ cannot exist in the case when the integrational domain $V_R$ of the three-dimensional integral has the form of one-piece-integrity domain. Indeed, the three-multiple integral in the left-hand side of eq. (\[ad2\]) has an integration domain of the volume $V_R$,then the both two-multiple integrals (the first and second ones on the right handside of the same equation) have a integration domain in form of surface of same volume (the outer skin $S_R$ and the inter skin $S_o$ of the volume $V_R$). Inasmuch as we don’t take into account the creation and annihilation of the FrthQntMicrPrt in the NrlQntMch, than the SchEl’s OrbWvFnc $\psi(r,t)$ may have no singularity within the volume $V_R$. Therefore the three-multiple integrals have the one-piece integrity domain of an integration without its inter skin surface $S_o$. Hence it is easily seen that both two-multiple integrals are canceled in the case when R go to $\infty$ and at the absence of any kind of singularity in the SchEl’s OrbWvFnc. Consequently eq.(\[ad1\]) becomes the form : $$\begin{aligned}
\label{ae}
\,\int\,\int_{V_\infty}\,\int\,(\nabla _j)^2\,{S_2}
\,\exp{\{-2\frac{S_2}{\hbar}\}}\;dV\,
=\,\frac{2}{\hbar}\,\int\int_{U_\infty}\,\int\,(\nabla _j {S_2}\,)^2
\,\exp{\{-2\frac{S_2}{\hbar}\}}\;dV\end{aligned}$$
Then in a result of the existence of the eqt.(\[ae\]) we may suppose the existence of the following equations between the values of both integrand functions : $$\label{af1}
\quad {\rm the\,first\,:} \quad
(\nabla_j)^2\,{S_2}\,\exp{\{-\,2\,\frac{S_2}{\hbar}\,\}}\,=
\,\frac{2}{\hbar}\,(\nabla _j {S_2})^2\,\exp{\{-\,2\,\frac{S_2}{\hbar}\,\}}\,$$
$$\label{af2}
\quad {\rm and\,the\,second\,:} \quad
(\nabla _j)^2 {S_2}\,=\,\frac{2}{\hbar}\,(\nabla _j {S_2})^2\,$$
Hence it is obviously seen that in a line with the existence of the eq. (\[af2\]) the equation (\[aa2\]) could been rewritten in the following transparent form : $$\label{ag1}
\,-\,\frac{\partial {S_1}}{\partial t}\,=
\,\frac{1}{2\,m}(\nabla _j {S_1})^2\,+
\,\frac{1}{2\,m}(\nabla _j {S_2})^2\,+\,U(r,t)$$
In such a way it is evidently that the right-hand side expressions of the equations of the continuity (\[ac1\]) and (\[ac2\]) are canceled by the virtue of the eq:(\[af2\]).Consequently we had an opportunity to shoe that the continuity partial differential equations are satisfied not only in the form (\[ab\]), but they are satisfied also in the forms (\[ac1\]) and (\[ac2\]). Furthermore the expression of the eq.(\[ag1\]) might been interpreted from my new point of view, that the kinetic energy $E_k$ of the SchrEl is formed by two differential parts. Really, if the first part $\frac{(\nabla_j\,{S_1})^2}{2\,m}$ describes the kinetic energy of its regular translation motion along some clear-cut thin smooth classical trajectory in an accordance with the laws of the NrlClsMch and ClsElcDnm with its current velocity $V_j\,=\,\frac{1}{m}\,\nabla _j{S_1}$ , then the second part $\frac{(\nabla_j\,{S_2})^2}{2\,m}$ mouth describe the kinetic energy of its Furthian quantum stochastic motion of the FrthQntMicrPrt with its probable velocity $U_j\,=\,\frac{1}{m}\,\nabla _j {S_1}$ in a total analogy with the Brownian classical stochastic motion of the BrnClsMicrPrt with its osmotic velocity.Therefore it is very helpfully to rewrite the expression (\[ag1\]) in the following well-known form : $$\label{ag2}
\,E\,=\,\frac{m\,V^2}{2}\,+\,\frac{m\,U^2}{2}\,=
\,\frac{(\langle \bar P \rangle)^2}{2m}\,+
\,\frac{\langle (\Delta P)^2 \rangle)}{2m}\$$
Indeed, some new facts have been brought to light. Therefore the upper investigation entitles us to make the explicit assertion that the most important difference between the quadratic differential wave equation in partial derivative of Schrodinger and the quadratic differential particle equation in partial derivative of Hamilton-Jacoby is exhibited by the existence of the kinetic energy of the QntMicrPrt’s Furthian trembling circular oscillations harmonic motion in the first one.
$$\label{g}
-\frac{\partial S_1}{\partial t}\,=\,\frac{(\nabla_j\,S_1)^2}{2m}\,+
\,\frac{(\nabla_j\,S_2)^2}{2m}\,+\,U\,;$$
As we can observe by cursory comparison there is a total coincidence of eq.(\[f1\]) with eq.(\[g\]). Hence we are able to proof that the QdrDfrPrtEqt with PrtDrv of Schrodinger may be obtained from the QdrDfrPrtEqt with PrtDrv of Hamilton-Jacoby by addition the part of the kinetic energy of the Furthian stochastic circular harmonic oscillations motion. Indeed, it is obviously to understand that the first term $\,\frac{(\nabla_l\,S_1)^2} {2m}\,$ in the eq.(\[g\]) describes the kinetic energy of the regular translation motion of the NtnClsPrt with its current velocity $\,V_l\,=\,\frac{\nabla_l\,S_1}{m}\,$ and the second term $\,\frac{(\nabla_l\,S_2)^2}{2m}\,$ describes the kinetic energy of the random trembling circular harmonic oscillations motion (RndTrmMtn) of the FrthQntPrt in a total analogous with BrnClsPrt with its osmotic velocity $U_l\,=\,\frac{\nabla_l\,S_2}{m}\,$. Therefore we can rewrite the expression (\[g\]) in the following form : $$\label{h}
\,E_t\,=\,\frac{m\,V^2}{2}\,+\,\frac{m\,U^2}{2}\,+\,U\,=
\,\frac{{\langle\,\bar P\,\rangle}^2}{2\,m}\,+
\,\frac{\langle\,(\Delta P)^2\,\rangle}{2\,m}\,+\,U\,;$$
After elementary physical obviously suppositions some new facts have been brought to light. Therefore the upper investigation entitles us to make the explicit assertion that the most important difference between the QntQdrDfr WvEqt with PrtDrv of Schodinger and the ClsQdrDfrPrtEqt with PrtDrv of Hamilton-Jacoby is exhibited by the existence of the kinetic energy of the FrthRndTrmCrcHrmOscsMtn in the first one. Therefore when the SchEl is appointed in the Coulomb’s potential of the atomic nucleus spotted like (SptLk) elementary electric charge (ElmElcChrg) $Ze$ its total energy may be written in the following form : $$\label{i}
\langle\,E_t\,\rangle\,=\,\frac{1}{2\,m}\,\left[(\langle P_r \rangle)^2\,+
\,\frac{(\langle L \rangle)^2}{(\langle r \rangle)^2}\,\right]\,+
\,\frac{1}{2\,m}\,\left[\langle(\Delta P_r)^2 \rangle\,+
\,\frac{\langle(\Delta L)^2 \rangle}{\langle r \rangle^2}\,\right]\,-
\,\frac{Z e^2}{\langle r \rangle}$$
As any SchEl has eigenvalues $n_r\,=\,0\,$ and $l\,=\,0\,$ in a case of its ground state, so it follows that $\langle P_r \rangle \,=\,0\,$ and $\langle L \rangle\,= \,0\,$. As a consistency with the eq.(\[k\]) the eigenvalue of the SchEl’s total energy $E_t^o$ in its ground state in some H-like atom is contained only by two parts : $$\label{j}
\langle\,E^o_t\,\rangle\,=
\,\frac{1}{2\,m}\,\left[\langle(\Delta P_r)^2 \rangle\,+
\,\frac{\langle(\Delta L)^2 \rangle}{\langle( r )^2\rangle}\,\right]\,-
\,\frac{Z e^2}{\langle r \rangle}$$
Further the values of the dispersions $\langle (\Delta P_r)^2\rangle$ and $\langle(\Delta L)^2\rangle$ can be determined by virtue of the Heisenberg Uncertainty Relations (HsnUncRlt): $$\label{k}
\,\langle (\Delta P_r)^2\rangle\,\times\,\langle (\Delta r)^2\rangle\,\ge
\,\frac{\hbar^2}{4}$$ $$\label{l}
\langle (\Delta L_x)^2\rangle\,\times\,\langle (\Delta L_y)^2\rangle\,\ge
\,\frac{\hbar^2}{4}\,\langle (\Delta L_z)^2\rangle\,$$
Thence the dispersion $\langle(\Delta P_r)^2\rangle$ will really have its minimal value at the maximal value of the $\langle(\Delta r)^2\rangle\,=
=\,\langle r \rangle^2$.In this way the minimal dispersion value of the $
\langle(\Delta P_r)^2\rangle$ can be determined by the following equation : $$\label{m}
\,\langle(\Delta P_r)^2\rangle\,=\,\frac{\hbar^2}{4\langle r^2\rangle}\,$$
As the SchEl’s ground state has a spherical symmetry at $l\,=\,0\,$,then the following equalities take place : $$\label{n}
\,\langle(\Delta L_x)^2\rangle\,=\,\langle(\Delta L_y)^2\rangle\,=
\,\langle(\Delta L_z)^2\rangle\,;$$
Hence we can obtain minimal values of the dispersions (\[n\]) through division of the eq.(\[k\]) with the corresponding equation from the eq. (\[n\]).In that a way we obtain the following result: $$\label{o}
\,\langle(\Delta L_x)^2\rangle\,+\,\langle(\Delta L_y)^2\rangle\,+
\,\langle(\Delta L_z)^2\rangle\,=\,\frac{3\hbar^2}{4}\;$$
Just now we are in a position to rewrite the expression (\[k\]) in the handy form as it is well-known : $$\label{p}
\,E_t^o\,=\,\frac{1}{2\,m}\,\left[\,\frac{\hbar^2}{4r^2}\,+
\,\frac{3\hbar^2}{4r^2}\,\right]\,-\,\frac{Z\,e^2}{r}\,=
\,\frac{1}{2}\,\frac{\hbar^2}{m\,r^2}\,-\,\frac{Z\,e^2}{r}\,;$$
It is extremely important to note here that we have used undisturbed ElcInt $E_j$ (\[HH\]) of the QntElcMgnFld of StchVrtPhtns from the FlcVcm by dint of the equations (\[GG\]) and (\[HH\]) in order to obtain constrain of dynamical mutual conjugated quantities $r_j$ (\[MM\]) and $P_x$ (\[NN\]) from the NrlClsMch in their operator forms ${\hat r}_j$ and ${\hat P}_j$ within NrlQntMch. The quantum behaviour of the SchEl within NrlQntMch is caused by the ElcIntAct between its WllSpr ElmElcChrg and the ElcInt $E_j$ of the undisturbed QntElcMgnFld of StchVrtPhtns from the FlcVcm. So in consequence of the continuous ElcIntAct of the SchEl’s WllSpr ElmElcChrg with the ElcInt of the QntElcMgnFld of StchVrtPhtns one participates in the Furthian quantized stochastic motion (FrthQntStchMtn), which is quite obviously analogous of the Brownian classical stochastic motion (BrnClsStchMtn). As it is well-known the BrnClsPrts have no classical wave properties (ClsWvPrp), but the FrthQntPrts have QntWvPrps and display them every where. The cause of this distinction consists of indifference between the liquid and FlcVcm. Indeed, if atoms and molecules within liquid have no ClsWvPrps,all excitations of the FlcVcm and one itself have QntWvPrp. Therefore the FlcVcm transfers its QntWvPrp over the SchEl at ones ElcIntAct with its WllSpr ElmElcChrg.
[99]{}
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---
abstract: 'Recently certain non-supersymmetric solutions of type IIb supergravity were constructed [@ross], which are everywhere smooth, have no horizons and are thought to describe certain non-BPS microstates of the D1-D5 system. We demonstrate that these solutions are all classically unstable. The instability is a generic feature of horizonless geometries with an ergoregion. We consider the endpoint of this instability and argue that the solutions decay to supersymmetric configurations. We also comment on the implications of the ergoregion instability for Mathur’s ‘fuzzball’ proposal.'
author:
- Vitor Cardoso
- 'Óscar J. C. Dias'
- 'Jordan L. Hovdebo'
- 'Robert C. Myers'
title: 'Instability of non-supersymmetric smooth geometries'
---
Introduction
============
String theory has made great progress in understanding the microphysics of black holes. In particular, for certain (nearly) supersymmetric black holes, one is able to show that the Bekenstein-Hawking entropy $S_{\rm BH}=A_{\rm hor}/4G$, as computed in the strongly-coupled supergravity description, can be reproduced in a weakly-coupled D-brane description as the degeneracy of the relevant microstates [@2] — for reviews, see [@revall]. The AdS/CFT correspondence [@big] provides further insights into these issues by providing a dictionary relating the geometric description of the physics in the near-horizon region with the physics of a dual conformal field theory — see [@adscft] for a review. In particular, the AdS/CFT indicates that Hawking evaporation should be a unitary process, in keeping with the basic tenets of quantum theory. The discussion of black holes in the context of the AdS/CFT correspondence makes evident that the path integral over geometries in the bulk may include multiple saddle-points, several classical supergravity solutions, as found in [@thermal1; @thermal2; @fairy].[^1] Another point that was realized early on is that the geometric description of individual microstates would not have a horizon [@pure; @amati].
In recent years, Mathur and collaborators have incorporated these ideas in a radical revision of the stringy description of black holes — for a review, see [@fuzzy]. They argue that each of the CFT microstates corresponds to a separate spacetime geometry with no horizon. The black hole is dual to an ensemble of such microstates and so the black hole geometry only emerges in a coarse-grained description which ‘averages’ over the $e^{S_{\rm
BH}}$ microstate geometries. In particular, this averaging should produce an effective horizon at a radius where the individual microstate geometries start to ‘differ appreciably’ from one another [@11; @10]. Therefore in this scenario, quantum gravity effects are not confined close to the black hole singularity, rather the entire interior of the black hole is ‘filled’ by fluctuating geometries — hence this picture is often referred to as the ‘fuzzball’ description of black holes. The first support for this proposal came from finding agreement between the propagation time of excitations in the throat of certain microstate geometries and in the dual brane description [@10; @19]. A further remarkable feature, that has drawn attention to these ideas, is that there is growing evidence that the microstate geometries may be smooth, as well as horizon-free.[^2] In the case of the D1-D5 system, smooth asymptotically flat geometries can be constructed corresponding to all of the RR ground states in the dual CFT [@10; @two]. Despite their large degeneracy, this two-charge system will not produce a macroscopic black hole horizon. However, a large horizon can be produced by introducing a third charge, Kaluza-Klein momentum [@early; @early1]. Recently progress has been made in constructing smooth microstate geometries in the D1-D5-P system [@three1; @three12; @three2]. While large families of such solitons are now known, a complete understanding of the three-charge case remains to be found. Further preliminary work on the four charge system of D1-D5-P-KK has also appeared [@four].
In general, the preceding discussion connecting microstates with smooth geometries focuses on supersymmetric configurations. This raises the interesting question of how the fuzzball proposal would be extended to non-supersymmetric black holes. In particular, are there non-supersymmetric versions of the smooth horizon-free geometries corresponding to non-BPS microstates? Remarkably, Jejjala, Madden, Ross and Titchener [@ross] recently extended the known set of D1-D5 microstate geometries with a family of non-supersymmetric solutions, hereafter referred to as solitons. The solutions comprise a five-parameter family of non-supersymmetric smooth geometries which are asymptotically flat.[^3] These solutions may be parameterized by the D1-brane and D5-brane charges, the (asymptotic) radius of the internal circle with Kaluza-Klein momentum, and by two integers $m$ and $n$ which fix the remaining physical parameters. These integers also determine a spectral flow in the CFT which allows the underlying microstate to be identified. For $m=n+1$, the solitons reduce to supersymmetric solutions found previously in [@two; @three1; @three12].
An important feature which distinguishes the solitons from any of the analogous supersymmetric solutions is the presence of an ergoregion. As a consequence, in these non-supersymmetric geometries, there is an inner region (that extends to the origin) where states of negative energy are allowed. This then leads naturally to the question of whether or not the ergoregion produces an instability of the background. One possibility is that the ergoregion may lead to superradiant scattering which can produce a catastrophic instability in some situations [@press; @dnr; @bhb]. However in the present case, this possibility is easily dismissed [@ross] because the solutions are horizon-free. Since the seminal work of Zel’dovich [@zel] on superradiant amplification of electromagnetic waves incident upon an absorbing cylinder, it has been known that the key ingredients for superradiance is the existence of an ergoregion [*and*]{} an absorbing surface. For black holes, the horizon plays the latter role but certainly the geometries lack such a surface.
Quite interestingly, there is another class of instabilities, which we simply refer to as ‘ergoregion instabilities’, that generically afflict spacetime geometries with an ergoregion, but no horizon. These instabilities were first discovered by Friedman [@friedman], who provided a very general discussion. Explicit computations of the instability were later made in [@cominsschutz; @compute] for the case of rotating stars with an ergoregion. There the existence of this instability was explicitly verified for a free scalar field in the background of a rotating star. According to Friedman’s general arguments however, the instability should also exist for electromagnetic and gravitational waves. Since the solutions [@ross] have an ergoregion but no horizon, one might suspect that a similar ergoregion instability would arise in these geometries. The present paper then explicitly verifies the presence of an ergoregion instability for the backgrounds with a variety of techniques. Further we consider the endpoint of the resulting decay and argue that it should be a smooth supersymmetric solution.
Our results have immediate consequences for the endpoint of tachyon decay discussed in [@rossTC]. There, Ross extended the discussion of [@horowitzTC] to D1-D5 black strings for which he identified tachyonic string modes in a particular winding sector. He argued that the condensation of these tachyons would transform the spacetime to a soliton. In conjunction with the above results, we see that these solutions cannot be the final endpoint of these decays but rather they should end with a supersymmetric microstate geometry. Our analysis and the ergoregion instability may also have interesting implications for Mathur’s fuzzball proposal more generally.
The remainder of our paper is organized as follows: Section \[general\] provides a brief exposition on Friedman’s analysis [@friedman]. In Section \[formalism\], we briefly review some of the features of the solutions and present the main equations used in the subsequent analysis, namely the radial and angular wave equations for a free massless scalar field, as well as some of their properties. In Section \[wkb\] we compute the details of the instability using a WKB approach [@cominsschutz]. We show explicitly that the instability exists for a general non-supersymmetric geometry of [@ross], and that it disappears for supersymmetric objects, as expected. In Section \[sec:Match\], we use an alternative method, that of matched asymptotic expansions, to investigate the instability and its properties. The methods of sections \[wkb\] and \[sec:Match\] are complementary, their regime of validity is different. We then perform a numerical analysis of the wave equation in Section \[numerical\] to complement the analytical calculations. We find that the results of both analytical analyses agree remarkably well with the numerical results. In section \[conclusion\], after summarizing the main properties of the ergoregion instability, we discuss various related topics: the endpoint of this instability; its consequences for Ross’s tachyon condensation [@rossTC]; general implications for the fuzzball picture of black holes.
\[general\][Ergoregion instabilities]{}
=======================================
There are two classes of instabilities that are of potential interest for the backgrounds [@ross] (or non-supersymmetric geometries in general), namely: the superradiant instability, and the ergoregion instability. In this section, we demonstrate why superradiance is not present in these geometries, as first noted in [@ross], and we introduce the general argument of [@friedman] that suggests an ergoregion instability is present. In the following sections, we verify the presence of the ergoregion instability with a complete analytic and numerical analysis of its properties.
Geometries with an ergoregion and horizon: Superradiance
--------------------------------------------------------
For a general (stationary asymptotically flat) black hole, the equations describing spin-$s$ fields can always be written as +V(,r)=0\[wave\]where $\omega$ was introduced with a Fourier transform with respect to the asymptotic time coordinate: $\Psi(t)=e^{-i\omega
t}\Psi(\omega)$. The radius $r_*$ is a convenient tortoise coordinate and in general one finds: $$\left\{
\begin{array}{ll}
r_* \sim r \,,& V \sim \omega ^2\,\,\, \,\,\,\,\quad
\quad {\rm as}\ r\rightarrow \infty \,, \\
e^{r_*} \sim (r-r_+)^{\alpha} \,,& V \sim (\omega-\Phi)^2\,\,\,
{\rm as}\ r\rightarrow r_+ \,,
\end{array}
\right. \label{bound}$$ where $\alpha$ is a positive constant. The potential $\Phi$ can be a rotational potential (in the Kerr geometry $\Phi=m\Omega$, with $m$ an azimuthal number, and $\Omega$ the angular velocity at the horizon) or a chemical potential (in the Reissner-N$\ddot{\rm
o}$rdstrom geometry, $\Phi=qQ$, where $q$ is the charge of the field and $Q$ the charge of the black hole).
For a wave scattering in this geometry, Eq. (\[wave\]) yields the following asymptotic behavior:[^4] $$\Psi _1 \sim\left\{
\begin{array}{ll}
\mathcal{T}\,(r-r_+)^{-\ii \alpha (\omega-\Phi)} & {\rm as}\
r\rightarrow r_+ \,,\label{bound2} \\
\mathcal{R}\,{\rm e}^{\ii {\omega r}}+ {\rm e}^{-\ii {\omega r}}&
{\rm as}\ r\rightarrow \infty\,.
\end{array}
\right.$$ These boundary conditions correspond to an incident wave of unit amplitude from $+\infty$ giving rise to a reflected wave of amplitude $\mathcal{R}$ going back to $+\infty$ and a transmitted wave of amplitude $\mathcal{T}$ at the horizon — the boundary condition introduces only ingoing waves at the horizon. Assuming a real potential (which is almost always the case) the complex conjugate of the solution $\Psi _1$ satisfying the boundary conditions (\[bound2\]) will satisfy the complex-conjugate boundary conditions: $$\Psi _2
\sim\left\{
\begin{array}{ll}
\mathcal{T}^*(r-r_+)^{\ii \alpha (\omega-\Phi)} & {\rm as}\ r\rightarrow r_+ \,, \\
\mathcal{R}^*{\rm e}^{-\ii {\omega r}}+ {\rm e}^{\ii {\omega r}}&
{\rm as}\ r\rightarrow \infty\,.
\end{array}
\right. \label{bound3}$$ Now, these two solutions are linearly independent, and the standard theory of ODE’s tells us that their Wronskian, $W=\Psi _1
\partial_{r_*}\Psi _2 -\Psi _2
\partial_{r_*}\Psi _1$, is a constant (independent of $r$). If we evaluate the Wronskian near the horizon, we get $W=
-2\ii(\omega-\Phi)|\mathcal{T}|^2$, and near infinity we find $W=2\ii \omega(|\mathcal{R}|^2-1)$. Equating the two we get $$|\mathcal{R}|^2=1-\frac{\omega -\Phi}{\omega}|\mathcal{T}|^2\,.$$ Now, in general $|\mathcal{R}|^2$ is less than unity, as is to be expected. However, for $\omega-\Phi<0$ we have that $|\mathcal{R}|^2>1$. Such a scattering process, where the reflected wave has actually been amplified, is known as superradiance. Of course the excess energy in the reflected wave must come from that of the black hole, which therefore decreases.
Superradiant scattering can lead to an instability if, we have a reflecting wall surrounding the black hole that scatters the returning wave back toward the horizon. In such a situation, the wave will bounce back and forth, between the mirror and the black hole, amplifying itself each time. The total extracted energy grows exponentially until finally the radiation pressure destroys the mirror. This is Press and Teukolsky’s black hole bomb, first proposed in [@press]. This instability can arise with an effective ‘mirror’ in a variety of situations: a scalar field with mass $\mu> \omega$ in a Kerr background creates a potential that can cause flux to scatter back toward the horizon [@detweiler]; infinity in asymptotically AdS spaces also provides a natural wall [@bhbAdS] that leads, for certain conditions, to an instability; a wave propagating around rotating black branes or rotating black strings may similarly find itself trapped [@cardoso].
\[free\]Geometries with ergoregion but horizon-free: Ergoregion instability
---------------------------------------------------------------------------
Suppose now there is no horizon in the background spacetime. What changes with respect to the former discussion is the boundary conditions: since there is no horizon and no absorption. For this case, the boundary condition (\[bound2\]) at the horizon is replaced by some kind of regularity condition at the origin. We suppose the radial coordinate $r$ now ranges from zero to infinity and we impose the following boundary condition: $$\Psi \sim A f(r)\,\,,\,\,\,r\rightarrow 0\,,$$ where $f(r)$ is some well-behaved [*real*]{} function. This ansatz encompasses for instance typical regularity requirements where, one chooses $f(r) \sim r^{\beta}$ with $\beta>0$. Repeating the above calculation, one gets $|\mathcal{R}|^2=1$. Therefore the absence of a horizon, which precludes any absorption, prevents superradiance and hence the superradiant instability.
Nevertheless, geometries with an ergoregion but without horizons are the arena of another class of instability. This ergoregion instability was discovered by Friedman [@friedman]. Even though his discussion was made in four-dimensions only, it is trivial to extend it to any number of dimensions. The instability arises because of the following [@friedman]: Given the test field energy-momentum tensor $T^{ab}$, we can associate a canonical energy $${\cal E}_S=\int_S t^a\, T_a{}^b dS_b\,,
\label{canon}$$ where $t^{a}$ is the background Killing vector which generates time translations in the asymptotic geometry. Now, because $t^a$ is space-like within an ergosphere, initial data can be chosen on a Cauchy surface S which makes ${\cal E}_S$ negative. Moreover, it is shown in [@friedman] that the energy can be negative only when the test field is time dependent. Then, since the field is time dependent, and since only positive energy can be radiated at future null infinity, the value of ${\cal E}_S$ can only decrease further from one asymptotically null hypersurface $S$ to another, say, $S'$, in the future of $S$. Thus, the energy ${\cal E}_S$ will typically grow negative without bound. This instability was computed analytically using a WKB approximation in [@cominsschutz] for rotating stars. There it was shown that the instability timescale is usually very large (typically larger than the age of the universe). The analysis of [@cominsschutz] was improved in [@compute] where further details of the instability were computed numerically.
A key assumption above is that the system can not settle down to a negative energy configuration which, while time dependent, is nonradiative. Friedman [@friedman] was able to rule out such marginal cases where ${\cal E}_S$ is negative but constant for a four-dimensional massless scalar or electromagnetic fields. However, in fact, one is able to identify negative energy bound states for the backgrounds — see Appendix \[sec:A2\] — and so a more thorough analysis is called for. Hence, in the following, we apply a variety of techniques to explicitly show that these microstate geometries suffer from an ergoregion instability.
\[formalism\] Formalism
=======================
We now consider wave propagation of a free massless scalar field in the backgrounds [@ross], and from this identify an ergoregion instability for these geometries in the subsequent sections. The solutions are described in detail in [@ross] and are quite involved. We will provide a brief discussion of some of the properties of these solutions here, but will refer the reader to [@ross] for the full details.
The solitons are solutions of type IIb supergravity corresponding to three-charge microstate geometries of the D1-D5-P system. The system is compactified to five dimensions on $T^4 \times
S^1$ with the D5-branes wrapping the full internal space and the D1-branes and KK-momentum on the distinguished $S^1$. The notation is best understood by considering the construction of these solutions. One begins with the general solutions of [@early1] which contain eight parameters: a mass parameter, $M$; spin parameters in two orthogonal planes, $a_1,a_2$; three boost parameters, $\delta_1,\delta_5,\delta_p$, which fix the D1-brane, D5-brane and KK-momentum charges, respectively; the radius of the $S^1$, $R$; the volume of the $T^4$ (which plays no role in the following). The geometry is described by the six-dimensional line element written in Eq. (2.12) of [@ross], which is parameterized by a time coordinate $t$; a radial coordinate $r$; three angular coordinates $\theta$, $\phi$, $\psi$; and the coordinate on the $S^1$, $y$.
One then imposes a series of constraints to ensure that the solutions are free of singularities, horizons and closed time-like curves. In particular, one focuses on a low-mass regime, $M^2<(a_1-a_2)^2$, in which no black holes exist. Then one finds solitonic solutions where an appropriate circle shrinks to zero at the origin and the constraints ensure that this happens smoothly. First, $M$ and $R$ can be fixed in terms of the remaining parameters — see Eqs. (3.15) and (3.20) of [@ross]. Two quantization conditions constrain the remaining parameters in terms of two integers $m,n$ [@ross]: $$\begin{aligned}
\frac{j+j^{-1}}{s+s^{-1}}=m-n\,, \qquad
\frac{j-j^{-1}}{s-s^{-1}}=m+n\,,
\label{ross 1a}\end{aligned}$$ where $j=\sqrt{\frac{a_2}{a_1}}\leq 1$ and $s=\sqrt{\frac{s_1 s_5
s_p}{c_1 c_5 c_p}}\leq 1$. We are using the notation here that $c_i \equiv \cosh \delta_i$ and $s_i \equiv \sinh \delta_i$. Without loss of generality, one assumes $a_1 \ge a_2\ge 0$ which implies $m>n\ge0$. We also note here that the special case $m=n+1$ corresponds to supersymmetric solutions.
This leaves a five-parameter family of smooth solitonic solutions. We can think of the independent parameters as the D1-brane and D5-brane charges, $Q_1,Q_5$; the (asymptotic) radius of the $y$-circle, $R$; and the two integers, $m$ and $n$, which fix the remaining physical parameters as [@ross] $$Q_P=nm\frac{Q_1Q_5}{R^2}\,,\quad J_\phi=-m\frac{Q_1Q_5}{R}\,,\quad
J_\psi=n\frac{Q_1Q_5}{R}\,. \label{simple}$$ Of course, depending on the specific application, it may be more appropriate and/or simpler to describe the solutions using a different set of quantities. In our case, when we make explicit calculations of the ergoregion instability, we will fix the parameters $n,m,a_1,c_1$ and $c_5$ or $c_p$. As we are interested in non-supersymmetric backgrounds, we also impose $m\geq n+2$. To conclude our discussion of notation, we add that the roots of $g^{rr}$, $r_+$ and $r_-$, will also appear in the following but they are determined by $M$ and the spin parameters — see Eq. (3.2) of [@ross].
The key ingredient producing the instability in the solutions is the existence of an ergoregion. To verify the presence of the ergoregion, one takes as usual the norm of the Killing vector $V=\partial_t$ and using Eq. (2.12) of [@ross], calculates $$\begin{aligned}
g_{\mu\nu}V^{\mu}V^{\nu} =-\frac{f-Mc_p^2}{\sqrt{\tilde{H}_1
\tilde{H}_5}}\,,
\label{ergoregionV}\end{aligned}$$ where $f(r)=r^2+a_1^2\sin^2\theta+a_2^2\cos^2\theta >0$ and $\tilde{H}_i(r)=f(r)+M s_i^2$, $i=1,5$. It is then clear that $V=\partial_t$ becomes space-like for $f(r)<M$ and thus an ergosphere appears at $f(r)=M$. An inspection of the metric also allows one to conclude the geometry rotates along $\phi$, $\psi$ and $y$ since $g_{t\phi}\neq 0$, $g_{t\psi}\neq 0$ and $g_{t
y}\neq 0$. The supersymmetric limit of the solitons corresponds to take the limit $M\rightarrow 0$ and $\delta_i
\rightarrow \infty$, while keeping the other parameters fixed, including the conserved charges $Q_i=M s_i c_i$ [@ross]. So, in the supersymmetric limit the norm becomes $|V|^2=-f /
\sqrt{\tilde{H}_1 \tilde{H}_5}$, which is always negative and thus the ergoregion is not present.
Now consider the Klein-Gordon equation for a massless scalar field propagating in the geometries, $$\frac{1}{\sqrt{-g}}\frac{\partial}{\partial x^{\mu}}
\left(\sqrt{-g}\,g^{\mu \nu}\frac{\partial}{\partial x^{\nu}}\Psi
\right)=0\,. \label{klein}$$ Implicitly, we are using the string-frame metric in which case one can think of eq. (\[klein\]) as the linearized equation of motion for the Ramond-Ramond scalar. As described above, these backgrounds can be thought of as special cases of the general D1-D5-P solutions found earlier [@early1] and so one may apply separation of variables following [@finnx]. Introducing the following ansatz [^5] $$\begin{aligned}
\Psi=\exp\left[-i\omega \frac{t}{R}-i\lambda \frac{y}{R}+i
m_{\psi} \psi +i m_{\phi} \phi \right]\, \chi(\theta)\,h(x) \,,
\label{separation ansatz}\end{aligned}$$ one gets an angular equation $$\begin{aligned}
\frac{1}{\sin{2\theta}}\frac{d}{d\theta}\left
(\sin{2\theta}\,\frac{d\chi}{d\theta}\right )+{\biggr
[}\Lambda-\frac{m_{\psi}^2}{\cos^2\theta}-\frac{m_{\phi}^2}{\sin
^2\theta}+\frac{\omega ^2-\lambda ^2}{R^2}(a_1 ^2\sin
^2\theta+a_2^2\cos^2\theta) {\biggr ]}\chi=0 \label{angeq}\, ,\end{aligned}$$ and a radial equation[^6] $$\begin{aligned}
\begin{aligned} & \frac{1}{r} \frac{d}{dr} \left[ \frac{g(r)}{r}
\frac{d}{dr} h \right]
- \Lambda h + \left[ \frac{(\omega^2 - \lambda^2)}{R^2} (r^2 + M s_1^2 + M
s_5^2) + (\omega c_p + \lambda s_p)^2 \frac{M}{R^2} \right] h
\\
& -(r_+^2-r_-^2)\frac{(\lambda - n m_\psi + m
m_\phi)^2}{(r^2-r_+^2)} \, h +
(r_+^2-r_-^2)\frac{(\omega \varrho + \lambda \vartheta - n m_\phi + m
m_\psi)^2}{(r^2- r_-^2)} \, h = 0 \label{radialeq-r}\,,
\end{aligned}\end{aligned}$$ where $g(r)=(r^2-r_+^2)(r^2-r_-^2)$, and we used $\sqrt{-g}=r\sin\theta\cos\theta \sqrt{{\tilde H_1}{\tilde H_5}}$ (this is the determinant of the metric (2.12) of [@ross]). If we introduce a dimensionless variable $$x = \frac{r^2 - r_+^2}{r_+^2 - r_-^2},$$ we can rewrite the radial equation in the form $$\partial_x[ x(x+1) \partial_x h]+ \frac{1}{4}
\left [ \kappa^{2}x + 1-\nu^2+
\frac{\xi^2}{x+1}-\frac{\zeta^2}{x}\right ] h =0 \,,
\label{rad eq0}$$ with $$\begin{aligned}
& &
\kappa^{2}=(\omega^2-\lambda^2)\frac{r_+^2-r_-^2}{R^2}\,, \nonumber \\
& & \xi=\omega\varrho+\lambda \vartheta-m_{\phi}n+ m_{\psi} m\,, \nonumber \\
& & \zeta=\lambda - m_{\psi}n +m_{\phi} m \,,\nonumber \\
& & \varrho=\frac{c_1^2 c_5^2 c_p^2-s_1^2 s_5^2 s_p^2}{s_1 c_1 s_5
c_5} \,,\nonumber \\
& & \vartheta =\frac{c_1^2 c_5^2 -s_1^2 s_5^2}{s_1 c_1 s_5
c_5}\,s_p c_p\,,
\label{rad eq parameters0}\end{aligned}$$ and $$\nu^2=1+\Lambda-\frac{\omega^2-\lambda^2}{R^2}(r_+^2+Ms_1^2+Ms_5^2)-(\omega
c_p+\lambda s_p)^2 \frac{M}{R^2}\,.
\label{nu0}$$ The quantities $\omega, \,\lambda, \, m_{\psi},\, m_{\phi}$ are all dimensionless — the last three being integers. Again, we refer the reader to [@ross] for a detailed account of the quantities appearing above. The reader should take note that our notation is not in complete accord with that of [@ross]. That is, to simplify our formulae in the following, we have defined $\kappa\equiv 1/\sigma$, the inverse of the quantity $\sigma$ used there.
Of critical importance in characterizing the solutions of the scalar wave equation is the sign of $\kappa^{2}$. The term $x
\kappa^2$ dominates at large $x$, determining the asymptotic behavior of the solution. In this paper we will mainly be interested in outgoing modes so we choose $\kappa^{2}$ to be positive. The two remaining possibilities: $\kappa^{2}=0$ and $\kappa^{2} < 0$, will be considered in the appendices.
The angular equation (\[angeq\]) (plus regularity requirements) is a Sturm-Liouville problem. We can label the corresponding eigenvalues $\Lambda$ with an index $l$, $\Lambda(\omega)=\Lambda_{lm} (\omega)$ and therefore the wavefunctions form a complete set over the integer $l$. In the general case, the problem at hand consists of two coupled second order differential equations: given some boundary conditions, one has to compute [*simultaneously*]{} both values of $\omega$ and $\Lambda$ that satisfy these boundary conditions. However, for vanishing $a_i ^2$ we get the (five-dimensional) flat space result, $\Lambda=l(l+2)$, and the associated angular functions are given by Jacobi polynomials. For non-zero, but small $\frac{\omega ^2-\lambda
^2}{R^2}a_i ^2$ we have $$\begin{aligned}
\Lambda=l(l+2)+\mathcal{O}\left (a_i^2\frac{\omega^2-\lambda
^2}{R^2}\right) \label{app} \,.\end{aligned}$$ The integer $l$ is constrained to be $l\geq
|m_{\psi}|+|m_{\phi}|$. We will always assume $a_i^2\frac{\omega^2-\lambda ^2}{R^2} \ll {\rm
max}(m_{\psi}^2,m_{\phi}^2)$ (with $i=1,2$), thus $\Lambda \simeq
l(l+2)$. Making this assumption implies we may neglect the terms proportional to $a_i$ in the angular equation, but given the way $\Lambda$ and $\omega$ appear in the radial equation, the corrections to $\Lambda$ may not be negligible when we determine $\omega$. To ensure that fixing $\Lambda=l(l+2)$ is consistent in both the angular and radial equations we must additionally require $$a_i^2 \ll \max \left ( | r_+^2+M(s_1^2+s_5^2) | , M c_p^2 \right ) \ ,$$ so that the contribution to $\nu$ from the $a_i$ dependent corrections of $\Lambda$ are negligible (see (\[nu0\])).
Taking the complex conjugate of Eq. (\[angeq\]) we can see that the exact solution to the angular equation has the symmetry \_[lm]{}(-\^\*)=\^\*\_[lm]{}().With this symmetry, one can also check the following: & & (\^[2]{})\^\*(,)=\^2(-\^\*,-), \[symmetry\]\
& &(\^[2]{})\^\*(,,m\_,m\_)=\^2(-\^\*,-,-m\_,-m\_),\
& &(\^2)\^\*(,m\_,m\_)=\^2(-,-m\_,-m\_) .Therefore, from the wave equation (\[rad eq0\]) it follows that if $\omega$ is an eigenvalue for given values of $m_{\psi},m_{\phi},\lambda$ with eigenfunction $h$, then $-\omega
^*$ is an eigenvalue for $-m_{\psi},-m_{\phi},-\lambda$ with eigenfunction $h^*$. Furthermore, if $he^{-i\omega t}$ is outgoing unstable, so is $h^*e^{i\omega ^*t}$. Since the symmetry simultaneously flips all the signs of $m_{\psi},m_{\phi},\lambda$, without loss of generality, we can only fix the sign of one, ${\mathcal Re}( \omega ) \leq 0$.
To conclude this section, we point out that the angular equation (\[angeq\]) can be recast in the somewhat more familiar form: ()+=0 \[angeqswsh\], where =+a\_1\^2 .This is just the equation for a five-dimensional scalar spheroidal harmonic [@teukolskyswsh] which arises, in the separation of Klein-Gordon equation in the background of a five-dimensional rotating black hole [@myersperry].
\[wkb\] WKB analysis
====================
We now explicitly show that the geometries [@ross] suffer from an ergoregion instability. As described above, this instability is due to the fact that the geometry has an ergoregion but no horizon. We shall identify modes of the scalar field that are regular at the origin, represent outgoing waves at infinity and grow with time. In this section, we follow the WKB analysis of [@cominsschutz] and show that it applies to the non-supersymmetric solutions, with the same qualitative conclusions.
To begin, we want to write the radial wave equation in the form of an effective Schr$\ddot{\rm o}$dinger equation. In order to do so, we first transform to a new ‘wavefunction’ $H$ defined with $$\begin{aligned}
h(x)=\frac{1}{\sqrt{x(1+x)}}\,H(x)\,.
\end{aligned}$$ Inserting this in (\[rad eq0\]), we get $$\begin{aligned}
-\partial ^2 _x H + U_{\rm eff}\,H=0 \label{Schrod 1}\,,\end{aligned}$$ where $$\begin{aligned}
U_{\rm eff} = -\frac{ \kappa^{2}x^3 + (1-\nu ^2+
\kappa^{2})x^2 +(1-\nu^2+\xi^2-\zeta^2)x
+1-\zeta^2}{4x^2(1+x)^2}\,. \label{rad eq01}\end{aligned}$$
Now in order to simplify our analysis, we choose: $\lambda=0$, $m_{\phi}=0$, and large $m_{\psi}$. With $\lambda\ne0$, the waves see a constant potential at infinity and thus the amplitude of the outgoing waves can be suppressed there. We also consider $l=m_{\psi}$ modes, which are expected to be the most unstable. Modes with $l \gg m_{\psi}$ must be similar to modes with $m_{\psi}=0$ for some $l$ and these are not unstable. With these choices, we have $$\begin{aligned}
\kappa^{2}&=&\omega ^2 \frac{r_+ ^2-r_-^2}{R^2}\,,\quad \zeta
^2 = n^2m_{\psi}^2\,, \quad \xi ^2 = m^2 m_{\psi}^2+\omega ^2\varrho
^2+2\omega \varrho mm_{\psi}\,,
\\
1-\nu ^2 &\simeq & - m_{\psi}^2+\omega
^2\frac{r_+^2+Ms_1^2+Ms_5^2+Mc_p^2}{R^2} \,.
\end{aligned}$$ Instead of working directly with the frequency of the wave, it will be convenient to work with the pattern speed along the $\psi$ direction, which is the angular velocity at which surfaces of constant phase rotate. This velocity is proportional to $$\begin{aligned}
\Sigma_{\psi} = \frac{\omega}{m_{\psi}}\,,
\label{Sigma}\end{aligned}$$ where the proportionality constant $R^{-1}$ is always positive. It is important to compare the sign of the pattern speed along $\psi$ with the sign of the angular velocity of the geometry along $\psi$ defined as usual by[^7] $$\begin{aligned}
\Omega_{\psi}=-\frac{g_{t\psi}}{g_{\psi\psi}}&=&
-\frac{2M s_p\, c_p \,\cos^2\theta}{\sqrt{\tilde{H}_1
\tilde{H}_5}}\frac{ R /n}{g_{\psi\psi}} \nonumber \\
&=& -\frac{2 Q_p \,\cos^2\theta}{\sqrt{\tilde{H}_1
\tilde{H}_5}}\frac{\cos^2\theta R /n}{g_{\psi\psi}} < 0, \quad
\forall \, x>0 \,,
\label{Omega}\end{aligned}$$ where $Q_p=M s_p\, c_p$ is the Kaluza-Klein momentum charge. So, when $\Sigma_{\psi}$ is negative, the wave is propagating in the same sense as the geometry.
Now it is useful to introduce the polynomial $$\begin{aligned}
{\cal P}=Bx^3+(A+B)x^2+(\varrho ^2+A)x\,,\end{aligned}$$ which is positive definite in the range of interest (positive $x$). We also define $$\begin{aligned}
T=-\frac{U_{\rm eff}}{m_{\psi}^{\,2}} \,, \quad
A \equiv \frac{r_+ ^2+M(s_1^2+s_5^2+c_p^2)}{R^2}\,,\quad B \equiv
\frac{r_+ ^2-r_-^2}{R^2}\,.\end{aligned}$$ Then, we can write the effective Schr$\ddot{\rm o}$dinger equation (\[Schrod 1\]) as $$\begin{aligned}
\partial ^2 _x H + m_{\psi}^{\,2}\, T\,H=0 \label{Schrod 2}\,,\end{aligned}$$ with $$T=\frac{\cal P}{4x^2(1+x)^2} {\biggl [}\Sigma_{\psi} ^2+\frac{2\varrho m x}{{\cal P}}\Sigma_{\psi}
-\frac{x^2-x(m^2-n^2-1)+n^2}{\cal P} {\biggr ]}\,,
\label{Def T}$$ where we have dropped certain small contributions to $T$.[^8] Now it is straightforward to factorize the potential $T$ and write it in the form T=(\_-V\_+)(\_-V\_-) \[factorize T\],with $$\begin{aligned}
V_{\pm}=-\frac{\varrho m x}{{\cal P}} \pm
{\biggl [} \left ( \frac{\varrho m x}{{\cal P}} \right )^2+
\frac{x^2-x(m^2-n^2-1)+n^2}{\cal P} {\biggr ]}^{\frac12} \label{potentials}.
\end{aligned}$$ For general $m\,,n$ the behavior of the potentials $V_+$ and $V_-$ (see Fig. \[fig:potential\]) is exactly the same as the one studied in [@cominsschutz], so we do expect an instability to arise, as will be shown below. However, and this is a key point, for the case $m=n+1$ which is the supersymmetric case, we have $$\begin{aligned}
V_{+}=-\frac{\varrho m x}{{\cal P}} +
{\biggl [} \left ( \frac{\varrho m x}{{\cal P}} \right )^2+
\frac{(x-n)^2}{\cal P} {\biggr ]}^{\frac12}\,,\qquad m=n+1\,,
\label{potential SUSY}\end{aligned}$$ which is always positive. Thus, this WKB analysis indicates that the supersymmetric solutions are stable, as expected.
Hence our radial equation has been reduced to the Schr$\ddot{\rm
o}$dinger form (\[Schrod 2\]) with an interesting potential (\[factorize T\]), which depends on the pattern speed (\[Sigma\]). Now the problem becomes to tune this potential by adjusting $\Sigma_{\psi}$ in order that a ‘zero-energy’ solution can be found with the appropriate boundary conditions: regular at the origin and outgoing waves at infinity. Note that in a region where $\Sigma_{\psi}$ is above $V_+$ or below $V_-$ (allowed regions), the solutions have an oscillatory behavior. In those intervals where $\Sigma_{\psi}$ is in between the curves of $V_+$ and $V_-$ (forbidden regions), the solutions have a real exponential behavior.
We proceed following [@cominsschutz] and study the scattering of waves in the effective potential constructed above. Consider a wave that comes from infinity with an amplitude $C_{\rm in}$, scatters in the ergoregion and returns to infinity with an amplitude $C_{\rm out}$. In particular, we introduce the scattering amplitude defined as $$\begin{aligned}
S\equiv \frac{C_{\rm out}}{C_{\rm in}}.
\label{Def S}\end{aligned}$$ The presence of a pole in $S$ (of a resonance) signals the existence of an instability. Indeed, a pole in $S$ occurs when $C_{\rm in}=0$ and $C_{\rm out} \neq 0$, and this means that we have finite outgoing radiation for zero incoming radiation. Near the pole frequency $\omega_{\rm p}$, the scattering amplitude can be written to lowest order as [@cominsschutz] $$\begin{aligned}
S\simeq e^{i 2\delta_0}\, \frac{\omega-\omega_{\rm
p}^{\ast}}{\omega-\omega_{\rm p}} \,,
\label{Def S near pole}\end{aligned}$$ where $\delta_0$ is a constant scattering phase shift and $\omega_{\rm p}^{\ast}$ is the complex conjugate of $\omega_{\rm p}$. Note that this expression guarantees that when the frequency of the wave is real, one has $S(\omega)[S(\omega)]^{\ast}=1$ as required by energy conservation. Generically, we can write the pole or resonant frequency as $$\begin{aligned}
\omega_{\rm p}=\omega_{\rm r}+i/\tau \,, \label{frequency}\end{aligned}$$ where $\omega_{\rm r}$ and $1/\tau$ are, respectively, the real and imaginary parts of $\omega_{\rm p}$. With this convention, a mode with positive $\tau$ represents an instability, and $\tau<0$ represents a damping mode, since the time dependence[^9] of the resonant wave is given by $e^{-i\omega_{\rm p}
t}=e^{-i\omega_{\rm r} t}e^{t/\tau}$. We can then write $$\begin{aligned}
S\simeq e^{i 2\delta_0}\,
\frac{\omega-\omega_{\rm r}+i/\tau} {\omega-\omega_{\rm r}-i/\tau} \,.
\label{Def S tau}\end{aligned}$$
To relate the amplitudes $C_{\rm in}$ and $C_{\rm out}$ we apply a WKB analysis. As we shall learn later on, the unstable modes are those whose pattern speed $\Sigma_{\psi}$ is negative and approaches the minimum of $V_+$ from above (see Fig. \[fig:potential\]). The scattering problem has then four distinct regions, namely: I, the innermost forbidden region ($0<x<x_0$); II, the allowed region where $V_+$ is below $\Sigma_{\psi}$ ($x_0<x<x_1$); III, the potential barrier region where $V_+$ is above $\Sigma_{\psi}$ ($x_1<x<x_2$); and finally the external allowed region where $\Sigma_{\psi}$ is below $V_-$ ($x_2<x<\infty$). The unstable modes are those that have $\Sigma_{\psi}
<0$. Thus, they are nearly bound states of the potential well in $V_+$ that can however tunnel out to infinity through $V_-$. In region I, the WKB wavefunction that vanishes at the origin $x=0$ is $$\begin{aligned}
H_{\rm I}\simeq \frac{C_1}{m_{\psi}^{1/2}|T|^{1/4}} {\rm
exp}\left[ -m_{\psi}\int_x^{x_0} \sqrt{|T|}\,dx \right]\,,
\label{H1}\end{aligned}$$ where $C_{1}$ is an amplitude constant. Then, the usual WKB connection formulae and WKB wavefunctions allow us to relate $H_{\rm I}$ with the wavefunctions of the other regions and, in particular, with the incoming and outgoing contributions of the wavefunction $H_{\rm IV}$ in region IV, which can be written as $$\begin{aligned}
H_{\rm IV}\simeq \frac{C_6}{m_{\psi}^{1/2}T^{1/4}} {\rm exp}\left[
i\,m_{\psi}\int_{x_2}^{x} \sqrt{T}\,dx \right]
+\frac{C_7}{m_{\psi}^{1/2}T^{1/4}} {\rm exp}\left[
-i\,m_{\psi}\int_{x_2}^{x} \sqrt{T}\,dx \right]\,. \label{H4}\end{aligned}$$ The WKB analysis yields the relation between the amplitudes $C_6$, $C_7$ and $C_1$ (see Appendix \[sec:A0\]): $$\begin{aligned}
C_1 e^{i \gamma}&=& \frac{1}{2} \left [ \left ( 2\eta
+\frac{1}{2\eta}\right )C_6 +i \left ( 2\eta
-\frac{1}{2\eta}\right )C_7 \right ] \nonumber \\
C_1 e^{-i \gamma}&=& \frac{1}{2} \left [ -i \left ( 2\eta
-\frac{1}{2\eta}\right )C_6 + \left ( 2\eta +\frac{1}{2\eta}\right
)C_7 \right ]\,,
\label{connectionWKB}\end{aligned}$$ where $$\begin{aligned}
\gamma&\equiv& m_{\psi}\int_{x_0}^{x_1} \sqrt{T}\,dx
-\frac{\pi}{4} \,, \label{WKBparameters1}\end{aligned}$$ $$\begin{aligned}
\ln \eta &\equiv& m_{\psi} \int_{x_1}^{x_2} \sqrt{|T|}\,dx\,.
\label{WKBparameters2}\end{aligned}$$ The identification of the ingoing and outgoing contributions in (\[H4\]) depends on the sign of $\Sigma_{\psi}$. Indeed, one has $\Psi\propto e^{-i\omega t} H_{\rm IV}(x)$. If $\Sigma_{\psi}$ is negative the term $C_6 e^{-i(\omega t-\gamma(x))}$ represents the ingoing contribution, while the term $C_7 e^{-i(\omega
t+\gamma(x))}$ describes the outgoing contribution (if $\Sigma_{\psi}>0$, the terms proportional to $C_6$ and $C_7$ in $H_{\rm IV}(x)$ represent, respectively, the outgoing and ingoing modes). Henceforth we consider the $\Sigma_{\psi}<0$ case (since this will be the unstable case), for which the scattering amplitude can be written as $$\begin{aligned}
S=\frac{C_7}{C_6}=\frac{i(4\eta^2-1)e^{i \gamma}+(4\eta^2+1)e^{-i
\gamma}}{(4\eta^2+1)e^{i \gamma}-i(4\eta^2-1)e^{-i \gamma}}\,.
\label{WKB S}\end{aligned}$$ The resonance peaks in the scattering amplitude occur at a frequency $\omega_N$ for which $e^{-i \gamma}+ie^{i \gamma}=0$, when $\gamma(\omega)=\gamma_N$ where $$\begin{aligned}
\gamma_N (\omega_N) \equiv N\pi +\frac{\pi}{4}\,
\label{ressonance freq}\end{aligned}$$ with $N$ being an integer usually referred to as the ‘harmonic’. The easiest way to see that the resonance peaks must be near these (real) frequencies is to note that $S(\gamma_N)=-i$ while for $\eta\rightarrow \infty$, one has $S(\gamma \neq \gamma_N)=+i$. So when $\eta\rightarrow \infty$, one has generally $S(\gamma)=+i$, but when $\gamma=\gamma_N$ a peak occurs that changes the value of $S$ from $+i$ to $-i$.
We can now do a Taylor expansion of the functions that appear in $S$ around $\gamma=\gamma_N$. Defining $$\begin{aligned}
\alpha = \frac{d \gamma}{d\omega}{\biggl |}_{\omega=\omega_N}=
\frac{d}{d\Sigma_{\psi}} \left [\int_{x_0}^{x_1} \sqrt{T}\,dx
\right]_{\Sigma_{\psi}=\Sigma_{\psi, N} }\,,
\label{WKB taylor alpha}\end{aligned}$$ the scattering amplitude can be written as $$\begin{aligned}
S \simeq \frac{-\alpha(\omega-
\omega_N)+\frac{1}{4\eta^2}-i\left[\alpha(\omega-
\omega_N)+\frac{1}{4\eta^2}\right]}{-\alpha(\omega-
\omega_N)+\frac{1}{4\eta^2}+i\left[\alpha(\omega-
\omega_N)+\frac{1}{4\eta^2}\right]}
\label{WKB S2}\end{aligned}$$ which, using $(1+i)/(1-i)=i$, can be cast in the form $$\begin{aligned}
& & S \simeq i\,\frac{\omega- \omega_N +i \frac{1}{4\eta^2
\alpha}}{\omega- \omega_N -i \frac{1}{4\eta^2 \alpha}}\,.
\label{WKB S4}\end{aligned}$$ This result takes the form (\[Def S tau\]). Hence the discrete spectrum of resonance frequencies $\omega_N$ is selected by condition (\[ressonance freq\]). Further comparing (\[Def S tau\]) with (\[WKB S4\]), one has that the growth or damping timescale is given by $$\begin{aligned}
\tau= 4\eta^2 \alpha\,.
\label{tau}\end{aligned}$$ Now, $\alpha$ defined in (\[WKB taylor alpha\]) is always positive since as $\Sigma_{\psi}$ increases so does $T$ and $\gamma$ defined in (\[WKBparameters1\]) (the area of the region in between the $\Sigma_{\psi}$ line and the $V_+$ curve, and in between $\Sigma_{\psi}$ line and the $V_-$ curve both increase when $\Sigma_{\psi}$ increases). So, we are guaranteed to have a positive $\tau$ and thus the negative $\Sigma_{\psi}$ modes are unstable. If we redo the computations to consider the $\Sigma_{\psi}>0$ case, the only difference is that in (\[H4\]) the ingoing and outgoing waves are given instead by the terms proportional to $C_7$ and $C_6$, respectively. This changes the scattering amplitude from $S$ to $S^{-1}$ and thus $\tau$ to $-\tau$ implying that the positive $\Sigma_{\psi}$ modes are damped.
Though the resonance frequencies and growth timescales can be computed with numerical methods from (\[ressonance freq\]) and (\[tau\]), as we shall do in Section \[num res\], we can still make some further progress analytically by approximating the well of $V_+$ by a parabola. Near the well, the potential $V_+$ behaves generally as $$\begin{aligned}
V_+ \simeq \frac{(x-x_{\rm m})^2}{P_{\rm m}}+a_{\rm m}\,,
\label{parabola}\end{aligned}$$ with $a_{\rm m}<0$. The boundaries $x_0$ and $x_1$ are the roots of $\Sigma_{\psi}-V_+$, namely: $x_0=x_{\rm m}-[P_{\rm m}(\Sigma_{\psi}-a_{\rm
m})]^{1/2}$ and $x_1=x_{\rm m}+[P_{\rm m}(\Sigma_{\psi}-a_{\rm
m})]^{1/2}$. Since $\sqrt{T}$ vanishes at these boundaries one has $$\begin{aligned}
\alpha = \int_{x_0}^{x_1} \frac{d \sqrt{T} }{d\Sigma_{\psi}}\,dx \,.
\label{alpha 2}\end{aligned}$$ Moreover, near the bottom of the well, only $\Sigma_{\psi}-V_+$ varies significantly with $x$, and we can assume that all the other quantities that appear in the integral of $\alpha$ are approximately constants given by their value at $x=x_{\rm m}$ (the accuracy of this assumption increases as $\Sigma_{\psi}$ approaches $a_{\rm m}$). One then has $$\begin{aligned}
\alpha \simeq \frac{\Sigma_{\psi}+\frac{\varrho m x_{\rm m}}{{\cal P}(x_{\rm
m})} } { \sqrt{\Sigma_{\psi} - V_-(x_{\rm m})} } \frac{\sqrt{{\cal
P}(x_{\rm m})}} {2x_{\rm m}(1+x_{\rm m})} \int_{x_0}^{x_1} \left
[\Sigma_{\psi}-V_+ \right ]^{-\frac12}\,dx \,,
\label{alpha 3}\end{aligned}$$ with $V_+$ given by (\[parabola\]), which yields for $\alpha$ the value $$\begin{aligned}
\alpha =\pi \sqrt{P_{\rm m}}\left [\Sigma_{\psi}+\frac{\varrho m x_{\rm
m}}{{\cal P}(x_{\rm m})} \right ]\left [\Sigma_{\psi} - V_-(x_{\rm m})
\right ]^{-1/2}\frac{\sqrt{{\cal P}(x_{\rm m})}} {2x_{\rm
m}(1+x_{\rm m})} \label{alpha 4}\,.\end{aligned}$$
Let us illustrate the use of the WKB method we have described in this section to compute the instability parameters in a particular configuration. Take, $$\begin{aligned}
m&=&14\,;\,\,n=10\,;\,\,a_1=32\,;\,\,c_1=5\,;\,\,c_p=5\,; \label{house} \\
& & \lambda=m_{\phi}=0\,;\,\,l=m_{\psi}=10\,.\label{house2}
\end{aligned}$$ By approximating the well in $V_+$ by a parabola, as in (\[parabola\]), we get $$\begin{aligned}
a_{\rm m}=-0.17894\,;\,\,x_{\rm m}=9.1537\,;\,\,P_{\rm
m}=2759.4\,.\end{aligned}$$ The resonant frequencies are those that satisfy condition (\[ressonance freq\]) with $\gamma(\omega)$ given by (\[WKBparameters1\]). For the fundamental harmonic ($N=0$), we get $$\begin{aligned}
\Sigma_{\psi}=-0.173 \,.
\label{numeric sigma 0}\end{aligned}$$ The growth timescale of the instability is given by (\[tau\]) with $\eta(\omega_N)$ given by (\[WKBparameters2\]). Again, for $N=0$ we get $$\begin{aligned}
\tau \sim 10^{47}\,.
\label{numeric tau}\end{aligned}$$
Independently of the parameters of the geometry, we note that as $m_{\psi}$ grows, $\Sigma_{\psi}$ approaches $a_{\rm m}$, the value of the $V_+$ at its minimum. For the particular geometry parameters described in (\[house\]) we have (for $\lambda=m_{\phi}=0$): $$\begin{aligned}
m_{\psi}&=&10:\,\,\Sigma_{\psi}=-0.173\,, \nonumber \\
m_{\psi}&=&20:\,\,\Sigma_{\psi}=-0.176\,, \nonumber \\
m_{\psi}&=&40:\,\,\Sigma_{\psi}=-0.177\,.
\label{numeric sigma}\end{aligned}$$ This feature can be proved analytically, as was done in [@cominsschutz].
Let us verify consistency of our results. We have assumed that $a_i^2\frac{\omega^2-\lambda ^2}{R^2}\ll 1$ in order to do the approximation $\Lambda \simeq l(l+2)$. Now, for the cases we listed above one has $ a_i^2\frac{\omega^2-\lambda ^2}{R^2} \sim
10^{-2}$, which is inside the range of validity for our approximations. A different combination of parameters yields a different instability timescale, and resonant frequency, so there are geometries more unstable than others. The following refers to the fundamental harmonic, and are computed within the parabolic approximation. We work with the following parameters: $$\begin{aligned}
c_1=5\,;\,\,c_p=5\,;\,\,\lambda=m_{\phi}=0\,;\,\,
l=m_{\psi}=10\,.\end{aligned}$$ For the fundamental harmonic we then get m&=&1400;n=10;a\_1=32; \_=-0.40502,\~310\^[82]{}\
m&=&12;n=10;a\_1=32; \_=-0.104,\~310\^[52]{}\
m&=&14;n=10;a\_1=3200; \_=-0.1728,\~310\^[48]{}\
m&=&3;n=1;a\_1=32; \_=-0.0148,\~31.710\^[44]{} It also evident that the instability is much stronger for small values of $m_{\psi}$, where the WKB is expected to break down.
To conclude this section, let us consider the regime of validity of the WKB approximation with more detail. A standard analysis of Eq. (\[Schrod 1\]) suggests the WKB approximation is valid for $\left |\partial_xU_{\rm eff} \right | \ll |U_{\rm eff}|^2$, which can be rewritten as $\left |
\partial_x T /T^2(x) \right | \ll m_{\psi}^2$. So, for large $m_{\psi}$, the WKB approximation seems to be valid quite generally. However, we must sound a note of caution. As we already remarked, Eq. (\[numeric sigma\]) shows that as $m_{\psi}$ grows, $\Sigma_{\psi}$ approaches $a_{\rm m}$, the value of the $V_+$ at its minimum — this can be proved analytically [@cominsschutz]. So when $m_{\psi}$ becomes very large, the two turning points are very close and the WKB analysis breaks down because $T(x)\rightarrow
0$. So we conclude that the WKB approximation used in this section should be valid in a regime with large $m_{\psi}$, but not exceedingly large. In any event, it is clear that the instability is strongest for small values of $m_{\psi}$, when the WKB analysis is certainly not valid. So, in the next two sections we will compute the features of the instability using complementary methods valid for small values of $m_{\psi}$. We will also find remarkable agreement between all three approaches.
\[sec:Match\]Matched asymptotic expansion analysis
==================================================
The WKB analysis described in the last section appears to be strongest when describing solutions for which $\kappa ^{-1}\sim
\zeta,\xi$, but in general this corresponds to solutions with high angular momentum. In the sense that the timescale of the instability due to these modes is largest, they are the least unstable. Conversely, the matched asymptotic expansion that we use in this section becomes valid when $\kappa ^{-1}> \zeta,\xi$, they are the dominant decay modes. As an additional bonus, the eigenvalues are determined explicitly through algebraic constraints. Having both approximations at our disposal allows us to accurately calculate the eigenvalues for most of the allowed parameters.
We follow a matching procedure introduced in [@staro1], which has previously been used for studying scalar fields in three-charge geometries by Giusto, Mathur and Saxena [@three2], in the backgrounds [@ross] and also in [@bhb; @detweiler; @bhbAdS; @cardoso; @CardDiasYosh]. The space is divided into two parts: a near-region, $x\ll \beta$, and a far-region, $x\gg \alpha$, such that $\alpha \ll \beta$. The radial equation is then solved approximately and the appropriate boundary conditions applied in each of the two regions. Finally, we match the near-region and the far-region solutions in the area for which they are both valid, $\alpha \ll r \ll \beta$. This gives a set of constraints, the solution of which gives the eigenvalues. Performing this analysis for the radial equation (\[rad eq0\]), we shall see that the only solutions which are regular at the origin and purely outgoing at infinity are finite as $x \rightarrow \infty$, and lead to instabilities. Except when otherwise stated, the analysis in this section will hold for general values of $m_{\psi}$, $m_{\phi}$ and $\lambda$.
\[sec:BH Near region\]The near region solution
----------------------------------------------
In the near-region, $\kappa^{2} x\ll |1-\nu^2|$, one can neglect the $\kappa^{2}x$ term, and the radial equation (\[rad eq0\]) is approximated by $$x(1+x)\partial_x^2 h+ (1+2x)
\partial_x h +\frac{1}{4} \left [ 1-\nu^2+
\frac{\xi^2}{x+1}-\frac{\zeta^2}{x}\right ] h=0\,.
\label{near wave eq}$$ With the definition $h=x^{|\zeta|/2} (1+x)^{\xi/2}\,w$, the near-region radial equation becomes a standard hypergeometric equation [@abramowitz] of the form $$x(1+x)\partial_x^2 w+[c+(a+b+1)x]\partial_x w+ab \, w=0,$$ where $$a=\frac{1}{2}\left (1+|\zeta|+\xi+\nu \right ) \,, \qquad
b=\frac{1}{2}\left (1+|\zeta|+\xi-\nu \right )\,, \qquad c=1+
|\zeta| \,.
\label{hypergeometric parameters}$$ The full solution to the above is given in terms of hypergeometric functions as $w = A\, F(a,b,c,-x)+B\, x^{1-c}
F(a-c+1,b-c+1,2-c,-x)$, which allows us finally to write the solution of the radial equation in the near region as $$\begin{aligned}
h &=& A\,x^{|\zeta|/2}(1+x)^{\xi/2} F(a,b,c,-x) \nonumber \\
& & +B \, x^{-|\zeta|/2}(1+x)^{\xi/2} F(a-c+1,b-c+1,2-c,-x) \,.
\label{hypergeometric solution}\end{aligned}$$ At this point we impose the first boundary condition: the solution must be regular at $x=0$ since the geometry is smooth at the origin of the “core". The term proportional to $x^{-|\zeta|/2}$ diverges at $x=0$ and must be discarded, its coefficient, $B$, must be set to zero.
To perform the matching we need to know the large $x$ behavior behavior of the regular near-region solution. To this end, one uses the $x \rightarrow 1/x$ transformation law for the hypergeometric function [@abramowitz] $$\begin{aligned}
F(a,b,c,-x)&=&\frac{\Gamma(c)\Gamma(b-a)}{\Gamma(b)\Gamma(c-a)}\,
x^{-a}
\,F(a,1\!-\!c\!+\!a,1\!-\!b\!+\!a,-\!1/x) \nonumber \\
& & + \frac{\Gamma(c)\Gamma(a-b)}{\Gamma(a)\Gamma(c-b)}\,x^{-b}
\,F(b,1\!-\!c\!+\!b,1\!-\!a\!+\!b,-\!1/x)\,,
\label{transformation law}\end{aligned}$$ and the property $F(a,b,c,0)=1$. Note that this expression for the transformation is only valid when $a-b=\nu$ is non-integer. This is an assumption we will continue to make throughout this section. In the end, we shall derive a condition determining the allowed eigenvalues that will not be dependent upon this assumption and therefore we may extend our results to integer values of $\nu$ by continuity.
The large $x$ behavior of the near-region solution is then given by $$\begin{aligned}
& & \hspace{-0.5cm} h \sim A\,\Gamma(1+|\zeta|) {\biggl [}
\frac{\Gamma(-\nu)}
{ \Gamma\left [\frac{1}{2}\left (1+|\zeta|+\xi-\nu \right )
\right ]
\Gamma\left [\frac{1}{2}\left (1+|\zeta|-\xi-\nu \right )
\right ] }\:
x^{-\frac{\nu+1}{2}}\nonumber \\
& & \hspace{2 cm}
+\frac{\Gamma(\nu)}
{ \Gamma\left [\frac{1}{2}\left (1+|\zeta|+\xi+\nu \right )
\right ]
\Gamma\left [\frac{1}{2}\left (1+|\zeta|-\xi+\nu \right )
\right ] }\:
x^{\frac{\nu-1}{2}} {\biggr ]}.\nonumber \\
& &
\label{near field large r}\end{aligned}$$
\[sec:BH Far region\]The far region solution
--------------------------------------------
In the far-region, $\kappa x^2\gg {\rm max}\{ \xi^2-1,\zeta^2 \}$, the terms $\xi^2/(x+1)$ and $\zeta^2/x$ can be neglected, and the radial equation can be approximated by $$\begin{aligned}
\partial_x^2 (x h)+ \left [ \frac{\kappa^{2}}{4x}-\frac{\nu^2-1}{4x^2}
\right ] (x h)=0\,.
\label{far wave eq}\end{aligned}$$ The most general solution of this equation when $\nu$ is non-integer is a linear combination of Bessel functions of the first kind [@abramowitz], $$\begin{aligned}
h=x^{-1/2}\left [ C J_{\,\nu}(\kappa \sqrt{x})+ D
J_{\,-\nu}(\kappa \sqrt{x})\right ]\,.
\label{far field}\end{aligned}$$ This form does not lend itself easily to application of the boundary conditions. Instead, for large $\kappa \sqrt{x}$, the solution may be expanded as [@abramowitz] $$\begin{aligned}
h \sim \frac{x^{-3/4}}{\sqrt{2\pi \kappa}}
{\biggl [} e^{i\kappa \sqrt{x}} e^{-i\frac{\pi}{4}}\left (C e^{-i
\frac{\pi\nu}{2}} +D e^{i \frac{\pi\nu}{2}} \right )
+ e^{-i\kappa \sqrt{x}}
e^{i \frac{\pi}{4}}\left (C e^{i \frac{\pi\nu}{2}} +D e^{-i
\frac{\pi\nu}{2}} \right ){\biggr ]}\,.
\label{far field-large r}\end{aligned}$$ As in the WKB analysis, we assume that the real part of $\omega$ is negative, and therefore the positive and negative sign exponentials give, respectively, ingoing and outgoing waves. We require that there be purely outgoing waves at infinity and so impose the constraint that the coefficient of the positive exponential vanishes, yielding $$\begin{aligned}
C = -D e^{i \pi\nu}. \label{far amplitude relation}\end{aligned}$$ When $\omega$ becomes complex, so too does $\kappa$. Since the sign of the real part of $\omega$ is negative, the definition of $\kappa$ (\[rad eq parameters0\]) implies that its imaginary part has a sign opposite that of the imaginary part of $\omega$. Therefore, requiring additionally that the solution be finite as $x\rightarrow \infty$ implies that the imaginary part of $\omega$ must be positive. This is precisely the sign for the imaginary part of the frequency that leads to instabilities. Thus we see that simply requiring the solutions with complex frequency be finite at infinity automatically guarantees they lead to instabilities.
Now, to do the matching in the overlapping region, we will need to know how the far-region solution behaves for small values of $x$. More specifically, for small $\kappa \sqrt{x}$, and considering only the dominant terms, the solution behaves as [@abramowitz] $$\begin{aligned}
h \sim D\left [ \frac{(2/\kappa)^{-\nu}}{\Gamma(1+\nu)}\:
x^{\frac{\nu-1}{2}} - e^{i \pi\nu}
\frac{(2/\kappa)^{\nu}}{\Gamma(1-\nu)}\: x^{-\frac{\nu+1}{2}}
\right ].
\label{far field-small r}\end{aligned}$$
\[sec:MatchCond\]Matching conditions: Selection of frequencies
--------------------------------------------------------------
We will now determine the frequencies that can appear when the geometry is perturbed by a scalar field. The frequency spectrum is not arbitrary: only those values that satisfy the matching conditions between the near-region and the far-region are allowed. We shall see that there are two solutions of the matching equations, yet only one will lead to instabilities.
Matching the powers of $x$ between the near (\[near field large r\]) and far-region solutions (\[far field-small r\]), and taking a ratio to eliminate the amplitudes $A$ and $D$, yields $$-e^{i \pi \nu} (\kappa/2)^{2\nu}
\frac{\Gamma(1-\nu)}{\Gamma(1+\nu)}
=\frac{\Gamma(\nu)}{\Gamma(-\nu)}
\frac{\Gamma(\frac{1}{2}(1-\nu+|\zeta|+\xi))}{\Gamma(\frac{1}{2}(1+\nu+|\zeta|+\xi))}
\frac{\Gamma(\frac{1}{2}(1-\nu+|\zeta| -
\xi))}{\Gamma(\frac{1}{2}(1+\nu+|\zeta|-\xi))} \ .
\label{eq:asymp2_finaleq}$$ The problem of finding the outgoing modes thus boils down to solving the single transcendental equation (\[eq:asymp2\_finaleq\]); we will do so by iteration. Note that the $\kappa$ dependence on the left hand side means that it is suppressed. For the equation to hold, a similar suppression must also occur on the right hand side. This is only possible if one of the gamma functions in the denominator of the right side is large. Since the gamma function diverges when its argument is a non-positive integer, we take as a first iteration the choice $$\nu+|\zeta| - \xi = -(2 N + 1) \ , \label{eq:asymp2_quantization}$$ where the non-negative integer $N$ will again be referred to as the harmonic. Note that we could also have chosen the above relation, but with the opposite sign for $\xi$. While this does indeed lead to a solution, one finds that the imaginary part of the frequency is always negative, the modes are exponentially damped in time.
This first estimate is obviously not the end of the story as it would cause the right side to completely vanish. To go beyond this approximation, we rewrite Eq. (\[eq:asymp2\_finaleq\]) in terms of $N$, then perturb $N \rightarrow N + \delta N$, where $\delta N
\ll N$. This deformation appears at leading order only for the $\Gamma$ function in the denominator on the right hand side that diverges, it may be neglected in all other factors. More concretely, to extract $\delta N$ from the $\Gamma$ function we use $\Gamma(z)\Gamma(1-z)= \pi/\sin(\pi z)$, and sine function identities to obtain the expansion $$\Gamma(-N-\delta N) \approx -\left [ (-1)^N N! \thinspace \delta
N \right ]^{-1}.$$ Substituting this into (\[eq:asymp2\_finaleq\]), and using a number of $\Gamma$ function identities, we solve for the imaginary part of the first correction $${\mathcal I m}(\delta N) = \pi \frac{ (\kappa/2)^{2\nu} }{\Gamma^2(\nu)}
[\nu]_N \thinspace [\nu]_{N+|\zeta|} \ , \label{deltaN}$$ where $[a]_n = \prod_{i=1}^n (1+a/i)$. Since $N$ is ${\mathcal O}(1)$ and $\delta N \sim \kappa^{2 \nu}$, we see that we may stop after the first iteration. As a function of $\nu$, this can have a single maximum near $\nu \sim \kappa$. In general we will have $\kappa \ll 1$ and $\nu \sim 1+l$, so will always be in a region where this is a monotonically decreasing function of $\nu$. For fixed $\nu$, the last two factors make this an increasing function of $N$ and $|\zeta|$, but the general behavior will be dominated by the effects of changing $\nu$.
The equation (\[eq:asymp2\_quantization\]) uniquely determining $\omega$ can be exactly solved $$(\varepsilon + \varrho^2) \thinspace \omega = - \left(\lambda
\frac{s_p c_p M}{R^2}+\varrho c \right ) + \sqrt{ \left(\lambda
\frac{s_p c_p M}{R^2}+\varrho c \right
)^2-(\varepsilon+\varrho^2)(c^2-\nu_0^2) } \ , \label{omega_solution}$$ where $$\varepsilon \equiv \frac{1}{R^2}(r_+^2 + M(s_1^2 + s_5^2 +
c_p^2)) \ , \qquad c \equiv \xi_0 - |\zeta| - (2N+1) \ ,$$ and a variable with a subscripted $0$ means we have set $\omega=0$. Note that as long as $m \geq n+2$, we have $\varepsilon/\varrho^2 \ll 1$ and both quantities are positive. When $m \rightarrow n+1$, though, $\varepsilon \rightarrow -\infty$ (since $M\rightarrow 0$, $r_+^2 \rightarrow -\infty$ and $R^2$ stays finite), ensuring that there can be no instability for the supersymmetric solutions. This extends to arbitrary modes the conclusion from the discussion associated to equation (\[potential SUSY\]) for modes with $m_\phi=\lambda=0$.
When evaluated on a solution, $\nu$ is given by $\nu=\omega \varrho +
c$. Since we are interested in solutions for which $\omega$ is negative, this means $c>0$. Then, requiring that $\omega$ be negative and real, gives three more conditions. The first ensures that the result is real while the second requires that the first term of (\[omega\_solution\]) is negative. Finally, the condition that appears to be the most difficult to satisfy ensures the contribution from the square root does not make the total result positive, $$c^2-\nu_0^2 > 0 \ . \label{constraint}$$ When $\lambda \neq 0$, these conditions must also be supplemented by the requirement that $\omega^2-\lambda^2>0$, which ensures the asymptotic behavior of the solution is correct. With these satisfied, we may determine the effect of the correction.
The imaginary contribution to $N$ is taken as resulting from a small imaginary correction to $\omega$. Then, the two are related through $$\begin{aligned}
\delta N & = & \left . \frac{\delta \omega}{2} \frac{d}{d\omega} (\xi-\omega) \right |_N \nonumber \\
& = & \frac{\delta \omega}{2\nu}
\left [ (\varrho^2+\varepsilon)\omega+(\lambda s_p c_p M/R^2+\varrho c) \right ] \nonumber \\
& = & \frac{\delta \omega}{2\nu} \sqrt{ \left(\lambda \frac{s_p
c_p M}{R^2}+\varrho c \right )^2-(\varepsilon+\varrho^2)(c^2-\nu_0^2) }
\ .\end{aligned}$$ In the final line we have used the solution (\[omega\_solution\]) to show that the sign of $\delta N$ determines the sign of the correction to $\omega$. Since ${\mathcal Im}(\delta N)$ is always positive when evaluated on the solution of (\[eq:asymp2\_quantization\]), the corresponding imaginary part of $\omega$ is positive.
To summarize, whenever the constraints, in particular (\[constraint\]), are satisfied there is a corresponding outgoing mode of the scalar field equation. Further, the imaginary part of the frequency of this mode is guaranteed to be positive, indicating that it leads to an instability. The timescale for the instability generated by the mode is a monotonically increasing function of $\nu$, which is given by $$\begin{aligned}
(\varepsilon + \varrho^2) \thinspace \nu & = & \varepsilon c -
\lambda \varrho s_p c_p \frac{M}{R^2} +
\sqrt{ \left ( \varepsilon c - \lambda \varrho s_p c_p \frac{M}{R^2} \right )^2
+ (\varepsilon + \varrho^2)(2 c \varrho \lambda s_p c_p \frac{M}{R^2} + \nu_0^2 \varrho^2 - \varepsilon c^2) }
\ . \label{nu_solution}\end{aligned}$$ A similar argument, based on the solution of equation (\[eq:asymp2\_quantization\]), but with the opposite sign for $\xi$ would lead to a set of outgoing modes with an amplitude that decays in time.
As an example, consider the particular background geometry and scalar field solution described by $$\begin{aligned}
m&=&5\,;\,\,n=1\,;\,\,a_1=19.1\,;\,\,c_1=5\,;\,\,c_p=1.05\,;\nonumber
\\& & \lambda=m_{\phi}=0\,;\,\,l=m_{\psi}=2\,.\end{aligned}$$ The first two iterations with $N=0$ gives $$\begin{aligned}
\omega & = & -2.8717 \ , \nonumber \\
\tau^{-1}={\mathcal Im (\delta \omega)} & = & 4.42 \times 10^{-11}
\ ,\end{aligned}$$
The results obtained here are consistent with the WKB analysis of the last section, there are outgoing modes that rotate in the same sense as the background geometry whose amplitude grows exponentially in time. What we have gained is an explicit set of relations that allows the unstable mode frequencies to be calculated. In particular, one can now make definite statements about the relative timescales for unstable modes just by looking at equation (\[nu\_solution\]). We leave the precise details of this to Appendix \[beyond\] and just give the results here. The most unstable modes are those which minimize $\nu$. Since $\varepsilon
\ll \varrho^2$ this generally means that the modes which maximize $c$ or minimize $\nu_0$ will be the most unstable. In general this means we should consider the lowest possible $l$ for which the constraints can be satisfied when setting $m_\psi = l$, $m_\phi=0$ and $N=0$.
A second benefit of this analysis is an improvement in accuracy for the most unstable modes. For comparison, performing the WKB analysis and not neglecting any terms in the potentials or approximating the bottom of the well with a parabola gives $\omega = -3.129+4.00
\times 10^{-10}i$. From the full numerical solution we have $\omega=2.8718+4.46 \times 10^{-11}i$. For values of $\omega$ in this range we have $\kappa^{-2} \sim 1900$, so we are well within the range for which we should trust this solution. As $\kappa
^{-1}$ approaches $\max(|\zeta|,\xi)$, this analysis begins to break down, but it appears that the WKB approach becomes increasingly accurate. In the next section we will present a more detailed list of eigenvalues corresponding to instabilities and discuss the results.
\[numerical\] Numerical Results
===============================
We will now solve the radial equation (\[rad eq0\]) numerically to extract the instability. We begin with an exposition of the numerical algorithm. The only approximation used in this section concerns the angular eigenvalue $\Lambda$, that we shall assume to be well described by (\[app\]). At the end of the calculation we always make sure the result fits the regime of validity of this approximation. Note, however, solutions can still be found even when outside this range. The easiest way to do this is by treating $\Lambda$ and $\omega$ respectively as eigenvalues of the angular and radial equations. The coupled system may then be solved by first assuming the approximation to hold and solving the radial equation for $\omega$, this is then fed into the angular equation to obtain an improved value of $\Lambda$. This process may be iterated until the desired level of convergence is achieved.
\[num proc\] Numerical procedure
--------------------------------
The method of finding solutions numerically is very much like performing the matched expansions. We use Eqs. (\[hypergeometric solution\]) and (\[far field\]) to fix the initial conditions for two integrations of the exact radial equation. Since the equation of motion is linear, we may immediately match the two solutions at a point in the interior region by rescaling. This leaves two more conditions to be satisfied, that matching of the derivatives of the real and imaginary parts. Fixing all other parameters, we vary the real and imaginary parts of $\omega$ to satisfy these conditions.
Given the small size of the expected imaginary part, it is most straightforward to use a package like Mathematica [@mathematica] with its software based arbitrary precision, to perform the calculations. Satisfying the matching conditions can be done by treating the difference in derivatives at the interior point as a complex valued function of $\omega$. A root may then be searched for using the built-in function [FindRoot]{} which, for a function without explicit derivatives, looks for the solution by constructing secants.
Since the imaginary part is expected to be far smaller than the real, gradients of the matching function in the imaginary $\omega$ direction will be large only when very near a solution, but negligible elsewhere. The initial guesses at the solution are therefore very important for ensuring that iterations converge to a solution. It was found empirically that solutions could consistently be found by choosing to start the search in a region around the real value of $\omega$ for which the inner solution vanishes at the matching point. Small changes in the imaginary part of $\omega$ near this point appear to be sufficient to bring about convergence.
In Fig. \[fig:sampleplot\] we show an example solution obtained in this manner. The solid line is the full numeric solution, with the integration starting at small $x$ in red to the left of the black dot and that starting at large $x$ on the right in blue. The dashed lines are the near (\[hypergeometric solution\]) and far (\[far field\]) approximations used to set the initial conditions for integrating the exact radial equation. The fact that the imaginary part of $\omega$ is in general very small raises non-trivial problems, related to the number of digits of precision used and the exact way in which boundary conditions are applied. A discussion of these aspects is deferred to Appendix \[sec:A1\].
\[num res\] Numerical results
-----------------------------
Our numerical results are summarized in Fig. \[fig:fixNvaryl\_compare\_methods\] and Table \[tab:efficiency3\]. In Fig. \[fig:fixNvaryl\_compare\_methods\] on the left we present the numerical solutions obtained for $$m = 5\,,\,\, n = 1 \,,\,\, c_1 = 1.1\,,\,\,c_5=1.52\,,\,\, a_1 =
262.7 \,,\,\, \lambda = m_{\phi} = 0 \ .$$ where we consider only the lowest harmonic, $N=0$, but vary $l=m_\psi$. At $l=1$, $ \kappa ^{-1} \sim 40$ indicating the matched solution is valid, as $l$ a grows so do $\xi, \zeta$ while $\kappa ^{-1}$ shrinks, meaning the approximation should soon break down. At $l=5$, $\kappa^ {-2} \sim 10$ and the approximation is becoming no longer valid. Finally, when $l=13$, $\kappa^ {-2} \sim 1$ and differences between the matched and numerically determined eigenvalues are starting to become apparent. In Fig. \[fig:fixNvaryl\_compare\_methods\] on the right, we use the same parameters as before, but now fix $l=m_{\psi}=4$ and vary the harmonic from $N=0$ up to $4$. Increasing $N$ leads to smaller values of $\omega$ and therefore smaller values of $\kappa$, so that the matched solutions are valid throughout. It should also be noted that if the approximation $a_1^2 \omega^2/R^2 \ll m_{\psi}^2$ is valid for a given $m_{\psi}$, then it should be valid for all $m_{\psi}$. This is because $\omega$ scales with $m_{\psi}$, as we observed within the WKB approximation.
In Table \[tab:efficiency3\] we present and compare the numerical results with those obtained through the approximate analyical approaches. The values labeled as WKB$_{\rm NUM}$ (numerical WKB) stand for values obtained using the full WKB approximation, formulae (\[WKBparameters1\]), (\[WKBparameters2\]), (\[ressonance freq\]) and (\[tau\]), which have been handled numerically. The values obtained using the parabolic approximation, formulae (\[parabola\])-(\[alpha 4\]), for the potential are denoted by WKB$_{\rm AN}$.
Notice first that all the different approaches yield consistent and in fact very similar results: they are all rather accurate in their own regime of validity. As predicted by the analytic approaches, and verified numerically, the real part of the frequency scales with $m_{\psi}$, whereas the logarithm of the imaginary part scales with $m_{\psi}$, see Eq. (\[WKBparameters2\]). Thus the instability timescale increases rapidly as a function of $m_{\psi}$.
$m_\psi$
---------- -------------------------------------- ----------------------------------------- ---------------------------------- ---------------------------------------
1 $ -0.184 + 3.83 \times 10^{-8}i $ $ - $ $-$ $ -0.184 + 3.83 \times 10^{-8}i $
2 $ -0.744 + 2.51 \times 10^{-8}i $ $ -0.812 + 1.61 \times 10^{ -7}i $ $-0.826+1.89\times 10^{-7}i$ $ -0.744 + 2.64 \times 10^{-8}i $
3 $ -1.312 + 3.73 \times 10^{ -9}i $ $ -1.359 + 1.20 \times 10^{ -8}i $ $-1.371+1.48\times 10^{-8}i$ $ -1.312 + 3.53 \times 10^{ -9}i$
4 $-1.883 + 3.69 \times 10^{ -10}i $ $ -1.919 + 9.33 \times 10^{ -10}i $ $-1.932+1.17\times 10^{-9}i$ $ -1.882 + 3.63 \times 10^{ -10}i$
5 $-2.456 + 3.55 \times 10^{ -11}i $ $ -2.486 + 7.37 \times 10^{ -11}i $ $-2.499+9.39 \times 10^{-11}i $ $ -2.454 + 3.39 \times 10^{ -11}i $
6 $ -3.030 + 3.22 \times 10^{ -12}i $ $ -3.055 + 5.89 \times 10^{ -12}i $ $-3.072+7.62\times 10^{-12}i$ $ -3.028 + 3.02 \times 10^{ -12}i $
7 $ -3.605 + 2.77 \times 10^{ -13}i $ $ -3.626 +4.73 \times 10^{ -13}i $ $-3.647+6.23 \times 10^{-13}i$ $ -3.602 + 2.63 \times 10^{ -13}i $
8 $ -4.180 + 2.47 \times 10^{ -14}i $ $ -4.199 + 3.82 \times 10^{ -14}i $ $-4.216+4.88\times 10^{-14}i$ $ -4.176 + 2.24 \times 10^{ -14}i $
9 $ -4.755 + 2.05 \times 10^{ -15}i $ $ -4.772 + 3.09 \times 10^{ -15}i $ $-4.794+4.03\times 10^{-15}i$ $ -4.751 + 1.89 \times 10^{ -15}i $
10 $ -5.331 + 1.76 \times 10^{ -16}i $ $-5.346 + 2.51 \times 10^{ -16}i $ $-5.369+3.26\times 10^{-16}i$ $ -5.326 + 1.58 \times 10^{ -16}i $
11 $-5.907 + 1.49 \times 10^{ -17}i$ $-5.921 + 2.03 \times 10^{ -17}i $ $-5.947+2.65 \times 10^{-17}i$ $ -5.902 + 1.32 \times 10^{ -17}i $
12 $ -6.483 + 1.22 \times 10^{ -18}i$ $-6.496 + 1.65 \times 10^{ -18}i $ $-6.516+ 2.07\times 10^{-18}i $ $-6.477 + 1.09 \times 10^{ -18}i $
13 $-7.059 + 1.04 \times 10^{ -19}i $ $-7.071 + 1.34 \times 10^{ -19}i $ $-7.102+1.81\times 10^{-19}i $ $ -7.053 + 8.97 \times 10^{ -20}i $
: \[tab:efficiency3\] Some numerical values of the instability for a geometry with $m=5,\, n=1,\, c_1=1.10,\,
c_5=1.52,\, a_1=262.7,\, \lambda=m_{\phi}=0$ and $l=m_{\psi}$. In the second column, we have the results of the full numerical analysis; in the third column, WKB$_{\rm NUM}$ (numerical WKB) stands for values obtained using the full WKB approximation, formulae (\[WKBparameters1\]), (\[WKBparameters2\]), (\[ressonance freq\]) and (\[tau\]); and in the fourth column, labelled as WKB$_{\rm AN}$, the values obtained using the parabolic approximation for the potential, formulae (\[parabola\])-(\[alpha 4\]), are given. In the final column, we present the results of the matching procedure (\[eq:asymp2\_quantization\]),(\[deltaN\]). Notice the close agreement between all the different methods. For $m_{\psi}=l=1$ and for these particular values of the parameters, the WKB analysis, as done here, breaks down. Indeed, for $m_{\psi}=1$, the potential $V_+$ has no minimum.
\[conclusion\] Discussion
=========================
In this paper, we have shown that the non-supersymmetric solitons [@ross] are classically unstable. The relevant instabilities are quite generic to spacetimes which have an ergoregion but are horizon-free [@friedman]. However, as noted in Section \[free\], the general proof does not strictly apply to the solutions since the latter support nonradiative negative energy modes as shown in Appendix \[sec:A2\]. Hence we have explicitly shown that the ergoregion instabilities are active in the geometries using three different approaches, which in the end show a remarkable agreement — see Fig. \[fig:fixNvaryl\_compare\_methods\] and Table \[tab:efficiency3\]. Perhaps the most physically intuitive method is the WKB analysis carried on in Sec. \[wkb\]. This approach allows us to clearly identify the nature and physical properties of the instability. However, this analysis is only expected to be valid for large angular momentum quantum numbers, $m_\psi\gg0$, which is not where the instability is strongest. The more unstable modes were studied using the matched asymptotic expansion method [@staro1] in Sec. \[sec:Match\]. As a final consistency check of these analytical results, we made a numerical analysis of the wave equation in Sec. \[numerical\].
In passing we note by considering orbifolds, the solutions were extended to a six-parameter family which includes a third integer $k$ characterizing the orbifold group $\mathbb{Z}_k$ [@ross]. Of course, it is straightforward to adapt our instability analysis so that the modes respect this orbifold symmetry in the covering space and so one concludes that the ergoregion instability arises in these orbifold geometries as well.
Let us now summarize some of the features of the ergoregion instability found for the solutions:
\(i) The general shape of the WKB potentials $V_\pm$ are sketched in Fig. \[fig:potential\] for the case in which an instability is present. The key point is that when the ergoregion is present the bottom of the potential well in $V_+$ reaches negative values. The unstable modes are those whose pattern speed $\Sigma_{\psi}$ is negative and approaches the minimum of $V_+$ from above (see Fig. \[fig:potential\]). Thus, they are nearly bound states of the potential well in $V_+$ that can however tunnel out to infinity through $V_-$.
\(ii) The fact that the unstable modes are those with negative phase velocity, $\Sigma_{\psi}<0$, has a clear physical interpretation. As in the discussion of Eqs. (\[Sigma\]) and (\[Omega\]), modes with $\Sigma_{\psi}<0$ are those that propagate in the same sense as geometry’s rotation $\Omega_{\psi}$. Therefore at infinity these modes carry positive angular momentum (same sense as $\Omega_{\psi}$), as well as positive energy. Hence by conservation of energy and angular momentum, with the onset of the ergoregion instability, the solutions are shedding both energy and angular momentum by an amount that increases exponentially.
\(iii) The instability can be quite strong, depending on the particular combination of parameters that define the geometry. More importantly, the instability is robust, in the sense that it exists for a wide range of parameters.
\(iv) With $m=n+1$, the solutions are supersymmetric and so must be stable. It is a consistency check of our analysis then that we find no instability in this case. As commented in Section \[wkb\], when $m=n+1$ the potential $V_+$, as given by Eq. (\[potential SUSY\]), is always positive. Hence there are no negative $\Sigma_{\psi}$ modes which could intersect the potential well of $V_+$ and the SUSY geometry is stable as required.
In our analysis, we have focused on the special case $\lambda=0$ and $m_\phi=0$, to simplify the relevant equations. In fact, the ergoregion instability persists when either or both of these parameters are nonvanishing. A discussion of the general situation is given in Appendix \[beyond\]. The result is most simply understood from the point of view of the WKB approach. Then all of the additional contributions to the effective potential (\[rad eq01\]) introduced by a nonvanishing $m_\phi$ or $\lambda$ are suppressed by inverse powers of $m_\psi$ and so can certainly be neglected in the limit of large $m_\psi$. One can further check that the instability exists over some range even when $m_\psi$ does not dominate the other two. One distinguishing feature of $\lambda\ne0$ is that asymptotically the scalar modes have an effective mass in five dimensions. In our analysis, this is reflected in the fact that asymptotically $V_\pm\rightarrow\pm|\lambda/m_\psi|$ and so there is an additional barrier for the modes to tunnel out to infinity. However, for sufficiently large $m_\psi$, such tunnelling is possible. One other interesting point about the large $m_\psi$ regime is that unstable modes appear with either sign of $m_\phi$ and $\lambda$. Hence while for the modes on which we have focussed, the instability is ‘powered’ by $J_\psi$ and results in decreasing this angular momentum, there are unstable modes which may at the same time increase $|J_\phi|$ and/or $P$.
Adding a mass for the scalar field modifies the potentials $V_\pm$ in essentially the same way as having nonvanishing $\lambda$. Hence we expect the ergoregion instability will even appear for massive fields, at least in modes with sufficiently large angular momentum. As described in section \[free\], the arguments given by Friedmann [@friedman] are quite general and so we expect the ergoregion instability to appear for higher spin fields as well. In particular, we expect the fields of the low energy type IIb supergravity will generically experience this instability. Having said that the ergoregion instability is robust, we must also add that it can be suppressed in certain parameter regions. In particular, one finds that the instability timescale becomes extremely long in the regime where $Q_1$ and $Q_5$ are much larger than the other scales. Further we add that in the decoupling limit where one isolates an asymptotically AdS$_3$ core [@ross], the ergoregion instability is absent. The simplest way to understand this result is that the AdS$_3$ core has a globally timelike Killing vector [@ross] and so there is a ‘rotating’ frame where we can define all energies to be positive. One can also explicitly verify the absence of an ergoregion instability in the core solutions by directly applying the analysis used in this paper to those backgrounds.
The geometries (both supersymmetric and non-supersymmetric) also have damped modes, modes (\[separation ansatz\]) for which the imaginary part of $\omega$ is negative. As per the WKB analysis, these are modes with positive $\Sigma_{\psi}$ below the local maximum of $V_+$ that tunnel out to infinity through $V_+$ — see Fig. \[fig:potential damped\].
![ Damped modes are those that have positive $\Sigma_{\psi}$. []{data-label="fig:potential damped"}](potential_damped){width="8" height="6"}
As emphasized previously, we can also find purely bound states (nonradiative modes) with $\kappa^2\propto\omega^2-\lambda^2<0$. With some fine-tuning, it may also be possible to find geometries which support bound states with $\kappa=0$. These nonradiative modes are described in Appendix \[sec:A2\]. The typical situation for such modes is sketched in Fig. \[fig:potential bound\]. As already noted above when $\lambda\ne0$, asymptotically $V_\pm\rightarrow\pm|\lambda/m_\psi|$ and so there is a finite potential barrier at infinity. If this barrier is sufficiently large relative to $\Sigma_{\psi}=\omega/m_\psi$, bound states can arise. These bound states can also be negative energy states, as can be seen with the energy integral (\[canon\]). The absence of such negative energy modes which do not radiate at infinity was central to Friedman’s general argument for the ergoregion instability. In [@friedman], he did not find any such nonradiative modes because he only considered the massless fields for which there is no potential barrier at infinity. Note, however that the current situation is more complicated because the KK-momentum of the background, as well as the angular momenta, contribute to the presence of an ergoregion.
![ Qualitative shape of the potentials $V_+$ and $V_-$ when $\omega^2-\lambda^2<0$. These are the purely bound states that are discussed in Appendix \[sec:A2\]. []{data-label="fig:potential bound"}](potential_bound){width="8" height="6"}
The appearance of negative energy states in the presence of an ergoregion can be anticipated from a geodesic analysis [@cominsschutz]. By definition, the Killing vector $t^a$, which generates asymptotic time translations, becomes space-like inside the ergosphere. Hence (time-like or null) geodesics can have either positive or negative energy, $e=-t\cdot u$, in this region. However, asymptotically only positive energy (future-oriented) geodesics are physical. Therefore any negative energy geodesics must be confined to circulate within the ergoregion. Of course, in a black hole background, such geodesics would ‘disappear’ behind the event horizon. However, for horizon-free geometries, such as the solutions, they are stable bound orbits and so it is natural to find bound states in the context of a field theory analysis. However, the question then becomes whether the analogous modes of the field ‘fit’ inside the ergoregion or whether they ‘leak’ out to infinity, whether a negative energy bound state or an ergoregion instability results. A more thorough examination of the bound states shows that the negative energy bound states states are characterized by having $\Sigma_y=\omega/\lambda<0$ while the ergoregion instability is associated with modes where $\Sigma_{\psi}=\omega/m_\psi$ and/or $\Sigma_{\phi}=\omega/m_\phi$ are negative – see Appendices \[beyond\] and \[sec:A2\]. Hence as the geodesic analysis would suggest the negative energy modes have a negative pattern speed or phase velocity, but the KK-momentum modes tend to lead to bound states while the spinning modes are related to instabilities.
The presence of negative energy bound states can also be expected to enhance the decay of these horizon-free geometries. The analysis of the ergoregion instability (considered in this paper) is only at the level of linearized test fields. Generically any theory coupling to gravity will also have nonlinear interactions (even the free scalar considered here has nonlinear couplings with gravitons). These nonlinear couplings might be expected to lead to processes, where positive energy modes are radiated at infinity while negative energy modes are populated within the ergoregion. However, one should note that the negative energy modes are exponentially decaying at large radius — see Appendix \[sec:A2\] — while the positive energy modes are power-law suppressed inside the ergoregion. Hence the overlap of these modes is expected to be small, which will suppress this nonlinear contribution to the decay.
We now turn to consider the endpoint of the ergoregion instabilities. As emphasized before, the presence of these instabilities relies on two key ingredients, namely, the geometry has an ergoregion but it does not have an event horizon. Hence the resulting decay process could be terminated either by the disappearance of the ergoregion or the appearance of a horizon. However, the unstable modes radiate with a positive energy density asymptotically which is compensated for by a negative energy density inside the ergoregion — as could be seen in Eq. (\[canon\]). Hence the onset of the ergoregion instability produces a(n exponential) build-up of negative energy near the core of the solutions. Therefore it seems unlikely that an event horizon will form since the latter is typically associated with a large build-up of positive energy density. This reasoning then suggests that the decay must terminate with the disappearance of the ergoregion. The supersymmetric D1-D5-P microstate geometries [@two; @three1; @three12; @three2] are all free of an ergoregion and hence it is natural to suppose that these are at the endpoint of the ergoregion instabilities. Of course, these solutions offer a huge family of possible endpoints and the precise one that forms will depend on the details of the decay process, beyond the linear regime considered here — although as we are only considering the classical evolution, it is in principle possible given a certain set of initial conditions. Of course, we can expect that the final mass should be close to the BPS mass determined by the charges of the initial solution, $E=\pi/4G_5\,[Q_1+Q_5+Q_P]$. Although even here, we can only say ‘close’ as we know that the unstable modes with $\lambda\ne0$ (and [*either*]{} sign of $\lambda$) occur which may modify the final value of $Q_P$. Similar comments apply for the angular momenta, $J_\psi$ and $J_\phi$. We also observe that there is no reason to expect that the decay process will lead to an endpoint within the family of supersymmetric solutions. Of course, at the level of the present discussion, we cannot rule out that the endpoint is only a nearly supersymmetric solution (or that this would be the effective endpoint). Our expectation is that such solutions will have a ‘small’ ergoregion and that the instability might be eliminated (or strongly suppressed) before the ergoregion precisely vanishes.[^10]
The stability analysis of the solitons [@ross] is relevant for the stringy tachyon decays discussed recently in [@rossTC]. Originally, [@horowitzTC] considered tachyon condensation in certain D1-D5 black string backgrounds where tachyonic string winding modes can occur if one chooses antiperiodic boundary conditions for the fermions around the circle on which the black string is compactified. The latter choice necessarily restricts the scenario to a non-supersymmetric sector of string theory which already suffers from various instabilities [@negative]. Ref. [@rossTC] considered adding angular momentum to the black strings. In this case, it was shown that string modes winding certain compact circles near the horizon can be tachyonic even when the asymptotic fermion boundary conditions are supersymmetric. The relevant point for the present discussion is that the endpoint of the tachyon condensation is in general one of the non-supersymmetric solitons. Now, in this paper, we have shown that these solitons are themselves unstable and so they will not be the final endpoint of these decays. Instead, the ergoregion instability will continue the decay process and as suggested above, will likely terminate with a supersymmetric microstate geometry.
We would now like to consider the implications of ergoregion instabilities for Mathur’s fuzzball program of describing black holes in terms of an ensemble of microstate geometries. If this program is to succeed it must supply a description of both supersymmetric and also non-supersymmetric black holes. At first sight, it may appear that constructing non-BPS microstate geometries is not possible. In particular, non-BPS states will decay and so it is not clear that there should be stationary geometries to describe them. However, the solutions provide an explicit example indicating that this is not really a problem. In fact, the decay of non-BPS microstates was already considered in the D-brane description of nonextremal black holes [@revall]. In that context, it was seen as a success of the string theoretic approach as this instability had an interpretation in terms of Hawking radiation [@curt; @malstrom; @sdsm]. Of course, Hawking radiation is a quantum effect in the black hole background and so presents no obstacle to the construction of classical supergravity solutions which are static or stationary.
It is perhaps useful to remind ourselves as to how this distinction arises. The classical limit can be understood as the limit where the string coupling $g_s$ is vanishingly small [@malstrom]. However, the interesting classical solutions are those which correspond to states where the various quantum numbers are extremely large. That is, $n_1,n_5\propto 1/g_s$ and $n_p,J_\psi,J_\phi\propto1/g_s^2$ while $g_s\rightarrow0$. These scalings are chosen to ensure that the gravitational ‘footprint’, $Q_1$, $Q_5$, $Q_P$, $a_1$ and $a_2$, associated with each of these quantum numbers remains finite. However, in this limit, the ADM energy of the system diverges with $E\propto 1/g_s^2$. As the energy is a dimensionful quantity, this can be accommodated by changing the units with which energies are measured in the classical limit. Essentially this divergence is associated with the divergence of the Planck mass, which does not serve as a useful reference scale in classical gravity. Now the decay rate of the nonextremal D1-D5-P black holes can be computed in a straightforward manner [@malstrom; @sdsm]. The key point, however, is that the final expression for $dE/dt$ is expressed in terms of geometric quantities and is independent of $g_s$. Therefore in the classical limit, the rate of energy loss remains finite in stringy units but becomes vanishingly small when measured against the fiducial energy scale that was established for classical physics. Hence the non-BPS black holes become stable in the classical regime.
We note that the ‘straightforward’ calculations of the decay rate referred to above can be performed either in the framework of a microscopic D-brane perspective or of the gravitational perspective of Hawking radiation. The suprising result is that the results of both analyses agree precisely [@malstrom; @sdsm], including greybody factors, at least in the so-called ‘dilute gas’ approximation [@dilute]. However, even though suggestive arguments can be made in this regime [@poor], this remarkable agreement remains poorly understood. As the solutions are horizon-free, the gravitational calculation of the decay rate would have to be modified. Using the connection between absorption and emission rates, it is possible that absorption calculations along the lines of those presented in [@ross] could be extended to yield the desired decay rate. On the other hand, the underlying microscopic states for the solutions were already identified in [@ross]. Hence one can use microscopic techniques to estimate the decay rate expected for these solutions. The result is $dE/dt\sim Q_1Q_5(m-n)^6/R^6$ and again this quantity remains finite as $g_s\rightarrow0$. Therefore we can again ignore this decay channel for the classical solutions.
However, the ergoregion instability investigated in this paper is a classical instability and so should not be associated with the decay discussed above. We should also note that the form of these two instabilities differs. Above one is considering the spontaneous decay of the system while the classical instability really corresponds to a decay that results when the initial data does not precisely match that of the solutions. Of course, in the quantum regime, the same modes associated with the ergoregion instability will give rise to spontaneous decay due to quantum fluctuations of the background.[^11] However, the latter will again be suppressed in the $g_s\rightarrow0$ limit. This reflects the fact that the background can be prepared with arbitrarily accurate precision in the classical limit and so it should be possible to produce an arbitrary suppression of ergoregion instability. Alternatively, working in the classical limit, we can regard the ergoregion instability as a property of how the solutions interact with external sources. That is, generically if an external wave packet impinges on one of the non-supersymmetric configurations, it will produce a dramatic decay of the original background. Hence this instability seems to present a major challenge for the fuzzball description of black holes.
We have argued that the ergoregion instability is a robust feature of the non-supersymmetric solutions over a wide range of parameters. Given general arguments along the lines of [@friedman], we also expect that this instability will be a generic feature of any smooth horizon-free geometries which describe microstates which are non-BPS and carry significant angular momentum (and hence have a macroscopic ergoregion). Therefore if a nonextremal D1-D5-P black hole is to be described by a coarse-grained ensemble fuzzball, it seems that that the classical black holes must suffer from an analogous instability. While the presence of an event horizon eliminates the possibility of an explicit ergoregion instability, there are, in fact, a number of potential instabilities which might afflict these black holes and possibly reproduce the same physics:
[**a) Superradiant Instability**]{}: Spinning nonextremal black holes will exhibit superradiant scattering, where an incident wave packet can be reflected with a stronger amplitude. Superradiance by itself does not provide a classical instability, but an instability can arise if the scattered modes are reflected back to rescatter, as described in Section \[general\]. This scattering was considered for higher dimensional spinning black branes [@cardoso] and there it was found that when the noncompact space has more than four dimensions, this instability does not arise. Explicitly analyzing the present D1-D5-P black string again seems to indicate the absence of an instability [@more2].
[**b) Gyration Instability**]{}: Considering supersymmetric D1-D5-P black strings, it was found that above a certain critical angular momentum a straight black string is unstable towards carrying the angular momentum in gyrations of the horizon [@donny]. This instability should also appear in non-supersymmetric configurations and so would present an instability at large values of the angular momentum.
[**c) Gregory-Laflamme Instability**]{}: The relevant configurations are black strings and so are expected to suffer from the Gregory-Laflamme instability [@grelaf] in two ways. The first is the usual instability of long wavelength modes along the string. Of course, this instability can be eliminated by reducing the radius of the compactification along the string. For a fixed radius, it is also suppressed by the boosting along this direction which induces the KK-momentum [@boost]. This instability is not related to the angular momentum carried by the black string or the presence of an ergoregion, but we list it here for completeness.
[**d) Ultra-spin Instability**]{}: In six or higher spacetime dimensions, one can find black hole solutions with an arbitrarily large spin per unit mass [@myersperry]. However, it was argued [@ultra] that a Gregory-Laflamme-like instability will arise to dynamically enforce a Kerr-like bound in these cases. While this analysis does not directly apply in five dimensions, entropy arguments suggest an analogous instability still exists and will lead to the formation of a black ring if the angular momentum is too large [@ring].
While there are several possibilities for instabilities of a black string in six dimensions, it seems that none of these can reproduce the physics of the ergoregion instability which will afflict the non-BPS microstate geometries. This observation relies on the fact that these instabilities have a different character at a very basic level. The ergoregion instability might be termed a radiative instability, in that, the instability is by definition connected to modes that radiate at infinity. In contrast, the four instabilities considered above for black strings can be termed internal instabilities. That is, these instabilities are primarily associated with a rearrangement of the internal or near-horizon structure of the black string. While these instabilities will be accompanied with some radiation at infinity, this will be a secondary effect with these instabilities. Therefore it seems that emulating the ergoregion instability in a nonextremal black string background will require the discovery of a new kind of instability. While we are performing a detailed analysis of the nonextremal D1-D5-P black string, our preliminary results indicate that no such instability arises [@more2].
We also note in passing that at the same time the microstate geometries should be able to emulate any instabilities found in the black string backgrounds. In particular, the Gregory-Laflamme instability is a robust instability that will afflict these backgrounds for sufficiently large $R$. In the microstate geometries, one should then find unstable modes carrying KK-momentum which are confined near the core of the soliton. We have studied bound states for a test field in the solutions, as described in Appendix \[sec:A2\]. While the modes we identified only arise for nonvanishing KK-momentum as desired, they are all stable, they have real frequencies. Hence they can not serve as the analog of the Gregory-Laflamme instability in the non-supersymmetric solutions. However, the latter would be a gravitational instability, it should not be expected to appear as a scalar test field, and so this question requires further investigation.
A possible reconciliation of these ideas with the fuzzball proposal would be that the microstate geometries could provide an accurate description of a black hole but only over a long but finite time. In the context of the AdS$_3$/CFT$_2$ duality, some evidence for such a picture has recently been found [@vjtime]. With this new point of view, a key question is to determine the timescale over which microstate geometries cannot be distinguished from black holes. One suggestion [@vjtime] is that it should be of the order of the recurrence time, which would be exponential in the relevant quantum numbers. An alternative suggestion might be that the timescale is associated with Hawking evaporation which would involve (inverse) powers of the quantum numbers. However, note that both of these suggestions diverge in the classical limit. Hence the ergoregion instability found here seems to be in conflict with both of these suggestions. While the instability timescale is certainly very long in certain parameter regimes, it is a classical timescale, it is finite in the classical limit. Hence our results would suggest that spinning microstate geometries and black holes should be distinguishable on a large but classically finite timescale.
However, one must ask how characteristic our results for the solutions will be of generic microstate geometries. In particular, we note that the CFT states corresponding to the solutions are exclusively in the untwisted sector [@ross; @three12; @cft]. On the other hand, the majority of microstates accounting for the entropy of the black strings are expected to be in a (maximally) twisted sector [@fuzzy]. From a geometric point of view, we would observe that the solutions have all the same Killing symmetries as the D1-D5-P black holes, while the generic microstate geometry is expected to have a complex nonsymmetric core. Therefore it is not unreasonable to expect that the ergoregion instability timescales found for the solutions will not be characteristic of the microstate geometries that make up ‘most’ of the black hole.
One possibility might be the generic non-BPS geometries do not have ergoregions despite the fact that they carry angular momentum. However, we argue that such a scenario is implausible as follows: The fuzzball description would now require that both the horizon and the ergosphere arise as effective surfaces in ‘coarse-graining’. However, quantum fluctuations must then extend out to the ergosphere. In particular, these fluctuations extend to regions of the spacetime which should be causally accessible to asymptotic observers on finite classical timescales. Hence it seems inconsistent to say that the underlying microstate geometries are hidden from asymptotic observers in this scenario.
Hence as argued above, if the non-BPS microstate geometries are horizon-free with an ergoregion, they should expect an ergoregion instability. However, it may be that instability timescales calculated for the solutions are not representative of those for typical microstate spacetimes. In particular, the latter should have complicated throats — as seen in their supersymmetric counterparts [@10; @19; @two] — which would emulate the absorptive behavior of a black hole horizon. Hence it might be expected that the relevant timescales are extremely long. An important question is then whether the instability timescale is classically finite or not. That is, will this timescale diverge as the quantum numbers grow as described above. Certainly finding more generic non-BPS microstate geometries is an essential step towards resolving this issue.
In closing, we note that in the context of the AdS/CFT, a complete description has been produced for half-BPS microstate geometries with AdS$_5$ [@lin] and AdS$_3$ [@ads3] asymptotics. This framework has given rise to an interesting program of semi-classical quantization [@vjtime; @semi; @semi3] and a coarse-graining description of spacetime geometry [@emerge]. With this program in mind, it is useful to recall the role of the smooth horizon-free microstate geometries in Mathur’s ‘fuzzball’ program [@fuzzy].
The BPS microstate geometries for the D1-D5 system can be derived by studying the F1-P geometries and applying a series of duality transformations [@two]. There the winding and wave numbers might be quantized by the geometry but classically the amplitudes of the string excitations are continuous variables. Solutions where select modes are excited with a large amplitude can then be seen as ‘coherent states’ of the underlying quantum theory. Such solutions may be further useful to understand certain properties of typical microstates, their transverse size [@fuzzy]. However, ultimately a generic state will have a vast number of modes excited with very few quanta and hence the corresponding ‘spacetime’ will not be accurately described by a classical geometry. However, the family of classical geometries still serve as a guide to the classical phase space which must be quantized [@semi3].
In the present context, we wish to go beyond the BPS sector where program is much less developed. In particular, we still face the challenge of constructing a more or less complete family microstate geometries. The existence of the solutions indicate that at least certain non-BPS states can be described by classical geometries. However, it is not at all clear how large a class of nonsupersymmetric smooth horizon-free geometries exists. Going beyond the present special class of solutions will probably call for the development of new solution-generating techniques, but the geometries offer hope that a broader class of nonsupersymmetric solutions can be found. This will certainly be an intriguing direction for further research and will undoubtedly lead to interesting new insights and discoveries.
Acknowledgements {#acknowledgements .unnumbered}
================
It is a pleasure to acknowledge Vijay Balasubramanian, Thomas Levi, Donald Marolf, Samir Mathur and Simon Ross for interesting comments and discussions. Research at the Perimeter Institute is supported in part by funds from NSERC of Canada and MEDT of Ontario. RCM is further supported by an NSERC Discovery grant. VC and OJCD acknowledge financial support from Fundação para a Ciência e Tecnologia (FCT) - Portugal through grant SFRH/BPD/2004. OJCD also acknowledges CENTRA - Centro Multidisciplinar de Astrofísica, Portugal for hospitality. JLH was supported by an NSERC Canada Graduate Scholarship.
\[sec:A0\] WKB matching
=======================
In this appendix we use the usual WKB wavefunctions and WKB connection formulae at the turning points to relate the amplitude of the wavefunctions in the four distinct regions of the scattering problem and, in particular, to derive (\[connectionWKB\]). The four WKB regions are (see Fig. \[fig:potential\]): Region I, the innermost forbidden region ($0<x<x_0$); Region II, the allowed region where $V_+$ is below $\Sigma_{\psi}$ ($x_0<x<x_1$); Region III, the potential barrier region where $V_+$ is above $\Sigma_{\psi}$ ($x_1<x<x_2$); and Region IV, the external allowed region where $\Sigma_{\psi}$ is below $V_-$ ($x_2<x<\infty$). The WKB wavefunctions in region I and in region IV were already written in (\[H1\]) and (\[H4\]), respectively, and in regions II and III they are given by $$\begin{aligned}
H_{\rm II}&\simeq& \frac{C_2}{m_{\psi}^{1/2}T^{1/4}} {\rm
exp}\left[ i\,m_{\psi}\int_{x_1}^{x} \sqrt{T}\,dx \right]
+\frac{C_3}{m_{\psi}^{1/2}T^{1/4}} {\rm exp}\left[
-i\,m_{\psi}\int_{x_1}^{x} \sqrt{T}\,dx \right], \label{H2} \\
H_{\rm III}&\simeq& \frac{C_4}{m_{\psi}^{1/2}|T|^{1/4}}
{\rm exp}\left[ -\,m_{\psi}\int_{x_1}^{x} \sqrt{|T|}\,dx \right] +
\frac{C_5}{m_{\psi}^{1/2}|T|^{1/4}} {\rm exp}\left[
\,m_{\psi}\int_{x_1}^{x} \sqrt{|T|}\,dx \right]. \label{H3}\end{aligned}$$ Using the WKB connection formulae in each turning point, $x_0$, $x_1$ and $x_2$, we can find the relations between the amplitudes $C_i$’s $(i=1,\cdots,7)$ of the several regions, yielding: $$C_2=C_1 e^{i \gamma}\,, \quad C_3=C_1 e^{-i \gamma}\,.
\label{connectionWKB-12}$$
$$C_4= \frac{1}{2} \left (C_2 e^{-i \pi/4}+C_3 e^{i \pi/4}\right)\,,
\quad C_5 =i \left (C_2 e^{-i \pi/4}-C_3 e^{i \pi/4}\right)\,,
\label{connectionWKB-23}$$
$$C_6 = \left ( \frac{i C_4}{2\eta}+ C_5 \eta \right )e^{-i
\pi/4}\,,\quad C_7 = \left ( -\frac{i C_4}{2\eta}+ C_5 \eta \right
)e^{i\pi/4}\,,
\label{connectionWKB-34}$$
with $\gamma$ and $\eta$ defined in (\[WKBparameters1\]) and (\[WKBparameters2\]), respectively. Finally, combining these three sets of relations yields (\[connectionWKB\]).
\[sec:A1\]Details of numerical analysis
=======================================
In this Appendix we discuss some issues related to the precision used in the numerical computations. Even though the very small imaginary parts of $\omega$ are well described by both approximations, for completeness we show that they are not a numerical artifact due to loss of precision in our numeric routines or a by-product of using the approximate solutions to specify the boundary conditions. In Fig. \[fig:compare\_precision\] we plot the imaginary part of $\omega$ for several values of the number of digits of precision used in the calculation. We use the same parameters as before and set $N=0$, $l=m_\psi=4$. We see, as one would expect if the imaginary part were actually non-zero, that the eigenvalue converges to a constant value when the number of digits is larger than the size of the imaginary part.
With only the asymptotic form of the solutions to specify the boundary conditions we are not actually setting the coefficient on the divergent term to zero. Instead, there will always be some amount of the divergent solution in the numerically defined solution. The suppression of the divergent term is dependent on how deep into the asymptotic region we choose to apply the boundary condition. To ensure that these small divergent terms are not causing any errors we study the effect of varying the point at which we apply the boundary conditions. This has been shown in Fig. \[fig:compare\_bcs\] on the left and right. In both cases, we again see that the eigenvalue converges to a constant value as we increase the accuracy of the calculation.
Detailed analysis of the instability {#beyond}
====================================
The existence of a solution to the matching procedure can be reduced to the requirement that a number of constraints be satisfied. The difficulty one runs into when trying to discuss the general properties of these solutions is that while all the parameters appearing in the various equations are uniquely determined by the set $\{Q_1,Q_5,R,m,n\}$, it is difficult to write explicit expressions for them. In this sense, the fact that the parameters (\[simple\]) can be written in such a simple form is really quite surprising since all are proportional to $M$, which can at best be defined implicitly in terms of the above parameters.
Hence it is useful to have an approximation for $M$ that allows one to understand the general behavior of the various parameters. Surprisingly there is quite a simple approximate solution given by $$M \approx 2(s^{-1}-s) \frac{Q_p}{1+nm \left ( \frac{Q_1 +
Q_5}{R^2} \right ) } \ , \label{m_approx}$$ where we recall that $Q_p = n m Q_1 Q_5/R^2$. For most parameter values, this expression is accurate on the order of a few percent. When one of the D-brane charges, say $Q_1$, grows much larger than $Q_5 \sim R^2$ this approximation can break down, though only by a few percent times $(m-(n+1))$. Similar problems appear when $R^2
\gg Q_1 \sim Q_5$, in this case the error appears to be of the same order. The important thing to note is that it gives the correct scaling of $M$ with the various parameters in all situations. In most cases, except those noted previously when $m
\gg n$, it also gives the correct order of magnitude. Treating $m$ as a continuous parameter, the approximation appears to produce the approach to the supersymmetric limit exactly.
Using this, one can approximate, or at least bound, the parameters appearing in the solutions. $$\begin{aligned}
\varrho & = & \frac{c_1^2 c_5^2 c_p^2-s_1^2 s_5^2 s_p^2}{s_1 c_1 s_5
c_5}
\approx \frac{s^{-1}+s}{2} \left (1 + n m \frac{Q_1+Q_5}{R^2} \right ) \\
\varepsilon & \leq & \frac{1}{R^2}
\left (Q_1 + Q_5 + Q_p + M \left ( 1-\frac{n m s^2}{(s^{-2}-s^2)^2} \right ) \right ) \nonumber \\
& \approx & \frac{1}{R^2}
\left (Q_1 + Q_5 + Q_p + \frac{2Q_p}{1+n m \frac{Q_1+Q_5}{R^2}}
\left [ \frac{ nm(1-s^4)^2-s^6}{n m s(1-s^4)(1+s^2)} \right ] \right ) \\
\vartheta & \leq & \frac{Q_p}{Q_1 Q_5} \left (Q_1 + Q_5 + M \right ) \nonumber \\
& \approx & \frac{Q_p}{Q_5} + \frac{Q_p}{Q_1}
+ 2 \frac{Q_p^2}{Q_1 Q_5} \frac{s^{-1}-s}{1+n m \frac{Q_1 +
Q_5}{R^2} } \label{background_parameters}\end{aligned}$$ In the above, the inequalities result from writing $s_i^2 \leq s_i
c_i = Q_i/M$, in particular they become exact for the extremal limit. From the expression for $\varepsilon$ one sees that it is finite, and in fact positive for all $m \geq n+2$. It is only in the extremal limit that $\varepsilon \rightarrow -\infty$, which precludes any possible instability. One may also check from these forms that $\varepsilon /\varrho^2 \ll 1$ for all values of the parameters, which can be verified numerically for sets of parameters in which the approximations are less trustworthy. In what follows then we will neglect $\varepsilon$ where consistent.
The timescale of the instability is an increasing function of $\nu$ which, given the above considerations, is given by $$\nu \approx - \lambda \frac{Q_p}{\varrho R^2}+
\sqrt{ \lambda \frac{Q_p}{\varrho R^2}
\left ( \lambda \frac{Q_p}{\varrho R^2}+2 c\right )
+ \nu_0^2 - \frac{\varepsilon}{\varrho^2}c^2 } \ .$$ Unfortunately, we cannot make any definite statements about the size of $Q_p/\varrho R^2$ like we did previously for $\varepsilon$ since it can be made arbitrarily large or small just by varying $R$. At this point we could use the explicit forms for $\nu_0^2$ and $c$ to discuss the general properties of the solutions. Instead we will for now set $\lambda=0$ to make the discussion more transparent. Non-zero $\lambda$ will not change the general features of the solutions.
Setting $\lambda = 0$, the above expression for $\nu$ simplifies quite a bit $$\nu \approx \sqrt{\nu_0^2 - \frac{\varepsilon}{\varrho^2}c^2 } \ .$$ We are now in a position to discuss the behavior of the timescale for various different solutions. Recall that the timescale for the instability is smallest when $\nu$ is smallest. Therefore the instability will be strongest when $\nu_0 = l+1$ is smallest. This, of course, does not mean that we should necessarily consider solutions with $l=0$, in fact we shall see in a moment that such solutions are not possible. More precisely the minimum value of $l$ for which all the constraints can be satisfied will lead to the most unstable solution.
Similarly, when $c^2$ or $\varepsilon/\varrho^2$ is largest the instability be the strongest. We shall deal with $c$ next, but for now it is sufficient to note that it is only dependent on $m$ and $n$. Observe from (\[background\_parameters\]) that for fixed $R$, $\varepsilon/\varrho^2$ varies roughly like the inverse of the charges, therefore when one considers limits in which the charges grow, the timescale of the instability diverges. Similar arguments hold when $R$ is vastly different from the charges, we find that $\varepsilon/\varrho^2$ shrinks and the lowering effect of $c^2$ is diminished. It appears then that the instability will be strongest when $Q_1 \sim Q_5 \sim R^2$.
To discuss the relative effect of $c$ we should return to the constraints. These also simplify when we set $\lambda=0$ and we may consider the simpler constraint $c-\nu_0 > 0$. The exact form that $c$ takes is dependent on the sign of $\zeta$. By studying the constraints, it turns out that solutions with $\zeta > 0$ will in general exist, but for larger values of $l$ than when $\zeta <
0$. Given the considerations above, the effect of these modes will be subdominant. We therefore focus on $\zeta = -n m_\psi + m
m_\phi< 0$ which implies that $m_\psi > m_\phi$. One can then write the constraint as $$\begin{aligned}
c-\nu_0 & = & \left [ (m-n)(m_\psi+m_\phi) - (2N+1) \right ] - \left [ l+1 \right ] \\
& = & (m-(n+1))(m_\psi + m_\phi) - (l-m_\psi-m_\phi) - 2(N+1) > 0
\ .\end{aligned}$$ Further, it can be shown that when this is satisfied, the other constraints follow automatically. The last two bracketed terms in the final line are positive, so a solution requires that $m_\psi +
m_\phi > 0$, implying that $m_\psi$ must be positive. This is a general result that is also obtained when $\zeta >0$ or $\lambda
\neq 0$. When $c$ is largest, the timescale will be shortest, therefore the lowest harmonic $N=0$ will lead to the strongest instability. One can also make $c$ large by choosing $m_\psi$ and $m_\phi$ as large as large as possible, $m_\psi+m_\phi=l$, but taking $l$ large will not necessarily give us a very unstable mode because as noted before it will cause $\nu_0$ to rise which has an opposing effect. Since $c^2$ enters weighted by $\varepsilon/\varrho^2$, the more important contribution will be that from $\nu_0$ and the net effect is a less unstable mode. Finally, note that $m_\phi$ and $m_\psi$ appear symmetrically in $c$, so that the value of $\nu$ will be independent of the partition of $l$ into $m_\phi$ and $m_\psi$. This does not mean that the timescale will be independent of this partition since it is a weakly shrinking function of $|\zeta|$ for fixed $\nu$. When $|\zeta|$ is maximized the timescale will be the shortest, which is the case when $m_\psi=l$, $m_\phi=0$.
To summarize then, for a fixed mode that solves the constraints, the instability will be strongest when $Q_1 \sim Q_5 \sim R^2$. On the other hand, when we fix a particular background, the instability with $\lambda=0$ will be strongest when $l=m_\psi$ is as small as possible and $m_\phi=0$.
Finally then we may discuss the solutions for which $\lambda \neq
0$. It turns out that the various scalings of the other parameters appears not to be changed. When $\lambda \neq 0$, the constraint $c^2 - \nu_0^2 > 0$ becomes easier to satisfy since $c$ picks up a contribution proportional to $\vartheta \lambda$ while the contribution to $\nu_0$ is smaller. The tougher constraint to satisfy is then the one that implies $\omega^2 > \lambda^2$. When all other parameters are fixed, this places upper and lower bounds on (we allow negative $\lambda$) $\lambda$. We will not go into detail here, but instead note one can always find solutions with non-zero $\lambda$ by going to sufficiently large angular momentum, $l$.
When studying the characteristic time for the instabilities, one finds that the timescale decreases as $\lambda$ is raised, but reaches a maximum shortly before reaching the upper bound. For negative values, on the other hand, the timescale is a constant decreasing function of $\lambda$. As mentioned, solutions with non-zero $\lambda$ require larger values of $l$ than when $\lambda=0$. Though larger $l$ tends to increase the timescale, the overall effect of going to larger $l$ to accommodate non-zero $\lambda$ can still lead to shorter timescales.
\[sec:A2\]Bound States
======================
The general radial dependence of the scalar field at large distances from the core is determined by the sign of $\kappa^{2}$. When it is positive, the general solution oscillates with a power-law falloff. This is the behavior that led to the in and outgoing waves at infinity which we have already discussed. The other two possibilities, where $\kappa^2$ is zero or negative, can lead to quite different behavior. For the former there is an exact solution, while the latter may again be solved with a matched expansion.
${\bm \kappa^{2}=0}$: Marginally Bound States
---------------------------------------------
By considering the special mode with $\omega^2=\lambda^2$, both the angular and radial equations simplify sufficiently that an exact solution may be found. Such a choice removes all $\omega$ dependence from the angular equation allowing it to be solved independently. The result is the eigenvalue equation for the harmonics on an $S^3$. The exact eigenvalue is $\Lambda=l(l+2)$.
For the radial equation, this choice of mode removes the $\kappa^2
x$ term; the same condition that previously led to the simplification in the near region. The previous solution in the near region (\[hypergeometric solution\]) therefore becomes the exact solution in the entire spacetime. This means that asymptotically the equation has a basis of solutions in terms of $r^{-1 \pm \nu}$. Ignoring for now the part dependent on the KK momentum, these become $r^{l}$ and $r^{-2-l}$. These are simply the terms one expects from a Laplace series in four flat spatial dimensions where the angular momentum creates an effective radial potential.
Asymptotic regularity requires that the $r^{-1 + \nu}$ component vanish whenever $\nu > 1$, leaving a field that falls off as $r^{-1-\nu}$. The natural generalization of Friedmann’s analysis of ergoregion instabilities to five-dimensions would involve studying fields that fall off as $r^{-2}$, therefore these modes will evade that analysis as long as $\nu > 1$. The requirement that removes the divergent term is similar to that for outgoing modes, except now it is an exact result $$\begin{aligned}
\nu+|\zeta| \mp \xi & = & -(2N+1) \ , \label{eq:mbound-state-quantized}\end{aligned}$$ where $N$ is a non-negative integer. Here, however, we allow for either of the $\Gamma$ functions in the denominator to diverge in eq. (\[near field large r\]), leading to both possibilities for the sign before $\xi$. This is in contrast to the search for unstables modes in which we could neglect one of the possibilities since it was found to corresponded to ingoing damped modes. Indeed, since (\[eq:mbound-state-quantized\]) contains terms linear in both $\lambda$ and $\omega$, one must consider both signs above in order to be consistent with the symmetry under flipping signs as in equation (\[symmetry\]).
In total then we have three constraints that must be satisfied for these modes. The first, $\omega^2=\lambda^2$, fixes $\omega$ to be an integer, meaning that there are no remaining continuous parameters characterizing the scalar field. For a general background then it is unlikely that the remaining constraints, in particular the one defining $N$, can be solved by a judicious choice of the integer eigenvalues. On the other hand, fixing the set $m_{\psi},m_{\phi}$ and $\lambda$, there may be families of backgrounds for which these marginally bound states exist.
${\bm \kappa^{2} < 0}$: Bound States
------------------------------------
The final possibility for solutions of the radial equation is $\omega^2 < \lambda^2$, or $\kappa^2 < 0$. As in the case where $\kappa^2$ is positive, we are unable to find an exact solution, though progress can be made through approximation. In particular, since the effect of the sign of $\kappa$ is only relevant at large distances from the core, we need only make slight modifications to the matched asymptotic expansion analysis presented earlier.
To begin, we factor out the sign of $\kappa^2$ by redefining $\kappa \rightarrow i\kappa$, giving solutions that are real valued exponentials asymptotically. Requiring regularity therefore leaves only the exponentially damped “bound states”, localized near the core region. Explicitly, after having made the redefinition in (\[far wave eq\]), a convenient basis of solutions is in terms of modified Bessel functions of the first and second kind. $$\begin{aligned}
h & = & \frac{1}{\sqrt{x}} \left [ A_1 I_\nu(\kappa \,\sqrt{x}) +
A_2 K_\nu (\kappa\, \sqrt{x} ) \right ] \label{eq:hout_solution} \
.\end{aligned}$$ The first of these diverges at large $x$ and so we require $A_1=0$ for regularity. For now though, we leave $A_1$ arbitrary, setting it to vanish only after we have performed the matching.
In the matching region expanding $I_\nu$ and $K_\nu$ in the $x^{\pm \nu/2}$ basis gives $$h \approx \frac{1}{\sqrt{x}} \left [
\left ( \frac{A_1}{\Gamma(1+\nu)} +\frac{A_2 \Gamma(-\nu)}{2} \right )
\left (\frac{\sqrt{x}\,\kappa}{2} \right )^\nu
+ \frac{A_2 \Gamma(\nu)}{2} \left (\frac{\sqrt{x}\,\kappa}{2} \right )^{-\nu}
\right ] \ . \label{eq:hout_overlap}$$ Note that $K_\nu$ contains both of these powers of $x$ when expanded in the overlap region. While the contribution of the positive power to $K_\nu$ is relatively small, we will keep this contribution until after we perform the matching so that we may see how the approximate solution comes about.
By construction, the solution in the near region (\[hypergeometric solution\]) is unaffected by the redefinition of $\kappa$. Immediately then we may proceed to matching the coefficients on powers of $x$ in the overlap region. Setting $A=1$ in the near region solution, we determine the coefficients $A_1,A_2$ in the outer region $$\begin{aligned}
\frac{A_1 (\kappa/2)^\nu}{\Gamma(1+\nu) } & = &
\frac{\Gamma(\nu)\Gamma(1+|\zeta|)}{\Gamma(\frac{1}{2}(1+\nu+|\zeta|+\xi))
\Gamma(\frac{1}{2}(1+\nu+|\zeta|-\xi))}
- \frac{\Gamma^2(-\nu)\Gamma(1+|\zeta|)(\kappa/2)^{2\nu}}{\Gamma(\nu)\Gamma(b)
\Gamma(\frac{1}{2}(1-\nu+|\zeta|-\xi))} \ , \label{eq:c1-eq}\\
\frac{A_2\Gamma(\nu)}{2(\kappa/2)^\nu} & = &
\frac{\Gamma(-\nu)\Gamma(1+|\zeta|)}{\Gamma(b)\Gamma(\frac{1}{2}(1-\nu+|\zeta|-\xi))}
\label{eq:c2-eq}\ ,\end{aligned}$$ As before, finding the spectrum of solutions now requires that we find values of the free parameters for which these equations are consistent with the boundary conditions. In particular, we now set $A_1=0$ and therefore ask that the right hand side of (\[eq:c1-eq\]) vanishes. Again, rather than find such parameters numerically there is an accurate approximation that comes from noting that consistency requires $A_2$ be non-zero. This implies that the second term in (\[eq:c1-eq\]) must be non-zero and therefore any solution must come from cancellation between the two terms. Since the second term is suppressed by the factor $\kappa^{2\nu}$, a comparable suppression must occur in the first term, again requiring the divergence of a $\Gamma$ function in the denominator. This gives a quantization condition similar to that found previously $$\nu+|\zeta| \mp \xi \approx -(2N+1)\ .
\label{eq:bound-state-quantized}$$ Here again, the terms linear in $\omega$ and $\lambda$ implicit in the above equation – see definitions in eq. (\[rad eq parameters0\]) — imply that both possibilities are required for consistency with the symmetry (\[symmetry\]), though in practice both may not lead to solutions for which $\omega^2 < \lambda^2$.
When $\nu$ is real, this appears to give solutions for $\omega$ which are purely real. Note, however, we must be careful in solving the constraint since, given the right combination of background charges, $\nu^2$ could become negative. For an arbitrary frequency in this range, Eq. (\[eq:c1-eq\]) will be complex so solutions where $\omega$ has both real and imaginary parts may be possible. Such solutions cannot be found with the sort of perturbative expansion used in studying the outgoing modes since now it is the real part of $\nu$ which gains a small correction, while the imaginary part is large. We can therefore no longer consider the behavior near the pole on the negative real axis defined by the real part of Eq. (\[eq:bound-state-quantized\]). Instead, we have resorted to searching for these solutions by solving (\[eq:c1-eq\]) numerically.
Generically, the root finding algorithm will produce a complex value of $\omega$ that sets the equation to zero within a specified precision. Since the imaginary part is many orders of magnitude smaller than the real part one should ensure that it really is non-zero and not a numerical artifact. In Figure \[fig:unstable\_bs\] we show the variation in the size of the imaginary part as a function of the tolerance used in finding the root of (\[eq:c1-eq\]). From this plot we see that the imaginary contribution is indeed just an artifact of trying to solve the complex equation. Surprisingly then it appears we can satisfy (\[eq:c1-eq\]) with a real value of $\omega$, even if that value causes $\nu^2 <0$. That value corresponds to the solution of the equation resulting from taking the real part of the quantization conditions (\[eq:bound-state-quantized\]).
![Variation of the size of the imaginary part of $\omega$ resulting from the numerical solution of (\[eq:c1-eq\]) as the precision is increased.[]{data-label="fig:unstable_bs"}](unstable_bs){width="8" height="6"}
Since the condition (\[eq:bound-state-quantized\]) is the same as for the outgoing modes, much of the analysis in Appendix \[beyond\] about the existence of solutions applies. The situation is somewhat more complicated in that one now allows modes with positive $\omega$ and there are two possible solutions corresponding to the two signs in (\[eq:bound-state-quantized\]), but the general characteristics of the solutions are the same. In particular, for the outgoing modes it was found that there are upper and lower bounds on the allowable values of $\lambda$ beyond which $\omega^2 - \lambda^2$ changes sign. In light of these bound states, we see that the full space of solutions may be considered as split into distinct regions based on the value of $\lambda$. There is a small-$|\lambda|$ regime, in which one finds the outgoing unstable modes. This is surrounded, at larger values of $|\lambda|$, by a regime where the bound states arise.
This separation of the two types of modes according to the parameter $\lambda$ makes clear the difference in their origin. In particular, one can always find outgoing unstable modes that do not carry KK momentum, they need only be supplied with sufficient angular momentum. This is in accord with our interpretation of these solutions as the unstable modes predicted by Friedman which result from the existence of the ergoregion. In contrast, bound states will always result as long as $|\lambda|$ is large enough. This includes modes which carry no angular momentum, thus indicating the important characteristic of these solutions is their KK momentum and the effective five-dimensional mass it induces.
Having established the existence of these bound states we should question just how close to the core region they are bound. The solutions are damped exponentially and so have characteristic size $$\begin{aligned}
x_{bs}^{-1} \sim \kappa^2 & = & (\lambda^2 - \omega^2) \frac{r_+^2-r_-^2}{R^2} \ , \\
& \approx & 2(\lambda^2-\omega^2) \frac{Q_1 Q_5}{(s^{-1}+s)R^2(R^2+n m(Q_1+Q_5))} \ .\end{aligned}$$ To arrive at the final line we have used the approximation for $M$ (\[m\_approx\]) found in Appendix \[beyond\]. The boundary of the ergoregion, on the other hand, is given by the vanishing of the norm of the Killing vector $\partial_t$ (\[ergoregionV\]). We ignore the $a_1,a_2$ dependent contributions appearing in $f$ to give an outer bound on the size of the ergoregion, given approximately by $r_{er}^2 \sim M c_p^2$. In terms of the variable $x$ this means $$\begin{aligned}
x_{er}^{-1} \gtrsim \frac{s^{-2}-s^2}{n m(s^{-2}-s^2)^2 c_p^2-s^2} \ .\end{aligned}$$
Whenever $Q_1$ and $Q_5$ are much smaller than $R^2$, the size of the bound state scales as $x_{bs}^{-1} \sim Q_1 Q_5/R^4 \ll 1$. On the other hand, for large $Q_1$ and $Q_5$ we have $x_{bs}^{-1} \sim Q_i/R^2$ where $Q_i$ is the smaller of the two. In other words, the size of the bound state is strongly dependent on the background. When the charges are large, the bound state will be mostly contained within the ergoregion, while for small charges the exponential tail of the bound state can extend far outside.
Finally we can consider the possibility that the bound states have negative energy, which requires a detailed analysis of the energy integral (\[canon\]). Examining the integrand evaluated on bound state solutions, we see that it may become negative near where the modulus of the scalar field peaks if the latter occurs inside the ergoregion. Though there are bound states for arbitrarily large values of $|\lambda|$, the total energy will not be negative for all of these. Instead, the modes that tend to exhibit negative energy densities (in the ergoregion) only appear for a limited range of $|\lambda|$, which is just beyond the small-$|\lambda|$ regime discussed above. That is, for values of $\lambda$ near where the ergoregion instability appears. When this is the case, the maximum of the modulus of the scalar field is inside the ergoregion and the phase velocity in the compactified direction $\Sigma_y = \omega/\lambda$ is negative, in the direction opposite to which the background is boosted.
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[^1]: Of course, in these examples, a single saddle-point typically dominates the path integral.
[^2]: ‘Smooth’ means the curvature is finite everywhere up to orbifold singularities. The curvatures in the throat may also be very large.
[^3]: By considering orbifolding, this family can extended by a third integer [@ross] but we will focus on the original five-parameter solutions.
[^4]: Implicitly, we consider a massless field here but the discussion is generalized to massive fields in a straightforward way.
[^5]: Note that the negative sign for $\lambda$ corrects a typo found in [@ross]
[^6]: Note the factor $(r_+^2-r_-^2)$ that appears in the two last terms of the lhs of (\[radialeq-r\]), which are necessary for dimensional consistency, corrects the typo appearing in Eq. (6.4) of [@ross]
[^7]: Note that the geometry rotates simultaneously along the $\psi$, $\phi$ and $y$ directions. We find $\Omega_{\psi}$ using of (2.1), (3.17) and (3.19) of [@ross].
[^8]: More precisely, we have dropped a term $1/(m_{\psi}^{\,2} {\cal P})$. This remains a very good approximation in the high-$m_{\psi}$ limit in which we are working. As an example, for $n=10$ and $m_{\psi}=10$ the factor that we dropped is $10^{-4}$ smaller than the last term of (\[Def T\]).
[^9]: Our conventions differ slightly from those of [@cominsschutz]. There waves carry a time dependence $e^{i\omega t}$ while we follow [@ross] which introduces the separation ansatz (\[separation ansatz\]) with a time dependence $e^{-i\omega
t}$.
[^10]: We should note that the solutions begin in a low-mass regime where $M^2<(a_1-a_2)^2$, however, if the ergoregion instability sheds the background angular momentum efficiently then the system will evolve to a regime where black holes can form. Hence we can not rule out the appearance of an event horizon – we thank Simon Ross for correspondence on this point.
[^11]: In [@ross] it was erroneously assumed that all of these geometries have an AdS$_3$ core to argue that such emissions would not occur.
|
---
abstract: 'The Motzkin numbers can be derived as coefficients of hybrid polynomials. Such an identification allows the derivation of new identities for this family of numbers and offers a tool to investigate previously unnoticed links with the theory of special functions and with the relevant treatment in terms of operational means. The use of umbral methods opens new directions for further developments and generalizations, which leads, e.g., to the identification of new Motzkin associated forms.'
author:
- |
M. Artioli\
ENEA - Bologna Research Center\
Via Via Martiri di Monte Sole, 4, 40129, Bologna, Italy\
[email protected]
- |
G. Dattoli\
ENEA - Frascati Research Center,\
Via Enrico Fermi 45, 00044, Frascati, Rome, Italy\
[email protected]
- |
S. Licciardi\
Dep. of Mathematics and Computer Science,\
University of Catania, Viale A. Doria 6, 95125, Catania, Italy\
ENEA - Frascati Research Center,\
Via Enrico Fermi 45, 00044, Frascati, Rome, Italy\
[email protected]
- |
S. Pagnutti\
ENEA - Bologna Research Center,\
Via Via Martiri di Monte Sole, 4, 40129, Bologna, Italy\
[email protected]
title: 'Motzkin Numbers: an Operational Point of View'
---
Introduction
============
The telephone numbers $(T_{n})$, also called convolution numbers, provide a very well known example of link between special numbers and special polynomials. The $(T_{n})$ can be expressed in terms of Hermite polynomials coefficients $ (h_{s}) $ [@Riordan]. Two of the present authors (M.A and G.D.) have recently pointed out in ref. [@Artioli] that the Padovan and Perrin numbers [@Perrin; @Padovan] can be recognized to be associated with particular values of two variable Legendre polynomials [@Dattoli].
Weinstein has discussed in [@Weisstein] the connection between Motzkin numbers and a family of hybrid polynomials, and Blasiak et al. and Dattoli et al. have studied, in [@Blasiak; @Lorenzutta], the relevant properties of Motzkin numbers.
The hybrid polynomials are indeed defined as [@Lorenzutta]
$$\label{Kn}
P_{n}^{(q)}(x,y)=n!\sum_{r=0}^{\lfloor\frac{n}{2}\rfloor }\dfrac{x^{n-2r}y^{r}}{(n-2r)!r!(r+q)!},$$
and the relevant generating function reads
$$\label{genKn}
\sum_{n=0}^{\infty}\dfrac{t^{n}}{n!}P_{n}^{(q)}(x,y)=\dfrac{I_{q}(2 \sqrt{y}\; t)}{(\sqrt{y}\;t)^{q}}e^{xt},$$
where $I_{q}(x)$ is the modified Bessel function of the first kind of order $ q $.
Within the present framework, the Motzkin numbers sequence can be specified as [@Blasiak]
$$\begin{split}\label{key}
& m_{n} = P_{n}^{(1)}(1,1)=\sum_{s=0}^{n}m_{n,s},\\
& m_{n,s}=\binom{n}{s}\;f_{s},\\
& f_{s} = \dfrac{s!}{\Gamma\left( \dfrac{s}{2}+2\right) \Gamma\left( \dfrac{s}{2}+1\right) }\left| \cos\left( s\dfrac{\pi}{2}\right) \right|,
\end{split}$$
where the coefficients $m_{n,s}$ can be represented as the triangle reported in the following table, in which $m_{n,2}$ corresponds, in OEIS, to the sequence $A000217$, $m_{n,4}$ to $A034827$, $m_{n,6}$ to $A000910$ and so on.\
$ m_{n}\; \textbf{Motzkin} $
-------------- ------------------------------ ------- ------- ------- ------- ------- ------- ------- ------- ---------
(lr)[3-10]{} **0** **1** **2** **3** **4** **5** **6** **7**
**0** 1 **1**
**1** 1 0 **1**
**2** 1 0 1 **2**
**3** 1 0 3 0 **4**
**4** 1 0 6 0 2 **9**
**5** 1 0 10 0 10 0 **21**
**6** 1 0 15 0 30 0 5 **51**
**7** 1 0 21 0 70 0 35 0 **127**
… … … … … … … … … …
: Motzkin Numbers and their Coefficients.[]{data-label="table1"}
According to eq. , the Motzkin numbers can also be defined as the coefficients of the following series expansion
$$\label{genMn}
\sum_{n=0}^{\infty}\dfrac{t^{n}}{n!}m_{n}=\dfrac{I_{1}(2 t)}{t}e^{t}.$$
In the following we will show how some progresses in the study of the relevant properties can be done by the use of a formalism of umbral nature.
Motzkin Numbers and Umbral Calculus
===================================
In order to simplify most of the algebra associated with the study of the properties of the Motzkin numbers and to get new relevant identities, we introduce a formalism successfully exploited elsewhere [@Borel] based on methods of umbral nature [@Roman]. To this aim we note that the function
$$\label{Tric}
C_{q}(x)=\dfrac{I_{q}(2 \sqrt{x})}{(\sqrt{x})^{q}}=\sum_{r=0}^{ \infty}\dfrac{x^{r}}{r!(q+r)!}$$
can be cast in the form
$$\label{TricOp}
C_{q}(x)=\hat{c}^{q} \circ e^{\hat{c}x},$$
where $\hat{c}$ is an umbral operator defined according to
$$\label{actOp}
\hat{c}^{\mu}=\dfrac{1}{\Gamma(\mu+1)},$$
with $\mu$ not necessarily integer and real.We define the following composition rule
$$\label{opop}
\hat{c}^{\mu} \circ \hat{c}^{\nu}=\hat{c}^{\mu+\nu}$$
and we let $\hat{C}=\{\hat{c}^{\alpha},\alpha\in \mathbb{C}\}$ denote the set of $\hat{c}$-operators. Then, the pair $(\hat{C},\circ)$ satisfying the Abelian-group property. The mathematical foundations of the theory of $\hat{c}$-operators can be traced back to those underlying the Borel transform and have been carefully discussed in ref. [@Roman].
The use of this formalism allows to restyle the hybrid polynomials in the form
$$\label{KnOp}
P_{n}^{(q)}(x,y)=\hat{c}^{q} \circ H_{n}(x,\hat{c}\;y),$$
where
$$\label{Herm}
H_{n}(x,y)=n!\sum_{r=0}^{\lfloor\frac{n}{2}\rfloor }\dfrac{x^{n-2r}y^{r}}{(n-2r)!r!}$$
are the two variable Hermite-Kampé de Fériét polynomials of order $ 2 $.We can accordingly use the wealth of properties of this family of polynomials to derive further and new relations regarding those of the Motzkin numbers family.
By recalling indeed the generating function [@Lorenzutta]
$$\label{genHermnl}
\sum_{n=0}^{\infty}\dfrac{t^{n}}{n!}H_{n+l}(x,y)=H_{l}(x+2yt,y)e^{xt+yt^{2}},$$
we find
$$\label{genmnl}
\sum_{n=0}^{\infty}\dfrac{t^{n}}{n!}m_{n+l}=\hat{c} \circ H_{l}(1+2\hat{c}t,\hat{c})e^{t+\hat{c}t^{2}},$$
which, after using eqs. , , , finally yields
$$\begin{split}\label{mu}
& \sum_{n=0}^{\infty}\dfrac{t^{n}}{n!}m_{n+l}=\mu_{l}(t)\;e^{t},\\
& \mu_{l}(t)=l!\sum_{r=0}^{\lfloor\frac{l}{2}\rfloor }\dfrac{1}{r!}\sum_{s=0}^{l-2r}\dfrac{2^{s}}{s!(l-2r-s)!}\dfrac{I_{s+r+1}(2t)}{t^{r+1}}.
\end{split}$$
Furthermore, the same procedure and the use of the Hermite polynomials duplication formula [@Andrews]
$$\label{Hd}
H_{2n}(x,y)=\sum_{r=0}^{n}\binom{n}{r}^{2}r!\;(2y)^{r}\left( H_{n-r}(x,y)\right)^{2},$$
yields the following identity for Motzkin numbers
$$\begin{split}\label{mdupl}
m_{2n}& =\hat{c} \circ \sum_{r=0}^{n}r!\;\binom{n}{r}^{2}(2\hat{c})^{r} \circ H_{n-r}(1,\hat{c}) \circ H_{n-r}(1,\hat{c})=\\
& = \sum_{r=0}^{n}\binom{n}{r}^{2}2^{r}r!(n-r)!\sum_{s=0}^{\lfloor\frac{n-r}{2}\rfloor }\dfrac{m_{n-r}^{(r+s+1)}}{(n-r-2s)!s!},
\end{split}$$
where
$$\label{mnmop}
m_{n}^{(q)}=P_{n}^{(q)}(1,1)=\hat{c}^{q} \circ H_{n}(1,\hat{c})$$
are associated Motzkin numbers [@Blasiak].The identification of Motzkin numbers as in eq. , along with the use of the recurrences of Hermite polynomials, yields, e.g., the identities
$$\begin{split}\label{recmn}
& m_{n+1}^{(q)}=m_{n}^{(q)}+2\;n\;m_{n-1}^{(q+1)},\\
& m_{n+p}=\sum_{s=0}^{\min[n,p]}2^{s}s!\;\binom{p}{s}\;\binom{n}{s}M_{p-s,\;n-s,\;s},\\
& M_{p,\;n,\;t}=p!\sum_{r=0}^{\lfloor\frac{p}{2}\rfloor}\dfrac{m_{n}^{(t+r+1)}}{(p-2r)!r!},
\end{split}$$
in which, the second identity, has been derived from the Nielsen formula for $H_{n+m}(x,y)$ [@Nielsen].
Final Comments
==============
In this paper we have shown that a fairly straightforward extension of the formalism put forward in ref. [@Blasiak], allows non trivial progresses in the theory of Motzkin numbers. Further relations can be easily obtained by applying the method we have envisaged as, e.g.,
$$\label{prmn}
\sum_{s=0}^{n}m_{n-s}\;m_{s}=2\;(n+1)\;m_{n}^{(2)},$$
which represents a discrete self-convolution of Motzkin numbers.
We have also mentioned the existence of the associated Motzkin numbers
$$\label{mnq}
m_{n}^{(q)}=P_{n}^{(q)}(1,1),$$
touched on in ref. [@Blasiak]. In the present context they have been introduced on purely algebraic grounds. Strictly speaking they are not integers and therefore they are not amenable for a combinatorial interpretation however, redefining them as
$$\label{tildem}
\tilde{m}_{n}^{(q)}=\dfrac{(n+q)!}{n!}P_{n}^{(q)}(1,1),$$
we obtain for $q=2$ the sequences in OEIS $(A014531)$, while for $q=3$ the sequences $(A014532)$ and so on.\
A more appropriate interpretation in combinatorial terms can be obtained by following, e.g., the procedures indicated in ref. [@Banderier] and deserves further investigations, out of the scope of the present paper.
We have mentioned in the introduction the theory of telephone numbers $T(n)$ [@Knuth], whose importance in chemical Graph theory has been recently emphasized in ref. [@Hatz]. As well known, they can be expressed in terms of ordinary Hermite polynomials, however the use of the two variable extension is more effective. They can indeed be expressed as $T(n)=H_{n}(1,\frac{1}{2})$ .The use of Hermite polynomials properties, like the index duplication formula, yields
$$\label{T2n}
T(2n)=\sum_{r=0}^{n}\;\binom{n}{r}^{2}\;r!\;T(n-r)^{2}.$$
The use of the Hermite numbers $h_{s}$ [@Germano] allows the derivation of the following further expression
$$\begin{split}\label{hs}
& T(n)=\sum_{s=0}^{n}t_{n,s},\\
& t_{n,s}=\binom{n}{s}\;h_{s}\left( \dfrac{1}{2}\right) ,\\
& h_{s}(y)=y^{\frac{s}{2}}\Gamma\left( \dfrac{s}{2}+2\right)f_{s}=\dfrac{y^{\frac{s}{2}}s!}{\Gamma\left( \dfrac{s}{2}+1\right) }\left| \cos\left( s\;\dfrac{\pi}{2}\right) \right|.
\end{split}$$
The coefficients $t_{n,s}$ of the telephone numbers can be arranged in the following triangle, in which, the numbers belonging to the column $s=4$ , $(3, 15, 45, 105, 210, 378,\dots)$, are identified, in OEIS, with the sequence $A050534$ and the column in $s=6$, $(15, 105, 420,1260, 3150, \dots)$, is just a multiple of $A00910$.
$T{(n)}\; \textbf{telephone numbers}$
-------------- --------------------------------------- ------- ------- ------- ------- ------- ------- ------- ------- ---------
(lr)[3-10]{} **0** **1** **2** **3** **4** **5** **6** **7**
**0** 1 **1**
**1** 1 0 **1**
**2** 1 0 1 **2**
**3** 1 0 3 0 **4**
**4** 1 0 6 0 3 **10**
**5** 1 0 10 0 15 0 **26**
**6** 1 0 15 0 45 0 15 **76**
**7** 1 0 21 0 105 0 105 0 **232**
… … … … … … … … … …
: Telephone Number Coefficients[]{data-label="table2"}
The use of the identification with two variable Hermite polynomials opens further perspectives, by exploiting indeed the polynomials (see [@Riordan] and references therein)
$$\label{Hnm}
H_{n}^{(m)}(x,y)=n!\sum_{r=0}^{\lfloor \frac{n}{m}\rfloor }\dfrac{x^{n-mr}y^{r}}{(n-mr)!r!},$$
we can introduce the following generalization of telephone numbers
$$\label{Tnm}
T_{n}^{(m)}=H_{n}^{(m)}\left( 1,\dfrac{1}{m}\right),$$
with generating function
$$\label{gentn}
\sum_{n=0}^{\infty}\dfrac{t^{n}}{n!}T_{n}^{(m)}=e^{t+\frac{1}{m}t^{m}},$$
which satisfy the recurrence
$$\label{tnmp}
T_{n+1}^{(m)}=T_{n}^{(m)}+\dfrac{n!}{(n-m+1)!}T_{n-m+1}^{(m)}.$$
In the case of $m=3$ the numbers $T_{n}^{(3)}= (1, 1, 1, 3, 9, 21, 81, 351, 1233,\dots)$ are identified with OEIS $A001470$, while for $m=4$, the series $(1, 1, 1, 1, 7, 31, 91, 211, 1681, 12097, \dots)$, corresponds to $A118934$. For $m=5$ the associated series appears to be $A052501$ but should be more appropriately identified with the coefficients of the expansion , finally the sequence $m=6$ is not reported in OEIS.
A more accurate analysis of this family of numbers and the relevant interplay with Motzkin will be discussed elsewhere.
J. Riordan, *Introduction to combinatorial analysis*, Dover (2002), 85–86.
M. Artioli and G. Dattoli, Geometric interpretation of Perrin and Padovan numbers, http://demonstrations.wolfram.com/ GeometricInterpretationOfPerrinAndPadovanNumbers/, *Wolfram Demonstrations Project*, Sept. 14, (2016).
E. W. Weisstein, Perrin sequence, *Wolfram MathWorld-A Wolfram Web Resource*, mathworld.wolfram.com/PerrinSequence.html.
E. W. Weisstein, Padovan sequence, *Wolfram MathWorld-A Wolfram Web Resource*, mathworld.wolfram.com/PadovanSequence.html.
M. Artioli and G. Dattoli, Geometry of two-variable Legendre polynomials, *Wolfram Demonstrations Project-A Wolfram Web Resource*, demonstrations.wolfram.com/GeometryOfTwoVariableLegendrePolynomials.
E. W. Weisstein, Motzkin number, *MathWorld-A Wolfram Web Resource*, http://mathworld.wolfram.com/MotzkinNumber.html.
P. Blasiak, G. Dattoli, A. Horzela, K. A. Penson and K. Zhukovsky, Motzkin numbers, central trinomial coefficients and hybrid polynomials, *J. Integer Seq*. **11**, Art. 08.1.1, (2008).
G. Dattoli , S. Lorenzutta , P. E. Ricci and C. Cesarano , On a family of hybrid polynomials, *J. Integral Transforms and Special Functions* **15** (2004), 485–490.
G. Dattoli, E. di Palma, E. Sabia, K. Górska, A. Horzela and K. A. Penson, Operational versus umbral methods and the Borel transform, *Int. J. Appl. and Comput. Math.* (2017), DOI 10.1007/s40819-017-0315-7.
S. Roman, The theory of the umbral calculus, *J. Math. Anal. Appl.* **87** (1982), 58.
L. C. Andrews, *Special functions for engineers and applied mathematicians*, Macmillan, New York (1985).
N. Nielsen, Recherches sur les polynomes d’Hermite, *Mathematisk-Fysiske Meddelelser* **1** (1918), 79, Det. Kgl, Danske Videnskabernes Selskab.
C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the Kernel method, *Dev. Math.*, Springer, (2017).
D. E. Knuth, The art of computer programming **3** (1973), *Sorting and Searching*, Reading, Mass.: Addison-Wesley, 65–67, MR 0445948.
R. Hatz, M. Korpinen, V. Hanninen and L. Halonen, Generalized intermolecular interaction tensor applied to long-range interactions in hydrogen and coinage metal (Cu, Ag, and Au) clusters, *J. Phys. Chem. A* **119** (2015), 11729–11736.
G. Dattoli, B. Germano, M. R. Martinelli and P. E. Ricci, Lacunary generating functions of Hermite polynomials and symbolic methods, *Ilirias J. of Math.* **4** (2015), 16–23 .
|
---
abstract: |
[Many complex systems can be represented as networks, and the problem of network comparison is becoming increasingly relevant. There are many techniques for network comparison, from simply comparing network summary statistics to sophisticated but computationally costly alignment-based approaches. Yet it remains challenging to accurately cluster networks that are of a different size and density, but hypothesized to be structurally similar. In this paper, we address this problem by introducing a new network comparison methodology that is aimed at identifying common organizational principles in networks. The methodology is simple, intuitive and applicable in a wide variety of settings ranging from the functional classification of proteins to tracking the evolution of a world trade network.]{} [networks $|$ network comparison $|$ machine learning $|$ earth mover’s distance $|$ network topology]{}\
author:
- |
[Anatol E. Wegner]{}$ˆ*$,\
University College London, Department of Statistical Science, Gower Street, London WC1E 6BT, UK\
University of Oxford, Department of Statistics, 24-29 St. Giles’, Oxford, OX1 3LB, UK\
$ˆ*$\
[Luis Ospina-Forero]{}\
University of Oxford, Department of Statistics, 24-29 St. Giles’, Oxford, OX1 3LB, UK\
[[email protected]]{}\
[Robert E. Gaunt]{}\
The University of Manchester, School of Mathematics, Manchester M13 9PL, UK\
University of Oxford, Department of Statistics, 24-29 St. Giles’, Oxford, OX1 3LB, UK\
[[email protected]]{}\
[Charlotte M. Deane]{}\
University of Oxford, Department of Statistics, 24-29 St. Giles’, Oxford, OX1 3LB, UK\
[[email protected]]{}\
[and]{}\
[Gesine Reinert]{}\
University of Oxford, Department of Statistics, 24-29 St. Giles’, Oxford, OX1 3LB, UK\
[[email protected]]{}
bibliography:
- 'sample.bib'
title: Identifying networks with common organizational principles
---
Introduction
============
Many complex systems can be represented as networks, including friendships, the World Wide Web, global trade flows and protein-protein interactions [@newmanbook]. The study of networks has been a very active area of research in recent years, and in particular, network comparison has become increasingly relevant e.g$.$ [@wilson2008study; @netal; @2014waqar; @2014yaveroglu]. Network comparison itself has many wide-ranging applications, for example, comparing protein-protein interaction networks could lead to increased understanding of underlying biological processes [@2014waqar]. Network comparison can also be used to study the evolution of networks over time and for identifying sudden changes and shocks.
Network comparison methods have attracted increasing attention in the field of machine learning, where they are mostly referred to as graph kernels, and have numerous applications in personalized medicine e.g$.$ [@borgdisease], computer vision and drug discovery e.g$.$ [@NCI]. In the machine learning setting, the problem of interest is to obtain classifiers that can accurately predict the class membership of graphs.
Methods for comparing networks range from comparison of summary statistics to sophisticated but computationally expensive alignment-based approaches [@migraal; @netal; @sana]. Real-world networks can be very large and are often inhomogeneous, which makes the problem of network comparison challenging, especially when networks differ significantly in terms of size and density. In this paper, we address this problem by introducing a new network comparison methodology that is aimed at comparing networks according to their common organizational principles.
The observation that the degree distribution of many real world networks is highly right skewed and in many cases approximately follows a power law has been very influential in the development of network science [@barabasi1999emergence]. Consequently, it has become widely accepted that the shape of the degree distribution (for example, binomial vs power law) is indicative of the generating mechanism underlying the network. In this paper, we formalize this idea by introducing a measure that captures the shape of distributions. The measure emerges from the requirement that a metric between forms of distributions should be invariant under rescalings and translations of the observables. Based on this measure, we then introduce a new network comparison methodology, which we call $NetEmd$.
Although our methodology is applicable to almost any type of feature that can be associated to nodes or edges of a graph, we focus mainly on distributions of small connected subgraphs, also known as graphlets. Graphlets form the basis of many of the state of the art network comparison methods [@2007gdda; @2014waqar; @2014yaveroglu] and hence using graphlet based features allows for a comparative assessment of the presented methodology. Moreover, certain graphlets, called network motifs [@2002milo], occur much more frequently in many real world networks than is expected on the basis of pure chance. Network motifs are considered to be basic building blocks of networks that contribute to the function of the network by performing modular tasks and have therefore been conjectured to be favoured by natural selection. This is supported by the observation that network motifs are largely conserved within classes of networks [@milo2004superfamilies; @wegner].
Our methodology provides an effective tool for comparing networks even when networks differ significantly in size and density, which is the case in most applications. The methodology performs well on a wide variety of networks ranging from chemical compounds having as few as 10 nodes to tens of thousands of nodes in internet networks. The method achieves state of the art performance even when it is based on rather restricted sets of inputs that can be computed efficiently and hence scales favourably to networks with millions and even billions of nodes. The method also behaves well under network sub-sampling as described in @SS. The methodology further meets the needs of researchers from a variety of fields, from the social sciences to the biological and life sciences, by being computationally efficient and simple to implement.
We test the presented methodology in a large number of settings, starting with clustering synthetic and real world networks, where we find that the presented methodology outperforms state of the art graphlet-based network comparison methods in clustering networks of different sizes and densities. We then test the more fine grained properties of $NetEmd$ using data sets that represent evolving networks at different points in time. Finally, we test whether $NetEmd$ can predict functional categories of networks by exploring machine learning applications and find that classifiers based on $NetEmd$ outperform state-of-the art graph classifiers on several benchmark data sets.
A measure for comparing shapes of distributions
===============================================
Here we build on the idea that the information encapsulated in the shape of the degree distribution and other network properties reflects the topological organization of the network. From an abstract point of view we think of the shape of a distribution as a property that is invariant under linear deformations i.e$.$ translations and re-scalings of the axis. For example, a Gaussian distribution always has its characteristic bell curve shape regardless of its mean and standard deviation. Consequently, we postulate that any metric that aims to capture the similarity of shapes should be invariant under linear transformations of its inputs.
Based on these ideas we define the following measure between distributions $p$ and $q$ that are supported on $\mathbb{R}$ and have non-zero, finite variances: $$\label{emdmet}
EMD^*(p,q)=\mathrm{inf}_{c\in\mathbb{R}}\left( EMD\big(\tilde{p}(\cdot+c),\tilde{q}(\cdot)\big)\right),$$ where $EMD$ is the earth mover’s distance and $\tilde{p}$ and $\tilde{q}$ are the distributions obtained by rescaling $p$ and $q$ to have variance 1. More precisely, $\tilde{p}$ is the distribution obtained from $p$ by the transformation $x\rightarrow \frac{x}{\sigma(p)}$, where $\sigma(p)$ is the standard deviation of $p$. Intuitively, $EMD$ (also known as the 1st Wasserstein metric [@emd1998] can be thought of as the minimal work, i.e$.$ mass times distance, needed to “transport” the mass of one distribution onto the other. For probability distributions $p$ and $q$ with support in $\mathbb{R}$ and bounded absolute first moment, the $EMD$ between $p$ and $q$ is given by $EMD(p,q)=\int_{-\infty}^\infty|F(x)-G(x)|\,\mathrm{d}x$, where $F$ and $G$ are the cumulative distribution functions of $p$ and $q$ respectively.
In principle, $EMD$ in Equation (\[emdmet\]) can be replaced by almost any other probability metric $d$ to obtain a corresponding metric $d^*$. Here we choose $EMD$ because it is well suited to comparing shapes, as shown by its many applications in the area of pattern recognition and image retrieval [@emd1998]. Moreover, we found that $EMD$ produces superior results to classical $L^1$ and Kolmogorov distances, especially for highly irregular distributions that one frequently encounters in real world networks.
For two networks $G$ and $G'$ and given network feature $t$, we define the corresponding $NetEmd_t$ measure by: $$NetEmd_t (G,G')=EMD^*(p_t(G),p_t(G')),$$ where $p_t(G)$ and $p_t(G')$ are the distributions of $t$ on $G$ and $G'$ respectively. $NetEmd_t$ can be shown to be a pseudometric between graphs for any feature $t$ (see Sec$.$ \[metric\]), that is it is non-negative, symmetric and satisfies the triangle inequality. Figure \[fig:DD\] gives examples where $t$ is taken to be the degree distribution, and $p_t(G)$ is the degree distribution of $G$.
Measures that are based on the comparison of multiple features can be expected to be more effective at identifying structural differences between networks than measures that are based on a single feature $t$, because for two networks to be considered similar they must show similarity across multiple features. Hence, for a given set $T=\{t_1,t_2,...,t_m\}$ of network features, we define the $NetEmd$ measure corresponding to $T$ simply as: $$\label{eq:def_netemd}
NetEmd_T(G,G')=\frac{1}{m}\sum_{j=1}^{m} NetEmd_{t_j} (G,G').$$
Although $NetEmd$ can in principle be based on any set $T$ of network features to which one can associate distributions, we initially consider only features that are based on distributions of small connected subgraphs, also known as graphlets. Graphlets form the basis of many state of the art network comparison methods and hence allow for a comparative assessment of the proposed methodology.
First, we consider graphlet degree distributions ($GDD$s) [@gdda] as our set of features. For a given graphlet $m$, the graphlet degree of a node is the number of graphlet-$m$ induced subgraphs that are attached to the node. One can distinguish between the different positions the node can have in $m$, which correspond to the automorphism orbits of $m$, see Figure \[fig.subgraphs\_and\_orbits\]. For graphlets up to size 5 there are 73 such orbits. We initially take the set of 73 $GDD$s corresponding to graphlets up to size 5 to be the default set of inputs, for which we denote the metric as $NetEmd_{G5}$.
Later we also explore alternative definitions of subgraph distributions based on ego networks, as well as the effect of varying the size of subgraphs considered in the input. Finally, we consider the eigenvalue spectra of the graph Laplacian and the normalized graph Laplacian as inputs.
![Graphlets on two to four nodes. The different shades in each graphlet represent different automorphism orbits, numbered from 0 to 14.[]{data-label="fig.subgraphs_and_orbits"}](motifs3_4_v2.png){width="85.00000%"}
Results
=======
In order to give a comparative assessment of $NetEmd$, we consider to other graphlet based network comparison methods, namely $GDDA$ [@gdda], $GCD$ [@2014yaveroglu] and Netdis [@2014waqar]. These represent the most effective alignment-free network comparison methodologies in the existing literature. While $GDDA$ directly compares distributions of graphlets up to size 5 in a pairwise fashion, $GCD$ is based on comparing rank correlations between graphlet degrees. Here we consider both default settings of GCD [@2014yaveroglu], namely $GCD11$, which is based on a non-redundant subset of 11 graphlets up to size 4, and $GCD73$ which uses all graphlets up to size 5. $Netdis$ differs from $GDDA$ and $GCD$ in that it is based on subgraph counts in ego-networks of nodes. Another important distinction is that $Netdis$ first centers these raw counts by comparing them to the counts that could be expected under a particular null model before computing the final statistics. In our analysis, we consider two null models: an Erdös-Rényi random graph and a duplication divergence [@DD1] graph which has a scale-free degree distribution as well as a high clustering coefficient. We denote these two variants as $Netdis_{ER}$ and $Netdis_{SF}$ respectively.
Clustering synthetic and real world networks
--------------------------------------------
We start with the classical setting of network comparison where the task is to identify groups of structurally similar networks. The main challenge in this setting is to identify structurally similar networks even though they might differ substantially in terms of size and density.
Given a set $S=\{G_1,G_2,...,G_n\}$ of networks consisting of disjoint classes $C=\{c_1,c_2,...,c_m\}$ one would like a network comparison measure $d$ to position networks from the same class closer to each other when compared to networks from other classes. Given a network $G$, this can be measured in terms of the empirical probability $P(G)$ that $d(G,G_1)<d(G,G_2)$ where $G_1$ is a randomly selected network from the same class as $G$ (excluding itself) and $G_2$ is a randomly selected network from outside the class of $G$ and $d$ is the network comparison statistic. Consequently, the performance over the whole data set is measured in terms of the quantity $\overline{P}=\frac{1}{|S|}\sum_{G\in S}P(G)$. It can be shown that $\overline{P}$ is equivalent to the average area under the receiver operator characteristic curve of a classifier that for a given network $G$ classifies the $k$ nearest neighbours of $G$ with respect to $d$ as being similar to $G$. Hence, a measure that positions networks randomly has an expected $\overline{P}$ of 0.5 whereas $\overline{P}=1$ corresponds to perfect separation between classes. Other measures are discussed in the Appendix. Conclusions reached in this paper hold regardless of which performance measure one uses.
We first test $NetEmd$ on synthetic networks corresponding to realizations of eight random graph models, namely the Erdős-Rényi random graphs [@1960er], the Barabasi Albert preferential attachment model [@barabasi1999emergence], two duplication divergence models [@DD1; @DD2], the geometric gene duplication model [@2008higham], 3D geometric random graphs [@2003penrose], the configuration model [@1995molloy], and Watts-Strogatz small world networks [@1998watts] (see Sec$.$ \[models\] in the Appendix for details).
For synthetic networks we consider three experimental settings of increasing difficulty, starting with the task of clustering networks that have same size $N$ and average degree $k$ according to generating mechanism - a task that is relevant in a model selection setting. For this we generate 16 data sets, which collectively we call $RG_1$, corresponding to combinations of $N\in\{1250,2500,5000,10000\}$ and $k\in\{10,20,40,80\}$, each containing 10 realizations per model, i.e. 80 networks. This is an easier problem than clustering networks of different sizes and densities, and in this setting we find that the $\overline{P}$ scores (see Table \[tab:clustering\]) of top performing measures tend to be within one standard deviation of each other. We find that $NetEmd_{G5}$ and $GCD73$ achieve the highest scores, followed by $GCD11$ and $Netdis_{SF}$.
Having established that $NetEmd$ is able to differentiate networks according to generating mechanism, we move on to the task of clustering networks of different sizes and densities. For this we generate two data sets: $RG_2$ in which the size $N$ and average degree $k$ are increased independently in linear steps to twice their initial value ($N\in\{2000,3000,4000\}$ and $k\in\{20,24,28,32,36,40\}$) and $RG_3$ in which the size and average degree are increased independently in multiples of 2 to 8 times their initial value ($N\in\{1250,2500,5000,10000\}$ and $k\in\{10,20,40,80\}$). In $RG_3$, the number of nodes and average degrees of the networks both vary by one order of magnitude, and therefore clustering according to model type is challenging. Both $RG_2$ and $RG_3$ contain 10 realizations per model parameter i.e. contain $3\times6\times8\times10=1440$ and $4\times4\times8\times10=1280$ networks, respectively. Finally, we consider a data set consisting of networks from 10 different classes of real world networks (RWN) as well as a data set from [@2014waqar] that consists of real world and synthetic networks from the larger collection compiled by Onnela $et$ $al.$ [@onnela].
\
We find that $NetEmd_{G5}$ outperforms all of the other three methods at clustering networks of different sizes and densities on all data sets. The difference can also be seen in the heatmaps of $NetEmd_{G5}$ and $GCD73$, the second best performing method for $RG_2$, given in Figures \[rg2netemd\] and \[rg2gcd\]. While the heatmap of $NetEmd_{G5}$ shows eight clearly identifiable blocks on the diagonal corresponding to different generative models, the heatmap of $GCD73$ shows signs of off-diagonal mixing. The difference in performance becomes even more pronounced on more challenging data sets, i.e$.$ on $RG_3$ (see Fig$.$ \[fig:RG3\] in the Appendix) and the Onnela $et$ $al.$ data set.
Time ordered networks
---------------------
A network comparison measure should ideally not only be able to identify groups of similar networks but should also be able to capture structural similarity at a finer local scale. To study the behavior of $NetEmd$ at a more local level, we consider data sets that represent a system measured at different points in time. Since such networks can be assumed to evolve gradually over time they offer an ideal setting for testing the local properties of network comparison methodologies.
We consider two data sets, named AS-caida and AS-733 [@as], that represent the topology of the internet at the level of autonomous systems and a third data set that consists of bilateral trade flows between countries for the years 1962–2014 [@feenstra2005world; @comtrade]. Both edges and nodes are added and deleted over time in all three data sets. As was noted in [@as] the time ranking in evolving networks is reflected to a certain degree in simple summary statistics. Hence, recovering the time ranking of evolving networks should be regarded as a test of consistency rather than an evaluation of performance.
In order to minimize the dependence of our results on the algorithm that is used to rank networks, we consider four different ways of ranking networks based on their pairwise distances as follows. We assume that either the first or last network in the time series is given. Rankings are then constructed in a step-wise fashion. At each step one either adds the network that is closest to the last added network (Algorithm 1), or adds the network that has smallest average distance to all the networks in the ranking constructed so far (Algorithm 2). The performance of a measure in ranking networks is then measured in terms of Kendall’s rank correlation coefficient $\tau$ between the true time ranking and the best ranking obtained by any of the 4 methods.
\
We find that $NetEmd_{G5}$ successfully recovers the time ordering for all three data sets, as can be seen in the time ordered heatmaps given in Figure \[time\] which all show clear groupings along the diagonal. The red regions in the two internet data sets correspond to outliers which can also be identified as sudden jumps in summary statistics e.g. the number of nodes. The two large clusters in the heatmap of world trade networks (Figure \[WTN\]) coincide with a change in the data gathering methodology in 1984 [@feenstra2005world]. Although $NetEmd_{G5}$ comes second to $Netdis_{SF}$ on AS-733 and to $GCD11$ on AS-caida, $NetEmd_{G5}$ has the highest overall score and is the only measure that achieves consistently high scores on all three data sets.
NetEmd based on different sets of inputs
----------------------------------------
We examine the effect of reducing the size of graphlets considered in the input of $NetEmd$, which is also relevant from a computational point of view, since enumerating graphlets up to size 5 can be challenging for very large networks. We consider variants based on the graphlet degree distributions of graphlets up to size 3 and 4, which we denote as $NetEmd_{G3}$ and $NetEmd_{G4}$. We also consider $NetEmd_{DD}$ which is based only on the degree distribution as a baseline. Results are given in Table \[tab:variants\].
We find that reducing the size of graphlets from 5 to 4 does not significantly decrease the performance of $NetEmd$ and actually produces better results on three data sets ($RG_3$, Real world and Onnela et $al.$). Even when based on only graphlets up to size 3, i.e. just edges, 2-paths and triangles, $NetEmd$ outperforms all other non-$NetEmd$ methods that we tested on at least 6 out of 8 data sets.
Given that the complexity of enumerating graphlets up to size $s$ in a network on $N$ nodes having maximum degree $k_{mak}$ is $O(Nk_{max}^{s-1})$, $NetEmd_{G4}$ offers an optimal combination of performance and computational efficiency in most cases. The even less computationally costly $NetEmd_{G3}$ scales favourably even to networks of billions of edges for which enumerating graphlets of size 4 can be computationally prohibitive. This opens the door for comparing very large networks which are outside the reach of current methods while still retaining state of the art performance. Furthermore, the $NetEmd$ measures perform well under sub-sampling of nodes [@SS] (see Appendix D) which can be leveraged to further improve computational efficiency.
We find that in some cases restricting the set of inputs actually leads to an increase in the performance of $NetEmd$. This indicates that not all graphlet distributions are equally informative in all settings [@2017_maugis]. Consequently, identifying (learning) which graphlet distributions contain the most pertinent information for a given task might lead to significant improvements in performance. Such generalizations can be incorporated into $NetEmd$ in a straightforward manner, for instance by modifying the sum in Equation (\[eq:def\_netemd\]) to incorporate weights. $NetEmd$ is ideally suited for such metric learning [@metriclearning] type generalizations since it constructs an individual distance for each graphlet distribution. Moreover, such single feature $NetEmd$ measures are in many cases highly informative even on their own. For instance $NetEmd_{DD}$, which only uses the degree distribution, outperforms the non-$NetEmd$ measures we tested individually on more than half the data sets we considered.
We also considered counts of graphlets up to size 4 in 1-step ego networks of nodes ($NetEmd_{E4}$) [@2014waqar] as an alternative way of capturing subgraph distributions, for which we denote the measure as $NetEmd_{E4}$. Although we find that $NetEmd_{E4}$ achieves consistently high scores, we find that variants based on graphlet degree distributions tend to perform better on most data sets.
Finally, we consider spectral distributions of graphs as a possible alternative to graphlet based features. The spectra of various graph operators are closely related to topological properties of graphs [@chung1997spectral; @mohar1991laplacian; @banerjee2008spectrum] and have been widely used to characterize and compare graphs [@gu2016spectral; @wilson2008study]. We used the spectra of the graph Laplacian and normalized graph Laplacian as inputs for $NetEmd$ for which we denote the measure as $NetEmd_S$. For a given graph the Laplacian is defined as $L=D-A$ where $A$ is the adjacency matrix of the graph and $D$ is the diagonal matrix whose diagonal entries are the node degrees. The normalized Laplacian $\hat{L}$ is defined as $D^{-\frac{1}{2}}LD^{-\frac{1}{2}}$. Given the eigenvalue distributions $S(L)$ and $S(\hat{L})$ of $L$ and $\hat{L}$ we define $NetEmd_S$ to be $\frac{1}{2}(NetEmd_{S(L)}+NetEmd_{S(\hat{L})})$.
We find that in general $NetEmd_S$ performs better in clustering random graphs of different sizes and densities when compared to graphlet based network comparison measures. However, on the RWN and Onnela et al. data sets graphlet based $NetEmd$ measures tend to perform better than the spectral variant which can be attributed to the prevalence of network motifs in real world networks, giving graphlet based measures an advantage. The spectral variant is also outperformed on the time ordering of data sets which in turn might be a result of the sensitivity of graph spectra to small changes in the underlying graph [@wilson2008study].
Functional classification of networks
-------------------------------------
One of the primary motivations in studying the structure of networks is to identify topological features that can be related to the function of a network. In the context of network comparison this translates into the problem of finding metrics that can identify functionally similar networks based on their topological structure.
In order to test whether $NetEmd$ can be used to identify functionally similar networks, we use several benchmarks from the machine learning literature where graph similarity measures, called graph kernels, have been intensively studied over the past decade. In the context of machine learning the goal is to construct classifiers that can accurately predict the class membership of unknown graphs.
We test $NetEmd$ on benchmark data sets representing social networks [@yanardag2015deep] consisting of Reddit posts, scientific collaborations and ego networks in the Internet Movie Database (IMDB). The Reddit data sets Reddit-Binary, Reddit-Multi-5k and Reddit-Multi-12k consist of networks representing Reddit treads where nodes correspond to users and two users are connected whenever one responded to the other’s comments. While for the Reddit-Binary data sets the task is to classify networks into discussion based and question/answer based communities, in the data sets Reddit-Multi-5k and Reddit-Multi-12k the task is to classify networks according to their subreddit categories. COLLAB is a data set consisting of ego-networks of scientists from the fields High Energy Physics, Condensed Matter Physics and Astro Physics and the task is to determine which of these fields a given researcher belongs to. Similarly, the data sets IMDB-Binary and IMDB-Multi represent collaborations between film actors derived from the IMDB and the task is to classify ego-networks into different genres i.e. action and romance in the case of IMDB-Binary and comedy, action and Sci-Fi genres in the case of IMDB-Multi.
We use C - support vector machine (C-SVM) [@csvm] classifiers with a Gaussian kernel $K(G,G')=\exp (-\alpha NetEmd(G,G')^2)$, where $\alpha$ is a free parameter to be learned during training. Performance evaluation is carried out by 10 fold cross validation, where at each step of the validation 9 folds are used for training and 1 fold for evaluation. Free parameters of classifiers are learned via 10 fold cross validation on the training data only. Finally, every experiment is repeated 10 fold and average prediction accuracy and standard deviation are reported.
Table \[usvm\] gives classification accuracies obtained using $NetEmd$ measures based on graphlets up to size five ($NetEmd_{G5}$) and spectra of Laplacian operators ($NetEmd_{S}$) on the data sets representing social networks. We compare $NetEmd$ based kernels to graphlet kernels [@GK] and deep graphlet kernels [@yanardag2015deep] as well as two non-SVM classifiers namely the random forest classifier introduced in [@featurebased] and the convolutional neural network based classifier introduced in [@PCSN].
On the Reddit data sets and the COLLAB data set, $NetEmd_{G5}$ significantly outperforms other state-of-the-art graph classifiers. On the other hand, we find that $NetEmd_{G5}$ performs poorly on the IMDB data sets. This can be traced back to the large number of complete graphs present in the IMDB data sets: 139 out of the 1000 graphs in IMDB-Binary and 789 out of 1500 graphs in IMDB-Multi are complete graphs which correspond to ego-networks of actors having acted only in a single film. By definition, $NetEmd_{G5}$ cannot distinguish between complete graphs of different sizes since all graphlet degree distributions are concentrated on a single value in complete graphs. The spectral variant $NetEmd_S$ is not affected by this and we find that $NetEmd_S$ is either on par with or outperforms the other non-$NetEmd$ graph classifiers on all six data sets.
We also tested $NetEmd$ on benchmark data sets representing chemical compounds and protein structures. Unlike the social network data sets, in these data sets nodes and edges are labeled to reflect domain specific knowledge such as atomic number, amino acid type and bond type. Although $NetEmd$, in contrast to the other graph kernels, does not rely on domain specific knowledge in the form of node or edge labels, we found that $NetEmd$ outperforms many of the considered graph kernels coming only second to the Weisfeiler-Lehman [@WL] type kernels in terms of overall performance (see Appendix \[MLC\]).
Discussion
==========
Starting from basic principles, we have introduced a general network comparison methodology, $NetEmd$, that is aimed at capturing common generating processes in networks. We tested $NetEmd$ in a large variety of experimental settings and found that $NetEmd$ successfully identifies similar networks at multiple scales even when networks differ significantly in terms of size and density, generally outperforming other graphlet based network comparison measures. Even when based only on graphlets up to size 3 (i.e. edges, 2-paths and triangles), $NetEmd$ has performance comparable to the state of the art, making $NetEmd$ feasible even for networks containing billions of edges and nodes.
By exploring machine learning applications we showed that $NetEmd$ captures topological similarity in a way that relates to the function of networks and outperforms state-of-the art graph classifiers on several graph classification benchmarks.
Although we only considered variants of $NetEmd$ that are based on distributions of graphlets and spectra of Laplacian operators in this paper, $NetEmd$ can also be applied to other graph features in a straightforward manner. For instance, distributions of paths and centrality measures might capture larger scale properties of networks and their inclusion into $NetEmd$ might lead to a more refined measure.
Data availability {#data-availability .unnumbered}
=================
The source code for $NetEmd$ is freely available at:www.opig.ox.ac.uk/resources
Acknowledgements {#acknowledgements .unnumbered}
================
This work was in part supported by EPSRC grant EP/K032402/1 (A.W, G.R, C.D and R.G) and EPSRC grants EP/G037280/1 and EP/L016044/1 (C.D). L.O acknowledges the support of Colciencias through grant 568. R.G. acknowledges support from the COST Action CA15109 and is currently supported by a Dame Kathleen Ollerenshaw Research Fellowship. C.D. and G.R. acknowledge the support of the Alan Turing Institute (grant EP/NS10129/1).
We thank Xiaochuan Xu and Martin O’Reilly for useful discussions.
Implementation
==============
Graphlet distributions.
-----------------------
In the main paper, both the graphlet degree distribution and graphlet counts in 1-step ego networks were used as inputs for $NetEmd$.
#### Graphlet degree distributions
The graphlet degree [@gdda] of a node specifies the number of graphlets (small induced subgraphs) of a certain type the node appears in, while distinguishing between different positions the node can have in a graphlet. Different positions within a graphlet correspond to the orbits of the automorphism group of the graphlet. Among graphs on two to four nodes, there are 9 possible graphs and 15 possible orbits. Among graphs on two to five nodes there are 30 possible graphs and 73 possible orbits.
#### Graphlet distributions based on ego-networks.
Another way of obtaining graphlet distributions is to consider graphlet counts in ego-networks [@2014waqar]. The $k$-step ego-network of a node $i$ is defined as the subgraph induced on all the nodes that can be reached from $i$ (including $i$) in less than $k$ steps. For a given $k$, the distribution of a graphlet $m$ in a network $G$ is then simply obtained by counting the occurrence of $m$ as an induced subgraph in the $k$-step ego-networks of each individual node.
Step-wise implementation
------------------------
In this paper, for integer valued network features such as graphlet based distributions, we base our implementation on the probability distribution that corresponds to the histogram of feature $t$ with bin width 1 as $p_t(G)$. $NetEmd$ can also be defined on the basis of discrete empirical distributions i.e. distributions consisting of point masses (See Section \[pr\]).
Here we summarise the calculation of the $NetEmd_T(G,G')$ distance between networks $G$ and $G'$ (with $N$ and $N'$ nodes respectively), based on the comparison of the set of local network features $T=\{t_1,\ldots,t_m\}$ of graphlet degrees corresponding to graphlets up to size $k$.
1. First one computes the graphlet degree sequences corresponding to graphlets up to size $k$ for networks $G$ and $G'$. This can be done efficiently using the algorithm ORCA [@hovcevar2014combinatorial]. For the graphlet degree $t_1$ compute a histogram across all $N$ nodes of $G$ having bins of width 1 of which the centers are at their respective values. This histogram is then normalized to have total mass 1. We then interpret the histogram as the (piecewise continuous) probability density function of a random variable. This probability density function is denoted by $p_{t_1}(G)$. The standard deviation of $p_{t_1}(G)$ is then computed, and is used to rescale the distribution so that it has variance 1. This distribution is denoted by $\widetilde{p_{t_1}(G)}$.
2. Repeat the above step for network $G'$, and denote the resulting distribution by $\widetilde{p_{t_1}(G')}$. Now compute $$NetEmd_{t_1}^*(G,G')=\mathrm{inf}_{c\in\mathbb{R}}\big(EMD\big(\widetilde{p_{t_1}(G)}(\cdot+c),\widetilde{p_{t_1}(G')}(\cdot)\big)\big).$$ In practice, this minimisation over $c$ is computed using a suitable optimization algorithm. In our implementation we use the Brent-Dekker algorithm [@brent1971algorithm] with an error tolerance of 0.00001 and with the number of iterations upper bounded by 150.
3. Repeat the above two steps for the network features $t_2,\ldots,t_m$ and compute $$NetEmd_T(G,G')=\frac{1}{m}\sum_{j=1}^m NetEmd_{t_j}^*(G,G').$$
Example: $EMD^*$ for Gaussian distributions
-------------------------------------------
Suppose that $p$ and $q$ are $N(\mu_1,\sigma_1^2)$ and $N(\mu_2,\sigma_2^2)$ distributions, respectively. Then
$$\begin{aligned}
EMD^*(p,q) &=\inf_{c\in\mathbb{R}}\Big(EMD\big(\tilde{p}(\cdot+c),\tilde{q}(\cdot)\big)\Big)\\
&=EMD \Big(\tilde{p}(\cdot-\frac{\mu_1}{\sigma_1}+\frac{\mu_2}{\sigma_2}),\tilde{q}(\cdot)\Big)\\
&=EMD\big(\tilde{q}(\cdot),\tilde{q}(\cdot)\big)=0.\end{aligned}$$
Here we used that if $X\sim N(\mu_1,\sigma_1^2)$ and $Y\sim N(\mu_2,\sigma_2^2)$, then $\frac{X}{\sigma_1}+c\sim N(\frac{\mu_1}{\sigma_1}-c,1)$ and $\frac{Y}{\sigma_2}\sim N(\frac{\mu_2}{\sigma_2},1)$, and these two distributions are equal if $c=\frac{\mu_1}{\sigma_1}-\frac{\mu_2}{\sigma_2}$.
Spectral NetEmd
---------------
When using spectra of graph operators, which take real values instead of the integer values one has in the case of graphlet distributions, we use the empirical distribution consisting of point masses for computing $NetEmd$. For more details see Section \[pr\] of this appendix.
Computational complexity
------------------------
The computational complexity of graphlet based comparison methods is dominated by the complexity of enumerating graphlets. For a network of size $N$ and maximum degree $d$, enumerating all connected graphlets up to size $m$ has complexity $O(Nd^{m-1})$, while counting all graphlets up to size $m$ in all $k$-step ego-networks has complexity $O(Nd^{k+m-1})$. Because most real world networks are sparse, graphlet enumeration algorithms tends to scale more favourably in practice than the worst case upper bounds given above.
In the case of spectral measures, the most commonly used algorithms for computing the eigenvalue spectrum have complexity $O(N^3)$. Recent results show that the spectra of graph operators can be approximated efficiently in $O(N^2)$ time [@thune2013eigenvalues].
Given the distribution of a feature $t$, computing $EMD_t^*(G,G')$ has complexity $O(k(s+s')\mathrm{log}(s+s'))$, where $s$ and $s'$ are the number of different values $t$ takes in $G$ and $G'$ respectively and $k$ is the maximum number function calls of the optimization algorithm used to align the distributions. For node based features such as motif distributions, the worst case complexity is $O(k(N(G)+N(G'))\mathrm{log}(N(G)+N(G')))$, where $N(G)$ is the number of nodes of $G$, since the number of different values $t$ can take is bounded by the number of nodes.
Proof that $NetEmd$ is a distance measure {#metric}
=========================================
We begin by stating a definition. A *pseudometric* on a set $X$ is a non-negative real-valued function $d:X\times X\rightarrow [0,\infty)$ such that, for all $x,y,z\in X$,
1. $d(x,x)= 0$;
2. $d(x,y)=d(y,x)$ (symmetry);
3. $d(x,z)\leq d(x,y)+d(y,z)$ (triangle inequality).
If Condition 1 is replaced by the condition that $d(x,y)=0\iff x=y$ then $d$ defines a *metric*. Note that this requirement can only be satisfied by a network comparison measure that is based on a complete set of graph invariants and hence network comparison measures in general will not satisfy this requirement.
#### Proposition
Let $M$ denote the space of all real-valued probability measures supported on $\mathbb{R}$ with finite, non-zero variance. Then the $EMD^*$ distance between probability measures, $\mu_X$ and $\mu_Y$ in $M$ defined by $$EMD^*(\mu_X,\mu_Y)=\inf_{c\in\mathbb{R}}EMD(\tilde{\mu}_X(\cdot),\tilde{\mu}_Y(\cdot+c)),$$ defines a pseudometric on the space of probability measures $M$.
#### Proof
We first note that if $\mu_X\in M$ then $\tilde{\mu}_X(\cdot+c)\in M$ for any $c\in\mathbb{R}$. Let us now verify that $EMD^*$ satisfies all properties of a pseudometric. Clearly, for any $\mu_X\in M$, we have $0\leq EMD^*(\mu_X,\mu_X)\leq EMD(\tilde{\mu}_X(\cdot),\tilde{\mu}_X(\cdot))=0$, and so $EMD^*(\mu_X,\mu_X)=0$. Symmetry holds, since for, any $\mu_X$ and $\mu_Y$ in $M$, $$\begin{aligned}
EMD^*(\mu_X,\mu_Y)&=\inf_{c\in\mathbb{R}}EMD(\tilde{\mu}_X(\cdot),\tilde{\mu}_Y(\cdot+c))\\&=\inf_{c\in\mathbb{R}}EMD(\tilde{\mu}_Y(\cdot+c),\tilde{\mu}_X(\cdot))\\
&=\inf_{c\in\mathbb{R}}EMD(\tilde{\mu}_Y(\cdot),\tilde{\mu}_X(\cdot+c))\\&=EMD^*(\mu_Y,\mu_X).\end{aligned}$$ Finally, we verify that $EMD^*$ satisfies the triangle inequality. Suppose $\mu_X$, $\mu_Y$ and $\mu_Z$ are probability measures from the space $M$, then so are $\tilde{\mu}_X(\cdot+a)$, $\tilde{\mu}_Y(\cdot+b)$ for any $a,b\in\mathbb{R}$. Since EMD satisfies the triangle inequality, we have, for any $a,b\in\mathbb{R}$, $$\label{emdeqn1}EMD(\tilde{\mu}_X(\cdot+a),\tilde{\mu}_Y(\cdot+b))\leq EMD(\tilde{\mu}_X(\cdot+a),\tilde{\mu}_Z(\cdot))+EMD(\tilde{\mu}_Y(\cdot+b),\tilde{\mu}_Z(\cdot)).$$ Since the above inequality holds for all $a,b\in\mathbb{R}$, we have that $$\begin{aligned}
EMD^*&(\mu_X,\mu_Y)=\inf_{c\in\mathbb{R}}EMD(\tilde{\mu}_X(\cdot+c),\tilde{\mu}_Y(\cdot))\\
&=\inf_{a,b\in\mathbb{R}}EMD(\tilde{\mu}_X(\cdot+a),\tilde{\mu}_Y(\cdot+b)) \\
&\leq \inf_{a,b\in\mathbb{R}}\big[EMD(\tilde{\mu}_X(\cdot+a),\tilde{\mu}_Z(\cdot))+EMD(\tilde{\mu}_Y(\cdot+b),\mu_Z(\cdot))\big] \\
&=\inf_{a\in\mathbb{R}}\big[EMD(\tilde{\mu}_X(\cdot+a),\tilde{\mu}_Z(\cdot))+\inf_{b\in\mathbb{R}}EMD(\tilde{\mu}_Y(\cdot+b),\tilde{\mu}_Z(\cdot))\big] \\
&=\inf_{a\in\mathbb{R}}EMD(\tilde{\mu}_X(\cdot+a),\tilde{\mu}_Z(\cdot))+\inf_{b\in\mathbb{R}}EMD(\tilde{\mu}_Y(\cdot+b),\tilde{\mu}_Z(\cdot))\\
&=EMD^*(\mu_X,\mu_Z)+EMD^*(\mu_Y,\mu_Z),\end{aligned}$$ as required. We have thus verified that $EMD^*$ satisfies all properties of a pseudometric. $\Box$
Generalization of $EMD^*$ to point masses {#pr}
=========================================
Although in the case of graphlet based features we based our implementation of $NetEmd$ on probability distribution functions that correspond to normalized histograms havning bin width 1 $NetEmd$ can also be based on empirical distributions consisting of collections of point masses located at the observed values.
The definition of $EMD^*$ can be generalized to include distributions of zero variance, i.e. unit point masses. Mathematically, the distribution of a point mass at $x_0$ is given by the Dirac measure $\delta_x(x_0)$. Such distributions are frequently encountered in practice since some graphlets do not occur in certain networks.
First, we note that unit point masses are always mapped onto unit point masses under rescaling operations. Moreover, for a unit point mass $\delta_x(x_0)$ we have that $\mathrm{inf}_{c\in\mathbb{R}}(EMD(\tilde{p}(\cdot+c),\delta_x(x_0)))$ $=\mathrm{inf}_{c\in\mathbb{R}}\left(EMD(\tilde{p}(\cdot+c),\delta_x(kx_0))\right)$ for all $p\in M$ and $k>0$. Consequently, $EMD^*$ can be generalized to include unit point masses in a consistent fashion by always rescaling them by 1: $$\label{emdmet2}
EMD^*(p,q)=\mathrm{inf}_{c\in\mathbb{R}}\big(EMD(\hat{p}(\cdot+c),\hat{q})\big),$$ where $\hat{p}=\tilde{p}$ (as in Eq. \[emdmet\]) if $p$ has a non-zero variance, and $\hat{p}=p$ if $p$ has variance zero.
Sub-sampling
============
$NetEmd$ is well suited for the sub-sampling procedure from [@SS]. Following this procedure we base the graphlet distributions used as an input of $NetEmd$ on a sample of nodes rather than the whole network.
Figure \[SS\] shows the $\overline{P}$ scores for variants of $NetEmd$ on a set of synthetic networks and the Onnela et al. data set. We find that the performance of $NetEmd$ is stable under sub-sampling and that in general using a sample of only $10\%$ of the nodes produces results comparable to the case where all nodes are used.
Results for data sets of chemical compounds and proteins {#MLC}
========================================================
We also tested $NetEmd$ on benchmark data sets representing chemical compounds (MUTAG, NCI1 and NCI109) and protein structures (ENZYMES and D&D). MUTAG [@mutag] is a data set of 188 chemical compounds that are labelled according to their mutagenic effect on Salmonella typhimurium. NCI1 and NCI109 represent sets of chemical compounds which are labelled for their activity against non-small cell lung cancer and ovarian cancer cell lines, respectively [@NCI]. Nodes and edges in MUTAG, NCI1 and NCI109 are labeled by atomic number and bound type, respectively. ENZYMES and D&D [@borgpro] consist of networks representing protein structures at the level of tertiary structure and amino acids respectively. While networks in ENZYMES are classified into six different enzyme classes, networks in D&D are classified according to whether or not they correspond to an enzyme. Nodes in ENZYMES are labelled according to structural element type and according to amino acid types in D&D.
Classification accuracies obtained using $NetEmd$ on the data sets of chemical compounds and protein structures are given in Table \[CSVM\], along with results for other graph kernels reported in [@WL]. For a detailed description of these kernels we refer to [@WL] and the references therein. Note that, in contrast to all other kernels in Table \[CSVM\], $NetEmd$ does not use any domain specific knowledge in the form of node or edge labels. Node and edge labels are highly informative for all five classification tasks - as shown in [@halting].
On MUTAG, $NetEmd$ achieves an accuracy that is comparable to the Weisfeiler-Lehman (WL) shortest path kernel, but is outperformed by the shortest path kernel and the kernel by Ramon & Gärtner. While on NCI1, NCI109 and ENZYMES, $NetEmd$ is outperformed only by WL kernels, on D&D $NetEmd$ achieves a classification accuracy that is comparable to the best performing kernels. Notably, on D&D $NetEmd$ also outperforms the vector model by Dobson and Doig [@DD] (classification accuracy: 76.86$\pm$1.23) which is based on 52 physical and chemical features without using domain specific knowledge i.e. solely based on graph topology.
Implementation of C-SVMs
------------------------
Following the procedure in [@WL] we use 10-fold cross validation with a C-SVM [@csvm] to test classification performance. We use the python package scikit-learn [@pedregosa2011scikit] which is based is build on libsvm implementation [@chang2011libsvm]. The $C-value$ of the C-SVM and the $\alpha$ for the Gaussian kernel is tuned independently for each fold using training data from that fold only. Each experiment is repeated 10 times, and average prediction accuracies and their standard deviations are reported.
We also note that note for all values of $\alpha$ is the Gaussian NetEmd kernel is positive semidefinite (psd) [@jayasumana2015kernel]. The implication is that the C-SVM converges to a stationary point that is not always guaranteed to be global optimum. Although there exist alternative algorithms [@luss2008support] for training C-SVMs with indefinite kernels which might result in better classification accuracy, here we chose to use the standard libsvm-algorithm in order to ensure a fair comparison between kernels. For a discussion of support vector machines with indefinite kernels see [@haasdonk2005feature].
Detailed description of data sets and models
=============================================
Synthetic networks and random graph models {#models}
------------------------------------------
$RG_1$ consists of 16 sub data sets corresponding to combinations of $N\in\{1250,2500,5000,10000\}$ and $k\in\{10,20,40,80\}$ containing 10 realizations for each model i.e. contain 80 networks each.
In $RG_2$ the size $N$ and average degree $k$ are increased independently in linear steps to twice their initial value ($N\in\{2000,3000,4000\}$ and $k\in\{20,24,28,32,36,40\}$) and contains 10 realizations per model parameter combination, resulting in a data set of $3\times6\times8\times10=1440$ networks.
In $RG_3$ the size $N$ and average degree $k$ are increased independently in multiples of 2 to 8 times their initial value ($N\in\{1250,2500,5000,10000\}$ and $k\in\{10,20,40,80\}$) and again contains 10 realizations per model parameter combination, resulting in a data set of $4\times4\times8\times10=1280$ networks. The models are as follows.
### The Erdős-R[é]{}nyi model
We consider the Erdős-R[é]{}nyi (ER) model [@1960er] $G(N,m)$ where $N$ is the number of nodes and $m$ is the number of edges. The edges are chosen uniformly at random without replacement from the $\binom{N}{2}$ possible edges.
### The configuration model
Given a graphical degree sequence, the configuration model creates a random graph that is drawn uniformly at random from the space of all graphs with the given degree sequence. The degree sequence of the configuration models used in the paper is taken to be degree sequence of a duplication divergence model that has the desired average degree.
### The Barab[á]{}si Albert preferential attachment model
In the Barab[á]{}si-Albert model [@barabasi1999emergence] a network is generated starting from a small initial network to which nodes of degree $m$ are added iteratively and the probability of connecting the new node to an existing node is proportional to the degree of the existing node.
### Geometric random graphs
Geometric random graphs [@1961gilbert] are constructed under the assumption that the nodes in the network are embedded into a $D$ dimensional space, and the presence of an edge depends only on the distance between the nodes and a given threshold $r$. The model is constructed by placing $N$ nodes uniformly at random in an $D$-dimensional square $[0,1]^D$. Then edges are placed between any pair of nodes for which the distance between them is less or equal to the threshold $r$. We use $D=3$ and set $r$ to be the threshold that results in a network with the desired average degree, while the distance is the Euclidean distance.
### The geometric gene duplication model
The geometric gene duplication model is a geometric model [@2008higham] in which the nodes are distributed in 3 dimensional Euclidean space $\mathbb{R}^3$ according to the following rule. Starting from an small initial set of nodes in three dimensions, at each step a randomly chosen node is selected and a new node is placed at random within a Euclidean distance $d$ of this node. The process is repeated until the desired number of nodes is reached. Nodes within a certain distance $r$ are then connected. We fix $r$ to obtain the desired average degree.
### The duplication divergence model of Vázquez et al.
The duplication divergence model of Vázquez et al. [@DD1] is defined by the following growing rules: (1) Duplication: A node $v_i$ is randomly selected and duplicated ($v_i'$) along with all of its interactions. An edge between $v_i$ and $v_i'$ is placed with probability $p$. (2) Divergence: For each pair of duplicated edges $\{(v_i,v_k);(v_i',v_k)\}$; one of the duplicated edges is selected uniformly at random and then deleted with probability $q$. This process is followed until the desired number of nodes is reached. In our case we fix $p$ to be 0.05 and adjust $q$ through a grid search to obtain a network that on average has the desired average degree.
### The duplication divergence of Ispolatov et al.
The duplication divergence model of Ispolatov et al. [@DD2] starts with an initial network consisting of a single edge and then at each step a random node is chosen for duplication and the duplicate is connected to each of the neighbours of its parent with probability $p$. We adjust $p$ to obtain networks that have on average the desired average degree.
### The Watts-Strogatz model
The Watts-Strogatz model, [@1998watts] creates graphs that interpolate between regular graphs and ER graphs. The model starts with a ring of $n$ nodes in which each node is connected to its $k$-nearest neighbours in both directions of the ring. Each edges is rewired with probability $p$ to a node which is selected uniformly at random. While $k$ is adjusted to obtain networks having the desired average degree we take $p$ to be 0.05.
Real world data sets
--------------------
Summary statistics of the data sets are given in Table \[RW\].
### Real world networks from different classes (RWN)
We compiled a data set consisting of 10 different classes of real world networks: social networks, metabolic networks, protein interaction networks, protein structure networks, food webs, autonomous systems networks of the internet, world trade networks, airline networks, peer to peer file sharing networks and scientific collaboration networks. Although in some instances larger versions of these data sets are available, we restrict the maximum number of networks in a certain class to 20 by taking random samples of larger data sets in order to avoid scores being dominated by larger network classes.
The class of social networks consists of 10 social networks from the Pajek data set which can be found at http://vlado.fmf.uni-lj.si/pub/networks/data/default.htm (June 12th 2015) (Networks: ’bkfrat’, ’bkham’, ’bkoff’, ’bktec’, ’dolphins’, ’kaptailS1’, ’kaptailS2’, ’kaptailT1’, ’kaptailT2’, ’karate’, ’lesmis’, ’prison’) and a sample of 10 Facebook networks from [@traud2012social] (Networks:’Auburn71’, ’Bucknell39’, ’Caltech36’, ’Duke14’, ’Harvard1’, ’JMU79’, ’MU78’, ’Maine59’, ’Maryland58’, ’Rice31’, ’Rutgers89’, ’Santa74’, ’UC61’, ’UC64’, ’UCLA26’, ’UPenn7’, ’UVA16’, ’Vassar85’, ’WashU32’, ’Yale4’). The class of metabolic networks consists of 20 networks taken [@jeong2000large] (Networks: ’AB’, ’AG’, ’AP’, ’AT’, ’BS’, ’CE’, ’CT’, ’EF’, ’HI’, ’MG’, ’MJ’, ’ML’, ’NG’, ’OS’, ’PA’, ’PN’, ’RP’, ’TH’, ’TM’, ’YP’). The class of protein interaction networks consists of 6 networks from BIOGRID [@stark2006biogrid] (Arabidopsis thaliana, Caenorhabditis elegans, Drosophila melanogaster, Homo sapiens, Mus musculus and Saccharomyces cerevisiae downloaded: October 2015) and 5 networks from HINT [@das2012hint] (Arabidopsis thaliana, Caenorhabditis elegans, Drosophila melanogaster, Homo sapiens and Mus musculus (Version: June 1 2014)) and the protein interaction network of Echeria coli by Rajagopala et al. [@rajagopala2014binary]. The class of protein structure networks consists of a sample of 20 networks from the data set D&D (Networks: 20, 119, 231, 279, 335, 354, 355, 369, 386, 462, 523, 529, 597, 748, 833, 866, 990, 1043, 1113, 1157). The class of food webs consists of 20 food webs from the Pajek data set: http://vlado.fmf.uni-lj.si/pub/networks/data/default.htm (June 10th 2015) (Networks: ’ChesLower’, ’ChesMiddle’, ’ChesUpper’, ’Chesapeake’, ’CrystalC’, ’CrystalD’, ’Everglades’, ’Florida’, ’Michigan’, ’Mondego’, ’Narragan’, ’StMarks’, ’baydry’, ’baywet’, ’cypdry’, ’cypwet’, ’gramdry’, ’gramwet’, ’mangdry’, ’mangwet’). The class of internet networks consists of 10 randomly chosen networks from AS-733 [@as] (Networks:’1997/11/12’, ’1997/12/28’, ’1998/01/01’, ’1998/06/06’, ’1998/08/13’, ’1998/12/04’, ’1999/03/30’, ’1999/04/17’, ’1999 /06/18’, ’1999/08/30’) and 10 randomly chosen networks from AS-caida [@as] (Networks: ’2004/10/04’, ’2006/01/23’, ’2006/03/27’, ’2006/07/10’, ’2006/09/25’, ’2006/11/27’, ’2007/01/15’, ’2007/04/30’, ’2007/05/28’, ’2007/09/24’). Both datasets are from SNAP [@snapnets](June 1 2016). The class of world trade networks is a sample of 20 networks of the larger data set considered in [@feenstra2005world; @comtrade] (Networks: 1968, 1971, 1974, 1975, 1976, 1978, 1980, 1984, 1989, 1992, 1993, 1996, 1998, 2001, 2003, 2005, 2007, 2010, 2011, 2012). The airline networks were derived from the data available at: http://openflights.org/ (June 12 2015). For this we considered the 50 largest airlines from the database in terms of the number of destinations that the airline serves. For each airline a network is obtained by the considering all airports that are serviced by the airlines which are connected whenever there is direct flight between a pair of nodes. We then took a sample of 20 networks from this larger data set (Airline codes of the networks: ’AD’, ’AF’, ’AM’, ’BA’, ’DY’, ’FL’, ’FR’, ’JJ’, ’JL’, ’MH’, ’MU’, ’NH’, ’QF’, ’SU’, ’SV’, ’U2’, ’UA’, ’US’, ’VY’, ’ZH’). The class of peer to peer networks consist of 9 networks of the Gnutella file sharing platform measured at different dates which are available at [@snapnets]. The scientific collaboration networks consists of 5 networks representing different scientific disciplines which were obtained from [@snapnets] (June 1 2015).
### Onnela et al. data set
The Onnela et al. data set consists of all undirected and unweighted networks from the larger collection analysed in [@onnela]. A complete list of networks and class membership can be found in the supplementary information of [@2014waqar].
### Time ordered data sets
The data sets AS-caida and AS-733 each represent the internet measured at the level of autonomous systems at various points in time. Both data sets were downloaded from [@snapnets](June 1 2015).
The World Trade Networks data set is based on the data set [@feenstra2005world] for the years 1962-2000 and on UN COMTRADE [@comtrade] for the years 2001-2015. Two countries are connected in the network whenever they import or export a commodity from a each other within the given calendar year. The complete data set was downloaded from : http://atlas.media.mit.edu/en/resources/data/ on July 12 2015.
### Machine learning benchmarks
A short description of the social networks datasets was given in the main text. A more detailed description can be found in [@yanardag2015deep]. The social network data sets were downloaded from https://ls11-www.cs.tu-dortmund.de/staff/morris/graphkerneldatasets on September 2 2016.
A short short description of the chemical compound and protein structure data sets was given in Section \[MLC\]. A more detailed description of the data set can be found in [@WL]. These data sets were downloaded from: https://www.bsse.ethz.ch/mlcb/research/machine-learning/graph-kernels.html on June 12 2016.
Performance evaluation via area under precision recall curve ?
==============================================================
The area under precision recall curve (AUPRC) was used as a performance metric for network comparison measures by Yaveroglu et al. [@2014yaveroglu]. The AUPRC is based on a classifier that for a given distance threshold $\epsilon$ classifies pairs of networks to be similar whenever $d(G,G')<\epsilon$.
A pair satisfying $d(G,G')<\epsilon$ is taken to be a true positive whenever $G$ and $G'$ are from the same class. The AUPRC is then defined to be the area under the precision recall curve obtained by varying $\epsilon$ in small increments. However, AUPRC is problematic, especially in settings where one has more than two classes and when classes are separated at different scales.
Figure \[auprcf\] gives three examples of metrics for a problem that has three classes: a) shows a metric $d_1$ (AUPRC=0.847) that clearly separates the 3-classes which, however, has a lower AUPRC than the metrics given in b) (AUPRC=0.902) which confuses half of Class-1 with Class-2 and c) (PRC=0.896) which shows 2 rather than 3 classes. The colour scale in the figure represents the magnitude of a comparison between a pair of individuals according to the corresponding metric.
Some of the problems of AUPRC are the following. First, AUPRC is based on a classifier that identifies pairs of similar networks and hence is only indirectly related to the problem of separating classes. Moreover, the classifier uses a single global threshold $\epsilon$ for all networks and classes, and hence implicitly assumes that all classes are separated on the same scale. The AUPRC further lacks a clear statistical interpretation, which complicates its use especially when one has multiple classes and when precision recall curves of different measures intersect.
Despite its problems we give AUPRC values for all measures we considered in the main text in Table \[auprct\] for the sake of completeness. Note that $NetEmd$ measures achieve the highest AUPRC on all data sets.
[lccccc]{} &$RG_1$&$RG_2$&$RG_3$&RWN&Onnela et al.\
$NetEmd_{G3}$&0.917$\pm$ 0.039&0.869&0.702&**0.800**&0.756\
$NetEmd_{G4}$&0.959$\pm$ 0.030&0.930&0.759&0.774&**0.786**\
$NetEmd_{G5}$&**0.981$\pm$ 0.018**&0.957&0.766&0.722& 0.757\
************
$NetEmd_{S}$&0.967$\pm$0.015&**0.958**&**0.833**&0.702&0.672\
$NetEmd_{E4}$&0.966$\pm$0.030&0.945&0.801&0.777&0.739\
$NetEmd_{DD}$&0.756$\pm$0.044&0.708&0.516&0.655&0.612\
$Netdis_{ER}$&0.867 $\pm$0.044 &0.579&0.396&0.607&0.621\
$Netdis_{SF}$&0.852$\pm$0.028&0.657&0.437&0.522&0.592\
$GCD11$ &0.888$\pm$0.084&0.709&0.478&0.713&0.693\
$GCD73$&0.966$\pm$0.052&0.858&0.571&0.736&0.743\
$GGDA$&0.815$\pm$0.176&0.740&0.481&0.500&0.625\
********
|
---
author:
- Guillaume Lagubeau
- Silvia Tecpan
- Carla Hernandez
title: |
Active Learning reduce academic risk of students with non-formal reasoning skills.\
Evidence from an introductory physics massive course in a Chilean public university. Supplementary material
---
Supplementary Material {#supplementary-material .unnumbered}
======================
In Table \[Table:content\] we reference the learning activities in chronological order. Evaluations were taken after themes :
- Units and measure, Vectors and Forces
- Torque and structure
- Distributed Load and Hydrostatics
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Theme Learning resource Reference
-------------------- ----------------------------------------------------------------------------------------------------------- ------------------------------------------- --
Units and measures Classroom Lawson Test of Scientific Reasoning [@lawson2010teaching]
Units and measures PEER instruction: orders of magnitude Own creation
Units and measures Collaborative Solving Problems: “Purchase order" & Own creation\
Units and measures & Worksheet: classification of scalar and vector quantities & Own creation\
Vectors & Sense making tasks: Classify and identify vectors. A2-QRT01 & [@hieggelke2015tipers]\
Vectors & PEER instruction : unit vector & [@barniol2014students]\
Vectors & Tutorial: Vector components & [@barniol2015calculation]\
Force & Tutorial: force diagrams & Own creation\
Force & Tutorial: force equilibrium & [@mcdermott1997tutoriales]\
Force & PEER instruction: action and reaction pair & Own creation\
Force & Collaborative solving problems: “hanging sculpture" & Own creation, based on [@serway2005fisica]\
Torque & Balancing Act: PhET simulation, act 1.1.16 & [@lehtinen2017guidance]\
Torque & Tutorial: torque & [@van2006physics]\
Torque & Collaborative Solving Problems: firefighter & Own creation, based on [@serway2005fisica]\
Torque & Collaborative Solving Problems: flexed person & [@van2006physics]\
Torque & Collaborative Solving Problems: suspended beam & [@van2006physics]\
Structure & Video: “types of structures" and discussion & [^1]\
Structure & Tutorial: structure elements & Own creation based on [@beer2016statics]\
Structure & Video: “tipos de Apoyos" and discussion & [^2]\
Structure & Tutorial: structure equilibrium & Own creation based on [@beer2016statics]\
Distributed Load & Video: “puente colgante soporta pesada carga" and discussion & [^3]\
Distributed Load & Tutorial: Distributed Load & Own creation, based on [@beer2016statics]
Distributed Load Peer instruction: Distributed Load Own creation, based on [@beer2016statics]
Hydrostatics Sensmaking tasks: C2 Fluids, C2-RT01, C2-RT03 [@hieggelke2015tipers]
Hydrostatics Sensmaking tasks: C2 Fluids, C2-RT05, C2-RT09 [@hieggelke2015tipers]
Hydrostatics Sensmaking tasks, C2 Fluids, C2-CT11, C2-CT12 [@hieggelke2015tipers]
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\[Table:content\]
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12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [**]{} (, ) @noop [**]{} (, ) @noop [****, ()]{} @noop [****, ()]{} @noop [**]{} (, ) @noop [**]{}, Vol. (, ) @noop [****, ()]{} @noop [**]{} (, ) @noop [**]{} (, )
[^1]: https://www.youtube.com/watch?v=VnBel53lrag
[^2]: https://www.youtube.com/watch?v=\_Xf4s0terv4
[^3]: https://www.youtube.com/watch?v=hApCZ0iE6RU
|
---
abstract: 'The convergence of Levenberg-Marquard method is discussed for the inverse problem to reconstruct the storage modulus and loss modulus for the so called scalar model by single interior measurement. The scalar model is the most simplest model for data analysis used as the modeling partial differential equation in the diagnosing modality called the magnetic resonance elastography which is used to diagnose for instance lever cancer. The convergence of the method is proved by showing that the measurement map which maps the above unknown moduli to the measured data satisfies the so called the tangential cone condition. The argument of the proof is quite general and in principle can be applied to any similar inverse problem to reconstruct the unknown coefficients of the model equation given as a partial differential equation of divergence form by one single interior measurement. The performance of the method is numerically tested for the two layered picewise homogneneous scalar model in a rectangular domain.'
author:
- |
Yu Jiang$^{1}$, Gen Nakamura$^{2}$\
$^1$School of Mathematics, Shanghai University of Finance and Economics,\
Shanghai 200433, P.R. China\
E-mail: [email protected]\
$^2$Department of Mathematics, Hokkaido University,\
Sapporo 060-0810, Japan\
E-mail: [email protected]
title: 'Convergence of Lebenberg-Marquard method for the Inverse Problem with an Interior Measurement'
---
[**Key words**]{} Levenberg-Marquard method, convergence, tangential cone condition, interior measurement, magnetic resonance elastography, storage modulus, loss modulus
[**Key words**]{} 35R30, 65N21
Introduction {#Introduction}
============
Let $\Omega\subset \mathbb{R}^n$ ($n=2\,\mbox{or}\,3)$ be a bounded domain with Lipschitz smooth boundary $\partial \Omega$ and $$\label{eq::gamma}
\gamma(x):=G'(x)+iG''(x)\,\text{with }i=\sqrt{-1},$$ where $G'(x)$ and $G''(x)$ are real valued bounded measurable functions on $\Omega$ which satisfy the positivity conditions: $$\label{eq::positivity}
0<\lambda_1\leq G',\,G''\leq \lambda_2\,\,(\mbox{a.e.}\,x\in\Omega)$$ with some positive constants $\lambda_1$ and $\lambda_2$. Further, let $\rho(x)\in L^\infty(\Omega)$ satisfy $0<\delta_1\leq \rho(x)\leq \delta_2$ $(\mbox{a.e.}\,x\in \Omega$) with some positive constants $\delta_1$ and $\delta_2$.
It is well known that for any given Dirichlet data $g(x)\in H^{1/2}(\partial \Omega)$, there exists a unique $u(x)\in H^1(\Omega)$ to the boundary value problem $$\label{eq::fp}
\left \{
\begin{array}{ll}
\nabla\cdot\big [\gamma(x)\nabla u(x)\big ]+\rho\omega^2u(x)=0 &\mbox{in } \Omega\\
u(x)=g(x) &\mbox{on } \partial \Omega,
\end{array}
\right.$$ where $\omega>0$ is a given constant. We note that this follows from the positivity of $-\nabla\cdot(G''\nabla\cdot)$ with Dirichlet boundary condition. It could be the positivity of $-\nabla\cdot(G'\nabla\cdot)$ with Dirichlet condition if there is no lower order term in the equation of . We will refer this as [*positivity*]{}. For further argument on this see Chapter 3 of [@Mizohata] or [@jiangsiam] for even more details. Also concerning the Lipschitz smoothness of $\partial\Omega$ for our boundary value problem, see [@Grisvard].
This boundary value problem is the simplest model called the [*scalar model*]{} for a recent diagnosing modality called MRE (Magnetic Resonance Elastography, see for example [@muth; @Mad] ) in which $u(x)$ describes a component of the displacement vector of a shear wave with attenuation in a human tissue. The equation of is sometimes called scalar model for MRE. $G'$ and $G''$ are called the [*storage*]{} and [*loss*]{} moduli of the tissue. Further $\rho$ describes the density of the tissue which can be taken equal to that of water i.e. $\rho=1000 \,\mbox{\rm kg}/\mbox{m}^3$ and $\omega,\,g$ are the frequency and a component of the displacement vector input to the human body. In the rest of this paper we assume that $\rho$ is equal to the above constant for simplicity.
The hardware of MRE consists of a MRI and vibration system. A time harmonic vibration excited by this vibration system is syncronized to a pulse sequence of MRI so that MRE can measure the displacement vector of a shear wave inside a human tissue. The above $u$ and $g$ in are the component of the displacement vector of time harmonic vibration in $\Omega$ and at $\partial\Omega$, respectively. Especially the diplacement $g$ at $\partial\Omega$ is given by a probe attached at some part of $\partial\Omega$ connected with a bar made of glass fiber-reinforced plastic (GFRP) which propagates the vibration excited by the vibration system of MRI (see [@fujisaki]). Away from the place the probe is attached we should have to give Neumann boundary condition. But to simplify the description, we just consider Dirichlet boundary condition given on the whole $\partial\Omega$. This so called elastogram of MRE is to recover $G^\prime$ from the [*MRE measured data*]{} $u(x)\,(x\in\Omega)$. This is an inverse problem with single interior measurement. A similar inverse problem can be seen in mathematically ideal form of inverse problem for ground water hydrology ([@hanke]).
The importance of MRE is that it can realize doctors’ palpation inside a human body which had been dreamed by doctors for a long time. Although the hardware of MRE is developing very quickly, the elastogram has not yet developed enough and there are so many challenging questions for elastogram. For further details of MRE and its elastogram, we can refer to, for example, [@jiangsiam; @Amm] for mathematical modeling, [@NJ; @Bal; @AmmS; @Hig] for theorical inversion analysis and [@AmmN; @Seo; @houten04; @JiangP; @mcn2] for numerical reconstruction schemes.
The precise formulation of this inverse problem is as follows.
[**Inverse Problem**]{}:
Recover $\gamma$ (i.e. the storage modulus $G' $ and loss modulus $G''$) from the [*MRE measured data*]{} $u(x)$ excited by given boundary input $g$ when the frequency $\omega$ is known.
Here it should be remarked that our MRE measured data is a single interior measurment. Likewise for any inverse problem, the basic questions for this inverse problem are the uniqueness, stability and reconstruction of identifying $G'$ from the MRE measured data. There isn’t any complete uniqueness and stability. What have been known for them so far is as follows for the case $\gamma\in C^1(\overline\Omega)$ with given non-identically zero input $g\in H^{3/2}(\partial\Omega)$ and when $\gamma\big|_{\partial\Omega}$ is known. That is suppose there are finitely disjoint closed analytic manifolds of codimension one which are curves for $n=2$ and surfaces for $n=3$ compactly embedded inside $\Omega$ and $\gamma\in C^1(\overline\Omega)$ is analytic inside and outside these manifolds. Further, $\gamma$ can be extended analytically up to these manifolds from inside and outside of them. If a perturbed $\widetilde\gamma$ of $\gamma$ satisfies the admissibility condition given by $$\label{admissibility condition}
|\mbox{Im}(\gamma-\widetilde\gamma)(x)|\le\tan(\kappa)|\mbox{Re}(\gamma-\widetilde\gamma)(x)|\,\,(x\in\overline\Omega),$$ where $\kappa<(\pi-\sigma)/2$ with a small $\sigma>0$, a local Hölder conditional stability estimate has been recently proved in [@HMN]. The Hölder exponent and constant in the estimate only depend on $\gamma$ and $\Vert\widetilde\gamma\Vert_{C^1(\overline\Omega)}$. As a corollary of this result, if $G''$ is known, then the admissibility condition is satisfied and hence the global uniqueness for identifying $G'$ follows.
Despite the lack of complete uniqueness which exactly fit to our case, we are particulary interesting in a mathematically rigorous reconstruction of $\gamma$ even in the case $\gamma\not\in C^1(\overline\Omega)$ from the practical point of view. As a recostruction scheme to identify $\gamma$ from single interior measurement i.e. just measure $u\big|_\Omega$, we will give a Newton type regularization scheme called Levenberg-Marquardt iterate and prove that the so-called tangential cone condition ([@hanke; @KNO]) holds, which is the key to show the convergence of this scheme. We would like to emphasize here that the inverse problem with a single interior measurement for any elliptic equation whose boundary value problem has the above mentioned [*positivity*]{} satisfies the tangential cone condition. We will also provide several numerical tests of our scheme for different cases.
The rest of this paper is organized as follows. In Section 2 we review the Levenbert-Marquardt method for the operator equation $F(x)=y$ in $x$ for a given $y$ from [@hanke; @KNO]. After this in Section 3, we put our inverse problem into such an operator equation and compute the Fréchet derivative of $F$ in Subsection 3.1. Then in Subsection 3.2, we show that $F$ satisfies the tangential cone condition. Section 4 is devoted to the numerical study of our reconstruction scheme for our inverse problem based on the convergence of Levenberg-Marquadt method. In final section, some discussions and a conclusion are given.
The Levenberg - Marquardt Method {#sec::LMM}
================================
Any nonlinear inverse problem can be treated as a nonlinear operator equation $$\label{eq::nonlinear op}
F(x)=y$$ with respect to $x$, where $F:D(F)\subset \mathcal{X}\to \mathcal{Y}$ is a differentiable operator between Hilbert space $\mathcal{X}$ and $\mathcal{Y}$. In practice, we can only measure a noisy data $y^{\delta}$ which satisfies $$\label{eq::noisy level}
\|y^{\delta}-y\|\leq \delta$$ with a noise level $\delta$. By knowing $y^{\delta}$, we always need to get a good approximation of the true solution $x^{\dag}$ which satisfies $F(x^{\dag})=y$.
In practice, it is necessary to find some fast method to solve it. Newton type iterative methods are good for this purpose. Having $x_k$ after $k$ iterations, this iteration updates $x_k$ to $x_{k+1}=x_k+h$, by solving $$\label{eq::linearized}
F'(x_k)h=y^{\delta}-F(x_k)$$ with respect to $h$, where $F'(\gamma_k)$ is the Fréchet derivative of $F$ at $x_k$. Here, we have to concern that these linearized problems are ill-posed. If the Tikhonov regularization is applied to this ill-posed linearized problem by adding the regularization term $\alpha_k\|h\|^2$, we have the [*Levenberg - Marquardt method*]{} (LM method) with the following iteration procedure (cf. [@hanke; @KNO]) $$\label{eq::LMmethod}
x_{k+1}=x_k+\big (F'(x_k)^*F'(x_k)+\alpha_k I\big )^{-1}F'(x_k)^*\big(y^{\delta}-F(x_k)\big)\quad (k\in\{0\}\cup \mathbb{N}),$$ where $x_k$ is uniquely determined by the Morozov descrepancey principle i.e. $x_{k+1}-x_k$ is the minimum norm solution of $$\label{eq::Morozov}
\Vert y^\delta-F(x_k)-F'(x_k)(x_{k+1}-x_k)\Vert=q\Vert y^\delta-F(x_k)\Vert$$ with any fixed $0<q<1$.
Assume that a true solution $x^{\dag}$ exists in an open ball $B_R(x_0)\subset D(F)$ with radius $R>0$ centered at $x_0$ for an [*initial guess $x_0$*]{}. Further, we assume that $F'$ is uniformly bounded in $B_R(x_0)$ and satisfies the [*tangential cone condition*]{}: $$\label{eq::tcc}
\|F(x)-F(\widetilde{x})-F'(x)(x-\widetilde{x})\|\leq c\|x-\widetilde{x}\|\|F(x)-F(\widetilde{x})\|,$$ for any $x$ and $\widetilde{x}\in B_R(x_0)\subset D(F)$. Then, we have the following theorem.
\[thm::convergence1\] The Levenberg - Marquardt method with exact data $y^{\delta}=y$, $\|x_0-x^{\dag}\|<q/c$ converges to a solution of $F(x)=y$ as $k \to \infty$.
This theorem means that a true solution $x^{\dag}\in B_R(x_0)$ of the equation with exact data $y$ can be recovered by the Levenberg - Marquardt method. For the noisy data $y^{\delta}$, we have to set up some stopping rule to terminate the iteration appropriately, i.e. as soon as the step index $k=k_*$ satisfies the following discrepancy principle $$\label{eq::stopping rule}
\|y^{\delta}-F(x_{k_*})\|\leq \tau\delta<\|y^{\delta}-F(x_k)\|,\quad 0\leq k<k_*,$$ with a constant $\tau>1/q$, stop the iteration. The following theorem gives a convergence of the Levenberg - Marquardt method for noisy data.
\[thm::convergence2\] Let $k_*=k_*(\delta,y^{\delta})$ be chosen according to the stopping rule with $\tau>1/q$. Then starting from the initial guess $x_0$ which satisfies $\|x_0-x^{\dag}\|\leq (q\tau-1)/(c(1+\tau))$, the discrepancy principle terminates the Levenberg - Marquardt method with $\alpha_k$ determined from after finitely many iterations $k_*$ and we have $$k_*(\delta,y^{\delta})=O(1+|\ln \delta|).$$ Moreover, the sequence $x_k$ $(k=0,1,\cdots)$ of the Levenberg - Marquardt method converges to a solution of the equation $F(x) = y$ as $\delta\to 0$.
Application to Inverse Problem of MRE
=====================================
Let $u^{\delta}:=u^{obs}\in H^1(\Omega)$ be the MRE measured datum which may have a noise with a noise level $\delta$. Also, let $\mathcal{P}$ be the set defined by $$\begin{array}{rl}
\mathcal{P}:=&\{\gamma=G'+iG''|(G',\,G'')\in [L^\infty(\Omega)]^2\}\\
&\cap\{\gamma=G'+iG''|0<\widetilde{G'}\leq G'\leq \widehat{G'},\,0<\widetilde{G''}\leq G''\leq \widehat{G''}\}
\end{array}$$ with some positive constants $\widetilde{G'},\,\,\widehat{G'}$. We consider the operator equation $$\label{eq::operator equation}
F(\gamma)=u,$$ where $F$ is defined by $$\label{eq::operator}
F:\, D(F):=\mathcal{P} \subset L^{\infty}(\Omega) \subset \mathcal{X}:=L^2(\Omega) \to \mathcal{Y}:=H^1(\Omega),$$ and $u$ is the solution to with $\gamma=G'+i G''\in \mathcal{P}$.
Under this setting we will use the Levenberg-Marquardt method given in Section \[sec::LMM\] which considered the following iteration procedure: $$\label{eq::newton}
\gamma_{k+1}=\gamma_k+\big(F'(\gamma_k) ^*F'(\gamma_k) +\alpha_k I \big)^{-1}F'(\gamma_k)^*
\big(u^{\delta}-F(\gamma_k)\big)\quad (k\in \mathbb{N}\cup \{0\}).$$ From what we gave in Section \[sec::LMM\], we can have the convegence of this iteration scheme, if we show the Fréchet derivative $F'(\gamma)$ at $\gamma=\gamma_k\in \mathcal{P}$ exists and is uniformly bounded in $B_R(\gamma_0)$, and the tangential cone condition holds in $B_{2R}(\gamma_0)\subset \mathcal{P}$ for some $R>0$ and an initial guess $\gamma_0$.
Fréchet Derivative
------------------
In this subsection we will compute the Fréched derivative $F'(\gamma)$ of $F$ at $\gamma$. Let $$\gamma^{\delta}:=\gamma+\delta \gamma\,\mbox{with}\,\,\delta \gamma\in L^\infty(\Omega),
\Vert \delta \gamma\Vert_{L^\infty(\Omega)}<<1$$ be the perturbation of $\gamma$ and $F(\gamma^{\delta}):=u^{\delta}$ be the corresponding output, where $u^{\delta}$ is the solution to $$\label{eq::pfp}
\left \{
\begin{array}{ll}
\nabla\cdot\big [\gamma^{\delta}\nabla u^{\delta}(x)\big ]+\rho\omega^2u^{\delta}(x)=0 &\mbox{in } \Omega\\
u^{\delta}(x)=g(x) &\mbox{on } \partial \Omega
\end{array}
\right.$$ with $\gamma^{\delta}=\gamma+\delta \gamma$. Note that $u^\delta$ here is not the noisy data of $u$ corresponding to $\gamma$, it is the exact solution corresponding to $\gamma^\delta=\gamma+\delta\gamma$.
Then the Fréchet derivative $F'(\gamma): \mathcal{P}\subset L^\infty(\Omega)\rightarrow H^1(\Omega)$ is given as follows.
\[lem::frechet\] $u'=F'(\gamma)\delta \gamma\in H^1(\Omega)$ is a solution to $$\label{eq::pbvp2}
\left \{
\begin{array}{ll}
\nabla\cdot\big [\gamma\nabla u'\big ]+\rho\omega^2 u'=-\nabla\cdot\big [\delta \gamma\nabla u\big ]&\mbox{\rm in } \Omega\\
u'=0 &\mbox{\rm on } \partial \Omega.
\end{array}
\right.$$
By comparing and , it is easy to find that $\delta u:=u^{\delta}-u\in H^1(\Omega)$ is the solution to following boundary value problem: $$\label{eq::pbvp}
\left \{
\begin{array}{ll}
\nabla\cdot\big [(\gamma+\delta \gamma)\nabla \delta u\big ]+\rho\omega^2\delta u=-\nabla\cdot\big [\delta \gamma \nabla u\big ]&\mbox{in } \Omega\\
\delta u=0 &\mbox{on } \partial \Omega
\end{array}
\right.$$ with $\delta\gamma=\gamma^{\delta}-\gamma$. By the standard estimate of solutions of the boundary value problem for elliptic equation satisfying the [*positivity*]{} which is the positivity of $G''$ if it is not zero and that of $G'$ if $G''=0$ and $\omega$ is small relative to $G'$, $\delta u $ satisfies the estimate $$\|\delta u\|_{H^1(\Omega)}\lesssim \|\delta \gamma\nabla u\|_{L^2(\Omega)}\lesssim \|g\|_{H^{1/2}(\partial \Omega)}\|\delta \gamma\|_{L^\infty(\Omega)}.$$ Hereafter in this paper, the notation “$\lesssim$” denotes the inequality “$ \le$” modulo a multiplication by a positive constant which depends only on $\widetilde{G'}$, $\widehat{G'}$, $\widetilde{G''}$, $\widehat{G''}$, $\lambda_1$, $\lambda_2$, $g$, $\rho$, $\omega$ and $\Omega$. From and , $v:=u^{\delta}-u-u'=\delta u-u'\in H^1(\Omega)$ is a solution to $$\label{eq::pbvp3}
\left \{
\begin{array}{ll}
\nabla\cdot\big [\gamma\nabla v\big ]+\rho\omega^2 v=-\nabla\cdot\big [\delta \gamma\nabla \delta u\big ]&\mbox{in } \Omega\\
v=0 &\mbox{on } \partial \Omega.
\end{array}
\right.$$ Again by the standard estimate of solutions of boundary value problem for elliptic equations with [*positivity*]{}, $v$ satisfies the estimate $$\label{eq::esdeltau}
\|v\|_{H^1(\Omega)}\lesssim \|\delta \gamma\nabla \delta u\|_{L^2(\Omega)}\lesssim \|\delta \gamma\|_{L^\infty(\Omega)}\|\delta u\|_{H^1(\Omega)}\lesssim \|g\|_{H^{1/2}(\partial \Omega)}\|\delta \gamma\|^2_{L^\infty(\Omega)}.$$ Thus we have $$\|u^{\delta}-u-u'\|_{H^1(\Omega)}=O(\|\delta \gamma\|^2_{L^\infty(\Omega)}),$$ which implies $F'(\gamma)\delta \gamma=u'$.
Moreover, by the regularity estimate, we have $$\|F'(\gamma)\delta \gamma\|_{H^1(\Omega)}\lesssim\|\delta\gamma\|_{L^\infty(\Omega)}\|g\|_{H^{1/2}(\partial \Omega)},$$ and this implies $F'$ is uniformly bounded near $\gamma$.
Tangential Cone Condition
-------------------------
We can have the tangential cone condition as follows:
\[lem::cone\] There is an open ball $B_R(\gamma)\subset \mathcal {P}$ about $\gamma$ of radius $R>0$ such that for all $\widetilde{\gamma},\,\widehat{\gamma}\in B_R(\gamma)$, $$\label{eq::tccm}
\|F(\widetilde{\gamma})-F(\widehat{\gamma})-F'(\widehat{\gamma})(\widetilde{\gamma}-\widehat{\gamma})\|_{H^1(\Omega)}\leq c \|\widetilde{\gamma}-\widehat{\gamma}\|_{L^\infty(\Omega)}\|F(\widetilde{\gamma})-F(\widehat{\gamma})\|_{H^1(\Omega)}.$$ Here we note that $c$ is proportional to the reciprocal of the lower bound of the positivity mentioned in the proof of Lemma \[lem::frechet\].
First of all, by and , we have: $$\label{eq::pbvp4}
\left \{
\begin{array}{ll}
\nabla\cdot\big [(\gamma+\delta \gamma)\nabla \delta u\big ]+\rho\omega^2\delta u=\nabla\cdot\big [\gamma\nabla u'\big ]+\rho\omega^2u'&\mbox{in } \Omega\\
\delta u=0 &\mbox{on } \partial \Omega.
\end{array}
\right.$$ This implies the estimate $$\|u^{\delta}-u\|_{H^1(\Omega)}=\|\delta u\|_{H^1(\Omega)}\lesssim \|\gamma\|_{L^\infty(\Omega)}\|u'\|_{H^1(\Omega)}.$$ By inserting this into , we have the estimate $$\label{eq::udelta}
\|u^{\delta}-u-u'\|_{H^1(\Omega)}\lesssim \|\gamma\|_{L^\infty(\Omega)} \|\delta \gamma\|_{L^\infty(\Omega)}\|u'\|_{H^1(\Omega)}.$$
Similar to , we find that $$\begin{array}{rl}
&\|F(\widetilde{\gamma})-F(\widehat{\gamma})-F'(\widehat{\gamma})\delta \gamma\|_{H^1(\Omega)}\\
\leq& C \|\widehat{\gamma}\|_{L^\infty(\Omega)} \|\delta \gamma\|_{L^\infty(\Omega)}\|F'(\widehat{\gamma})\delta \gamma\|_{H^1(\Omega)}\\
\leq &C( \|\gamma\|_{L^\infty(\Omega)}+r_0)\|\delta \gamma\|_{L^\infty(\Omega)}\|F'(\widehat{\gamma})\delta \gamma\|_{H^1(\Omega)},
\end{array}$$ which holds for all $\widehat{\gamma} \in B_{r_0}(\gamma)$ where $r_0>0$ is such that $B_{r_0}(\gamma)\subset \mathcal{P}$ and $\widetilde{\gamma}=\widehat{\gamma}+\delta \gamma \in \mathcal{P}$. Here $C$ is a positive constant independent of $\widetilde{\gamma}$, $\widehat{\gamma}$ and $\gamma$. If $\|\delta \gamma\|_{L^\infty(\Omega)}<[2C( \|\gamma\|_{L^\infty(\Omega)}+r_0)]^{-1}$ then we have $$\|F'(\widehat{\gamma})\delta \gamma\|_{H^1(\Omega)}\leq 2\|F(\widetilde{\gamma})-F(\widehat{\gamma})\|_{H^1(\Omega)}$$ and it follows that the estimate $$\begin{array}{rl}
&\|F(\widetilde{\gamma})-F(\widehat{\gamma})-F'(\widehat{\gamma})(\widetilde{\gamma}-\widehat{\gamma})\|_{H^1(\Omega)}\\
\leq &2C( \|\gamma\|_{L^\infty(\Omega)}+r_0)\|\widetilde{\gamma}-\widehat{\gamma}\|_{L^\infty(\Omega)}\|F(\widetilde{\gamma})-F(\widehat{\gamma})\|_{H^1(\Omega)}.
\end{array}$$
Therefore, for any positive $R<\min \left \{r_0,\,[2C( \|\gamma\|_{L^\infty(\Omega)}+r_0)]^{-1}\right \}$, the tangential cone condition always holds true for $c=2C( \|\gamma\|_{L^\infty(\Omega)}+r_0)$.
Before closing this section, we formulate our main result which we can have by combining Theorem \[thm::convergence1\] , Theorem \[thm::convergence2\] , Lemma \[lem::frechet\] and Lemma \[lem::cone\].
\[thm::main\] The Levenberg-Marquad method can be applied to give a reconstruction scheme for our inverse problem formulated as solving the operator equation both for the cases that the data is exact and is inexact with an error.
Numerical test of Lebenberg-Marquard method for MRE
===================================================
In this section we will numerically test the performance of our Levenberg-Marquad method. The set up for this numerical test is based on our MRE experiments done in Hokkaido University [@fujisaki].
![Some setup of parameters for 2D simulations.[]{data-label="fig::fw0"}](fw.eps){width="7cm"}
Concerning the spacial resolution of our 0.3 T micro-MRI, which is 1 mm, and the typical size of our two layered agarose gel phantom, which is $120 \times 120$ grids of a 120 mm $\times$ 120 mm rectangular domain $\Omega$ (see Figure \[fig::fw0\]), we consider the following boundary value problem as a special case of : $$\label{eq::fpnum}
\left \{
\begin{array}{ll}
(G_1'+iG_1'')\Delta u_1(x_1,x_2)+\rho\omega^2u_1(x_1,x_2)=0\\
\hspace{3cm} 0<x_1<120,\, x_L < x_2<120,\\
(G_2'+iG_2'')\Delta u_2(x_1,x_2)+\rho\omega^2u_2(x_1,x_2)=0\\
\hspace{3cm} 0<x_1<120,\, 0<x_2<x_L,\\
u_1(x_1,x_L)=u_2(x_1,x_L) &0<x_1<120,\\
(G_1'+iG_1'')\partial_{x_2}u_1(x_1,x_L)=(G_2'+iG_2'')\partial_{x_2}u_2(x_1,x_L)&0<x_1<120,\\
u(x_1,120)=u_1(x_1,120)=0.02\sin\left (\dfrac{\pi x_1}{120}\right ) &0<x_1<120,\\
u(x_1,0)=u_2(x_1,0)=0 &0<x_1<120,\\
u(0,x_2)=u_1(0,x_2)=0,\,u(120,x_2)=u_1(120,x_2)=0 &x_L \leq x_2<120,\\
u(0,x_2)=u_2(0,x_2)=0,\,u(120,x_2)=u_2(120,x_2)=0 &0<x_2<120,
\end{array}
\right.$$ with a interface along $x_2=x_L$. Here $(G_1'\,G_1'')$ and $(G_2'\,G_2'')$ are constants.
By applying the method of separation of variables, the solution of is given as $$\label{eq::solution}
u(x_1,x_2)=v(x_2)\times 0.02\sin\left (\dfrac{\pi x_1}{120}\right )$$ with $$v(x_2)=\left \{
\begin{array}{ll}
c_1e^{i\beta_1 x_2}+c_2 e^{-i\beta_1 x_2} &x_L < x_2<120,\\
d_1e^{i\beta_2 x_2}+d_2 e^{-i\beta_2 x_2} &0<x_2<x_L,
\end{array}
\right.$$ where $(c_1,c_2,d_1,d_2)^{\rm{T}}$ is the solution of linear system: $$\left [\begin{array}{cccc}
e^{120i\beta_1 } & e^{-120i\beta_1 } & 0 & 0 \\
e^{i\beta_1x_L } & e^{-i\beta_1x_L } & -e^{i\beta_2x_L } & -e^{-i\beta_2x_L } \\
\beta_1e^{i\beta_1x_L } & -\beta_1e^{-i\beta_1x_L } & -\beta_2e^{i\beta_2x_L } & \beta_2e^{-i\beta_2x_L } \\
0 & 0 & 1 & 1
\end{array} \right]
\left [
\begin{array}{c}
c_1 \\
c_2 \\
d_1 \\
d_2
\end{array}\right]=\left [
\begin{array}{c}
1 \\
0 \\
0\\
0
\end{array} \right]$$ with $$\beta_j=\sqrt{\dfrac{\rho\omega^2}{G'_j+iG''_j}+\left(\dfrac{\pi}{120}\right)^2}, \quad j=1,2.$$ The other assumptions of parameters used in our numerical simulation are given in Table \[table::setup\],
Layer 1 ($x_L<x_2<120$) $G'$ 20 kPa $G''$ 0.4 Pa$\cdot$s$\times \omega$
------------------------- -------- ----------------------------- ------- -------------------------------
Layer 2 ($0<x_2<x_L$) $G'$ 10 kPa $G''$ 0.3 Pa$\cdot$s$\times \omega$
Others $\rho$ 1.0$\times$ 10$^3$ kg/m$^3$ $x_L$ 60 mm$\times 2 \pi$
: Physical parameters in MRE experiments.[]{data-label="table::setup"}
Consider the inverse problem stated in Section \[Introduction\] for the boundary value problem under the assumption that we know $x_L$. The numerical test of the performance of our Lebenberg-Marquard method will be given below for this inverse problem.
[**Example 1: elastic case:**]{}
In this example, we assume $G''= 0$. Recall the uniqueness and stability results of [@HMN] given in Section \[Introduction\]. It does not exactly fit to the case we have right now. However, due to the fact that we do know the interface and $\Omega$ is just a rectangle, we can easily adapt the argument given in [@HMN] to have the uniqueness for our inverse problem in this case. By using the simulated data generated by , we will test our Levenberg-Marquad method in a finite dimensional space obtained by discretizing for the above setup. We used Matlab$\circledR$ inner-embedded program for the numerical implementation of the method. The progam can adjust automatically the regularizing parameters to ensure the convergence of $\{\gamma_k\}$.
It should be noticed that the iterative sequence may converge to some local minimal point for unsuitable initial iteration guess according to Theorem \[thm::convergence1\]. Because the constant $c$ in Theorem \[thm::convergence1\] is highly dependent on $G'$ and $\omega$ when $G''= 0$, we either choose an initial guess of $G'$ which is quite close to the exact value in high frequency case, or can choose an initial guess of $G'$ which is not so close to the exact value in low frequency case.
[**Example 1.1: low frequency case:**]{}
In this case, we assume the angular frequency $\omega=20$ Hz and consider the two simulated data. The one is without noise (see Figure \[fig::eobs\](a)) and the other is with 20$\%$ relative Gaussian noise (see Figure \[fig::eobs\](b) and Figure \[fig::eu60\] ).
(a)![Simulated data: (a) without noise (mm); (b) with 20$\%$ relative noise (mm).[]{data-label="fig::eobs"}](eobs.eps "fig:"){width=".45\textwidth"} (b)![Simulated data: (a) without noise (mm); (b) with 20$\%$ relative noise (mm).[]{data-label="fig::eobs"}](eobs20.eps "fig:"){width=".45\textwidth"}
![Simulated data along $x_1=60$ mm (blue line: without noise, red line: with 20$\%$ relative noise).[]{data-label="fig::eu60"}](eu60.eps){width=".6\textwidth"}
By applying our Levenberg-Marquad method, we recovered $\gamma$ from noisy data shown in Table \[table::ereconver\] and the reconstructed wave fields are shown in Figure \[fig::eobsr\], \[fig::eur60\].
Initial guess $G'$ 30 kPa
------------------------- ------ -------------
Layer 1 ($x_L<x_2<120$) $G'$ 20.1370 kPa
Layer 2 ($0<x_2<x_L$) $G'$ 9.9888 kPa
: Recovery of $G'$.[]{data-label="table::ereconver"}
![Reconstructed simulated data $u$ (mm).[]{data-label="fig::eobsr"}](eobsr.eps){width=".45\textwidth"}
![Reconstructed data along $x_1=60$ mm (blue line: without noise, red dot dash line: reconstructed).[]{data-label="fig::eur60"}](eur60.eps){width=".6\textwidth"}
[**Example 1.2: high frequency case:**]{} Let the angular frequency $\omega=250$ Hz and consider the two simulated data. The one is without noise (see Figure \[fig::heobs\](a)) and the other is with 20$\%$ relative Gaussian noise (see Figure \[fig::heobs\](b) and Figure \[fig::heu60\] ).
(a)![Simulated data: (a) without noise (mm); (b) with 20$\%$ relative noise (mm).[]{data-label="fig::heobs"}](heobs.eps "fig:"){width=".45\textwidth"} (b)![Simulated data: (a) without noise (mm); (b) with 20$\%$ relative noise (mm).[]{data-label="fig::heobs"}](heobs20.eps "fig:"){width=".45\textwidth"}
![Simulated data along $x_1=60$ mm (blue line: without noise, red line: with 20$\%$ relative noise).[]{data-label="fig::heu60"}](heu60.eps){width=".6\textwidth"}
By applying our Levenberg-Marquad method, we recovered $\gamma$ from the noisy data which is shown in Table \[table::hereconver\]. The reconstructed wave fields using the recovered $\gamma$ are shown in Figure \[fig::eobsr\] and Figure \[fig::heur60\]. We need to emphasize here that the initial guess of $G'$ must be quite close to the exact value.
Initial guess $G'$ 21 kPa (layer 1) and 9.5 kPa (layer 2)
------------------------- ------ ----------------------------------------
Layer 1 ($x_L<x_2<120$) $G'$ 19.9987 kPa
Layer 2 ($0<x_2<x_L$) $G'$ 9.9999 kPa
: Recovery of $G'$.[]{data-label="table::hereconver"}
![Reconstructed simulated data $u$ (mm).[]{data-label="fig::heobsr"}](heobsr.eps){width=".45\textwidth"}
![Reconstructed data along $x_1=60$ mm (blue line: without noise, red dot dash line: reconstructed).[]{data-label="fig::heur60"}](heur60.eps){width=".6\textwidth"}
[**Example 2: viscoelastic case:**]{}
In this example, we assume $G''\neq 0$ as shown in Table \[table::setup\]. Due to the positivity of $G''$, we can choose an initial guess of $\gamma$ which is not so close to the exact value for any frequency case according to Theorem \[thm::convergence1\] and Lemma \[lem::cone\].
[**Example 2.1: low frequency case:**]{}
Let the angular frequency $\omega=20$ Hz. Use the simulated data without noise (see Figure \[fig::lobs\]) and the noisy simulated data with 20$\%$ relative Gaussian noise (see Figure \[fig::lobs20\] and Figure \[fig::lu60\] ).
(a)![Simulated data without noise: (a) real part of $u$ (mm); (b) imaginary part of $u$ (mm).[]{data-label="fig::lobs"}](lobs_real.eps "fig:"){width=".45\textwidth"} (b)![Simulated data without noise: (a) real part of $u$ (mm); (b) imaginary part of $u$ (mm).[]{data-label="fig::lobs"}](lobs_imag.eps "fig:"){width=".45\textwidth"}
(a)![Noisy simulated data: (a) real part of $u$ (mm); (b) imaginary part of $u$ (mm).[]{data-label="fig::lobs20"}](lobs20_real.eps "fig:"){width=".45\textwidth"} (b)![Noisy simulated data: (a) real part of $u$ (mm); (b) imaginary part of $u$ (mm).[]{data-label="fig::lobs20"}](lobs20_imag.eps "fig:"){width=".45\textwidth"}
![Simulated data along $x_1=60$ mm (blue line: without noise, red line: with 20$\%$ relative noise).[]{data-label="fig::lu60"}](lu60.eps){width=".6\textwidth"}
By applying our Levenberg-Marquad method, we recovered $\gamma$ from noisy data which is shown in Table \[table::lreconver\] and the associated reconstructed wave fields are shown in Figure \[fig::lobsr\] and Figure \[fig::lur60\]. The result shows that both the storage modulus $G'$ and loss modulus $G''$ are recovered very well.
Initial guess $G'$ 30 kPa $G''$ 0.5 Pa$\cdot$s$\times \omega$
------------------------- ------ ------------- ------- ----------------------------------
Layer 1 ($x_L<x_2<120$) $G'$ 19.8202 kPa $G''$ 0.3849 Pa$\cdot$s$\times \omega$
Layer 2 ($0<x_2<x_L$) $G'$ 9.9829 kPa $G''$ 0.2990 Pa$\cdot$s$\times \omega$
: Recovery of $\gamma$.[]{data-label="table::lreconver"}
(a)![Reconstructed simulated data: (a) real part of $u$ (mm); (b) imaginary part of $u$ (mm).[]{data-label="fig::lobsr"}](lobsr_real.eps "fig:"){width=".45\textwidth"} (b)![Reconstructed simulated data: (a) real part of $u$ (mm); (b) imaginary part of $u$ (mm).[]{data-label="fig::lobsr"}](lobsr_imag.eps "fig:"){width=".45\textwidth"}
![Reconstructed data along $x_1=60$ mm (blue line: without noise, red dot dash line: reconstructed).[]{data-label="fig::lur60"}](lur60.eps){width=".6\textwidth"}
![Simulated data along $x_1=60$ mm (real part of $u$, blue line: $G''=0$, red line: $G''\neq 0$).[]{data-label="fig::uev60"}](uev60.eps){width=".6\textwidth"}
It is interesting to observe here that the imaginary part of simulated data is quite small compared with the real part of simulated data which is almost the same as the one in Example 1.1 (see Figure \[fig::uev60\]) . This is because the loss modulus $G''=0.4$ Pa$\cdot$s$\times\omega$$\approx$ 0.0503 kPa for layer 1 ($0.3$ Pa$\cdot$s$\times\omega$$\approx$ 0.0377 kPa for layer 2) is small while the storage modules is 20 kPa for layer 1 (10 kPa for layer 2). However, the existence of loss modulus, no matter how small it is, enables us to choose some initial guess of $\gamma$ in Table \[table::lreconver\] not so close to the exact value due to Theorem \[thm::convergence1\] and Lemma \[lem::cone\]. Further it is impossible to recover neither $G'$ nor $G''$ reasonably well by using other existing methods, such as the modified integral method in [@JiangP] since there is less than half wave in each layer which cannot meet the requirement of our modified integral method. Similar result can either find in [@AmmN].
[**Example 2.2: high frequency case:**]{} Let the angular frequency $\omega=250$ Hz. Use the simulated data without noise (see Figure \[fig::obs\]) and the noisy simulated data with 20$\%$ relative Gaussian noise (see Figure \[fig::obs20\] and Figure \[fig::u60\]).
(a)![Simulated data without noise: (a) real part of $u$ (mm); (b) imaginary part of $u$ (mm).[]{data-label="fig::obs"}](obs_real.eps "fig:"){width=".45\textwidth"} (b)![Simulated data without noise: (a) real part of $u$ (mm); (b) imaginary part of $u$ (mm).[]{data-label="fig::obs"}](obs_imag.eps "fig:"){width=".45\textwidth"}
(a)![Noisy simulated data: (a) real part of $u$ (mm); (b) imaginary part of $u$ (mm).[]{data-label="fig::obs20"}](obs20_real.eps "fig:"){width=".45\textwidth"} (b)![Noisy simulated data: (a) real part of $u$ (mm); (b) imaginary part of $u$ (mm).[]{data-label="fig::obs20"}](obs20_imag.eps "fig:"){width=".45\textwidth"}
![Simulated data along $x_1=60$ mm (blue line: without noise, red line: with 20$\%$ relative noise).[]{data-label="fig::u60"}](u60.eps){width=".6\textwidth"}
The recovery of $\gamma$ from noisy data is shown in Table \[table::reconver\] and the reconstructed wave fields are shown in Figure \[fig::obsr\] and Figure \[fig::ur60\]. Again the recovery is very well.
Initial guess $G'$ 30 kPa $G''$ 0.5 Pa$\cdot$s$\times \omega$
------------------------- ------ ------------- ------- ----------------------------------
Layer 1 ($x_L<x_2<120$) $G'$ 19.9951 kPa $G''$ 0.3948 Pa$\cdot$s$\times \omega$
Layer 2 ($0<x_2<x_L$) $G'$ 9.9997 kPa $G''$ 0.3040 Pa$\cdot$s$\times \omega$
: Recovery of $\gamma$.[]{data-label="table::reconver"}
(a)![Reconstructed simulated data: (a) real part of $u$ (mm); (b) imaginary part of $u$ (mm).[]{data-label="fig::obsr"}](obsr_real.eps "fig:"){width=".45\textwidth"} (b)![Reconstructed simulated data: (a) real part of $u$ (mm); (b) imaginary part of $u$ (mm).[]{data-label="fig::obsr"}](obsr_imag.eps "fig:"){width=".45\textwidth"}
![Reconstructed data along $x_1=60$ mm (blue line: without noise, red dot dash line: reconstructed).[]{data-label="fig::ur60"}](ur60.eps){width=".6\textwidth"}
Here, we applied the modified integral method to recover $\gamma$ from noisy data (see Figure \[fig::mim60\]). The recovery of $G'$ is good, meanwhile the recovery of $G''$ is quite poor.
(a)![Recovery of $\gamma$ by using modified integral equation method along $x_1=60$ mm ((1) $G'$, (b) $G''$, blue line: recovery, red dot dash line: exact value).[]{data-label="fig::mim60"}](mu60.eps "fig:"){width=".45\textwidth"} (b)![Recovery of $\gamma$ by using modified integral equation method along $x_1=60$ mm ((1) $G'$, (b) $G''$, blue line: recovery, red dot dash line: exact value).[]{data-label="fig::mim60"}](eta60.eps "fig:"){width=".45\textwidth"}
Discussions and conclusion
==========================
We have shown that the convergence of Levenberg-Marquard method for an inverse problem with single interior measurement which arises in the data analysis for the magnetic resonance elastography (MRE). The key for this was to show that the measurement map of MRE satisfies the tangential cone condition. A similar result can be obtained also for other coefficients identification problem by single interior measurement of solution to the boundary value problem for partial differential equation of divergence form such that the real part or the imaginary part of the associated sesquilinear form is positive. The numerical performance of this method was given for several cases and observed that it is quite well. In particular the recovery of the loss modulus was very good compared with the other existing methods. Based on these we conclude that this method has a strong potential to become one of a standard method for elastogram not only recovering the storage modulus but also the loss modulus.
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---
abstract: 'We study the effects of strangeness on the quark sector of a hybrid star equation of state. Since the model we use to describe quarks is the same as the one we use to describe hadrons, we can also study the effects of strangeness on the chiral symmetry restoration and deconfinement phase transitions (first order or crossover). Finally, we analyze combined effects of hyperons and quarks on global properties of hybrid stars, like mass, radius and cooling profiles. It is found that a large amount of strangeness in the core is related to the generation of twin-star solutions, which can have the same mass as the lower or zero strangeness counterpart, but with smaller radii.'
author:
- 'V. Dexheimer'
- 'R. Negreiros'
- 'S. Schramm'
bibliography:
- 'apssamp.bib'
title: The role of strangeness in hybrid stars and possible observables
---
\[sec:level1\]Introduction
==========================
Recently, it has been understood that any realistic calculation for neutron star equations of state must take into account hyperons and/or quark degrees of freedom, since densities of several times nuclear saturation density can be reached in the center of the star. Moreover, many models that include hyperon and quark degrees of freedom are able to describe massive stars with masses around $2M_\odot$ (see [@Takatsuka:2004ch; @Buballa:2014jta] and references therein). This means that the role played by strangeness at high densities has again become the focus of intense discussions. The interplay between the appearance of hyperons and quarks has also been discussed, as for example in Refs. [@Zdunik:2012dj; @Dexheimer:2012eu].
In addition, recent studies of neutron star radii have indicated that these objects might have smaller radii than previously expected, around $10$ km or less [@Psaltis:2013fha; @Guillot:2013wu]. Although recent small radii measurements have been criticized (see for instance Refs. [@Steiner:2010fz; @Heinke:2014xaa]), the idea of small radii stars together with the constraint of stars with $2M_\odot$ [@Demorest:2010bx; @Antoniadis:2013pzd] pushes toward equations of state very close to the causal limit, beyond which the speed of sound is larger than the speed of light. The so called “hyperon puzzle", discussed in Ref [@Zdunik:2012dj], refers to the larger radii of massive stars containing hyperons. A possible solution, in this case, is the coexistence of strange hadronic and quark stars, which would separately fulfill mass and radius constraints. Such a possibility is achieved through another family of stars, referred to as “twin stars" or tertiary stars [@Gerlach:1968zz; @Kampfer:1981yr; @Glendenning:1998ag; @Schertler:2000xq; @SchaffnerBielich:2002ki; @Schramm:2013pma; @Alvarez-Castillo:2013cxa; @Blaschke:2013ana; @Pagliara:2013gya; @Pagliara:2014gja; @Benic:2014jia]. These are stars containing quarks and with smaller radii than the respective stars containing hadrons. In this work, we investigate this possibility but in the context of stars with different strangeness content.
We make use of a self-consistent approach that includes hadrons and quarks in the same model to study the interplay between different degrees of freedom. We do that by changing the strength of the strange meson coupling to the quarks. This makes it straightforward to study the effect of the appearance of strangeness in neutron stars and allows, in addition, the study of the effect of different kinds of phase transitions in the system. Note that the effect of the strength of the strange meson couplings to baryons has recently been studied in detail, for example, in Refs. [@Weissenborn:2011ut; @Gusakov:2014ota; @Lopes:2013cpa; @2014arXiv1404.2428M]. Within our framework, we study the possibility of smooth crossovers and first order phase transitions from hadronic dominant to quark dominant matter and from non-strange dominant to strange dominant matter. We then use our equation of state to obtain observables such as mass, radius and cooling profiles for neutron stars.
\[sec:level1\]The Model
=======================
Chiral sigma models are effective quantum relativistic models that describe hadrons interacting via meson exchange and, most importantly, are constructed from symmetry relations. They are constructed in a chirally invariant manner as the particle masses originate from interactions with the medium and, therefore, go to zero at high density and/or temperature. The nonlinear realization of the sigma model is an alternative approach to the widely used linear sigma model [@Papazoglou:1997uw; @Lenaghan:2000ey; @Nahrgang:2011mg] and it is in good agreement with nuclear physics results [@Papazoglou:1998vr; @Bonanno:2008tt].
The Lagrangian density of the SU(3) non-linear realization of the sigma model constrained further by astrophysics data can be found in Refs. [@Dexheimer:2008ax; @Schurhoff:2010ph]. A recent extension of this model which includes quarks as dynamical degrees of freedom [@Dexheimer:2009hi; @2013PhRvC..88a4906H; @Negreiros:2010hk] is described in the following. The Lagrangian density of the model in the mean field approximation reads $$\begin{aligned}
L = L_{Kin}+L_{Int}+L_{Self}+L_{SB}\,,\end{aligned}$$ where, besides the kinetic energy term for hadrons, quarks, and leptons, the terms: $$\begin{aligned}
L_{Int}&=&-\sum_i \bar{\psi_i}[\gamma_0(g_{i\omega}\omega+g_{i\phi}\phi+g_{i\rho}\tau_3\rho)+M_i^*]\psi_i,\nonumber\\
%
L_{Self}&=&\frac{1}{2}(m_\omega^2\omega^2+m_\rho^2\rho^2+m_\phi^2\phi^2)\nonumber\\
&+&g_4\left(\omega^4+\rho^4+\alpha^2\frac{\phi^4}{2}+3\alpha(\omega^2+\rho^2)\phi^2\right)\nonumber\\&-&k_0(\sigma^2+\zeta^2+\delta^2)-k_1(\sigma^2+\zeta^2+\delta^2)^2\nonumber\\&-&k_2\left(\frac{\sigma^4}{2}+\frac{\delta^4}{2}
+3\sigma^2\delta^2+\zeta^4\right)
-k_3(\sigma^2-\delta^2)\zeta\nonumber\\&-&k_4\ \ \ln{\frac{(\sigma^2-\delta^2)\zeta}{\sigma_0^2\zeta_0}}\,,\nonumber\\
%
L_{SB}&=&-m_\pi^2 f_\pi\sigma-\left(\sqrt{2}m_k^ 2f_k-\frac{1}{\sqrt{2}}m_\pi^ 2 f_\pi\right)\zeta\,,\end{aligned}$$ represent the interactions between baryons (and quarks) and vector and scalar mesons, the self interactions of scalar and vector mesons, and an explicit chiral symmetry breaking term (responsible for producing the masses of the pseudo-scalar mesons). The underlying flavor symmetry of the model is SU(3) and the index $i$ denotes the baryon octet and the three light quarks. The mesons included are the vector-isoscalars $\omega$ and $\phi$ (strange quark-antiquark state), the vector-isovector $\rho$, the scalar-isoscalars $\sigma$ and $\zeta$ (strange quark-antiquark state) and the scalar-isovector $\delta$, with $\tau_3$ being twice the isospin projection of each particle. The isovector mesons affect isospin-asymmetric matter and are consequently important for neutron star physics. Note, that different self-interaction schemes for the vector mesons ($g_4$ term) can be included in the model, as long as they conform with chiral symmetry [@Dexheimer:2008ax]. The parameter $\alpha$ results from a renormalization of the vector fields in order to obtain their correct vacuum masses. Assuming equal contributions of the $\omega$ and $\rho$ meson, these constants can be absorbed in the coupling $g_4$ and only the $\phi$ field terms are affected (see Ref. [@Papazoglou:1997uw] for a detailed discussion).
In the model presented in Ref. [@Dexheimer:2009hi; @2013PhRvC..88a4906H], the degrees of freedom change due to the effective masses of the baryons and quarks. Here, we adopt a different formalism, explained in the following, and the effective masses for baryons and quarks are simply generated by the scalar mesons, except for a small explicit mass term $m_0$ $$\begin{aligned}
M_{B}^*&=&g_{B\sigma}\sigma+g_{B\delta}\tau_3\delta+g_{B\zeta}\zeta+m_{0_B}\,,\nonumber\\
%
M_{q}^*&=&g_{q\sigma}\sigma+g_{q\delta}\tau_3\delta+g_{q\zeta}\zeta+m_{0_q}\,.\end{aligned}$$
The coupling constants of the model are shown in Table \[tabela1\]. They were fitted to reproduce the vacuum masses of the baryons and mesons, nuclear saturation properties (density $\rho_0=0.15$ fm$^{-3}$, binding energy per nucleon $B/A=-15.65$ MeV, nucleon effective mass $M^*_N=0.67$ $M_N$, compressibility $K=318.76$ MeV), asymmetry energy ($E_{sym}=32.43$ MeV with slope $L=102.77$ MeV), and reasonable values for the hyperon potentials ($U_\Lambda=-30.44$ MeV, $U_\Sigma=2.47$ MeV, $U_\Xi=-26.28$ MeV). The vacuum expectation values of the scalar mesons are constrained by reproducing the pion and kaon decay constants $f_\pi$ and $f_\kappa$.
The slope of the symmetry energy has become a very prominent constraint for the equation of state in the past years, as its measurements (through, for example, neutron skin experiments) have become more accurate. These results seem to indicate low values for this quantity ($L \sim 50-70$ MeV). See Refs. [@Lattimer:2012xj; @Tsang:2012se; @Li:2013ola] or [@Horowitz:2014bja] for a recent review on the topic. Notice, however, that some works such as the one in Ref. [@Cozma:2013sja] found much higher values for the slope of the symmetry energy ($L \sim 100$ MeV). A detailed work on the role of the symmetry energy in the Lagrangian density of the SU(3) non-linear realization of the sigma model is in progress and it will be available soon.
---------------------------------- ------------------------------- -----------------------------
$ g_{N\omega}=11.46 $ $ g_{N\rho}=3.83 $ $ g_{N\phi}=0 $
$ g_{N\sigma}=-9.83 $ $ g_{N\delta}=0 $ $ g_{N\zeta}=1.22$
$ g_{\Lambda\omega}=7.64 $ $ g_{\Lambda\rho}=0 $ $ g_{\Lambda\phi}=7.06 $
$ g_{\Lambda\sigma}=-5.39 $ $ g_{\Lambda\delta}=0 $ $ g_{\Lambda\zeta}=-2.21$
$ g_{\Sigma\omega}=7.64 $ $ g_{\Sigma\rho}=7.64 $ $ g_{\Sigma\phi}=7.06 $
$ g_{\Sigma\sigma}=-3.88 $ $ g_{\Sigma\delta}=0 $ $ g_{\Sigma\zeta}=-4.36 $
$ g_{\Xi\omega}=3.82 $ $ g_{\Xi\rho}=3.82 $ $ g_{\Xi\phi}=14.11 $
$ g_{\Xi\sigma}=-1.54 $ $ g_{\Xi\delta}=0 $ $ g_{\Xi\zeta}=-7.66 $
$ g_{u\omega}=g_{d\omega}=4.70 $ $ g_{u\rho}=g_{d\rho}=-2.00 $ $ g_{u\phi}=g_{d\phi}=0 $
$ g_{u\sigma}=g_{d\sigma}=3.80 $ $ g_{u\delta}=g_{d\delta}=0 $ $ g_{u\zeta}=g_{d\zeta}=0 $
$ g_{s\omega}=0 $ $ g_{s\rho}=0 $ $ g_{s\phi}=$variable
$ g_{s\sigma}=0 $ $ g_{s\delta}=0 $ $ g_{s\zeta}=-3.80 $
$ g_4=38.5 $ $ k_0=1.19 \chi^2 $ $ k_1=-1.40 $
$ k_2=5.55 $ $ k_3=2.66 \chi $ $ k_4=-0.07 \chi^4 $
$ m_{0_u}=m_{0_d}=6$ MeV $ m_{0_s}=72$ MeV $ m_{0_N}=150$ MeV
$ m_{0_\Lambda}=376.58$ MeV $ m_{0_\Sigma}=376.58$ MeV $ m_{0_\Xi}=376.58$ MeV
---------------------------------- ------------------------------- -----------------------------
: \[tabela1\]Coupling constants for the model, using $\chi = 401.93$ MeV.
In order to suppress the hadrons at high density and/or temperature and allow the quarks to dominate, we introduce an excluded volume for the baryons. The use of such a technique was proposed long ago in Refs. [@Baacke:1976jv; @Hagedorn:1980kb; @Gorenstein:1981fa; @Kapusta:1982qd; @Hagedorn:1982qh; @Rischke:1991ke; @Lattimer:1991nc; @Cleymans:1992jz; @Shen:1998gq; @Bugaev:2000wz; @Bugaev:2008zz; @Satarov:2009zx; @Hempel:2011kh] and recently used in Refs. [@Steinheimer:2010ib; @Steinheimer:2011ea; @Dexheimer:2012eu]. This is done by introducing the volume occupied by baryons and quarks as $$\begin{aligned}
v_{B} = 0.64 {\rm \ fm}^{3} \ \ \ \ {\rm and} \ \ \ \ v_{q} = 0\,.\end{aligned}$$ The $v_{B}$ value is chosen to represent the effect of the repulsive baryonic hard core with a corresponding radius of $r = 0.34$ fm. In this case, the chemical potential for baryons and quarks needs to be further modified in order to maintain thermodynamical consistency $$\begin{aligned}
\hat{\mu_i}=\mu_i^* - v_i P\,,\end{aligned}$$ with $$\begin{aligned}
\mu_i^*= Q_{B_i} \mu_B - Q_i \mu_Q - g_{i\omega} \omega - g_{\phi} \phi - g_{i\rho} \tau_3 \rho\,,\end{aligned}$$ with $P$ being the total pressure of the system, $Q_{B_i}$ the baryon number of each particle, $\mu_B$ the baryon chemical potential, $Q_i$ the electric charge of each particle and $\mu_Q$ the charge chemical potential. In this way, the chemical potentials of the baryons are decreased by the appearance of quarks, but not vice versa. Furthermore, to be thermodynamically consistent, all particle densities, i.e., number density, energy density, entropy density, etc., have to be multiplied by a volume correction factor $f$, defined as the ratio of the unoccupied (excluded) volume $V'$ and the total volume $V$ $$\begin{aligned}
f = \frac{V'}{V} = \left(1 + \sum_i v_i Q_{B_i} \rho_i \right)^{-1}\,,\end{aligned}$$ where $\rho_i$ is the density of each particle. In this way, the quarks effectively suppress the baryons by changing their chemical potential, while the quarks are only affected through the volume correction factor $f$.
![(Color online) Population (particle density normalized by baryon number) as a function of baryon density for symmetric matter.\[sum\]](Popsym.pdf){width="8.7cm"}
As a result, for symmetric matter, the model used in this work predicts that protons and neutrons are only suppressed at high densities, when a significant amount of quarks appear. This can be seen in Fig. \[sum\]. Due to the zero strangeness constraint, only the up and down quarks slowly appear at about $2.3$ times the saturation density. The strangeness constraint is necessary in order to compare theoretical results with nuclear and particle experimental results, which take place in a time interval much shorter than the weak equilibration time.
\[sec:level1\]Stellar Matter and Structure
==========================================
In order to study neutron stars, we take into account charge neutrality and chemical equilibrium. Strangeness is not constrained since, for neutron stars, the time scale is large enough for strangeness not to be conserved. The equation of state for such a system is shown in Fig. \[eos\]. Depending on the strength of the quark couplings, the stiffness of the equations of state and the kind of phase transitions obtained are different.
![(Color online) Equation of state for different strengths of the quark couplings.\[eos\]](EOS.pdf){width="8.2cm"}
We show curves for different strengths of the quark coupling to the strange vector meson compared to the non-strange vector meson defined as $\xi=g_{q\phi}/g_{q\omega}$, keeping $g_{q\omega}$ constant. Larger values of the parameter $\xi$ reproduce stiffer equations of state due to extra repulsion at intermediate and/or larger densities. This can be seen in Fig. \[eos\]. While most values of $\xi$ reproduce smooth equations of state, negative values below $\xi=-1.10$ reproduce first order phase transitions. So far, it is not understood what kind of deconfinement and chiral restoration phase transitions take place at small and zero temperatures. In Refs. [@Baym:2008me; @Lourenco:2012yv], for example, it has been proposed that the phase transition in such a limit is a smooth crossover, as in the high temperature and low density regime.
In order to understand better the results from Fig. \[eos\], we show the variation of the scalar meson and the strange scalar meson fields for different strengths of the quark couplings in Figs. \[sigma\] and \[zeta\]. The sigma field, usually referred to as the chiral condensate, is intrinsically related to the restoration of chiral symmetry and it is shown here also for the symmetric case. The zeta field is related to the quantity of strange particles in the system, meaning that a larger deviation from the vacuum value enhances the amount of hyperons (specially $\Xi$’s, which have larger strangeness) and strange quarks.
Now, we discuss the particle population present in the stars reproduced by the parametrizations discussed above. For $\xi=3.62$, Fig. \[pop1\] shows that the down quark appears very early in the system, followed by the hyperons $\Lambda$, $\Sigma^-$, $\Xi^-$, $\Xi^0$, $\Sigma^0$ and $\Sigma^+$, followed by the up and strange quarks. As the density increases, the down quark is temporarily suppressed by the appearance of hyperons, but it becomes dominant at high densities. The strange quark only appears at very high densities, as it not only has a larger bare mass, but is also suppressed by the strong positive coupling to the strange vector meson.
Note that a sequential occurrence of deconfined quarks with different flavors at high densities has already been discussed in Ref. [@Blaschke:2008gd]. In that work, the chiral symmetry restoration (associated with the quark deconfinement) is calculated in the NJL model by solving the gap equations and taking into account charge neutrality in such a way that, similar to our case, it depends on the quark chemical potentials. These chemical potentials are in turn dependent on the electric charge and bare mass of particles. In this scenario, the down quark drips out of the baryons first, followed by the up and strange quarks. At high temperature and zero chemical potential, sequential quark deconfinement has also been seen in lattice QCD calculations [@Bellwied:2013cta].
![(Color online) Scalar meson field (normalized by vacuum value) as a function of chemical potential for different strengths of the quark couplings.\[sigma\]](Sigma.pdf){width="9.5cm"}
Fig. \[pop2\] shows that the down quark appears once more very early in the system for $\xi=0$, followed by the hyperon $\Lambda$ together with the strange quark, and then the hyperons $\Xi^-$, $\Xi^0$, $\Sigma^+$, $\Sigma^0$ and $\Sigma^-$, followed by the up quarks. Note that the early presence of the strange quarks in this case (without the influence of the $\phi$ meson) suppresses the negative charged hyperons, which would normally appear before the others to fulfill charge neutrality (now taken care of by the strange quark).
Note also, that we could have used any positive value for the parameter $\xi$. The choices of zero and $\xi=3.62$ were only made to show the largest possible span of effects. After $\xi=3.62$, the effect of different $\phi$ coupling with the strange quark saturates, not showing further changes. This is, because at $\xi=3.62$, the strange quarks appear after all of the other possible particles, and in a small amount.
For negative values of the $\xi$ parameter, there is no change in the order of the appearance of the quarks and hyperons with respect to density, since the strange quark appears basically at the same low density as in the $\xi=0$ case. Nevertheless, for negative enough values of the $\xi$ parameter ($\xi<-1.10$), the transition from hadronic-dominant to quark-dominant matter, and the transition from non-strange-dominant to strange-dominant matter (which so far have been smooth crossovers) become a first order phase transition.
This had already been shown in Fig. \[eos\] but can be better understood in Fig. \[pop3\] plotted for $\xi=-1.29$, where a jump in the baryon density (which is a first derivative of the baryon chemical potential) can be observed. The grey shaded area in the figure represents an unphysical region, which would collapse to a point in the presence of gravity, as it corresponds to a single value of pressure.
![(Color online) Strange scalar meson field (normalized by vacuum value) as a function of chemical potential for different strengths of the quark couplings.\[zeta\]](Zeta.pdf){width="9.5cm"}
The equations of state for the three cases already discussed are analyzed under the influence of gravity in Fig. \[tov\], from the solutions of the Tolman-Oppenheimer-Volkoff equations [@Tolman:1939jz; @Oppenheimer:1939ne]. When $\xi=3.62$, a maximum mass of $2.31 M_\odot$ is obtained with a corresponding radius of $12.58$ km. When $\xi=0$, a maximum mass of $2.05 M_\odot$ is obtained with a corresponding radius of $13.07$ km. Finally, when $\xi=-1.29$, a maximum mass of $1.97 M_\odot$ is obtained with a corresponding radius of $13.61$ km.
An interesting feature of negative values of the $\xi$ parameter is that a secondary family of stars appears (at $\xi=-1.26$) with significantly smaller radii. This feature is related to the baryon density discontinuity at the phase transition and it is often called “twin stars solution". These stars have equivalent masses (to the normal branch), but different radii.
As $\xi$ becomes more negative, the amount of stars in the twin family increases. When $\xi=-1.29$, the first order phase transition to dominant quark strange matter reproduces twin stars with radii spanning from $9.60$ to $10.22$ km and a maximum mass of $1.68 M_\odot$. The discontinuity, across the phase transition that allows for such a configuration is quite small, $\Delta\rho_B=0.18$ fm$^{-3}$ (equivalent to an energy density discontinuity of $\Delta\epsilon=238.40$ MeV/fm$^3$).
Note that the star branch with smaller radii in Fig. \[tov\], with hyperon and quark contributions, has significantly more strangeness than the normal branch. Observation of such configurations would point to the confirmation of first order phase transitions in stars, as already pointed out by Ref. [@Glendenning:1998ag; @Schertler:2000xq; @Alvarez-Castillo:2013cxa; @Benic:2014jia]. In the context of hadronic matter, the possibility of smooth and strong phase transitions to strange matter has been explored in Ref. [@Gulminelli:2013qma]. For $\xi=-1.29$, for example, a star mass of $1.68 M_\odot$, corresponding to radii of $14.00$ km and $9.60$ km in different branches, contains strangeness of $f_s = 0.01$ and $f_s = 1.68$, respectively, at the center. The total strangeness is defined as $f_s = \sum_i \rho(i) Q_{S_i} / \rho_B$, where $Q_{S_i}$ is the strangeness of each particle. For values of $\xi$ more negative than $-1.29$, the twin star solutions exist, but their equations of state becomes supraluminal for high enough densities. Therefore, we will not discuss them in this work. For a complete review on the topic of the behavior of models containing the excluded volume technique at high densities, see Ref. [@Satarov:2014voa]. For a review on the topic of the sound speed in neutron stars see Ref. [@Bedaque:2014sqa].
![(Color online) Population (particle density normalized by baryon number) as a function of baryon density for $\xi=3.62$.\[pop1\]](Pop17.pdf){width="8.7cm"}
![(Color online) Population (particle density normalized by baryon number) as a function of baryon density for $\xi=0$.\[pop2\]](Pop0.pdf){width="8.7cm"}
![(Color online) Population (particle density normalized by baryon number) as a function of baryon density for $\xi=-1.29$.\[pop3\]](Popminus606.pdf){width="8.7cm"}
Figure \[lattimer\] (modified from Refs. [@Lattimer:2004sa; @Lattimer:2012nd]) shows the relation between the maximum mass star and the corresponding central density predicted by several classes of models from the literature, such as nonrelativistic (NR) potential or Skyrme-like, relativistic (R) field theoretical, EOS’s containing significant amounts of quarks, and EOS’s with significant contributions from hyperons or meson condensates. Note that the EOS’s from Refs. [@Dexheimer:2008ax; @Dexheimer:2009hi], which derive from another version of the model discussed in this work, are also shown marked by the letters “A" and “B". Finally, we analyze the compactness of the equation of state generated from the parametrization $\xi=-1.29$ discussed in this work (marked by “C" and “D" in the figure). The letter “C" denotes the maximum mass star of the normal branch, and the letter “D" the one of the twin branch. Note that “D" is beyond the limiting curve “s=1/3", which delimits the supraluminal behavior for quark stars when modeled by the simple MIT bag model. It is interesting to note that this line is also a limit to many models which reproduce a substantial amount of quarks in stars. The distance between “C" and “D" points to the difference between the compactness of both star families reproduced (normal and twin).
In addition, we would like to compare our results with Ref. [@Alford:2013aca] by Alford and Han, which discusses different categories of twin stars assuming a Maxwell construction between different phases of matter and a constant speed of sound. Although our case with $\xi=-1.29$ reproduces twin star solutions, it shows no connected branch (stars with density beyond the phase transition jump but still in the stable part of the main star branch in the mass-radius diagram) within our numerical accuracy. We believe that we might reproduce a connected branch in agreement with Ref. [@Alford:2013aca], however, in practice, it may be too small to be seen. We reproduce a ratio of $\Delta\epsilon$/$\epsilon_{\rm trans}=0.33$, where $\Delta\epsilon$ is the jump in energy density across the phase transition and $\epsilon_{\rm trans}$ the energy density at which the transition takes place, and we reproduce a ratio $P_{\rm trans}$/$\epsilon_{\rm trans}=0.16$, where $P_{\rm trans}$ is the pressure at which the transition takes place. These values are important as they are connected to the ability of the low density phase to be supported by a large enough high density phase in the core. An intermediate size for the high density phase in the core necessarily turns the star unstable, originating disconnected solutions for stars. In other words, the existence of the twin branch depends on the size of the discontinuity across the phase transition and the stiffness of the equations of state of both phases.
\[sec:level1\]Thermal Evolution
===============================
![(Color online) Diagram showing different models for neutron stars placed with respect to their compactness. Figure adapted from Refs. [@Lattimer:2004sa; @Lattimer:2012nd] by Lattimer and Prakash.\[lattimer\]](vd.pdf){width="9.7cm"}
We now turn our attention to the thermal evolution of the objects whose structures and compositions were discussed above. The thermal behavior of a compact star strongly depends on its microscopic and macroscopic properties, thus, when combined with observational data, it is a formidable way of probing the characteristics of these objects. The thermal evolution of a compact star is governed by the general relativistic thermal balance and transport equations, given by ($G = c = 1$) [@Weber] $$\begin{aligned}
\frac{ \partial (l e^{2\phi})}{\partial m}& =
&-\frac{1}{\rho \sqrt{1 - 2m/r}} \left( \epsilon_\nu
e^{2\phi} + c_v \frac{\partial (T e^\phi) }{\partial t} \right) \, ,
\label{coeq1} \\
\frac{\partial (T e^\phi)}{\partial m} &=& -
\frac{(l e^{\phi})}{16 \pi^2 r^4 \kappa \rho \sqrt{1 - 2m/r}}
\label{coeq2}
\, ,\end{aligned}$$ where the macroscopic dependence is given by the variables $r$, $\rho(r)$ and $m(r)$, that represent the radial distance, the energy density, and the stellar mass, respectively. The thermal properties are represented by the temperature $T(r,t)$, luminosity $l(r,t)$, neutrino emissivity $\epsilon_\nu(r,T)$, thermal conductivity $\kappa(r,T)$ and specific heat $c_v(r,T)$. Furthermore, the boundary conditions of Eqs. (\[coeq1\]) and (\[coeq2\]) are provided by the vanishing heat flux at the center of the star and the luminosity at its surface, defined by the relationship between the mantle and photosphere temperature [@Gudmundsson1982; @Gudmundsson1983; @Page2006].
![(Color online) Gap energy for the neutron triplet pairing shown for different temperatures. \[Delta\]](Delta.pdf){width="9.cm"}
In our study, we consider all state of the art neutrino emission processes for the thermal evolution. In the hadronic phase, we take into account the direct Urca, modified Urca and bremsstrahlung processes, as well as the Pair-Breaking-Formation process (PBF) that accompanies pairing (described below) [@Yakovlev2001a]. For the quarks we consider the quark direct Urca, quark modified Urca, and quark bremsstrahlung processes. Furthermore, we consider the possibility of hadronic as well as quark pairing. For the hadronic model, we consider proton singlet pairing ($^1S_0$), neutron triplet pairing ($^3P_2$) for the core region and neutron singlet ($^1S_0$) pairing for the crustal region. To illustrate gap values, we show in Fig. \[Delta\] the neutron triplet gap energy as a function of density, for several values of temperature. Note that the proton pairing has a similar behavior, except for much stronger pairing, which is necessary to explain the behavior of Cas A [@Page2011a; @Yakovlev2011]. For the quarks, we consider pairing with a gap $\Delta = 10$ MeV.
We start by showing the results for the thermal evolution of stars generated using the parameter $\xi=-1.29$ for the strength of the quark coupling, which we display in Fig. \[cool1\]. In the case of neutron stars of relatively low masses (1.4 and 1.6 $M_\odot$) their composition is similar enough that their thermal evolution is almost indistinguishable. As the star masses increase, however, the thermal evolution starts to exhibit a faster behavior, as expected in this case. Such a qualitative behavior is also exhibited in the thermal evolution of stars generated using the parameters $\xi=0$ and $\xi=3.62$, as shown in Figs. \[cool2\] and \[cool3\].
The similarity among the thermal evolution of different model parametrizations of the quark coupling to the strange vector meson compared to the non-strange vector meson $\xi$ is not surprising, in particular for the low mass stars, where quark matter is not strongly present. A possible way of differentiating between the parametrizations studied is to investigate the cooling of the maximum mass star for each of them. These objects have the highest possible quark matter content (in the normal branch) allowed by each parametrization and, thus, should exhibit differences in their cooling. This is shown in Fig. \[cool4\]. Note that, in this way, we can also obtain information about the strangeness (through the mass of the star) in the cooling curves. Although we do not have observational information regarding the mass of isolated stars, we could at the very least infer some constraints by using the fact that we know that larger strangeness implies slower star cooling, among other things, due to the smaller underlying stellar mass.
![(Color online) Thermal evolution (red-shifted surface temperature as a function of time) of stars for $\xi=-1.29$.\[cool1\]](cool_graph1.pdf){width="10.2cm"}
![(Color online) Thermal evolution (red-shifted surface temperature as a function of time) of stars for $\xi=0$.\[cool2\]](cool_graph3.pdf){width="10.2cm"}
As shown in Fig. \[cool4\], although all parametrizations exhibit the same qualitative behavior, the parametrization $\xi=-1.29$ seems to be slightly better if one is to interpret recent thermal observations such as in Cas A. The core-crust thermal coupling in this case (signaled by the sudden drop in surface temperature) takes place a later ages (when compared to the other parametrizations). This might facilitate the interpretation of Cas A data as this object exhibits a surface temperature drop at around $\sim 300$ years. In practice, there are other factors in play, in particular the pattern of the superconducting/superfluidity phases. A thorough investigation of this subject will be performed in a future investigation.
\[sec:level1\]Conclusions
=========================
![(Color online) Thermal evolution (red-shifted surface temperature as a function of time) of stars for $\xi=3.62$.\[cool3\]](cool_graph2.pdf){width="10.2cm"}
We studied the dependance of neutron star properties on the strength of the quark couplings (strange vector to vector quark coupling ratio). The choice of values for this quantity affects the stiffness of the equation of state and the strength of the phase transition to (dominantly) deconfined strange matter, ranging from crossovers to first order phase transitions. We performed this task in a controlled way by making use of a self-consistent model that includes hadrons and quarks degrees of freedom. In this model, the interactions determine the density at which deconfinement and chiral symmetry restoration take place and the inclusion of an excluded volume for the hadrons ensures that they are not present at high densities.
The effect of strangeness in neutron stars emerges in the mass-radius relation, where a large amount of strangeness is related to the generation of twin-stars, which can have the same mass as the lower or zero strangeness counterpart, but with smaller radii. The measurement of such stars would be a clear indication of a first order phase transition taking place at high densities and low temperatures, at least for charge neutral and beta-equilibrated matter. Nevertheless, this would provide us a priceless insight in the understanding of the QCD phase diagram.
![(Color online) Thermal evolution of the maximum mass star from Figures 11, 12 and 13.\[cool4\]](cool_graph4.pdf){width="10.2cm"}
With respect to the thermal evolution of stars, we have shown that the three quark coupling parametrizations studied present distinct behavior for massive stars, where the quark phase manifests itself differently for the three cases investigated. All three parametrizations, however, seem to be in agreement with the current cooling picture, in which high mass objects present a relatively faster thermal evolution than those objects with less mass and pairing in the hadronic and quark phases is necessary if one is to agree with recent data such as that obtained for Cassiopeia A. Among the three parametrizations studied, in the one with the smaller strange vector to vector quark coupling ratio, the core-crust thermal coupling (indicated by the sudden drop in the surface temperature) occurs at later times. This indicates agreement with Cas A data, but a more detailed study of the topic will be performed in future investigations.
In the future, we intend to extend our calculations to include relativistic excluded volume effects. Such a consistent approach will provide us with a better idea of the stiffness of the equation of state around deconfinement, and also at higher densities. Work on relativistic versions of excluded volume techniques have been already pursued, for example in Ref. [@Zhang:1995ux; @Zeeb:2002xn; @Bugaev:2006pt], but so far with no guarantee of a physical speed of sound. We also intend to extend our calculation to finite temperature and include neutrino trapping. In this way, we will also be able to study the behavior of strangeness in supernova explosions.
Aknowledgements {#aknowledgements .unnumbered}
===============
We thank J. Lattimer, M. Alford and S. Han for fruitful discussions on the physics of compact stars and possible observables. V. D. acknowledges financial support from Helmholtz International Center for FAIR. R.N. acknowledges financial support from CAPES and CNPQ.
|
---
abstract: 'In this work we investigate the spectral and transport properties of a single correlated layer attached to two metallic leads, with particular focus on the low-energy physics. A steady state current is driven across the layer by applying a bias voltage between the leads. Extending previous work we introduce a nonzero temperature in the leads, which enables us to study the influence of quasiparticle excitations on the transport characteristics in detail. Even though the system is clearly three dimensional we obtain current-voltage curves that closely resemble those of single quantum dots. Furthermore, a splitting of the quasiparticle excitation with bias voltage is observed in the spectral function.'
address: 'Institute of Theoretical and Computational Physics, Graz University of Technology, 8010 Graz, Austria'
author:
- 'Antonius Dorda, Irakli Titvinidze, and Enrico Arrigoni'
bibliography:
- 'ndmft\_QP.bib'
title: Quasiparticle excitations in steady state transport across a correlated layer
---
Introduction
============
Correlated systems out of equilibrium and in particular electronic transport through quantum dots [@go.sh.98; @cr.oo.98; @kr.sh.12; @zh.ka.13] and correlated heterostructures [@an.ga.99; @is.og.01; @oh.mu.02; @oh.hw.04; @ga.ah.02; @zh.wa.12] have recently attracted increasing interest. Related model systems of paramount importance are the single impurity Anderson model (SIAM) [@ande.61] and the Hubbard model [@hubb.63; @gutz.63; @kana.63]. At the present time the equilibrium properties of these systems are understood to large extent [@hews.93; @bu.co.08; @le.an.15u; @voll.12; @geor.04]. The so-called dynamical mean field theory (DMFT) [@ge.ko.96; @me.vo.89; @voll.12; @geor.04] was a key step in the theoretical description and understanding of Hubbard-like models and furthermore, established a link between correlated lattice systems which exhibit a Mott transition and the Kondo physics of a SIAM. Within this framework, the coherent quasiparticle excitations are described as a self-consistent Kondo effect [@geor.04]. The close relation between the SIAM and the Hubbard model poses the question whether analogous behavior is seen in the transport characteristics of the two systems. Exactly this question is the topic of the investigation presented here.
At the heart of DMFT lies the self-consistent solution of a quantum impurity model, the SIAM. An accurate description of the nonequilibrium physics of the SIAM and the related Kondo model is challenging by itself and currently intensively studied with different methods. To mention just a few, central aspects could be established with the noncrossing approximation [@wi.me.94; @le.sc.01; @ro.kr.01], perturbative renormalization group (RG) [@ro.pa.03; @sh.ro.06], flow equations [@kehr.05; @fr.ke.10], real-time RG [@pl.sc.12; @re.pl.14], time-dependent density matrix RG [@he.fe.09; @ho.mc.09; @nu.ga.13; @nu.ga.15], numerical RG [@ande.08; @sc.an.11] and Monte Carlo methods [@we.ok.10; @ha.he.07; @co.gu.14]. A method recently introduced by some of us, which is well-suited for an application within nonequilibrium DMFT, is the so-called auxiliary master equation approach (AMEA) [@ar.kn.13; @do.nu.14]. AMEA has proven to be an accurate method for the study of the nonequilibrium steady state physics of the SIAM [@do.nu.14; @do.ga.15]. Special emphasis was laid on investigating the evolution of the Kondo resonance upon driving the impurity model out of equilibrium by applying a bias voltage, and we found a transition from a single peak structure to a linear splitting of the Kondo resonance with bias. Furthermore, the current-voltage characteristics obtained for different temperatures showed clear signatures of the Kondo effect [@do.ga.15].
In the last years, rapid progress was made in the treatment of correlated lattice models out of equilibrium within DMFT, in explicitly time-dependent [@fr.tu.06; @ec.ko.09; @ao.ts.14; @ba.wo.15] and periodic or steady state situations [@sc.mo.02u; @jo.fr.08; @okam.08; @ar.ko.12]. In the study presented here we consider the special case of transport through a correlated heterostructure, consisting of a single correlated layer attached to two metallic leads at different chemical potentials. A similar setup was already treated in earlier studies with AMEA [@ar.kn.13; @ti.do.15u], however, the influence of temperature was not investigated. Here we consider the transport and spectral properties of the system starting from a lowest temperature, which can still be well-resolved with the employed impurity solver and results in a strong quasiparticle excitation, and successively extending to larger values of $T$ up to the quasigapped regime, analogous to a Mott insulator. Besides the bias-dependent spectral function, the experimentally well-accessible current-voltage characteristics is presented.
The work is organized as follows: In [Sec.]{}\[sec:model\] the investigated model is defined, in [Sec.]{}\[sec:method\] we briefly introduce the nonequilibrium DMFT approach together with AMEA, and in [Sec.]{}\[sec:results\] the obtained results are presented and discussed. Concluding remarks are given in [Sec.]{}\[sec:conclusio\].
Model {#sec:model}
=====
![Sketch of the investigated heterostructure, consisting of a single correlated layer of infinite size (red) with local Hubbard interaction $U$, on-site energy $\varepsilon_c$ and in-plane hopping amplitude $t_c$, sandwiched between two semi-infinite metallic leads (green). The hopping in the leads $l$ and $r$ is isotropic with amplitudes $t_l$ and $t_r$, respectively, with $v_l$ and $v_r$ denoting the hybridizations between the respective lead and the layer. An applied bias voltage $\phi$ shifts the on-site energies $\varepsilon_l$ and $\varepsilon_r$, as well as the chemical potentials $\mu_l$ and $\mu_r$ anti-symmetrically, which ensures together with $\varepsilon_c=-U/2$ particle-hole symmetry. All hoppings are for nearest neighbor terms only, and the temperature of the leads is labeled by $T = T_l = T_r$. We take $t_c$ as unit of energy and consider the case with $U = 10\,t_c$, $v_l = v_r = t_c$ and $t_l = t_r = 2\,t_c$. We discuss results for different values of $T$ and $\phi$. []{data-label="fig:model"}](fig1){width="60.00000%"}
The model system considered in this work is schematically depicted in [Fig.\[fig:model\]]{}. The corresponding Hamiltonian is given by $$\label{eq:H}
{H}={H}_c + \sum_{\alpha=l,r}{H}_\alpha +{H}_{\rm coup} \, ,$$ consisting of a part for the central system ${H}_c$, a part for each decoupled lead ${H}_{l/r}$ and a coupling between the leads and the correlated layer ${H}_{\rm coup}$. In the detail the Hamiltonian reads $$\begin{aligned}
{H}_c&=& -t_c \sum_{\langle ij\rangle, \sigma}c_{i\sigma}^\dagger c_{j\sigma}^{\phantom\dagger} +U\sum_i n_{i\uparrow}n_{i_\downarrow} +\varepsilon_c\sum_{i,\sigma}n_{i\sigma} \,,\\
{H}_\alpha&=& -t_\alpha \sum_{\langle i j\rangle \sigma}c_{\alpha i \sigma }^\dagger c_{\alpha j \sigma }^{\phantom\dagger}
+\varepsilon_\alpha\sum_{i \sigma}c_{\alpha i\sigma}^\dagger c_{\alpha i \sigma }^{\phantom\dagger} \,,\\
{H}_{\rm coup}&=&\sum_{\langle i j\rangle \alpha \sigma} v_\alpha\left(c_{i\sigma}^\dagger c_{ \alpha j\sigma}^{\phantom\dagger} + h.c.\right) \, ,\end{aligned}$$ where $ \langle i,j\rangle$ indicates neighboring sites, $c_{i,\sigma}^\dagger$ creates an electron at the $i$-th site of the correlated layer with spin $\sigma=\uparrow,\downarrow$, and $n_{i\sigma}=c_{i \sigma}^\dagger c_{i \sigma}^{\phantom\dagger}$ denote the corresponding occupation number operators. The analogous fermionic creation/annihilation operators of lead $\alpha$ are labeled by $c_{ \alpha i \sigma}^\dagger/c_{ \alpha i \sigma}^{{\phantom{\dagger}}}$. For the particular parameters see [Fig.\[fig:model\]]{}.
Method {#sec:method}
======
In the following we outline the method only briefly and for details we refer to [Ref. [@ar.kn.13; @do.nu.14; @ti.do.15u]]{}.
Nonequilibrium dynamics are conveniently formulated in terms of Keldysh Green’s functions [@kad.baym; @schw.61; @keld.65; @ha.ja; @ra.sm.86; @le.da.06], whereby for the steady state limit it suffices to consider $2\times2$ objects on the Keldysh contour $$\underline G =\left(
\begin{array}{cc}
G^R & G^K \\
0 & G^A
\end{array}
\right) \,,
\label{eq:KeldyshGF}$$ which we denote by an underscore. Here, the retarded $G^R$ and the Keldysh component $G^K$ are independent functions in a generic nonequilibrium situation, and the advanced part is given by $G^A=(G^R)^\dagger$. The spectral function is defined as usual: $A = i/2\pi\,(G^R-G^A)$.
Since the model outlined in [Sec.]{}\[sec:model\] is translationally invariant in the in-plane direction, it is convenient to introduce the corresponding momentum variable ${\bf k}_{||}$. Furthermore, time translational invariance applies in the steady state limit and the governing equations can be formulated in the frequency domain $\omega$. With this the Green’s function of the correlated layer is given in terms of Dyson’s equation by $${\underline G}^{-1}(\omega, {\bf k}_{||}) ={\underline g}^{-1}_0(\omega, {\bf k}_{||}) - \sum_{\alpha=l,r}v_\alpha^2~{\underline g}_\alpha(\omega, {\bf k}_{||}) - {\underline{\Sigma}}(\omega, {\bf k}_{||}) \,.
\label{eq:Dyson}$$ Here, the decoupled non-interacting Green’s function of the layer is denoted by ${\underline g}_0(\omega, {\bf k}_{||})$, and those of the leads by ${\underline g}_\alpha(\omega, {\bf k}_{||})$. The non-interacting Green’s functions are known exactly but the determination of the self-energy of the correlated layer ${\underline{\Sigma}}(\omega, {\bf k}_{||})$ is challenging and one has to resort to approximations. In particular we neglect spatial correlations and restrict ourselves to a local self-energy ${\underline{\Sigma}}(\omega, {\bf k}_{||}) ={\underline{\Sigma}}(\omega)$, as usually done in the context of DMFT [@ge.ko.96; @me.vo.89; @geor.04; @voll.12; @sc.mo.02u; @fr.tu.06]. Within DMFT, the local quantity ${\underline{\Sigma}}(\omega)$ is determined by mapping the lattice problem onto an equivalent quantum impurity model, with the same local parameters $U$ and $\varepsilon_c$. However, the bath degrees of freedom of the impurity model depend on ${\underline{\Sigma}}(\omega)$, such that a self-consistent solution is needed, which is commonly obtained in an iterative manner. The bath for the impurity model is fully specified by the hybridization function $$\underline{\Delta}_\mathrm{ph}(\omega)=\underline{g}_0^{-1}(\omega) - \underline{G}^{-1}_{\rm loc}(\omega)-\underline{\Sigma}(\omega)\,,
\label{eq:D_ph}$$ where $\underline{g}_0^{-1}(\omega)$ is the non-interacting Green’s function of the disconnected impurity and the local Green’s function is obtained by $$\underline{G}_{\rm loc}(\omega)=\int\limits_{\rm BZ} \frac{d{\bf k}_{||}}{(2\pi)^2}\underline{G}(\omega,{\bf k}_{||}) \,.
\label{eq:G_loc}$$ In order to solve the nonequilibrium impurity problem we resort to AMEA, cf. [@ar.kn.13; @do.nu.14], which maps the original impurity problem onto an auxiliary one, with a finite number of bath sites $N_B$ and additional Markovian environments. The resulting open quantum system is described by a Lindblad equation and is small enough to be solved accurately by numerical techniques. In contrast to other approaches, the parameters of the Lindblad equation are not obtained perturbatively but through an optimization procedure. In particular, we consider the hybridization function of the auxiliary system ${\underline{\Delta}_\mathrm{aux}(\omega)}$ and vary the auxiliary bath parameters in order to minimize the difference to the physical hybridization function ${\underline{\Delta}_\mathrm{ph}(\omega)}$, [Eq.(\[eq:D\_ph\])]{}. In the limit of large $N_B$ the approach becomes exact but even for small values of $N_B$ we obtain ${\underline{\Delta}_\mathrm{aux}(\omega)}\approx {\underline{\Delta}_\mathrm{ph}(\omega)}$ to very good approximation. Typically, an exponential convergence with increasing $N_B$ is achieved. After the mapping procedure, which can be done in a $U=0$ calculation, the interacting impurity problem is solved. For this we introduced two different strategies in previous work: On the one hand, an implementation of AMEA which makes use of Krylov space methods, cf. [Ref. [@do.nu.14]]{}, and on the other hand, a matrix product states based solution, cf. [Ref. [@do.ga.15]]{}. The latter allows for a highly accurate solution of the impurity problem but requires a rather large amount of CPU time. The former is not as accurate at low temperatures but faster in many cases and is used in the present work. The Krylov space solver enables us to consider up to $N_B=6$ bath sites, which suffices to treat cases with strong Kondo or quasiparticle excitations reliably, cf. [Ref. [@do.nu.14; @ti.do.15u]]{}. Once the many-body problem of the auxiliary system is solved one obtains an approximation for the self-energy $$\underline{\Sigma}(\omega)=\underline{G}_{{\rm aux},0}^{-1}(\omega)-\underline{G}_{\rm aux}^{-1}(\omega) \,,
\label{eq:Sigma}$$ from the knowledge of the non-interacting and the interacting auxiliary Green’s functions. By inserting $\underline{\Sigma}(\omega)$ from [Eq.(\[eq:Sigma\])]{} into [Eqs.(\[eq:Dyson\])]{},(\[eq:D\_ph\]),(\[eq:G\_loc\]) we close the DMFT cycle and iterate until a self-consistent point is reached.
Results {#sec:results}
=======
{width="47.00000%"} {width="47.00000%"}
{width="95.00000%"}
{width="95.00000%"}
In the following we present results for the transport and spectral properties of a correlated layer in a nonequilibrium steady state situation. The particular model is defined in [Sec.]{}\[sec:model\] and depicted in [Fig.\[fig:model\]]{}. We place special emphasis on the low-energy properties and consider cases with rather low bias voltages $\phi$. To investigate the role of resonant quasiparticle excitations, different temperatures $T = T_l = T_r$ are introduced in the leads. In contrast to previous work [@ti.do.15u], a nonzero temperature is considered in the leads and additionally, the hopping parameters are chosen such that each correlated site has an equal hopping amplitude to neighboring sites ($v_l = v_r = t_c$). By this we expect to be in a regime in which a competition between the physics of an isolated 2D layer and the one of single quantum dots occurs.
In [Fig.\[fig:current\]]{} the current-voltage characteristics [$j(\phi)$ ]{}together with the differential conductance [$\partial j / \partial \phi$ ]{}are displayed for various $T$. At low bias voltages $\phi \lesssim 2\,t_c$ the temperature has large influence on [$j(\phi)$ ]{}and [$\partial j / \partial \phi$]{}, whereas for larger voltages $\phi \gtrsim 2.5\,t_c$ we find quantitatively similar current values for all of the temperatures. This is in close analogy to what was observed in [Ref. [@do.ga.15]]{} for the nonequilibrium properties of a quantum dot. In [$\partial j / \partial \phi$ ]{}we obtain for $\phi \lesssim 1\,t_c$ a strong dependence on $T$. Again, the behavior is similar to what is known from Kondo systems [@kr.sh.12; @zh.ka.13; @pl.sc.12; @re.pl.14]. However, the accuracy of the present calculations does not permit us to investigate the scaling with $\phi$ in detail, in particular if a logarithmic dependence as in quantum dots is present. Still, one observes that the experimentally well-accessible quantity [$\partial j / \partial \phi$ ]{}exhibits a strong temperature dependence and clear signatures of a Kondo-like behavior are visible for low $T$. From a technical point of view one should note that slight kinks or jumps in [$j(\phi)$ ]{}are present, at different values of $\phi$ for the various $T$. These artefacts originate in the mapping procedure and appear at values $\phi = \phi_c$ at which more than one parameter set for the auxiliary system minimizes the difference to the physical hybridization function. Usually, one of these minima is better for $\phi<\phi_c$ while the other one for $\phi>\phi_c$. Therefore, such a crossing of minima leads to an abrupt change in parameters of the auxiliary system and may result in a slight shift of spectral weight, cf. [Ref. [@do.nu.14]]{}. In general, this effect is smallest in situations in which the difference between ${\underline{\Delta}_\mathrm{aux}(\omega)}$ and ${\underline{\Delta}_\mathrm{ph}(\omega)}$ is small in any case and thus, is reduced upon increasing the number of bath sites $N_B$.
A more detailed picture of the state of the system is obtained by investigating the spectral function, see [Fig.\[fig:spectral\_function\]]{}. In the equilibrium case, $\phi=0$, we find a strong quasiparticle excitation at $\omega=0$ for $T = 0.2\,t_c$, which is attenuated with increasing temperature ($T = 0.4\,t_c$ and $T = 0.8\,t_c$) and is completely suppressed for $T = 1.6\,t_c$.[@footnote_friedel] Especially interesting are the low temperature situations $T = 0.2\,t_c$ and $T = 0.4\,t_c$ in which a strong zero frequency excitation is visible in equilibrium, and for which we observe with increasing $\phi$ at first a reduction of the peak height before a splitting sets in. Similar to a quantum dot system [@do.nu.14; @do.ga.15], we find two resonant excitations at the Fermi-energies of the two leads and thus a linear splitting with $\phi$. For the case of $T = 0.2\,t_c$ the excitations are still clearly visible at rather large voltages up to $\phi \approx 5\,t_c$. The results for $T = 0.8\,t_c$ and $T = 1.6\,t_c$ reveal dissimilar spectral functions at low bias voltages, but surprisingly, the obtained current values in [Fig.\[fig:current\]]{} are comparable. The reason for this is that the high temperatures average out details in $A(\omega)$ to large extent.
The presence of resonant excitations is also clearly visible in the retarded part of the self-energy, displayed in [Fig.\[fig:self\_energy\]]{}. For $\phi=0$ we find for temperatures up to $T = 0.8\,t_c$ a local minimum in $-\Im\{\Sigma^R(\omega)\}$ at $\omega=0$, indicating a quasiparticle excitation. But only in the cases $T = 0.2\,t_c$ and $T = 0.4\,t_c$ the temperature induced decoherence is weak enough to obtain a splitting of the single minimum when increasing the bias voltage. In contrast, the self-energy for $T = 1.6\,t_c$ is rather featureless and only weakly dependent on the bias voltage.
Conclusions {#sec:conclusio}
===========
In this work we investigated the steady state properties of a correlated layer sandwiched between two metallic leads at different chemical potentials, induced by an externally applied bias voltage. For this we made use of a nonequilibrium DMFT approach together with AMEA as impurity solver. In addition to previous work [@ti.do.15u], we studied the influence of temperature on the transport characteristics and on the bias-dependent spectral function, with focus on the low-bias regime. The parameters of the system were chosen such that a certain direction was not preferred in advance. In particular, all of the hopping amplitudes of a correlated site to its neighbors were of equal size. From investigating the spectral function and the differential conductance as a function of bias voltage and for various temperatures, we found that the considered system bore close analogy to the case of a single quantum dot. A result like this could be expected when considering the limit in which the hopping parallel to the 2D layer is much smaller than the longitudinal one, regarding the layer and the leads. But, since the hoppings to correlated sites were isotropic the result is not intuitive nor trivial.
Acknowledgements {#ac .unnumbered}
================
We acknowledge valuable discussions with Martin Nuss, Markus Aichhorn, Michael Knap, and Wolfgang von der Linden. This work was supported by the Austrian Science Fund (FWF): P24081, P26508, as well as SFB-ViCoM project F04103, and NaWi Graz. The calculations were partly performed on the D-cluster Graz and on the VSC-3 cluster Vienna.
References {#references .unnumbered}
==========
|
---
abstract: 'Decision taking is discussed in the context of the role it may play for various types of agents, and it is contrasted with action determination. Some remarks are made about the role of decision taking and action determination in the ongoing debate concerning the reverse polder development of the hertogin Hedwige polder.'
author:
- |
Jan A. Bergstra\
[Section Theory of Computer Science, Informatics Institute,]{}\
[Faculty of Science, University of Amsterdam, The Netherlands.]{}[^1]
title: Decision Taking versus Action Determination
---
Introduction {#sec:Intro}
============
In [@Bergstra2011a] I have proposed an explanation of decision taking (DT) and the way it is embedded in and differs from decision making (DM). In [@Bergstra2012a] the distinction between decision taking and various forms of voting and promising is elaborated in the context of an investigation of the concept of decision taking as a service.[^2] In this paper I will make a modest attempt to link the work of [@Bergstra2011a] and [@Bergstra2012a] with some topics of Frans Groen’s research. Frans Groen’s work mostly had a focus on methods and techniques that an artificial agent can use in order to choose an appropriate behavior in a physical environment. For instance in [@JansenMHG2005] colour vision is used in such a way that it may improve the obstacle avoidance capability of an artificially controlled car, an obvious necessity for automated off-road driving, and in [@DorpGroen2003] radar is used to improve upon the detection of walking humans above the capabilities that human agents have to that end. The technical focus of his research work has been on a wide range of sensor systems and on corresponding processing methods for sensor data.
Each of these techniques can be, and ultimately will be, incorporated in autonomous systems governed by software agents. This perspective was a fundamental theme for Frans Groen’s work for the last 20 years. Software agents may operate individually or in groups. Software agents need to trigger actions and to make choices and may be involved in collective decision making such as described in [@GroenSKP2007]. These choice processes are comparable to natural human action taking place in real time, see e.g. [@MahalelZK1985].
Decisions are scarce
--------------------
Now I hold that when operating in real time both artificial and human agents don’t engage in decision taking in spite of the fact that they may use sensor data and background information to compute what action to perform next. By consequence I hold that each of the contributions of Frans Groen to the theory, the technology, and the practice of autonomous systems can be profitably incorporated in applications without playing a role in decision taking. Decision taking may even be considered a redundant concept for artificial autonomous systems. Further decision taking is very difficult for groups of human agents. In addition I hold that animals are incapable of decision taking. Together this means that as a fraction of choice processes on future behavior decision taking is relatively scarce.
These positions can be defended only on the basis of a sufficiently exotic definition of what it means to take a decision. In [@Bergstra2011a] I have argued that (i) a decision is an act of decision taking, performed by an agent (decision taker), operating in a specified role, having explicit intentions, and equipped with an explicit expectation of how its decision will contribute to a realization of these intentions, (ii) a decision produces a decision outcome, which is a tangible piece of information, (iii) the decision outcome may trigger agents in its scope to put it into effect, thus leading to the consequences of the decision outcome, (iv) decision taking constitutes a final phase of decision making, (v) decision taking plausibly involves carrying out a protocol, the preparation of which is a task of the decision making process, (vi) determination of the content of the decision outcome and of parts of it belongs to decision making (and in particular to what is termed decision preparation in [@Bergstra2011a]) rather than to decision taking.[^3]
The garbage can model of decision making of [@CohenMO1972] fits well in this view of decision taking. In that model decision making produces plans that may constitute candidate solutions to forthcoming problems, while decision taking singles out and activates candidate solutions that are considered proper solutions for actual problems, at appropriate moments.
Action determination, a universal behavioral pattern
----------------------------------------------------
If an agent engages in an action on the basis of some reflection about past and recent data as well as a portfolio of rules of conduct it can be said that the agent has determined a plan or an action for immediate effectuation. Determining what to do next is a frequent process for any agent, including artificial agents. I will speak of action determination. For a human agent it is common to say that the agent makes up his or her mind concerning an action to come.
Action determination can be phrased in different ways: choosing an option, where the option is chosen from a menu of actions; or selecting an alternative from a set of alternatives. Action determination leads to the determination of an action. The action that is determined may also be referred to as the determination outcome.[^4] Unlike with decision taking as conceived in [@Bergstra2011a], the consequences of natural decision taking are in general not brought about through the intermediate stage of a decision outcome.
For a choice process proceeding in real time (such as the driving activity of [@MahalelZK1985]) the phrase “natural decision making” has been coined (see e.g. [@ThwaitesWilliams2006] for an application or [@ZsambokKlein1997] for an introduction of the notion). The phrase “natural decision taking” provides better consistency with the terminology of [@Bergstra2011a]. But I prefer to speak of action determination rather than of natural decision taking, because action determination suitably highlights the real time character of the choice process involved.[^5]
Some examples of the use of “action determination” may be helpful: (i) a car driving agent who notices the approach of a traffic light that has just turned red must determine whether to stop (and if so whether to stop abruptly (and if so whether stop before the white demarcation line linked with the traffic light or to tolerate passing over that line some fraction of a car length), or to stop gently), or to proceed (and if so, at an accelerated speed, at a constant speed or at a lower speed). (ii) When no service is being offered in a shop to an agent acting in the role of a candidate customer the agent must determine whether to wait (and if so, to perform that determination repeatedly until either the agent has been served or in determines that the waiting must come to an end), or to leave the shop.
Real time action determination
------------------------------
Action determination mainly takes place in real time. Real time requirements often preclude the construction of an information carrying outcome as an intermediate stage for action determination.
Sensor data from devices monitoring the physical environment can play a role in decision preparation, probably after sampling and subsequent statistical processing. Instead during decision taking, which is a relatively slow process heading for an outcome consisting of a piece of data, the usage of real time collected sensor data is less plausible. The process is too slow for making relevant use of real time measurements and too fast for systematic sampling with subsequent statistical evaluation. Nevertheless, real time data from social media may be taken into account in a decision taking protocol, for instance by blocking a decision if there is a large number of opponents online.
Action determination in real time can be performed rationally if it is based on some theory about what is to be optimized. Limitations of an agent’s computational capacity will give rise to bounded rationality in real time action determination.
Action determination and social choice
--------------------------------------
Action determination need not involve decision taking but in some cases decision taking may be used as a way to achieve action determination. Action determination can be performed on the basis of rational rules in which case one may speak of rational action determination.
Action determination may also be implemented by means of social choice. Action determination is common to all forms of agents and also it is also a common process for groups of humans. For all types of agents action determination seems to take place more frequently than decision taking. Phrased differently, action determination is less scarce than decision taking.
Mechanical aspects of decision taking
=====================================
Following [@Bergstra2011a] it is essential for a decision that it produces a decision outcome as an intermediate product, which by itself, and solely based on the role of the decision taker, not on the decision taker’s subsequent actions, influences its environment to enact consequences that are in conformance with the decision taker’s expectations and from which the decision taker expects a positive contribution to the realization of its objectives.
As was mentioned in [@Bergstra2011a] I consider it implausible that an animal agent produces an intermediate piece of information encoding its intentions and expectations, and I doubt that large groups of humans can entertain intentions or have expectations, from which I infer that decision taking is less plausible for large human groups. the case of small human groups is different. I assume that such groups can have intentions and objectives and that it makes perfect sense to think of their decision taking activity.
Software agents choose without deciding
---------------------------------------
For software agents the argument against decision taking being a prominent behavioral pattern is slightly different than the arguments given for animals and for large human groups. Taking decisions requires an explicit maintenance of intentions and expectations, and a setting where its own actions as well as that of other agents are based upon inspection of a portfolio of decision outcomes. Explicit maintenance of decision outcomes as information objects is certainly an option for the design of a software agent, but it is definitely not an essential feature for artificially intelligent behavior.
Further a decision taker must operate in the context of a role which provides weight to its decision outcomes for other agents. That an artificial agent acts within a role with authority is hard to imagine, unless it impersonates a human agent acting within such a role.[^6]
For these reasons I consider decision taking to be an advanced feature for artificial agents, though not an inaccessible feature. Action determination or making choices from menus of alternatives, which is often viewed as the core of decision making (a viewpoint that I do not share), however constitutes an essential feature of artificial agent behavior.
Software agent control: a multi-scale phenomenon {#Msp}
------------------------------------------------
The behavioral control of a software agent is a phenomenon which requires a multi-scale approach: (i) at the hardware level some controls of a software controlled agent constitute the lowest scale which is out of reach of the software, (ii) at the software level action determination may be programmed by means of simple algorithms, and (iii) at a higher level (scale) action determination may involve explicit reasoning, (iv) action determination may depend on the generation of intermediate pieces of information thus featuring artificial decision taking, (v) at the largest (highest) scale of its functionality a software agent interact with other software agents in order to participate in artificial social choice mechanisms, and (vi) at yet another scale a human engineer takes decisions about the deployment of software for an embedded software agent (and so on).
I have come to suspect that several concepts in computing, among which the notion of a computer program, are essentially multi-scale notions in the sense that an account of those concepts in terms of a single scale turns out to be defective invariably. An intriguing example of a multi-scale concept is money. To see this one may notice that at the scale of a single household it is clear that there may be a lack of money, but at the level of an entire economy speaking of that kind of shortage makes much less sense.
Multi-threading and short-circuit logic
---------------------------------------
I consider it to be a realistic assumption that a decision taker is involved in a plurality of decision taking threads simultaneously. That form of concurrency can be described adequately with strategic interleaving as specified in [@BergstraMiddelburg2007]. If a protocol for taking a particular decision is applied that application may amount to the effectuation of an instruction sequence (e.g. as in [@BergstraLoots2002a; @BergstraMiddelburg2012]) in a suitable execution architecture (see [@BergstraPonse2007a]), involving the evaluation of composite conditions (as in [@BergstraPonse2011a]), thus generating a sequential thread.
The evaluation of composite conditions in real time may bring with it a dynamic setting where for evaluating a single composite condition the same boolean value needs to be successively evaluated several times and takes different values. That situation leads to so called reactive valuations as proposed in the short-circuit logic of [@BergstraPonse2012a].
An important consequence of a decision taking agent being engaged in a plurality of decision taking threads concurrently is that a serious failure in one thread may induce a premature exit from other threads as a side-effect. This typically happen with agents in a political role and who may loose support concerning their handling of a particular case thereby negatively impacting on their competences in other areas of decision taking.
Protocols and modularization
----------------------------
I view decision taking as a form of modularization of organizational behavior. Stated differently: decision taking is an organizational pattern or feature. As a feature it may have different degrees of visibility in an organization’s operation. For instance, the preparation for the so-called “instellingsaccreditatie” (institutional accreditation), at the time of writing ongoing for the University of Amsterdam, requires an effort to increase the amount of decision taking that University’s processes at many levels.
Changing an organization by installing new patterns of decision making and decision taking can only be done if a prepared portfolio of decision taking protocols is available. The design of decision taking protocols in terms of instruction sequences making use of short-cicuit logic, and of strategic interleavings for threads resulting from the effectuation of such protocols, is a challenge for forthcoming research. This task is especially interesting in connection with human decision taking. For instance when buying a second hand car it may be practical to have a prepared protocol at hand for checking all relevant matters before signing.[^7]
While decision taking can be understood as an optional architectural feature for organizational design, action determination is unavoidable and to propose its understanding as an aspect of a software agent’s software architecture would be misleading.
Group decisions, a controversial example {#HP}
========================================
Decision taking and social choice plays a central role in the behavior of human groups. For instance regarding certain themes in Dutch politics it is essential to be able to analyze the situation in terms of social choice, decision making, and decision taking. An interesting case is the long standing debate about whether or not to return the hertogin Hedwige polder to the influence of tidal movements in the Westerschelde. I will look into that case in some detail.
Reverse polder development of the Hedwige Polder
------------------------------------------------
For the plan of this paper it would be preferable to use an example which (i) contrasts natural decision taking as mentioned above that makes use of real time sensor data with decision taking as meant in [@Bergstra2011a], and which (ii) at the same time illustrates the contrast between decision taking and events of social choice. Unfortunately the example discussed is rather skewed towards the second aspect, but its timeliness makes up for that disadvantage.
The case that I will discuss in some detail concerns a long standing policy issue between Belgium and the Netherlands: the accessibility of the port of Antwerp. Nowadays the matter is significantly complicated by three aspects: (i) the need to accommodate larger ships, especially container carriers, (ii) the need to take ecological consequences into account much more seriously then before, and (iii) the abundant use of social computing that produces immediate impact figures on any decision taker’s popularity.
In [@Floor2009] and [@Tilburg2010] meticulous accounts are presented regarding the remarkable number of actions, choices, decisions, and events of social choice and action determination that have taken place since the so-called second improvement ([*tweede uitdieping*]{}) of the fairway to Antwerp has been requested by the Belgian government some 20 years ago. The key issue is that improving the accessibility of the Westerschelde for larger ships leads to ecological costs which can, and according to important voices, must be compensated for by reverse polder development ([*ontpolderen*]{} in Dutch), that is moving an existing polder outside the dykes which are protecting it against the open waters of the Westerschelde, in order to obtain a specific type of wetlands considered to be of essential ecological value, thus preserving the volume of such wetlands in the Schelde estuary. This is technically achieved by placing new dykes behind the polder and subsequently removing the dykes in front of it at least partially. The hertogin Hedwige polder in Zeeuws Vlaanderen has become a focus of attention for this matter. Local farmers and land owners strongly oppose the process of inundating a polder currently providing valuable farmland, mainly on emotional grounds, and claiming that returning a polder to the sea goes against a tradition of centuries as well as against local and even regional economic interests.
A cascade of so-called decisions has produced a state in which it has become almost impossible to understand the merits of reverse polder development relative to the merits of keeping the polders and their usage unchanged. At the time of writing this paper it is unknown what the fate of the Hedwige polder will be. It is probably difficult to predict what will happen for all agents involved. The matter is obviously problematic for the Dutch government and it may lead to a serious conflict with the port authority in Antwerp. Such conflicts have in fact existed for centuries in the past, only to be resolved by Napoleon in 1792. A treaty dating back as a long as 1839 about enduring guarantees for the accessibility of the harbor of Antwerp and its maintenance still impacts on these matters.
Justification of some candidate decision outcomes
-------------------------------------------------
Regarding this case I have come to several conclusions in connection with decision taking and action determination, in particular regarding the possible justification of some candidate decision outcomes. These are my personal and subjective viewpoints in a case where other persons have chosen different positions about various aspects of the matter.
1. For a clear analysis of the political situation concerning the Hedwige polder and its past it is useful to distinguish action, choice, and decision, as well as to maintain a distinction between decision taking, decision making, and social choice, and to distinguish decision outcome from social choice outcome.
2. A recent development is that some forms of social choice can be realized in real time by means of statistical and automatic analysis of social media traffic. This is “socio-technical sensor technology”$\!$, a part of “socio-technical informatics” (or “socio-technical informaticology”). Socio-tech-nical informatics might be productively grouped together as a theme with classical sensor technology that has been the focus of work of Frans Groen. Results of socio-technical sensing, which in The Netherlands often take the form of quite arbitrary and disputable polls about hypothetical voting outcomes for the Dutch parliament, seem to impact decision processes and events of social choice aimed at real time action determination.[^8]
3. It seems to be a hidden assumption made by many actors relevant to this issue that full or partial reverse polder development of the Hedwige polder can only take place after a decision to do so has been taken, and has not been itself reversed in subsequent legal action initiated by its opponents. This assumption is wrong if the concept of decision of [@Bergstra2011a] is used. It is wrong because it might be the case that only events of social choice come into play which may qualify as action determinations but do not qualify as decisions.
The occurrence of an event of social choice does not imply the existence of an agent with corresponding intentions and expectations, and for that reason an event of social choice may not count as a decision. Human groups tend to produce instances and corresponding outcomes of social choice rather than decisions and corresponding decision outcomes. For human groups (like the Dutch parliament) taking decisions is difficult.
4. It can be maintained, however, that action determination will be required as a precondition for each form of reverse polder development in the Netherlands. The relevant action determination phase may, however, comprise no more than one or more events of social choice each failing to qualify as decision taking.
5. Taking a decision that amounts to reverse polder development of the entire Hedwige polder is justified at this stage.[^9] If the government had been able to insist on taking decisions (that is ministers taking decisions strictly corresponding to their own intentions and expectations and getting the resulting decision outcomes subsequently ratified in parliament), rather than to have their intentions cluttered up by modifications proposed by MP’s with various backgrounds, then a decision outcome of this kind would probably have been obtained at this stage already.
6. Thus I consider a decision outcome that amounts to reverse polder development of the entire Hedwige polder justified. Let full rpdHP abbreviate that (potential) decision outcome, and let partial rpdHP denote an outcome by which a part of the Hedwige polder is preserved. Let no-rpdHP abbreviate the decision outcome that no full or partial reverse polder development of the Hedwige polder is carried out and no further measures are taken to compensate for that. Now I wish to state that:
1. I already had a prior preference for full rdpHP over no-rdpHP and over partial rdpHP based on a “green sentiment” before reading a selection of the available documentation about the matter.
2. I consider the presence (in my mind) of a prior preference for one potential decision outcome over another potential decision outcome not to be in contradiction with the imperative of impartial analysis. The case seems to be comparable to the existence of prior odds in subjective probability theory and Bayesian statistics. By becoming increasingly aware of pieces of information one’s subjective justification of a particular decision outcome as compared to a different candidate decision outcome may be adapted in successive stages. The presence of a prior preference for a possible decision outcome is to be expected if one accepts a method of subjective justification of potential decision outcomes.
3. I do not understand in detail the entire chain of arguments, both legal and ecological, that leads to the claimed necessity of full or partial rdpHP; these arguments are quite technical and have a multi-disciplinary nature, in particular the expected negative ecological consequences of the “tweede uitdieping” are not so easy to grasp.[^10] The latter difficulty perhaps indicates a shortcoming in the communication of the matter by the proponents of full or partial rdpHP.
4. There may be a speculative link of this theme with sensor technology as follows: by performing a detailed investigation, probably technically supported with extensive use of sensing devices, of water and sand movements in the estuary of the Schelde, a method might conceivably be discovered which allows to make the fairway to Antwerp accessible for larger ships and at the same time to prevent degradation of the ecosystem. Mobile elements in open water, reacting on the presence of ships, as well as dynamic and natural reconfiguration of tidal flows may be instrumental for such solutions. At present, however, such speculative options cannot be brought forward as an argument against full or partial rpdHP.
5. As an owner of a modest amount of farmland some 7 kilometers to the west of the Hedwige polder I may have a personal interest in the matter. However, I am not sure about the implications of that connection; my stated preference deviates from the ZLTO[^11] position to which I might supposedly be attracted in the mentioned capacity, and the implications of full or partial rpdHP on the value of farmland in the area are difficult to estimate, and so are the implications of choosing a position that deviates from that of the local agricultural sector which, like ZLTO, opposes to both full and partial rdpHP.
7. Given the incredible costs of decision making in this matter, secretary Bleker’s recent proposal for a partial reverse polder development of the Hedwige polder, proposed in combination with a range of other probably less problematic developments, constitutes a sounds step of (true) decision preparation.
Regarding that proposal it is amazing that an additional cost of some 150.000.000 Euro is considered acceptable in order to avoid the reverse polder development of some 2.000.000 square meters, thus protecting two thirds of the Hedwige polder against inundation. It is hardly conceivable that the opponents of reverse polder development would consider spending that much money on the preservation of agricultural area if they could not force the state to pay this excessive sum. Rather than being dissatisfied with the result (if Bleker’s plan came true), its opponents should be grateful for such a formidable investment made by the Dutch government to meet their complaints, if only partially.
Bleker’s proposal has subsequently been rejected in the Dutch parliament in May 2012 and the issue has been postponed until after the next elections thus leaving matters wide open again.
Decision taking as a service (DTaaS)
------------------------------------
In [@Bergstra2012a] I have outlined that in some cases decision taking can be offered as a service by a service provider to a customer who was in charge of the same kind of decisions before outtasking that part of its decision taking load. Perhaps it will be helpful to have the decision taking concerning the future of the Hedwige polder outtasked from political functionaries to some external agents.
The search for useful majorities in parliament has become infected with issues like the so-called Euro crisis which are very distant from finding convincing solutions to ecological matters that don’t reach beyond Belgium and the Netherlands. By using DTaaS accidental dependencies of DT concerning the Hedwige polder on unrelated issues may be prevented.[^12]
Decision preparation as a service (DPaaS)
-----------------------------------------
Using the equation DM = DT + DP (decision making = decision taking + decision preparation) from [@Bergstra2011a] it is also plausible to consider the service casting (see [@Bergstra2012a]) of decision preparation (DP): “decision preparation as a service” (DPaaS) as an option besides service casting of DT (DTaaS).
The following can be remarked about DPaaS: (i) DP can be outtasked and in contrast with DT it can also be outsourced, (ii) DPaaS is very common, outtasking of DP involves the services of consultants of various kinds, outsourcing of DP involves moving teams of specialists to other organizations from where they will provide their expertise wrapped in a service, (iii) if DPaaS is used in combination with DTaaS it seems very important to see to it that different, and independent, service providers are used, (iv) DPaaS in case it comes about through outsourcing of DP activity may lead to a provider lock in, (v) DPaaS has occurred in the Hedwige polder issue to the extent that participants of the DT process may have lost contact with the argumentation used by the relevant DP professionals, (vi) the latter effect, if true, might explain a tendency which seems to exist to drift towards the use of legal arguments by various actors in the DT process.
Some risks for prospective decision takers
------------------------------------------
Two groups in Dutch politics have chosen the position of definite opponents of full or partial rpdHP: SP and PVV. Apparently their “negative view”$\!$, which I consider to be remarkably shallow in its motivation and analysis of the issue, is currently very powerful in terms of its effects in the media and the polls. This influence through various social media is so strong that for other political groups it is not without risk to field any actor who proposes tho opt in favor of full or partial rpdHP simply as a decision outcome compliant with his/her own points of view about the matter when stripped from its legal history.[^13]
The perception of this risk induces a preference for members of other political groups to view the whole issue as being implied by the legal positions obtained in the matter after various episodes of negotiation with the Belgian government and the EU. This preference also opens the door for a display of the anti EU mentality of SP and PVV in the context of this particular case. Making that connection is not very convincing because the negotiations about the fairway in the Westerschelde and its maintenance have been properly conducted.[^14]
Now it is likely that SP and PVV will not take responsibility for actually refusing full or partial rdpHP if it appears that their opponents can see to it that this policy of refusal (when effectuated) will be followed by an assignment of responsibility for its consequences. According to the opponents of SP and PVV these consequences amount to the effects of international legal action by Belgian authorities that will eventually force The Netherlands to act as promised and to compensate in financial terms for significant and unnecessary delays in doing so.
Protected by the expectation that SP and PVV are unlikely to have government responsibility soon, they take the liberty to maintain disputable viewpoints regarding the future of the Hedwige polder. This mechanism must be challenged by their political opponents.
Concluding remarks
==================
Decision taking (as viewed in [@Bergstra2011a]) provides a perspective on the behavior of human agents operating alone or in small groups. Decision taking has been contrasted with action determination. Both action determination and decision taking need to make use of data, either serving as parameters for determination outcomes or decision outcomes, the real time collection and processing of which has been the research focus of Frans Groen, to whom I want to express my gratitude for many years of effective and pleasant cooperation in “het IvI”.
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[^1]: Author’s email address: [[email protected]]{}. This paper was written in connection with the ceremony at the first of June 2012 in the “Aula” of the University of Amsterdam highlighting the retirement of Frans Groen as a professor of informatics with the Informatics Institute of the Faculty of Sciences of the University of Amsterdam.
[^2]: In [@Bergstra2011a] it is found that at least in principle decision taking can be outtasked (see [@BDV2011c]), but that it cannot be outsourced, thus contradicting a suggestion to that end made in [@Valdman2010].
[^3]: An example of a protocol element of a decision taking thread occurs when selling a home (that is just before transferring economic ownership in a formal session): (i) one needs to check that the property has been properly insured by the buyer, (ii) one needs to check having available all keys, (iii) one needs to have available all current metering data concerning water and various forms of energy, (iv) the property has been brought in the required state, (v) one needs to have passports or other means of identification available for all sellers, or otherwise have transferred their representation to someone else who will attend the session, (vi) a bank account has been provided to the solicitor which can accommodate the sum that will be transferred once the legal ownership has been transferred as well.
[^4]: Unlike a decision outcome an action determination outcome is not primarily a piece of information. Perhaps a more systematic terminology is obtained when “action determining” is used instead of “action determination”. Then it may be said that a determination is an act of action determining, just as a decision is an act of deciding (decision taking).
[^5]: Process algebra (see [@BaetenBastenReniers2009]) can be understood as a theory of action determination with a primary focus on (i) immediate effects of actions in terms of process interaction (communication), (ii) external visibility of actions (abstraction), (iii) enabling of actions (encapsulation), and (iv) preferences between actions (priorities).
[^6]: I am limiting attention to artificial agents that are perceived as being artificial by all other agents which they are interacting with.
[^7]: Signing a contract for buying a car certainly qualifies as taking a decision, with the contract playing the role of the decision outcome.
[^8]: In [@DorpGroen2003] radar is used to see walking persons through a wall, an instance of observation outside the range of capabilities of human observers. That issue may not be so distant after all from measuring a persons preferences about a policy issue by asking questions about seemingly unrelated matters.
[^9]: Some recent results of social choice that have been produced by the Dutch parliament in this matter have an interesting complexity from a logical point of view. For instance the following package: (i) to start with improving the waterway, (ii) and at the same time to prepare for reverse development of the Hedwige polder, and (iii) concurrently to look for alternatives which allow to preserve the Hedwige polder, which (iv) are given priority over reverse polder development when considered sufficiently attractive. Abstraction from the many internal steps involved in the threads of these activities is an implicit assumption when working at the level of abstraction of a legislator. A process theory sufficiently powerful to “specify” the plan prescribed by this outcome of social choice must combine concurrency, composition of alternatives, sequential composition, conditions, priorities, and abstraction. Such process theories are rare, an example of a process model which meets these requirements, so-called orthogonal bisimulation semantics, has been given in [@BergstraPonseZ2003a].
[^10]: Ecosystem preservation seems to be a multi-scale concept (see \[Msp\]), a feature that creates significant confusion.
[^11]: Zeeuwse Land en Tuinbouw Organisatie.
[^12]: At the time of writing all EU related policy matters, including compliance with EU regulations on ecosystem preservation, are overshadowed by the so-called Euro crisis. We move through “Grexiting times”$\!$, with significant excitement generated by a possible forthcoming exit of Greece (called Grexit by some) from the Eurozone. The multi-scale aspect of money (see \[Msp\]) seems to be one of the causes of this excitement. Those who claim that Greece simply must get its budget in order (looking at Greece at the scale of an ordinary household) hardly understand those who claim that the Euro system at large is at stake (with Greece constituting no more than a temporary center and highlight of problems that are much more widespread) and that artificial restrictions on the availability and usage of money need to be addressed with higher priority.
[^13]: The appearance of such actors is required if decisions and decision outcomes as meant in [@Bergstra2011a] are to be produced.
[^14]: As it seems SP and PVV make use of an interesting argument in order to preserve the consistency of their position: the introduction of a labeling of the individuals who negotiated about these matters in the last 15 years. These are supposed to have been urban intellectuals who displayed an insufficient grasp of the business logic of farming.
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---
abstract: 'The pseudo-periodicity of voiced speech can be exploited in several speech processing applications. This requires however that the precise locations of the Glottal Closure Instants (GCIs) are available. The focus of this paper is the evaluation of automatic methods for the detection of GCIs directly from the speech waveform. Five state-of-the-art GCI detection algorithms are compared using six different databases with contemporaneous electroglottographic recordings as ground truth, and containing many hours of speech by multiple speakers. The five techniques compared are the Hilbert Envelope-based detection (HE), the Zero Frequency Resonator-based method (ZFR), the Dynamic Programming Phase Slope Algorithm (DYPSA), the Speech Event Detection using the Residual Excitation And a Mean-based Signal (SEDREAMS) and the Yet Another GCI Algorithm (YAGA). The efficacy of these methods is first evaluated on clean speech, both in terms of reliabililty and accuracy. Their robustness to additive noise and to reverberation is also assessed. A further contribution of the paper is the evaluation of their performance on a concrete application of speech processing: the causal-anticausal decomposition of speech. It is shown that for clean speech, SEDREAMS and YAGA are the best performing techniques, both in terms of identification rate and accuracy. ZFR and SEDREAMS also show a superior robustness to additive noise and reverberation.'
author:
- 'Thomas Drugman, Mark Thomas, Jon Gudnason, Patrick Naylor, Thierry Dutoit'
bibliography:
- 'bare\_jrnl.bib'
title: 'Detection of Glottal Closure Instants from Speech Signals: a Quantitative Review'
---
Speech Processing, Speech Analysis, Pitch-synchronous, Glottal Closure Instant
Introduction {#sec:Intro}
============
-synchronous speech processing is a field of speech science in which the pseudoperiodicity of voiced speech is exploited. Research into the tracking of pitch contours has proven useful in the field of phonetics [@Catford1977] and speech quality assessment [@ITU_T_P862]; however more recent efforts in the detection of Glottal Closure Instants (GCIs) enable the estimation of both pitch contours and, additionally, the boundaries of individual cycles of speech. Such information has been put to practical use in applications including prosodic speech modification [@Moulines1990], speech dereverberation [@Gaubitch2007], glottal flow estimation [@Wong1979], speech synthesis [@HNM], [@DSM], data-driven voice source modelling [@Thomas2009] and causal-anticausal deconvolution of speech signals [@MixedPhase].
Increased interest in glottal-synchronous speech processing has brought about a corresponding demand for automatic and reliable detection of GCIs from both clean speech and speech that has been corrupted by acoustic noise sources and/or reverberation. Early approaches that search for maxima in the autocorrelation function of the speech signal [@Strube1974a] were found to be unreliable due to formant frequencies causing multiple maxima. More recent methods search for discontinuities in the linear production model of speech [@Rabiner1988] by deconvolving the excitation signal and vocal tract filter with linear predictive coding (LPC) [@Makhoul1975]. Preliminary efforts are documented in [@Wong1979]; more recent algorithms use known features of speech to achieve more reliable detection [@Plumpe1999; @Naylor2007a; @Thomas2010b]. Deconvolution of the vocal tract and excitation signal by homomorphic processing [@Chytil2006] has also been used for GCI detection although its efficacy compared with LPC has not been fully researched. Various studies have shown that, while linear model-based approaches can give accurate results on clean speech, reverberation can be particularly detrimental to performance [@Gaubitch2007; @Thomas2007a].
Methods that use smoothing or measures of energy in speech signal are also common. These include the Hilbert Envelope [@Ananthapadmanabha1979], Frobenius Norm [@Ma1994], Zero-Frequency Resonator (ZFR) [@Murty2008] and SEDREAMS [@SEDREAMS]. Smoothing of the speech signal is advantageous because the vocal tract resonances, additive noise and reverberation are attenuated while the periodicity of the speech signal is preserved. A disadvantage lies in the ambiguity of the precise time instant of the GCI; for this reason LP residual can be used in addition to smoothed speech to obtain more accurate estimates [@Naylor2007a; @SEDREAMS]. Smoothing on multiple dyadic scales is exploited by wavelet decomposition of the speech signal with the Multiscale Product [@Bouzid2004] and Lines of Maximum Amplitudes (LOMA) [@Tuan1999a] to achieve both accuracy and robustness. The YAGA algorithm [@Thomas2010b] employs both multiscale processing and the linear speech model.
The aim of this paper is to provide a review and objective evaluation of five contemporary methods for GCI detection, namely Hilbert Envelope-based method [@Ananthapadmanabha1979], DYPSA [@Naylor2007a], ZFR [@Murty2008], SEDREAMS [@SEDREAMS] and YAGA [@Thomas2010b] algorithms. In their corresponding references, all these techniques reported interesting results and were shown to outperform other state-of-the-art methods. Besides, as described in Section \[sec:Methods\], they rely on different approaches: some are based on the speech signal while others focus on the residual signal or an estimate of the glottal source; some use dynamic programming while others exploit a smoothing process. As a consequence, these techniques may have different properties in terms of reliability, accuracy and robustness. They are here evaluated against reference GCIs provided by an Electroglottograph (EGG) signal on six databases, of combined duration 232 minutes, containing contemporaneous recordings of EGG and speech. Performance is also evaluated in the presence of additive noise and reverberation. A novel contribution of this paper is the application of the algorithms to causal-anticausal deconvolution [@MixedPhase], which provides additional insight into their performance in a real-world problem.
The remainder of this paper is organised as follows. In Section \[sec:Methods\] the algorithms under test are described. In Section \[sec:Assessment\] the evaluation techniques are described. Sections \[sec:ExpClean\] and \[sec:Robustness\] discuss the performance results on clean and noisy/reverberant speech respectively. Section \[sec:Complexity\] compares the methods in terms of computational complexity. Conclusions are given in Section \[sec:conclu\].
Methods Compared in this Work {#sec:Methods}
=============================
This Section presents five of the main representative state-of-the-art methods for automatically detecting GCIs from speech waveforms. These techniques are detailed here below and their reliability, accuracy and robustness will be compared in Sections \[sec:ExpClean\] and \[sec:Robustness\]. It is worth noting at this point that all methods assume a positive polarity of the speech signal. Polarity should then be verified and corrected if required, using an algorithm such as [@Polarity].
Hilbert Envelope-based method {#ssec:HE}
-----------------------------
Several approaches relying on the Hilbert Envelope (HE) have been proposed in the literature [@Yegnanarayana1979; @Cheng1989; @Rao2007]. In this article, a method based on the HE of the Linear Prediction (LP) residual signal (i.e the signal whitened by inverse filtering after removing an auto-regressive modeling of the spectral envelope) is considered.
Figure \[fig:HE\_illus\] illustrates the principle of this method for a short segment of voiced speech (Fig.\[fig:HE\_illus\](a)). The corresponding synchronized derivative of the ElectroGlottoGraph (dEGG) is displayed in Fig.\[fig:HE\_illus\](e), as it is informative about the actual positions of both GCIs (instants where the dEGG has a large positive value) and GOIs (instants of weaker negative peaks between two successive GCIs). The LP residual signal (shown in Fig.\[fig:HE\_illus\](b)) contains clear peaks around the GCI locations. Indeed the impulse-like nature of the excitation at GCIs is reflected by discontinuities in this signal. It is also observed that for some glottal cycles (particularly before 170 ms or beyond 280 ms) the LP residual also presents clear discontinuities around GOIs. The resulting HE of the LP residual, containing large positive peaks when the excitation presents discontinuities, and its Center of Gravity (CoG)-based signal are respectively exhibited in Figures \[fig:HE\_illus\](c) and \[fig:HE\_illus\](d). Denoting $H_e(n)$ the Hilbert envelope of the residue at sample index $n$, the CoG-based signal is defined as:
$$\label{eq:CoG}
CoG(n)=\frac{\sum_{m=-N}^N{m \cdot w(m)H_e(n+m)}}{\sum_{m=-N}^N{w(m)H_e(n+m)}}$$
where $w(m)$ is a windowing function of length $2N+1$. In this work a Blackman window whose length is 1.1 times the mean pitch period of the considered speaker was used. We empirically reported in our experiments that using this window length led to a good compromise between misses and false alarms (i.e to the best reliability performance). Once the CoG-based signal is computed, GCI locations correspond to the instants of negative zero-crossing. The resulting GCI positions obtained for the speech segment are indicated in the top of Fig.\[fig:HE\_illus\](e). It is clearly noticed that the possible ambiguity with the discontinuities around GOIs is removed by using the CoG-based signal.
![Illustration of GCI detection using the Hilbert Envelope-based method on a segment of voiced speech. *(a) :* the speech signal, *(b) :* the LP residual signal, *(c) :* the Hilbert Envelope (HE) of the LP residue, *(d) :* the Center of Gravity-based signal computed from the HE, *(e) :* the synchronized differenced EGG with the GCI positions located by the HE-based method.[]{data-label="fig:HE_illus"}](HE_illus2.eps){width="50.00000%"}
The DYPSA algorithm {#ssec:DYPSA}
-------------------
The Dynamic Programming Phase Slope Algorithm (DYPSA) [@Naylor2007a] estimates GCIs by the identification of peaks in the linear prediction residual of speech in a similar way to the HE method. It consists of two main components: estimation of GCI candidates with the group delay function of the LP residual and $N$-best dynamic programming. These components are defined as follows.
### Group Delay Function {#sssec:GD}
The group delay function is the average slope of the unwrapped phase spectrum of the short time Fourier transform of the LP residual [@Yegnanarayana1995] [@Brookes2006]. It can be shown to accurately identify impulsive features in a function provided their minimum separation is known. GCI candidates are selected based on the negative-going zero crossings of the group delay function. Consider an LP residual signal, $e(n)$, and an $R$-sample windowed segment $x_n(r)$ beginning at sample $n$ $$x_n(r)=w(r)e(n+r)~\text{for}~r=0,\dots,R-1$$ where $w(r)$ is a windowing function. The group delay of $x_n(r)$ is given by [@Yegnanarayana1995] $$\label{eq:grdel}
\tau_n(k)=\frac{-\text{d}\arg(X_n)}{\text{d}\omega}=\Re\left(\frac{\tilde{X}_n(k)}{X_n(k)}\right)$$ where $X_n(k)$ is the Fourier transform of $x_n(r)$ and $\tilde{X}_n(k)$ is the Fourier transform of $rx_n(r)$. If $x_n(r)=\delta(r-r_0)$, where $\delta(r)$ is a unit impulse function, it follows from (\[eq:grdel\]) that $\tau_n(k)\equiv r_0\forall k$. In the presence of noise, $\tau_n(k)$ becomes noisy, therefore an averaging procedure is performed over $k$. Different approaches are reviewed in [@Brookes2006]. The *Energy-Weighted Group Delay* is defined as $$d(n)=\frac{\sum^{R-1}_{k=0}|X_n(k)|^2\tau_n(k)}{\sum^{R-1}_{k=0}|X_n(k)|^2}-\frac{R-1}{2}.$$ Manipulation yields the simplified expression $$\label{eq:ew_grdel}
d(n)=\frac{\sum^{R-1}_{r=0}rx^2_n(r)}{\sum^{R-1}_{r=0}x^2_n(r)}-\frac{R-1}{2}$$ which is an efficient time-domain formulation and can be viewed as a centre of gravity of $x_n(r)$, bounded in the range $[-(R-1)/2, (R-1)/2]$. The location of the negative-going zero crossings of $d(n)$ give an accurate estimation of the location of a peak in a function.
It can be shown that the signal $d(n)$ does not always produce a negative-going zero crossing when an impulsive feature occurs in $e(n)$. In such cases, it has been observed that $d(n)$ consistently exhibits local minima followed by local maxima in the vicinity of the impulsive feature [@Naylor2007a]. A *phase-slope projection* technique is therefore introduced to estimate the time of the impulsive feature by finding the midpoint between local maxima and minima where no zero crossing is produced, then projecting a line onto the time axis with negative unit slope.
### Dynamic Programming
Erroneous GCI candidates are removed using known characteristics of voiced speech by minimising a cost function so as to select a subset of the GCI candidates which most likely correspond to true GCIs. The subset of candidates is selected according by minimising the following cost function $$\min_\Omega\sum_{r=1}^{|\Omega|}\boldsymbol{\lambda}^T\textbf{c}_\Omega(r),$$ where $\Omega$ is a subset with GCI candidates of size $|\Omega|$ selected to produce minimum cost, $\boldsymbol{\lambda}=[\lambda_{A}~\lambda_{P}~\lambda_{J}~\lambda_{F}~\lambda_{S}]^T=[0.8~0.5~0.4~0.3~0.1]^T$ is a vector of weighting factors, the choice of which is described in [@Naylor2007a], and $\textbf{c}(r)=[c_A(r)~c_P(r)~c_J(r)~c_F(r)~c_S(r)]^T$ is a vector of cost elements evaluated at the $r$th element of $\Omega$. The cost vector elements are:
- *Speech waveform similarity*, $c_A(r)$, between neighbouring candidates, where candidates not correlated with the previous candidate are penalised.
- *Pitch deviation*, $c_P(r)$, between the current and the previous two candidates, where candidates with large deviation are penalised.
- *Projected candidate cost*, $c_J(r)$, for the candidates from the phase-slope projection, which often arise from erroneous peaks.
- *Normalised energy*, $c_F(r)$, which penalises candidates that do not correspond to high energy in the speech signal.
- *Ideal phase-slope function deviation*, $c_S(r)$, where candidates arising from zero-crossings with gradients close to unity are favoured.
The Zero Frequency Resonator-based technique {#ssec:ZFR}
--------------------------------------------
The Zero Frequency Resonator-based (ZFR) technique relies on the observation that the impulsive nature of the excitation at GCIs is reflected across all frequencies [@Murty2008]. The GCI positions can be detected by confining the analysis around a single frequency. More precisely, the method focuses the analysis on the output of zero frequency resonators to guarantee that the influence of vocal-tract resonances is minimal and, consequently, that the output of the zero frequency resonators is mainly controlled by the excitation pulses. The zero frequency-filtered signal (denoted $y(n)$ here below) is obtained from the speech waveform $s(n)$ by the following operations [@Murty2008]:
1. Remove from the speech signal the dc or low-frequency bias during recording:
$$x(n)=s(n)-s(n-1)$$
2. Pass this signal two times through an ideal zero-frequency resonator:
$$y_1(n)=x(n)+2\cdot y_1(n-1)+y_1(n-2)$$
$$y_2(n)=y_1(n)+2\cdot y_2(n-1)+y_2(n-2)$$
The two passages are necessary for minimizing the influence of the vocal tract resonances in $y_2(n)$.
3. As the resulting signal $y_2(n)$ is exponentially increasing or decreasing after this filtering, its trend is removed by a mean-substraction operation:
$$\label{eq:MeanRemoval}
y(n)=y_2(n)-\frac{1}{2N+1}\sum_{m=-N}^N{y_2(n+m)}$$
where the window length $2N+1$ was reported in [@Murty2008] to be not very critical, as long as it is in the range of about 1 to 2 times the average pitch period $\bar{T}_{0,mean}$ of the considered speaker. Accordingly, we used in this study a window whose length is 1.5$\cdot$$\bar{T}_{0,mean}$. Note also that this operation of mean removal has to be repeated three times in order to avoid any residual drift of $y(n)$.
An illustration of the resulting zero frequency-filtered signal is displayed in Fig. \[fig:ZFR\_illus\](b) for our example. This signal is observed to possess two advantageous properties: 1) it oscillates at the local pitch period, 2) the positive zero-crossings of this signal correspond to the GCI positions. This is confirmed in Fig. \[fig:ZFR\_illus\](c), where a good agreement is noticed between the GCI locations identified by the ZFR technique and the actual discontinuities in the synchronized dEGG.
![Illustration of GCI detection using the Zero Frequency Resonator-based method on a segment of voiced speech. *(a) :* the speech signal, *(b) :* the zero frequency-filtered signal, *(c) :* the synchronized dEGG with the GCI positions located by the ZFR-based method.[]{data-label="fig:ZFR_illus"}](ZFR_illus2.eps){width="50.00000%"}
The SEDREAMS algorithm {#ssec:SEDREAMS}
----------------------
The Speech Event Detection using the Residual Excitation And a Mean-based Signal (SEDREAMS) algorithm was recently proposed in [@SEDREAMS] as a reliable and accurate method for locating both GCIs and GOIs from the speech waveform. Since the present study only focuses on GCIs, the determination of GOI locations by the SEDREAMS algorithm is omitted. The two steps involved in this method are: *i)* the determination of short intervals where GCIs are expected to occur and *ii)* the refinement of the GCI locations within these intervals. These two steps are described in the following subsections.
### Determining intervals of presence using a mean-based signal {#sssec:Mean-based}
As highlighted by the ZFR technique [@Murty2008], a discontinuity in the excitation is reflected over the whole spectral band, including the zero frequency. Inspired by this observation, the analysis is focused on a mean-based signal. Denoting the speech waveform as $s(n)$, the mean-based signal $y(n)$ is defined as:
$$\label{eq:Mean}
y(n)=\frac{1}{2N+1}\sum_{m=-N}^N{w(m)s(n+m)}$$
where $w(m)$ is a windowing function of length $2N+1$. While the choice of the window shape is not critical (a typical Blackman window is used in this study), it has been shown [@SEDREAMS] that its length, which influences the time response of this filtering operation, may affect the reliability of the method.
A segment of voiced speech and its corresponding mean-based signal using an appropriate window length are illustrated in Figs. \[fig:SEDREAMS\_illus\](a) and \[fig:SEDREAMS\_illus\](b). Interestingly it is observed that the mean-based signal oscillates at the local pitch period. If the window is too short, it causes the appearance of spurious extrema in the mean-based signal, giving rise to false alarms. On the other hand, too large a window smooths it, leading to some possible misses. It has been observed in [@SEDREAMS] that maximal reliability is obtained when the window length is between 1.5 and 2 times the average pitch period $\bar{T}_{0,mean}$ of the considered speaker. Accordingly, throughout the rest of this article a window whose length is 1.75$\cdot$$\bar{T}_{0,mean}$ is used for computing the mean-based signal of the SEDREAMS algorithm.
![Illustration of GCI detection using the SEDREAMS algorithm on a segment of voiced speech. *(a) :* the speech signal, *(b) :* the mean-based signal, *(c) :* intervals of presence derived from the mean-based signal, *(d) :* the LP residual signal, *(e) :* the synchronized dEGG with the GCI positions located by the SEDREAMS algorithm.[]{data-label="fig:SEDREAMS_illus"}](SEDREAMS_illus2.eps){width="45.00000%"}
However the mean-based signal is not sufficient in itself for accurately locating GCIs. Indeed, consider Fig. \[fig:RelativePosition\] where, for five different speakers, the distributions of the actual GCI positions (extracted from synchronized EGG recordings) are displayed within a normalized cycle of the mean-based signal. It turns out that GCIs may occur at a non-constant relative position within the cycle. However, once minima and maxima of the mean-based signal are located, it is straightforward to derive short intervals of presence where GCIs are expected to occur. More precisely, as observed in Fig. \[fig:RelativePosition\], these intervals are defined as the timespan starting at the minimum of the mean-based signal, and whose length is 0.35 times the local pitch period (i.e the period between two consecutive minima). Such intervals are illustrated in Fig.\[fig:SEDREAMS\_illus\](c) for our example.
![Distributions, for five speakers, of the actual GCI positions (plot *(b)*) within a normalized cycle of the mean-based signal (plot *(a)*).[]{data-label="fig:RelativePosition"}](RelativePosition.eps){width="45.00000%"}
### Refining GCI locations using the residual excitation {#sssec:Refinement}
Intervals of presence obtained in the previous step give fuzzy short regions where a GCI should happen. The goal of the next step is to refine, for each of these intervals, the precise location of the GCI occuring inside it. The LP residual is therefore inspected, assuming that the largest discontinuity of this signal within a given interval corresponds to the GCI location.
Figs. \[fig:SEDREAMS\_illus\](d) and \[fig:SEDREAMS\_illus\](e) show the LP residual and the time-aligned dEGG for our example. It is clearly noted that combining the intervals extracted from the mean-based signal with a peak picking method on the LP residue allows the accurate and unambiguous detection of GCIs (as indicated in Fig.\[fig:SEDREAMS\_illus\](e)).
It is worth noting that the advantage of using the mean-based signal is two-fold. First of all, since it oscillates at the local pitch period, this signal guarantees good performance in terms of reliability (i.e the risk of misses or false alarms is limited). Secondly, the intervals of presence that are derived from this signal imply that the GCI timing error is bounded by the depth of these intervals (i.e 0.35 times the local pitch period).
The YAGA algorithm {#ssec:YAGA}
------------------
The Yet Another GCI Algorithm (YAGA) [@Thomas2010b], like DYPSA, is an LP-based approach that employs $N$-best dynamic programming to find the best path through a set of candidate GCIs. The algorithms differ in the way in which the candidate set is estimated. Candidates are derived in DYPSA using a linear prediction residual, calculated by inverse-filtering a preemphasised speech signal with the LP coefficients. GCIs are manifest as impulsive features that may be detected with the group delay function. In YAGA, candidates are derived from an estimate of the voice source signal $u'(n)$ by using the same LP coefficients to inverse-filter the non-preemphasized speech signal. This differs crucially in that it exhibits discontinuities at both GCIs and GOIs, although GOIs are not considered in this paper. The speech signal $s(n)$ and voice source signal $u'(n)$ are shown for a short speech sample in Fig. \[fig:YAGA\] (a) and (b) respectively.
![Illustration of GCI detection using the YAGA algorithm on a segment of voiced speech. *(a) :* the speech signal, *(b) :* the corresponding voice source signal, *(c) :* the multiscale product of the voice source, *(d) :* the group-delay function, *(e) :* the synchronized dEGG with the GCI positions located by the YAGA algorithm.[]{data-label="fig:YAGA"}](yaga.eps){width="45.00000%"}
The impulsive nature of the LPC residual is well-suited to detection with the group delay method as discussed in Section \[ssec:DYPSA\]. In order for the group delay method to be applied to voice source signal, a discontinuity detector that yields an impulse-like signal is required. Such a detector might be achieved by a 1st-order differentiator, however it is known that GCIs and GOIs are not instantaneous discontinuities but are instead spread over time [@Bouzid2004]. The Stationary Wavelet Transform (SWT) is a multiscale analysis tool for the detection of discontinuities in a signal by considering the product of the signal at different scales [@Mallat1992]. It was first used in the context of GCI detection in [@Bouzid2004] by application to the speech signal. YAGA employs a similar approach on the voice source signal, which is expected to yield better results as it is free from unwanted vocal tract resonances. The SWT of signal $u'(n)$, $1\leq n\leq N$ at scale $j$ is $$\begin{aligned}
d^s_j(n)&=W_{2^j}u'(n), \nonumber \\
&=\sum_{k}g_j(k)a_{j-1}^s(n-k),\end{aligned}$$ where the maximum scale $J$ is bounded by $\log_2N$ and $j=1,2,\dots,J-1$. The approximation coefficients are given by $$a^s_j(n)=\sum_{k}h_j(k)a_{j-1}^s(n-k),$$ where $a^s_0(n)=u'(n)$ and $g_j(k)$, $h_j(k)$ are detail and approximation filters respectively that are upsampled by two on each iteration to effect a change of scale [@Mallat1992]. Filters are derived from a biorthogonal spline wavelet with one vanishing moment [@Mallat1992]. The multiscale product, $p(n)$, is formed by $$p(n)=\prod^{j_1}_{j=1}d_j(n) = \prod^{j_1}_{j=1}W_{2^j}u'(n),$$ where it is assumed that the lowest scale to include is always 1. The de-noising effect of the approximation filters each scale in conjunction with the multiscale product means that $p(n)$ is near-zero except at discontinuities across the first $j_1$ scales of $u'(n)$ where it becomes impulse-like. The value of $j_1$ is bounded by $J$, but in practice $j_1=3$ gives good localization of discontinuities in acoustic signals [@Sadler1999].
The multiscale product of the voice source signal in Fig. \[fig:YAGA\] (b) is shown in plot (c). Impulse-like features can be seen in the vicinity of discontinuities of $u'(n)$; such features are then detected by the negative-going zero-crossings of the group delay function in plot (d) that form the candidate set of GCIs. In order to distinguish between GCIs, GOIs and false candidates, an $N$-best dynamic programming algorithm is applied. The cost function employed is similar to that of DYPSA with an improved waveform similarity measure and an additional element to reliably differentiate between GCIs and GOIs.
Assessment of GCI Extraction Techniques {#sec:Assessment}
=======================================
Speech Material {#ssec:Material}
---------------
The evaluation of the GCI detection methods relies on ground-truth obtained from EGG recordings. The methods are compared on six large corpora containing contemporaneous EGG recordings whose description is summarized in Table \[tab:TabDba\]. The first three corpora come from the CMU ARCTIC databases [@Festvox]. They were collected at the Language Technologies Institute at Carnegie Mellon University with the goal of developing unit selection speech synthesizers. Each phonetically balanced dataset contains 1150 sentences uttered by a single speaker: BDL (US male), JMK (US male) and SLT (US female). The fourth corpus consists of a set of nonsense words containing all phone-phone transitions for English, uttered by the UK male speaker RAB. The fifth corpus is the KED Timit database and contains 453 utterances spoken by a US male speaker. These five first databases are freely available on the Festvox webpage [@Festvox]. The sixth corpus is the APLAWD dataset [@Lindsey1987] which contains ten repetitions of five phonetically balanced English sentences spoken by each of five male and five female talkers. For each of these six corpora, the speech and EGG signals sampled at 16 kHz are considered. The APLAWD database contains a square wave calibration signal for correcting low-frequency phase distortion, introduced in the recording chain, with an allpass equalization filter [@Hunt1978]. While this is particularly important in the field of voice source estimation and modelling [@Funaki1999], we have found GCI detection to be relatively insensitive to such phase distortion. An intuitive explanation is that the glottal excitation at the GCI excites many high-frequency bins such that low-frequency distortion does not have a significant effect upon the timing of the estimated GCI.
**Dataset** **Speaker(s)** **Approximative duration**
------------- --------------------- ---------------------------- --
BDL 1 male 54 min.
JMK 1 male 55 min.
SLT 1 female 54 min.
RAB 1 male 29 min.
KED 1 male 20 min.
APLAWD 5 males - 5 females 20 min.
Total 9 males - 6 females 232 min.
: Description of the databases.[]{data-label="tab:TabDba"}
Objective Evaluation {#ssec:Evaluation}
--------------------
The most common way to assess the performance of GCI detection techniques is to compare the estimates with the reference locations extracted from EGG signals (Section \[sssec:compEGG\]). Besides it is also proposed to evaluate their efficiency on a specific application of speech processing: the causal-anticausal deconvolution (Section \[sssec:MixedPhase\]).
### Comparison with Electroglottographic Signals {#sssec:compEGG}
Electroglottography (EGG), also known as electrolaryngography, is a non-intrusive technique for measuring the time-varying impedance between the vocal folds. The EGG signal is obtained by passing a weak electrical current between a pair of electrodes placed in contact with the skin on both sides of the larynx. This measure is proportionate to the contact area of the vocal folds. As clearly seen in the explanatory figures of Section \[sec:Methods\], true positions of GCIs can then be easily detected by locating the greatest positive peaks in the differenced EGG signal. Note that, for the automatic assessment, EGG signals need to be time-aligned with speech signals by compensating the delay between the EGG and the microphone. This was done in this work by a manual verification for each database (inside which the delay is assumed to remain constant).
![Characterization of GCI estimates showing three glottal cycles with examples of each possible outcome from GCI estimation [@Naylor2007a]. Identification accuracy is characterized by $\xi$.[]{data-label="fig:ErrorMeasures"}](ErrorMeasures.eps){width="45.00000%"}
Performance of a GCI detection method can be evaluated by comparing the locations that are estimated with the synchronized reference positions derived from the EGG recording. For this, we here make use of the performance measure defined in [@Naylor2007a], presented with the help of Fig. \[fig:ErrorMeasures\]. The first three measures describe how *reliable* the algorithm is in identifying GCIs:
- the Identification Rate (IDR): the proportion of glottal cycles for which exactly one GCI is detected,
- the Miss Rate (MR): the proportion of glottal cycles for which no GCI is detected,
- and the False Alarm Rate (FAR): the proportion of glottal cycles for which more than one GCI is detected.
For each correct GCI detection (i.e respecting the IDR criterion), a timing error $\xi$ is made with reference to the EGG-derived GCI position. When analyzing a given dataset with a particular method of GCI detection, $\xi$ has a probability density comparable to the histograms of Fig. \[fig:ErrorHisto\] (which will be detailed later in this paper). Such a distribution can be characterized by the following measures for quantifying the *accuracy* of the method [@Naylor2007a]:
- the Identification Accuracy (IDA): the standard deviation of the distribution,
- the Accuracy to $\pm$ 0.25 ms: the proportion of detections for which the timing error is smaller than this bound.
### A Speech Processing Application: the Causal-Anticausal Deconvolution {#sssec:MixedPhase}
The causal-anticausal decomposition (also known as mixed-phase decomposition) is a non-parametric technique of source-tract deconvolution known to be highly sensitive to GCI location errors [@MixedPhase]. It can therefore be employed as a framework for assessing our methods of GCI extraction on a speech processing application. The principle of this decomposition relies on the mixed-phase model of speech [@Doval-CALM], [@MixedPhase]. According to this model, voiced speech is composed of both minimum-phase (i.e causal) and maximum-phase (i.e anticausal) components. While the vocal tract response and the glottal *return phase* can be considered as minimum-phase signals, it has been shown [@Doval-CALM] that the glottal *open phase* is a maximum-phase signal. The key idea of the causal-anticausal (or mixed-phase) decomposition is then to separate both minimum and maximum-phase components of speech, where the latter is only due to the glottal contribution. By isolating the anticausal component of speech, causal-anticausal separation allows to estimate the glottal open phase.
Two algorithms have been proposed in the literature for achieving the causal-anticausal separation: the Zeros of the Z-Transform (ZZT, [@ZZT]) method and the Complex Cepstrum-based Decomposition (CCD, [@Drugman-CCD]). It has been shown [@Drugman-CCD] that both algorithms are functionally equivalent and lead to a reliable estimation of the glottal flow. However the use of the CCD technique was recommended for its much higher computational speed compared to ZZT. Besides it was also shown in [@Drugman-CCD] that windowing is crucial and dramatically conditions the efficiency of the causal-anticausal decomposition. It is indeed essential that the window applied to the segment of voiced speech respects some constraints in order to exhibit correct mixed-phase properties. Among these constraints, the window should be synchronized on a GCI, and have an appropriate shape and length (proportional to the pitch period). If the windowing is such that the speech segment respects the properties of the mixed-phase model, a correct deconvolution is achieved and the anticausal component gives a reliable estimate of the glottal flow (i.e which corroborates the models of the glottal source, such as the LF model [@Fant1985]), as illustrated in Fig. \[fig:GlottalFlow\](a). On the contrary, if this is not the case (possibly due to the fact that the window is not perfectly synchronized with the GCI), the causal-anticausal decomposition fails, and the resulting anticausal component generally contains an irrelevant high-frequency noise (see Fig.\[fig:GlottalFlow\](b)).
![Two cycles of the anticausal component isolated by mixed-phase decomposition *(*a): when the speech segment exhibits characteristics of the mixed-phase model, *(*b): when this is not the case.[]{data-label="fig:GlottalFlow"}](GlottalFlow.eps){width="45.00000%"}
As a simple (but accurate) criterion for deciding whether a frame has been correctly decomposed or not, the spectral center of gravity of the anticausal component is investigated. For a given dataset, this feature has a distribution as the one displayed in Fig. \[fig:COGhisto\]. A principal mode around 2 kHz clearly emerges and corresponds to the majority of frames for which a correct decomposition is carried out (as in Fig.\[fig:GlottalFlow\](a)). A second mode at higher frequencies is also observed. It is related to the frames where the causal-anticausal decomposition fails, leading to a maximum-phase signal containing an irrelevant high-frequency noise (as in Fig.\[fig:GlottalFlow\](b)). It can be noticed from this histogram that fixing a threshold at around 2.7 kHz optimally discriminate frames that are correctly and incorrectly decomposed.
![Example of distribution for the spectral center of gravity of the maximum-phase component. Fixing a threshold around 2.7kHz makes a good separation between correctly and incorrectly decomposed frames.[]{data-label="fig:COGhisto"}](COGhisto.eps){width="45.00000%"}
In conclusion, it is expected that the use of good GCI estimates reduces the proportion of frames that are incorrectly decomposed using the causal-anticausal separation.
Experiments on Clean Speech Data {#sec:ExpClean}
================================
Based on the experimental protocol described in Section \[sec:Assessment\], the performance of the five methods of GCI detection introduced in Section \[sec:Methods\] is now compared on the original clean speech utterances.
Comparison with Electroglottographic Signals {#ssec:compEGG}
--------------------------------------------
Results obtained from the comparison with electroglottographic recordings are presented in Table \[tab:TabClean\] for the various databases.
**Database** **Method** **IDR ($\%$)** **MR ($\%$)** **FAR ($\%$)** **IDA (ms)** **Accuracy to $\pm 0.25$ms ($\%$)**
-------------- ------------ ---------------- --------------- ---------------- -------------- -------------------------------------
HE 97.04 1.93 1.03 0.58 46.24
DYPSA 95.54 2.12 2.34 0.42 83.74
BDL ZFR 97.97 1.05 **0.98** 0.30 80.93
SEDREAMS 98.08 0.77 1.15 0.31 89.35
YAGA **98.43** **0.39** 1.18 **0.29** **90.31**
HE 93.01 3.94 3.05 0.90 38.66
DYPSA 98.26 0.88 0.86 0.46 77.26
JMK ZFR 96.17 3.43 **0.4** 0.60 41.62
SEDREAMS **99.29** **0.25** 0.46 0.42 80.78
YAGA 99.13 0.27 0.60 **0.40** **81.05**
HE 96.16 2.83 1.01 0.56 52.46
DYPSA 97.18 1.41 1.41 0.44 72.17
SLT ZFR **99.26** 0.15 **0.59** **0.22** 83.70
SEDREAMS 99.15 **0.12** 0.73 0.30 81.35
YAGA 98.90 0.20 0.90 0.28 **86.18**
HE 92.08 2.55 5.37 0.78 38.67
DYPSA 82.33 1.87 15.80 0.46 86.76
RAB ZFR 92.94 6.31 0.75 0.56 55.87
SEDREAMS **98.87** **0.63** **0.50** **0.37** **91.26**
YAGA 95.70 0.47 3.83 0.49 89.77
HE 94.73 1.75 3.52 0.56 65.81
DYPSA 97.24 1.56 1.20 0.34 89.46
KED ZFR 87.36 7.90 4.74 0.63 46.82
SEDREAMS **98.65** 0.67 **0.68** **0.33** 94.65
YAGA 98.21 **0.63** 1.16 0.34 **95.14**
HE 91.74 5.64 2.62 0.73 54.20
DYPSA 96.12 2.24 1.64 0.59 77.82
APLAWD ZFR **98.89** 0.59 0.52 0.55 57.87
SEDREAMS 98.67 0.82 **0.51** **0.45** 85.15
YAGA 98.88 **0.52** 0.60 0.49 **85.51**
In terms of *reliability* performance, SEDREAMS and YAGA algorithms generally give the highest identification rates. Among others, it turns out that SEDREAMS correctly identifies more than 98$\%$ of GCIs for any dataset. This is also true for YAGA, except on the RAB database where it reaches 95.70$\%$. Although the performance of ZFR is below these two techniques for JMK, RAB and KED speakers, its results are rather similar on other datasets, obtaining even the best reliability scores on SLT and APLAWD. As for the DYPSA method, its performance remains behind SEDREAMS and YAGA, albeit it reaches IDRs comprised between 95.54$\%$ and 98.26$\%$, except for the RAB speaker where the technique fails, leading to an important amount of false alarms (15.80$\%$). Finally the HE-based approach is outperformed by all other methods most of the time. However it achieves on all databases identification rates, comprised between 91.74$\%$ and 97.04$\%$.
In terms of *accuracy*, it is observed on all the databases, except for the RAB speaker, that YAGA leads the highest rates of frames for which the timing error is lower than 0.25 ms. The SEDREAMS algorithm gives almost comparable accuracy performance, just below the accuracy of YAGA. The DYPSA and HE algorithms, are outperformed by YAGA and SEDREAMS on all datasets. As it was the case for the reliability results, the accuracy of ZFR strongly depends on the considered speaker. It achieves very good results on the BDL and SLT speakers even though the overall accuracy is rather low especially for the KED corpus.
The accuracy performance is illustrated in Fig. \[fig:ErrorHisto\] for the five measures. The distributions of the GCI identification error $\xi$ is averaged over all datasets. The histograms for the SEDREAMS and YAGA methods are the sharpest and are highly similar. It is worth pointing out that some discrepancy is expected even if the GCI methods identify the acoustic events with high accuracy, since the delay between the speech signal, recorded by the microphone, and the EGG does not remain constant during recordings.
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In conclusion from the results of Table \[tab:TabClean\], the SEDREAMS and YAGA techniques, with highly similar performance, generally outperform other methods of GCI detection on clean speech, both in terms of reliability and accuracy. The ZFR method can also reach comparable (or even slightly better) results for some databases, but its performance is observed to be strongly sensitive to the considered speaker. In general, these three approaches are respectively followed by the DYPSA algorithm and the HE-based method.
Performance based on Causal-Anticausal Deconvolution {#ssec:MixedPhase}
----------------------------------------------------
As introduced in Section \[sssec:MixedPhase\], the Causal-Anticausal deconvolution is a well-suited approach for evaluating our techniques of GCI determination on a concrete application of speech processing. It was indeed emphasized that this method of glottal flow estimation is highly sensitive to GCI location errors. Besides we presented in Section \[sssec:MixedPhase\] an objective spectral criterion for deciding whether the mixed-phase separation fails or not. It is important to note at this point that the constraint of precise GCI-synchronization is a necessary, but not sufficient, condition for having a correct deconvolution.
Figure \[fig:Mixed-PhaseResults\] displays, for all databases and GCI estimation techniques, the proportion of speech frames that are incorrectly decomposed via mixed-phase separation (achieved in this work by the complex cepstrum-based algorithm [@Drugman-CCD]). It can be observed that for all datasets (except for SLT), SEDREAMS and YAGA outperform other approaches and lead again to almost the same results. They are closely followed by the DYPSA algorithm whose accuracy was also shown to be quite high in the previous section. The ZFR method turns out to be generally outperformed by these three latter techniques, but still gives the best results on the SLT voice. Finally, it is seen that the HE-based approach leads to the highest rates of incorrectly decomposed frames. Interestingly, these results achieved in the applicative context of the mixed-phase deconvolution corroborate the conclusions drawn from the comparison with EGG signals, especially regarding their accuracy to $\pm 0.25$ ms (see Section \[ssec:compEGG\]). This means that the choice of an efficient technique of GCI estimation, as those compared in this work, may significantly improve the performance of applications of speech processing for which a pitch-synchronous analysis or synthesis is required.
![Proportion of speech frames leading to an incorrect mixed-phase deconvolution using all GCI estimation techniques on all databases.[]{data-label="fig:Mixed-PhaseResults"}](Mixed-PhaseResults.eps){width="45.00000%"}
Robustness of GCI Extraction Methods {#sec:Robustness}
====================================
In some speech processing applications, such as speech synthesis, utterances are recorded in well controlled conditions. For such high-quality speech signals, the performance of GCI estimation techniques was studied in Section \[sec:ExpClean\]. For many other types of speech processing systems however, there is no other choice than capturing the speech signal in a *real world environment*, where noise and/or reverberation may dramatically degrade its quality. The goal of this section is to evaluate how GCI detection methods are affected by additive noise (Section \[ssec:Noise\]) and by reverberation (Section \[ssec:Reverb\]). Note that results presented here below were averaged over the six databases.
Robustness to an Additive Noise {#ssec:Noise}
-------------------------------
In a first experiment, noise was added to the original speech waveform at various Signal-to-Noise Ratio (SNR). Both a White Gaussian Noise (WGN) and a babble noise (also known as cocktail party noise) were considered. The noise signals were taken from the Noisex-92 database [@Noisex], and were added so as to control the segmental SNR without silence removal. Results for these two noise types are exhibited in Figs. \[fig:Robustness\_WhiteNoise\] and \[fig:Robustness\_BabbleNoise\] according to the measures detailed in Section \[sssec:compEGG\]. In these figures, miss rate and false alarm rate are in logarithmic scale for the sake of clarity. It is observed that, for both noise types, the general trends remain unchanged. However it turns out that the degradation of reliability is more severe with the white noise, while the accuracy is more affected by the babble noise.
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In terms of reliability, it is noticed that SEDREAMS and ZFR lead to the best robustness, since their performance is almost unchanged up to 0dB of SNR. Secondly, the degradation for YAGA and HE is almost equivalent, while it is noticed that DYPSA is strongly affected by additive noise. Among others, it is observed that HE is characterized by an increasing miss rate as the noise level increases, while the degradation is reflected by an increasing number of false alarms for DYPSA, and for YAGA in a lesser extent. This latter observation is probably due to the difficulty of the dynamic programing process to deal with spurious GCI candidates caused by the additive noise.
Regarding the accuracy capabilities, similar conclusions hold. Nevertheless the sensitivity of SEDREAMS is this time comparable to that of YAGA and HE. Again, the ZFR algorithm is found to be the most robust technique, while DYPSA is the one presenting the strongest degradation and HE displays the worst identification accuracy.
Good results of robustness for ZFR and SEDREAMS can be explained by the low sensitivity to an additive noise of respectively the zero-frequency resonators and the mean-based signal. In the case of ZFR, analysis is confined around 0 Hz, which tends to minimize not only the effect of the vocal tract, but of an additive noise as well. As for SEDREAMS, the mean-based signal is computed as in Equation \[eq:Mean\], which is a linear relation. In other words, the mean-based signal of the noise is added to the mean-based signal of the speech signal. On a duration of 1.75$\cdot$$\bar{T}_{0,mean}$, the white noise is assumed to be almost zero-mean. A similar conclusion is observed for the babble noise, which is composed of several sources of speech talking at the same time. It can indeed be understood that the higher the number of sources in the babble noise, the lesser its degradation on the target mean-based signal. Finally, the strong sensitivity of DYPSA and YAGA might be explained, among others, by the fact that they rely on some thresholds, which have been optimized for clean speech.
Robustness to Reverberation {#ssec:Reverb}
---------------------------
In many modern telecommunication applications, speech signals are obtained in enclosed spaces with the talker situated at a distance from the microphone. The received speech signal is distorted by reverberation, caused by reflected signals from walls and hard objects, diminishing intelligibility and perceived speech quality [@Bolt1949; @Kuttruff2000]. It has been further observed that the performance of GCI identification algorithms is degraded when applied to reverberant signals [@Gaubitch2007].
The observation of reverberant speech at microphone $m$ is $$\label{eq:revsp1}
x_m(n) = h_m(n)\ast {s}(n), \quad m=1,2,\ldots,M,$$ where $h_m(n)$ is the $L$-tap Room Impulse Response (RIR) of the acoustic channel between the source to the $m$th microphone. It has been shown that multiple time-aligned observations with a microphone array can be exploited for GCI estimation in reverberant environments [@Thomas2007a]; in this paper we only consider the robustness of single-channel algorithms to the observation at channel $x_1(n)$. RIRs are characterised by the value $T_{60}$, defined as the time for the amplitude of the RIR to decay to -60dB of its initial value. A room measuring 3x4x5 m and $T_{60}$ ranging {100, 200, …, 500} ms was simulated using the source-image method [@Allen1979] and the simulated impulse responses convolved with the clean speech signals described in Section \[sec:Assessment\].
{width="100.00000%"}
The results in Figure \[fig:Robustness\_Reverb\] show that the performance of the algorithms monotonically reduces with increasing reverberation, with the most significant change in performance occurring between $T_{60}=100$ and $200$ ms. They also reveal that reverberation has a particularly detrimental effect upon identification rate of the LP-based approaches, namely HE, DYPSA and YAGA. This is consistent with previous studies which have shown that the RIR results in additional spurious peaks in the LP residual of similar amplitude to the voiced excitation [@Brandstein2001; @Yegnanarayana2000], generally increasing false alarm rate for DYPSA and YAGA but increasing miss rate for HE. Although spurious peaks result in increased false alarms, the identification accuracy of the hits is much less affected. The non-LP approaches generally exhibit better identification rates in reverberation, in particular SEDREAMS. The ZFR algorithm appears to be the least sensitive to reverberation while providing the best overall performance. However, the challenge of GCI detection from single-channel reverberant observations remains an ongoing research problem as no single algorithm consistently provides good results for all five measures.
Computational Complexity of GCI Extraction Methods {#sec:Complexity}
==================================================
In the previous sections, methods of GCI estimation have been compared according to their reliability and accuracy both in clean conditions (Section \[sec:ExpClean\]) and noisy/reverberant environments (Section \[sec:Robustness\]). In order to provide a complete comparison, an investigation into computational complexity is described in this section. The algorithms described in Section \[sec:Methods\] are relatively complex and their computational complexity is highly data-dependent; it is therefore difficult to find a closed-form expression for computational complexity. In this section we discuss those components that present a high computational load and provide a quantitative analysis based upon empirical measurements.
For HE, ZFR and SEDREAMS, the most time-consuming step is the computation of the oscillating signal which they rely on. For the HE method, the CoG-based signal is computed from Equation \[eq:CoG\] and requires, for each sample, around $2.2\cdot F_s/\bar{T}_{0,mean}$ multiplications and the same number of additions. For ZFR, the mean removal operation (Equation \[eq:MeanRemoval\]) is repeated three times, and thus requires about $4.5\cdot F_s/\bar{T}_{0,mean}$ additions for each sample of the zero frequency-filtered signal. As for the SEDREAMS algorithm, the computation of each sample of the mean-based signal (Equation \[eq:Mean\]) requires $1.75\cdot F_s/\bar{T}_{0,mean}$ multiplications and the same number of additions.
However, it is worth emphasizing that the computation time requested by HE and SEDREAMS can be significantly reduced. Indeed these methods only exploit some particular points of the oscillating signal they rely on: the negative zero-crossings for HE, and the extrema for SEDREAMS. It is then not necessary to compute all the samples of these signals for finding these particular events. Based on this idea, a multiscale approach can be used. For example, the oscillating signals can be first calculated only for the samples multiple of $2^p$. From this downsampled signal, a first approximation of the particular points is obtained. This approximation is then refined iteratively using the $p$ successive smaller scales. The lower bounding value of $p$ means there are, for the first approximation, at least two samples per cycle. In the following, we used $p=4$ so that voices with pitch up to 570 Hz can be processed. The resulting methods are hereafter called *Fast HE* and *Fast SEDREAMS*. Notice that a similar acceleration cannot be transposed to ZFR as the operation of mean removal is applied 3 times successively.
In the case of DYPSA and YAGA, the signal conditioning stages present a relatively low computational load. The LPC residual, Group Delay Function and Multiscale Product scale approximately $\mathcal{O}(N^2)$, $\mathcal{O}(N\log_2 N)$ and $\mathcal{O}(N)$ respectively, where $N$ is the total number of samples in the speech signal. Computational load is significantly heavier in the dynamic programming stages due to the large number of erroneous GCI candidates that must be removed. In particular, the waveform similarity measure, used to determine the similarity of two neighbouring cycles, presents a high computational load due to the large number of executions required to find the optimum path. At present this is calculated on full-band speech although it is expected that calculation of waveform similarity on a downsampled signal may yield similar results for a much-reduced computational load. A second optimization lies in the length of the group delay evaluation window, which is inversely proportional to the number of candidates generated. At present this takes a fixed value based upon the maximum expected $f_0$; far fewer erroneous candidates could be generated by dynamically varying the length based upon a crude initial estimate of $f_0$.
So as to compare their computational complexity, the *Relative Computation Time* (RCT) of each GCI estimation method is evaluated on all databases:
$$\label{eq:RCT}
RCT (\%) = 100\cdot \frac{\text{CPU time (s)}}{\text{Sound duration (s)}}$$
Table \[tab:TabComplexity\] shows, for both male and female speakers, the averaged RCT obtained for our Matlab implementations and with a Intel Core 2 Duo T7500 2.20 GHz CPU with 3GB of RAM. First of all, it is observed that results are ostensibly the same for both genders. Regarding the non-accelerated versions of the GCI detection methods, it turns out that DYPSA is the fastest (with a RCT around 20%), followed by SEDREAMS and YAGA, which both have a RCT of about 28%. The HE-based technique gives a RCT of around 33%, and ZFR, due to its operation of mean removal which has to be repeated three times, is the slowest method with a RCT of 75%. Interestingly, it is noticed that the accelerated versions of HE and SEDREAMS reduce the computation time by about 5 times on male voices, and by around 4 times for female speakers. This leads to the fastest GCI detection algorithms, reaching a RCT of around 6% for Fast SEDREAMS, and about 8% for Fast HE. Note finally that these results could be highly reduced by using, for example, a C-implementation of these techniques, albeit the conclusions remain identical.
**Method** **Male** **Female**
--------------- ---------- ------------
HE 35.0 31.8
Fast HE 7.6 7.8
DYPSA 19.9 19.4
ZFR 75.7 74.9
SEDREAMS 27.8 27.1
Fast SEDREAMS 5.4 6.9
YAGA 28.6 28.3
: Relative Computation Time (RCT), in %, for all methods and for male and female speakers. Results have been averaged across all databases.[]{data-label="tab:TabComplexity"}
Conclusion {#sec:conclu}
==========
This paper gave a comparative evaluation of five of the most effective methods for automatically determining GCIs from the speech waveform: Hilbert Envelope-based detection (HE), the Zero Frequency Resonator-based method (ZFR), DYPSA, SEDREAMS and YAGA. The performance of these methods was assessed on six databases containing several male and female speakers, for a total amount of data of approximately four hours. In our first experiments on clean speech, the SEDREAMS and YAGA algorithms gave the best results, with a comparable performance. For *any* database, they reached an identification rate greater than $98\%$ and more than $80\%$ of GCIs were located with an accuracy of $0.25$ ms. Although the ZFR technique can lead to a similar performance, its efficiency can also be rather low in some cases. In general, these three approaches were shown to respectively outperform DYPSA and HE. In a second experiment on clean speech, the impact of the performance of these five methods was studied on a concrete application of speech processing: the causal-anticausal deconvolution. Results showed that adopting a GCI detection with high performance could significantly improve the proportion of correctly deconvolved frames. In the last experiment, the robustness of the five techniques to additive noise, as well as to reverberation was investigated. The ZFR and SEDREAMS algorithms were shown to have the highest robustness, with an almost unchanged reliability. DYPSA was observed to be especially affected, which was reflected by a high rate of false alarms. Although the degradation of accuracy was relatively slow with the level of additive noise, it was noticed that reverberation dramatically affects the precision GCI detection methods. In addition, the computational complexity of the algorithms was studied. A method for accelerating the GCI location using HE and SEDREAMS was proposed. This led, for our Matlab implementation, to a computation time about 6% real-time for the fast version of SEDREAMS.
Depending on the speech application to design, some GCI methods could be preferred to some others, based on their performance for the criteria studied in this article. However, if the application is placed in an unknown environment, we suggest the use of SEDREAMS for the following reasons: *i)* it gave the best results with YAGA on clean speech, *ii)* it was the best performing technique in noisy conditions, *iii)* it led with ZFR to the best robustness in a reverberant environment, and *iv)* it was the most suited method for a real-time implementation.
Acknowledgment {#acknowledgment .unnumbered}
==============
Thomas Drugman is supported by the Belgian Fonds National de la Recherche Scientifique (FNRS). Authors also would like to thank the reviewers for their fruitful comments.
|
---
abstract:
- 'Seemingly unrelated regression models generalize linear regression models by considering multiple regression equations that are linked by contemporaneously correlated disturbances. Robust inference for seemingly unrelated regression models is considered. MM-estimators are introduced to obtain estimators that have both a high breakdown point and a high normal efficiency. A fast and robust bootstrap procedure is developed to obtain robust inference for these estimators. Confidence intervals for the model parameters as well as hypothesis tests for linear restrictions of the regression coefficients in seemingly unrelated regression models are constructed. Moreover, in order to evaluate the need for a seemingly unrelated regression model, a robust procedure is proposed to test for the presence of correlation among the disturbances. The performance of the fast and robust bootstrap inference is evaluated empirically in simulation studies and illustrated on real data.'
- 'In the supplementary material we introduce functionals corresponding to MM-estimators and discuss important properties of these MM-functionals such as equivariance, influence function and asymptotic variance. Also influence functions and asymptotic distributions are derived for the proposed robust test statistics. Power curves are included for a situation which is less deviating from diagonality than the equicorrelation matrix. Furthermore, we construct bootstrap confidence intervals based on fast and robust bootstrap and evaluate their performance in a simulation study. In addition, we illustrate these confidence intervals on Grunfeld data. The appendix also contains expressions for the partial derivatives required in the fast and robust bootstrap procedure, a verification of the consistency conditions for the robust test on regression coefficients, and the proofs of the theorems.'
author:
- |
Kris Peremans and Stefan Van Aelst\
Department of Mathematics, KU Leuven, 3001 Leuven, Belgium\
- |
Kris Peremans and Stefan Van Aelst\
Department of Mathematics, KU Leuven, 3001 Leuven, Belgium\
bibliography:
- 'ms\_references.bib'
title:
- '**Robust Inference for Seemingly Unrelated Regression Models**'
- '**Robust Inference for Seemingly Unrelated Regression Models: Supplementary Material**'
---
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[0]{}
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**Robust Inference for Seemingly Unrelated Regression Models**
KEYWORDS: Diagonality test; Fast and robust bootstrap; MM-estimator; Robust testing
© 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license <http://creativecommons.org/licenses/by-nc-nd/4.0/>
Introduction
============
Many scientists have investigated statistical problems involving multiple linear regression equations. Unconsidered factors in these equations can lead to highly correlated disturbances. In such cases, estimating the regression parameters equation-by-equation by, e.g., least squares is not likely to yield efficient estimates. Therefore, seemingly unrelated regression (SUR) models have been developed. SUR models take the underlying covariance structure of the error terms across equations into account. Applications in econometrics and related fields include demand and supply models [@Kotakou2011; @Martin2007], capital asset pricing models [@Hodgson2002; @Pastor2002], chain ladder models [@Hubert2017; @Zhang2010], vector autoregressive models [@Wang2010], household consumption and expenditure models [@Kuson2012; @Lar2011], environmental sciences [@Olaolu2011; @Zaman2011], natural sciences [@Cadavez2012; @Hasenauer1998] and many more.
A SUR model, introduced by @Zellner1962, consists of $m>1$ dependent linear regression equations, also called blocks. Denote the $j$th block in matrix form by $$y_j = X_j \beta_j + \varepsilon_j,$$ where $y_j = (y_{1j},\ldots,y_{nj})^\top$ contains the $n$ observed values of the response variable and $X_j$ is an $n \times p_j$ matrix containing the values of $p_j$ input variables. Note that the number of predictors does not need to be the same for all blocks. The vector $\beta_j = (\beta_{1j},\ldots,\beta_{p_jj})^\top$ contains the unknown regression coefficients for the $j$th block and $\varepsilon_j = (\varepsilon_{1j},\ldots,\varepsilon_{nj})^\top$ constitutes its error term. The error term $\varepsilon_j$ is assumed to have ${\rm E}[\varepsilon_j] = 0$ and ${\rm Cov}[\varepsilon_j] = \sigma_{jj} I_n$ where $\sigma_{jj}$ is the unknown variance of the errors in the $j$th block, and $I_n$ represents the identity matrix of size $n$. In the SUR model blocks are connected by the assumption of contemporaneous correlation. That is, the $i$th element of the error term of block $j$ may be correlated with the $i$th element of the error term of block $k$. With $i$ and $\ell$ observation numbers and $j$ and $k$ block numbers, the covariance structure of the disturbances can be summarized as $$\begin{aligned}
{\rm E}[\varepsilon_{ij} \varepsilon_{ik}] &= \sigma_{jk}, \quad i=1,\ldots,n \text{ and } j,k=1,\ldots,m; \\
{\rm E}[\varepsilon_{ij} \varepsilon_{\ell j}] &= 0, \quad i \neq \ell; \\
{\rm E}[\varepsilon_{ij} \varepsilon_{\ell k}] &= 0, \quad j \neq k \text{ and } i \neq \ell.\end{aligned}$$ Note that each regression equation in a SUR model is a linear regression model in its own right. The different blocks may seem to be unrelated at first sight, but are actually related through their error terms.
The regression equations in a SUR model can be combined into two equivalent single matrix form equations. Let bdiag$()$ denote the operator that constructs a block diagonal matrix from its arguments. Moreover, let $\kron$ denote the Kronecker product and let $\Sigma$ be a symmetric matrix with elements $\sigma_{jk}$. First, the SUR model can be rewritten as a single linear regression model $$y = X \beta + \varepsilon,$$ where $y = (y_1^\top,\ldots,y_m^\top)^\top$, $X = {\rm bdiag}(X_1,\ldots,X_m)$ a $nm \times p$ block diagonal matrix with $p = \sum_{j=1}^m p_j$, and $\beta = (\beta_1^\top,\ldots,\beta_m^\top)^\top$. For the error term $\varepsilon = (\varepsilon_1^\top,\ldots,\varepsilon_m^\top)^\top$ it then holds that ${\rm Cov}[\varepsilon] = \Sigma \kron I_n$. Secondly, the SUR model can be represented as a multivariate linear regression model $$Y = \tilde{X} \mathcal{B} + \mathcal{E},$$ where $Y = (y_1,\ldots,y_m)$, $\tilde{X} = (X_1,\ldots,X_m)$, $\mathcal{B} = {\rm bdiag}(\beta_1,\ldots,\beta_m)$ and $\mathcal{E} = (\varepsilon_1,\ldots,\varepsilon_m)$. Equivalently, we can write the error matrix as $\mathcal{E} = (e_1,\ldots,e_n)^\top$ with $e_i = (\varepsilon_{i1},\ldots,\varepsilon_{im})^\top$ which satisfies ${\rm Cov}[e_i] = \Sigma$. Hence, the covariance of the error matrix $\mathcal{E}$ is given by ${\rm Cov}[\mathcal{E}] = \Sigma \kron I_n$.
It is well-known that ordinary least squares which ignores the correlation patterns across blocks may yield inefficient estimators. Generalized least squares (GLS) is a modification of least squares that can deal with any type of correlation, including contemporaneous correlation. For the SUR model, the GLS estimator takes the form $$\label{beta_GLS}
\hat{\beta}_{\text{GLS}} = (X^\top (\Sigma^{-1} \kron I_n) X)^{-1} X^\top (\Sigma^{-1} \kron I_n) y.$$ GLS coincides with the separate least squares estimates if $\sigma_{jk}$ for $j \neq k$, or if $X_1 = \ldots = X_m$. GLS is more efficient than least squares estimator [@Zellner1962], but in most situations the covariance $\Sigma$ needed in GLS is unknown. Feasible generalized least squares (FGLS) estimates the elements of $\Sigma$ by $\hat{\sigma}_{jk} = \hat{\varepsilon}_j^\top \hat{\varepsilon}_k / n$ where $\hat{\varepsilon}_j$ is the residual vector of the $j$th block obtained from ordinary least squares and then replaces $\Sigma$ in GLS by the resulting estimator $\hat{\Sigma}$. The finite-sample efficiency of FGLS is smaller than for GLS, although the asymptotic efficiency of both methods is identical. Note that FGLS can be repeated iteratively.
Alternatively, maximum likelihood estimators (MLE) can be considered [see @Srivastava1987]. Assuming that the disturbances are normally distributed, the log-likelihood of the SUR model is given by $$\label{logLikelihood}
l(\beta,\Sigma|X,y) = - \frac{mn}{2} \ln(2 \pi) - \frac{n}{2} \ln({\lvert\Sigma\rvert}) - \frac{1}{2} (y - X \beta)^\top (\Sigma^{-1} \kron I_n) (y - X \beta).$$ Maximizing this log-likelihood with respect to $(\beta,\Sigma)$ yields the estimators $(\hat{\beta}_{\text{MLE}},\hat{\Sigma}_{\text{MLE}})$ which are the solutions of the following equations $$\begin{gathered}
\label{MLE}
\hat{\beta}_{\text{MLE}} = (X^\top (\hat{\Sigma}_{\text{MLE}}^{-1} \kron I_n) X)^{-1} X^\top (\hat{\Sigma}_{\text{MLE}}^{-1} \kron I_n) y \\
\hat{\Sigma}_{\text{MLE}} = (Y - \tilde{X} \hat{\mathcal{B}}_{\text{MLE}})^\top (Y - \tilde{X} \hat{\mathcal{B}}_{\text{MLE}}) / n
\end{gathered}$$ with $\hat{\mathcal{B}}_{\text{MLE}}$ the block diagonal form of $\hat{\beta}_{\text{MLE}}$. Hence, the maximum likelihood estimators correspond to the fully iterated FGLS estimators.
It is well-known that outliers in the data (observations which deviate from the majority of the data) can severely influence classical estimators such as LS, MLE and their modifications. Hence, FGLS and MLE are expected to yield non-robust estimates. Robust M-estimators for the SUR model have been proposed, but these estimators lack affine equivariance [@Koenker1990]. @Bilodeau2000 have introduced robust and affine equivariant S-estimators. Recently, @Hubert2017 developed an efficient algorithm for these estimators. Despite its remarkable robustness properties, S-estimators can have a low efficiency, which makes them less suitable for inference. Therefore, we introduce MM-estimators for the SUR model which can combine high robustness with a high efficiency. To obtain efficient and powerful robust tests, we also introduce an efficient MM-estimator of the error scale based on the residuals of the MM-estimates.
Asymptotic theory can be used to draw inference corresponding to the MM-estimates in the SUR model. However, these asymptotic results rely on assumptions that are hard to verify in practice. The bootstrap [@Efron1979] offers an alternative approach that does not require strict assumptions. However, the standard bootstrap lacks speed and robustness. Therefore, the fast and robust bootstrap (FRB) procedure of @Salibian2002 is adapted to the SUR setting. The FRB can be used to construct confidence intervals [@Salibian2006; @Salibian2002] as well as to develop hypothesis tests [@Salibian2005; @Salibian2016; @VanAelst2011]. In particular, one of our main goals is to develop a robust test for diagonality of the covariance matrix $\Sigma$ to evaluate the need for using a SUR model.
To set the scene, MM-estimators for the SUR model are introduced in Section \[Robust Estimators for the SUR Model\] as an extension of S-estimators. Section \[Fast and Robust Bootstrap\] focuses on the fast and robust bootstrap procedure to develop robust inference. In Section \[Robust Tests for the Regression Parameters\] the MM-estimator of scale is introduced and hypothesis tests concerning the regression coefficients are studied. In Section \[Robust Test for Diagonality of the Covariance Matrix\] we investigate a robust procedure to test for diagonality of the covariance matrix $\Sigma$, i.e., to test whether a SUR model is really needed. The finite-sample performance of the FRB inference procedures is investigated by simulation in Section \[Finite-Sample Performance\]. Section \[Example: Grunfeld Data\] illustrates the robust inference on a real data example from economics and Section \[Conclusion\] concludes. The supplementary material includes properties of MM-estimators and the proposed test statistics, and contains some extra results on robust confidence intervals.
Robust Estimators for the SUR Model {#Robust Estimators for the SUR Model}
===================================
S-estimators
------------
We first introduce S-estimators for the SUR model as proposed by @Bilodeau2000. Consider so-called $\rho$-functions which satisfy the following conditions:
1. $\rho$ is symmetric, twice continuously differentiable and satisfies $\rho(0)=0$
2. $\rho$ is strictly increasing on $[0,c]$ and constant on $[c,\infty[$ for some $c>0$.
The most popular family of $\rho$-functions is the class of Tukey bisquare $\rho$-functions given by $\rho(u) = {\rm min} (u^2/2 - u^4/2 c^2 + u^6/6 c^4,c^2/6)$ where $c>0$ is a tuning parameter.
\[Sestimator\] Let $(X_j,y_j) \in \mathbb{R}^{n \times (p_j + 1)}$ for $j=1,\ldots,m$ and let $\rho_0$ be a $\rho$-function with parameter $c_0$ in (C2). Then, the S-estimators of the SUR model $(\tilde{\mathcal{B}},\tilde{\Sigma})$ are the solutions that minimize ${\lvertC\rvert}$ subject to the condition $$\frac{1}{n} \sum_{i=1}^n \rho_0 \left( \sqrt{e_i(B)^\top C^{-1} e_i(B)} \right) = \delta_0,$$ where the minimization is over all $B = {\rm bdiag}(b_1,\ldots,b_m) \in \mathbb{R}^{p \times m}$ and $C \in \text{PDS}(m)$ with PDS$(m)$ the set of positive definite and symmetric matrices of dimension $m \times m$. The determinant of $C$ is denoted by $|C|$ and $e_i(B)^\top$ represents the $i$th row of the residual matrix $Y - \tilde{X} B$.
The constant $\delta_0$ can be chosen as $\delta_0 = {\rm E}_F[\rho_0({\lVerte\rVert})]$ to obtain a consistent estimator at an assumed error distribution $F$. Usually, the errors are assumed to follow a normal distribution with mean zero and then we can take $F \sim \mathcal{N}_m(0,I_m)$. As before, the regression coefficient estimates in the matrix $\tilde{\mathcal{B}}$ can also be collected in the vector $\tilde{\beta} = (\tilde{\beta}_1^\top,\ldots,\tilde{\beta}_m^\top)^\top$.
The first-order conditions corresponding to the above minimization problem yield the following fixed-point equations for S-estimators $$\begin{gathered}
\label{EstimatingEquationS}
\tilde{\beta} = (X^\top (\tilde{\Sigma}^{-1} \kron \tilde{D}) X)^{-1} X^\top (\tilde{\Sigma}^{-1} \kron \tilde{D}) y \\
\tilde{\Sigma} = m (Y - \tilde{X} \tilde{\mathcal{B}})^\top \tilde{D} (Y - \tilde{X} \tilde{\mathcal{B}}) \left( \sum_{i=1}^n v_0(\tilde{d}_i) \right)^{-1}
\end{gathered}$$ with diagonal matrix $\tilde{D} = {\rm diag}(w_0(\tilde{d}_1),\ldots,w_0(\tilde{d}_n))$ where $\tilde{d}_i^2 = e_i(\tilde{\mathcal{B}})^\top \tilde{\Sigma}^{-1} e_i(\tilde{\mathcal{B}}),$ $w_0(u) = \psi_0(u)/u$, $\psi_0(u) = \rho_0'(u)$ and $v_0(u) = \psi_0(u) u - \rho_0(u) + \delta_0$. Note the similarities with the GLS in and the MLE in . The factor $w_0(\tilde{d}_i)$ can be interpreted as the weight that the estimator gives to the $i$th observation. A small (large) residual distance $\tilde{d}_i$ leads to a large (small) weight $w_0(\tilde{d}_i)$. The smaller the weight of an observation, the smaller its contribution to the SUR fit. To compute the S-estimates efficiently, @Hubert2017 developed the fastSUR algorithm based on the ideas of @Salibian2006b.
The breakdown point of an estimator is the smallest fraction of the data that needs to be contaminated in order to drive the bias of the estimator to infinity. S-estimators with a bounded loss function, as we consider here, have a positive breakdown point [@Lopuhaa1991; @VanAelst2005]. Their asymptotic breakdown point equals $\varepsilon^* = \delta_0/\rho_0(c_0)$. The constant $\delta_0$ has been fixed to guarantee consistency, but the parameter $c_0$ can be tuned to obtain any desired breakdown point $0 < \varepsilon^* \leq 0.5$. Hence, S-estimators can attain the maximal breakdown point of $50\%$. S-estimators with a smaller value of $c_0$ downweight observations more heavily and correspond to a higher breakdown point.
S-estimators satisfy the first-order conditions of M-estimators [see @Huber1981], so they are asymptotically normal. However, the choice of the tuning parameter $c_0$ involves a trade-off between breakdown point (robustness) and efficiency at the central model [@Bilodeau2000]. For this reason, S-estimators are less adequate for robust inference. MM-estimators [@Yohai1987] avoid this trade-off by computing an efficient M-estimator starting from a highly robust S-estimator [see, e.g., @Kudraszow2011; @Tatsuoka2000; @VanAelst2013]. We now introduce MM-estimators for the SUR model.
MM-estimators
-------------
Let $\tilde{\Sigma}$ denote the S-estimator of covariance in Definition \[Sestimator\]. Decompose $\tilde{\Sigma}$ into a scale component $\tilde{\sigma}$ and a shape matrix $\tilde{\Gamma}$ such that $\tilde{\Sigma} = \tilde{\sigma}^2 \tilde{\Gamma}$ with ${\lvert\tilde{\Gamma}\rvert}=1$.
\[MMestimator\] Let $(X_j,y_j) \in \mathbb{R}^{n \times (p_j + 1)}$ for $j=1,\ldots,m$ and let $\rho_1$ be a $\rho$-function with parameter $c_1$ in (C2). Given the S-scale $\tilde{\sigma}$, MM-estimators of the SUR model $(\hat{\mathcal{B}},\hat{\Gamma})$ minimize $$\frac{1}{n} \sum_{i=1}^n \rho_1 \left( \frac{\sqrt{e_i(B)^\top G^{-1} e_i(B)}}{\tilde{\sigma}} \right),$$ over all $B = {\rm bdiag}(b_1,\ldots,b_m) \in \mathbb{R}^{p \times m}$ and $G \in \text{PDS}(m)$ with ${\lvertG\rvert} = 1$. The MM-estimator for covariance is defined as $\hat{\Sigma} = \tilde{\sigma}^2 \hat{\Gamma}$.
As before, the MM-estimator of the regression coefficients $\hat{\mathcal{B}}$ can also be written in vector form $\hat{\beta}=(\hat{\beta}_1^\top,\ldots,\hat{\beta}_m^\top)^\top$. Similarly as for S-estimators, the first-order conditions corresponding to the above minimization problem yield a set of fixed-point equations: $$\begin{gathered}
\label{EstimatingEquationMM}
\hat{\beta} = (X^\top (\hat{\Sigma}^{-1} \kron D) X)^{-1} X^\top (\hat{\Sigma}^{-1} \kron D) y \\
\hat{\Sigma} = m (Y - \tilde{X} \hat{\mathcal{B}})^\top D (Y - \tilde{X} \hat{\mathcal{B}}) \left( \sum_{i=1}^n \psi_1(d_i) d_i \right)^{-1}
\end{gathered}$$ with $D = {\rm diag}(w_1(d_1),\ldots,w_1(d_n))$ where $d_i^2 = e_i(\hat{\mathcal{B}})^\top \hat{\Sigma}^{-1} e_i(\hat{\mathcal{B}})$, $w_1(u) = \psi_1(u)/u$ and $\psi_1(u) = \rho_1'(u)$. Starting from the initial S-estimates, the MM-estimates are calculated easily by iterating these estimating equations until convergence.
MM-estimators inherit the breakdown point of the initial S-estimators. Hence, they can attain the maximal breakdown point if initial high-breakdown point S-estimators are used. Moreover, since MM-estimators also satisfy the first-order conditions of M-estimators, they are asymptotically normal. In the supplementary material it is shown that the asymptotic efficiency of $\hat{\beta}$ does not depend on the $\rho$-function $\rho_0$ of the initial S-estimator. Therefore, the breakdown point and the efficiency of MM-estimators can be tuned independently. That is, the tuning constant $c_0$ in $\rho_0$ can be chosen to obtain an S-scale estimator with maximal breakdown point, while the constant $c_1 (> c_0)$ in $\rho_1$ is tuned to attain a desired efficiency, e.g., $90\%$, at the central model with normal errors. Note that while MM-estimators have maximal breakdown point, there is some loss of robustness because the bias due to contamination is generally higher as compared to S-estimators [see, e.g., @Berrendero2007].
Fast and Robust Bootstrap {#Fast and Robust Bootstrap}
=========================
The asymptotic distribution of MM-estimators can be used to obtain inference for the parameters in the SUR model based on their MM-estimates. However, these asymptotic results are only reasonable for sufficiently large samples and rely on the assumption of elliptically symmetric errors which does not necessarily hold in practice. The bootstrap offers an alternative approach that requires less assumptions. Unfortunately, for robust estimators the standard bootstrap procedure lacks speed and robustness. The standard bootstrap is computer intensive because many bootstrap replicates are needed and the fastSUR algorithm is itself already computationally intensive. Moreover, classical bootstrap does not yield robust inference results. Indeed, due to the resampling with replacement, the proportion of outlying observations varies among bootstrap samples. Some bootstrap samples thus contain a majority of outliers, resulting in breakdown of the estimator. These estimates affect the bootstrap distribution leading to unreliable inference. Therefore, we use the fast and robust bootstrap introduced by @Salibian2002 and generalized in e.g., @Salibian2006 and @Peremans2017.
Consider an estimator $\hat{\theta}$ of a parameter $\theta$ that satisfies the fixed-point equations $g(\hat{\theta})=\hat{\theta}$ where the function $g$ depends on the given sample. For a bootstrap sample it equivalently holds that $g^*(\hat{\theta}^*)=\hat{\theta}^*$. Now, consider $g^*(\hat{\theta})$ as a first-step approximation of the bootstrap estimate $\hat{\theta}^*$. These first-step approximations underestimate the variability of the bootstrap distribution since the starting value is the same for all bootstrap approximations. To remedy this deficiency a linear correction factor can be derived from a Taylor expansion of $g^*(\hat{\theta}^*)$. This yields the fast and robust bootstrap (FRB) estimator, given by $$\hat{\theta}^{R*} = \hat{\theta} + (I - \nabla g(\hat{\theta}) )^{-1} (g^*(\hat{\theta}) - \hat{\theta}),$$ with $\nabla g(\hat{\theta})$ the gradient of $g$ evaluated at $\hat{\theta}$. Consistency of $\hat{\theta}^{R*}$ has been discussed in detail by @Salibian2002 [@Salibian2006]. The FRB estimator is computationally much more efficient because the first-step approximations are easy to compute and the linear correction term needs to be calculated only once, since it depends only on the original sample. Moreover, for a robust estimator the fixed-point equations usually correspond to a weighted version of the corresponding equations for the non-robust MLE or generalized least squares estimator. The weights in the equations downweight outlying observations. In such case, the FRB estimator is robust because no matter how many times an outlying observation appears in a bootstrap sample, it receives the same low weight as in the original sample since the weights depend on the estimate $\hat{\theta}$ corresponding to the original sample.
To apply the FRB to the S and MM-estimators for the SUR model, we rewrite the estimating equations of S-estimators in as $$\begin{aligned}
g_4(\tilde{\beta},\tilde{\Sigma}) \quad &= \quad (X^\top ( \tilde{\Sigma}^{-1} \kron \tilde{D}) X)^{-1} X^\top ( \tilde{\Sigma}^{-1} \kron \tilde{D}) y \\
g_3(\tilde{\beta},\tilde{\Sigma}) \quad &= \quad m (Y - \tilde{X} \tilde{\mathcal{B}})^\top \tilde{D} (Y - \tilde{X} \tilde{\mathcal{B}}) \left( \sum_{i=1}^n v_0(\tilde{d}_i) \right)^{-1}\end{aligned}$$ where $\tilde{D} = {\rm diag}(w_0(\tilde{d}_1),\ldots,w_0(\tilde{d}_n))$, $\tilde{d}_i^2 = \tilde{e}_i( \tilde{\mathcal{B}})^\top \tilde{\Sigma}^{-1} \tilde{e}_i( \tilde{\mathcal{B}})$. Similarly, we rewrite the estimating equations of MM-estimators as $$\begin{aligned}
g_1(\hat{\beta},\hat{\Gamma},\tilde{\Sigma}) \quad &= \quad (X^\top ( \hat{\Gamma}^{-1} \kron D) X)^{-1} X^\top ( \hat{\Gamma}^{-1} \kron D) y \\
g_2(\hat{\beta},\hat{\Gamma},\tilde{\Sigma}) \quad &= \quad \phi ( (Y - \tilde{X} \hat{\mathcal{B}})^\top D (Y - \tilde{X} \hat{\mathcal{B}}) )\end{aligned}$$ where $D = {\rm diag}(w_1(d_1),\ldots,w_1(d_n))$, $d_i^2 = {\lvert\tilde{\Sigma}\rvert}^{-1/m} e_i(\hat{\mathcal{B}})^\top \hat{\Gamma}^{-1} e_i(\hat{\mathcal{B}})$, and $\phi(A) = {\lvertA\rvert}^{-1/m} A$ for an $m \times m$ matrix $A$. Now, let $\hat{\theta}=(\hat{\beta}^\top,{\rm vec}(\hat{\Gamma})^\top,{\rm vec}(\tilde{\Sigma})^\top,\tilde{\beta}^\top)^\top$ be the vector which combines the S and MM-estimates for the SUR model and let $$\label{FRBgfunctionMMestimator}
g(\hat{\theta}) = (g_1(\hat{\beta},\hat{\Gamma},\tilde{\Sigma})^\top,g_2(\hat{\beta},\hat{\Gamma},\tilde{\Sigma})^\top,g_3(\tilde{\beta},\tilde{\Sigma})^\top,g_4(\tilde{\beta},\tilde{\Sigma}))^\top.$$ Then, we have that $g(\hat{\theta})=\hat{\theta}$. Expressions for the partial derivatives in $\nabla g$ can be found in the supplementary material.
Based on the FRB estimates $\hat{\theta}^{R*}$ confidence intervals for the model parameters can be constructed by using standard bootstrap techniques. This is shown in more detail in the supplementary material. In the next sections we construct robust test procedures for the SUR model and show how FRB can be used to estimate their null distribution.
Robust Tests for the Regression Parameters {#Robust Tests for the Regression Parameters}
==========================================
Consider the following general null and alternative hypothesis with respect to the regression parameters in the SUR model $$\label{HypothesisTestbeta}
H_0: R \beta = q \quad \text{vs} \quad H_1: R \beta \neq q,$$ for some $R \in \mathbb{R}^{r \times p}$ and $q \in \mathbb{R}^r$. Here $r \leq p$ represents the number of linear restrictions on the regression parameters under the null hypothesis. For example, for $R = (0,\ldots,0,1)$ and $q = 0$ the null hypothesis simplifies to $\beta_{p_mm} = 0$. Note that the null hypothesis can restrict regression parameters of different blocks, e.g., $H_0: \beta_{11} = \beta_{12}$.
For maximum likelihood estimation, the standard test statistic is the well-known likelihood-ratio statistic. With the log-likelihood in it is given by $$\Lambda_{\text{MLE}} = -n \ln \left( \frac{{\lvert\hat{\Sigma}_{\text{MLE}}\rvert}}{{\lvert\hat{\Sigma}_{\text{MLE},r}\rvert}} \right),$$ where $\hat{\Sigma}_{\text{MLE}}$ is the MLE in the full model and $\hat{\Sigma}_{\text{MLE},r}$ the MLE in the restricted model under the null hypothesis. Under the null hypothesis the test statistic is asymptotically chi-squared distributed with $r$ degrees of freedom. See, e.g., @Henningsen2007 for more details on standard test statistics (such as Wald and F-statistics) in SUR models.
A robust likelihood-ratio type test statistic corresponding to MM-estimators can be obtained by using the plug-in principle. Let $\hat{\Sigma}$ denote the unrestricted scatter MM-estimator and $\hat{\Sigma}_r$ the restricted MM-estimator. Then, the robust likelihood-ratio statistic becomes $$\label{RobustLikelihoodRatioS}
\Lambda_{\text{S}} = -n \ln \left( \frac{{\lvert\hat{\Sigma}\rvert}}{{\lvert\hat{\Sigma}_r\rvert}} \right) = - 2nm \ln \left( \frac{\tilde{\sigma}}{\tilde{\sigma}_r} \right),$$ with $\tilde{\sigma}$ and $\tilde{\sigma}_r$ the scale S-estimators of the full and null model, respectively. Similarly to $\Lambda_{\text{MLE}}$, the test statistic $\Lambda_{\text{S}}$ is nonnegative, since $\tilde{\sigma} \leq \tilde{\sigma}_r$ by definition of the S-estimators.
The test statistic $\Lambda_{\text{S}}$ in only depends on S-scale estimators. Hence, the low efficiency of S-estimators may affect the efficiency of tests based on $\Lambda_{\text{S}}$. In the linear regression context, @VanAelst2013b recently introduced an efficient MM-scale estimator corresponding to regression MM-estimators. Analogously, we propose to update the S-estimator of scale $\tilde{\sigma}$ in the SUR model by a more efficient M-scale $\hat{\sigma}$, defined as $$\hat{\sigma} = \tilde{\sigma} \sqrt{\frac{1}{n \delta_1} \sum_{i=1}^n \rho_1 \left( \frac{\sqrt{e_i(\hat{\mathcal{B}})^\top \hat{\Gamma}^{-1} e_i(\hat{\mathcal{B}})}}{\tilde{\sigma}} \right)}.$$ Similarly to $\delta_0$, the constant $\delta_1$ can be chosen as $\delta_1 = {\rm E}_F [\rho_1({\lVerte\rVert})]$ to obtain a consistent estimator at the assumed error distribution $F$, e.g., $F \sim \mathcal{N}_m(0,I_m)$. The likelihood-ratio type test statistic corresponding to this MM-scale estimator is then defined as $$\label{RobustLikelihoodRatioM}
\Lambda_{\text{MM}} = - 2nm \ln \left( \frac{\hat{\sigma}}{\hat{\sigma}_r} \right).$$
Results on the asymptotic distribution and influence function of these test statistics are provided in the supplementary material. Since the asymptotic distribution is only useful for sufficiently large samples, we consider FRB as an alternative to estimate the null distribution of the test statistics. However, since likelihood-ratio type test statistics converge at a higher rate than the estimators themselves, a standard application of FRB leads to an inconsistent estimate of the null distribution of the test statistic [@VanAelst2011]. To overcome this issue, the test statistic $\Lambda_{\text{S}}$ in is rewritten as $$\label{RobustLikelihoodRatioS_consistent}
\Lambda_{\text{S}} = - 2nm \ln \left( \frac{\tilde{s}(\tilde{\mathcal{B}},\tilde{\Gamma})}{\tilde{s}(\tilde{\mathcal{B}}_r,\tilde{\Gamma}_r)} \right),$$ where $(\tilde{\mathcal{B}},\tilde{\Gamma})$ and $(\tilde{\mathcal{B}}_r,\tilde{\Gamma}_r)$ are the S-estimators in the full and null model respectively and where $\tilde{s}(B,G)$ is the multivariate M-estimator of scale corresponding to a given $B \in \mathbb{R}^{p \times m}$ and $G \in \text{PDS}(m)$ with ${\lvertG\rvert}=1$. That is, $\tilde{s}(B,G)$ is the solution of $$\label{MultivariateMscale}
\frac{1}{n} \sum_{i=1}^n \rho_0 \left( \frac{\sqrt{e_i(B)^\top G^{-1} e_i(B)}}{\tilde{s}(B,G)} \right) = \delta_0.$$ Similarly, the MM-based test statistic $\Lambda_{\text{MM}}$ in is rewritten as $$\label{RobustLikelihoodRatioM_consistent}
\Lambda_{\text{MM}} = - 2nm \ln \left( \frac{\hat{s}(\tilde{\mathcal{B}},\tilde{\Gamma},\hat{\mathcal{B}},\hat{\Gamma})}{\hat{s}(\tilde{\mathcal{B}}_r,\tilde{\Gamma}_r,\hat{\mathcal{B}}_r,\hat{\Gamma}_r)} \right),$$ where $$\label{GeneralizedMultivariateMscale}
\hat{s}(\tilde{\mathcal{B}},\tilde{\Gamma},\hat{\mathcal{B}},\hat{\Gamma}) = \tilde{s}(\tilde{\mathcal{B}},\tilde{\Gamma}) \sqrt{\frac{1}{n \delta_1} \sum_{i=1}^n \rho_1 \left( \frac{\sqrt{e_i(\hat{\mathcal{B}})^\top \hat{\Gamma}^{-1} e_i(\hat{\mathcal{B}})}}{\tilde{s}(\tilde{\mathcal{B}},\tilde{\Gamma})} \right)}.$$
Let $\hat{\theta} = (\hat{\beta}^\top,{\rm vec}(\hat{\Gamma})^\top,{\rm vec}(\tilde{\Gamma})^\top,\tilde{\beta}^\top)^\top$ contain the S and MM-estimators of the regression coefficients and error shape matrices for the full model and let $\hat{\theta}_r$ contain the corresponding estimators for the reduced model. Denote $\hat{\Theta} = (\hat{\theta},\hat{\theta}_r)$, then both test statistics can be written in the general form $$\Lambda_. = h(\hat{\Theta}),$$ where the dot in the subscript can be either S or MM and the function $h$ is determined by - or -, respectively. The FRB approximation for the null distribution of this test statistic then consists of the values $$\Lambda_.^{R*} = h^{*}(\hat{\Theta}^{R*}),$$ where $\hat{\Theta}^{R*} = (\hat{\theta}^{R*},\hat{\theta}_r^{R*})$ are the FRB approximations for the regression and shape estimates in the bootstrap samples. It can be checked that the function $h$ satisfies the condition $$\label{consistencycondition}
\nabla h(\hat{\Theta}) = o_p(1),$$ so the partial derivatives of $h$ vanish asymptotically. This condition guarantees that the FRB procedure consistently estimates the null distribution of the test statistic, as shown in @VanAelst2011. Note that the FRB procedure for hypothesis tests is computationally less efficient than for the construction of confidence intervals (see supplementary material) because the S-scales of the full and null model have to be computed by an iterative procedure for each of the bootstrap samples. However, the increase in computation time is almost negligible compared to the time needed by the standard (non-robust) bootstrap for these robust estimators.
Bootstrapping a test statistic to estimate its null distribution requires that the bootstrap samples follow the null hypothesis, even when this hypothesis does not hold in the original data. Therefore, we first construct null data that approximately satisfy the null hypothesis, regardless of the hypothesis that holds in the original data. According to @Salibian2016, for the linear constraints in null data for $\Lambda_{\text{MM}}$ can be constructed as $$(\tilde{X}^{(0)},Y^{(0)}) = (\tilde{X},\tilde{X} \hat{\mathcal{B}}_r + E),$$ with $E = Y - \tilde{X} \hat{\mathcal{B}}$ the residuals in the full model. Bootstrap samples are now generated by sampling with replacement from the null data $(\tilde{X}^{(0)},Y^{(0)})$. Let $(\hat{\mathcal{B}}^{(0)},\hat{\Sigma}^{(0)})$ denote the MM-estimates for the null data in the full model and let $(\hat{\mathcal{B}}_r^{(0)},\hat{\Sigma}_r^{(0)})$ denote the MM-estimates for the null data in the restricted model. Due to affine equivariance we have that $(\hat{\mathcal{B}}^{(0)},\hat{\Sigma}^{(0)}) = (\hat{\mathcal{B}}_r,\hat{\Sigma})$, so these estimates can be obtained without extra computations. However, the estimates for the reduced model cannot be derived from equivariance properties and need to be computed from the transformed data. Similarly, null data can be constructed for $\Lambda_{\text{S}}$. Finally, when $N$ FRB recalculated values $\Lambda_.^{R*}$ of the test statistic have been calculated based on the null data, then the corresponding FRB p-value is given by $$\label{p-value}
\text{p-value}= \frac{(\# \Lambda_.^{R*} > \Lambda_.) + 1}{N + 2},$$ where $\Lambda_.$ is the value of the test statistic at the original sample.
Robust Test for Diagonality of the Covariance Matrix {#Robust Test for Diagonality of the Covariance Matrix}
====================================================
The key feature of the SUR model is the existence of contemporaneous correlation, corresponding to a non-diagonal covariance matrix $\Sigma$. If the covariance matrix is diagonal the SUR model simplifies to $m$ unrelated regression models. Therefore, by testing for diagonality of $\Sigma$ the necessity of a SUR model is evaluated.
Consider the following hypotheses $$\label{HypothesisTestSigma}
H_0: \Sigma \text{ is diagonal } \quad \text{vs} \quad H_1: \Sigma \text{ is not diagonal}.$$ A popular diagonality test for the standard SUR model is the Breusch-Pagan test [@Breusch1980] which is based on the Lagrange multiplier idea [@Baltagi2008]. It measures the total sum of squared correlations: $$\text{LM}_{\text{MLE}} = n \sum_{j<k} r_{jk}^2,$$ with $r_{jk}$ the elements of the sample correlation matrix of the residual vectors $\hat{\varepsilon}_j$, $j=1,\ldots,m$. Here, each $\hat{\varepsilon}_j$ is the residual vector corresponding to a single-equation LS fit in block $j$. Under the null hypothesis $\text{LM}_{\text{MLE}}$ is asymptotically chi-squared distributed with $m(m-1)/2$ degrees of freedom. Evidently, the LS based Breusch-Pagan test is vulnerable to outliers in the data. Therefore, we introduce robust Breusch-Pagan type tests.
Contrary to the classical estimators, the S and MM-estimators in a SUR model do not simplify to their univariate analogues under the null hypothesis. However, to calculate the restricted estimates the S and MM-estimators and corresponding fastSUR algorithm can be adapted such that the equations for the off-diagonal elements of the covariance matrix are excluded. For example, in case of MM-estimators the estimating equations become $$\begin{gathered}
\hat{\beta}_r = (X^\top (\hat{\Sigma}_r^{-1} \kron D) X)^{-1} X^\top (\hat{\Sigma}_r^{-1} \kron D) y \\
\hat{\sigma}_{r,jj} = m \left( \sum_{i=1}^n w_1(d_i) e_{ij}^2(\hat{\mathcal{B}}_r) \right) \left( \sum_{i=1}^n \psi_1(d_i) d_i \right)^{-1}
\end{gathered}$$ for $j=1,\ldots,m$ and with $D = {\rm diag}(w_1(d_1),\ldots,w_1(d_n))$ where $d_i^2 = e_i(\hat{\mathcal{B}}_r)^\top \hat{\Sigma}_r^{-1} e_i(\hat{\mathcal{B}}_r)$. The restricted covariance matrix estimates $\tilde{\Sigma}_r$ and $\hat{\Sigma}_r$ under $H_0$ then become diagonal matrices as needed. Since the tuning constants of the $\rho$-functions are kept fixed, the reduced estimators $(\tilde{\beta}_r,\tilde{\Sigma}_r,\hat{\beta}_r,\hat{\Sigma}_r)$ also have the same breakdown-point and efficiency level as their counterparts in the full model. Moreover, the multivariate structure is not lost, i.e., we still obtain a single weight for each observation across all blocks.
Based on the restricted estimators, we now estimate the correlation between the errors of block $j$ and $k$ as $$r_{jk} = \frac{\sum_{i=1}^n w_1(d_i) e_{ij}(\hat{\mathcal{B}}_r) e_{ik}(\hat{\mathcal{B}}_r)}{\sqrt{\left( \sum_{i=1}^n w_1(d_i) e_{ij}^2(\hat{\mathcal{B}}_r) \right) \left( \sum_{i=1}^n w_1(d_i) e_{ik}^2(\hat{\mathcal{B}}_r) \right)}},$$ with $d_i^2 = e_i( \hat{\mathcal{B}}_r)^\top \hat{\Sigma}_r^{-1} e_i( \hat{\mathcal{B}}_r)$. Based on these correlation estimates we propose a robust Breusch-Pagan test statistic: $$\label{RobustLM}
\text{LM}_{\text{MM}} = n \sum_{j<k} r_{jk}^2.$$ Note that $\text{LM}_{\text{MM}}$ is nonnegative. Similarly, a robust Breusch-Pagan test based on S-estimators, denoted by $\text{LM}_{\text{S}}$, can be defined as well, but it will not benefit from the gain in efficiency of MM-estimators.
From their asymptotic chi-squared distribution (see the supplementary material) non-robust p-values may be derived. Alternatively, FRB can again be used to estimate the null distribution of the test statistics. Note that the robust Breusch-Pagan test statistic only requires the estimates in the restricted model as can be expected for a Lagrange multiplier test. Let $\hat{\theta}_r$ denote the vector that collects all S and MM-estimators in the restricted model. Based on the FRB approximations $\hat{\theta}_r^{R*}$, bootstrap replications for the null distribution of $\text{LM}_{\text{MM}}$ can be generated as $$\text{LM}_{\text{MM}}^{R*} = n \sum_{j<k} (r_{jk}^{R*})^2,$$ with $$r_{jk}^{R*} = \frac{\sum_{i=1}^n w_1(d_i^{R*}) e_{ij}(\hat{\mathcal{B}}_r^{R*}) e_{ik}(\hat{\mathcal{B}}_r^{R*})}{\sqrt{\left( \sum_{i=1}^n w_1(d_i^{R*}) e_{ij}^2(\hat{\mathcal{B}}_r^{R*}) \right) \left( \sum_{i=1}^n w_1(d_i^{R*}) e_{ik}^2(\hat{\mathcal{B}}_r^{R*}) \right)}},$$ where $(d_i^{R*})^2 = e_i(\hat{\mathcal{B}}_r^{R*})^\top (\hat{\Sigma}_r^{R*})^{-1} e_i( \hat{\mathcal{B}}_r^{R*})$, and similarly for $\text{LM}_{\text{S}}$. It is straightforward to check that the consistency condition in holds under $H_0$ for these test statistics, where $h$ is now defined through . Hence, the FRB procedure consistently estimates the null distribution of the test statistics.
To make sure that the bootstrap samples satisfy the null hypothesis, we generate bootstrap samples from the following transformed data $$(\tilde{X}^{(0)},Y^{(0)}) = (\tilde{X},\tilde{X} \hat{\mathcal{B}} + E \hat{\Sigma}^{-1/2}),$$ with $E = Y - \tilde{X} \hat{\mathcal{B}}$ the residuals in the full model. The residuals $E$ of the full SUR model are possibly correlated across blocks. By transforming these residuals with $\hat{\Sigma}^{-1/2}$, this correlation is removed and it can be expected that for the transformed data $$\label{nullmatrix}
\hat{\Sigma}^{(0)} \approx I_m \quad \text{and} \quad \hat{\Sigma}_r^{(0)} \approx I_m,$$ regardless of the hypothesis that holds in the original data. Note that in the SUR model we cannot rely on equivariance properties to obtain the identity matrix exactly because the model is only affine equivariant for transformations within blocks. However, extensive empirical investigation confirmed that holds for the transformed data, and the corresponding value of the test statistic $\text{LM}_{\text{MM}}^{(0)}$ indeed becomes approximately zero. Similarly, null data can be created for $\text{LM}_{\text{S}}$ as well.
Finite-Sample Performance {#Finite-Sample Performance}
=========================
We now investigate by simulation the performance of FRB tests based on the robust likelihood-ratio test statistics $\Lambda_{\text{S}}$ and $\Lambda_{\text{MM}}$ and the robust Breusch-Pagan statistics $\text{LM}_{\text{S}}$ and $\text{LM}_{\text{MM}}$. The tests are performed at the $5\%$ significance level. We study both the efficiency of the tests under the null hypothesis and the power under the alternative as well as their robustness.
In the SUR model, bootstrap samples can be obtained by either case (row) resampling from the original sample $(\tilde{X},Y)$ or by resampling the $m$-dimensional residuals $e_i$, $i=1,\ldots,n$. While the results in the previous sections hold for both types of bootstrapping, in this paper we use case resampling which is a more nonparametric approach than the model based error resampling.
Consider first the following hypothesis test in a SUR model: $$\label{HypothesisTestbeta_simulation}
H_0: \beta_{p_mm} = 0 \quad \text{vs} \quad H_1: \beta_{p_mm} \neq 0.$$ To investigate the efficiency of the test procedures, data are simulated under the null hypothesis. Observations are generated according to a SUR model with three blocks ($m=3$) and two predictors (as well as an intercept) in each block. Hence, there are $p=9$ regression coefficients in the model. The predictor variables are generated independently from a standard normal distribution. The $p$-dimensional vector of regression coefficients equals $\beta = (1,\ldots,1,0)^\top$ such that the null hypothesis holds. The covariance matrix $\Sigma$ is taken to be a correlation matrix with all correlations equal to 0.5. The multivariate errors are generated from either $\mathcal{N}_m(0,\Sigma)$ or $t_m(0,\Sigma)$ (a multivariate elliptical t-distribution with mean zero and scatter $\Sigma$) with 3 degrees of freedom. To investigate the robustness of the procedure we also considered contaminated data. We have generated the worst possible type of outliers, namely bad leverage points, by replacing in each block all the regressors of the first 10% or 30% of the observations by uniform values between -10 and -5 and by adding to each of the corresponding original responses a value that is normally distributed with mean 20 and variance 1.
Robust S-estimators and MM-estimators with maximal breakdown point of 50% are computed. The MM-estimator is tuned to have 90% efficiency. The null distribution of both $\Lambda_{\text{S}}$ and $\Lambda_{\text{MM}}$ are estimated by FRB as explained in Section \[Robust Tests for the Regression Parameters\], using $N=1000$ bootstrap samples. The corresponding p-values are obtained as in . For each simulation setting 1000 random samples are generated for sample sizes $n=25,50,75,100,150,200,250$ and 300 (recall that $n$ represents the number of observations per block). Figure \[paper\_simulationFRBsurLR\_betatestH0\] shows the empirical level of the two tests for both clean and contaminated data.
It can be seen that the empirical levels are close to the 5% nominal level in most cases. The difference between $\Lambda_{\text{S}}$ and $\Lambda_{\text{MM}}$ is mainly seen when the sample size is small. Indeed, for $n=25$, the test using $\Lambda_{\text{MM}}$ performs better than when $\Lambda_{\text{S}}$ is used. Note that outliers in the data only have a limited effect on the rejection rates, showing robustness of the level of the FRB tests.
To investigate the power of the robust tests, we have simulated data sets under the alternative hypothesis. In Figure \[paper\_simulationFRBsurLR\_betatestH1\] we show the power of the tests for samples of size $n=100$ with $\beta = (1,\ldots,1,d)^\top$ where $d$ ranges from 0 to 0.5 with step length 0.1.
From the left plot we see that the power increases quickly when $d$ becomes larger. The power of the robust tests is only slightly lower than for the classical test in the non-contaminated setting. Moreover, the power of the $\Lambda_{\text{MM}}$ test is (slightly) higher than for the $\Lambda_{\text{S}}$ test. The plot on the right shows that the classical test completely fails if the data is contaminated with 10% of bad leverage points. On the other hand, the robust tests are not affected much by the contamination and yield similar power curves as in the case without contamination.
Let us now consider the test for diagonality of the covariance matrix in . First, data are generated under the null hypothesis, i.e., data are simulated as in the previous section, but the multivariate errors are generated from either $\mathcal{N}_m(0,\Sigma)$ or from $t_m(0,\Sigma)$ with $\Sigma$ the identity matrix. The LM test statistic corresponding to both S and MM-estimators is computed. As before, 1000 data sets were generated for each setting. In Figure \[paper\_simulationFRBsurLM\_diagtestH0\] the rejection rates are plotted as a function of sample size for the four cases considered (normal errors, t-distributed errors, 10% contamination and 30% contamination).
The rejection rates in the different cases behave similar. The lower efficiency of S-estimators becomes apparent as the empirical levels of $\text{LM}_{\text{S}}$ are lower in all (but one) cases. For small sample sizes the nominal level is clearly underestimated, but for MM-estimation the nominal level is already reached for $n \geq 75$. The efficiency of the tests is not much affected by heavy tailed errors or contamination which confirms their robustness under the null hypothesis.
To investigate the power of the test procedures, data were simulated under the alternative hypothesis as well. To this end, $\Sigma$ was set equal to an equicorrelation matrix with correlation $\tau$ taking values from 0 to 0.5 with step length 0.1 for the case $n=100$.
The left plot in Figure \[paper\_simulationFRBsurLM\_diagtestH1\] shows the resulting power curves of the classical and robust Breusch-Pagan tests. We see that the test based on MM-estimators performs almost as well as the classical Breusch-Pagan test. For $\tau=0.4$ the empirical level of $\text{LM}_{\text{MM}}$ reaches almost one. The test based on S-estimators performs less well in this setting with $m=3$ blocks. However, we have noted that the performance of $\text{LM}_{\text{S}}$ increases with the number of blocks $m$ in the SUR model. For larger block sizes the difference with $\text{LM}_{\text{MM}}$ becomes negligible. The right plot in Figure \[paper\_simulationFRBsurLM\_diagtestH1\] shows that the classical Breusch-Pagan test cannot handle contamination, resulting in a drastic loss of power. On the other hand, the power of the robust tests is not affected much by the bad leverage points, resulting in power curves that are similar to the uncontaminated case. This setting where $\Sigma$ is an equicorrelation matrix can be considered to be a strong deviation from diagonality because the deviation is present in all covariance elements. Therefore, we also investigated the power of the diagonality test for other structures of $\Sigma$. It turns out that the comparison between the three tests remains the same for other settings. The power curves for the case where only one covariance deviates from zero are given in the supplementary material.
Example: Grunfeld Data {#Example: Grunfeld Data}
======================
As an illustration we consider the well-known Grunfeld data (see, e.g., @Bilodeau2000). This dataset contains information on the annual gross investment of 10 large U.S. corporations for the period 1935-1954. The recorded response is the annual gross investment of each corporation (Investment). Two predictor variables have been measured as well, which are the value of outstanding shares at the beginning of the year (Shares) and the beginning-of-year real capital stock (Capital). One may expect that within the same year the activities of one corporation can affect the others. Hence, the SUR model seems to be appropriate. Unfortunately, the classical and robust estimators of the covariance matrix become singular when all 10 companies are considered. Therefore, we only focus on the measurements of three U.S. corporations: General Electric (GE), Westinghouse (W) and Diamond Match (DM). General Electric and Westinghouse are active in the same field of industry and thus their activities can highly influence each other. Since the interest is in modeling dependencies between the corporations within the same year, a SUR model with three blocks is considered. The model is given by $$\label{Grunfeld_model}
\text{Investment}_{ij} = \beta_{0j} + \beta_{1j} \text{ Shares}_{ij} + \beta_{2j} \text{ Capital}_{ij} + \varepsilon_{ij},$$ with ${\rm Cov}[\varepsilon_{ij},\varepsilon_{ik}]=\sigma_{jk}$ for $i=1,\ldots,20$ and $j,k=1,2,3$.
We consider inference corresponding to the standard MLE and robust MM-estimators. MM-estimates are obtained with 50% breakdown point and a normal efficiency of 90%. For the MLE, inference is obtained by using asymptotic results and standard bootstrap. For MM-estimators, robust inference is based on the asymptotic results as well as on FRB using $N=1000$ bootstrap samples generated by case resampling. Given the small sample size, we may expect that the bootstrap inference is more reliable than the asymptotic inference according the simulation results in the previous section.
Table \[Grunfeld\_beta\_se\] contains the estimates for the regression coefficients and corresponding standard errors (between brackets) based on bootstrap for the SUR model in .
----------------- ----------- --------- --------- ----------- --------- ---------
**Corporation**
Intercept Shares Capital Intercept Shares Capital
-42.270 0.049 0.122 -30.661 0.033 0.152
(27.559) (0.016) (0.034) (26.679) (0.014) (0.026)
-3.684 0.067 0.018 -6.320 0.059 0.117
(8.293) (0.016) (0.074) (10.779) (0.022) (0.102)
-0.716 0.016 0.453 -0.855 0.002 0.614
(1.394) (0.022) (0.144) (0.608) (0.009) (0.093)
----------------- ----------- --------- --------- ----------- --------- ---------
: Estimated regression coefficients and bootstrap standard errors (between brackets) for the MLE and MM-estimator applied to the SUR model for the Grunfeld data. Standard errors have been obtained by classical bootstrap (MLE) or FRB (MM-estimates).[]{data-label="Grunfeld_beta_se"}
We can clearly see that there are differences between the estimates of both procedures. Focusing on the slope estimates, we see that the MM-estimator yields larger effects of Capital (beginning-of-year real capital stock) and smaller effects of Shares (value of outstanding shares at beginning of the year) on annual gross investments than the MLE. The largest differences can be seen in the estimates $\hat{\beta}_{22}$, $\hat{\beta}_{13}$, and $\hat{\beta}_{23}$ and their standard errors. The estimates for the scatter matrix $\Sigma$ and corresponding correlation matrix $R$ are given by $$\hat{\Sigma}_{\text{MLE}} =
\left[
\begin{array}{r@{}l r@{}l r@{}l}
784&.2 & 224&.2 & 19&.4 \\
& & 97&.8 & 6&.5 \\
& & & & 1&.0 \\
\end{array} \right],
\quad
R_{\text{MLE}} =
\left[
\begin{array}{r r r}
1 & 0.81 & 0.69 \\
& 1 & 0.65 \\
& & 1 \\
\end{array} \right],$$ and $$\hat{\Sigma}_{\text{MM}} = \left[
\begin{array}{r@{}l r@{}l r@{}l}
520&.9 & 194&.6 & 6&.1 \\
& & 110&.1 & 2&.6 \\
& & & & 0&.2 \\
\end{array} \right],
\quad
R_{\text{MM}} =
\left[
\begin{array}{r r r}
1 & 0.81 & 0.56 \\
& 1 & 0.52 \\
& & 1 \\
\end{array} \right],$$ respectively. The robust covariance estimates are generally smaller than the classical estimates. Both estimators find large correlations between the errors of the different blocks. The largest correlation occurs between the first two blocks, which correspond to the equations of General Electric and Westinghouse.
Since there are several differences between the non-robust MLE and the robust MM-estimates, we investigate the data for the presence of outliers. Outliers can be detected by constructing a multivariate diagnostic plot as in @Hubert2017. This plot displays the residual distances of the observations versus the robust distance of its predictors. Based on the SUR estimates the residual distances are computed as $$d_i = \sqrt{e_i(\hat{\mathcal{B}}_{\text{MM}})^\top \hat{\Sigma}_{\text{MM}}^{-1} e_i(\hat{\mathcal{B}}_{\text{MM}})}.$$ Similarly, to measure how far an observations lies from the majority in the predictor space, robust distances can be calculated as $$\text{RD}_i = \sqrt{(\tilde{X}_i - \hat{m}_{\text{MM}})^\top \hat{C}_{\text{MM}}^{-1} (\tilde{X}_i - \hat{m}_{\text{MM}})},$$ with $\tilde{X}_i$ the $i$th row of $\tilde{X}$ and where $\hat{m}_{\text{MM}}$ and $\hat{C}_{\text{MM}}$ are MM-estimates of the location and scatter of $\tilde{X}$ [@Tatsuoka2000]. Note that contributions of intercept terms have been removed from $\tilde{X}$ so that only the actual predictors are taken into account. For non-outlying observations with normal errors, the squared residual distances are asymptotically chi-squared distributed with $m$ degrees of freedom as usual. Therefore, a horizontal line at cut-off value $\sqrt{\chi_{m,0.975}^2}$ (the square root of the 0.975 quantile of a chi-squared distribution with $m$ degrees of freedom) is added to the plot to flag outliers. Observations that exceed this cut-off are considered to be outliers. Similarly, if the predictors of the regular observations are approximately normally distributed, then asymptotically the squared robust distances are approximately chi-squared distributed with $p$ degrees of freedom. Therefore, we add a vertical line to the plot at cut-off value $\sqrt{\chi_{p,0.975}^2}$ to identify outliers in the predictor space, i.e., leverage points. An observation is called a vertical outlier if its residual distance exceeds the cut-off but it is not outlying in the predictor space. If the observation is also outlying in the predictor space, it is called a bad leverage point. Observations with small residual distance which are outlying in the predictor space are called good leverage points because they still follow the SUR model. Similarly, a diagnostic plot can be constructed based on the initial S-estimates for the SUR model or even based on the MLE, although the latter will not reliably identify outliers due to the non-robustness of the estimates.
Multivariate diagnostic plots corresponding to our analysis of the Grunfeld data are shown in Figure \[paper\_Grunfeld\_diagnostic\], based on both the MLE and MM-estimates.
![Multivariate diagnostic plots based on the classical estimates (left panel) and robust estimates (right panel) for three companies in the Grunfeld data[]{data-label="paper_Grunfeld_diagnostic"}](paper_Grunfeld_diagnostic.eps){width="\textwidth"}
The diagnostic plot corresponding to the classical non-robust estimates does not reveal any clear outliers. It seems that all observations follow the SUR model. However, outliers may have affected the estimates to the extent that the outliers are masked. Therefore, we consider the robust diagnostic plot corresponding to the MM-estimates. This plot indeed shows a different picture. Three vertical outliers and one bad leverage point are identified, as well as one good leverage point. The three vertical outliers correspond to the years 1946, 1947 and 1948, while the bad leverage point corresponds to the year 1954. Further exploration of the data indicates that the three vertical outliers are mainly due to exceptionally high investments in those three post World War II years. For the final year 1954, the measurements for all variables are rather extreme, most likely due to the postwar booming economy, which explains why this year is flagged as a bad leverage point in the robust analysis. These four outliers may potentially influence the inference results based on MLE, leading to misleading conclusions. To verify the effect of the outliers on the MLE estimates of the parameters, we also calculated the MLE estimates based on the data without the outliers. The results (not shown) confirmed that the outliers and especially the bad leverage point affect the MLE estimates, because without these outliers the MLE estimates highly resemble the MM-estimates in Table \[Grunfeld\_beta\_se\].
The large correlation estimates between the errors of the different blocks already suggested that these correlations should not be ignored, and thus that the SUR model is indispensable. We can now formally test whether it is indeed necessary to use the SUR model. Therefore, we apply the diagonality test in Section \[Robust Test for Diagonality of the Covariance Matrix\] to test the hypotheses in . Table \[Grunfeld\_LM\] shows the results for the Breusch-Pagan test as well as our robust Breusch-Pagan test. The table contains the values of both test statistics, as well as the corresponding asymptotic p-values and bootstrap p-values. The proportionality constant for the asymptotic chi-squared distribution is estimated by using the empirical distribution to calculate the expected value.
**Estimator** **LM** **AS p-value** **B p-value**
--------------- -------- ---------------- ---------------
23.482 0.001 0.003
14.825 0.003 0.019
: Results of the classical and robust Breusch-Pagan test for the hypothesis test in using the Grunfeld data.[]{data-label="Grunfeld_LM"}
We immediately see that at the $5\%$ significance level, the null hypothesis of diagonality is rejected in all cases. Hence, the outliers in this example do not affect the MLE estimates in such a way that the covariance structure of the SUR model is completely hidden.
From an econometric point of view it can now be interesting to investigate whether the predictors Shares and Capital have the same effect on investments for the two energy companies General Electric and Westinghouse. Hence, we test $$\label{Grunfeld_HypothesisTestbeta}
H_0: \beta_{11} = \beta_{12} \text{ and } \beta_{21} = \beta_{22} \quad \text{vs} \quad H_1: \beta_{11} \neq \beta_{12} \text{ or } \beta_{21} \neq \beta_{22}.$$ Table \[Grunfeld\_LR\] contains the values of the likelihood-ratio statistics and corresponding asymptotic and bootstrap p-values.
**Estimator** **$\Lambda$** **AS p-value** **B p-value**
--------------- --------------- ---------------- ---------------
6.728 0.035 0.168
7.255 0.057 0.086
: Classical and robust test results for the hypothesis test in using the Grunfeld data.[]{data-label="Grunfeld_LR"}
If we consider a $5\%$ significance level, then the conclusion is not completely clear for the MLE. The commonly used asymptotic p-value does reject the null hypothesis, but based on the bootstrap p-value we cannot reject the null hypothesis anymore. On the other hand, the robust test yields asymptotic and bootstrap p-values that lie closer together and which do not reject the null hypothesis. Hence, the presence of outliers does not affect the outcome of the robust hypothesis test while it seems to have caused instability for the classical test based on the MLE. Indeed, if we remove the bad leverage point, then the asymptotic p-value corresponding to the MLE already increases to $0.061$ which is in line with the p-value based on the MM-estimator for the full data set.
Conclusion {#Conclusion}
==========
In this paper we have introduced MM-estimators for the SUR model as an extension of S-estimators. MM-estimators combine high robustness (breakdown point) with high efficiency at the central model. Based on these MM-estimators robust inference for the SUR model has been developed based on the FRB principle. We considered likelihood ratio type statistics to test the existence of linear restrictions among the regression coefficients. While MM-estimators update the S-estimates of the regression coefficients and shape matrix, they do not automatically update the S-scale estimate. However, it turns out that more accurate and powerful tests are obtained if a more efficient MM-scale estimator is used.
An important question is whether it is necessary to use a joint SUR model rather than individual linear regression models for each of the blocks. To evaluate the need for a SUR model we proposed a robust alternative for the well-known Breusch-Pagan test. The FRB was used again to obtain a highly reliable test for diagonality of the covariance matrix, i.e., for existence of contemporaneous correlation among the errors in the different blocks of the SUR model.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research has been partially supported by grant C16/15/068 of International Funds KU Leuven and the CRoNoS COST Action IC1408. The computational resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation - Flanders (FWO) and the Flemish Government - department EWI.
Supplementary Material {#supplementary-material .unnumbered}
======================
In the supplementary material we introduce functionals corresponding to MM-estimators and discuss important properties of these MM-functionals such as equivariance, influence function and asymptotic variance. Also influence functions and asymptotic distributions are derived for the proposed robust test statistics. Power curves are included for a situation which is less deviating from diagonality than the equicorrelation matrix. Furthermore, we construct bootstrap confidence intervals based on FRB and evaluate their performance in a simulation study. In addition, we illustrate these confidence intervals on Grunfeld data. The appendix also contains expressions for the partial derivatives required in the FRB procedure, a verification of the consistency conditions for the robust test on regression coefficients, and the proofs of the theorems.
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**Robust Inference for Seemingly Unrelated Regression Models: Supplementary Material**
KEYWORDS: Asymptotic normal efficiency; Influence function; Fast and robust bootstrap; Robust confidence interval
Properties of MM-estimators {#Properties of MM-estimators}
===========================
We investigate the properties of MM-estimators in more detail. To this end, we first introduce MM-functionals corresponding to the MM-estimators introduced in the manuscript. We state equivariance properties of these MM-functionals and investigate their robustness and efficiency by deriving their influence function and asymptotic variance. We present results for both the estimator of the regression coefficients $\hat{\beta}$ and the estimator of the scatter $\hat{\Sigma}$.
Functionals
-----------
Functional versions of S and MM-estimators for the SUR model can be defined as follows.
Let $H: \mathbb{R}^{p+m} \longrightarrow \mathbb{R}$ be the distribution function of $(\tilde{X}^\top,Y^\top)^\top$ and let $\rho_0$ be a $\rho$-function as before. Then, the S-functionals of the SUR model $(\tilde{\mathcal{B}}(H),\tilde{\Sigma}(H))$ are the solutions that minimize ${\lvertC\rvert}$ subject to the condition $${\rm E}_H \left[ \rho_0 \left( \sqrt{e(B)^\top C^{-1} e(B)} \right) \right] = \delta_0,$$ over all $B = {\rm bdiag}(b_1,\ldots,b_m) \in \mathbb{R}^{p \times m}$ and $C \in \text{PDS}(m)$ with $e(B)=Y - B^\top \tilde{X}$.
To define the MM-functionals, we again decompose the scatter matrix functional into a scale and a shape component, i.e., $\tilde{\Sigma}(H) = \tilde{\sigma}^2(H) \tilde{\Gamma}(H)$ such that ${\lvert\tilde{\Gamma}(H)\rvert}=1$.
Let $H: \mathbb{R}^{p+m} \longrightarrow \mathbb{R}$ be the distribution function of $(\tilde{X}^\top,Y^\top)^\top$ and let $\rho_1$ be a $\rho$-function as before. Given the S-scale functional $\tilde{\sigma}(H)$, the MM-functionals of the SUR model $(\hat{\mathcal{B}}(H),\hat{\Gamma}(H))$ minimize $${\rm E}_H \left[ \rho_1 \left( \frac{ \sqrt{e(B)^\top G^{-1} e(B)}}{\tilde{\sigma}(H)} \right) \right],$$ over all $B = {\rm bdiag}(b_1,\ldots,b_m) \in \mathbb{R}^{p \times m}$ and $G \in \text{PDS}(m)$ with ${\lvertG\rvert} = 1$. The MM-functional for covariance is defined as $\hat{\Sigma}(H) = \tilde{\sigma}^2(H) \hat{\Gamma}(H)$.
Note that the S and MM-estimators can be obtained by the choice $H=\hat{H}_n$, the empirical distribution function corresponding to the data.
Equivariance
------------
Similarly as for S-estimators [@Bilodeau2000], it can easily be shown that the MM-functionals in the SUR model are equivariant under affine transformations of the regressors, regression transformations and blockwise scale transformations of the responses.
For ease of notation, let us write the MM-functionals as $\hat{\beta}(X,Y)$ and $\hat{\Sigma}(X,Y)$ with $X = {\rm bdiag} (X_1^\top, \ldots, X_m^\top)$ with $X_j \in \mathbb{R}^{p_j}$. Then, the MM-functionals satisfy the following equivariance properties:
(a) Affine equivariance of regressors: $$\hat{\beta}(X A,Y) = A^{-1} \hat{\beta}(X,Y) \quad \text{and} \quad \hat{\Sigma}(XA,Y) = \hat{\Sigma}(X,Y),$$ with $A= {\rm bdiag}(A_1,\ldots,A_m)$ where the blocks $A_j$ are of size $p_j \times p_j$.
(b) Regression equivariance: $$\hat{\beta}(X,Y + Xa) = \hat{\beta}(X,Y) + a \quad \text{and} \quad \hat{\Sigma}(X,Y + Xa) = \hat{\Sigma}(X,Y),$$ for any $a \in \mathbb{R}^p$.
(c) Scale equivariance of responses: $$\hat{\beta}(X,AY) = \tilde{A} \hat{\beta}(X,Y) \quad \text{and} \quad \hat{\Sigma}(X,AY) = A \hat{\Sigma}(X,Y) A,$$ for any diagonal matrix $A={\rm diag}(a_{11},\ldots,a_{mm})$ with diagonal matrix $\tilde{A} = {\rm diag}(a_{11},\ldots,a_{11},\ldots,a_{mm},\ldots,a_{mm})$ in which each diagonal element $a_{jj}$ of $A$ is repeated $p_j$ times.
Influence Function
------------------
We now derive the influence functions of the MM-functionals introduced above. Since MM-functionals reduce to S-functionals when $\rho_1=\rho_0$, we only have to consider influence functions for MM-functionals. While the breakdown point is a global measure of robustness, the influence function is a local measure of robustness. The influence function of a functional $T$ measures the effect on $T$ of an infinitesimal amount of contamination at a point $z=(\tilde{x}^\top,y^\top)^\top \in \mathbb{R}^{p+m}$. Consider the contaminated distribution $$H_{\epsilon,\Delta_z} = (1 - \epsilon) H + \epsilon \Delta_z,$$ with $\Delta_z$ the point mass distribution at $z$ and $0<\epsilon<1$. Then, the influence function of $T$ is defined as $$\text{IF}(z;T,H) = \lim_{\epsilon \to 0} \frac{T(H_{\epsilon,\Delta_z}) - T(H)}{\epsilon} = \diff(,\epsilon) \left( T(H_{\epsilon,\Delta_z}) \right) \Big{|}_{\epsilon = 0}.$$
To derive the influence function, we consider the SUR model $$Y = B^\top \tilde{X} + \mathcal{E} = X \beta + \mathcal{E} ,$$ where the $p$-dimensional vector $\tilde{X}$ has distribution $K$ and is independent of the $m$-dimensional error variable $\mathcal{E}$. We assume that $\mathcal{E}$ follows a unimodal elliptically symmetric distribution $F_\Sigma$ with density $$f_\Sigma(u)= {\lvert\Sigma\rvert}^{-1/2} g(u^\top\Sigma^{-1}u),$$ where $\Sigma \in \text{PDS}(m)$ and the function $g$ has a strictly negative derivative. The error distribution is thus symmetric around the origin. Let $H_{\beta,\Sigma}$ denote the resulting distribution of $Z=(\tilde{X}^\top,Y^\top)^\top$. The following theorem gives the influence functions of the regression and scatter MM-functionals for model distributions $H_{\beta,\Sigma}$.
\[TheoremInfluenceFunctions\] If $Z=(\tilde{X}^\top,Y^\top)^\top$ has model distribution $H_{\beta,\Sigma}$ as defined above, then the influence functions of the MM-estimators for the SUR model are given by $$\label{InfluenceFunctionbeta}
\text{IF}(z;\hat{\beta},H_{\beta,\Sigma}) = \frac{1}{\eta_1} w_1({\lVerte\rVert}_{\Sigma}) {\rm E}_K [X^\top \Sigma^{-1} X]^{-1} x^\top \Sigma^{-1} e,$$ and $$\text{IF}(z;\hat{\Sigma},H_{\beta,\Sigma}) = \frac{m}{\pi_1} \psi_1({\lVerte\rVert}_{\Sigma}) {\lVerte\rVert}_{\Sigma} \left( \frac{e e^\top}{{\lVerte\rVert}_{\Sigma}^2} - \frac{1}{m} \Sigma \right) + \frac{2}{\gamma_0} (\rho_0({\lVerte\rVert}_{\Sigma}) - \delta_0) \hspace{0.5mm} \Sigma,$$ with $e=y-x\beta$ and where we use the notation ${\lVerta\rVert}_{C}^2 = a^\top C^{-1} a$ for $a \in \mathbb{R}^{m}$ and $C \in \text{PDS}(m)$. With $F:= F_\Sigma $ the constants are given by $$\begin{aligned}
\label{eta_1}
\eta_1 &= {\rm E}_F \left[ \left( 1-\frac{1}{m} \right) w_1({\lVert\mathcal{E}\rVert}_{\Sigma}) + \frac{1}{m} \psi_1'({\lVert\mathcal{E}\rVert}_{\Sigma}) \right], \\
\label{pi_1}
\pi_1 &= \frac{1}{m+2} {\rm E}_F [(m+1) \psi_1({\lVert\mathcal{E}\rVert}_{\Sigma}) {\lVert\mathcal{E}\rVert}_{\Sigma} + \psi_1'({\lVert\mathcal{E}\rVert}_{\Sigma}) {\lVert\mathcal{E}\rVert}_{\Sigma}^2], \\
\label{gamma_0}
\gamma_0 &= {\rm E}_F [\psi_0({\lVert\mathcal{E}\rVert}_{\Sigma}) {\lVert\mathcal{E}\rVert}_{\Sigma}].\end{aligned}$$
Note that the influence function of the regression functional $\hat{\beta}$ is bounded in $e$ but unbounded in $x$. Hence, contamination in the direction of the response has a bounded influence on $\hat{\beta}$. The effect becomes zero for far away outliers because the weight function $w_1({\lVerte\rVert}_{\Sigma})$ becomes zero for large values of its argument. On the other hand, contamination in the predictor space can have an infinitely large effect on the estimator, but only if the corresponding residual is sufficiently small. This means that the point is a good leverage point since it does not deviate from the SUR model. Moreover, the influence function of the scatter functional $\hat{\Sigma}$ only depends on $e$ and is bounded. Hence, contamination in the predictor space does not affect the scatter functional while the effect of contamination in the response remains bounded.
Asymptotic Variance
-------------------
Following @Hampel1986, the asymptotic variance of a functional $T$ is obtained by $$\text{ASV} (T,H) = {\rm E}_H [\text{IF}(z;T,H) \text{IF}(z;T,H)^\top].$$ By using the expressions for the influence functions in Theorem \[TheoremInfluenceFunctions\] we immediately obtain the asymptotic variances of the MM-estimators for the SUR model in Theorem \[TheoremAsymptoticVariance\] below. We use the notation $K_m$ for the commutation matrix of size $m^2 \times m^2$ such that $K_m {\rm vec} (A) = {\rm vec} (A)^\top$ for any matrix $A \in \mathbb{R}^{m \times m}$. Note that vec denotes the vector operator, stacking all columns of its matrix argument into one vector.
\[TheoremAsymptoticVariance\] If $Z=(\tilde{X}^\top,Y^\top)^\top$ has model distribution $H_{\beta,\Sigma}$, then the asymptotic variances of the MM-estimators for the SUR model are given by $$\label{AsymptoticVariancebeta}
\text{ASV} (\hat{\beta},H_{\beta,\Sigma}) = \frac{\alpha_1}{m \eta_1^2} {\rm E}_K [X^\top \Sigma^{-1} X]^{-1},$$ and $$\text{ASV} (\hat{\Sigma},H_{\beta,\Sigma}) = \sigma_1 (I_{m^2} + K_m) (\Sigma \kron \Sigma) + \sigma_2 {\rm vec} (\Sigma) {\rm vec} (\Sigma)^\top.$$ With $F:= F_\Sigma $ as before, the constants $\alpha_1$, $\sigma_1$ and $\sigma_2$ are equal to $$\begin{aligned}
\label{alpha_1}
\alpha_1 &= {\rm E}_F [\psi_1^2({\lVert\mathcal{E}\rVert}_{\Sigma})], \\
\nonumber \sigma_1 &= \frac{m}{\pi_1^2 (m+2)} {\rm E}_F [\psi_1^2({\lVert\mathcal{E}\rVert}_{\Sigma}) {\lVert\mathcal{E}\rVert}_{\Sigma}^2], \\
\nonumber \sigma_2 &= \frac{4}{\gamma_0^2} {\rm E}_F [(\rho_0({\lVert\mathcal{E}\rVert}_{\Sigma})-\delta_0)^2] - \frac{2}{m} \sigma_1,\end{aligned}$$ and $\eta_1$, $\pi_1$ and $\gamma_0$ are given by , and , respectively.
In case of S-estimators ($\rho_1=\rho_0$), these expressions correspond to the asymptotic variances of S-estimators in @Bilodeau2000. Moreover, the asymptotic variance of the scatter $\hat{\Sigma}$ coincides with that in @Lopuhaa1989 and @Salibian2006.
The asymptotic relative efficiency (ARE) for the regression coefficients $\hat{\beta}$, relative to the MLE $\hat{\beta}_{\text{MLE}}$, becomes $$\text{ARE}(\hat{\beta},H_{\beta,\Sigma}) = \frac{\text{ASV}(\hat{\beta}_{\text{MLE}},H_{\beta,\Sigma})}{\text{ASV}(\hat{\beta},H_{\beta,\Sigma})} = \frac{m \eta_1^2}{\alpha_1}.$$ Note that the ARE does not depend on the number of predictors $p$ in the SUR model nor on the distribution of $\tilde{X}$, but only depends on the number of blocks $m$ in the model and the distribution of the errors. Moreover, it can immediately be seen that the ARE of the MM-estimator $\hat{\beta}$ does not depend on the initial loss function $\rho_0$ for the S-estimator, but only depends on the loss function $\rho_1$. Hence, the constant $c_1$ in $\rho_1$ can indeed be tuned to guarantee a desired efficiency at the central model, independently of the breakdown point which is determined by the constant $c_0$ in $\rho_0$.
Asymptotic Results of the Proposed Test Statistics {#Asymptotic Results of the Proposed Test Statistics}
==================================================
Furthermore, we present some asymptotic results of the robust test statistics $\Lambda_.$ and $\text{LM}_.$ (see Sections \[Robust Tests for the Regression Parameters\] and \[Robust Test for Diagonality of the Covariance Matrix\] of the manuscript respectively).
Robust Tests for the Regression Parameters {#robust-tests-for-the-regression-parameters}
------------------------------------------
Under the null hypothesis in the asymptotic distributions of the test statistics $\Lambda_{\text{S}}$ and $\Lambda_{\text{MM}}$ are proportional to a chi-squared distribution with $r$ degrees of freedom. Denote $$\begin{aligned}
\eta_\ell &= {\rm E}_F \left[ \left( 1-\frac{1}{m} \right) w_\ell({\lVert\mathcal{E}\rVert}_{\Sigma}) + \frac{1}{m} \psi_\ell'({\lVert\mathcal{E}\rVert}_{\Sigma}) \right], \\
\gamma_\ell &= {\rm E}_F [\psi_\ell({\lVert\mathcal{E}\rVert}_{\Sigma}) ({\lVert\mathcal{E}\rVert}_{\Sigma})], \\
\alpha_\ell &= {\rm E}_F [\psi_\ell^2({\lVert\mathcal{E}\rVert}_{\Sigma})],\end{aligned}$$ for $\ell=0,1$ and with $F = F_{\Sigma}$. Remark that the constants $\eta_1$, $\gamma_0$ and $\alpha_1$ are already defined in , and respectively. Then, we have the following result.
\[TheoremAsymptoticDistributionLambda\] Let $Z=(\tilde{X}^\top,Y^\top)^\top$ have model distribution $H_{\beta,\Sigma}$. Under the null hypothesis $H_0: R \beta = q$ it holds that $$\Lambda_{\text{S}} \quad \overset{d}{\longrightarrow} \quad \frac{\alpha_0}{\eta_0 \gamma_0} \hspace{1mm} \chi_r^2,$$ and $$\Lambda_{\text{MM}} \quad \overset{d}{\longrightarrow} \quad \frac{\alpha_1}{\eta_1 \gamma_1} \hspace{1mm} \chi_r^2.$$
These asymptotic null distributions can be used to obtain p-values corresponding to the test statistics in the finite-sample case. However, this standard approach requires a sufficiently large sample size and also accurate estimates of the expectations in the proportionality factors to obtain reliable results.
Robustness of these test statistics is investigated through their influence functions. The (first-order) influence function of these test statistics equals zero. Therefore, we consider their second-order influence function [@Croux2008], which is defined as $$\text{IF2}(z;T,H) = \diff(^2,\epsilon^2) \left( T(H_{\epsilon,\Delta_z}) \right) \Big{|}_{\epsilon = 0}.$$ Boundedness of this influence function guarantees stability of the asymptotic level and power of the asymptotic test in presence of contamination [@Heritier1994]. The next theorem yields the second-order influence functions of $\Lambda_{\text{S}}$ and $\Lambda_{\text{MM}}$ at model distribution $H_{\beta,\Sigma}$ under the null hypothesis $H_0$.
\[TheoremInfluenceFunctionsLambda\] If $Z=(\tilde{X}^\top,Y^\top)^\top$ has model distribution $H := H_{\beta,\Sigma}$ and if $H_0$ is true, then the second-order influence functions of $\Lambda_{\text{S}}$ and $\Lambda_{\text{MM}}$ are given by $$\text{IF2}(z;\Lambda_{\text{S}},H) = - \frac{2m \eta_0}{\gamma_0} (R \hspace{0.5mm} \text{IF}(z;\tilde{\beta},H) - q)^\top \left( R \hspace{0.5mm} {\rm E}_K [X^\top \Sigma^{-1} X]^{-1} R^\top \right)^{-1} (R \hspace{0.5mm} \text{IF}(z;\tilde{\beta},H) - q),$$ and $$\text{IF2}(z;\Lambda_{\text{MM}},H) = \left( 1 - \frac{\gamma_1}{2\delta_1} \right) \text{IF2}(z;\Lambda_{\text{S}},H) \\
- \frac{m \eta_1}{\delta_1} (R \hspace{0.5mm} \text{IF}(z;\hat{\beta},H) - q)^\top \left( R \hspace{0.5mm} {\rm E}_K [X^\top \Sigma^{-1} X]^{-1} R^\top \right)^{-1} (R \hspace{0.5mm} \text{IF}(z;\hat{\beta},H) - q).$$
The second-order influence functions in Theorem \[TheoremInfluenceFunctionsLambda\] are unbounded in $x$ but bounded in $e$. The redescending nature of the functions $w_0$ and $w_1$ guarantees that contamination in the response does not affect the test statistics when ${\lVerte\rVert}_{\Sigma}$ becomes large. Hence, only good leverage points can have a large effect on the test statistics. Since $\gamma_j < 2b_j$ and the constants $\eta_j$ and $\gamma_j$ are always positive, the impact of contamination is larger for $\Lambda_{\text{MM}}$ than for $\Lambda_{\text{S}}$. The increased efficiency of MM-estimators thus implies some loss in robustness.
Robust Test for Diagonality of the Covariance Matrix {#robust-test-for-diagonality-of-the-covariance-matrix}
----------------------------------------------------
The following theorem shows that under the null hypothesis in the asymptotic distribution of the robust test statistics $\text{LM}_{\text{S}}$ and $\text{LM}_{\text{MM}}$ is proportional to a chi-squared distribution.
\[TheoremAsymptoticDistributionLM\] Let $Z=(\tilde{X}^\top,Y^\top)^\top$ have model distribution $H_{\beta,\Sigma}$. Assume that $\Sigma$ is a diagonal matrix. Then, it holds that $$\text{LM}_{\text{S}} \quad \overset{d}{\longrightarrow} \quad \frac{m}{(m+2) \gamma_0^2} {\rm E}_F [\psi_0^2({\lVert\mathcal{E}\rVert}_{\Sigma}) {\lVert\mathcal{E}\rVert}_{\Sigma}^2] \hspace{1mm} \chi_{m(m-1)/2}^2,$$ and $$\text{LM}_{\text{MM}} \quad \overset{d}{\longrightarrow} \quad \frac{m}{(m+2) \gamma_1^2} {\rm E}_F [\psi_1^2({\lVert\mathcal{E}\rVert}_{\Sigma}) {\lVert\mathcal{E}\rVert}_{\Sigma}^2] \hspace{1mm} \chi_{m(m-1)/2}^2.$$
To investigate the robustness of the resulting tests, we again derive the second-order influence function of the test statistics under the null hypothesis.
\[TheoremInfluenceFunctionsLM\] If $Z=(\tilde{X}^\top,Y^\top)^\top$ has model distribution $H_{\beta,\Sigma}$ and if $\Sigma$ is a diagonal matrix, then the second-order influence function of $\text{LM}_{\text{S}}$ and $\text{LM}_{\text{MM}}$ are given by $$\text{IF2}(z;\text{LM}_{\text{S}},H) = \frac{2 m^2}{\gamma_0^2} w_0^2({\lVerte\rVert}_{\Sigma}) \sum_{j<k} \frac{e_j^2 e_k^2}{\sigma_{jj} \sigma_{kk}},$$ and $$\text{IF2}(z;\text{LM}_{\text{MM}},H) = \frac{2 m^2}{\gamma_1^2} w_1^2({\lVerte\rVert}_{\Sigma}) \sum_{j<k} \frac{e_j^2 e_k^2}{\sigma_{jj} \sigma_{kk}}.$$
This theorem shows that leverage points do not influence the test statistic and that large response outliers have zero influence as well. The boundedness of the second-order influence functions ensures the stability of the asymptotic level and power of these diagonality tests [@Heritier1994].
Finite-Sample Performance of diagonality test (continued)
=========================================================
In Section \[Finite-Sample Performance\] of the manuscript we have investigated the power of the diagonality test for the situation where $\Sigma$ is an equicorrelation matrix with correlation $\tau$ ranging from 0 to 0.5 with step length 0.1. While for this setting the deviation from diagonality was present in all covariance elements, we now consider a situation that is less diverging from diagonality. In particular, we consider the same simulation setting as in the final paragraph of Section \[Finite-Sample Performance\] but now set $\Sigma$ equal to $$\begin{bmatrix}
1 & \tau & 0 \\
\tau & 1 & 0 \\
0 & 0 & 1 \\
\end{bmatrix},$$ where $\tau$ takes values from 0 to 0.5 with step length 0.1. For data simulated under this alternative hypothesis, the resulting power curves are shown in Figure \[paper\_simulationFRBsurLM\_diagtestH1\_extra\].
The left plot corresponds to the situation with normal errors without outliers. The right panel shows the power curves in case 10% contamination is added to the data as in Section \[Finite-Sample Performance\] of the manuscript. Compared to the equicorrelation setting considered in the manuscript, all power curves increase at a slower pace because we are now considering a difficult case where only one correlation is responsible for the deviation from diagonality. However, when comparing the classical and robust Breusch-Pagan tests, the same conclusions can be drawn as in the manuscript. In absence of contamination the test based on MM-estimators performs almost as well as the classical test. Moreover, in contrast to the classical test its performance is not much affected in the presence of bad leverage points.
Robust Confidence Intervals
===========================
The results in Theorem \[TheoremAsymptoticVariance\] can be used to construct confidence intervals for the parameters in the SUR model based on their MM-estimates. For example, a $100(1-\alpha)\%$ confidence interval for a regression parameter $\beta_{kl}$ can be obtained as $$\label{AsymptoticConfidenceInterval}
[\hat{\beta}_{kl} - z_{1-\alpha/2} \sqrt{\text{ASV}(\hat{\beta}_{kl},\hat{H}_n)/n}, \hat{\beta}_{kl} + z_{1-\alpha/2} \sqrt{\text{ASV}(\hat{\beta}_{kl},\hat{H}_n)/n}],$$ with $z_{\alpha}$ the $\alpha$ quantile of the standard normal distribution and $\text{ASV}(\hat{\beta}_{kl},\hat{H}_n)$ an estimate of the asymptotic variance in based on the empirical distribution corresponding to the data.
Alternatively, regular bootstrap confidence intervals are constructed as follows. Let $(\hat{\beta}_{kl}^*)_1,\ldots,(\hat{\beta}_{kl}^*)_N$ be a set of $N$ parameter estimates based on bootstrap samples. Then a $100(1-\alpha)\%$ percentile confidence interval for $\beta_{kl}$ is obtained as $$[(\hat{\beta}_{kl}^*)_{((N+1)\alpha_L)}, (\hat{\beta}_{kl}^*)_{((N+1)\alpha_R)}],$$ where $(\hat{\beta}_{kl}^*)_{(.)}$ denotes the order statistics corresponding to the bootstrap estimates. Basic percentile (BP) confidence intervals select $\alpha_L=\alpha/2$ and $\alpha_R=1-\alpha/2$. To improve the accuracy of the confidence intervals, the bias-corrected and accelerated (BCa) method [@Efron1987] can be used to determine the confidence levels $\alpha_L$ and $\alpha_R$. See, e.g., @Davison1997 for more details on percentile methods.
As explained in Section \[Fast and Robust Bootstrap\] of the manuscript, confidence intervals based on standard bootstrap are not attractive because they are not robust. Therefore, we propose to construct bootstrap confidence intervals based on the FRB estimates. For example, a $100(1-\alpha)\%$ FRB percentile confidence interval for $\beta_{kl}$ is computed as $$\label{FRBPercentileConfidenceInterval}
[(\hat{\beta}_{kl}^{R*})_{((N+1)\alpha_L)}, (\hat{\beta}_{kl}^{R*})_{((N+1)\alpha_R)}],$$ where $(\hat{\beta}_{kl}^{R*})_1,\ldots,(\hat{\beta}_{kl}^{R*})_N$ is a set of $N$ FRB bootstrap replicates.
Finite-Sample Performance for Confidence Intervals
==================================================
The performance of confidence intervals obtained by FRB based on robust S and MM-estimators for the SUR model is investigated by simulation. We focus on intervals for the regression coefficients $\beta$ with 95% confidence level. The performance is measured by their coverage and their average length.
We consider the same simulation setting as in Section \[Finite-Sample Performance\] of the manuscript. Robust S-estimators and MM-estimators are computed with maximal breakdown point of 50% and the MM-estimator has 90% efficiency. $N=1000$ bootstrap samples are generated for the FRB. Three different confidence intervals are calculated for the slopes in the model: asymptotic confidence intervals (AS) given by , and BP and BCa confidence intervals according to based on FRB. For each simulation setting the coverage is estimated by the fraction of the confidence intervals that contains the true value of the parameter. The reported coverage and average lengths of the confidence intervals are the average results for all slopes in the model. In Figure \[paper\_simulationFRBsurMM\_cover\] the coverage is depicted as a function of sample size, while the average interval lengths are given in Tables \[simulation\_FRBsurMM\_meanlength\_normal\] and \[simulation\_FRBsurMM\_meanlength\_cont10\] for data with normal errors, containing 0% or 10% of contamination, respectively.
![Coverage results of 95% confidence intervals obtained by the AS (dashed), BP (dash-dotted) and BCa (dotted) methods. The solid (red) line represents the nominal level of 95%.[]{data-label="paper_simulationFRBsurMM_cover"}](paper_simulationFRBsurMM_coverage.eps){width="\textwidth"}
--------------- ---------- ------- ------- ------- ------- ------- ------- ------- -------
**Estimator** **Type**
25 50 75 100 150 200 250 300
AS 0.467 0.470 0.406 0.359 0.298 0.260 0.234 0.214
S BP 1.315 0.786 0.538 0.437 0.336 0.283 0.250 0.226
BCa 1.319 0.787 0.539 0.438 0.338 0.284 0.251 0.227
AS 0.533 0.458 0.380 0.331 0.272 0.236 0.212 0.194
MM BP 1.064 0.572 0.431 0.362 0.288 0.245 0.218 0.198
BCa 1.126 0.573 0.433 0.364 0.289 0.246 0.219 0.199
--------------- ---------- ------- ------- ------- ------- ------- ------- ------- -------
: Average length of 95% confidence intervals obtained with the AS, BP and BCa methods for normal errors and without contamination.[]{data-label="simulation_FRBsurMM_meanlength_normal"}
--------------- ---------- ------- ------- ------- ------- ------- ------- ------- -------
**Estimator** **Type**
25 50 75 100 150 200 250 300
AS 0.387 0.349 0.298 0.261 0.216 0.188 0.169 0.155
S BP 1.506 0.751 0.526 0.430 0.334 0.282 0.250 0.226
BCa 1.543 0.754 0.528 0.432 0.336 0.283 0.251 0.227
AS 0.426 0.340 0.283 0.246 0.202 0.175 0.157 0.144
MM BP 1.084 0.587 0.444 0.374 0.297 0.254 0.226 0.205
BCa 1.108 0.590 0.446 0.376 0.298 0.255 0.227 0.206
--------------- ---------- ------- ------- ------- ------- ------- ------- ------- -------
: Average length of 95% confidence intervals obtained with the AS, BP and BCa methods for normal errors and 10% bad leverage points.[]{data-label="simulation_FRBsurMM_meanlength_cont10"}
From Figure \[paper\_simulationFRBsurMM\_cover\] we can see that the coverage for S and MM-estimators is very similar for all settings. These results clearly show that the FRB confidence intervals reach the nominal 95% coverage level much sooner (for $n \geq 50$ already) then the asymptotic confidence intervals, which for $n=300$ still haven’t completely reached the nominal level. Moreover, there is almost no difference between the two types of FRB percentile confidence intervals. Hence, the more complex BCa intervals do not seem to offer any gain over the more simple basic percentile intervals in this case. For the situations with normal and t-distributed errors, the coverage converges to the nominal level for all three methods. On the other hand, for data contaminated with 10% or 30% bad leverage points in each block, the asymptotic confidence intervals fail to get close to 95% coverage while the FRB confidence intervals still reach the nominal level quickly. This clearly shows the robustness of the FRB based confidence intervals.
Using MM-estimators does not yield confidence intervals with better coverage compared to S-estimators. However, as can be seen from Table \[simulation\_FRBsurMM\_meanlength\_normal\] and \[simulation\_FRBsurMM\_meanlength\_cont10\], confidence intervals based on MM-estimators are generally shorter than those based on S-estimators. The increased efficiency of the MM-estimators thus leads to more informative confidence intervals. These tables also show that the asymptotic confidence intervals are much shorter than the FRB intervals in all cases. However, these intervals are too short, resulting in (severe) under-coverage as seen in Figure \[paper\_simulationFRBsurMM\_cover\]. Note that in terms of average length there is again little difference between the BCa and BP confidence intervals. Finally, by comparing the two tables it can be seen that 10% of bad leverage points does not affect the average length of the FRB confidence intervals much in this setting.
Similarly as for the regression coefficients, confidence intervals for the elements of the scatter matrix $\Sigma$ or shape matrix $\Gamma$ and scale $\sigma$ can be constructed. For the shape matrix, the behavior of the confidence intervals is the same as for the regression coefficients. For the scale and the elements of the scatter matrix the performance is generally worse in presence of contamination. The reason is that the scale S-estimator is not redescending and thus contamination has a persistent effect on the scale estimate which also affects the confidence intervals.
In summary, we can conclude that asymptotic confidence intervals only yield reliable results for clean data with large sample size while FRB confidence intervals remain reliable for contaminated data and smaller sample sizes. Moreover, MM-estimators yield more informative inference than S-estimators.
Example: Grunfeld Data (Continued) {#Example: Grunfeld Data (Continued}
==================================
As in Section \[Example: Grunfeld Data\] of the manuscript we use the Grunfeld data and consider a SUR model with three blocks corresponding to the U.S. corporations General Electric (GE), Westinghouse (W), and Diamond Match (DM). The SUR model is given in . As before, the MM-estimates are calculated with 50% breakdown point and a normal efficiency of 90%. The robust coefficient estimates (and their bootstrap standard errors) are presented in Table \[Grunfeld\_beta\_se\] of the manuscript.
We consider the construction of confidence intervals corresponding to the robust MM-estimators. Confidence intervals are computed based on asymptotic results and the fast and robust bootstrap. For the FRB $N=999$ bootstrap samples are generated using case resampling.
We now compare inference results for regression coefficients in the SUR model. As an example, we first focus on $\beta_{22}$, the slope for predictor Capital in the regression equation for Westinghouse. A histogram of the FRB replications of $\hat{\beta}_{22}$ is presented in Figure \[paper\_Grunfeld\_bootstraphist\_MMbeta22\].
![Histogram of FRB replications of $\hat{\beta}_{22}$ in the SUR model for the Grunfeld data. The solid (red) line corresponds to the MM-estimate $\hat{\beta}_{22}$. Three $95\%$ confidence intervals based on the MM-estimate are shown. The dashed (orange) lines represent the boundaries of the asymptotic confidence interval. The dash-dotted (purple) and dotted (blue) lines show the bound of the BP and BCa confidence intervals, respectively.[]{data-label="paper_Grunfeld_bootstraphist_MMbeta22"}](paper_Grunfeld_bootstraphist_MMbeta22.eps){width="\textwidth"}
The solid (red) vertical line corresponds to the MM-estimate of this coefficient as reported in Table \[Grunfeld\_beta\_se\]. The dashed (orange) lines represent the asymptotic $95\%$ confidence interval based on the MM-estimate as given by . The dash-dotted (purple) and dotted (blue) lines represent the BP and BCa confidence intervals as given by , respectively. It can immediately be seen that the asymptotic confidence interval which relies on the assumption that the distribution of $\hat{\beta}_{22}$ is a normal distribution is much narrower than the other two. However, from the histogram of the FRB replications it is clear that the bootstrap distribution is skewed, which indicates that the normality assumption is not realistic. The two bootstrap confidence intervals do not rely on the normality assumption and they can also better resist the effect of outliers, which makes them more reliable in this case. When checking significance of this regression coefficient, the bootstrap confidence intervals yield a different conclusion than the asymptotic confidence interval. Indeed, both bootstrap percentile confidence intervals contain zero, implying that the coefficient is non-significant, but based on the asymptotic confidence interval the coefficient would be considered significant. However, as seen in the simulations, the asymptotic confidence interval is most likely too small, leading to under-coverage and too optimistic conclusions.
The three confidence intervals for each of the regression coefficients are reported in Table \[Grunfeld\_beta\_conf\].
----------------- ----------------- ----- ----- --- ----- ----- ----- ---- ----- ----- ----- ---- -----
**Corporation** **Coefficient**
$\beta_{01}$ -68 941 7 619 -84 541 21 659 -89 872 17 221
$\beta_{11}$ 0 015 0 051 0 010 0 069 0 012 0 071
$\beta_{21}$ 0 117 0 187 0 083 0 184 0 096 0 186
$\beta_{02}$ -19 106 6 465 -25 379 18 015 -34 574 10 736
$\beta_{12}$ 0 035 0 082 0 017 0 105 0 027 0 121
$\beta_{22}$ 0 029 0 204 -0 117 0 282 -0 123 0 280
$\beta_{03}$ -2 253 0 543 -2 187 0 110 -2 167 0 170
$\beta_{13}$ -0 016 0 020 -0 011 0 021 -0 013 0 019
$\beta_{23}$ 0 554 0 674 0 411 0 773 0 390 0 757
----------------- ----------------- ----- ----- --- ----- ----- ----- ---- ----- ----- ----- ---- -----
: Three $95\%$ confidence intervals (AS, BP and BCa) for the regression coefficients in the SUR model for the Grunfeld data, based on their MM-estimates.[]{data-label="Grunfeld_beta_conf"}
As already seen in the simulations, both percentile confidence intervals are very similar, while the asymptotic confidence intervals are generally much shorter. This illustrates again that asymptotic confidence intervals may lead to unreliable conclusions.
Appendix {#appendix .unnumbered}
========
**Partial derivatives of $g$.** In order to apply the fast and robust bootstrap procedure the partial derivatives $\nabla g$ need to be computed. The Jacobian of $g = (g_1^\top, g_2^\top, g_3^\top, g_4^\top)^\top$ given in equation has the following form $$\nabla g =
\begin{bmatrix}
\diff(g_1,\hat{\beta}) & \diff(g_1,\hat{\Gamma}) & \diff(g_1,\tilde{\Sigma}) & 0 \\[3mm]
\diff(g_2,\hat{\beta}) & \diff(g_2,\hat{\Gamma}) & \diff(g_2,\tilde{\Sigma}) & 0 \\[3mm]
0 & 0 & \diff(g_3,\tilde{\Sigma}) & \diff(g_3,\tilde{\beta}) \\[3mm]
0 & 0 & \diff(g_4,\tilde{\Sigma}) & \diff(g_4,\tilde{\beta})
\end{bmatrix}.$$ Note that the two upper rows in this gradient correspond to the estimating equations of the MM-estimator, while the two bottom rows correspond to those of the S-estimator. The expressions for the S-estimator are omitted because these are similar to the derivatives for the MM-estimator. Consider the matrices $$A_i = {\rm bdiag}(a_i,\ldots,a_i) \quad i=1,\ldots,n$$ where the vector $a_i$ is repeated $m$ times. The vector $a_i$ has length $n$ and is defined as $a_i = (0, \ldots, 0, 1, 0, \ldots, 0)^\top$ with the 1 at the $i$th entry of the vector. Write $\tilde{y}_i = A_i^\top y$ and $x_i = A_i^\top X$, that is, $\tilde{y}_i$ and $x_i$ contain the information of the $i$th observation across all blocks. Introduce the following notation as well: $$\begin{aligned}
{2}
U \quad &= \quad X^\top ( \hat{\Sigma}^{-1} \kron D) X \quad && (p \times p) \\
W \quad &= \quad X^\top ( \hat{\Sigma}^{-1} \kron D) y \quad && (p \times 1) \\
T \quad &= \quad (W \kron I_p)^\top (U^{-1} \kron U^{-1}) \quad && (p \times p^2) \\
V \quad &= \quad (Y - \tilde{X} \hat{\mathcal{B}})^\top D (Y - \tilde{X} \hat{\mathcal{B}}) \quad && (m \times m) \\
S \quad &= \quad {\lvertV\rvert}^{-1/m} \left( I_{m^2} - \frac{1}{m} {\rm vec} (V) {\rm vec} (V^{-1})^\top \right) \quad && (m^2 \times m^2)\end{aligned}$$ Straightforward derivations then lead to the following expressions: $$\begin{aligned}
\diff(g_1,\hat{\beta}) &= T \Bigg( \sum_{i=1}^n \frac{w_1'(d_i)}{d_i \tilde{\sigma}^2} {\rm vec} (x_i^\top \hat{\Gamma}^{-1} x_i) (x_i^\top \hat{\Gamma}^{-1} e_i)^\top \Bigg) - U^{-1} \Bigg( \sum_{i=1}^n \frac{w_1'(d_i)}{d_i \tilde{\sigma}^2} (x_i^\top \hat{\Gamma}^{-1} \tilde{y}_i) (x_i^\top \hat{\Gamma}^{-1} e_i)^\top \Bigg), \\
\diff(g_1,\hat{\Gamma}) &= T \Bigg( \sum_{i=1}^n w_1(d_i) (x_i^\top \kron x_i^\top) (\hat{\Gamma}^{-1} \kron \hat{\Gamma}^{-1}) + \frac{w_1'(d_i)}{2 d_i \tilde{\sigma}^2} {\rm vec} (x_i^\top \hat{\Gamma}^{-1} x_i) {\rm vec} (\hat{\Gamma}^{-1} e_i e_i^\top \hat{\Gamma}^{-1})^\top \Bigg) \\
& \hspace{5mm} - U^{-1} \Bigg( \sum_{i=1}^n w_1(d_i) (\tilde{y}_i^\top \kron x_i^\top) (\hat{\Gamma}^{-1} \kron \hat{\Gamma}^{-1}) + \frac{w_1'(d_i)}{2 d_i \tilde{\sigma}^2} (x_i^\top \hat{\Gamma}^{-1} \tilde{y}_i) {\rm vec} (\hat{\Gamma}^{-1} e_i e_i^\top \hat{\Gamma}^{-1})^\top \Bigg), \\
\diff(g_1,\tilde{\Sigma}) &= T \left( \sum_{i=1}^n \frac{w_1'(d_i) d_i}{2 m \tilde{\sigma}^2} {\rm vec} (x_i^\top \hat{\Gamma}^{-1} x_i) {\rm vec} (\tilde{\Gamma}^{-1})^\top \right) - U^{-1} \left( \sum_{i=1}^n \frac{w_1 (d_i) d_i}{2 m \tilde{\sigma}^2} (x_i^\top \hat{\Gamma}^{-1} \tilde{y}_i) {\rm vec} (\tilde{\Gamma}^{-1})^\top \right), \\
\diff(g_2,\hat{\beta}) &= - S \Bigg( \sum_{i=1}^n w_1(d_i) \big( x_i \kron \tilde{y}_i + \tilde{y}_i \kron x_i - (x_i \kron x_i) (I_p \kron \hat{\beta} + \hat{\beta} \kron I_p) \big) + \frac{w_1'(d_i)}{d_i \tilde{\sigma}^2} {\rm vec} (e_i e_i^\top) (x_i^\top \hat{\Gamma}^{-1} e_i)^\top \Bigg), \\
\diff(g_2,\hat{\Gamma}) &= - S \left( \sum_{i=1}^n \frac{w_1'(d_i)}{2 d_i \tilde{\sigma}^2} {\rm vec} (e_i e_i^\top) {\rm vec} (\hat{\Gamma}^{-1} e_i e_i^\top \hat{\Gamma}^{-1})^\top \right), \\
\diff(g_2,\tilde{\Sigma}) &= - S \left( \sum_{i=1}^n \frac{w_1'(d_i) d_i}{2 m \tilde{\sigma}^2} {\rm vec} (e_i e_i^\top) {\rm vec} (\tilde{\Gamma}^{-1})^\top \right).\end{aligned}$$
**Consistency condition of $\Lambda_{\text{S}}$ and $\Lambda_{\text{MM}}$.** Consider the $h$ function of $\Lambda_{\text{S}}$ defined through equations and (a similar derivation holds for $\Lambda_{\text{MM}}$). In order for the partial derivatives of $h$ to vanish it is sufficient to show that the partial derivatives of $\tilde{s}(b,G)$ converge to zero for $(\tilde{\beta},\tilde{\Gamma})$. Differentiating with respect to $b$ leads to $$\frac{1}{n} \sum_{i=1}^n \frac{\psi_0 (d_i(b,G))}{\tilde{s}^2(b,G)} \left( \diff(,b) \left( \sqrt{e_i^\top(b) \phi(G^{-1}) e_i(b)} \right) \tilde{s}(b,G) - d_i \diff({\tilde{s}(b,G)},b) \right) = 0.$$ Rearranging terms and evaluating the inner derivative we obtain $$\diff({\tilde{s}(b,G)},b) = - \left( \sum_{i=1}^n \frac{w_0(d_i(b,G))}{\tilde{s}^2(b,G)} X_i^\top \phi(G^{-1}) e_i(b) \right) \left( \sum_{i=1}^n \frac{\psi_0 (d_i(b,G)) d_i(b,G)}{\tilde{s}(b,G)} \right)^{-1},$$ which is exactly zero for $(b,G) = (\tilde{\beta},\tilde{\Gamma})$ due to the estimating equations of $\tilde{\beta}$.
Differentiating with respect to $G$ leads to $$\frac{1}{n} \sum_{i=1}^n \frac{\psi_0 (d_i(b,G)) d_i(b,G)}{\tilde{s}(b,G)} \diff({\tilde{s}(b,G)},G) = \frac{1}{n} \sum_{i=1}^n \frac{\psi_0 (d_i(b,G))}{2 \tilde{s}^2(b,G) d_i(b,G)} \diff(,G) \left( \sqrt{e_i^\top(b) \phi(G^{-1}) e_i(b)} \right).$$ The right hand-side of this equality can be simplified to $$- {\lvertG^{-1}\rvert}^{-1/m} G^{-1} \left( \sum_{i=1}^n \frac{\psi_0 (d_i(b,G))}{2 \tilde{s}^2(b,G) d_i(b,G)} e_i(b) e_i^\top(b) \right) G^{-1} + \frac{1}{m} {\lvertG^{-1}\rvert}^{-1/m} G^{-1} \sum_{i=1}^n \frac{\psi_0 (d_i(b,G)) d_i(b,G)}{2}.$$ By evaluating the previous line at $(b,G) = (\tilde{\beta},\tilde{\Gamma})$ and using the estimating equations for $\tilde{\Sigma}$, it can be shown that the right hand-side reduces to zero.
Let $Z=(\tilde{X}^\top,Y^\top)^\top$ have model distribution $H=(K,F)$ with $K$ the distribution of $\tilde{X}$ and $F:=F_\Sigma$ the elliptically symmetric distribution of $Y$. Let us denote $H_{\epsilon} = H_{\epsilon,\Delta_{z}}$ to simplify the notation. The influence function of $\hat{\beta}(H)$ is obtained by differentiating the estimating equations for $\hat{\beta}(H_{\epsilon})$ w.r.t. $\epsilon$ and evaluating the result at $\epsilon = 0$. The derivation of these equations is similar as in the finite-sample case. For a general distribution function $H$ of $Z=(\tilde{X}^\top,Y^\top)^\top$, the estimating equations of the MM-functionals $\hat{\beta}(H)$ and $\hat{\Sigma}(H)$ are given by $$\begin{gathered}
\int w_1(d(H)) x^\top \hat{\Sigma}^{-1}(H) e(H) \hspace{0.5mm} dH(z) = 0 \\
\hat{\Sigma}(H) \int \psi_1(d(H)) d(H) \hspace{0.5mm} dH(z) = m \int w_1(d(H)) e(H) e(H)^\top \hspace{0.5mm} dH(z)\end{gathered}$$ where $d^2(H) = e(H)^\top \hat{\Sigma}^{-1}(H) e(H)$ and $e(H) = y - x \hat{\beta}(H)$. We thus have $$\diff(,\epsilon) \left[ \int w_1(d(H_{\epsilon})) x^\top \hat{\Sigma}^{-1}(H_{\epsilon}) e(H_{\epsilon}) \hspace{0.5mm} dH_{\epsilon}(z) \right] \Bigg{|}_{\epsilon = 0} = 0,$$ which can be rewritten as $$\diff(,\epsilon) \bigg[ (1-\epsilon) \int w_1(d(H_{\epsilon})) x^\top \hat{\Sigma}^{-1}(H_{\epsilon}) e(H_{\epsilon}) \hspace{0.5mm} dH(z) + \epsilon \int w_1(d(H_{\epsilon})) x^\top \hat{\Sigma}^{-1}(H_{\epsilon}) e(H_{\epsilon}) \hspace{0.5mm} d\Delta_{z}(z) \bigg] \Bigg{|}_{\epsilon = 0} = 0.$$ Applying the chain rule and using the estimating equation at $H$ yields $$\diff(,\epsilon) \left[ \int w_1(d(H_{\epsilon})) x^\top \hat{\Sigma}^{-1}(H_{\epsilon}) e(H_{\epsilon}) \hspace{0.5mm} dH(z) \right] \Bigg{|}_{\epsilon = 0} + \int w_1(d(H)) x^\top \hat{\Sigma}^{-1}(H) e(H) \hspace{0.5mm} d\Delta_{z} (z) = 0.$$ The second term simplifies to $w_1({\lVerty\rVert}_{\Sigma}) x^\top \Sigma^{-1} y$. Differentiation of the first term and symmetry of $F$ yields $$\int \diff(,\epsilon) \left( w_1(d(H_{\epsilon})) \right) \Big{|}_{\epsilon = 0} x^\top \Sigma^{-1} y \hspace{0.5mm} dH(z) + \int w_1({\lVerty\rVert}_{\Sigma}) x^\top \Sigma^{-1} \diff(,\epsilon) \left( e(H_{\epsilon}) \right) \Big{|}_{\epsilon = 0} \hspace{0.5mm} dH(z).$$ Computing the inner derivatives and simplifying the result leads to $$- \int \frac{w_1'({\lVerty\rVert}_{\Sigma})}{{\lVerty\rVert}_{\Sigma}} y^\top \Sigma^{-1} x \text{IF}(z;\hat{\beta},H) x^\top \Sigma^{-1} y \hspace{0.5mm} dH(z) - \int w_1({\lVerty\rVert}_{\Sigma}) x^\top \Sigma^{-1} x \hspace{0.5mm} dH(z) \hspace{0.5mm} \text{IF}(z;\hat{\beta},H).
\label{A.1}$$ Splitting the first integral into a $\tilde{x}$ and a $y$ component yields $$\int x^\top \Sigma^{-1/2} \left( \int \frac{w_1'({\lVerty\rVert}_{\Sigma})}{{\lVerty\rVert}_{\Sigma}} \Sigma^{-1/2} y y^\top \Sigma^{-1/2} \hspace{0.5mm} dF(y) \right) \Sigma^{-1/2} x \hspace{0.5mm} dK(\tilde{x}) \hspace{0.5mm} \text{IF}(z;\hat{\beta},H).$$ Using symmetry and results in [@Lopuhaa1999] this can be rewritten as $$\int x^\top \Sigma^{-1/2} \left( \frac{1}{m} \int w_1'({\lVerty\rVert}_{\Sigma}) {\lVerty\rVert}_{\Sigma} \hspace{0.5mm} dF(y) \hspace{0.5mm} I_m \right) \Sigma^{-1/2} x \hspace{0.5mm} dK(\tilde{x}) \hspace{0.5mm} \text{IF}(z;\hat{\beta},H).$$ Combining both integrals in now yields $$- {\rm E}_F \left[ \frac{1}{m} w_1'({\lVertY\rVert}_{\Sigma}) {\lVertY\rVert}_{\Sigma} + w_1({\lVertY\rVert}_{\Sigma}) \right] {\rm E}_K [X^\top \Sigma^{-1} X] \hspace{0.5mm} \text{IF}(z;\hat{\beta},H) \\
+ w_1({\lVerty\rVert}_{\Sigma}) x^\top \Sigma^{-1} y = 0.$$ Rearranging terms leads to the result in .
Consider $Z=(\tilde{X}^\top,Y^\top)^\top$ with model distribution $H:=H_{0,\Sigma}$. The asymptotic variance of $\hat{\beta}$ is given by $$\text{ASV} (\hat{\beta},H) = \int \text{IF}(z;\hat{\beta},H) \text{IF}^\top(z;\hat{\beta},H) \hspace{0.5mm} dH(z).$$ Using the expression for the influence function in , we obtain $$\int \frac{1}{\eta_1^2} w_1^2({\lVerty\rVert}_{\Sigma}) {\rm E}_K [X^\top \Sigma^{-1} X]^{-1} x^\top \Sigma^{-1} y y^\top \Sigma^{-1} x {\rm E}_K [X^\top \Sigma^{-1} X]^{-1} \hspace{0.5mm} dH(z).$$ Splitting the remaining integral yields $$\int x^\top \Sigma^{-1/2} \left( \int w_1^2({\lVerty\rVert}_{\Sigma}) \Sigma^{-1/2} y y^\top \Sigma^{-1/2} \hspace{0.5mm} dF(y) \right) \Sigma^{-1/2} x \hspace{0.5mm} dK(\tilde{x}),$$ which by symmetry can be rewritten as $$\frac{\alpha_1}{m} {\rm E}_K [X^\top \Sigma^{-1} X].$$ Combining the results yields . For a general distribution $H_{\beta,\Sigma}$ this result is obtained by using the affine equivariance property.
To proof Theorem \[TheoremAsymptoticDistributionLambda\], we need the following lemma.
\[Lemmafirstorderapprox\] Under $H_0: R \beta = q$ and the conditions of Theorem \[TheoremAsymptoticDistributionLambda\], it holds that $$\label{firstorderapprox}
\sqrt{n} (\tilde{\beta} - \tilde{\beta}_r) = \sqrt{n} {\rm E}_K [X^\top \Sigma^{-1} X]^{-1} R^\top \left( R \hspace{0.5mm} {\rm E}_K [X^\top \Sigma^{-1} X]^{-1} R^\top \right)^{-1} ( R \tilde{\beta} - q ) + o_p(1).$$
We prove the lemma for the simple case $H_0: \beta_{p_mm} = 0$, that is, $R=(0,\ldots,0,1)$ and $q=0$. First, application of the delta method yields the following first-order approximation $$\sqrt{n} (\tilde{\beta} - \beta) = \frac{1}{\sqrt{n} \eta_0} \sum_{i=1}^n w(d_i(\beta,\tilde{\Sigma})) \Omega^{-1} x_i^\top \tilde{\Sigma}^{-1} e_i(\beta) + o_p(1),$$ where $d_i^2(\beta,\Sigma) = e_i^\top(\beta) \Sigma^{-1} e_i^\top(\beta)$, $e_i^\top(\beta) = \tilde{y}_i - x_i \beta$ and $\Omega = {\rm E}_K [X^\top \Sigma^{-1} X]$. If we replace $\tilde{\Sigma}$ with its true value $\Sigma$, we obtain an asymptotic equivalent expression. A similar expression is true for $\tilde{\beta}_r$. Decompose $\beta=(\beta^{(1)^{\scriptstyle t}},\beta^{(2)})^\top$ with $\beta^{(2)} = \beta_{p_mm}$ and similarly for $x_i$ and other variables. Then a first-order approximation for $\tilde{\beta}_r^{(1)}$ is given by $$\sqrt{n} (\tilde{\beta}_r^{(1)} - \beta^{(1)}) = \frac{1}{\sqrt{n} \eta_0} \sum_{i=1}^n w(d_i(\beta,\tilde{\Sigma}_r)) (\Omega^{(1)})^{-1} x_i^{(1)^{\scriptstyle t}} \tilde{\Sigma}_r^{-1} e_i(\beta) + o_p(1),$$ with $\Omega^{(1)} = {\rm E}_K [X^{(1)^{\scriptstyle t}} \Sigma^{-1} X^{(1)}]$, since under $H_0$ it holds that $e_i(\beta) = \tilde{y}_i - x_i^{(1)} \beta^{(1)}$.
In this simple case it is easy to show that the $p$th component of the right hand-side of equation is equal to $\tilde{\beta}^{(2)}$. Therefore, we only need to prove the lemma for the remaining components. Denote $P$ as the $(p-1) \times p$ elimination matrix, i.e., $P=(I_{p-1},(0,\ldots,0))$. Then, using the first-order approximations we obtain $$\sqrt{n} (\tilde{\beta}^{(1)} - \tilde{\beta}_r^{(1)}) = \frac{1}{\sqrt{n} \eta_0} \sum_{i=1}^n w(d_i(\beta,\Sigma)) \left[ P \Omega^{-1} x_i^\top - (\Omega^{(1)})^{-1} x_i^{(1)^{\scriptstyle t}} \right] \Sigma^{-1} e_i(\beta) + o_p(1).$$ Considering the general expression for the inverse of a block matrix, the terms between brackets reduce to $$P \Omega^{-1} R^\top (R \Omega^{-1} R^\top)^{-1} R \Omega^{-1} x_i^\top.$$ Consequently, we have that $$\sqrt{n} (\tilde{\beta}^{(1)} - \tilde{\beta}_r^{(1)}) = P \Omega^{-1} R^\top (R \Omega^{-1} R^\top)^{-1} R \frac{1}{\sqrt{n} \eta_0} \sum_{i=1}^n w(d_i(\beta,\Sigma)) \Omega^{-1} x_i^\top \Sigma^{-1} e_i(\beta) + o_p(1).$$ In the last line we recognize the first-order approximation of $\tilde{\beta}$. Hence, $$\sqrt{n} (\tilde{\beta}^{(1)} - \tilde{\beta}_r^{(1)}) = \sqrt{n} P \Omega^{-1} R^\top (R \Omega^{-1} R^\top)^{-1} \tilde{\beta}^{(2)} + o_p(1).$$ By combining these results the lemma is proven for the case $H_0: \beta_{p_mm} = 0$. To obtain the general result, we need to obtain a first-order approximation for $\tilde{\beta}_r$ starting from its (general) estimating equation and continue as above, but this derivation is quite lengthy and therefore is omitted.
Consider $\Lambda_{\text{S}}$ first. Expectations in the proof are with respect to $K$. Write $\tilde{y}_i = A_i^\top y$ and $x_i = A_i^\top X$ as above. Application of the delta method permits us to rewrite the test statistic as $$\Lambda_{\text{S}} = -n \ln \left( \frac{{\lvert\tilde{\Sigma}\rvert}}{{\lvert\tilde{\Sigma}_r\rvert}} \right) = - nm \ln \left( \frac{\tilde{\sigma}^2}{\tilde{\sigma}_r^2} \right) = nm \frac{\tilde{\sigma}_r^2 - \tilde{\sigma}^2}{\tilde{\sigma}_r^2} + o_p(1).$$ An alternative to is to define $\tilde{s}(b,G)$ as the solution of $$\label{MultivariateMscale2}
\frac{1}{n} \sum_{i=1}^n \rho_0 \left( \frac{\sqrt{e_i(b)^\top \phi(G^{-1}) e_i(b)}}{\tilde{s}(b,G)} \right) = \delta_0,$$ with $\phi(A) = {\lvertA\rvert}^{-1/m} A$ for a $m \times m$ matrix $A$ and where $e_i(b) = A_i^\top (y - X b) = \tilde{y}_i - x_i b$. Now, $G$ can be any positive definite matrix of size $m \times m$ (without imposing the restriction that ${\lvertG\rvert} = 1$). Moreover, it holds that $\tilde{s}(\tilde{\beta},\tilde{\Gamma}) = \tilde{\sigma}$ and $\tilde{s}(\tilde{\beta}_r,\tilde{\Gamma}_r) = \tilde{\sigma}_r$. Hence, $$\Lambda_{\text{S}} = nm \frac{\tilde{s}^2(\tilde{\beta}_r,\tilde{\Gamma}_r) - \tilde{s}^2(\tilde{\beta},\tilde{\Gamma})}{\tilde{s}^2(\tilde{\beta}_r,\tilde{\Gamma}_r)} + o_p(1).$$ A Taylor expansion of $\tilde{s}^2(\tilde{\beta}_r,\tilde{\Gamma}_r)$ around $(\tilde{\beta},\tilde{\Gamma})$ yields $$\label{eq1}
\tilde{s}^2(\tilde{\beta}_r,\tilde{\Gamma}_r) - \tilde{s}^2(\tilde{\beta},\tilde{\Gamma}) = \frac{1}{2} (\tilde{\beta}_r - \tilde{\beta})^\top \left( \frac{\partial^2 \tilde{s}^2(b,G)}{\partial b^\top \partial b} \right) \Big{|}_{(\tilde{\beta}^{*},\tilde{\Gamma}^{*})} (\tilde{\beta}_r - \tilde{\beta}) + o_p(1/n),$$ where $\tilde{\beta}^{*}$ is an intermediate point between $\tilde{\beta}_r$ and $\tilde{\beta}$ and $\tilde{\Gamma}^{*}$ is an intermediate point between $\tilde{\Gamma}_r$ and $\tilde{\Gamma}$. Due to the definition of the S-estimator, the first-order derivatives vanish (see also the consistency condition of $\Lambda_{\text{S}}$). The second-order derivative of $G$ and the mixed derivative can be shown to be of order $o_p(1/n)$. Then we simplify the second-order derivative w.r.t. $b$. It can be shown that $$\left( \sum_{i=1}^n \frac{\psi_0(d_i(b,G)) d_i(b,G)}{\tilde{s}(b,G)} \right) \diff({\tilde{s}(b,G)},b) = - \left( \sum_{i=1}^n \frac{w_0(d_i(b,G))}{\tilde{s}^2(b,G)} x_i^\top \phi(G^{-1}) e_i(b) \right),$$ with $d_i^2(b,G) = e_i^\top(b) \phi(G^{-1}) e_i(b) / \tilde{s}^2(b,G)$. Taking derivatives w.r.t. $b$, we obtain $$\left( \frac{\partial^2 \tilde{s}(b,G)}{\partial b^\top \partial b} \right) \left( \sum_{i=1}^n \frac{\psi_0(d_i(b,G)) d_i(b,G)}{\tilde{s}(b,G)} \right) + \left( \diff({\tilde{s}(b,G)},b) \right) \left( \diff(,b) \sum_{i=1}^n \frac{\psi_0(d_i(b,G)) d_i(b,G)}{\tilde{s}(b,G)} \right)^\top,$$ for the left hand side and $$\begin{gathered}
\sum_{i=1}^n \frac{w_0(d_i(b,G))}{\tilde{s}^2(b,G)} x_i^\top \phi(G^{-1}) x_i + \sum_{i=1}^n \frac{w_0'(d_i(b,G))}{d_i(b,G) \tilde{s}^4(b,G)} x_i^\top \phi(G^{-1}) e_i(b) e_i^\top(b) \phi(G^{-1}) x_i \\
+ \sum_{i=1}^n \frac{w_0(d_i(b,G)) + w_0'(d_i(b,G)) d_i(b,G)/2}{\tilde{s}^4(b,G)} x_i^\top \phi(G^{-1}) e_i(b) \left( \diff({\tilde{s}^2(b,G)},b) \right)^\top,\end{gathered}$$ for the right hand side. Since $\tilde{\beta}$ and $\tilde{\beta}_r$ are consistent estimators (under $H_0$), also $\tilde{\beta}^{*}$ is consistent. Similarly for $\Gamma$. Therefore, since it is true that $$\diff({\tilde{s}(b,G)},b) \Big{|}_{(\tilde{\beta}^{*},\tilde{\Gamma}^{*})} \overset{a.s.}{\longrightarrow} 0,$$ we have $$\left( \frac{\partial^2 \tilde{s}(b,G)}{\partial b^\top \partial b} \right) \Big{|}_{(\tilde{\beta}^{*},\tilde{\Gamma}^{*})} \overset{a.s.}{\longrightarrow} \frac{\sigma \eta_0}{\gamma_0} {\rm E}_K [X^\top \Sigma^{-1} X].$$ Then reduces to $$\label{eq2}
\tilde{s}^2(\tilde{\beta}_r,\tilde{\Gamma}_r) - \tilde{s}^2(\tilde{\beta},\tilde{\Gamma}) = \frac{\sigma^2 \eta_0}{\gamma_0} (\tilde{\beta}_r - \tilde{\beta})^\top {\rm E}_K [X^\top \Sigma^{-1} X] (\tilde{\beta}_r - \tilde{\beta}) + o_p(1/n),$$ and the test-statistic can be rewritten as $$\Lambda_{\text{S}} = \frac{nm \eta_0}{\gamma_0} (\tilde{\beta}_r - \tilde{\beta})^\top {\rm E}_K [X^\top \Sigma^{-1} X] (\tilde{\beta}_r - \tilde{\beta}) + o_p(1).$$ Using the result of lemma \[Lemmafirstorderapprox\] this expression for $\Lambda_{\text{S}}$ becomes $$\Lambda_{\text{S}} = \frac{nm \eta_0}{\gamma_0} ( R \tilde{\beta} - q )^\top \left( R \hspace{0.5mm} {\rm E}_K [X^\top \Sigma^{-1} X]^{-1} R^\top \right)^{-1} ( R \tilde{\beta} - q ) + o_p(1).$$ Using the results from Theorem \[TheoremAsymptoticVariance\], we can rewrite this as $$\Lambda_{\text{S}} = \frac{n \alpha_0}{\eta_0 \gamma_0} ( R \tilde{\beta} - q )^\top \left( R \hspace{0.5mm} \text{ASV}(\tilde{\beta},H_{\beta,\Sigma}) R^\top \right)^{-1} ( R \tilde{\beta} - q ) + o_p(1).$$ Finally, the result follows by applying Slutzky’s theorem.
Now, consider $\Lambda_{\text{MM}}$. The proof is similar as above, therefore, we only give a sketch of the proof. Write $\Lambda_{\text{MM}}$ as the difference of $\hat{\sigma}_r^2$ and $\hat{\sigma}^2$. Consider these estimates as a function of $\hat{\beta}$, $\hat{\Gamma}$ and $\tilde{\sigma}^2$ (and their restricted versions respectively). Then a Taylor expansion leads to similar expressions as in and . Since the result of Lemma \[Lemmafirstorderapprox\] can be generalized to MM-estimators, a similar derivation as above ends the proof.
Let $Z=(\tilde{X}^\top,Y^\top)^\top$ have model distribution $H=(K,F_\Sigma)$ with $K$ the distribution of $\tilde{X}$ and $F_\Sigma$ the elliptically symmetric distribution of the error terms. Let us denote $H_{\epsilon} = H_{\epsilon,\Delta_{z}}$ to simplify the notation. We only derive the second-order influence function of $\Lambda_{\text{S}}$ and $\Lambda_{\text{MM}}$ (see and ) under $H_0: \beta_{p_mm}=0$.
First, consider $\Lambda_{\text{S}}$. We introduce its functional version as $$\Lambda_{\text{S}}(H) = - 2m \ln \left( \frac{\tilde{\sigma}(H)}{\tilde{\sigma}_r(H)} \right),$$ with $\tilde{\sigma}(H)$ the population version of $\tilde{\sigma}$. Under the null hypothesis we obtain $$\text{IF2}(z;\Lambda_{\text{S}},H) = \frac{2m}{\sigma} \left( \text{IF2}(z;\tilde{\sigma}_r,H) - \text{IF2}(z;\tilde{\sigma},H) \right),$$ by taking the second-order derivative with respect to $\epsilon$. Hence, the proof requires the second-order influence function of $\tilde{\sigma}$. For the scale functional, the following equation holds: $$\int \rho_0 \left( \frac{\sqrt{e(H)^\top \hat{\Gamma}^{-1}(H) e(H)}}{\tilde{\sigma}(H)} \right) dH(z) = \delta_0,$$ with $e(H) = y - x \hat{\beta}(H)$. To find $\text{IF2}(z;\tilde{\sigma},H)$, we consider this equation for $H=H_{\epsilon}$ and differentiate it twice. Since we need the difference of $\text{IF2}(z;\tilde{\sigma},H)$ and $\text{IF2}(z;\tilde{\sigma}_r,H)$, we only mind about the terms that are different, i.e., only the terms involving $\beta$ since these are different. Performing similar steps as in the proof of Theorem \[TheoremInfluenceFunctions\] we obtain $$\text{IF2}(z;\tilde{\sigma},H) = \frac{\eta_0 \sigma}{\gamma_0} \text{IF}(z;\tilde{\beta},H)^\top \hspace{0.5mm} \Omega \hspace{0.5mm} \text{IF}(z;\tilde{\beta},H) + R(\tilde{\Sigma}),$$ with $\Omega = {\rm E}_K [X^\top \Sigma^{-1} X]$ and where $R(\tilde{\Sigma})$ contains the remaining terms not involving $\beta$. Remark that $R(\tilde{\Sigma})$ has an explicit expression, but to save space we do not show it. Likewise, such an expression can be obtained for $\tilde{\sigma}_r$. Consequently, we have $$\text{IF2}(z;\Lambda_{\text{S}},H) = - \frac{2m \eta_0}{\gamma_0} \text{IF}(z;\tilde{\beta},H)^\top \hspace{0.5mm} \Omega \hspace{0.5mm} \text{IF}(z;\tilde{\beta},H) + \frac{2m \eta_0}{\gamma_0} \text{IF}(z;\tilde{\beta}_r^{(1)},H)^\top \hspace{0.5mm} \Omega^{(1)} \hspace{0.5mm} \text{IF}(z;\tilde{\beta}_r^{(1)},H),$$ with $\Omega^{(1)} = {\rm E}_K [X^{(1)^{\scriptstyle t}} \Sigma^{-1} X^{(1)}]$, where $X^{(1)}$ is defined as in Lemma \[Lemmafirstorderapprox\]. Plugging in the results of Theorem \[TheoremInfluenceFunctions\] we get $$\text{IF2}(z;\Lambda_{\text{S}},H) = - \frac{2m}{\eta_0 \gamma_0} w_0^2({\lVerte(H)\rVert}_{\Sigma}) e(H)^\top \Sigma^{-1} \left[ x \Omega^{-1} x^\top - x^{(1)} (\Omega^{1})^{-1} x^{(1)^{\scriptstyle t}} \right] \Sigma^{-1} e(H),$$ where $x^{(1)}$ is defined similarly as $X^{(1)}$. A comparable reasoning as in Lemma \[Lemmafirstorderapprox\] shows that the previous line reduces to $$- \frac{2m}{\eta_0 \gamma_0} w_0^2({\lVerte(H)\rVert}_{\Sigma}) e(H)^\top \Sigma^{-1} x \Omega^{-1} R^\top \left( R \Omega^{-1} R^\top \right)^{-1} R \Omega^{-1} x^\top \Sigma^{-1} e(H).$$ Now we recognize the influence function of $\tilde{\beta}$ and find $$\text{IF2}(z;\Lambda_{\text{S}},H) = - \frac{2m \eta_0}{\gamma_0} \text{IF}(z;\tilde{\beta},H)^\top R^\top \left( R \Omega^{-1} R^\top \right)^{-1} R \hspace{0.5mm} \text{IF}(z;\tilde{\beta},H),$$ as was to be proven.
Then, consider $\Lambda_{\text{MM}}$ with functional version $$\Lambda_{\text{MM}}(H) = - 2m \ln \left( \frac{\hat{\sigma}(H)}{\hat{\sigma}_r(H)} \right),$$ with $\hat{\sigma}(H)$ the population version of $\hat{\sigma}$. Under the null hypothesis the second-order influence function of $\Lambda_{\text{MM}}$ is again the difference of the second-order influence functions of $\hat{\sigma}$. An identical derivation verifies the result.
We proof the result for $\text{LM}_{\text{MM}}$. Consider a SUR model with two blocks ($m=2$) for ease of notation. The results can be generalized to $m>2$.
Since the estimating equations of $\hat{\Sigma}_r$ are the diagonal parts of equations , $\text{LM}_{\text{MM}}$ can be rewritten as $$\text{LM}_{\text{MM}} = \frac{n m^2 \left( \sum_{i=1}^n w_1(d_i) e_{i1} e_{i2} \right)^2}{\hat{\sigma}_{r,11} \hat{\sigma}_{r,22} \left( \sum_{i=1}^n \psi_1(d_i) d_i \right)^2},$$ with $\hat{\sigma}_{r,jj}$ the $j$th diagonal element of $\hat{\Sigma}_r$. According to the null hypothesis we have $${\rm E}_F [w_1(d) e_1 e_2] = 0,$$ with $d^2=e^\top \Sigma^{-1} e$ and $e=(e_1,e_2)^\top \sim F = F_{\Sigma}$. Moreover, due to a result of @Lopuhaa1999, for the variance we obtain $${\rm Var}_F [w_1(d) e_1 e_2] = \frac{\sigma_{11} \sigma_{22}}{m(m+2)} {\rm E}_F [\psi_1^2({\lVerte\rVert}_{\Sigma}) {\lVerte\rVert}_{\Sigma}^2].$$ Since $w_1(d_i) e_{i1} e_{i2}$, $i=1,\ldots,n$ are independent identically distributed random variables, the central limit theorem gives $$\sqrt{n} \left( \frac{1}{n} \sum_{i=1}^n w_1(d_i) e_{i1} e_{i2} \right) \quad \overset{d}{\longrightarrow} \quad N(0,{\rm Var}_F [w_1(d) e_1 e_2]),$$ or equivalently $$\frac{nm(m+2)}{\sigma_{11} \sigma_{22} {\rm E}_F [\psi_1^2({\lVerte\rVert}_{\Sigma}) {\lVerte\rVert}_{\Sigma}^2]} \left( \frac{1}{n} \sum_{i=1}^n w_1(d_i) e_{i1} e_{i2} \right)^2 \quad \overset{d}{\longrightarrow} \quad \chi_1^2.$$ Since for $j=1,2$, $\hat{\sigma}_{r,jj}$ is a consistent estimator under $H_0$ and $$\frac{1}{n} \sum_{i=1}^n \psi_1(d_i) d_i \quad \overset{a.s.}{\longrightarrow} \quad \gamma_1,$$ the result now follows.
Let $Z=(\tilde{X}^\top,Y^\top)^\top$ have model distribution $H=(K,F_\Sigma)$ with $K$ the distribution of $\tilde{X}$ and $F_\Sigma$ the elliptically symmetric distribution of $Y$. Let us denote $H_{\epsilon} = H_{\epsilon,\Delta_{z}}$ to simplify the notation. We derive the second-order influence function of $\text{LM}_{\text{MM}}$. Again we assume $m=2$ for simplicity.
We introduce the functional version as $$\text{LM}_{\text{MM}}(H) = \frac{\left( \int w_1(d(H)) e_1(H) e_2(H) \hspace{0.5mm} dH(z) \right)^2}{\left( \int w_1(d(H)) e_1^2(H) \hspace{0.5mm} dH(z) \right) \left( \int w_1(d(H)) e_2^2(H) \hspace{0.5mm} dH(z) \right)},$$ with $d^2(H)=e^\top(H) \hat{\Sigma}^{-1}(H) e(H)$ and $e=(e_1,e_2)^\top$. Since the first-order influence function is exactly zero and $$\int w_1(d(H)) y_1 y_2 \hspace{0.5mm} dH(z) = 0,$$ under the null hypothesis, we obtain $$\text{IF2}(z;\text{LM}_{\text{MM}},H) = \frac{2m^2}{\gamma_1^2 \sigma_{11} \sigma_{22}} \left( \diff(,\epsilon) \left( \int w_1(d(H_{\epsilon})) e_1(H_{\epsilon}) e_2(H_{\epsilon}) \hspace{0.5mm} dH_{\epsilon}(z) \right) \Bigg{|}_{\epsilon = 0} \right)^2.$$ Following the same steps as in the proof of Theorem \[TheoremInfluenceFunctions\] the above derivative becomes $$\int \frac{w_1'({\lVerty\rVert}_{\Sigma})}{2{\lVerty\rVert}_{\Sigma}} (-2y \Sigma^{-1} x \text{IF}(z;\hat{\beta},H) + y^\top \text{IF}(z;\hat{\Sigma}^{-1},H) y) y_1 y_2 \hspace{0.5mm} dH(z) - \int w_1(d(H)) y_1 y_2 \hspace{0.5mm} dH(z) + w_1({\lVerty\rVert}_{\Sigma}) y_1 y_2.$$ Due to symmetry and under $H_0$ the integrals vanish. Combining the results, proves the theorem.
|
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author:
- |
B. Ferrario\
Dipartimento di Matematica - Università di Pavia
title: 'Well posedness of a stochastic hyperviscosity-regularized 3D Navier-Stokes equation'
---
Introduction
============
We call stochastic Navier–Stokes problem the following: $$\label{stochNS}
\left\{
\begin{array}{l}
\dfrac{du}{dt} -\nu \Delta u+\left( u\cdot\nabla\right) u+\nabla p
= f+ n
\\[2mm]
\text{div }u =0
\\
u|_{t=0} =u_{0}
\end{array}
\right.$$ Here $u=u(t,x)$ is the 3-dimensional velocity vector field defined for $t\geq 0$ and $x\in D\subseteq \mathbb R^3$, $p=p(t,x)$ is the scalar pressure field, $\nu>0$ is the coefficient of kinematic viscosity, $u_0=u_0(x)$ is the initial velocity, $f=f(t,x)$ and $n=n(t,x)$ are, respectively, the deterministic and stochastic forcing terms. If the spatial domain has a boundary, we assume that $u$ vanishes on $\partial D$.
When there is no noise term $n$, this reduces to the deterministic Navier–Stokes problem which models the motion of viscous fluids. For the 3-dimensional setting, both in the deterministic and in the stochastic case we know the existence of a weak solution but uniqueness is proved in a smaller class, where existence is not known. The question of proving existence and uniqueness on any finite time interval and with any initial data for the deterministic Navier–Stokes equation is one of the Millennium Prize problems (see, e.g., [@feff]). However, for the 3-dimensional problem there are results for small initial data or locally in time, whereas the 2-dimensional problem is well posed (see, e.g., [@temam; @temam2] for the deterministic problem and [@Fla; @funiq] for the 2-dimensional stochastic problem with additive noise).
There have been many attempts to modify the 3-dimensional Navier–Stokes equation in order to prove an existence and uniqueness result. The first models go back to Lions [@lions]. For more recent results, we focus on two particular cases: [@ms] and [@sri]. In [@ms] the Laplacian operator $-\Delta$ is replaced with $(-\Delta)^\alpha$ (for $\alpha >1$) in the [*deterministic*]{} Navier–Stokes equation; the [*stochastic*]{} problem with a similar modification ($-\nu \Delta$ replaced with $-\nu_0\Delta+\nu_1(-1)^\alpha\Delta^\alpha$) is studied in [@sri]. Setting $\alpha>1$ we obtain a model for hyperviscous fluids (see [@sri] and references therein).
Our aim is to analyse the well posedness of the stochastic version of the modified Navier–Stokes equation considered by Mattingly–Sinai in [@ms], that is $$\label{alpha}
\left\{
\begin{array}{l}
\dfrac{du}{dt}+ \nu (-\Delta)^\alpha u+\left( u\cdot\nabla\right) u+\nabla p
=
n
\\[2mm]
\text{div }u =0
\\
u|_{t=0} =u_{0}
\end{array}
\right.$$ We shall consider the model of additive noise, i.e. $n$ is independent of $u$. This is the simplest case, which reduces the technicalities. However, the case of multiplicative noise can be treated in a similar way.
In Section 3, we shall prove an existence and uniqueness result for $\alpha \ge \frac 54$, as conjectured in [@f-gir]. The bound $\alpha > \frac 54$ appeared first in [@ms] for the deterministic problem. Regularity results will be given in Section 4.
Finally, we point out that our technique is different from that of [@ms] or [@sri]; indeed, we use tools from [@temam] and [@Fla].
Notations and preliminaries {#prelim}
===========================
Let the spatial domain be a torus, i.e. the spatial variable $x$ belongs to $\mathcal T=[0,L]^3$ and periodic boundary conditions are assumed.
We introduce the classical spaces for the Navier–Stokes equation (see, e.g., [@temam2; @temam] for all the results in this section). $\mathcal{D}^{\infty}$ is defined as the space of infinitely differentiable divergence free periodic fields $u:\mathcal{T} \to \mathbb R^3$, with zero mean ($ \int_{\mathcal{T}} u(x) dx=0$). Let $H^0$ be the closure of $\mathcal{D}^{\infty}$ in the $[L^2(\mathcal T)]^3$-topology; it is the subspace of $[L^2(\mathcal T)]^3$ of all fields $u$ such that $\mbox{div}\, u=0$, the normal component of $u$ on the boundary is periodic, $\int_{\mathcal{T}}u\left( x\right) dx=0$. We endow $H^0$ with the inner product $
\left\langle u,v\right\rangle=
\int_{\mathcal{T}} u(x)\cdot v(x)\ dx
$ and the associated norm $\left| \cdot \right|$.\
Similarly, for $m \in \mathbb N$ let $H^m$ be the closure of $\mathcal{D}^{\infty}$ in the $[H^m(\mathcal T)]^3$-topology.
Let $A:D(A)\subset H^0\rightarrow H^0$ be the operator $Au=-\Delta u$ (componentwise) with $D(A)=H^2$. This is called Stokes operator and it is a strictly positive unbounded self-adjoint operator in $H^0$, whose eigenvectors $h_j$ form a complete orthonormal basis of the space $H^0$; the eigenvalues $\lambda_j$ are strictly positive and $0<\lambda_1\le \lambda_2 \le \dots$ with $\lambda_j \sim j^{2/3}$ for $j \to \infty$. Since the spatial domain is the torus, we know the expressions of the eigenvectors with their eigenvalues (see, e.g., [@FF]).
The power operators $A^\alpha$ are defined for any $\alpha \in \mathbb R$. If $u=\sum_j u_j h_j$, then $$A^\alpha u=\sum_{j=1}^\infty \lambda^\alpha_j u_j h_j
\quad\text{ and }\quad
|A^\alpha u|^2 = \sum_{j=1}^\infty \lambda_j^{2\alpha}|u_j|^2.$$ We set $|u|_{2\alpha}=|A^\alpha u|$ and the space $H^s$ can be defined (for any $s \in \mathbb R$) as the closure of $\mathcal{D}^{\infty}$ in the $|\cdot|_s$-metric. For $s<0$ the space $H^s$ is the dual space of $H^{-s}$ with respect to the $H^0$-topology. The space $H^{s+r}$ is dense and compactly embedded in $H^s$ for any $r>0$.
Notice that $|u|_m$ is equivalent to the usual $[H^m(\mathcal T)]^3$-norm.
In particular $$|u|_1^2= \langle u, Au\rangle=\sum_{i,j=1}^{3}\int_{\mathcal T}
(\partial_j u_i(x))^2 dx.$$
We have $H^0 =\text{span} \{ h_{k}\} $ and we set $H_n=\text{span} \{h_{k}: |k|\le n\}$; moreover, we denote by $\pi_n$ the projection operator from $H^0$ onto $H_n$. The operators $A$ and $\pi_n$ commute. By $\Pi$ we denote the projector operator from $[L^2(\mathcal T)]^3$ onto $H^0$.\
The operator $-A$ generates in $H^0$ (and in any $H^s$) an analytic semigroup of negative type $\{e^{-tA}\}_{t\ge 0}$ of class $C_0$.
Let $B\left( \cdot,\cdot\right) : H^1\times H^1\rightarrow H^{-1}$ be the bilinear operator defined as $$\label{defB}
\left\langle w,B\left( u,v\right) \right\rangle
=
\sum_{i,j=1}^{3}\int_{\mathcal T} u_i (\partial_i v_j) w_j dx$$ for every $u,v,w\in H^1$. By the incompressibility condition, we have $$\label{incomp}
\langle B\left( u^1,u^2\right),u^2 \rangle =0, \qquad
\langle B\left( u^1,u^2\right),u^3 \rangle =
-\langle B\left( u^1,u^3\right),u^2 \rangle .$$ We shall use the following estimates (see Lemma 2.1 in [@temam2]): $$\label{Bcon4}
|\langle B(u^1,u^2),u^3\rangle|
\le \ c \ |u^1|\ |u^2|_\alpha \ |u^3|_{\alpha}
\qquad \text{ for }\alpha\ge \frac 54,$$ $$\label{BconA}
\begin{split}
|\langle B(u^1,u^2),Au^3\rangle|
&\le
c |u^1|_\alpha |u^2|_1 |Au^3|_{\alpha-1}
\qquad \text{ for }\alpha \ge \frac 54
\\
&=
c |u^1|_\alpha |u^2|_1 |u^3|_{\alpha+1}
\end{split}$$ and similarly $$\label{BconA-2}
\begin{split}
|\langle B(u^1,u^2),Au^3\rangle|
&\le
c |u^1|_1 |u^2|_\alpha |Au^3|_{\alpha-1}
\qquad \text{ for }\alpha \ge \frac 54
\\
&=
c |u^1|_1 |u^2|_\alpha |u^3|_{\alpha+1}.
\end{split}$$ Here and in the following, we denote by $c$ a positive constant, which may vary from place to place.
Main theorem
============
We apply the projection operator $\Pi$ to the first equation in . We get an Itô equation in an infinite dimensional Hilbert space: $$\label{sns}
\left\{
\begin{array}{l}
du(t) +\left[ \nu A^\alpha u(t)+B\big(u(t),u(t)\big)\:\right]\; dt
=A^{-\gamma} \; dw(t)\\
u(0)=u_0
\end{array}
\right.$$ assuming the noise is of white type in time and with spatial covariance independent of $u$. This means that $w$ is a cylindrical Wiener process in $H^0$ defined on a complete probability space with filtration $(\Omega,\mathcal F,\{\mathcal F_t\}_{t\ge 0},\mathbb P)$ (i.e. given a sequence $\{\beta_j\}_j$ of i.i.d. standard Wiener processes, we represent the Wiener process in series as $w(t)= \sum_j \beta_j(t) h_j$). For simplicity we consider the operator in front of $w$ to be a power of the Stokes operator; this is the model studied in [@f-gir].\
For $\alpha=1$, this corresponds to the stochastic Navier–Stokes equation as analysed for instance in [@Fla] for the 2-dimensional setting.\
The technique to study equation comes from [@bens; @vishik; @dpz; @Fla]. First we consider the linear equation, that is the modified stochastic Stokes equation $$\label{ou}
\left\{
\begin{array}{l}
dz(t) +\nu A^\alpha z(t) dt =A^{-\gamma} \; dw(t)\\
z(0)=0
\end{array}
\right.$$ Then, we define $v:=u-z$; this unknown solves the following equation, obtained subtracting equation to equation and bearing in mind the bilinearity of the operator $B$: $$\label{eq-v}
\left\{
\begin{array}{l}
\dfrac{d\;}{dt}v(t) +\left[ \nu A^\alpha v(t)+B\big(v(t),v(t)\big) +
B\big(z(t),v(t)\big) +B\big(v(t),z(t)\big) \:\right]
\\\hspace*{3cm}=-B\big(z(t),z(t)\big) \\
v(0)=u_0
\end{array}
\right.$$ The noise term $A^{-\gamma} \; dw(t)$ has disappeared.
Let $[0,T]$ be any finite time interval. We now state our main result.
\[teo1\] Let $\alpha \ge \frac 54$.\
For any $u_0\in H^1$, if $\gamma>\frac 34$ then there exists a unique process $u$ which is a strong solution of such that $$u \in C([0,T];H^1)\cap L^{\frac {2\alpha}{\alpha-1}}(0,T;H^{\alpha}) \qquad {{\mathbb P}}-a.s.;$$ $u$ is progressively measurable in these topologies and is a Markov process in $H^1$.
Existence
---------
We study pathwise the problems for the unknowns $z$ and $v$. This will imply an existence result for $u$.
For the linear problem we have (see, e.g., Proposition 4.1 in [@f-gir], based on [@dpz])
\[lemma-z\] If $$\label{conv-z}
\alpha+2\gamma>\theta+\frac 32,$$ then equation has a unique strong solution $z$ such that $$\label{nsz:2p}
{{\mathbb P}}\{z\in C([0,T];H^{\theta})\}=1.$$
Now, we work pathwise for the equation satisfied by $v$ and therefore also for $u$.
\[propo-v\] Let $\alpha \ge \frac 54$.\
For any $u_0\in H^1$, if $\gamma>\frac 34$ then there exists a process $v$ which is a strong solution of such that $$v \in C([0,T];H^1)\cap L^2(0,T;H^{1+\alpha}) \qquad {{\mathbb P}}-a.s.$$ $v$ is progressively measurable in these topologies.
From Lemma \[lemma-z\] we have that $z \in C([0,T];H^\alpha)$ ${{\mathbb P}}$-a.s., since $\gamma>\frac 34$. We take the scalar product of equation with $v$ and use -: $$\begin{split}
\frac 12 \frac {d\;}{dt}|v|^2+\nu|v|^2_{\alpha}
& = - \langle B(v,v)+B(v,z)+B(z,v)+B(z,z), v\rangle
\\
&=-
\langle B(v,z)+B(z,z), v\rangle
\\
&\le c |v| |z|_\alpha |v|_\alpha + c |z|_\alpha^2 |v|_\alpha
\\
& \le \frac \nu 2 |v|_\alpha^2+c_\nu
\big(|z|^2_\alpha |v|^2 +|z|_\alpha^4\big) .
\end{split}$$ Then $$\frac {d\;}{dt}|v|^2
\le
c |z|^2_\alpha |v|^2 +c |z|_\alpha^4$$ and from Gronwall lemma: $\displaystyle\sup_{0\le t\le T} |v(t)|^2 <\infty$. Moreover, integrating in time the first inequality above, we have $\int_0^T |v(t)|^2_\alpha dt <\infty$.
Now we take the scalar product of equation with $Av$: $$\frac 12 \frac {d\;}{dt}|v|_1^2+\nu|v|^2_{1+\alpha}=-
\langle B(v,v)+B(v,z)+B(z,v)+B(z,z), Av\rangle .$$ We use and Young inequality to obtain $$\label{st-per-v}
\frac 12 \frac {d\;}{dt}|v|_1^2+\nu |v|^2_{1+\alpha}
\le
\frac \nu 2 |v|^2_{1+\alpha} + c_\nu (|z|_\alpha^2+|v|_\alpha^2)|v|_1^2
+c_\nu |z|_\alpha^4.$$ As usual, from $$\frac {d\;}{dt}|v|_1^2 \le 2 c_\nu (|z|_\alpha^2+|v|_\alpha^2)|v|_1^2
+ 2 c_\nu |z|_\alpha^4,$$ Gronwall inequality, with the result $v \in L^2(0,T;H^\alpha)$ proved before, implies $\displaystyle \sup_{0\le t \le T}|v(t)|^2_1 <\infty$ and integrating in time we get $\int_0^T |v(t)|^2_{1+\alpha}dt <\infty$.
The technique to prove existence is classical (see [@temam]). We consider first the finite dimensional problem in the unknown $v_n$, obtained projecting equation onto $H_n$. This is the Galerkin approximation, for any $n=1,2,\dots$. The above estimates hold uniformly also for the Galerkin sequence: for any $n$ $$\sup_{0\le t \le T}|v_n(t)|^2_1 <c_1, \qquad
\int_0^T |v_n(t)|^2_{1+\alpha}dt < c_2$$ for constants $c_1$ and $c_2$ independent of $n$.\
Any finite dimensional (Galerkin) problem has a solution. By passing to the limit as $n \to \infty$ we get an existence result for . We also need that $\frac{dv_n}{dt}$ is uniformly bounded in $ L^2(0,T;H^{1-\alpha})$. We verify easily that $\frac{dv_n}{dt}\in L^2(0,T;H^{1-\alpha})$ and the norm is bounded uniformly for all $n$, since $A^\alpha v \in L^2(0,T;H^{1-\alpha})$ and according to all the bilinear terms in are in $L^2(0,T;H^0)$. Therefore, we have a compact embedding (see Theorem 2.1, Ch. III in [@temam]) so the Galerkin sequence stays in a compact subset of $L^2(0,T;H^1)$. Therefore there exists a subsequence converging to $v$ as follows: $$\begin{split}
& v_m \to v \quad \text{ weakly in } L^2(0,T;H^{1+\alpha}),\\
& v_m \to v \quad\star-\text{weakly in } L^\infty(0,T;H^1) ,\\
& v_m \to v \quad\text{ strongly in } L^2(0,T;H^1).
\end{split}$$ The strong convergence allows one to pass to the limit in the bilinear term (see Lemma 3.2, Ch. III in [@temam]). Finally, $v \in C([0,T];H^1)$ (see Lemma 1.2, Ch. III in [@temam]). The limit $v$ fulfils all the estimates found above: $v\in C([0,T];H^1)\cap L^2(0,T;H^{1+\alpha})$. $\hfill \Box$
We conclude noting that by interpolation $L^\infty(0,T;H^1)\cap L^2(0,T;H^{1+\alpha})\subset
L^{\frac {2\alpha}{\alpha-1}}(0,T;H^\alpha)$. Since the paths of $z$ belong to $C([0,T];H^\alpha)$ and those of $v$ to $L^\infty(0,T;H^1)\cap L^2(0,T;H^{1+\alpha})$, then $u=v+z \in C([0,T];H^1)\cap L^{\frac {2\alpha}{\alpha-1}}(0,T;H^\alpha)$ ${{\mathbb P}}$-a.s. This concludes the existence result of Theorem \[teo1\]\
The measurability property is inherited by from the Galerkin sequence.
The spatial covariance of the noise can be taken of a more general form. Indeed, what is needed is that pathwise we have $z \in C([0,T];H^\alpha)$. Hence, we can prove the same result of Theorem \[teo1\] when instead of $A^{-\gamma} dw(t)$ the noise is $Gdw(t)$ assuming that the linear operator $G:H^0\to H^0$ is a Hilbert–Schmidt operator. This allows to consider any finite noise, that is acting on a finite number of components $h_k$ of the space $H^0$.\
More generally, Lemma \[lemma-z\] is true if the range of the operator $G$ is a subset of $D(A^\gamma)$ with $\gamma$ fulfilling .
Pathwise uniqueness {#unic}
-------------------
We consider two solutions $u_1$ and $u_2$ of obtained in the previous section; we have that, for $\alpha \ge \frac 54$, $$u_1, u_2 \in C([0,T];H^1)
\cap L^2(0,T;H^\alpha) \; {{\mathbb P}}-a.s.$$ since $\frac {2\alpha}{\alpha-1}>2$. The difference $U=u_1-u_2$ satisfies $$\frac{d\;}{dt}U(t) +\nu A^\alpha U(t)+B\big(u_1(t),u_1(t)\big)
-B\big(u_2(t),u_2(t)\big) = 0.$$ Since the operator $B$ is bilinear, this becomes $$\label{diffU}
\frac{d\;}{dt}U(t) +\nu A^\alpha U(t)+B\big(u_1(t),U(t)\big)
+B\big(U(t),u_2(t)\big) = 0.$$ Taking the scalar product of with $AU$ in $H^0$, we get $$\frac 12 \frac{d\;}{dt}|U(t)|_1^2+\nu |U(t)|^2_{1+\alpha}=
-\langle B\big(u_1(t),U(t)\big)+B\big(U(t),u_2(t)\big), AU(t)\rangle$$ with $U(0)=0$.
We estimate the r.h.s. according to - $$\frac 12 \frac{d\;}{dt}|U(t)|_1^2+\nu |U(t)|^2_{1+\alpha}
\le c |u_1(t)|_\alpha |U(t)|_1 |U(t)|_{1+\alpha} + c
|u_2(t)|_\alpha |U(t)|_1 |U(t)|_{1+\alpha} .$$ Thus, by Young inequality: $$\frac{d\;}{dt}|U(t)|_1^2
\le c \big(|u_1(t)|_\alpha^2+|u_2(t)|_\alpha^2 \big)|U(t)|_1^2$$ and, by Gronwall inequality $$\label{sGro}
|U(t)|_1^2\le |U(0)|_1^2 \; e^{\textstyle\int_0^t c
(|u_1(s)|_\alpha^2+|u_2(s)|_\alpha^2 ) ds}.$$ Then $U(t)=0$ for all $t \ge 0 $, because $U(0)=0$. This means that pathwise we have $u_1(t)=u_2(t)$ for all $t \ge 0 $.
i\) The Markov property of $u$ comes from the same properties for the Galerkin approximations and from the uniqueness result (see, e.g., [@Fla]).\
ii) The pathwise estimate implies also the Feller property in $H^1$, that is given a sequence of solutions $u^j$ with initial data $u^j_0$, if $\displaystyle\lim_j u_0^j=u_0$ in $H^1$ then $\displaystyle\lim_j \mathbb E \phi (u^j(t))=\mathbb E \phi(u(t))$ for any $t \in [0,T]$ and any continuous bounded function $\phi:H^1 \to\mathbb R$.
Regularity results
==================
Considering as phase space other Hilbert spaces $H^s$, we get different bounds on $\alpha$ to obtain that the dynamics of the stochastic Navier–Stokes equation is well posed in such $H^s$. As it has been pointed out in [@ms], the smaller is $s$ (with $u_0\in H^s$) the bigger is $\alpha$. Theorem \[teo1\] deals with $s=1$. In this section, we show that problem is well-posed in the space $H^0$ if $\alpha>\frac 32$, and in the spaces $H^s$ with $s\ge 2$ if $\alpha \ge
\frac 54$.\
$\boxed{\mathbf H^0\text{\bf -regularity}}$\
We have the following result
Let $\alpha>\frac 32$.\
For any $u_0\in H^0$, if $\gamma>\frac 34$ then there exists a unique process $u$ which is a strong solution of such that $$u \in C([0,T];H^0)\cap
L^2(0,T;H^{\alpha}) \qquad {{\mathbb P}}-a.s.;$$ $u$ is progressively measurable in these topologies and a Markov process in $H^0$.
Existence is proved by means of a priori estimates as in the proof of Proposition \[propo-v\]; to be precise, for $\alpha \ge \frac 54$ we get that there exists a solution $u \in C([0,T];H^0)\cap
L^2(0,T;H^\alpha)$ ${{\mathbb P}}$-a.s. if $z$ has paths in $C([0,T];H^\alpha)$, i.e. if $\gamma > \frac 34$.
Pathwise uniqueness is obtained according to the result by Prodi [@prodi], requiring $u \in L^{s}(0,T;[L^q(\mathcal T)]^3) \ {{\mathbb P}}$-a.s. for $\frac 2s +\frac 3 q \le 1$. However, using an interpolation result and Sobolev embedding we have $$L^\infty(0,T;H^0)\cap L^2(0,T;H^\alpha) \subset L^s(0,T;H^{2\frac \alpha s})
\subset L^s(0,T;[L^q(\mathcal T)]^3)$$ for $2< s < \infty$ and $\frac 1q=\frac 12 -\frac {2\alpha}{3s}$. The condition $\frac 2s +\frac 3 q \le 1$ holds if $\alpha>\frac 32$. $\hfill \Box$
For $\alpha \ge 1$ we can prove that there exists a solution of equation such that $u \in C_w([0,T];H^0)\cap L^\infty(0,T;H^0)\cap
L^2(0,T;H^\alpha)$ ${{\mathbb P}}$-a.s. But uniqueness is unknown.
Consider, for instance, $\alpha=1$. We require that $z \in C([0,T];H^{\frac 32})$ ${{\mathbb P}}$-a.s. and equation is treated as in the deterministic setting.\
For this, change the first estimate in the proof of Proposition \[propo-v\] as follows: $$\begin{split}
\frac 12 \frac{d}{dt}|v|^2+\nu|v|_1^2
& =-\langle B(v+z,z),v\rangle
\\
&\le c |v+z| |z|_{\frac 32} |v|_1
\\
& \le c |v| |z|_{\frac 32} |v|_1+ |z|^2_{\frac 32} |v|_1
\\
&\le \frac \nu 2 |v|_1^2
+ c_\nu |z|_{\frac 32}^2 |v|^2 + c_\nu |z|_{\frac 32}^4.
\end{split}$$ Then $\sup_{0\le t \le T} |v(t)|<\infty$, $\int_0^T |v(t)|_1^2 dt<\infty$.
Moreover, from Lemma 2.1 in [@temam2] we have that $B:H^0\times H^1 \to H^{-\sigma}$ for any $\sigma
>\frac 32$. Then $\dot v= -\nu A v-B\big(v+z,v+z\big)
\in L^2(0,T;H^{-\sigma})$. This gives a compact embedding and therefore there exists a subsequence of the Galerkin sequence that converges to $v$ as follows: $$\begin{split}
& v_m \to v \;\;\text{ weakly in } L^2(0,T;H^1),\\
& v_m \to v \;\;\star-\text{weakly in } L^\infty(0,T;H^0) ,\\
& v_m \to v \;\;\text{ strongly in } L^2(0,T;H^0).
\end{split}$$ Finally $v \in C_w([0,T];H^0)$. Notice that the previous result $v \in C([0,T];H^0)$ came from $v \in L^2(0,T;H^\alpha), \dot v \in L^2(0,T;H^{-\alpha})$ (see Lemma 1.2, Ch. III in [@temam]).
To compare our result with [@sri], we have that [@sri], for its model, deals with the phase space $H^0$ assuming $\alpha\ge 2$.
$\boxed{\mathbf {H^s\text{\bf -regularity with } s \ge 2}}$\
We need the following estimates:
$$\label{B1}
|B(u,\tilde u)|_m \le c |u|_{m+1}|\tilde u|_{m+1}
\qquad \text{ for } m =1,2,3,\dots$$
First, consider for $m=1$. We have $$\begin{split}
|B(u,\tilde u)|_1^2 &=|(u\cdot \nabla)\tilde u|_1^2
\\&=
\sum_{k,l=1}^3\big|\partial_k(\sum_{i=1}^3 u_i\partial_i\tilde u_l)\big|_{L^2}^2
\\
&\le 2 \sum_{k,l=1}^3\big|\sum_{i=1}^3 \partial_k u_i\partial_i \tilde u_l\big|_{L^2}^2
+ 2 \sum_{k,l=1}^3\big|\sum_{i=1}^3 u_i\partial_k\partial_i \tilde u_l\big|_{L^2}^2
\\
&\le 6 \sum_{k,l,i}\big| \partial_k u_i\big|^2_{L^4}
\big|\partial_i \tilde u_l\big|_{L^4}^2
+ 6 \sum_{k,l,i}\big| u_i\big|_{L^\infty}^2
\big|\partial_k\partial_i \tilde u_l\big|_{L^2}^2.
\end{split}$$ Then use the continuous embeddings $H^1(\mathcal T) \subset L^4(\mathcal T)$ and $H^2(\mathcal T)\subset L^\infty(\mathcal T)$.
For $m=2,3,\dots$ we use that $H^m$ is a multiplicative algebra if $m >\frac 32$; then $$|B(u,\tilde u)|_m \le c |u|_m|\tilde u|_{m+1} \text{ for } m =2,3,\dots$$ which is even stronger than . $\hfill \Box$
We have the following result
Let $\alpha\ge \frac 54$ and $s\ge 2$.\
For any $u_0\in H^s$, if $\alpha+2\gamma>s+\frac 32$ then there exists a unique process $u$ which is a strong solution of such that $$u \in C([0,T];H^s)$$ ${{\mathbb P}}$-a.s.\
$u$ is progressively measurable in these topologies and is a Markov process in $H^s$.
For simplicity, we provide the proof for $s=2$. In this way we show the difference with respect to the case $s=1$ considered in the previous section. However, the proof would go along the same lines for $s>2$ using .
Set $s=2$. Then almost every path of $z$ is in $C([0,T];H^2)$.\
We prove existence for $\alpha \ge \frac 54$. We use with $m=1$ and take the scalar product of equation with $A^2v$: $$\begin{split}
\frac 12 \frac {d\;}{dt}|v|_2^2+\nu|v|^2_{2+\alpha}
=-
\langle B(v+z,v+z), A^2v\rangle
&
= -\langle A^{\frac 12}B(v+z,v+z), A^{\frac32}v\rangle \\
&
\le |B(v+z,v+z)|_1 |v|_3\\
&
\le c |v+z|_2^2 |v|_3\\
&
\le c |v+z|_2^2 |v|_{2+\alpha}\\
&\le
\frac \nu 2 |v|^2_{2+\alpha}+ c_\nu |v|_2^4+c_\nu |z|^4_2.
\end{split}$$ Since we already know from Proposition \[propo-v\] that $v \in L^2(0,T;H^{1+\alpha})
\subset L^2(0,T;H^2)$, it follows as usual by Gronwall lemma that $v \in L^\infty(0,T;H^2)\cap L^2(0,T;H^{2+\alpha})$. From now on, the proof goes as in Proposition \[propo-v\].\
Pathwise uniqueness: the estimates hold for any $\alpha \ge 1$ but the regularity required on $u_i$ holds for $\alpha \ge \frac 54$. This shows that in $H^2$ it is ”easier” to prove uniqueness than existence.\
Set $U=u_1-u_2$ as in Section \[unic\]; now $u_1, u_2 \in C([0,T];H^2)$. Taking the scalar product of with $A^2U$ in $H^0$, we get $$\frac 12 \frac{d\;}{dt}|U(t)|_2^2+\nu |U(t)|^2_{2+\alpha}=
-\langle B\big(u_1(t),U(t)\big)+B\big(U(t),u_2(t)\big), A^2U(t)\rangle$$ with $U(0)=0$. As before, we estimate the r.h.s. by means of , and get $$\begin{split}
\frac 12 \frac{d\;}{dt}|U(t)|_2^2+\nu |U(t)|^2_{2+\alpha}
&\le c |u_1(t)|_2 |U(t)|_2 |U(t)|_3 + c
|u_2(t)|_2 |U(t)|_2 |U(t)|_3
\\
&\le
c |u_1(t)|_2 |U(t)|_2 |U(t)|_{2+\alpha} + c
|u_2(t)|_2 |U(t)|_2 |U(t)|_{2+\alpha}
\\
&\le
\frac \nu 2 |U(t)|_{2+\alpha}^2+
c_\nu(|u_1(t)|_2^2+|u_2(t)|_2^2)|U(t)|_2^2.
\end{split}$$ From $$\frac{d\;}{dt}|U(t)|_2^2\le 2 c_\nu(|u_1(t)|_2^2+|u_2(t)|_2^2)|U(t)|_2^2$$ we conclude that $|U(t)|_2=0$ for any $t \in [0,T]. \hfill \Box$
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|
---
abstract: 'A lot of effort has been invested into characterizing the convergence rates of gradient based algorithms for non-linear convex optimization. Recently, motivated by large datasets and problems in machine learning, the interest has shifted towards distributed optimization. In this work we present a distributed algorithm for strongly convex constrained optimization. Each node in a network of $n$ computers converges to the optimum of a strongly convex, $L$-Lipchitz continuous, separable objective at a rate $O\left( \frac{\log{(\sqrt{n} T )}}{T}\right)$ where $T$ is the number of iterations. This rate is achieved in the online setting where the data is revealed one at a time to the nodes, and in the batch setting where each node has access to its full local dataset from the start. The same convergence rate is achieved in expectation when the subgradients used at each node are corrupted with additive zero-mean noise.'
author:
- |
\
\
and\
\
bibliography:
- '../PhDThesis/References.bib'
title: Distributed Strongly Convex Optimization
---
Introduction
============
In this work we focus on solving optimization problems of the form $$\begin{aligned}
\label{eq:general_optimization_problem}
\underset{w \in \mathcal{W}}{\operatorname{minimize}}\ F(w) = \frac{1}{T} \sum_{t=1}^T f^t(w)\end{aligned}$$ where each function $f^1(w), f^2(w), \dots,$ is convex over a convex set $\mathcal{W} \subseteq \mathds{R}^d$. This formulation applies widely in machine learning scenarios, where $f^t(w)$ measures the loss of model $w$ with respect to data point $t$, and $F(w)$ is the average loss over $T$ data points. In particular, we are interested in the behavior of online distributed optimization algorithms for this sort of problem as the number of data points $T$ tends to infinity. We describe a distributed algorithm which, for strongly convex functions $f^t$, converges at a rate $O\left( \frac{\log(\sqrt{n} T)}{T}\right)$. To the best of our knowledge this is the first distributed algorithm to achieve this converge rate for constrained optimization without relying on smoothness assumptions on the objective or non-trivial communication mechanisms between the nodes. The result is true both in the online and the batch optimization setting.
When faced with a non-linear convex optimization problem, gradient-based methods can be applied to find the solution. The behavior of these algorithms is well-understood in the single-processor (centralized) setting. Under the assumption that the objective is $L$-Lipschitz continuous, projected gradient descent-type algorithms converge at a rate $O(\frac{1}{\sqrt{T}})$ [@zinkOnlineConvexOpt; @nesterovDualOpt]. This rate is achieved both in an online setting where the $f^t$’s are revealed to the algorithm sequentially and in the batch setting where all $f^t$ are known in advance. If the cost functions are also strongly convex then gradient algorithms can achieve linear rates, $O\left(\frac{1}{T}\right)$, in the batch setting [@largeScaleSGD] and nearly-linear rates, $O\left(\frac{\log(T)}{T}\right)$, in the online setting [@logRegRepeatedGames]. Under additional smoothness assumptions, such as Lipschitz continuous gradients, the same rate of convergence can also be achieved by second order methods in the online setting [@LogRegretOCOot; @AdaptiveOGD], while accelerated methods can achieve a quadratic rate in the batch setting; see [@TsengAccelerated] and references therein.
The aim of this work is to extend the aforementioned results to the distributed setting where a network of processors jointly optimize a similar objective. Assuming the network is arranged as an expander graph with constant spectral gap, for general convex cost functions that are only $L$-Lipschitz continuous, the rate at which existing algorithms on a network of $n$ processors will all reach the optimum value is $O(\frac{\log(T \sqrt{n})}{\sqrt{T}})$, i.e., similar to the optimal single processor algorithms up to a logarithmic factor [@dualAveraging; @ContrainedDistrOpt]. This is true both in a batch setting and in an online setting, even when the gradients are corrupted by noise. The technique proposed in [@DekelMiniBatches] makes use of mini-batches to obtain asymptotic rates $O\left( \frac{\sqrt{\log{(n)}}}{\sqrt{n T}}\right)$ for online optimization of smooth cost functions that have Lipschitz continuous gradients corrupted by bounded variance noise, and $O\left( \frac{1}{nT}\right)$ for smooth strongly convex functions. However, this technique requires that each node exchange messages with every other node at the end of each iteration. Finally, if the objective function is strongly convex and three times differentiable, a distributed version of Nesterov’s accelerated method [@FastDistributedGradMethods] achieves a rate of $O\left(\frac{\log(T)}{T}\right)$ for unconstrained problems in the batch setting, but the dependence on $n$ is not characterized. The algorithm presented in this paper achieves a rate $O\left( \frac{\log{(\sqrt{n} T )}}{T}\right)$ for strongly convex functions. Our formulation allows for convex constraints in the problem and assumes the objective function is Lipschitz continuous and strongly convex; no higher-order smoothness assumptions are made. Our algorithm works in both the online and batch setting and it scales nearly-linearly in number of iterations for network topologies with fast information diffusion. In addition, at each iteration nodes are only required to exchange messages with a subset of other nodes in the network (their neighbors).
The rest of the paper is organized as follows. Section \[sec:into\_convex\_opt\] introduces notation and formalizes the problem. Section \[sec:algorithm\] describes the proposed algorithm and states our main results. These results are proven in Section \[sec:analysis\], and Section \[sec:stochastic\_opt\] extends the analysis to the case where gradients are noisy. Section \[sec:experiments\] presents the results of numerical experiments illustrating the performance of the algorithm, and the paper concludes in Section \[sec:future\].
Online Convex Optimization {#sec:into_convex_opt}
==========================
Consider the problem of minimizing a convex function $F(w)$ over a convex set $\mathcal{W} \subseteq \mathds{R}^d$. Of particular interest is the setting where the algorithm sequentially receives noisy samples of the (sub)gradients of $F(w)$. This setting arises in online loss minimization for machine learning when the data arrives as a steam and the (sub)gradient is evaluated using an individual data point at each step [@zinkOnlineConvexOpt]. Suppose the $t$th data point $x(t) \in \mathcal{X} \subseteq \mathds{R}^d$ is drawn i.i.d. from an unknown distribution $\mathcal{D}$, and let $f^t(w) = f(w, x(t))$ denote the loss of this data point with respect to a particular model $w$. In this setting one would like to find the model $w$ that minimizes the expected loss $\mathds{E}_{\mathcal{D}}[f(w,x)]$, possibly with the constraint that $w$ be restricted to a model space $\mathcal{W}$. Clearly, as $T \rightarrow \infty$, the objective $F(w) = \frac{1}{T} \sum_{t=1}^T f^t(w) \rightarrow \mathds{E}_{\mathcal{D}}[f(w,x)]$, and so if the data stream is finite this motivates minimizing the empirical loss $F(w)$.
An online convex optimization algorithm observes a data stream $x(1), x(2), \dots$, and sequentially chooses a sequence of models $w(1), w(2), \dots$, after each observation. Upon choosing $w(t)$, the algorithm receives a subgradient $g(t) \in \partial f^t(w(t))$. The goal is for the sequence $w(1), w(2), \dots$ to converge to a minimizer $w^*$ of $F(w)$.
The performance of an online optimization algorithm is measured in terms of the *regret*: $$\begin{aligned}
R(T) = \sum_{t=1}^T f^t(w(t)) - \min_{w \in \mathcal{W}} \sum_{t=1}^T f^t(w).\end{aligned}$$ The regret measures the gap between the cost accumulated by the online optimization algorithm over $T$ steps and that of a model chosen to simultaneously minimize the total regret over all $T$ cost terms. If the costs $f^t$ are allowed to be arbitrary convex functions then it can be shown that the best achievable rate for any online optimization algorithm is $\frac{R(T)}{T} = \Omega(\frac{1}{\sqrt{T}})$, and this bound is also achievable [@zinkOnlineConvexOpt]. The rate can be significantly improved if the cost functions has more favourable properties.
Assumptions
-----------
\[ass:stronglyConvex\] We assume for the rest of the paper that each cost function $f^t(w) = f(w, x(t))$ is $\sigma$-strongly convex for all $x(t) \in \mathcal{X}$; i.e., there is a $\sigma > 0$ such that for all $\theta \in [0,1]$ and all $u,w \in \mathcal{W}$ $$\begin{aligned}
f^t(\theta u +& (1- \theta) w ) \leq \notag \\
&\theta f^t(u) + (1-\theta) f^t(w) - \frac{\sigma}{2} \theta (1 - \theta) {\left\lVertu - w \right\rVert}^2.\end{aligned}$$
If each $f^t(w)$ is $\sigma$-strongly convex, it follows that $F(w)$ is also $\sigma$-strongly convex. Moreover, if $F(w)$ is strongly convex then it is also strictly convex, and so $F(w)$ has a unique minimizer which we denote by $w^*$.
\[ass:boundedGradients\] We also assume that the subgradients ${g}(t)$ of each cost function $f^t$ are bounded by a known constant $L > 0$; i.e., ${\left\lVert{g}(t) \right\rVert} \leq L$ where ${\left\lVert\cdot \right\rVert}$ is the ($\ell_2$) Euclidean norm.
Example: Training a Classifier
------------------------------
For a specific example of this setup, consider the problem of training an SVM classifier using a hinge-loss with $\ell_2$ regularization [@logRegRepeatedGames]. In this case, the data stream consists of pairs $\{x(t), y(t)\}$ such that $x(t) \in \mathcal{X}$ and $y(t) \in \{-1, +1\}$. The goal is to minimize the misclassification error as measured by the $\ell_2$-regularized hinge loss. Formally, we wish to find the $w^* \in \mathcal{W} \subseteq \mathds{R}^d$ that solves $$\begin{aligned}
\label{eq:svm}
\operatorname{minimize}_{w \in \mathcal{W}} \frac{\sigma}{2} {\left\lVertw \right\rVert}^2 + \frac{1}{m} \sum_{t=1}^{m} \max\{0, 1 - y(t) \langle w, x(t)\rangle\}\end{aligned}$$ which is $\sigma$-strongly convex[^1]. For these types of problems, using a single-processor stochastic gradient descent algorithm, one can achieve $\frac{R(T)}{T} = O(\frac{\log{T}}{T})$[@logRegRepeatedGames] or $\frac{R(T)}{T} = O(\frac{1}{T})$ [@StochOptStrognlyConvex] by using different update schemes.
Distributed Online Convex Optimization
--------------------------------------
In this paper, we are interested in solving online convex optimization problems with a network of computers. The computers are organized as a network $G=(V,E)$ with ${\left\lvertV \right\rvert} = n$ nodes, and messages are only exchanged between nodes connected with an edge in $E$.
\[ass:connected\] In this work we assume that $G$ is connected and undirected.
Each node $i$ receives a stream of data $x_i(1), x_i(2), \dots$, similar to the serial case, and the nodes must collaborate to minimize the network-wide objective $$\begin{aligned}
F(w) = \frac{1}{nT} \sum_{t=1}^T \sum_{i=1}^n f_i^t(w),\end{aligned}$$ where $f_i^t(w) = f(w, x_i(t))$ is the cost incurred at processor $i$ at time $t$. In the distributed setting, the definition of regret is naturally extended to $$\begin{aligned}
R(T) = \sum_{t=1}^T \sum_{i=1}^n f(w_i(t), x_i(t)) - \min_{w \in \mathcal{W}} \sum_{t=1}^T \sum_{i=1}^n f(w, x_i(t)).\end{aligned}$$ For general convex cost functions, the distributed algorithm proposed in [@dualAveraging] has been proven to have an average regret that decreases at a rate $\sqrt{T}$, similar to the serial case, and this result holds even when the algorithm receives noisy, unbiased, observations of the true subgradients at each step. In the next section, we present a distributed algorithm that achieves a nearly-linear rate of decrease of the average regret (up to a logarithmic factor) when the cost functions are strongly convex.
Algorithm {#sec:algorithm}
=========
Nodes must collaborate to solve the distributed online convex optimization problem described in the previous section. To that end, the network is endowed with a $n \times n$ consensus matrix $P$ which respects the structure of $G$, in the sense that $[P]_{ji} = 0$ if $(i,j) \notin E$. We assume that $P$ is doubly stochastic, although generalizations to the case where $P$ is row stochastic or column stochastic (but not both) are also possible [@TsianosCDC2012; @TsianosACC2012].
A detailed description of the proposed algorithm, *distributed online gradient descent* (DOGD), is given in Algorithm \[alg:dogd\]. In the algorithm, each node performs a total of $T$ updates. One update involves processing a single data point $x_i(t)$ at each processor. The updates are performed over $k$ rounds, and $T_s$ updates are performed in round $s \leq k $. The main steps within each round (lines 9–11) involve updating an accumulated gradient variable, $z_i^k(t)$, by simultaneously incorporating the information received from neighboring nodes and taking a local gradient-descent like step. The accumulated gradient is projected onto the constraint set to obtain $w_i^k(t)$, where $$\begin{aligned}
{ \Pi_{\mathcal{W}}\left[z\right]} = {\underset{w \in \mathcal{W}}{\operatorname{argmin}}} {\left\lVertw - z \right\rVert}\end{aligned}$$ denotes the Euclidean projection of $z$ onto $\mathcal{W}$, and then this projected value is merged into a running average $\hat{w}_i(r)$. The step size parameter $a_k$ remains constant within each round, and the step size is reduced by half at the end of each round. The number of updates per round doubles from one round to the next.
Note that the algorithm proposed here differs from the distributed dual averaging algorithm described in [@dualAveraging], where a proximal projection is used rather than the Euclidean projection. Also, in contrast to the distributed subgradient algorithms described in [@distrStochSubgrOpt], DOGD maintains an accumulated gradient variable in $z_i^k(t+1)$ which is updated using $\{z_j^k(t)\}$ as opposed to the primal feasible variables $\{w_j^k(t)\}$. Finally, key to achieving fast convergence is the exponential decrease of the learning rate after performing an exponentially increasing number of gradient steps together with a proper initialization of the learning rate.
The next section provides theoretical guarantees on the performance of DOGD.
$\operatorname{Initialize:} T_1 = \left\lceil \frac{2 }{\sigma} \right\rceil, a_1 = 1, k=1, z_i^1(1) = w_i^1(1) = 0$ Send/receive $z_i^k(t)$ and $z_j^k(t)$ to/from neighbors Obtain next subgradient ${g}_i(t) \in \partial_w f_i^t(w_i^k(t))$ $z_i^k(t+1) = \sum_{j=1}^n p_{ij} z_j^k(t) - a_k {g}_i(t)$ $w_i^k(t+1) = { \Pi_{\mathcal{W}}\left[z_i^k(t+1)\right]}$ $w_i^{k+1}(1) = w_i^k(T_k)$ $z_i^{k+1}(1) = w_i^{k+1}(1)$ $\hat{w}_i^{k+1} = \frac{1}{T_k} \sum_{t=1}^{T_k} w_i^k(t) $ $T_{k+1} \leftarrow 2 T_k$ $a_{k+1} \leftarrow \frac{a_k}{2}$ $k = k + 1$
Convergence Analysis {#sec:analysis}
====================
Our main convergence result, stated below, guarantees that the average regret decreases at a rate which is nearly linear.
\[thm:dogd\_convergence\] Let Assumptions \[ass:stronglyConvex\]–\[ass:connected\] hold and suppose that the consensus matrix $P$ is doubly stochastic with constant $\lambda_2$. Let $w^*$ be the minimizer of $F(w)$. Then the sequence $\{\hat{w}_i^k\}$ produced by nodes running DOGD to minimize $F(w)$ obeys $$\begin{aligned}
F(\hat{w}_i^{k+1}) - F(w^*) = O\left( \frac{\log{(\sqrt{n} T )}}{T} \right),\end{aligned}$$ where $k = \lfloor \log_2(T/2 + 1)\rfloor$ is the number of rounds executed during a total of $T$ gradient steps per node, and $\hat{w}_i^{k}$ is the running average maintained locally at each node.
*Remark 1:* We state the result for the case where $\lambda_2$ is constant. This is the case when $G$ is, e.g., a complete graph or an expander graph [@kRegExpanders]. For other graph topologies where $\lambda_2$ shrinks with $n$ and consensus does not converge fast, the convergence rate dependence on $n$ is going to be worse due to a factor $1 - \sqrt{\lambda_2}$ in the denominator; see the proof of Theorem \[thm:dogd\_convergence\] below for the precise dependence on the spectral gap $1 - \sqrt{\lambda_2}$.
*Remark 2:* The theorem characterizes performance of the online algorithm DOGD, where the data and cost functions $f^t_i$ are processed sequentially at each node in order to minimize an objective of the form $$\begin{aligned}
F(w) = \frac{1}{n} \sum_{i=1}^n \frac{1}{T} \sum_{t=1}^T f_i^t(w).\end{aligned}$$ However, as pointed out in [@logRegRepeatedGames], if the entire dataset is available in advance, we can use the same scheme to do batch minimization by effectively setting $f_i^t(w) = f_i^1(w)$, where $f_i^1(w)$ is the objective function accounting for the entire dataset available to node $i$. Thus, the same result holds immediately for a batch version of DOGD.
The remainder of this section is devoted to the proof of Theorem \[thm:dogd\_convergence\]. Our analysis follows arguments that can be found in [@zinkOnlineConvexOpt; @StochOptStrognlyConvex; @dualAveraging] and references therein. We first state and prove some intermediate results.
Properties of Strongly Convex Functions
---------------------------------------
Recall the definition of $\sigma$-strong convexity given in Assumption \[ass:stronglyConvex\]. A direct consequence of this definition is that if $F(w)$ is $\sigma$-strongly convex then $$\begin{aligned}
F(w) - F(w^*) \geq \frac{\sigma}{2} {\left\lVertw - w^* \right\rVert}^2.\end{aligned}$$
Strong convexity can be combined with the assumptions above to upper bound the difference $F(w)-F(w^*)$ for an arbitrary point $w \in \mathcal{W}$.
\[lem:startingPoint\] Let $w^*$ be the minimizer of $F(w)$. For all $w \in \mathcal{W}$, we have $F(w) - F(w^*) \leq \frac{2 L^2}{\sigma}$.
For any subgradient ${g}$ of $F$ at $w$, by convexity we know that $F(w) - F(w^*) \leq \langle {g}, w - w^* \rangle$. It follows from Assumption \[ass:boundedGradients\] that $F(w) - F(w^*) \leq L {\left\lVertw- w^* \right\rVert} $. Furthermore, from Assumption \[ass:stronglyConvex\], we obtain that $\frac{\sigma}{2} {\left\lVertw - w^* \right\rVert}^2 \leq L {\left\lVertw - w^* \right\rVert}$ or ${\left\lVertw - w^* \right\rVert} \leq \frac{2 L}{ \sigma}$. As a result, $F(w) - F(w^*) \leq \frac{2 L^2}{\sigma}$.
The Lazy Projection Algorithm
-----------------------------
The analysis of DOGD below involves showing that the average state, $\frac{1}{n}\sum_{i=1}^n w_i^k(t)$, evolves according to the so-called (single processor) *lazy projection* algorithm [@zinkOnlineConvexOpt], which we discuss next. The lazy projection algorithm is an online convex optimization scheme for the serial problem discussed at the beginning of Section \[sec:into\_convex\_opt\]. A single processor sequentially chooses a new variable $w(t)$ and receives a subgradient $g(t)$ of $f(w(t), x(t))$. The algorithm chooses $w(t+1)$ by repeating the steps $$\begin{aligned}
z(t+1) = & z(t) - a {g}(t) \label{eqn:lazyProjection1} \\
w(t+1) = & { \Pi_{\mathcal{W}}\left[z(t+1)\right]} \label{eqn:lazyProjection2}.\end{aligned}$$ By unwrapping the recursive form of , we get $$\begin{aligned}
z(t+1) = -a \sum_{s=1}^t g(t) + z(1). \label{eqn:lazyProjectionUnwrapped}\end{aligned}$$
The following is a typical result for subgradient descent-style algorithms, and is useful towards eventually characterizing how the regret accumulates. Its proof can be found in the appendix of the extended version of [@zinkOnlineConvexOpt].
\[thm:zink\] Let $w(1) \in \mathcal{W}$, let $a > 0$, and set $z(1) = w(1)$. After $T$ rounds of the serial lazy projection algorithm –, we have $$\begin{aligned}
\sum_{t=1}^T \langle {g}(t), w(t) - w^* \rangle \leq \frac{{\left\lVertw(1) - w^* \right\rVert}^2}{2 a} + \frac{T a L^2}{2}.\end{aligned}$$
Theorem \[thm:zink\] immediately yields the same bound for the regret of lazy projection [@zinkOnlineConvexOpt].
Evolution of Network-Average Quantities in DOGD
-----------------------------------------------
We turn our attention to Algorithm \[alg:dogd\]. A standard approach to studying convergence of distributed optimization algorithms, such as DOGD, is to keep track of the discrepancy between every node’s state and an average state sequence defined as $$\begin{aligned}
\overline{z}^k(t) = \frac{1}{n} \sum_{i=1}^n z_i^k(t) \quad \text{and} \quad \overline{w}^k(t) = { \Pi_{\mathcal{W}}\left[\overline{z}^k(t)\right]}.\end{aligned}$$ Observe that $\overline{z}^k(t)$ evolves in a simple recursive manner, $$\begin{aligned}
\overline{z}^k(t+1) = &\frac{1}{n} \sum_{i=1}^n z_i^k(t+1) \\
= & \frac{1}{n} \sum_{i=1}^n \left[ \sum_{j=1}^n p_{ij} z_j^k(t) - a_k {g}_i(t) \right] \\
= & \frac{1}{n} \sum_{j=1}^n z_j^k(t) \sum_{i=1}^n p_{ij} - \frac{ a_k}{n} \sum_{i=1}^n {g}_i(t) \\
= & \overline{z}(t) - \frac{ a_k}{n} \sum_{i=1}^n {g}_i(t) \label{eq:doubly_stoch_P}\\
= & -a_k \sum_{s=1}^t \frac{1}{n} \sum_{i=1}^n {g}_i(s) + \frac{1}{n}\sum_{i=1}^n z_i^k(1) \label{eq:zbar}\end{aligned}$$ where equation holds since $P$ is doubly stochastic. Notice (cf. eqn. ) that the states $\{\overline{z}^k(t), \overline{w}^k(t)\}$ evolve according to the lazy projection algorithm with gradients $\overline{g}(t)= \frac{1}{n} \sum_{i=1}^n {g}_i(t) $ and learning rate $a_k$. In the sequel, we will also use an analytic expression for $z_i^k(t)$ derived by back substituting in its recursive update equation. After some algebraic manipulation, we obtain $$\begin{aligned}
z_i^k(t) = & - a_k \sum_{s=1}^{t-1} \sum_{j=1}^n \left[P^{t-s+1}\right]_{ij} {g}_j(s-1) - a_k {g}_i(t-1) \nonumber \\
& + \sum_{j=1}^n [P^t]_{ij} z_j^k(1), \label{eqn:DOGD_unwrapped}\end{aligned}$$ and since the projection in non-expansive and $z_i^1(1) = 0, \forall i$, $$\begin{aligned}
{\left\lVertz_i^{k+1}(1) \right\rVert}= & {\left\lVertw_i^{k+1}(1) \right\rVert} = {\left\lVertw_i^k(T_k) \right\rVert} = {\left\lVert{ \Pi_{\mathcal{W}}\left[z_i^k(T_k)\right]} \right\rVert} \\
\leq &{\left\lVertz_i^{k}(T_k) \right\rVert} \\
\leq & {\left\lVert - a_{k} \sum_{t=1}^{T_{k}-1} \sum_{i=1}^n \left[ P^{T_k - s + 1}\right]_{ij} g_i(s-1) \right\rVert} \nonumber \\
& + {\left\lVert-a_k g_i(T_k - 1) \right\rVert} + \sum_{j=1}^n \left[P^{T_k} \right]_{ij} {\left\lVertz_j^k(1) \right\rVert} \\
\leq & a_k T_k L + \sum_{j=1}^n \left[P^{T_k} \right]_{ij} {\left\lVertz_j^k(1) \right\rVert} \\
\leq & \cdots \\
\leq & L \sum_{s=1}^{k} a_{s} T_{s}. \label{eq:znorm_bound} \end{aligned}$$
Analysis of One Round of DOGD
-----------------------------
Next, we focus on bounding the amount of regret accumulated during the $k$th round of DOGD (lines 5–12 of Algorithm 1) during which the learning rate remains fixed at $a_k$. Using Assumptions \[ass:stronglyConvex\], \[ass:boundedGradients\], and the triangle inequality we have that $$\begin{aligned}
\sum_{t=1}^{T_k} & [ F(w_i^{k}(t)) - F(w^*)] \notag \\
= & \sum_{t=1}^{T_k}\left[F\left(\overline{w}^k(t) \right) - F(w^*) + F(w_i^{k}(t)) - F\left( \overline{w}^k(t) \right) \right] \\
\leq & \sum_{t=1}^{T_k} \left[ F( \overline{w}^k(t) ) - F(w^*) + L{\left\lVert w_i^{k}(t) - \overline{w}^k(t) \right\rVert} \right] \\
\leq & \sum_{t=1}^{T_k} \frac{1}{n} \sum_{i=1}^n \left[ f_i(w_i^k(t)) - f_i(w^*) \right] \notag \\
& + \sum_{t=1}^{T_k} \frac{1}{n} \sum_{i=1}^n \left[ f_i(\overline{w}^k(t))- f_i(w_i^k(t)) \right] \notag \\
& + \sum_{t=1}^{T_k} L {\left\lVertw_i^k(t) - \overline{w}^k(t) \right\rVert} \\
\leq & \underbrace{\sum_{t=1}^{T_k} \frac{1}{n} \sum_{i=1}^n \langle g_i(t), w_i^k(t) - w^*\rangle}_{A_1} \notag \\
& + \sum_{t=1}^{T_k} \frac{1}{n} \sum_{i=1}^n L {\left\lVert\overline{w}^k(t) - w_i^k(t) \right\rVert} \notag \\
& + \sum_{t=1}^{T_k} L {\left\lVertw_i^k(t) - \overline{w}^k(t) \right\rVert}. \end{aligned}$$ For the first summand we have $$\begin{aligned}
A_1 = &\sum_{t=1}^{T_k} \frac{1}{n} \sum_{i=1}^n \langle g_i(t), w_i^k(t) - w^*\rangle \\
\leq & \sum_{t=1}^{T_k} \frac{1}{n} \sum_{i=1}^n \langle {g}_i(t) , \overline{w}^k(t) - w^*\rangle \notag \\
& + \sum_{t=1}^{T_k} \frac{1}{n} \sum_{i=1}^n \langle {g}_i(t) , w_i^k(t) - \overline{w}^k(t)\rangle \\
\leq & \underbrace{\sum_{t=1}^{T_k} \frac{1}{n} \sum_{i=1}^n \langle {g}_i(t) , \overline{w}^k(t) - w^*\rangle}_{A_2} \notag \\
& + \sum_{t=1}^{T_k} \frac{1}{n} \sum_{i=1}^n L {\left\lVertw_i^k(t) - \overline{w}^k(t) \right\rVert}. \label{eq:optimization_error_term}\end{aligned}$$
To bound term $A_2$ we invoke Theorem \[thm:zink\] for the average sequences $\{\overline{w}^k(t)\}$ and $\{\overline{z}^k(t)\}$. $$\begin{aligned}
A_2 = & \sum_{t=1}^{T_k} \frac{1}{n} \sum_{i=1}^n \langle {g}_i(t) , \overline{w}^k(t) - w^*\rangle \label{eq:bound_starting_point} \\
= & \sum_{t=1}^{T_k} \Big\langle \frac{1}{n} \sum_{i=1}^n {g}_i(t) , { \Pi_{\mathcal{W}}\left[ \overline{z}^k(t)\right]} - w^* \Big\rangle \\
= & \sum_{t=1}^{T_k} \Big\langle \overline{{g}}(t) , { \Pi_{\mathcal{W}}\left[ \overline{z}^k(t)\right]} - w^* \Big\rangle \\
\leq & \frac{{\left\lVert\overline{w}^k(1) - w^* \right\rVert}^2}{2 a_k} + \frac{{T_k} a_k {\left\lVert\frac{1}{n} \sum_{i=1}^n {g}_i(t) \right\rVert}^2 }{2} \\
= & \frac{{\left\lVert\overline{w}^k(1) - w^* \right\rVert}^2}{2 a_k} + \frac{{T_k} a_k L^2 }{2}.\end{aligned}$$ Collecting now all the partial results and bounds, so far we have shown that $$\begin{aligned}
\sum_{t=1}^{T_k} [ F(w_i^{k}(t)) & - F(w^*) ]\leq \frac{{\left\lVert\overline{w}^k(1) - w^* \right\rVert}^2}{2 a_k} + \frac{{T_k} a_k L^2 }{2} \notag \\
& +\sum_{t=1}^{T_k} \frac{2}{n} \sum_{i=1}^n L {\left\lVertw_i^k(t) - \overline{w}^k(t) \right\rVert} \notag \\
& + \sum_{t=1}^{T_k} L{\left\lVertw_i^k(t) - \overline{w}^k(t) \right\rVert} .\end{aligned}$$ and since the projection operator is non-expansive, we have $$\begin{aligned}
\sum_{t=1}^{T_k} [ F(w_i^{k}(t)) & - F(w^*) ] \leq \frac{{\left\lVert\overline{w}^k(1) - w^* \right\rVert}^2}{2 a_k} + \frac{{T_k} a_k L^2 }{2} \nonumber \\
& +\sum_{t=1}^{T_k} \frac{2}{n} \sum_{i=1}^n L {\left\lVertz_i^k(t) - \overline{z}^k(t) \right\rVert} \label{eqn:oneRoundBound} \\
& + \sum_{t=1}^{T_k} L {\left\lVertz_i^k(t) - \overline{z}^k(t) \right\rVert} . \nonumber\end{aligned}$$ The first two terms are standard for subgradient algorithms using a constant step size. The last two terms depend on the error between each node’s iterate $z_i^k(t)$ and the network-wide average $\overline{z}^k(t)$, which we bound next.
Bounding the Network Error
--------------------------
What remains is to bound the term ${\left\lVertz_i^k(t) - \overline{z}^k(t) \right\rVert}$ which describes an error induced by the network since the different nodes do not agree on the direction towards the optimum. By recalling that $P$ is doubly stochastic and manipulating the recursive expressions and for $z_i(t)$ and $\overline{z}^k(t)$ using arguments similar to those in [@dualAveraging; @TsianosACC2012], we obtain the bound, $$\begin{aligned}
{\left\lVertz_i^k(t) - \overline{z}^k(t) \right\rVert} \leq & a_k L \sum_{s=1}^{t-1} \sum_{j=1}^n {\left\lvert \frac{1}{n} {\boldsymbol{1}}^{T} - \left[ P^{t-s-1}\right]_{ij} \right\rvert}_1 + 2 a_k L \notag \\
& + \sum_{j=1}^n {\left\lvert \frac{1}{n} - [P^t]_{ij} \right\rvert} {\left\lVert z_j^k(1) \right\rVert} \\
= & a_k L \sum_{s=1}^{t-1} {\left\lVert\frac{1}{n} {\boldsymbol{1}}^{T} - \left[ P^{t-s-1}\right]_{i,:} \right\rVert}_1 + 2 a_k L \notag \\
& + \sum_{j=1}^n {\left\lvert \frac{1}{n} - [P^t]_{ij} \right\rvert} {\left\lVert z_j^k(1) \right\rVert}.\end{aligned}$$ The $\ell_1$ norm can be bounded using Lemma \[lem:P\_tv\_convergence\], which is stated and proven in the Appendix, and using we arrive at $$\begin{aligned}
{\left\lVertz_i^k(t) - \overline{z}^k(t) \right\rVert} \leq &2 a_k L \frac{\log{(T_k \sqrt{n})}}{1 - \sqrt{\lambda_2}} + 3 a_k L + \frac{L \sum_{s=1}^{k-1} a_{s} T_{s} }{T_k}\end{aligned}$$ where $\lambda_2$ is the second largest eigenvalue of $P$. Using this bound in equation , along with the fact that $F(w)$ is convex, we conclude that $$\begin{aligned}
F(\hat{w}_i^{k+1}) - F(w^*) = & F\left(\frac{1}{T_k} \sum_{t=1}^{T_k} w_i^{k}(t) \right) - F(w^*)\\
\leq & \frac{1}{T_k} \sum_{t=1}^{T_k} \left[ F( w_i^{k}(t) ) - F(w^*) \right] \\
\leq & \frac{{\left\lVert\overline{w}^k(1) - w^* \right\rVert}^2}{2 a_k T_k} + \frac{ a_k L^2 }{2} \notag \\
& + L^2 a_k \left[ 6 \frac{\log{(T_k \sqrt{n})}}{1 - \sqrt{\lambda_2}} + 9 \right] \notag \\
& + \frac{3 L^2 \sum_{s=1}^{k-1} a_{s} T_{s} }{T_k},\end{aligned}$$ where $\overline{w}^k(1) = { \Pi_{\mathcal{W}}\left[\frac{1}{n} \sum_{i=1}^{n} z_i^k(1) \right]}$.
Analysis of DOGD over Multiple Rounds
-------------------------------------
As our last intermediate step, we must control the learning rate and update of $T_k$ from round-to-round to ensure linear convergence of the error. From strong convexity of $F$ we have $$\begin{aligned}
{\left\lVert\overline{w}^k(1) - w^* \right\rVert}^2 \leq 2 \frac{ F(\overline{w}^k(1)) - F(w^*)}{\sigma}\end{aligned}$$ and thus $$\begin{aligned}
F(\hat{w}_i^{k+1}) & - F(w^*) \leq \frac{ F(\overline{w}^k(1)) - F(w^*)}{\sigma a_k T_k} \notag \\
& + \frac{L^2 a_k}{2} \left[ 12 \frac{\log{(T_k \sqrt{n})}}{1 - \sqrt{\lambda_2}} + 19\right] \notag \\
& + \frac{3 L^2 \sum_{s=1}^{k-1} a_{s} T_{s} }{T_k}\label{eq:Fwhat_bound}.\end{aligned}$$ Now, from Theorem $3$ in [@zinkOnlineConvexOpt] which is a direct consequence of Theorem \[thm:zink\] for the average sequence $\overline{w}$ viewed as a single processor lazy projection algorithm, we have that after executing $T_{k-1}$ gradient steps in round $k-1$, $$\begin{aligned}
F(\overline{w}^k(1)) - F(w^*) \leq \frac{{\left\lVert\overline{w}^{k-1}(1) - w^* \right\rVert}^2}{2 a_{k-1} T_{k-1}} + \frac{ a_{k-1} L^2}{2}\end{aligned}$$ and by repeatedly using strong convexity and Theorem \[thm:zink\] we see that $$\begin{aligned}
F(\overline{w}^k(1)) - F(w^*) \leq & \frac{F(\overline{w}^{k-1}(1)) - F(w^*)}{\sigma a_{k-1} T_{k-1}} + \frac{ a_{k-1} L^2}{2} \\
\leq & \cdots \\
\leq & \frac{F(\overline{w}^{1}(1)) - F(w^*)}{\prod_{j=0}^{k-1} (\sigma a_{k-j} T_{k-j})} \notag \\
& + \sum_{j=1}^{k-1} \frac{a_{k-j} L^2}{2 \prod_{s=1}^{j-1} (\sigma a_{k-s} T_{k-s})} \label{eq:Fwbar_bound}.\end{aligned}$$ Now, let us fix positive integers $b$ and $c$, and suppose we use the following rules to determine the step size and number of updates performed within each round: $$\begin{aligned}
a_k = &\frac{a_{k-1}}{b} = \cdots = \frac{a_1}{b^{k-1}} \\
T_k = & c T_{k-1} = \cdots = c^{k-1} T_1.\end{aligned}$$ Combining with and invoking Lemma \[lem:startingPoint\], we have $$\begin{aligned}
F(\hat{w}_i^{k+1}) & - F(w^*) \leq \frac{2 L^2}{\sigma \prod_{j=0}^{k-1} \left( \sigma a_1 T_1 \left( \frac{c}{b} \right)^{k-j-1} \right)} \notag \\
& + \sum_{j=1}^{k-1} \frac{a_{1} L^2}{2 b^{k-j-1} \prod_{s=0}^{j-1} \left(\sigma a_{1} T_{1} \left( \frac{c}{b} \right)^{k-s-1} \right)} \notag \\
& + \frac{L^2 a_1}{2 b^{k-1}} \left[ 12 \frac{\log{(T_1 c^{k-1} \sqrt{n})}}{1 - \sqrt{\lambda_2}} + 19\right] \notag \\
& + \frac{3L^2 \sum_{s=1}^{k-1} a_1 T_1 \left( \frac{c}{b}\right)^{s-1} }{T_1 c^{k-1}}.\end{aligned}$$ To ensure convergence to zero, we need $c \geq b$ and $\sigma a_1 T_1 > 1$ or $a_1 > \frac{1}{T_1 \sigma}$. Given these restrictions, let us make the choices $$\begin{aligned}
a_1 = 1,\ \ T_1 = \left\lceil \frac{2 }{\sigma} \right\rceil ,\ \ c = b = 2.\end{aligned}$$ To simplify the exposition, let us assume that $T_1 = \frac{2}{\sigma}$ is an integer. Using the selected values, we obtain $$\begin{aligned}
F(\hat{w}_i^{k+1}) & - F(w^*) \leq \frac{2 L^2}{\sigma \prod_{j=0}^{k-1} \left( 2 \left( \frac{2}{2} \right)^{k-j-1} \right)} \notag \\
& + \sum_{j=1}^{k-1} \frac{L^2}{ \cdot 2 \cdot 2^{k-j-1} \prod_{s=0}^{j-1} \left(2 \left( \frac{2}{2} \right)^{k-s-1} \right)} \notag \\
& + \frac{L^2 }{2 \cdot 2^{k-1}} \left[ 12 \frac{\log{(\frac{2 }{\sigma} \cdot 2^{k-1} \sqrt{n})}}{1 - \sqrt{\lambda_2}} + 19\right] \notag \\
& + \frac{3L^2 \sum_{s=1}^{k-1} \left( \frac{2}{2}\right)^{s-1} }{ 2^{k-1}} \\
\leq & \frac{2 L^2}{\sigma 2^k } + \sum_{j=1}^{k-1} \frac{L^2}{ 2^{k-j} 2^j } \notag \\
& + \frac{L^2 }{2^{k}} \left[ 12 \frac{\log{( \frac{ 2^{k} \sqrt{n} }{\sigma})}}{1 - \sqrt{\lambda_2}} + 19\right] + \frac{3L^2 (k-1) }{ 2^{k-1}} \\
\leq & \frac{2 L^2}{\sigma 2^k } + \frac{L^2 (k-1)}{ 2^{k} }\notag \\
& + \frac{L^2 }{ 2^{k}} \left[ 12 \frac{\log{(\frac{ 2^{k} \sqrt{n} }{\sigma})}}{1 - \sqrt{\lambda_2}} + 19\right] + \frac{6L^2 (k-1) }{ 2^{k}} \\
\leq & \frac{2 L^2}{\sigma 2^k } + \frac{L^2 (k-1)}{ 2^{k} }\notag \\
& + \frac{L^2 }{2^{k}} \left[ 12 \frac{\log{(\frac{ 2^{k} \sqrt{n} }{\sigma})}}{1 - \sqrt{\lambda_2}} + 19\right] + \frac{6L^2 (k-1) }{ 2^{k}}.\end{aligned}$$ Finally, we have all we need to complete the analysis of Algorithm \[alg:dogd\].
Proof of Theorem \[thm:dogd\_convergence\]
------------------------------------------
Suppose we run Algorithm \[alg:dogd\] for $T$ total steps at each node. This allows for $\tilde{k}$ rounds, where $\tilde{k}$ is determined by solving $$\begin{aligned}
\sum_{i=1}^{\tilde{k}} T_i \leq T \Longleftrightarrow \sum_{i=1}^{\tilde{k}} 2 \cdot 2^i \leq T \Longleftrightarrow \tilde{k} \leq \log_2{\left( \frac{T}{2} + 1\right)}.
$$ Using this value for $k$ we see that $$\begin{aligned}
F(\hat{w}_i^{\tilde{k}+1}) & - F(w^*) \leq \frac{L^2}{\sigma } 2^{\tilde{k}} + \frac{L^2 (\tilde{k}-1)}{ 2^{\tilde{k}}} \notag \\
& + \frac{L^2 }{ 2^{\tilde{k}}} \left[ 12 \frac{\log{( \frac{ 2^{\tilde{k}} \sqrt{n} }{\sigma})}}{1 - \sqrt{\lambda_2}} + 19\right] + \frac{6L^2 (\tilde{k}-1) }{ 2^{\tilde{k}}} \\
\leq & \frac{L^2}{\sigma \left( \frac{T}{2} + 1\right) } + \frac{L^2 (\log_2 \left( \frac{T}{2} + 1\right)-1)}{\left( \frac{T}{2} + 1\right) } \notag \\
& + \frac{ L^2 }{\left( \frac{T}{2} + 1\right)} \left[ 12 \frac{\log{\left( \frac{ \left( \frac{T}{2} + 1\right) \sqrt{n} }{\sigma}\right)}}{1 - \sqrt{\lambda_2}} + 19\right] \notag \\
& + \frac{6L^2 (\left( \frac{T}{2} + 1\right)-1) }{\left( \frac{T}{2} + 1\right)} \notag \\
= & O\left( \frac{ \log{(\sqrt{n} T)}}{ T (1 - \sqrt{\lambda_2})}\right) = O\left( \frac{\log(\sqrt{n} T)}{T} \right),\end{aligned}$$ when $\lambda_2$ is constant and does not scale with $n$, and this concludes the proof of Theorem \[thm:dogd\_convergence\].
Extension to Stochastic Optimization {#sec:stochastic_opt}
====================================
The proof presented in the previous section can easily be extended to the case where each node receives a random estimate $\hat{g}(t)$ of the gradient, satisfying $\mathds{E}[\hat{g}(t)] = g(t)$, instead of receiving $g(t)$ directly. We assume that noisy gradients still have bounded variance i.e., $\mathds{E}[ {\left\lVert\hat{g}_i(t) \right\rVert}^2 ] \leq L^2$. In this setting, instead of equation , we have $$\begin{aligned}
A_2 = & \sum_{t=1}^{T_k} \frac{1}{n} \sum_{i=1}^n \langle g_i(t), \overline{w}^k(t) - w^*\rangle \\
= & \sum_{t=1}^{T_k} \langle \frac{1}{n} \sum_{i=1}^n \hat{g}_i(t), \overline{w}^k(t) - w^*\rangle \notag \\
& + \sum_{t=1}^{T_k} \frac{1}{n} \sum_{i=1}^n \langle g_i(t) - \hat{g}_i(t), \overline{w}^k(t) - w^*\rangle.\end{aligned}$$ However, the proof of Theorem \[thm:zink\] does not depend on the gradients being correct; rather, it holds for noisy gradients $\hat{g}_(t)$ as well. Moreover, we have $\mathds{E}[{\left\lVert \hat{g}_i(t) \right\rVert}] \leq L$, and by Hölder’s inequality $\mathds{E}[ {\left\lVert\hat{g}_i(t) \right\rVert} {\left\lVert\hat{g}_j(t) \right\rVert} ] \leq L^2$. Thus, $$\begin{aligned}
\mathds{E} \left[ {\left\lVert \frac{1}{n} \sum_{i=1}^n \hat{g}_i(t) \right\rVert} \right] \leq \frac{1}{n^2} \sum_{i,j = 1}^n \mathds{E}[ {\left\lVert\hat{g}_i(t) \right\rVert} {\left\lVert\hat{g}_j(t) \right\rVert} ] \leq L^2.\end{aligned}$$ Thus, invoking Theorem \[thm:zink\], if the new data and thus the subgradients are independent of the past, and since $\mathds{E}[\hat{g}_i(t)] = g_i(t)$, we have $$\begin{aligned}
\mathds{E}[ A_2] \leq & \frac{{\left\lVert\overline{w}^k(1) - w^* \right\rVert}^2}{2 a_k} + \frac{{T_k} a_k L^2 }{2} \notag \\
& +\mathds{E}[\sum_{t=1}^{T_k} \frac{1}{n} \sum_{i=1}^n \langle g_i(t) - \hat{g}_i(t), \overline{w}^k(t) - w^*\rangle ] \\
= & \frac{{\left\lVert\overline{w}^k(1) - w^* \right\rVert}^2}{2 a_k} + \frac{{T_k} a_k L^2 }{2} \notag \\
&+ \sum_{t=1}^{T_k} \frac{1}{n} \sum_{i=1}^n \langle \mathds{E} \left[ g_i(t) - \hat{g}_i(t) \right], \overline{w}^k(t) - w^*\rangle \\
= &\frac{{\left\lVert\overline{w}^k(1) - w^* \right\rVert}^2}{2 a_k} + \frac{{T_k} a_k L^2 }{2}.\end{aligned}$$ Furthermore, the network error bound holds in expectation as well, i.e., $$\begin{aligned}
\mathds{E}[{\left\lVert\overline{w}^k(t) - w_i^k(t) \right\rVert}& ] \leq \mathds{E}[{\left\lVert\overline{z}^k(t) - z_i^k(t) \right\rVert}] \notag \\
\leq & 2 a_k L \frac{\log{(T_k \sqrt{n})}}{1 - \sqrt{\lambda_2}} + 3 a_k L + \frac{L \sum_{s=1}^{k-1} a_s T_s}{T_k} \end{aligned}$$ Collecting all these observations we have shown that, in expectation, $$\begin{aligned}
\mathds{E}\left[ F(\hat{w}_i^{k+1}) - F(w^*) \right] \leq &\frac{{\left\lVert\overline{w}^k(1) - w^* \right\rVert}^2}{2 a_k T_k} + \frac{ a_k L^2 }{2} \notag \\
& + L^2 a_k \left[ 6 \frac{\log{(T_k \sqrt{n})}}{1 - \sqrt{\lambda_2}} + 9 \right] \notag \\
& + \frac{3L^2 \sum_{s=1}^{k-1} a_s T_s}{T_k}\end{aligned}$$ which, after using the update rules for $a_k$ and $T_k$, is exactly the same rate as before. We note however that there may still be room for improvement in the distributed stochastic optimization setting since [@StochOptStrognlyConvex] describes a single-processor algorithm that converges at a rate $O\left( \frac{1}{T} \right)$.
Simulation {#sec:experiments}
==========
To illustrate the performance of DOGD we simulate online training of a classifier by solving the problem using a network of $10$ nodes arranged as a random geometric graph. Each node is given $T=600$ data points, and the input dimension is $d=100$. We set $\sigma = 0.1$ and generate the data from a standard normal distribution and classify them as $-1$ or $1$ depending on their relative position to a randomly drawn hyperplane in $\mathds{R}^d$. As we see in Figure \[fig:svml2\_minimization\], DODG minimizes the objective much faster than Distributed Dual Averaging (DDA) [@dualAveraging] which has a convergence rate of $O\left( \frac{\log(T \sqrt{n})}{\sqrt{T}}\right)$. DDA is simulated using the learning rate that is suggested in [@dualAveraging]. We have observed that boosting this learning rate may yield faster convergence, but still not as fast as DOGD. Figure \[fig:svml2\_minimization\] also shows the performance of a version of Fast Distributed Gradient Descent (FDGD) [@FastDistributedGradMethods]. As we can see, FDGD fails to converge in an online or stochastic setting and ends up oscillating.
![\[fig:svml2\_minimization\] Optimization of a $d=100$ dimensional problem of the form with a random network of $10$ nodes. Our proposed algorithm DOGD(red) converges faster than DDA(green) as expected from the $T$ instead of $\sqrt{T}$ in the denominator of the convergence rate bound. FDGD(black), is unable to converge in the online problem.](svml2_minimization.pdf){width="3.3in"}
Future Work {#sec:future}
===========
In this paper we have proposed and analyzed a novel distributed optimization algorithm which we call Distributed Online Gradient Descent (DOGD). Our analysis shows that DOGD converges at a rate $O(\frac{\log(\sqrt{n}T)}{T})$ when solving online, stochastic or batch constrained convex optimization problems if the objective function is strongly convex. This rate is optimal in the number of iterations for the online and batch setting and slower than a serial algorithm only by a logarithmic factor in the stochastic optimization setting.
In its current form, DOGD requires the nodes in the network to exchange gradient information at every iteration. Our preliminary investigation suggests that gradually performing more and more updates between each communication can speed up distributed optimization algorithms in the batch setting when one explicitly accounts for the time required to communicate data. Our future work will carry out a similar analysis for online and stochastic optimization algorithms.
Appendix {#appendix .unnumbered}
========
\[lem:P\_tv\_convergence\] If $P$ is a doubly stochastic matrix defined over a strongly connected graph $G=(V,E)$ with ${\left\lvertV \right\rvert}=n$ nodes so that $p_{ji} = 0$ if $(i,j) \not \in E$, then for any $t \leq T$, $$\begin{aligned}
\sum_{s=1}^{t-1 }{\left\lVert\frac{1}{n} {\boldsymbol{1}}^T - \left[ P^{t-s+1} \right]_{i,:} \right\rVert}_1 \leq 1 + \frac{\log{(T \sqrt{n})}}{1 - \sqrt{\lambda_2}}\end{aligned}$$ where $\lambda_2$ is the second largest eigenvalue of $P$.
If the consensus matrix $P$ is doubly stochastic it is straightforward to show that $P^t \rightarrow \frac{1}{n} {\boldsymbol{1}} {\boldsymbol{1}}^T$ as $t \rightarrow \infty$. Moreover, from standard Perron-Frobenius is it easy to show (see e.g., [@StookDiaconis]) $$\begin{aligned}
{\left\lVert\frac{1}{n} {\boldsymbol{1}}^T - \left[ P^t \right]_{i,:} \right\rVert}_1 = 2 {\left\lVert\frac{1}{n} {\boldsymbol{1}}^T - \left[ P^t \right]_{i,:} \right\rVert}_{TV} \leq \sqrt{n} \left(\sqrt{\lambda_2}\right)^t\end{aligned}$$ so in our case ${\left\lVert\frac{1}{n} {\boldsymbol{1}}^T - \left[ P^{t - s+1} \right]_{i,:} \right\rVert}_1 \leq \sqrt{n} \left(\sqrt{\lambda_2}\right)^{t -s + 1}$. Next, demand that the right hand side bound is less than $\sqrt{n} \delta$ with $\delta$ to be determined: $$\begin{aligned}
\sqrt{n} \left(\sqrt{\lambda_2}\right)^{t - s + 1} \leq \sqrt{n} \delta \Rightarrow t - s +1 \geq \frac{\log{(\delta^{-1})}}{\log{(\sqrt{\lambda_2}^{-1})}}.\end{aligned}$$ So with the choice $\delta^{-1} = \sqrt{n} T$, $$\begin{aligned}
{\left\lVert\frac{1}{n} {\boldsymbol{1}}^T - \left[ P^{t - s + 1} \right]_{i,:} \right\rVert}_1 \leq \sqrt{n} \frac{1}{\sqrt{n} T} = \frac{1}{T}\end{aligned}$$ if $t - s +1 \geq \frac{\log{(\delta^{-1})}}{\log{(\sqrt{\lambda_2}^{-1})}} = \hat{t}$. When $s$ is large and $t - s + 1 < \hat{t}$ we take ${\left\lVert\frac{1}{n} {\boldsymbol{1}}^T - \left[ P^{t - s +1} \right]_{i,:} \right\rVert}_1 \leq 2$. The desired bound is not obtained as follows $$\begin{aligned}
\sum_{s=1}^{t-1} {\left\lVert \frac{1}{n} {\boldsymbol{1}}^T - \left[ P^{ t- s + 1} \right]_{i,:} \right\rVert}_1 = & \sum_{s=1}^{t- \hat{t} -1 } {\left\lVert \frac{1}{n} {\boldsymbol{1}}^T - \left[ P^{ t- s +1} \right]_{i,:} \right\rVert}_1 \\
& + \sum_{s = t - \hat{t}}^{t - 1} {\left\lVert \frac{1}{n} {\boldsymbol{1}}^T - \left[ P^{t- s+1} \right]_{i,:} \right\rVert}_1 \notag \\
\leq &\sum_{s=1}^{t - \hat{t} -1 } \frac{1}{T} + \sum_{s = t - \hat{t}}^{t - 1} 2 \\
\leq & \frac{t - \hat{t} }{T} + 2 \hat{t} \leq 1 + 2 \hat{t}\end{aligned}$$ Since $t \leq T$ we know that $t - \hat{t} < T$. Moreover, $\log{(\sqrt{\lambda_2})^{-1}} \geq 1 - \sqrt{\lambda_2}$. Using there two fact we arrive at the result. The same bound is true for any individual entry of $P^t$ approaching $\frac{1}{n}$.
[^1]: Although the hinge loss itself is not strongly convex, adding a strongly convex regularizer makes the overall cost function strongly convex.
|
---
author:
- 'Kenji <span style="font-variant:small-caps;"><span style="font-variant:small-caps;">Yamada</span></span>'
title: |
Properties of the Ground-State $q \bar{q}$ Mesons and\
Possible Classification of Observed Mesons in the $\widetilde{U}(12)_{SF} \times O(3,1)_{L}$ Scheme
---
Introduction
=============
Recently, Ishida et al. have proposed the covariant $\widetilde{U}(12)_{SF}$-classification scheme of hadrons with $\widetilde{U}(12)_{SF} \times O(3,1)_{L}$,[@IIM2000] which gives covariant quark representations for composite hadrons with definite Lorentz and chiral transformation properties. The $\widetilde{U}(12)_{SF}$-classification scheme has a “static” unitary $U(12)_{SF}$ spin-flavor symmetry in the rest frame of hadrons,[@IIYMO2005] embedded in the covariant $\widetilde{U}(12)_{SF}$-representation space, which includes subgroups as $\widetilde{U}(12)_{SF} \supset \widetilde{U}(4)_{D} \times U(3)_{F}$ ($\widetilde{U}(4)_{D}$ being the pseudounitary homogeneous Lorentz group for Dirac spinors). Since
$$U(12)_{SF} \supset U(4)_{D} \times U(3)_{F}$$
with $$U(4)_{D} \supset SU(2)_{\rho} \times SU(2)_{\sigma},
\label{eq:1.1b}$$
the static $U(12)_{SF}$ symmetry includes as its subgroup both the nonrelativistic spin-flavor $SU(6)_{SF}$ and the chiral $U(3)_{L} \times U(3)_{R}$ symmetry as
$$U(12)_{SF} \supset SU(6)_{SF} \times SU(2)_{\rho}$$
and $$U(12)_{SF} \supset U(3)_{L} \times U(3)_{R} \times SU(2)_{\sigma},$$
where $SU(2)_{\rho}$ and $SU(2)_{\sigma}$ are the Pauli-spin groups concerning the boosting and intrinsic spin rotation, respectively, of constituent quarks (being connected with decomposition of Dirac $\gamma$-matrices, $\gamma \equiv \rho \otimes \sigma$). This implies that the $\widetilde{U}(12)_{SF}$-classification scheme is able to incorporate effectively the effects of chiral symmetry and its spontaneous breaking, essential for understanding of properties of the low-lying hadrons, into what is called a constituent quark model.
Experimental candidates for the ground-state $q \bar{q}$ mesons
================================================================
Essential features of the $\widetilde{U}(12)_{SF}$-classification scheme
-------------------------------------------------------------------------
An essential feature of the $\widetilde{U}(12)_{SF}$-classification scheme is to have the static $U(4)_{D}$-spin symmetry in Eq. (\[eq:1.1b\]) for light $u, d, s$ quarks confined inside hadrons. The degree of freedom on the $\rho$-spin, being indispensable for covariant description of spin $1/2$ particles, offers a basis to define the rule of chiral transformation for quark-composite hadrons.[@IIM2000; @IIYMO2005]
Since we have the $\rho$-spin degree of freedom, which is discriminated by the eigenvalues $r=\pm$ of $\rho _{3}$, in addition to the ordinary Pauli-spin, the ground states of light-quark $q\bar{q}$ mesons are composed of eight $SU(3)_{F}$ multiplets with respective $J^{PC}$ quantum numbers, two pseudoscalars $\{P^{(N)}(0^{-+}), P^{(E)}(0^{-+})\}$, two scalars $\{S_{A}^{(N)}(0^{++}), S_{B}^{(E)}(0^{+-})\}$, two vectors $\{V^{(N)}(1^{--}), V^{(E)}(1^{--})\}$, and two axial-vectors $\{A^{(N)}(1^{++}), B^{(E)}(1^{+-})\}$ ($N$ and $E$ denoting “normal” and “extra”), where each $N (E)$ even-parity multiplet is the chiral partner of the corresponding $N (E)$ odd-parity multiplet and they form linear representations of the chiral $U(3)_{L} \times U(3)_{R}$ symmetry.
For heavy-light mesons we have two heavy-spin multiplets $\{P(0^{-}), V(1^{-})\}$ and $\{S(0^{+}), A(1^{+})\}$, which are the chiral partner of each other, since the eigenstates only with the $\rho _{3}$-eigenvalue of $r=+$ are taken for heavy quarks. For heavy-heavy mesons we have the same $\{P(0^{-}), V(1^{-})\}$-spin multiplets as in the conventional quark model. For both the heavy-light and heavy-heavy systems, their spin wave functions are the perfect mixtures, equally weighted sum, of the $N$ and $E$ states concerning the respective $J^{P}$ states.
Experimental candidates for the ground-state $q\bar{q}$ mesons in the $\widetilde{U}(12)_{SF}$-classification scheme {#expcand}
--------------------------------------------------------------------------------------------------------------------
We try to assign some of the known mesons to the predicted ground-state $q\bar{q}$ multiplets, resorting to their $J^{PC}$ quantum numbers and masses. The experimental data are taken from the Particle Data Group 2004 edition.[@PDG2004] The resulting assignments, though some of them are ambiguous, are shown in the Table \[table:1\].
------------ ----------------- -------------------- -------------- --------------- ---------------- ---------------- --------------- ---------------
$P^{(N)}$ $S_{A}^{(N)}$ $P^{(E)}$ $S_{B}^{(E)}$ $V^{(N)}$ $A^{(N)}$ $V^{(E)}$ $B^{(E)}$
$q\bar{q}$
$0^{-+}$ $0^{++}$ $0^{-+}$ $0^{+-}$ $1^{--}$ $1^{++}$ $1^{--}$ $1^{+-}$
$n\bar{n}$ $\pi$ $a_{0}(980)$ $\pi(1300)$ $\rho(770)$ $a_{1}(1260)$ $b_{1}(1235)$
$\eta$ $\sigma$ $\eta(1295)$ $\omega(782)$ $f_{1}(1285)$ $h_{1}(1170)$
$s\bar{s}$ $\eta '(958)$ $f_{0}(980)$ $\eta(1475)$ $\phi(1020)$ $f_{1}(1420)$ $h_{1}(1380)$
$s\bar{n}$ $K$ $\kappa$ $K(1460)$ $K^{*}(892)$ $K_{1}(1270)$ $K^{*}(1410)$ $K_{1}(1400)$
$c\bar{n}$ $D$ – – $D^{*}$ – –
$c\bar{s}$ $D_{s}$ $D^{*}_{sJ}(2317)$ – – $D^{*}_{s}$ $D_{sJ}(2460)$ – –
$b\bar{n}$ $B$ – – $B^{*}$ – –
$b\bar{s}$ $B_{s}$ – – $B^{*}_{s}$ – –
$c\bar{c}$ $\eta _{c}(1S)$ – – – $J/\psi(1S)$ – – –
$b\bar{b}$ $\eta _{b}(1S)$ – – – $\Upsilon(1S)$ – – –
------------ ----------------- -------------------- -------------- --------------- ---------------- ---------------- --------------- ---------------
: Experimental candidates for the ground-state $q\bar{q}$ mesons in the $\widetilde{U}_{SF}(12)$-classification scheme. The data are taken from Ref. .[]{data-label="table:1"}
Here we make some comments on these assignments as follows:
1. The light scalar mesons $\{a_{0}(980), \sigma, f_{0}(980),
\kappa\}$ are assigned to the $S_{A}^{(N)}(0^{++})$ nonet as a chiral partner of the $\pi$-meson $P^{(N)}(0^{-+})$ nonet. Recently the existence of the $\kappa$ meson has been confirmed in two independent partial-wave analyses of the decay $J/\psi \rightarrow \bar{K}^{*}(892)^{0}K^{+}\pi^{-}$ by the BES Collaboration.[@BES]
2. The $B^{(E)}(1^{+-})$ nonet is composed of the $b_{1}(1235)$, $h_{1}(1170)$, $h_{1}(1380)$ and $K_{1}(1400)$ mesons. The low-mass vector meson $K^{*}(1410)$ is assigned as a member of the $V^{(E)}(1^{--})$ nonet which is a chiral partner of the $B^{(E)}(1^{+-})$ nonet.
3. The axial-vector mesons $\{a_{1}(1260), f_{1}(1285),
f_{1}(1420), K_{1}(1270)\}$ are assigned to the $A^{(N)}(1^{++})$ nonet which is a chiral partner of the $\rho(770)$-meson $V^{(N)}(1^{--})$ nonet. The $a_{1}(1260)$ and $f_{1}(1285)$ mesons are tentatively assigned to this nonet, while their masses seem to be higher than could be expected and also their observed properties of radiative transitions does not seem to be consistent with those expected in the $\widetilde{U}(12)_{SF}$-classification scheme.[@Maeda]
4. The recent discovered mesons $D_{sJ}^{*}(2317)$ and $D_{sJ}(2460)$ are just assigned to the $\{S(0^{+}), A(1^{+})\}$-spin multiplet as a chiral partner of the $\{P(0^{-}), V(1^{-})\}$-spin multiplet, $\{D_{s}, D_{s}^{*}\}$.[@Ishida2003] These newly observed states, together with the $\sigma$-meson nonet, are the best candidates for the hadronic states with the $\rho _{3}$-eigenvalue of $r=-$ whose existence is expected in the $\widetilde{U}(12)_{SF}$-classification scheme.
5. The $N$ and $E$ states with the same $J^{PC}$, that is, the vector $V^{(N,E)}$ and pseudoscalar $P^{(N,E)}$ states generally mix together, due to the spontaneous as well as explicit breaking of chiral symmetry, and so do the strange scalar $\kappa^{(N,E)}$ and axial-vector $K_{1}^{(N,E)}$ states. However, as far as the pseudoscalar $P^{(N,E)}$ octet is concerned, it is assumed that no mixing occurs so as to preserve the property of both the $\pi$-meson octet, the Nambu-Goldstone bosons associated with the spontaneous breaking of the axial $SU(3)_{A}$ symmetry, and the $\sigma$-meson nonet belonging to the same chiral multiplet. As for the flavor-singlet $P^{(N)}$ and $P^{(E)}$ states, the $\eta'$-like mesons, there would be mixing between them according to a common understanding that the axial $U(1)_{A}$ is not a true symmetry in the strong interactions.
Masses and $N$-$E$ mixings of the ground-state $q \bar{q}$ mesons {#masses}
=================================================================
In the following we examine the respective $N$- and $E$-state mixings of $V^{(N,E)}$, $K_{1}^{(N,E)}$ and $\kappa^{(N,E)}$ in a simple phenomenological model. Through this analyses the flavor mixing between $n \bar{n}$ and $s \bar{s}$ states is neglected for the isoscalar channels and the masses of $n \bar{n}(I=0)$ states are taken to be equal to those of $n \bar{n}(I=1)$ states. It is also assumed that the $N$- and $E$-state masses of different flavor states are related to each other as $$M_{(N,E)}(s \bar{s}) - M_{(N,E)}(s \bar{n}) =
M_{(N,E)}(s \bar{n}) - M_{(N,E)}(n \bar{n})
\equiv \Delta m_{s},
\label{eq:deltams}$$ where $\Delta m_{s}$ is taken to be 120 MeV.
Phenomenological mixing scheme of the N and E states
-----------------------------------------------------
In the normal and extra basis $\left( |E \rangle, |N \rangle \right)$, the mass-squared matrix describing the $N$-$E$ mixing can be written as $$\boldsymbol{M}^{2} =
\begin{pmatrix}
M_{E}^{2} & \Delta \\
\Delta & M_{N}^{2}
\end{pmatrix},$$ where $M_{N}$ and $M_{E}$ are the masses of the $N$ and $E$ states, respectively, before mixing and $\Delta$ is a phenomenological parameter corresponding to the mixing strength. Assuming that the physical-state basis $\left( |L \rangle, |H \rangle \right)$ ($L$ and $H$ denoting “low” and “high”) is an eigenvector of the mass-squared matrix $\boldsymbol{M}^{2}$ with the eigenvalues $M_{L}^{2}$ and $M_{H}^{2}$, we can diagonalize the matrix $\boldsymbol{M}^{2}$ as $$U^{-1}\boldsymbol{M}^{2}U =
\begin{pmatrix}
M_{L}^{2} & 0 \\
0 & M_{H}^{2}
\end{pmatrix}$$ by the unitary transformation
$$\begin{pmatrix}
|L \rangle \\ |H \rangle
\end{pmatrix}
= U^{-1}
\begin{pmatrix}
|E \rangle \\ |N \rangle
\end{pmatrix}$$
with $$U^{-1} =
\begin{pmatrix}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
\end{pmatrix},$$
in which the $N$-$E$ mixing angle is defined. It is noted that the low- and high-mass states have dominantly $|E \rangle$ and $|N \rangle$ components, respectively, if the mixing angle is $|\theta| < 45^{\circ}$.
Masses and mixing properties of the vector $V^{(N)}$ and $V^{(E)}$ nonets {#vector}
-------------------------------------------------------------------------
In the analysis of the vector meson nonets $V^{(N)}(1^{--})$ and $V^{(E)}(1^{--})$ we assume that the $V^{(N)}$ and $V^{(E)}$ states are mixed maximally, that is, the $N$-$E$ mixing angle is $|\theta| = 45^{\circ}$, in accord with the fact that the lowest-lying vector mesons $\rho(770)$, $\omega(782)$, $\phi(1020)$, $K^{*}(892)$ are well described as nonrelativistic $q\bar{q}$ states and the analysis of their radiative transitions in the $\widetilde{U}(12)_{SF}$-classification scheme.[@Maeda] These maximally mixed low- and high-mass vector states, $V^{(NR)}$ and $V^{(ER)}$, mean to be nonrelativistic (NR) and extremely relativistic (ER) states which have the $\rho_{3}$-eigenvalues of $(r_{q}, r_{\bar q}) = (+, +)$ and $(-, -)$, respectively.
First, we consider the $K^{*}$ system, whose mass-squared matrix relation is given by $$U^{-1}
\begin{pmatrix}
M_{K^{*(E)}}^{2} & \Delta_{K^{*}} \\
\Delta _{K^{*}} & M_{K^{*(N)}}^{2}
\end{pmatrix}U =
\begin{pmatrix}
M_{K^{*}(892)}^{2} & 0 \\
0 & M_{K^{*}(1410)}^{2}
\end{pmatrix},$$ where $K^{*}(892)$ and $K^{*}(1410)$ are members of the $V^{(N)}(1^{--})$ and $V^{(E)}(1^{--})$ nonets as mentioned in §\[expcand\]. Using the mixing angle $|\theta| = 45^{\circ}$, which means $M_{K^{*(N)}} = M_{K^{*(E)}}$, and $M_{K^{*}(892)} = 894\ \mathrm{MeV}$ and $M_{K^{*}(1410)} = 1414\ \mathrm{MeV}$, we obtain $$M_{K^{*(N,E)}} = 1183\ \mathrm{MeV}, \ \
|\Delta _{K^{*}}| = 0.6001\ \mathrm{GeV}^{2}.$$ Then, from the assumption in Eq. (\[eq:deltams\]) we have $M_{\rho^{(N,E)}} = M_{\omega^{(N,E)}} = 1063\ \mathrm{MeV}$ for the $\rho$ and $\omega$ systems. For the $\rho$ system the mass-squared matrix relation is given by $$U^{-1}
\begin{pmatrix}
M_{\rho^{(E)}(1063)}^{2} & \Delta _{\rho} \\
\Delta_{\rho} & M_{\rho^{(N)}(1063)}^{2}
\end{pmatrix}U =
\begin{pmatrix}
M_{\rho(770)}^{2} & 0 \\
0 & M_{\rho^{\prime}}^{2}
\end{pmatrix},$$ which gives, taking the mass value of $M_{\rho(770)} = M_{\omega(782)} = 776\ \mathrm{MeV}$, $$M_{\rho^{\prime}} = M_{\omega^{\prime}} = 1290\ \mathrm{MeV}, \ \
|\Delta _{\rho}| = |\Delta _{\omega}| = 0.5284\ \mathrm{GeV}^{2}.$$ In a similar way, we obtain
$$M_{\phi^{(N,E)}} = 1303\ \mathrm{MeV}, \ \
M_{\phi^{\prime}} = 1535\ \mathrm{MeV},$$
$$|\Delta _{\phi}| = 0.6585\ \mathrm{GeV}^{2}$$
for the $\phi$ system. These results are tabulated in Table \[table:2\].
Here we point out that there exist observed candidates suitable for the predicted low-mass vector mesons $\rho'(1290)$ and $\omega'(1290)$. In fact, there is some experimental evidence for the $\rho(1250)$ reported by the LASS[@LASS] and OBELIX[@OBELIX] Collaborations and the existence of $\omega(1250)$ is claimed in the analysis of the $e^{+}e^{-} \to \pi^{+}\pi^{-}\pi^{0}$ cross section by the SND[@SND] and BABAR[@BABAR] Collaborations. Furthermore, a recent reanalysis of the BABAR[@BABAR], SND[@SND; @SNDop] and CMD-2[@CMD2] data on the $e^{+}e^{-} \to \pi^{+}\pi^{-}\pi^{0}$ and $e^{+}e^{-} \to \omega\pi^{0}$ cross sections indicates the existence of these two vector states.[@Komada]
It is worthwhile to mention that only the $V^{(N)}(1^{--})$ states and not the $V^{(E)}(1^{--})$ can be produced in the $e^{+}e^{-}$ annihilation process of $e^{+}e^{-} \to \gamma^{*} \to V$ due to the chirality conservation of quarks,[^1] since both the constituent quark and antiquark of $V^{(N)}(1^{--})$ have the same chirality, while those of $V^{(E)}(1^{--})$ have opposite one in the $\widetilde{U}(12)_{SF}$-classification scheme. Therefore we can expect that the predicted $\rho'(1290)$ and $\omega'(1290)$ mesons, which have the same amount of $V^{(N)}(1^{--})$ component as the $\rho(770)$ and $\omega(782)$, are seen in the above $e^{+}e^{-}$ annihilation process.
[cccccc]{} & & Predicted mass & Observed mass$^{\textrm{a)}}$ & Mixing angle $|\theta|$ & Mixing strength $|\Delta|$\
$q\bar{q}$ & State & (MeV) & (MeV) & & $(\mathrm{GeV}^{2})$\
$n\bar{n}$ & $\rho(770)$ & $\underline{776}$ & $775.8\pm 0.5$ & $45^{\circ}$ & 0.5284\
& $\rho'$ & 1290 & – &\
& $\omega(782)$ & 776 & $782.59\pm 0.11$\
& $\omega'$ & 1290 & – &\
$s\bar{s}$ & $\phi(1020)$ & $\underline{1019}$ & $1019.456\pm 0.020$ & $45^{\circ}$ & 0.6585\
& $\phi'$ & 1535 & – &\
$s\bar{n}$ & $K^{*}(892)$ & $\underline{894}$ & $893.9\pm 2.5$ & $45^{\circ}$ & 0.6001\
& $K^{*}(1410)$ & $\underline{1414}$ & $1414\pm 15$\
\
Masses and mixing properties of the axial-vector $A^{(N)}$ and $B^{(E)}$ nonets
--------------------------------------------------------------------------------
In the axial-vector meson nonets, $A^{(N)}(1^{++})$ and $B^{(E)}(1^{+-})$, their mixing could occur only for the strange $K_{1}$ system and therefore the $N$ and $E$ states are physical ones for the isoscalar and isovector channels. In this analysis we select, as input data, the $b_{1}(1235)$, $K_{1}(1270)$ and $K_{1}(1400)$ mesons from among the assigned mesons in §\[expcand\].
For the $K_{1}$ system we obtain $M_{K_{1B}^{(E)}} = 1350\ \mathrm{MeV}$ from the assumption in Eq. (\[eq:deltams\]) with the measured mass of $M_{b_{1}(1235)} = 1230\ \mathrm{MeV}$ and the mass-squared matrix relation is given by $$U^{-1}
\begin{pmatrix}
M_{K_{1B}^{(E)}(1350)}^{2} & \Delta _{K_{1}} \\
\Delta _{K_{1}} & M_{K_{1A}^{(N)}}^{2}
\end{pmatrix}U =
\begin{pmatrix}
M_{K_{1}(1270)}^{2} & 0 \\
0 & M_{K_{1}(1400)}^{2}
\end{pmatrix}.$$ Taking the mass values of $M_{K_{1}(1270)} = 1273\ \mathrm{MeV}$ and $M_{K_{1}(1400)} = 1402\ \mathrm{MeV}$, we derive
$$M_{K_{1A}^{(N)}} = 1328\ \mathrm{MeV},$$
$$|\Delta _{K_{1}}| = 0.1700\ \mathrm{GeV}^{2}, \ \
|\theta _{K_{1}}| = 49.9^{\circ}$$
and then
$$\begin{aligned}
M_{a_{1}} &= M_{f_{1}(n\bar{n})} = 1210\ \mathrm{MeV}, \ \
M_{f_{1}'(s\bar{s})} = 1450\ \mathrm{MeV}, \\
M_{b_{1}} &= M_{h_{1}(n\bar{n})} = 1230\ \mathrm{MeV}, \ \
M_{h_{1}'(s\bar{s})} = 1470\ \mathrm{MeV}
\label{eq:3.13b}
\end{aligned}$$
for the isoscalar and isovector channels. These results are given in Table \[table:3\].
It should be noted here that $K_{1A}^{(N)}$ and $K_{1B}^{(E)}$ are the $1^{3}P_{1}$ and $1^{1}P_{1}$ states respectively in the conventional quark model, while they are relativistic $S$-wave states in which each $q$ and $\bar{q}$ has the opposite $\rho _{3}$-eigenvalue in the $\widetilde{U}(12)_{SF}$-classification scheme. The resulting mixing angle $|\theta _{K_{1}}| = 49.9^{\circ}$ means that the dominant components of $K_{1}(1270)$ and $K_{1}(1400)$ are the $K_{1A}^{(N)}$ and $K_{1B}^{(E)}$ states, respectively.
[cccccc]{} & & Predicted mass & Observed mass$^{\textrm{a)}}$ & Mixing angle $|\theta|$ & Mixing strength $|\Delta|$\
$q\bar{q}$ & State & (MeV) & (MeV) & & $(\mathrm{GeV}^{2})$\
$n\bar{n}$ & $a_{1}(1260)$ & 1210 & $1230\pm 40$ & – & –\
& $f_{1}(1285)$ & 1210 & $1281.8\pm 0.6$ & – & –\
& $b_{1}(1235)$ & $\underline{1230}$ & $1229.5\pm 3.2$ & – &–\
& $h_{1}(1170)$ & 1230 & $1170\pm 20$ & – & –\
$s\bar{s}$ & $f_{1}(1420)$ & 1450 & $1426.3\pm 0.9$ & – & –\
& $h_{1}(1380)$ & 1470 & $1386\pm 19$ & – & –\
$s\bar{n}$ & $K_{1}(1270)$ & $\underline{1273}$ & $1273\pm 7$ & $49.9^{\circ}$ & 0.1700\
& $K_{1}(1400)$ & $\underline{1402}$ & $1402\pm 7$\
\
Masses and mixing properties of the scalar $S_{A}^{(N)}$ and $S_{B}^{(E)}$ nonets
----------------------------------------------------------------------------------
The scalar $S_{A}^{(N)}(0^{++})$ and $S_{B}^{(E)}(0^{+-})$ states could mix only for the strange $\kappa$ system, as in the $K_{1}$ system. For lack of experimental information and theoretical understanding on the light scalar mesons, we assume here that the mixing strength between $\kappa _{A}^{(N)}$ and $\kappa _{B}^{(E)}$ is equal to that of the $K_{1}$ system, that is, $$\Delta _{\kappa} = \Delta _{K_{1}},$$ which might be realized, provided that the $\kappa _{A}^{(N)}$-$\kappa _{B}^{(E)}$ mixing originates from the spontaneous breaking of chiral symmetry. As input data in this analysis we take the mass values of the $a_{0}(980)$ and $\kappa$ mesons to be $985\ \mathrm{MeV}$ and $875\ \mathrm{MeV}$, respectively.
For the $\kappa$ system we have $M_{\kappa _{A}^{(N)}} = 1105\ \mathrm{MeV}$ from the assumption in Eq. (\[eq:deltams\]) with the $a_{0}(980)$ mass and the mass-squared matrix relation is given by $$U^{-1}
\begin{pmatrix}
M_{\kappa _{B}^{(E)}}^{2} & \Delta _{\kappa} \\
\Delta _{\kappa} & M_{\kappa _{A}^{(N)}(1105)}^{2}
\end{pmatrix}U =
\begin{pmatrix}
M_{\kappa(875)}^{2} & 0 \\
0 & M_{\kappa'}^{2}
\end{pmatrix}$$ with $|\Delta _{\kappa}| = 0.1700\ \mathrm{GeV}^{2}$. This relation gives
$$M_{\kappa _{B}^{(E)}} = 911\ \mathrm{MeV}, \ \
M_{\kappa'} = 1135\ \mathrm{MeV},$$
$$|\theta _{\kappa}| = 20.5^{\circ}$$
and then
$$\begin{aligned}
M_{a_{0}} &= M_{f_{0}(n\bar{n})} = 985\ \mathrm{MeV}, \ \
M_{f_{0}'(s\bar{s})} = 1225\ \mathrm{MeV}, \\
M_{b_{0}} &= M_{h_{0}(n\bar{n})} = 790\ \mathrm{MeV}, \ \
M_{h_{0}'(s\bar{s})} = 1030\ \mathrm{MeV}
\label{eq:3.17b}
\end{aligned}$$
for the isoscalar and isovector channels. These results are given in Table \[table:4\].
Here it is noticeable that a dominant component of the $\kappa(875)$ is $\kappa _{B}^{(E)}$ while that of the $\kappa'(1135)$ is $\kappa _{A}^{(N)}$, due to the mixing angle of $|\theta _{\kappa}| = 20.5^{\circ}$. This implies that the unknown $\kappa'(1135)$ rather than the $\kappa(875)$ is a member of the light scalar $\sigma$-meson nonet. In this connection, it is of great interest that a recent lattice-QCD study of light scalar mesons with the interpolating field $\bar{\psi}\psi$ gives the results of the lightest $a_{0}$ meson having a mass of $1.01\pm 0.04\ \mathrm{GeV}$ and the $K_{0}^{*}$ meson $100$-$130\ \mathrm{MeV}$ heavier than the $a_{0}$ meson.[@UKQCD] These results are in conformity with our prediction, since the $\kappa _{A}^{(N)}$ and $\kappa _{B}^{(E)}$ are considered to be states corresponding to the interpolating fields[^2] $\bar{\psi}\psi$ and $\bar{\psi}i\gamma _{\mu}\partial_{\mu}\psi$, respectively.
[cccccc]{} & & Predicted mass & Observed mass$^{\textrm{a)}}$ & Mixing angle $|\theta|$ & Mixing strength $|\Delta|$\
$q\bar{q}$ & State & (MeV) & (MeV) & & $(\mathrm{GeV}^{2})$\
$n\bar{n}$ & $a_{0}(980)$ & $\underline{985}$ & $ 984.7\pm 1.2$ & – & –\
& $\sigma$ & 985 & $\sim$ 400-600 & – & –\
& $b_{0}$ & 790 & & – & –\
& $h_{0}$ & 790 & & – & –\
$s\bar{s}$ & $f_{0}(980)$ & 1225 & $ 980\pm 10$ & – & –\
& $h_{0}'$ & 1030 & & – & –\
$s\bar{n}$ & $\kappa'$ & 1135 & & $20.5^{\circ}$ & 0.1700\
& $\kappa$ & $\underline{875}$ & $878\pm 23^{+64}_{-55}$ &\
Mass spectra and possible assignments for observed mesons in the $\widetilde{U}(12)_{SF} \times O(3,1)_{L}$-classification scheme
=================================================================================================================================
Mass spectra of $n\bar{n}$ mesons
---------------------------------
We examine excited states of the respective ground-state sectors in the extended $\widetilde{U}(12)_{SF}$-classification scheme with $\widetilde{U}(12)_{SF} \times O(3,1)_{L}$, in which the degree of freedom concerning the orbital motion of quarks is incorporated. In this classification scheme it is predicted that there exist some $q\bar{q}$ exotic states with $J^{PC}$ quantum numbers, such as $0^{+-}$ in the ground states, $0^{--}$ and $1^{-+}$ in the excited $P$-wave states, $2^{+-}$ in the excited $D$-wave states, which never appear in any nonrelativistic quark model. There is presently the experimental observation of two exotic mesons with $J^{PC} = 1^{-+}$, the $\pi _{1}(1400)$ and $\pi _{1}(1600)$.[@PDG2004] Since, as far as the $1^{-+}$ exotic state is concerned, we have just two states in the $P$-wave excitation in this scheme, the $\pi _{1}(1400)$ and $\pi _{1}(1600)$ mesons are expected to be promising experimental candidates for these predicted exotics.
In the $\widetilde{U}(12)_{SF} \times O(3,1)_{L}$-classification scheme the mass of excited states is given by $$\label{eq:4.1}
M_{N}^{2} = M_{0}^{2} + N\Omega, \ \
N = L + 2N',$$ where $M_{0}$ is the ground-state mass, $\Omega^{-1}$ the slope parameter of linear Regge trajectories, and $L$ ($N'$) the orbital-angular-momentum (radial) quantum number. We take here a value of $\Omega$ to be $1.136\ \mathrm{GeV}^{2}$ for the $n\bar{n}$ meson system, determined from the mass-squared distance between the $\rho(770)$ (775.8 MeV) and $a_{2}(1320)$ (1318.3 MeV) mesons and also take the respective ground-state masses obtained in §\[masses\] as values of the $M_{0}(n\bar{n})$. For the $P^{(E)}$ sector we use a mass, 1300 MeV, of the $\pi(1300)$ meson, which was assigned to the isovector $P^{(E)}$ state in §\[expcand\]. Since we have no appropriate ground state as input for the $P^{(N)}$ sector, due to its Nambu-Goldstone nature, we use a mass, 1880 MeV, of the $\pi _{2}(1880)$ meson which is considered as an experimental candidate for the excited $1^{1}D_{2}$ isovector state. Then we estimate masses of the excited $1P$, $1D$ and $2S$ states for the respective sectors of $n\bar{n}$ mesons. The results are presented in Table \[table:5\].
$N$ $L$ $P^{(N)}$ $S_{A}^{(N)}$ $P^{(E)}$ $S_{B}^{(E)}$ $V^{(NR)}$ $A^{(N)}$ $V^{(ER)}$ $B^{(E)}$
----- ----- ----------- --------------- ----------- --------------- ------------ ----------- ------------ -----------
0 0 $0^{-+}$ $0^{++}$ $0^{-+}$ $0^{+-}$ $1^{--}$ $1^{++}$ $1^{--}$ $1^{+-}$
–
$0^{++}$ $0^{--}$ $0^{++}$ $0^{-+}$
1 1 $1^{+-}$ $1^{--}$ $1^{+-}$ $1^{-+}$ $1^{++}$ $1^{--}$ $1^{++}$ $1^{-+}$
$2^{++}$ $2^{--}$ $2^{++}$ $2^{-+}$
1.55 1.45 1.68 1.33 1.32 1.61 1.67 1.63
2 0 $0^{-+}$ $0^{++}$ $0^{-+}$ $0^{+-}$ $1^{--}$ $1^{++}$ $1^{--}$ $1^{+-}$
1.88 1.80 1.99 1.70 1.70 1.93 1.98 1.95
$1^{--}$ $1^{++}$ $1^{--}$ $1^{+-}$
2 2 $2^{-+}$ $2^{++}$ $2^{-+}$ $2^{+-}$ $2^{--}$ $2^{++}$ $2^{--}$ $2^{+-}$
$3^{--}$ $3^{++}$ $3^{--}$ $3^{+-}$
1.80 1.99 1.70 1.70 1.93 1.98 1.95
: Estimated masses (in units of GeV) of the $1S$, $1P$, $2S$ and $1D$ states for the respective sectors of states in the $n\bar{n}$ system. Input values are underlined.[]{data-label="table:5"}
The $\widetilde{U}(12)_{SF} \times O(3,1)_{L}$ classification of observed mesons below $\sim 2$
------------------------------------------------------------------------------------------------
We seek experimental candidates for the $1S$, $1P$, $2S$ and $1D$ states out of the known mesons[^3] listed in the Particle Data Group 2004,[@PDG2004] in addition to the states assigned in the previous sections, based on their $J^{PC}$ quantum numbers, measured masses and decay modes. The resulting possible assignments are given in Table \[table:6\] for the mesons selected in this way, though very tentative. From this table it is found that we have a number of observed states which could be classified well in terms of the $\widetilde{U}(12)_{SF} \times O(3,1)_{L}$ scheme.
Since the $1^{-+}$ exotic states are in the $P$-wave excitation of the $S_{B}^{(E)}(0^{+-};1^{1}S_{0})$ and $B^{(E)}(1^{+-};1^{3}S_{1})$ ground states, their masses are given as $$M(1^{-+};1^{1}P_{1}\ S_{B}^{(E)}) = 1.33\ \mathrm{GeV}, \ \
M(1^{-+};1^{3}P_{1}\ B^{(E)}) = 1.63\ \mathrm{GeV},$$ corresponding to the respective ground-state masses, $0.790\ \mathrm{GeV}$ and $1.230\ \mathrm{GeV}$. We see that these masses are quite consistent with their measured values [@PDG2004] as $$M_{\pi _{1}(1400)} = 1376\pm 17\ \mathrm{MeV}, \ \
M_{\pi _{1}(1600)} = 1653^{+18}_{-15}\ \mathrm{MeV}.$$
Recently, the BES collaboration has reported the observation of a broad $K^{+}K^{-}$ resonance $X(1576)$ with $J^{PC}=1^{--}$ and the pole position of $1576^{+49}_{-55}(\mathrm{stat})^{+98}
_{-91}(\mathrm{syst}) -
i409^{+11}_{-12}(\mathrm{stat})^{+32}_{-67}(\mathrm{syst})$ MeV in the decay $J/\psi \rightarrow K^{+}K^{-}\pi^{0}$.[@BESX] Subsequently, it is shown that this resonance should be an isovector state.[@GS] In the $\widetilde{U}(12)_{SF} \times O(3,1)_{L}$-classification scheme, taking into account its measured mass of $\sim 1.5$-$1.6$ GeV, we have a proper place to assign the $X(1576)$, that is, $(1^{--};1^{3}P_{1}\ A^{(N)})$ in the $P$-wave excitation of the $A^{(N)}(1^{++};1^{3}S_{1})$. Since both the quark and antiquark of this state are the same chirality like $V^{(N)}(1^{--};1^{3}S_{1})$, the $X(1576)$ and its isoscalar partners might be expected to be seen in the $e^{+}e^{-}$ annihilation process, as was mentioned in §\[vector\], though in this case the production rates may be considerably suppressed as compared with the $S$-wave states due to their $P$-wave spatial wave functions.
We have further another $1^{--}$ vector multiplet in the $P$-wave excitation of the $S_{A}^{(N)}(0^{++};1^{1}S_{0})$ and vector mesons belonging to this multiplet is composed of a quark and an antiquark which have the opposite chirality each other the same as $S_{A}^{(N)}$. Therefore, it is expected that isoscalar and isovector members of this multiplet would be doubly hard to be produced in the $e^{+}e^{-}$ annihilation process.
$P^{(N)}$ $S_{A}^{(N)}$ $P^{(E)}$ $S_{B}^{(E)}$ $V^{(NR)}$ $A^{(N)}$ $V^{(ER)}$ $B^{(E)}$
------------------ ----------------------------- ------------------ ------------------ ------------------- ------------------------- -------------------- ------------------
$0^{-+}$ $0^{++}$ $0^{-+}$ $0^{+-}$ $1^{--}$ $1^{++}$ $1^{--}$ $1^{+-}$
$\pi$ $a_{0}(980)$ $\pi(1300)$ $[b_{0}(790)]$ $\rho(770)$ $a_{1}(1260)$ $\rho(1250)$ $b_{1}(1235)$
$\eta$ $\sigma$ $\eta(1295)$ $[h_{0}(790)]$ $\omega(782)$ $f_{1}(1285)$ $\omega(1250)$ $h_{1}(1170)$
$\eta'(958)$ $f_{0}(980)$ $\eta(1475)$ $[h_{0}'(1030)]$ $\phi(1020)$ $f_{1}(1420)$ $[\phi(1540)]$ $h_{1}(1380)$
$K$ $[\kappa'(1135)]$ $K(1460)$ $\kappa$ $K^{*}(892)$ $K_{1}(1270)$ $K^{*}(1410)$ $K_{1}(1400)$
$0^{++}$ $0^{--}$ $0^{++}$ $0^{-+}$
$a_{0}(1450)$
$f_{0}(1370)$ $f_{0}(1500)$ $\eta(1405)$
$f_{0}(1710)$
$K^{*}_{0}(1430)$
$1^{+-}$ $1^{--}$ $1^{+-}$ $1^{-+}$ $1^{++}$ $1^{--}$ $1^{++}$ $1^{-+}$
$\rho(1450)$ $\pi_{1}(1400)$ $X(1576)^{\textrm{a)}}$ $a_{1}(1640)$ $\pi_{1}(1600)$
$h_{1}(1595)$ $\omega(1420)$
$\phi(1680)$ $f_{1}(1510)$
$K_{1}(1650)$ $K^{*}(1680)$
$2^{++}$ $2^{--}$ $2^{++}$ $2^{-+}$
$a_{2}(1320)$ $a_{2}(1700)$ $\pi_{2}(1670)$
$f_{2}(1270)$ $f_{2}(1640)$ $\eta_{2}(1645)$
$f '_{2}(1525)$
$K^{*}_{2}(1430)$ $K_{2}(1580)$ $K_{2}(1770)$
$0^{-+}$ $0^{++}$ $0^{-+}$ $0^{+-}$ $1^{--}$ $1^{++}$ $1^{--}$ $1^{+-}$
$\pi(1800)$ $\rho(1700)$ $\rho(1900)$
$\eta(1760)$ $f_{0}(1790)^{\textrm{b)}}$ $\omega(1650)$
$K(1830)$ $K^{*}_{0}(1950) $
$1^{--}$ $1^{++}$ $1^{--}$ $1^{+-}$
$a_{1}(1930)$ $\rho(1965)$ $b_{1}(1960)$
$f_{1}(1970)$ $\omega(1960)$ $h_{1}(1965)$
$2^{-+}$ $2^{++}$ $2^{-+}$ $2^{+-}$ $2^{--}$ $2^{++}$ $2^{--}$ $2^{+-}$
$\pi_{2}(1880)$ $\pi_{2}(2005)$ $a_{2}(1990)$ $\rho_{2}(1940)$
$\eta_{2}(1870)$ $f_{2}(1810)$ $\eta_{2}(2030)$ $f_{2}1910$ $\omega_{2}(1975)$
$f_{2}(1950)$ $f_{2}(2010)$
$K^{*}_{2}(1980)$ $K_{2}(1820)$
: Possible assignments for the known mesons to the $1S$, $1P$, $2S$ and $1D$ states in the $\widetilde{U}(12)_{SF} \times O(3,1)_{L}$-classification scheme. Mesons in brackets are the unknown states whose masses are estimated in the present work.[]{data-label="table:6"}
[cccccccc]{} $P^{(N)}$ & $S_{A}^{(N)}$ & $P^{(E)}$ & $S_{B}^{(E)}$ & $V^{(NR)}$ & $A^{(N)}$ & $V^{(ER)}$ & $B^{(E)}$\
& & & &\
& & & & $3^{--}$ & $3^{++}$ & $3^{--}$ & $3^{+-}$\
& & & & $\rho_{3}(1690)$ & $a_{3}(1875)$ & $\rho_{3}(1990)$ & $b_{3}(2025)$\
& & & & $\omega_{3}(1670)$ & & $\omega_{3}(1945)$ & $h_{3}(2025)$\
& & & & $\phi_{3}(1850)$\
& & & & $K^{*}_{3}(1780)$\
\
Concluding remarks
==================
We have proposed the possible assignments for a number of observed mesons below $\sim 2$ GeV in the $\widetilde{U}(12)_{SF} \times O(3,1)_{L}$-classification scheme. Considering the phenomenological mixing scheme of the normal and extra states, we estimated the masses of missing members of the ground-state multiplets and also the masses of the excited $1P$, $1D$ and $2S$ states. We see that enigmatic states, such as the light scalar $\sigma$, $\kappa$, $a_{0}(980)$ and $f_{0}(980)$, the exotic $1^{-+}$ states $\pi _{1}(1400)$ and $\pi _{1}(1600)$, the low-mass vector $\rho(1250)$ and $\omega(1250)$, and the unexpectedly low-mass states $D_{sJ}^{*}(2317)$ and $D_{sJ}(2460)$ could be classified naturally as conventional $q\bar{q}$ states in the $\widetilde{U}(12)_{SF} \times O(3,1)_{L}$-classification scheme, without resort to more exotic or farfetched interpretations like a multiquark, molecule or low-mass hybrid. Since it goes without saying that only mass spectra are insufficient to establish their assignments, it is important to examine the production and decay properties, such as pionic and radiative transitions, of the assigned states in the $\widetilde{U}(12)_{SF} \times O(3,1)_{L}$-classification scheme.
Acknowledgements {#acknowledgements .unnumbered}
================
I am grateful to Shin Ishida, Kunio Takamatsu and other members of the sigma group for useful discussions.
[99]{} S. Ishida, M. Ishida and T. Maeda, .
S. Ishida, M. Ishida, K. Yamada, T. Maeda and M. Oda, hep-ph/0408136.
Particle Data Group, S. Eidelman et al., .
BES Collaboration, M.Ablikim et al., .
T. Maeda, K. Yamada, M. Oda and S. Ishida, these proceedings, *Proceedings of the Seminar on Perspectives for Studies of Chiral Particles at BES, IHEP, Beijing, 2006*, KEK Proceedings.
S. Ishida, in *HADRON SPECTROSCOPY*, edited by E. Klempt et al., AIP Conference Proceedings 717 (Melville, New York, 2004), 716.
LASS Collaboration, D. Aston et al., SLAC-PUB-5606 (1994).
OBELIX Collaboration, A. Bertin et al., .
M. N. Achasov et al., ; ; .
BABAR Collaboration, B. Aubert et al., .
M. N. Achasov et al., .
R. R. Akhmetshin et al., .
T. Komada, in *HADRON SPECTROSCOPY*, edited by A. Reis et al., AIP Conference Proceedings 814 (Melville, New York, 2006), 458; these proceedings, *Proceedings of the Seminar on Perspectives for Studies of Chiral Particles at BES, IHEP, Beijing, 2006*, KEK Proceedings.
UKQCD Collaboration, C. McNeile and C. Michael, .
L. Ya. Glozman, .\
Thomas D. Cohen and Xiangdong Ji, .
BES Collaboration, M.Ablikim et al., hep-ex/0606047.
Feng-Kun Guo and Peng-Nian Shen, hep-ph/0606273.
BES Collaboration, M.Ablikim et al., .
[^1]: A vertex operator for the quark-photon interaction is assumed to be proportional to $\gamma_{\mu}$.
[^2]: Chiral transformation properties of various interpolating fields of mesons have been examined in Ref. .
[^3]: They include meson states which are listed under the section ‘Further States’ in Ref. .
|
---
abstract: 'Nodes in a multiplex network are connected by multiple types of relations. However, most existing network embedding methods assume that only a single type of relation exists between nodes. Even for those that consider the multiplexity of a network, they overlook node attributes, resort to node labels for training, and fail to model the global properties of a graph. We present a simple yet effective unsupervised network embedding method for attributed multiplex network called , inspired by Deep Graph Infomax (DGI) that maximizes the mutual information between local patches of a graph, and the global representation of the entire graph. We devise a systematic way to jointly integrate the node embeddings from multiple graphs by introducing 1) the consensus regularization framework that minimizes the disagreements among the relation-type specific node embeddings, and 2) the universal discriminator that discriminates true samples regardless of the relation types. We also show that the attention mechanism infers the importance of each relation type, and thus can be useful for filtering unnecessary relation types as a preprocessing step. Extensive experiments on various downstream tasks demonstrate that outperforms the state-of-the-art methods, even though is fully unsupervised.'
author:
- |
Chanyoung Park^1^, Donghyun Kim^2^, Jiawei Han^1^, Hwanjo Yu^3^\
^1^Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA\
^2^Yahoo! Research, CA, USA\
^3^Department of Computer Science and Engineering, Pohang University of Science and Technology, Korea\
[email protected], [email protected], [email protected], [email protected]
bibliography:
- 'Bibliography-File.bib'
title: Unsupervised Attributed Multiplex Network Embedding
---
Introduction
============
Analyzing and mining useful knowledge in graphs have been an actively researched topic for decades both in academia and industry. Among various graph mining techniques, network embedding, which learns low-dimensional vector representations for nodes in a graph, is shown to be especially effective for various network-based tasks [@tang2015line; @wang2017community; @meng2019co].
However, most existing network embedding methods assume that only a single type of relation exists between nodes [@velivckovic2017graph; @velivckovic2018deep; @kipf2016semi], whereas in reality networks are *multiplex* [@de2013mathematical] in nature, i.e., with multiple types of relations. Taking the publication network as an example, two papers can be connected due to various reasons, such as authors (two papers are authored by a common author), citation (one paper cites the other), or keywords (two papers share common keywords). As another example, in a movie database network, two movies can be connected via a common director, or a common actor.
Although different types of relations can independently form different graphs, *these graphs are related*, and thus can mutually help each other for various downstream tasks. As a concrete example of the publication network, although it is hard to infer the topic of a paper only from its citations (citations can be diverse), also knowing other papers written by the same authors will help predict its topic, because authors usually work on a specific research topic. Furthermore, nodes in graphs may contain attribute information, which plays important roles in many applications [@zhang2018anrl]. For example, if we are additionally given the abstract of the papers in the publication network, it will be much easier to infer their topics. As such, the main challenge is to learn a consensus representation of a node that not only considers its *[multiplexity]{}*, but also its *[attributes]{}*.
Several recent studies have been conducted for multiplex network embedding, however, some issues remain that need further consideration. First, previous methods [@qu2017attention; @zhang2018scalable; @shi2018mvn2vec; @liu2017principled] focus on the integration of multiple graphs, but overlook **node attributes**. Second, even for those that consider node attributes [@schlichtkrull2018modeling; @wang2019heterogeneous], they require **node labels** for training. However, as node labeling is often expensive and time-consuming, it would be the best if a method can show competitive performance even without any label. Third, most of these methods fail to model the **global properties** of a graph, because they are based on random walk-based skip-gram model or graph convolutional network (GCN) [@kipf2016semi], both of which are known to be effective for capturing the local graph structure [@yadav2019lovasz]. More precisely, nodes that are “close” (i.e., within the same context window or neighborhoods) in the graph are trained to have similar representations, whereas nodes that are far apart do not have similar representations, even though they are structurally similar [@ribeiro2017struc2vec].
Keeping these limitations in mind, we propose a simple yet effective unsupervised method for embedding attributed multiplex networks. The core building block of our proposed method is Deep Graph Infomax (DGI) [@velivckovic2018deep] that aims to learn a node encoder that maximizes the mutual information between local patches of a graph, and the global representation of the entire graph. DGI is the workhorse method for our task, because it 1) naturally integrates the node attributes by using a GCN, 2) is trained in a fully unsupervised manner, and 3) captures the global properties of the entire graph. However, it is challenging to apply DGI, which is designed for embedding a single network, to a multiplex network in which the interactions among multiple relation types, and the importance of each relation type should be considered. In this paper, we present a systematic way to jointly integrate the embeddings from multiple types of relations between nodes, so as to facilitate them to mutually help each other learn high-quality embeddings useful for various downstream tasks. More precisely, we introduce the *consensus regularization framework* that minimizes the disagreements among the relation-type specific node embeddings, and the *universal discriminator* that discriminates true samples, i.e., ground truth “(graph-level summary, local patch)” pairs, regardless of the relation types. Moreover, we demonstrate that through the *attention mechanism*, we can infer the importance of each relation type in generating the consensus node embeddings, which can be used for filtering unnecessary relation types as a preprocessing step. Our extensive experiments demonstrate that our proposed method, Deep Multplex Graph Infomax (), outperforms the state-of-the-art attributed multiplex network embedding methods in terms of node clustering, similarity search, and especially, node classification even though is fully unsupervised.
Problem Statement
=================
**(Attributed Multiplex Network)** An attributed multiplex network is a network $\mathcal{G}=\{\mathcal{G}^1,\mathcal{G}^2,...,\mathcal{G}^{|\mathcal{R}|}\}=\{\mathcal{V},\mathcal{E},\textbf{X}\}$, where $\mathcal{G}^r=\{\mathcal{V},\mathcal{E}^{(r)},\textbf{X}\}$ is a graph of the relation type $r\in\mathcal{R}$, $\mathcal{V}$ is the set of $n$ nodes, $\mathcal{E}=\bigcup_{r\in \mathcal{R}} \mathcal{E}^{(r)}\subseteq \mathcal{V}\times\mathcal{V}$ is the set of all edges with relation type $r\in\mathcal{R}$, and ${\textbf{X}}\in\mathbb{R}^{n \times f}$ is a matrix that encodes node attributes information for $n$ nodes. Note that $|\mathcal{R}|>1$ for multiplex networks, and $|\mathcal{R}|=1$ for a single network. Given the network $\mathcal{G}$, $\mathcal{A}=\{\textbf{A}^{(1)},...,\textbf{A}^{(|\mathcal{R}|)}\}$ is a set of adjacency matrices, where $\textbf{A}^{(r)}\in\{0,1\}^{|V|\times|V|}$ is an adjacency matrix of the network $\mathcal{G}^r$.
**Task:** **Unsupervised Attributed Multiplex Network Embedding.** Given an attributed multiplex network $\mathcal{G}=\{\mathcal{V},\mathcal{E},\textbf{X}\}$, and the set of adjacency matrices $\mathcal{A}$, the task of unsupervised attributed multiplex network embedding is to learn a $d$-dimensional vector representation $\textbf{z}_i\in\mathcal{\textbf{Z}}\in\mathbb{R}^{n\times d}$ for each node $v_i\in \mathcal{V}$ without using any labels.
Unsupervised Attributed Multiplex Network Embedding
===================================================
We begin by introducing Deep Graph Informax (DGI) [@velivckovic2018deep], then we discuss about its limitations, and present our proposed method.
**Deep Graph Infomax (DGI).** @velivckovic2018deep proposed an unsupervised method for learning node representations, called DGI, that relies on the infomax principle [@linsker1988self]. More precisely, DGI aims to learn a low-dimensional vector representation for each node $v_i$, i.e., $\textbf{h}_i\in\mathbb{R}^d$, such that the average mutual information (MI) between the graph-level (global) summary representation $\textbf{s}\in\mathbb{R}^d$, and the representations of the local patches $\{\textbf{h}_1,\textbf{h}_2,...,\textbf{h}_n\}$ is maximized. To this end, DGI introduces a discriminator $\mathcal{D}$ that discriminates the true samples, i.e., $(\textbf{h}_i, \textbf{s})$, from its negative counterparts, i.e., $(\bm{\tilde{\textbf{h}}}_j,\textbf{s})$: $$\begin{split}
\mathcal{L}=\sum_{v_i\in\mathcal{V}}^{n} \log \mathcal{D}\left({\textbf{h}}_{i}, {\textbf{s}}\right)+\sum_{j=1}^{n}\log \left(1-\mathcal{D}\left({\bm{\tilde{\textbf{h}}}_j}, {\textbf{s}}\right)\right)
\end{split}
\label{eqn:dgi}$$ where $\textbf{h}_i=\sigma\left(\sum_{j\in N(i)} \frac{1}{c_{i j}} \textbf{x}_j\textbf{W}\right)$, $N(i)$ is the set of neighboring nodes of $v_i$ including $v_i$ itself, $\textbf{W}\in\mathbb{R}^{f\times d}$, and $c_{ij}$ is a normalizing constant for edge $(v_i,v_j)$, $\textbf{s}=\sigma\left(\frac{1}{n} \sum_{i=1}^{n} \textbf{h}_{i}\right)$, and $\sigma$ is the sigmoid nonlinearity. Negative patch representation $\bm{\tilde{\textbf{h}}}_j$ is obtained by row-wise shuffling the original attribute matrix $\textbf{X}$. @velivckovic2018deep theoretically proved that the binary cross entropy loss shown in Eqn. \[eqn:dgi\] amounts to maximizing the mutual information (MI) between $\textbf{h}_i$ and $\textbf{s}$, based on the Jensen-Shannon divergence [@velivckovic2018deep]. Refer to Section 3.3 of [@velivckovic2018deep] for the detailed proof. As the local patch representations $\{\textbf{h}_1,\textbf{h}_2,...,\textbf{h}_n\}$ are learned to preserve the MI with the graph-level representation $\textbf{s}$, each $\textbf{h}_i$ is expected to capture the global properties of the entire graph.
**Limitation.** Despite its effectiveness, DGI is designed for a single attributed network, and thus it is not straightforward to apply it to a multiplex network. As a naive extension of DGI to a multiplex attributed network, we can independently apply DGI to each graph formed by each relation type, and then compute the average of the embeddings obtained from each graph to get the final node representations. However, we argue that this fails to model the multiplexity of the network, because the interactions among the node embeddings from different relation types is not captured. Thus, we need a more systematic way to integrate multiple independent models to obtain the final consensus embedding that every model can agree on.
Deep Multiplex Graph Infomax:
------------------------------
We present our unsupervised method for embedding an attributed multiplex network. We first describe how to independently model each graph pertaining to each relation type, then explain how to jointly integrate them to finally obtain the consensus node embedding matrix.
**Relation-type specific Node Embedding.** For each relation type $r\in\mathcal{R}$, we introduce a relation-type specific node encoder $g_{r} : \mathbb{R}^{n \times f} \times \mathbb{R}^{n \times n} \rightarrow \mathbb{R}^{n \times d}$ to generate the relation-type specific node embedding matrix $\textbf{H}^{(r)}$ of nodes in $\mathcal{G}^{(r)}$. The encoder is a single–layered GCN: $$\small
\textbf{H}^{(r)}=g_{r}(\mathbf{X}, \mathbf{A}^{(r)}|\textbf{W}^{(r)})=\sigma\left(\hat{\mathbf{D}}_r^{-\frac{1}{2}} \hat{\mathbf{A}}^{(r)} \hat{\mathbf{D}}_r^{-\frac{1}{2}} \mathbf{X} \textbf{W}^{(r)}\right)
\label{eqn:gcn}$$ where $\hat{\mathbf{A}}^{(r)}=\mathbf{A}^{(r)}+w\mathbf{I}_{n}$, $\hat{D}_{i i}=\sum_{j} \hat{A}_{i j}$, $\textbf{W}^{(r)}\in\mathbb{R}^{f\times d}$ is a trainable weight matrix of the relation-type specific decoder $g_r$, and $\sigma$ is the ReLU nonlinearity. Unlike conventional GCNs [@kipf2016semi], we control the weight of the self-connections by introducing a weight $w\in\mathbb{R}$. Larger $w$ indicates that the node itself plays a more important role in generating its embedding, which in turn diminishes the importance of its neighboring nodes. Then, we compute the graph-level summary representation $\textbf{s}^{(r)}$ that summarizes the global content of the graph $\mathcal{G}^{(r)}$. We employ a readout function $\textsf{Readout}:\mathbb{R}^{n\times d} \rightarrow \mathbb{R}^d$: $$\small
\textbf{s}^{(r)}=\textsf{Readout}(\mathbf{H}^{(r)})=\sigma\left(\frac{1}{n} \sum_{i=1}^{n} \textbf{h}^{(r)}_{i}\right)
\label{eqn:readout}$$ where $\sigma$ is the logistic sigmoid nonlinearity, and $\textbf{h}_i^{(r)}$ denotes the $i$-th row vector of the matrix $\textbf{H}^{(r)}$. Next, given the relation-type specific node embedding matrix $\textbf{H}^{(r)}$, and its graph-level summary representation $\textbf{s}^{(r)}$, we compute the relation-type specific cross entropy: $$\small
\begin{split}
\mathcal{L}^{(r)}=\sum_{v_i\in\mathcal{V}}^{n} \log \mathcal{D}\left({\textbf{h}}^{(r)}_{i}, {\textbf{s}^{(r)}}\right)+\sum_{j=1}^{n}\log \left(1-\mathcal{D}\left({\bm{\tilde{\textbf{h}}}}^{(r)}_{j}, {\textbf{s}^{(r)}}\right)\right)
\end{split}
\raisetag{2.7\baselineskip}
\label{eqn:rel_loss}$$ where $\mathcal{D}:\mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$ is a discriminator that scores patch-summary representation pairs, i.e., $(\textbf{h}^{(r)}_i,\textbf{s}^{(r)})$. In this paper, we apply a simple bilinear scoring function as it empirically performs the best in our experiments: $$\small
\mathcal{D}\left({\textbf{h}}^{(r)}_{i}, {\textbf{s}^{(r)}}\right) = \sigma(\textbf{h}_i^{(r)T}\textbf{M}^{(r)}\textbf{s}^{(r)})
\label{eqn:disc}$$ where $\sigma$ is the logistic sigmoid nonlinearity, and $\textbf{M}^{(r)}\in\mathbb{R}^{d\times d}$ is a trainable scoring matrix. To generate the negative node embedding $\bm{\tilde{\textbf{h}}}_j^{(r)}$, we corrupt the original attribute matrix by shuffling it in the row-wise manner [@velivckovic2018deep], i.e., $\bm\tilde{\textbf{X}}\leftarrow \textbf{X}$, and reuse the encoder in Eqn. \[eqn:gcn\]. i.e. $\bm\tilde{\textbf{H}}^{(r)} = g_{r}(\bm\tilde{\textbf{X}}, \mathbf{A}^{(r)}|\textbf{W}^{(r)})$.
### Joint Modeling and Consensus Regularization.
Heretofore, by independently maximizing the average MI between the local patches $\{\textbf{h}^{(r)}_1,\textbf{h}^{(r)}_2,...,\textbf{h}^{(r)}_n\}$ and the graph-level summary $\textbf{s}^{(r)}$ pertaining to each graph $\mathcal{G}^{(r)} (\forall r\in\mathcal{R})$, we obtained relation-type specific node embedding matrix $\textbf{H}^{(r)}$ that captures the global information in $\mathcal{G}^{(r)}$. However, as each $\textbf{H}^{(r)}$ is trained independently for each $r\in\mathcal{R}$, these embedding matrices only contain relevant information regarding each relation type, and therefore fail to take advantage of the multiplexity of the network. This motivates us to develop a systematic way to jointly integrate the embeddings from different relation types, so as to facilitate them to mutually help each other learn high-quality embeddings.
To this end, we introduce the consensus embedding matrix $\textbf{Z}\in\mathbb{R}^{n\times d}$ on which every relation-type specific node embedding matrix $\textbf{H}^{(r)}$ can agree. More precisely, we introduce the *consensus regularization* framework that consists of 1) a regularizer minimizing the disagreements between the set of original node embeddings, i.e. $\{\textbf{H}^{(r)}\;|\;r\in\mathcal{R} \}$ and the consensus embedding $\textbf{Z}$, and 2) another regularizer maximizing the disagreement between the corrupted node embeddings, i.e., $\{\bm{\tilde{\textbf{H}}}^{(r)}\;|\;r\in\mathcal{R} \}$, and the consensus embedding $\textbf{Z}$, which are formulated as follows:
$$\small
\begin{split}
\ell_{\text{cs}}=
\left[\textbf{Z}-\mathcal{Q}\left(\{\textbf{H}^{(r)}\;|\;r\in\mathcal{R} \}\right)\right]^2-
\left[\textbf{Z}-\mathcal{Q}\left(\{\bm{\tilde{\textbf{H}}}^{(r)}\;|\;r\in\mathcal{R} \}\right)\right]^2
\end{split}
\raisetag{2.5\baselineskip}
\label{eqn:reg}$$
where $\mathcal{Q}$ is an aggregation function that combines a set of node embedding matrices from multiple relation types into a single embedding matrix. i.e., $\textbf{H}\in\mathbb{R}^{n\times d}$. $\mathcal{Q}$ can be any pooling method that can handle permutation invariant input, such as set2set [@vinyals2015order] or Set Transformer [@lee2018set]. However, considering the efficiency of the method, we simply employ average pooling, i.e., computing the average of the set of embedding matrices: $$\small
\textbf{H}=\mathcal{Q}\left(\{\textbf{H}^{(r)}\;|\;r\in\mathcal{R}\}\right)=\frac{1}{|\mathcal{R}|}\sum_{r\in\mathcal{R}}\textbf{H}^{(r)}
\label{eqn:avg}$$
It is important to note that the scoring matrix $\textbf{M}^{(*)}$ in Eqn. \[eqn:disc\] is shared among all the relations $r\in\mathcal{R}$. i.e., $\textbf{M}=\textbf{M}^{(1)}=\textbf{M}^{(2)}=...=\textbf{M}^{(|\mathcal{R}|)}$. The intuition is to learn the *universal discriminator* that is capable of scoring the true pairs higher than the negative pairs regardless the relation types. We argue that the universal discriminator facilitates the joint modeling of different relation types together with the consensus regularization.
Finally, we jointly optimize the sum of all the relation-type specific loss in Eqn. \[eqn:rel\_loss\], and the consensus regularization in Eqn. \[eqn:reg\] to obtain the final objective $\mathcal{J}$ as follows: $$\mathcal{J}=\sum_{r\in\mathcal{R}}\mathcal{L}^{(r)} + \alpha\ell_{\text{cs}} + \beta||\Theta||^2
\label{eqn:final_loss}$$ where $\alpha$ controls the importance of the consensus regularization, $\beta$ is a coefficient for l2 regularization on $\Theta$, which is a set of trainable parameters. i.e., $\Theta=\{\{\textbf{W}^{(r)}\;|\;{r\in\mathcal{R}}\}, \textbf{M}, \textbf{Z}\}$, and $\mathcal{J}$ is optimized by Adam optimizer. Figure \[fig:overall\] illustrates the overview of .
[c||c|>m[0.7cm]{}|>m[0.7cm]{}|>m[0.8cm]{}||>m[2.2cm]{}|c|c|c|c]{} &
-----------
Relations
(A-B)
-----------
& Num. A & Num. B & Num. A-B & Relation type &
-----------
Num.
relations
-----------
&
-----------------
Num.
node attributes
-----------------
&
--------------
Num.
labeled data
--------------
&
---------
Num.
classes
---------
\
& aper-uthor & 3,025 & 5,835 & 9,744 & -- & 29,281 & & & 3\
& aper-ubject & 3,025 & 56 & 3,025 & -- & 2,210,761 & & &\
& ovie-ctor & 3,550 & 4,441 & 10,650 & -- & 66,428 & & & 3\
& ovie-irector & 3,550 & 1,726 & 3,550 & -- & 13,788 & & &\
& aper-uthor & 7,907 & 1,960 & 14,238 & -- & 144,783 & & & 4\
& aper-aper & 7,907 & 7,907 & 10,522 & -- & 90,145 & & &\
& uthor-erm & 1,960 & 1,975 & 57,269 & ---- & 57,137,515 & & &\
& & & & 38,514 & Also-view & 266,237 & & & 4\
& & & &45,446 & Also-bought & 1,104,257 & & &\
& & & & 9,783 & Bought-together & 16,305 & &
\[tab:stats\]
![Overview of (Best viewed in color).[]{data-label="fig:overall"}](overall.pdf){width="\linewidth"}
**Discussion.** Despite its efficiency, the above average pooling scheme in Eqn. \[eqn:avg\] treats all the relations equally, whereas, as will be shown in the experiments, some relation type is more beneficial for a certain downstream task than others. For example, the co-authorship information between two papers plays a more significant role in predicting the topic of a paper compared with their citation information; eventually, these two information mutually help each other to more accurately predict the topic of a paper. Therefore, we can adopt the attention mechanism [@bahdanau2014neural] to distinguish between different relation types as follows: $$\small
\textbf{h}_i=\mathcal{Q}\left(\{\textbf{h}^{(r)}\;|\;r\in\mathcal{R}\}\right)=\sum_{r\in\mathcal{R}}a_i^{(r)}\textbf{h}^{(r)}
\label{eqn:attn}$$ where $a_i^{(r)}$ denotes the importance of relation $r$ in generating the final embedding of node $v_i$ defined as: $$\small
a_{i}^{(r)}=\frac{\exp \left({\textbf{q}}^{(r)} \cdot \mathbf{\textbf{h}}_{i}^{(r)}\right)}{\sum_{r^{\prime}\in\mathcal{R}} \exp \left({\textbf{q}}^{(r^{\prime})} \cdot {\textbf{h}}_{i}^{r^{\prime}}\right)}$$ where $\textbf{q}^{(r)}\in\mathbb{R}^d$ is the feature vector of relation $r$.
### Extension to Semi-Supervised Learning.
It is important to note that is trained in a *fully unsupervised* manner. However, in reality, nodes are sometimes associated with label information, which can guide the training of node embeddings even with a small amount [@kipf2016semi; @qu2017attention]. To this end, we introduce a *semi-supervised module* into our framework that predicts the labels of labeled nodes from the consensus embedding **Z**. More precisely, we minimize the cross-entropy error over the labeled nodes: $$\ell_{\text{sup}}=-\frac{1}{|\mathcal{Y}_{L}|}\sum_{l \in \mathcal{Y}_{L}} \sum_{i=1}^{c} Y_{l i} \ln \hat{Y}_{l i}$$ where $\mathcal{Y}_{L}$ is the set of node indices with labels, $Y\in\mathbb{R}^{n\times c}$ is the ground truth label, ${\hat{Y}}=\textsf{softmax}(f(\textbf{Z}))$ is the output of a softmax layer, and $f:\mathbb{R}^{n\times d}\rightarrow\mathbb{R}^{n\times c}$ is a classifier that predicts the label of a node from its embedding, which is a single fully connected layer in this work. The final objective function with the semi-supervised module is: $$\mathcal{J}_\textsf{semi}=\sum_{r\in\mathcal{R}}\mathcal{L}^{(r)} + \alpha\ell_{\text{cs}}+ \beta||\Theta|| + \gamma\ell_{\text{sup}}
\label{eqn:semi}$$ where $\gamma$ the coefficient of the semi-supervised module.
Experiments
===========
**Dataset.** To make fair comparisons with HAN [@wang2019heterogeneous], which is the most relevant baseline method, we evaluate our proposed method on the datasets used in their original paper [@wang2019heterogeneous], i.e., ACM, DBLP, and IMDB. We used publicly available ACM dataset [@wang2019heterogeneous], and preprocessed DBLP and IMDB datasets. For ACM and DBLP datasets, the task is to classify the papers into three classes (Database, Wireless Communication, Data Mining), and four classes (DM, AI, CV, NLP)[^1], respectively, according to the research topic. For IMDB dataset, the task is to classify the movies into three classes (Action, Comedy, Drama). We note that the above datasets used by previous work are not truly multiplex in nature because the multiplexity between nodes is inferred via intermediate nodes (e.g., ACM: Paper-Paper relationships are inferred via Authors and Subjects that connect two Papers. i.e., “PAP” and “PSP”). Thus, to make our evaluation more practical, we used Amazon dataset [@he2016ups] that genuinely contains a multiplex network of items, i.e., also-viewed, also-bought, and bought-together relations between items. We used datasets from four categories[^2], i.e., Beauty, Automotive, Patio Lawn and Garden, and Baby, and the task is to classify items into the four classes. For ACM and IMDB datasets, we used the same number of labeled data as in [@wang2019heterogeneous] for fair comparisons, and for the remaining datasets, we used 20 labeled data for each class. Table \[tab:stats\] summarizes the data statistics.
**Methods Compared.**
1. Embedding methods for a single network
- : **Deepwalk** [@perozzi2014deepwalk], **node2vec** [@grover2016node2vec]: They learn node embeddings by random walks and skip-gram.
- : **GCN** [@kipf2016semi], **GAT** [@velivckovic2017graph]: They learn node embeddings based on local neighborhood structures. As they perform similarly, we report the best performing method among them; **DGI** [@velivckovic2018deep]: It maximizes the MI between the graph-level summary representation and the local patches; **ANRL** [@zhang2018anrl]: It uses neighbor enhancement autoencoder to model the node attribute information, and skip-gram model to capture the network structure; **CAN** [@meng2019co]: It learns embeddings of both attributes and nodes in the same semantic space; **DGCN** [@zhuang2018dual]: It models the local and global properties of a graph by employing dual GCNs.
2. Multiplex embedding methods
- : **CMNA** [@chu2019cross]: It leverages the cross-network information to refine inter-vector for network alignment and intra-vector for other downstream tasks. We use the intra-vector for our evaluations; **MNE** [@zhang2018scalable]: It jointly models multiple networks by introducing a common embedding, and a additional embedding for each relation type.
- : **mGCN** [@ma2019multi], **HAN** [@wang2019heterogeneous]: They apply GCNs, and GATs on multiplex network considering the inter-, and intra-network interactions. For fair comparisons, we initialized the initial node embeddings of mGCN by using the node attribute matrix, although the node attributes information is ignored in the original mGCN; **[$_{\textsf{attn}}$]{}**: with the attention mechanism (Eqn. \[eqn:attn\]).
For the sake of fair comparisons with , which considers the node attributes, we concatenated the raw attribute matrix $\textbf{X}$ to the learned node embeddings $\textbf{Z}$ of the methods that ignore the node attributes. i.e., Deepwalk, node2vec, CMNA, and MNE. i.e., $\textbf{Z}\leftarrow[\textbf{Z};\textbf{X}]$. Moreover, regarding the embedding methods for a single network, i.e., the methods that belong to the first category in the above list, we obtain the final node embedding matrix $\textbf{Z}$ by computing the average of the node embeddings obtained from each single graph. i.e., $\textbf{Z}=\frac{1}{|\mathcal{R}|}\sum_{r\in\mathcal{R}}\textbf{H}^{(r)}$. We provide a summary of the properties of the compared methods in Table \[tab:property\].
**Evaluation Metrics.** Recall that is an unsupervised method that does not require any labeled data for training. Therefore, we evaluate the performance of in terms of **node clustering** and **similarity search**, both of which are classical performance measures for unsupervised methods. For node clustering, we use the most commonly used metric [@wang2019heterogeneous], i.e., Normalized Mutual Information (NMI). For similarity search, we compute the cosine similarity scores of the node embeddings between all pairs of nodes, and for each node, we rank the nodes according to the similarity score. Then, we calculate the ratio of the nodes that belong to the same class within top-5 ranked nodes (Sim@5). Moreover, we also evaluate on the performance in terms of **node classification**. More precisely, after learning the node embeddings, we train a logistic regression classifier on the learned embeddings in the training set, and then evaluate on the nodes in the test set. We use Macro-F1 (MaF1) and Micro-F1 (MiF1) [@wang2019heterogeneous].
**Experimental Settings.** We randomly split our dataset into train/validation/test, and we have the equal number of labeled data for training and validation datasets. We report the test performance when the performance on validation data gives the best result. For , we set the node embedding dimension $d=64$, self-connection weight $w=3$, tune $\alpha,\beta,\gamma \in \{0.0001,0.001,0.01,0.1\}$. We implement in PyTorch[^3], and for all other methods, we used the source codes published by the authors, and tried to tune them to their best performance. More precisely, apart from the guidelines provided by the original papers, we tuned learning rate, and the coefficients for regularization from {0.0001,0.0005,0.001,0.005} on the validation dataset. After learning the node embeddings, for fair comparisons, we conducted the evaluations within the same platform.
---------------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
MaF1 MiF1 MaF1 MiF1 MaF1 MiF1 MaF1 MiF1
Deepwalk 0.739 0.748 0.532 0.550 0.533 0.537 0.663 0.671
node2vec 0.741 0.749 0.533 0.550 0.543 0.547 0.662 0.669
GCN/GAT 0.869 0.870 0.603 0.611 0.734 0.717 0.646 0.649
DGI 0.881 0.881 0.598 0.606 0.723 0.720 0.403 0.418
ANRL 0.819 0.820 0.573 0.576 0.770 0.699 0.692 0.690
CAN 0.590 0.636 0.577 0.588 0.702 0.694 0.498 0.499
DGCN 0.888 0.888 0.582 0.592 0.707 0.698 0.478 0.509
CMNA 0.782 0.788 0.549 0.566 0.566 0.561 0.657 0.665
MNE 0.792 0.797 0.552 0.574 0.566 0.562 0.556 0.567
mGCN 0.858 0.860 0.623 0.630 0.725 0.713 0.660 0.661
HAN 0.878 0.879 0.599 0.607 0.716 0.708 0.501 0.509
**0.898** **0.898** **0.648** **0.648** 0.771 0.766 0.746 0.748
[[$_{\textsf{attn}}$]{}]{} 0.887 0.887 0.602 0.606 **0.778** **0.770** **0.758** **0.758**
---------------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
\[tab:classification\]
[ {width="\linewidth"} \[fig:distribution\] ]{}
Performance Analysis
--------------------
**Overall evaluation.** Table \[tab:clustering\] and Table \[tab:classification\] show the evaluation results on unsupervised and supervised task, respectively. We have the following observations: 1) Our proposed and [$_{\textsf{attn}}$]{} outperform all the state-of-the-art baselines not only on the unsupervised tasks, but also the supervised task, although the improvement is more significant in the unsupervised task as expected. This verifies the benefit of our framework that models the multiplexity and the global property of a network together with the node attributes within a single framework. 2) Although DGI shows relatively good performance, the performance is unstable (poor performance on Amazon dataset), indicating that multiple relation types should be jointly modeled. 3) Attribute-aware multiplex network embedding methods, such as mGCN and HAN, generally perform better than those that neglect the node attributes. i.e., CMNA and MNE, even though we concatenated node attributes to the node embeddings. This verifies not only the benefit of modeling the node attributes, but also that the attributes should be systematically incorporated into the model. 4) Multiplex network embedding methods generally outperform single network embedding methods, although the gap is not significant. This verifies that the multiplexity of a network should be carefully modeled, otherwise a simple aggregation of multiple relation-type specific embeddings learned from independent single network embedding methods may perform better.
[c|c|ccc|cc]{} &
-----
GCN
-----
: Performance of similarity search (Sim@5) of embedding methods for a single network. ( denotes the average of all the relation-type specific embeddings.)
& DGI & ANRL & \[2\][\*]{} & \[2\][\*]{}
[@c@]{}\
$_\textsf{attn}$
\
\[2\][\*]{}
------
Rel.
Type
------
: Performance of similarity search (Sim@5) of embedding methods for a single network. ( denotes the average of all the relation-type specific embeddings.)
& PAP & 0.822 & 0.875 & 0.795 & &\
& PSP & 0.721 & 0.675 & 0.694 & &\
& 0.867 & 0.889 & 0.814 & 0.898 & **0.901**\
&
-----
GCN
-----
: Performance of similarity search (Sim@5) of embedding methods for a single network. ( denotes the average of all the relation-type specific embeddings.)
& DGI & ANRL & \[2\][\*]{} & \[2\][\*]{}
[@c@]{}\
$_\textsf{attn}$
\
\[2\][\*]{}
------
Rel.
Type
------
: Performance of similarity search (Sim@5) of embedding methods for a single network. ( denotes the average of all the relation-type specific embeddings.)
& MAM & 0.485 & 0.484 & 0.495 & &\
& MDM & 0.548 & 0.562 & 0.520 & &\
& 0.566 & 0.578 & 0.527 & **0.605** & 0.586\
&
-----
GCN
-----
: Performance of similarity search (Sim@5) of embedding methods for a single network. ( denotes the average of all the relation-type specific embeddings.)
& DGI & ANRL & \[2\][\*]{} & \[2\][\*]{}
[@c@]{}\
$_\textsf{attn}$
\
\[3\][\*]{}
------
Rel.
Type
------
: Performance of similarity search (Sim@5) of embedding methods for a single network. ( denotes the average of all the relation-type specific embeddings.)
& PAP & 0.730 & 0.779 & 0.692 & &\
& PPP & 0.456 & 0.477 & 0.680 & &\
& PATAP & 0.431 & 0.409 & OOM & &\
& 0.724 & 0.786 & 0.720 & 0.766 & **0.799**\
&
-----
GCN
-----
: Performance of similarity search (Sim@5) of embedding methods for a single network. ( denotes the average of all the relation-type specific embeddings.)
& DGI & ANRL & \[2\][\*]{} & \[2\][\*]{}
[@c@]{}\
$_\textsf{attn}$
\
\[3\][\*]{}
------
Rel.
Type
------
: Performance of similarity search (Sim@5) of embedding methods for a single network. ( denotes the average of all the relation-type specific embeddings.)
& Also-V & 0.355 & 0.367 & 0.563 & &\
& Also-B & 0.357 & 0.381 & 0.516 & &\
& Bou.-T & 0.662 & 0.639 & 0.770 & &\
& 0.624 & 0.558 & 0.764 & 0.816 & **0.825**\
\[tab:attn\]
**Effect of the attention mechanism.** In Table \[tab:attn\], we show the performance of and [$_{\textsf{attn}}$]{}, together with the performance of single network embedding methods (GCN/GAT, DGI, and ANRL). We observe that [$_{\textsf{attn}}$]{} outperforms in most of the datasets but IMDB dataset. To analyze the reason for this, we first plot the distribution of the attention weights on DBLP dataset over the training epochs in Figure \[fig:distribution\]. The above graph in Figure \[fig:distribution\] demonstrates that the attention weights eventually end up in both extremes. i.e., close to 0 or close to 1, and the below graphs show that most of the attention weight is dedicated to a single relation type, i.e., “PAP”, which actually turns out to be the most important relation among the three (See Table \[tab:attn\]); This phenomenon is common in every dataset. Next, we look at the performance of the single network embedding methods, especially DGI, on each relation type in Table \[tab:attn\]. We observe that the performance differences among relation types in ACM, DBLP, and Amazon datasets are more biased to a single relation type, whereas in IMDB dataset, “MAM” and “MDM” relations relatively show similar performance. To summarize our findings, since the attention mechanism tends to favor the single most important relation type (“PAP” in ACM, “MDM” in IMDB, “PAP” in DBLP, and “Bought-together” in Amazon), [$_{\textsf{attn}}$]{} outperforms on datasets where one relation type significantly outperforms the other, i.e., ACM, DBLP, and Amazon, by removing the noise from other relations. On the other hand, for datasets where all the relations show relatively even performance, i.e., IMDB, extremely favoring a single well performing relation type (“MDM”) is rather detrimental to the overall performance because the relation “MAM” should also be considered to some extent.
We also note that since the attention mechanism of [$_{\textsf{attn}}$]{} can infer the importance of each relation type, we can filter out unnecessary relation types as a preprocessing step. To verify this, we evaluated on all possible combinations of relation types in DBLP dataset (Table \[tab:attnfilter\]). We observe that by removing the relation “PATAP”, which turned out to be the most useless relation type in Table \[tab:attn\], [$_{\textsf{attn}}$]{} obtains even better results than using all the relation types, whereas for GCN and DGI, still considering all the relation types shows the best performance. This indicates that the attention mechanism can be useful to filter out unnecessary relation types, which will especially come in handy when the number of relation types is large.
-------------------- ----------- ----------- ----------- -----------
\[2\][\*]{}[NMI]{} PAP+PPP 0.464 0.543 **0.565**
PAP+PATAP 0.458 0.535 0.017
PPP+PATAP 0.332 0.237 0.201
*All* **0.465** **0.551** 0.554
-------------------- ----------- ----------- ----------- -----------
: NMI on various combinations of relation types.
\[tab:attnfilter\]
**Ablation study.** To measure the impact of each component of [$_{\textsf{attn}}$]{}, we conduct ablation studies on the largest dataset, i.e., DBLP, in Table \[tab:ablation\]. We have the following observations: 1) As expected, the semi-supervised module specifically helps improve the node classification performance, which is a supervised task, whereas the performance on the unsupervised task remains on par. 2) Various readout functions including ones that contain trainable weights (Linear projection and SAGPool [@lee2019self]) do not have much impact on the performance, which promotes our use of average pooling. 3) The second term in Eqn. \[eqn:reg\] indeed plays a significant role in the consensus regularization framework. 4) The sharing of the scoring matrix $\textbf{M}$ facilitates to model the interaction among multiple relation types. 5) Node attributes are crucial for representation learning of nodes. 6) Shuffling adjacency matrix instead of attribute matrix deteriorates the model performance.
MaF1 NMI Sim@5
-- ------------------- ------- ------- -------
0.778 0.554 0.798
0.791 0.555 0.798
Random sample 0.774 0.555 0.797
Maxpool 0.778 0.552 0.802
Linear projection 0.783 0.565 0.803
SAGPool 0.797 0.563 0.797
0.749 0.448 0.787
0.645 0.076 0.677
0.377 0.053 0.763
0.364 0.156 0.504
: Result for ablation studies of [$_{\textsf{attn}}$]{}.
\[tab:ablation\]
Related Work
============
**Network embedding.** Network embedding methods aim at learning low-dimensional vector representation for nodes in a graph while preserving the network structure [@perozzi2014deepwalk; @grover2016node2vec; @tang2015line], and various other properties such as node attributes [@zhang2018anrl; @meng2019co], structural role [@ribeiro2017struc2vec], and node label information [@huang2017label]. **Multiplex Network embedding.** A multiplex network, which is also known as a multi-view network [@tang2015line; @shi2018mvn2vec] or a multi-dimensional network [@ma2018multi; @ma2019multi] in the literature, consists of multiple relation types among a set of single-typed nodes. It can be thought of as a special type of heterogeneous network [@dong2017metapath2vec; @fu2017hin2vec] with a single type of node and multiple types of edges. Therefore, a multiplex network calls for a special attention because there is no need to consider the semantics between different types of nodes, which is often addressed by the concept of meta-path [@sun2011pathsim]. Distinguished from heterogeneous network, a key challenge in the multiplex network embedding is to learn a consensus embedding for each node by taking into account the interrelationship among the multiple graphs. In this regard, existing methods mainly focused on how to integrate the information from multiple graphs. HAN [@wang2019heterogeneous] employed graph attention network [@velivckovic2017graph] on each graph, and then applied the attention mechanism to merge the node representations learned from each graph by considering the importance of each graph. However, the existing methods either require labels for training [@wang2019heterogeneous; @qu2017attention; @schlichtkrull2018modeling], or overlook the node attributes [@liu2017principled; @xu2017multi; @li2018multi; @shi2018mvn2vec; @zhang2018scalable; @ni2018co; @chu2019cross]. Most recently, @ma2019multi proposed a graph convolutional network (GCN) based method called mGCN, which is not only unsupervised, but also naturally incorporates the node attributes by using GCNs. However, since it is based on GCNs that capture the local graph structure [@yadav2019lovasz], it fails to fully model the global properties of a graph [@zhuang2018dual; @wang2016structural; @velivckovic2018deep]. **Attributed Network Embedding.** Nodes in a network are often affiliated with various contents, such as abstract text in the publication network, user profiles in social networks, and item description text in movie database or item networks. Such networks are called attributed networks, and have been extensively studied [@li2017attributed; @hamilton2017inductive; @yang2015network; @zhang2018anrl; @gao2018deep; @zhou2018prre; @velivckovic2018deep; @meng2019co]. Their goal is to preserve not only the network structure, but also the node attribute proximity in learning representations. Recently, GCNs [@kipf2016semi; @velivckovic2017graph; @velivckovic2018deep] have been widely praised for its seamless integration of the network structure, and node attributes into a single framework. **Mutual Information.** it has been recently made possible to compute the MI between high dimensional input/output pairs of deep neural networks [@belghazi2018mine]. Several recent work adopted the infomax principle [@linsker1988self] to learn the unsupervised representations in different domains, such as images [@hjelm2018learning], speech [@ravanelli2018learning] and graphs [@velivckovic2018deep]. More precisely, @velivckovic2018deep proposed Deep Graph Infomax (DGI) for learning representations of graph structured inputs by maximizing the MI between a high-level global representation, and the local patches of a graph.
Conclusion
==========
We presented a simple yet effective unsupervised method for embedding attributed multiplex network. can jointly integrate the embeddings from multiple types of relations between nodes through the consensus regularization framework, and the universal discriminator. Moreover, the attention mechanism of [$_{\textsf{attn}}$]{} can infer the importance of each relation type, which facilitates the preprocessing of the multiplex network. Experimental results on not only unsupervised tasks, but also a supervised task verify the superiority of our proposed framework.
[^1]: **DM**: KDD,WSDM,ICDM, **AI**: ICML,AAAI,IJCAI, **CV**: CVPR, **NLP**: ACL,NAACL,EMNLP
[^2]: We chose these categories because the three types of item-item relations from these categories are similar in number
[^3]: https://github.com/pcy1302/DMGI
|
---
abstract: 'We report microcanonical Monte Carlo simulations of melting and superheating of a generic, Lennard-Jones system starting from the crystalline phase. The isochoric curve, the melting temperature $T_m$ and the critical superheating temperature $T_{LS}$ obtained are in close agreement (well within the microcanonical temperature fluctuations) with standard molecular dynamics one-phase and two-phase methods. These results validate the use of microcanonical Monte Carlo to compute melting points, a method which has the advantage of only requiring the configurational degrees of freedom. Our findings show that the strict preservation of the Hamiltonian dynamics does not constitute a necessary condition to produce a realistic estimate of $T_{LS}$ and the melting point, which brings new insight on the nature of the melting transition. These results widen the use and applicability of the recently developed Z method for the determination of the melting points of materials.'
author:
- Sergio Davis
- Gonzalo Gutiérrez
bibliography:
- 'allrefs.bib'
title: Microcanonical Monte Carlo approach for computing melting curves by atomistic simulations
---
Introduction
============
Melting curves of materials at extreme conditions are fundamental pieces of knowledge in the fields of materials science [@Alfe2004], geology [@Belonoshko2000], planetary sciences [@Oganov2005; @Cavazzoni1999], mechanical engineering [@Padture2002], condensed matter physics [@Datchi2000], among others, not to mention the renewed interest in the melting mechanisms from the point of view of fundamental science [@Tallon89; @Jin2001; @Forsblom2005]. In both areas computer simulations play an increasingly important role, and development of new methods for computing melting points, together with further improvement of existing methods, is a crucial piece for future progress in the field. The current techniques used for the determination of melting curves via atomistic computer simulation (either from first-principles calculations or using semi-empirical interatomic potentials) can be divided into two categories: *coexistence* simulations and *one-phase* simulations.
The usual approach to simulating coexistence is the two-phase [@Belonoshko1994; @Morris1994] method. In this method a mixed sample, composed of different phases (in the case of melting, solid and liquid), is simulated, and the thermodynamic conditions for coexistence of the two phases are explored. For instance, in the microcanonical ensemble, the two-phase method proceeds by choosing a total energy for which coexistence is observed and computing the average temperature, which is then associated with the melting temperature $T_m$. In the variants of the two-phase method where temperature is controlled (i.e., where simulations are carried out in the canonical or isothermal-isobaric ensemble), an initially guessed temperature interval containing $T_m$ is narrowed down systematically in order to constrain $T_m$ between an upper bound $T_0$ and a lower bound $T_1$, $T_0$ being such that it leads to an homogeneous solid phase and $T_1$ to an homogeneous liquid phase.
Regarding the one-phase methods, thermodynamic integration can be used [@Sugino1995; @DeWijs1998; @DeKoning1999; @Donadio2010] at constant volume to compute the Helmholtz free energy differences $\Delta F$ between the solid and liquid phase as a function of temperature, and therefore to obtain the melting point the temperature $T_m$ for which $\Delta F(T_m)=0$. The same procedure can be applied in the isothermal-isobaric ensemble to find the melting point at constant pressure $P$, by means of equating the Gibbs free energies of the different phases ($\Delta G(P, T_m)=0$) or even in the microcanonical ensemble by equating their entropies, $\Delta S(E_m)=0$ where $E_m$ is the internal energy of melting.
Most of these methods are implemented in ensembles different from the microcanonical, due to the simplicity of fixing the temperature or pressure as control parameters to exactly their desired values. There are cases, however, where averages under such ensembles differ considerably from the (in principle) exact microcanonical averages. Ensemble equivalence is, in most cases, guaranteed in the thermodynamic limit (although there are examples of systems where does not hold [@Gulminelli2002; @Campa2009]), and therefore, for small enough systems the appropriate course is to compute microcanonical averages. It is mostly because of these limitations that the microcanonical approach [@Pearson1985; @Lustig1998; @Gross2000; @Gross2005] has regained interest when computing thermophysical properties and in the study of phase transitions for finite-size systems, such as metallic clusters [@Westergren2003] and proteins [@Junghans2006; @HernandezRojas2008; @Bereau2011].
It is well known that early one-phase simulations of melting (such as the somewhat naïve idea of just heating the solid using velocity rescaling until melting is observed) attempted before the use of two-phase simulations suffer the phenomenon of *superheating*, that is, the melting temperature is overestimated. A relatively recent approach to determine the melting point using atomistic computer simulations is the Z method [@Belonoshko2006], which is a microcanonical one-phase method taking into account (and in fact, based on) the superheating effect. *Empirically* it has been found that, when starting from the ideal crystalline structure and increasing the total energy at fixed volume $V$, there is a well defined maximum (for the solid phase), $E_S(T_{LS}; V)$ where $T_{LS}$ corresponds to the limit of superheating. Increasing the energy beyond $E_S$ by a small amount $\delta E$, the solid spontaneously melts at $E_S+\delta E \approx E_S$, but due to the increase in potential energy, namely the latent heat of fusion, temperature decreases. The interesting fact is that the final temperature after melting at $E_S$ seems to coincide with the melting point $T_m$ obtained from other methods. Thus the following equivalence is established in practice (so far without clear theoretical foundations, but nevertheless supported by ample evidence from numerical simulations),
$$E_S(T_{LS}; V) = E_L(T_m; V) \equiv E_{LS}.$$
The procedure for the Z method computation of the melting point is then as follows: at a fixed volume, the ($E$, $\big<T\big>$) points from different simulations draw a “Z” shape (hence the name of the method). In this Z-shaped curve the sharp inflection at the higher temperature corresponds to $T_{LS}$ and the one at the lower temperature to $T_m$. Thus, knowledge of the lower inflection point for different densities allows the determination of the melting curve for a particular range of pressures.
The Z method achieves the same precision in the determination of $T_m$ as the two-phase method, but using only half the atoms (only a single phase is simulated at any
It has been proposed that this success of the Z method relies on sampling from “genuine” Hamiltonian dynamics, without resorting to fictitious forces such as the ones arising from thermostat algorithms. In fact, the Z method has indeed been tried under the Nosé-Hoover thermostat, leading to a lower value of $T_{LS}$ relative to the microcanonical MD implementation. Moreover, $T_{LS}$ seems to be connected to anomalous diffusion time scales, as recently suggested [@Davis2011], which leads to the following question: *to which extent is $T_{LS}$ a dynamical phenomenon, and therefore dependent on the Hamiltonian dynamics?* Could the same results obtained in MD be also obtained following a stochastic dynamics, such as the one generated by microcanonical Monte Carlo (MC) methods? From the point of view of a basic understanding of the nature of $T_{LS}$ it should be useful to clarify its dependence on strictly following the deterministic Hamiltonian trajectories.
In a practical sense, if following those trajectories is not required for a reliable determination of the isochoric curve, it would widen the spectrum of possible methods for computing melting points. If we could disregard the momentum degrees of freedom it would be possible to afford larger system sizes, which is critical in first-principles atomistic simulations.
In this paper we attempt to answer these questions. We present results indicating than a fully stochastic implementation of the Z method is possible, being in close agreement with the standard molecular dynamics implementation.
The paper is organized as follows. First, the MC formulation for the microcanonical ensemble used in this work is presented, followed by the simulation details. Next, we describe the results obtained from comparison of MC and MD simulations. Finally we summarize our findings.
Microcanonical Monte Carlo
==========================
We will consider a classical system of $6N$ degrees of freedom (3N momenta, denoted collectively by $\mathbf{p}$, 3N coordinates denoted by $\mathbf{r}$), with Hamiltonian
$$\mathcal{H} = \frac{\mathbf{p}^2}{2m} + \Phi(\mathbf{r}).$$
The probability of the system having phase space coordinates $(\mathbf{r},
\mathbf{p})$ at total energy $E$ is given by
$$P(\mathbf{r}, \mathbf{p}; E) = \frac{1}{\Omega(E)}
\delta(E-\mathcal{H}(\mathbf{r},\mathbf{p})),
\label{ProbDelta}$$
where
$$\Omega(E) = \int d\mathbf{r} d\mathbf{p} \delta(E-\mathcal{H}(\mathbf{r},\mathbf{p}))$$
is the density of states having energy $E$. Given that the dependence of the Hamiltonian on $\mathbf{p}$ is fully known, those degrees of freedom can be integrated out explicitly [@Severin1978; @Pearson1985]. To do this, we separate $\mathcal{H}$ inside the delta function and use
$$\int d\mathbf{p} \delta(E-\mathbf{p}^2/2m-\Phi(\mathbf{r}))
\rightarrow \int_{\Sigma_p} \frac{d\Sigma_p}{|\nabla (\mathbf{p}^2/2m)|}$$
where the last integral is over the $(3N-1)$-dimensional surface $\Sigma_p$ defined by $$|\mathbf{p}|=\sqrt{2m(E-\Phi(\mathbf{r}))}.$$ After this we can rewrite the probability in Eq. \[ProbDelta\] as
$$P(\mathbf{r}; E) = \frac{1}{\Omega(E)}
\Theta(E-\Phi(\mathbf{r}))\sqrt{E-\Phi(\mathbf{r})}^{3N-2},
\label{ProbRay}$$
where now the density of states $\Omega(E)$ can be written as $$\Omega(E) = \int d\mathbf{r} \Theta(E-\Phi(\mathbf{r}))\sqrt{E-\Phi(\mathbf{r})}^{3N-2},$$ and $\Theta$ is Heaviside’s step function.
Equation \[ProbRay\] leads to the following Metropolis acceptance probability [@Ray1991],
$$P(\mathbf{r}_1 \rightarrow \mathbf{r}_2) = \min\left(1,
\sqrt{\frac{E-\Phi(\mathbf{r_2})}{E-\Phi(\mathbf{r_1})}}^{3N-2}\right).$$
This rule makes it possible to simulate a system in the microcanonical ensemble without incorporating the momentum degrees of freedom explicitly. It also avoids the use of a “demon” to impose conservation of energy (as it is done in Creutz’s version of microcanonical MC [@Creutz1983]).
Results
=======
We performed microcanonical MC simulations on highly compressed fcc crystals whose atoms interact via the Lennard-Jones pair potential truncated at a cutoff radius $r_c$,
$$\phi(r; \sigma, \epsilon, r_c) =
4\epsilon\Theta(r_c-r)\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^6 \right],$$
where $r$ is the distance between atoms $i$ and $j$ and the values considered for the parameters are $\sigma$=3.41 Å , $\epsilon/k_B$= 119.8 K and $r_c$=2.5 $\sigma$. The crystals simulated ranged from 3$\times$3$\times$3 to 6$\times$6$\times$6 unit cells (108 to 864 atoms) with a lattice constant $a$=4.2 Å. This value of $a$ corresponds to a point on the melting curve with 5133 K $< T_m <$ 5251 K and $P(T_m)=$ 70 GPa, as reported by Belonoshko from two-phase simulations [@Belonoshko2006]. In both MD and MC simulations we imposed periodic boundary conditions. We performed about 20 different simulations under each method and system size, with temperatures ranging from 5000K to 7000K. Averages were taken over the last 50 thousand steps in each simulation, after 50 thousand equilibration steps. For MD simulations, the time step used was $\Delta t$=1 fs.
As all MC methods based on the Metropolis rule, a reasonably low rejection rate for moves must be imposed, and the usual way is to employ a small enough atomic displacement when proposing a move. In our MC simulations, we always kept the rejection rate below 60%. Interestingly, we noted that failure to control rejection has a similar effect to failure of energy conservation in MD, namely, averages like temperature start to drift linearly with MC “time”.
Instantaneous temperatures $T_i$ during a MC simulation can be obtained from the kinetic energy (as usual in molecular dynamics simulations),
$$\frac{1}{k_BT_i}=\frac{3N-2}{2(E-\Phi)},$$
but also from derivatives of the potential energy, using the so-called configurational temperature [@Rugh1997; @Butler1998; @Rickayzen2001]) which is given by
$$\frac{1}{k_BT_i}=\frac{\nabla^2\Phi}{\vert\vec{\nabla}\Phi\vert^2} +
\mathcal{O}(\frac{1}{N}),$$
and from these, the equilibrium thermodynamical temperature is obtained as microcanonical averages, $T(E)=\big<T_i\big>_E$. Figure \[ConfigTemp\] shows a comparison of both configurational and kinetic definitions of instantaneous temperature for a typical MC run. This provides an additional consistency check for our MC simulations, in order to make sure the microcanonical ensemble is adequately sampled.
Figure \[ZRay\] shows a comparison between the isochoric curves obtained by standard, MD version of the Z method, and the MC version, for a system size $N$=864 atoms. The agreement between the two is perfect in the thermodynamic stability region (solid and liquid straight lines), and both methods yield the same $T_m$ and $T_{LS}$ within the statistical margin of error, as shown in table \[tbl\_temps\]. The same level of agreement is seen for all the smaller system sizes studied. The slight overestimation (about 4.5%) of $T_m$ as compared with Belonoshko’s two-phase simulations with $N$=32000 is a well known size effect, the Z method overestimates $T_m$ and $T_{LS}$ for small systems. Here we are only interested in comparing the two implementations of the method for equal conditions.
Method $T_{LS}$ (K) $T_m$ (K)
-------------------- ---------------- ----------------
Molecular Dynamics 6265 $\pm$ 148 5427 $\pm$ 103
Monte Carlo 6225 $\pm$ 138 5428 $\pm$ 123
: Values of the critical superheating temperature $T_{LS}$ and the melting temperature $T_m$ obtained for $N$=864 atoms by the MD Z method and the MC Z method.[]{data-label="tbl_temps"}
Figure \[ZFluct\] shows the evolution of the instantaneous temperature as a function of MC steps, for a total energy above $E_{LS}$. The system starts in the solid phase, melting spontaneously after the first 350 steps. Temperature fluctuations are significant, due to the limited size of the system. In this case we did not see the alternating behavior between solid and liquid phases expected in small systems (as reported by Alfè [@Alfe2011]), most probably because of the larger system size simulated (864 atoms instead of 96). In fact, for 72 atoms the alternation occurs, as shown in Fig. \[fig\_altern\]. The precision needed to find the energy $E_m$ at which dynamical coexistence is observed depends on the system size, this is due to the fact that there is a finite, non-zero probability that a small system could oscillate between phases even when their respective entropies are not exactly equal (i.e. when we are close but not exactly at $E_m$). In Ref. [@Alfe2011], the alternation effect is treated considering the fraction of time $\alpha$ spent in its solid or liquid phase, and from equilibrium microcanonical considerations, a relation connecting the fractions $\alpha$ to the entropy of melting $\Delta S$ is found. Interestingly, the analysis can be done without any reference to equilibrium, in the framework of Evans’ fluctuation theorem [@Evans2002],
$$P(S\rightarrow L)/P(L\rightarrow S) = e^{2N\Delta s(S\rightarrow L)/k_B}.
\label{eq_evans}$$
Here $P$ is the transition probability from one state to another, $L$ and $S$ represent liquid and solid states and $\Delta s$ is the entropy of melting per atom. From Eq. \[eq\_evans\] it can be seen that, for small systems ($N \rightarrow 0$), $\Delta s$ can be slightly larger and the right-hand side will still be close enough to 1 to allow transitions from solid to liquid and the reverse. In practice, finding this alternation in MC simulations could be more difficult also due to our use of simple local updates (one atom is displaced at a time on each trial move) which near the transition point could lead to an analog of the *critical slowing down* effect seen in lattice MC simulations [@Swendsen1987; @Wolff1989; @Binder2010], thus making alternation events extremely difficult to generate.
It is important to notice that reproducing $T_{LS}$ via a stochastic procedure does not contradict the notion of $T_{LS}$ being related to time scales [@Davis2011] (which are absent in an absolute sense in Monte Carlo simulations). Far from it, a correct prediction of $T_{LS}$ under stochastic dynamics seems to support the notion of it being related to random walk statistics with jump probabilities only dependent on (microcanonical) thermodynamic properties.
In fact, as an illustration consider simulations of the mean square displacement $\big<r^2(t)\big>$ (a) in liquids via molecular dynamics, and (b) via isotropic random walk simulations. In both cases we have $$\big<r^2(t)\big> \propto 6Dt$$ as $t \rightarrow \infty$, and this is not surprising even though $t$ is not a “real” time but a number of MC steps. In both cases a random walk is used to sample a thermodynamical quantity, namely $D=D(T)$, which happens to have dynamical consequences such as the diffusion rate. In the same way, we conjecture that $T_{LS}$ is a function of dynamical properties which in turn, depend only on features of the material’s potential energy landscape.
![Comparison between configurational and kinetic instantaneous temperatures during a MC simulation for $N$=864 atoms.[]{data-label="ConfigTemp"}](fig1.pdf)
![Comparison between the isochoric curves obtained by standard, microcanonical molecular dynamics and microcanonical MC methods, for a system of $N$=864 atoms.[]{data-label="ZRay"}](fig2.pdf)
![Evolution of the instantaneous temperature during a MC simulation where melting is observed, for $N$=108 atoms. Total simulation comprises 1 million MC steps, and the inset shows the first 2000 MC steps.[]{data-label="ZFluct"}](fig3.pdf)
![Alternation between solid and liquid phases as seen from the instantaneous temperature, for a MC simulation with $N$=72 atoms.[]{data-label="fig_altern"}](fig4.pdf)
Summary and conclusions
=======================
Belonoshko *et al* [@Belonoshko2007] attributed the success of the Z method in reaching the highest $T_{LS}$ among several molecular dynamics methods to the preservation of the natural dynamics of the system. However, we have shown this is not the case: using a MC algorithm which is completely oblivious to the equations of motion we have reproduced the same $T_{LS}$ and the same $T_m$ as in standard molecular dynamics. This suggests that the previously thought advantage of the Z method comes from a different direction, namely, that the preservation of the microcanonical condition is the important fact, and the specific trajectory followed by each atom is not important. Therefore, any method capable of computing microcanonical averages should be just as reliable in Z method computations.
Practical consequences of this finding are clear in terms of the efficiency for large or complex systems. The success of MC methods in the determination of the Z curve hints to the possibility of a fully-parallelizable version of the method.
Acknowledgements
================
SD acknowledges financial support from FONDECYT grant 1140514. GG and SD thank partial support from CONICYT-PIA grant ACT-1115, Chile.
|
---
abstract: 'We present the first intermediate-resolution ($\lambda$ / 3000) spectrum of the bright quasi-stellar object 3C273 at wavelengths between 900 and 1200 Å. Observations were performed with the Berkeley spectrograph aboard the -II mission [@Hetal98]. We detect Lyman $\beta$ counterparts to intergalactic Lyman $\alpha$ features identified by at $cz$ = 19900, 1600, and 1000 ; counterparts to other putative Lyman $\alpha$ clouds along the sight line are below our detection limit. The strengths of the two very low redshift Lyman $\beta$ features, which are believed to arise in Virgo intracluster gas, exceed preflight expectations [@WRWMH95], suggesting that the previous determination of the cloud parameters may underestimate the true column densities. A curve-of-growth analysis sets a minimum column density of $4 \times 10^{14}$ cm$^{-2}$ for the 1600 cloud. We find marginally significant evidence for Galactic along the sight line, with a total column density of about $10^{15}$ cm$^{-2}$. We detect the stronger interstellar doublet member unambiguously; the weaker member is blended with other features. If the Doppler $b$ value for is comparable to that determined for [@SST97] then the column density is $7 \pm 2 \times 10^{14}$ cm$^{-2}$, significantly above the only previous estimate [@D93]. The / ratio is about 10, consistent with the low end of the range observed in the disk (compilation of ). Additional interstellar species detected for the first time toward 3C273 (at modest statistical significance) include , , , and .'
author:
- 'Mark Hurwitz, Immo Appenzeller$^2$, Juergen Barnstedt$^3$, Stuart Bowyer, W. Van Dyke Dixon, Michael Grewing$^3$, Norbert Kappelmann$^3$, Gerhard Kraemer$^3$, Joachim Krautter$^2$, and Holger Mandel$^2$'
title: |
ORFEUS-II FAR-ULTRAVIOLET OBSERVATIONS OF 3C273:\
1. INTERSTELLAR AND INTERGALACTIC ABSORPTION LINES
---
internalcite ==
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OBSERVATIONS
============
We present the first intermediate-resolution far-ultraviolet spectrum of the bright quasi-stellar object 3C273 in the 900 – 1210 Å band, emphasizing absorption features arising along the line of sight. discuss the intrinsic spectrum of 3C273. These data were collected with the Berkeley spectrograph [@Hetal98] during the ORFEUS-SPAS II mission. The project and the [*ASTRO-SPAS*]{} platform are described in .
3C273 was observed five times, for a total of 10,797 s. All observations took place in late 1996, between 330/21:41 and 338/16:33 (Day of Year/HH:MM, GMT). The target coordinates were $\alpha$ = 12 29 06.7, $\delta$ = +02 03 09 (J2000). Absolute positioning of the 26$\arcsec$ diameter ORFEUS entrance aperture is accurate to $\pm$ 5$\arcsec$. Extraction of the spectrum and subtraction of airglow follow the discussion in . Statistical and systematic uncertainties associated with background and airglow subtraction are properly tracked. As a final step, we bin the data on 0.1 Å centers.
In Figure 1 we show the complete spectrum (solid line), the 1 $\sigma$ uncertainty associated with shot noise and detector flat-field effects (dotted line), and a continuum established by an automated fitting routine (dashed line). The automated fitting routine identifies absorption features deeper than a statistically determined threshold, replaces those data with a linear interpolation from nearby wavelengths, smoothes heavily, then iterates twice more at increasing sensitivity to absorption lines. The resulting continuum has the advantage of being determined objectively, and evidently tracks most of the true spectrum well at wavelengths greater than 1000 Å. Like many such routines, however, this one yields unreliable results when strong features are closely spaced on a curved continuum (as in the region of absorption lines 3 and 4) and will tend to underestimate the true value when the spectrum contains a cluster of weak absorption features below the statistical threshold (note the weak dip in the fitted continuum around 1050 Å).
In Table 1 we list the interstellar absorption features and blends that are sufficiently strong and isolated for direct equivalent width measurement using the continuum shown in Figure 1 (unless otherwise noted). Statistical noise in the spectrum (1$\sigma$) corresponds to an unresolved absorption feature with equivalent width of about 0.040 Å; continuum placement uncertainty corresponds to about half that value. The value in the fourth column includes the statistical uncertainty only. For species whose column densities have not previously been reported, we list the minimum column (if optically thin) based on the best fit equivalent width and the oscillator strenths of .
[llllll]{}
1 & 1206.44 & 0.573 & 0.056 & Si III 1206.50 & 13.4 2 & 1199.75 & 0.896 & 0.114 & N I blend & 3 & 1193.56 & 0.510 & 0.050 & Si II 1193.29 & 4 & 1190.58 & 0.532 & 0.051 & Si II 1190.42 & 5 & 1152.91 & 0.105 & 0.038 & P II 1152.82 & 13.6 6 & 1144.92 & 0.340 & 0.054 & Fe II 1144.94 & 7 & 1143.44 & 0.137 & 0.040 & Fe II 1143.22 & 8 & 1134.63 & 1.135 & 0.084 & N I blend & 15.1 9 & 1122.54 & 0.157 & 0.038 & Fe III 1122.53 & 14.3 10 & 1121.99 & 0.149 & 0.038 & Fe II 1121.97 & 11 & 1096.94 & 0.202 & 0.047 & Fe II 1096.88 & 12 & 1084.19 & 0.515 & 0.051 & N II 1083.99, N II\* blend & 13 & 1066.77 & 0.135 & 0.047 & Ar I 1066.66 & 14 & 1063.20 & 0.327 & 0.049 & Fe II 1063.18 + & 15 & 1048.19 & 0.096 & 0.035 & Ar I 1048.22 & 13.6 16 & 1039.18 & 0.433 & 0.044 & O I 1039.23 & 17 & 1031.87 & 0.363 & 0.038 & O VI 1031.93 & 18 & 1020.75 & 0.129 & 0.040 & Si II 1020.70 & 19 & 1012.53 & 0.083 & 0.036 & S III 1012.50 & 14.4 20 & 1008.51 & 0.165 & 0.039 & NOID &
The identification of most of the strong interstellar features listed in Table 1 is secure [@M91]. There are no obvious interstellar or intergalactic candidates for the unidentified line at 1008.5 Å, which may be spurious. Although the observed features at 1066.8 and 1048.2 Å presumably correspond to a pair of lines, their equivalent widths at first glance appear to be anomalous (the 1048.2 Å line should be stronger). As noted, however, the longer wavelength feature may be partially blended with an line at 1066.9 Å. The shorter wavelength feature probably suffers from local continuum depression by nearby . Using the linear continuum shown in Figure 2, we estimate a best fit equivalent width closer to 0.13 Å. These systematic effects, in combination with the statistical errors, do much to reconcile the anomaly (especially if the features are on the flat part of the curve-of-growth).
Below 1000 Å, the increasing noise in the spectrum makes quantitative analysis of absorption features difficult, and the continuum fitting routine fails. Galactic features that are clearly detected include a blend from (and ?) near 989 Å, (977.02 Å), Lyman $\delta$ (972.54 Å), and an blend near 953.8 Å. Higher series Lyman lines are likely to be filled in by diffuse emission (we correct for diffuse emission only through Lyman $\delta$). An intriguing dip near 982 Å may be associated with in the Virgo cluster, but this identification is speculative. Difficulties in continuum placement render the equivalent widths very uncertain; we attempt no quantitative analysis of absorption lines below 1000 Å.
INTERSTELLAR GAS
================
The equivalent widths of features at wavelengths of overlap with the bandpass of the Hubble Space Telescope are consistent with previous measurements from that instrument (, ). For several other interstellar species (such as ), parameters inferred from the GHRS measurements [@SLWMG93] yield predictions for lines in the ORFEUS band that are consistent with the features observed. Our spectrum represents the first detection of , , , and along this sight line. The statistical significance of these features is modest, as can be seen in Table 1. The minimum columns inferred for these elements (and for and , features of which have been observed previously but whose column densities have not been reported) are shown in Table 1 and are of moderate interest only. Our minimum column for is a factor of 2.5 below that established for [@SLWMG93]. The ratio of to [@SLWMG93] is at least 0.06 .
Interesting interstellar species unique to the ORFEUS band include and , on which we now focus our attention. In Figure 2 we show a portion of the spectrum between 1046 and 1066 Å. This region is comparatively clear of competing interstellar lines and provides a convenient hunting ground for features of . The solid line is the observed spectrum; the lowest dotted line is the 1 $\sigma$ error. Overlying the data is a synthetic spectrum consisting of a flat continuum plus interstellar features convolved with the instrument response (dashed line). Wherever possible we adopt column densities, velocities, and effective Doppler $b$-values of atomic species from . An atomic-only model (omitted for clarity) does not reproduce the observed modulation near 1050 Å nor the observed width of the the strong line at 1063.2 Å.
The total column density implied by our data is about $1 \times 10^{15}$ cm$^{-2}$, corresponding to a molecular fraction ($f_{H_2}$) of about $2 \times 10^{-5}$. This value is not exceptional for a low-reddening sight line [@SB82]. The rotational excitation temperature $T_{01}$ is poorly constrained but appears to be consistent with the range typical of interstellar clouds (50 – 80 K).
In Figure 3 we show a portion of the spectrum between 1023 and 1041 Å. The legend is as in Figure 2, except that we have not labeled a few weak features to avoid crowding. Features labeled Cld. 1 and Cld. 2 arise in intergalactic gas. The discrepancy between their synthetic and observed spectra is discussed below. The strong galactic Lyman $\beta$ line is well fit in the wings; at the core, the subtraction of bright diffuse emission introduces a significant uncertainty indicated by the large peak in the error tracing. The stronger doublet member (1031.9 Å) is resolved cleanly; the weaker member is blended but clearly present. To produce the fit shown in Figure 3, we found it necessary to set the column density of near the upper limit from . This modification to the column does not significantly affect the previous determination that the cooling rate per nucleon is substantially below (about 1/6 of) the Galactic average [@SLWMG93].
With a $b$ value of about 35 (estimated from the profile of ), our best estimate of the column density is $7 \pm 2 \times 10^{14}$ cm$^{-2}$. found an column of $8 \times 10^{13}$ cm$^{-2}$, yielding an / ratio of about 10. The ratio along disk sight lines varies from about 10 to 20 (compilation of ). Ionization in this component of the hot galactic gas toward 3C273 therefore appears to be roughly similar to corresponding conditions in the disk. The fact that our measurement is near the low end of the disk range is qualitatively consistent with a scenario in which the gas is heated impulsively then cools as it moves away from the plane. Whether the comparatively high total column density is significantly nonrepresentative of the high latitude sky [@HB96] will be determined with certainty only after study of many additional sight lines. The presence of Radio Loops I and IV [@BHS71] cautions at the very least that division of our projected column by the midplane density [@J178] should not be taken as a reliable estimate of the typical scale height. However, the fact that this first clear measurement of columns for both and yields a value comparable to that in the disk bodes well for future far-ultraviolet missions such as FUSE. The previous measurement of toward 3C273 [@D93] had set only a lower limit on the column density, corresponding to an / ratio substantially below the range of disk values.
INTERGALACTIC GAS
=================
Sixteen otherwise unidentified absorption features in the GHRS spectrum were attributed to intergalactic Lyman $\alpha$ clouds by . Most of these are sufficiently weak that their higher Lyman series lines would not be detectable in our spectrum. We do find probable Lyman $\beta$ counterparts to the strongest proposed Lyman $\alpha$ forest features, however. propose a cloud at $cz$ = 19900 ; from its column density and $b$ value we predict a Lyman $\beta$ equivalent width of 0.080 Å at 1093.93 Å. We find an otherwise unidentified feature within 0.2 Å of the expected position with an equivalent width of 0.060 $\pm$ 0.038 Åconfirming, at least to 1.6 $\sigma$, the intergalactic origin.
Of greater interest are a strong pair of features attributed to comparatively low redshift gas in the Virgo supercluster. Their Lyman $\alpha$ lines appear near +1000 and +1600 [@WRWMH95]. Lyman $\beta$ counterparts are expected at 1029.21 and 1031.16 Å. We detect absorption features at 1029.11 and 1031.14 Å; however, the equivalent widths are stronger than expected. employ profile fitting to constrain column densities and Doppler $b$ values. Based on those parameters, the Lyman $\beta$ equivalent widths are predicted to be 0.098 and 0.10 Å, respectively. We observe 0.145 ($\pm$ 0.037) and 0.241 ($\pm$ 0.032) Å. The disagreement is seen clearly in Figure 3, where the synthetic spectrum relies on the parameters.
The discrepancy is particularly pronounced for the +1600 cloud. Even with the parameters proposed by the Lyman $\alpha$ line is saturated ($\tau_0$ = 3.7), so determination of a column density based solely on profile fitting of that feature may be affected by unresolved velocity structure and/or uncertainties in the instrumental profile. We have analyzed the GHRS spectrum to extract the equivalent width of the Lyman $\alpha$ line. In combination with the Lyman $\beta$ line from the ORFEUS spectrum, we establish constraints on the logarithmic column density and effective $b$ value for the +1600 cloud using a curve-of-growth technique. We show the results, and the parameters proposed by , in Figure 4. Our results suggest that if the feature arises in a single cloud, its column density is higher by at least a factor of four compared to the value suggested by . The curve-of-growth results are consistent with $b$ values as low as 12 and column densities in excess of $10^{18}$ cm$^{-2}$. Although this extremum of parameter space is probably inconsistent with the Lyman $\alpha$ profile fitting, it is not difficult to construct a multiple-cloud system that closely approximates the theoretical Voigt profile inferred from the parameters while hiding a great deal of gas in the saturated core.
searched for 21 cm emission from Virgo gas near the velocities of the absorbing clouds, detecting no to a limit of about $2.8 \times 10^{19}$ cm$^{-2}$. By adopting a relationship between the volume density of clouds with columns near the radio limit and of clouds with columns near those of the absorption line systems [@T87], the radio survey constrains the minimum size of the absorbing systems. The minimum permitted cloud size grows slowly with column density ($r_{cloud} \propto N^{0.25}$), so our results increase the minimum cloud size from about 4 kpc [@GBJS93] only to about 6 kpc, at least for the smallest columns within our 95% confidence interval.
Other interstellar species that might conceivably be blended with the +1600 cloud include at 1030.88 Å, or ($J$ = 3) at 1030.89 Å. Based on the nondetection of other features from these species, we conservatively limit the equivalent width of their combined contribution to no more than 0.050 Å. A contribution from Galactic at high negative velocity is more difficult to rule out formally from our spectrum. Neither nor appears at high negative velocity, however [@SST97].
CONCLUSIONS
===========
We observed the bright quasi-stellar object 3C273 for 10,797 s with the Berkeley spectrograph aboard the -II mission [@Hetal98]. The resulting spectrum offers intermediate spectral resolution ($\lambda$ / 3000 FWHM) and a 1$\sigma$ noise corresponding to an absorption feature with equivalent width of about 0.040 Å.
The spectrum reveals, at modest statistical significance, a variety of previously undetected low ionization interstellar species, including , , , and . There is marginally significant evidence for , with a total column density of about $10^{15}$ cm$^{-2}$. The $J$ = 0 and $J$ = 1 rotational levels seem comparably populated, consistent with an excitation temperature $T_{01}$ typical for interstellar clouds (50 – 80 K).
We detect the stronger interstellar doublet member unambiguously; the weaker member is blended. If the Doppler $b$ value for is comparable to that determined for [@SST97], the column density is $7 \pm 2 \times 10^{14}$ cm$^{-2}$. The / ratio is about 10, consistent with the low end of the range observed in the disk (compilation of ). If impulsively heated gas cools as it moves away from the Galactic disk, the / ratio might be expected to decrease (compared to the disk value) along halo sight lines.
We detect Lyman $\beta$ counterparts to intergalactic Lyman $\alpha$ features identified by at $cz$ = 19900, 1600, and 1000 . Counterparts to other putative Lyman $\alpha$ clouds along the sight line are below our detection limit. The strengths of the two very low redshift Lyman $\beta$ features, which are believed to arise in Virgo intracluster gas, exceed preflight expectations [@WRWMH95]. A curve-of-growth analysis sets a minimum column density of $4 \times 10^{14}$ cm$^{-2}$ for the 1600 cloud. Following , our revised column density and nondetection of 21 cm radio emission discussed in that work can set a lower limit of about 6 kpc to the size of the Virgo cluster clouds.
We acknowledge our colleagues on the team and the many NASA and DARA personnel who helped make the -II mission successful. This work is supported by NASA grant NAG5-696.
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|
---
abstract: 'In this paper we consider a symmetric simple exclusion process (SSEP) on the $d$-dimensional discrete torus ${{\mathbb T}}^d_N$ with a spatial non-homogeneity given by a slow membrane. The slow membrane is defined here as the boundary of a smooth simple connected region $\Lambda$ on the continuous $d$-dimensional torus ${{\mathbb T}}^d$. In this setting, bonds crossing the membrane have jump rate $\alpha/N^\beta$ and all other bonds have jump rate one, where $\alpha>0$, $\beta\in[0,\infty]$, and $N\in {{\mathbb N}}$ is the scaling parameter. In the diffusive scaling we prove that the hydrodynamic limit presents a dynamical phase transition, that is, it depends on the regime of $\beta$. For $\beta\in[0,1)$, the hydrodynamic equation is given by the usual heat equation on the continuous torus, meaning that the slow membrane has no effect in the limit. For $\beta\in(1,\infty]$, the hydrodynamic equation is the heat equation with Neumann boundary conditions, meaning that the slow membrane ${\partial}\Lambda$ divides ${{\mathbb T}}^d$ into two isolated regions $\Lambda$ and $\Lambda^\complement$. And for the critical value $\beta=1$, the hydrodynamic equation is the heat equation with certain Robin boundary conditions related to the Fick’s Law.'
address:
- |
UFBA\
Instituto de Matemática, Campus de Ondina, Av. Adhemar de Barros, S/N. CEP 40170-110\
Salvador, Brazil
- |
UFBA\
Instituto de Matemática, Campus de Ondina, Av. Adhemar de Barros, S/N. CEP 40170-110\
Salvador, Brazil
author:
- Tertuliano Franco
- Mariana Tavares
bibliography:
- 'bibliografia.bib'
title: |
Hydrodynamic Limit for the SSEP\
with a Slow Membrane
---
[^1]
[^2]
Introduction {#s1}
============
A central question of Statistical Mechanics is about how microscopic interactions determine the macroscopic behavior of a given system. Under this guideline, an entire area on scaling limits of interacting random particle systems has been developed, see [@kl] and references therein.
In the last years, many attention has been given to scaling limits of (spatially) non-homogeneous interacting systems, see for instance [@Franco2010; @fgn1] among many others. Such an attention is quite natural due to the fact that a non-homogeneity may represent vast physical situations, as impurities, changing of density in the media etc. Among those interacting particles systems, processes of *exclusion type* have special importance: they are, at same time, mathematically tractable and have a physical interaction, leading to precise representation of many phenomena. Being more precise, a random process is called of *exclusion type* if it has the *hard-core interaction*, that is, at most one particle is allowed per site of a given graph. The random evolution of the system (in the symmetric case) can be described as follows: to each edge of the given graph, a Poisson clock is associated, all of them independent. At a ring time of some clock, the occupation values for the vertexes of the corresponding edge are interchanged.
In [@Franco2010], a quite broad setting for the one-dimensional symmetric exclusion process (SEP) in non-homogeneous medium has been considered, being obtained its hydrodynamic limit, that is, the law of large numbers for the time evolution of the spatial density of particles. The hydrodynamic equation there was given by a PDE related to a Krein-Feller operator. And in [@FARFAN2010], the fluctuations for the same model were obtained.
The scenario for the SEP in non-homogeneous medium in dimension $d\geq 2$ up to now is far less understood. In [@valentim2012], a generalization of [@Franco2010] to the $d$-dimensional setting was reached. However, the definition of model there was very specific to permit a reduction to the one-dimensional approach of [@Franco2010].
In [@hld], the hydrodynamic limit in the diffusive scaling for the following $d$-dimensional simple symmetric exclusion process (SSEP) in non-homogeneous medium was proved, where the term *simple* means that only jumps to nearest neighbors are allowed. The underlying graph is the discrete $d$-dimensional torus, and all bonds of the graph have rate one, except those laying over a $(d-1)$-dimensional closed surface, which have rate given by $N^{-1}$ times a constant depending on the angle between the edge and the normal vector to the surface, where $N$ is the scaling parameter. The hydrodynamic equation obtained was given by a PDE related to a $d$-dimensional Krein-Feller operator. Despite less broad in certain sense than the setting of [@valentim2012], the model in [@hld] cannot be approached by one-dimensional techniques, being truly $d$-dimensional.
=\[fill=blue,circle,scale=.25\]
(1,2) to\[out=45,in=225\] (3,3) to\[out=45,in=0\] (3,0) to\[out=180,in=225\] (1,2); (2.5,2.65)–(2.25,3.25); (2,3.5) node[$\vec{\zeta}(u)$]{}; (2.45,2.75) node\[below right\][$u$]{};
(2.75,1.25) node [$\Lambda$]{}; (4.75,2.25) node [$\Lambda^\complement$]{};
(0,-1) grid (5,4); (5.7,3.75) node\[above\][$N^{-1}{{\mathbb T}}^d_N$]{};
In the present paper, we consider a $d$-dimensional model close to the one in [@hld] and related to the *slow bond phase transition behavior* of [@fgn1; @fgn2; @fgn3]. It is fixed a $(d-1)$-dimensional smooth surface ${\partial}\Lambda$ in the continuous $d$-dimensional torus ${{\mathbb T}}^d$, see Figure \[Fig1\]. Edges have rates equal to one, except those intersecting ${\partial}\Lambda$, which have rate $\alpha/N^\beta$, where $\alpha>0$, $\beta\in[0,\infty]$ and $N\in {{\mathbb N}}$ is the scaling parameter. Here we prove the hydrodynamic limit, which depends on the range of $\beta$, namely, if $\beta\in[0,1)$, $\beta=1$ or $\beta\in(1,\infty]$.
For $\beta\in[0,1)$, the hydrodynamic equation is given by the usual heat equation: meaning that, in this regime, the slow bonds do not have any effect in the continuum limit. For $\beta\in (1,\infty]$, the hydrodynamic equation is the heat equation with the following Neumann boundary conditions over ${\partial}\Lambda$: $$\frac{\partial\rho(t,u^{+})}{\partial\vec{\zeta}(u)}=\frac{\partial\rho(t,u^{-})}{\partial\vec{\zeta}(u)}=0,\qquad \forall\, t\geq0, \, u\in {\partial}\Lambda,$$ where $\vec{\zeta}$ is the normal unitary vector to ${\partial}\Lambda$. This means that, in this regime, the slow bonds are so strong that there no flux of mass through ${\partial}\Lambda$ in the continuum, despite the existence of flux of particles in the discrete for each $N\in {{\mathbb N}}$. For the critical value $\beta=1$, the hydrodynamic equation is given by the heat equation with the following Robin boundary conditions: $$\label{BRcond}
\frac{\partial\rho(t,u^{+})}{\partial\vec{\zeta}(u)}=\frac{\partial\rho(t,u^{-})}{\partial\vec{\zeta}(u)}=\alpha\Big(\rho(t,u^{+})-\rho(t,u^{-})\Big)\sum_{j=1}^{d}|{\langle}\vec{\zeta}(u),e_j{\rangle}|, \quad t\geq 0, \, u\in {\partial}\Lambda\,,$$ where $u^-$ denotes the limit towards $u\in{\partial}\Lambda$ through points over $\Lambda$ while $u^+$ denotes the limit towards $u\in{\partial}\Lambda$ through points over $\Lambda^\complement$, and $\{e_1\ldots,e_d\}$ is the canonical basis of ${{\mathbb R}}^d$.
We observe that the Robin boundary condition above is in agreement with the *Fick’s Law*: the spatial derivatives are equal due to the conservation of particles, representing the rate at which the mass crosses the boundary. Such a rate is proportional to the difference of concentration on each side of the boundary, being the diffusion coefficient through the boundary at a point $u\in {\partial}\Lambda$ given by $D(u)=\alpha\sum_{j=1}^{d}|{\langle}\vec{\zeta}(u),e_j{\rangle}|$. Since $\vec{\zeta}(u)$ is a unitary vector, the reader can check via Lagrange multipliers that this diffusion coefficient satisfies $$\begin{aligned}
\alpha\;\leq \; D(u)\;\leq \; \alpha\sqrt{d}\end{aligned}$$ in dimension $d\geq 2$. Moreover, in this case $\beta=1$, the hydrodynamic equation exhibits the phenomena of *non-invariance for isometries*. Let us explain this notion. Consider an isometry $\mathbf{T}:{{\mathbb T}}^d\to{{\mathbb T}}^d$, an initial density profile $\rho_0:{{\mathbb T}}^d\to [0,1]$ and denote by $(S(t)\rho_0)(u)$ the solution of the usual heat equation with initial condition $\rho_0$. Then, $$\big(S(t)(\rho_0\circ \mathbf{T})\big)(u)\;=\; (S(t)\rho_0)\big(\mathbf{T}(u)\big)\,.$$ In other words, if we isometrically move the initial condition of the usual heat equation, the solution of the PDE under this new initial condition is the equal to the previous solution moved by the same isometry. On the other hand, as we can see in , the diffusion coefficient $D(u)$ depends on how the surface ${\partial}\Lambda$ is positioned with respect to the canonical basis. Hence the PDE for $\beta=1$ *is not invariant for isometries*, differently from the cases $\beta\in[0,1)$ and $\beta\in(1,\infty]$. Note that the diffusion coefficient also says that the underlying graph plays a role in the limit.
Besides the dynamical phase transition itself, this work has the following features. First of all, in contrast with some previous works, the hydrodynamic equations are characterized as classical PDEs, with clear interpretation. In the regime $\beta\in[0,1)$, the proof relies on a sharp replacement lemma which compares occupations of neighbor sites in opposite sides of ${\partial}\Lambda$. For $\beta=1$, the proof is based on a precise analysis of the surface integrals and the model drops the *ad hoc* hypothesis adopted in [@hld]: here the rates for bonds crossing ${\partial}\Lambda$ are all equal to $\alpha/N$, with no extra constant depending on the incident angle. Finally, a remark the uniqueness of weak solutions for the cases $\beta=1$ and $\beta\in(1,\infty]$. Uniqueness of weak solutions are in general a delicate and technical issue, specially for dimension higher than one. In Proposition \[prop72\] we provide a general statement which leads to the uniqueness of weak solutions in both cases $\beta=1$ and $\beta\in(1,\infty]$. The keystone of the proof is the notion of *Friedrichs extension* for strongly monotone symmetric operators. The uniqueness statement has the feature of being simple, $d$-dimensional and easily adaptable to many contexts. However, it is strictly limited to the uniqueness of weak solutions of parabolic linear PDEs with linear boundary conditions.
The paper is divided as follows: In Section \[s2\] we state definitions and results. In Section \[s3\] we draw the strategy of proof for the hydrodynamic limit. In Section \[s4\] is reserved to the proof of tightness of the processes. In Section \[s5\] we prove the necessary replacement lemmas and energy estimates. In Section \[s6\] we characterize limit points as concentrated on weak solutions of the respective PDEs, and in Section \[s7\] we assure uniqueness of those weak solutions.
Definitions and Results {#s2}
=======================
Let ${{\mathbb T}}^d$ be the continuous $d$-dimensional torus, which is $[0,1)^d$ with periodic boundary conditions, and let ${{\mathbb T}}^d_N$ be the discrete torus with $N^d$ points, which can naturally embedded in the continuous torus as $N^{-1} {{\mathbb T}}^d_N$, see Figure \[Fig1\]. We therefore will not distinguish notation for functions defined on ${{\mathbb T}}^d$ or $ N^{-1} {{\mathbb T}}_{N}^d$.
By $\eta = (\eta(x))_{x \in {{\mathbb T}}_{N}^d}$ we denote configurations in the state space $\Omega_N = \{0,1\}^{{{\mathbb T}}_{N}^d}$, where $\eta(x)=0$ means that the site $x$ is empty, and $\eta(x)=1$ means that the site $x$ is occupied. By a *symmetric simple exclusion process* we mean the Markov Process with configuration space $\Omega_N$ and exchange rates $\xi^{N}_{x,y} > 0$ for $x,y \in {{\mathbb T}}^d_N$ with $\Vert x-y\Vert_{1} = 1$. This process can be characterized in terms of the infinitesimal generator $\mathscr{L}_{N}$ acting on functions $f:\Omega_N \to {{\mathbb R}}$ as $$\label{ln}
(\mathscr{L}_{N}f)(\eta)\;=\;\sum_{x\in {{\mathbb T}}_{N}^d}\sum_{j=1}^d \,\xi^{N}_{x,x+e_j}\,\Big[f(\eta^{x,x+e_j})-f(\eta)\Big]\,,$$ where $\{e_1,\ldots,e_d\}$ is the canonical basis of ${{\mathbb R}}^d$ and $\eta^{x,x+e_j}$ is the configuration obtained from $\eta$ by exchanging the occupation variables $\eta(x)$ and $\eta(x+e_j)$, that is, $$\eta^{x,x+e_j}(y)\;=\;\left\{\begin{array}{cl}
\eta(x+e_j),& \mbox{if}\,\,\, y=x\,,\\
\eta(x),& \mbox{if} \,\,\,y=x+e_j\,,\\
\eta(y),& \mbox{otherwise.}
\end{array}
\right.$$ The Bernoulli product measures $\{\nu^N_\theta\,:\,\theta \in [0,1]\}$ are invariant and in fact, reversible, for the symmetric nearest neighbor exclusion process introduced above. Namely, $\nu^N_\theta$ is a product measure on $\Omega_N$ whose marginal at site $x\in {{\mathbb T}}^d_N$ is given by $$\nu^N_\theta\{ \eta:\eta(x)=1\}\;=\;\theta\,.$$
Fix now two parameters $\alpha>0$ and $\beta \in [0,\infty]$ and a simple connected closed region $\Lambda\subset{{\mathbb T}}^d$ whose boundary ${\partial}\Lambda$ is a smooth $(d-1)$-dimensional surface. The *symmetric simple exclusion process with slow bonds over* ${\partial}\Lambda$ (SSEP with slow bonds over ${\partial}\Lambda$) we define now is the particular simple symmetric exclusion process with exchange rates given by $$\label{rates}
\xi^N_{x,x+e_j}\;=\;\left\{\begin{array}{cl}
\dfrac{\alpha}{N^\beta}\,, & \mbox{if }~\dfrac{x}{N}\in\Lambda\textrm{ and }
\frac{x+e_j}{N}\in\Lambda^\complement, \textrm{ or } \dfrac{x}{N}\in\Lambda^\complement \textrm{ and } \dfrac{x+e_j}{N}\in\Lambda,\medskip\\
1\,, &\mbox{otherwise,}
\end{array}
\right.$$ for all $x \in {{\mathbb T}}^d_N$ and $j=1,\ldots,d$. That is, the *slow bonds* of the process will be the bonds in $N^{-1}{{\mathbb T}}^{d}_{N}$ for which one of its vertices belongs to $\Lambda$ and the other one belongs to $\Lambda^\complement$. See Figure \[Fig1\] for an illustration.
Note that, when $\beta=\infty$, there are no crossings of particles through the boundary ${\partial}\Lambda$. From now on, abusing of notation, we will call the generator of the SSEP with slow bonds over ${\partial}\Lambda$ by $\mathscr{L}_N$, being understood that jump rates will be given by .
Denote by $\{\eta_t : t\ge 0\}$ the Markov process with state space $\Omega_N$ and generator $N^2\mathscr{L}_N$, where the $N^2$ factor is the so-called *diffusive scaling*. This Markov process depends on $N$, but it will not be indexed on it to not overload notation. Let $D({{\mathbb R}}_+, \Omega_N)$ be the *Skorohod space* of càdlàg trajectories taking values in $\Omega_N$. For a measure $\mu_N$ on $\Omega_N$, denote by ${{\mathbb P}}^N_{\mu_N}$ the probability measure on $D({{\mathbb R}}_+, \Omega_N)$ induced by the initial state $\mu_N$ and the Markov process $\{\eta_t : t\ge 0\}$. Expectation with respect to ${{\mathbb P}}^N_{\mu_N}$ will be denoted by ${{\mathbb E}}^N_{\mu_N}$.
In the sequel, we present the partial differential equations governing the time evolution of the density profile for the different regimes of $\beta$, defining the notion of weak solution for each one of those equations. Denote by $\rho_t$ a function $\rho(t, \cdot)$ and denote by $C^n({{\mathbb T}}^d)$ the set of continuous functions from ${{\mathbb T}}^d$ to ${{\mathbb R}}$ with continuous derivatives of order up to $n$. Let ${\langle}\cdot , \cdot {\rangle}$ and $\| \cdot \|$ be the inner product and norm in $L^2({{\mathbb T}}^d)$, that is, $$\label{inner}
{\langle}f, g {\rangle}\;=\; \int_{{{\mathbb T}}^d} f(u)\, g(u)\, du\; \, \;\textrm{ and }\; \,
\| f \| = \sqrt{ {\langle}f, f {\rangle}} \; ,\quad \forall\, f, g \in L^2({{\mathbb T}}^d)\,.$$ Fix once and for all a measurable density profile $\rho_0: {{\mathbb T}}^d \rightarrow [0,1]$. Note that $\rho_0$ is bounded.
\[beta<1\] A bounded function $\rho : [0,T]\times {{\mathbb T}}^d \to {{\mathbb R}}$ is said to be a weak solution of the heat equation $$\label{edpheat}
\left\{
\begin{array}{ll}
{\displaystyle {\partial}_t \rho(t,u) \; =\; \Delta \rho(t,u)}, & t\geq0, \, u\in {{\mathbb T}}^d, \\
{\displaystyle \rho(0,u) \;=\; \rho_0(u)},& u\in{{\mathbb T}}^d .
\end{array}
\right.$$ if, for all functions $H\in C^2({{\mathbb T}}^d)$ and all $t\in[0,T]$, the function $\rho(t, \cdot)$ satisfies the integral equation $$\label{eqint1}
{\langle}\rho_t, H{\rangle}\;-\; {\langle}\rho_0 , H{\rangle}- \int_0^t {\langle}\rho_s , \Delta H {\rangle}\, ds\; \;=\; 0\,.$$
We recall next the definition of Sobolev Space from [@e]. Let $U$ be an open set of ${{\mathbb R}}^d$ or ${{\mathbb T}}^d$. The Sobolev Space ${{\mathcal H}}^{1}(U)$ consists of all locally summable functions $\kappa:U\rightarrow {{\mathbb R}}$ such that there exist functions ${\partial}_{u_{j}}\kappa\in L^{2}(U)$, $j=1,\ldots, d$, satisfying $$\int_{{{\mathbb T}}^d} {\partial}_{u_j}H(u)\kappa(u)\,du\;=\;-\int_{{{\mathbb T}}^d}H(u){\partial}_{u_{j}}\kappa(u)\,du$$ for all $H\in C^{\infty}(U)$ with compact support. Furthermore, for $\kappa\in {{\mathcal H}}^{1}(U)$, we define the norm $\Vert\kappa\Vert_{{{\mathcal H}}^1(U)}\;=\;\Big(\sum_{j=1}^d\int_U \big|{\partial}_{u_{j}}\kappa\big|^2\, du\Big)^{1/2}$. Finally, we define the space $L^2([0,T], {{\mathcal H}}^{1}(U))$, which consists of all measurable functions $\tau:[0,T]\rightarrow {{\mathcal H}}^{1}(U)$ such that $$\Vert\tau\Vert_{L^2([0,T], {{\mathcal H}}^1(U))}\;:=\; \Big(\int_0^T \Vert \tau_t\Vert^2_{{{\mathcal H}}^1(U)}\, dt\Big)^{1/2}\;<\;\infty\,.$$ Note that $U={{\mathbb T}}^d\backslash {\partial}\Lambda$ is an open subset of ${{\mathbb T}}^d$.
The following notation will be used several times along the text. Given a function $f:{{\mathbb T}}^d\backslash {\partial}\Lambda\to {{\mathbb R}}$ and $u\in {\partial}\Lambda$, we denote $$\label{maismenos}
f(u^+)\;:=\;\lim_{{\genfrac{}{}{0pt}{}{v\to u}{v\in \Lambda^\complement}}}f(v)\quad \text{ and } f(u^-)\;:=\;\lim_{{\genfrac{}{}{0pt}{}{v\to u}{v\in \Lambda}}}f(v)\,,$$ that is, $f(u^+)$ is the limit of $f(v)$ as $v$ approaches $u\in {\partial}\Lambda$ through *the complement of* $\Lambda$, while $f(u^-)$ is the limit of $f(v)$ as $v$ approaches $u\in {\partial}\Lambda$ through $\Lambda$. Let $\mathbf{1}_A$ be the indicator function of a set $A$, that is, $\mathbf{1}_A(a)=1$ if $a\in A$ and zero otherwise. Denote by $\vec{\zeta}(u)$ the normal unitary exterior vector to the region $\Lambda$ at the point $u\in{\partial}\Lambda$ and by ${\partial}/{\partial}\vec{\zeta}$ the directional derivative with respect to $\vec{\zeta}(u)$.
Below, by ${\langle}\vec{u},\vec{v}{\rangle}$ we denote the canonical inner product of two vectors $\vec{u}$ and $\vec{v}$ in ${{\mathbb R}}^d$, which shall not be misunderstood with the inner product in $L^2({{\mathbb T}}^d)$ as defined in . By $dS$ we indicate a surface integral.
\[beta=1\] A bounded function $\rho : [0,T]\times {{\mathbb T}}^d \to {{\mathbb R}}$ is said to be a weak solution of the following heat equation with Robin boundary conditions $$\label{edp12}
\left\{
\begin{array}{ll}
{\displaystyle {\partial}_t \rho(t,u) \; =\; \Delta \rho(t,u)}, &\hspace{-0.2cm} t\geq0, \, u\in {{\mathbb T}}^d, \\
{\displaystyle \frac{\partial\rho(t,u^{+})}{\partial\vec{\zeta}(u)}=\frac{\partial\rho(t,u^{-})}{\partial\vec{\zeta}(u)}=\alpha\Big(\rho(t,u^{+})-\rho(t,u^{-})\Big)\sum_{j=1}^{d}|{\langle}\vec{\zeta}(u),e_j{\rangle}|}, &\hspace{-0.2cm} t\geq0, \, u\in {\partial}\Lambda, \\
{\displaystyle \rho(0,u) \;=\; \rho_0(u)}, &\hspace{-0.2cm} u\in{{\mathbb T}}^d \, .
\end{array}
\right.$$ if $\rho\in L^2([0,T], {{\mathcal H}}^1({{\mathbb T}}^d\backslash {\partial}\Lambda))$ and, for all functions $H=h_1\mathbf{1}_{\Lambda}+h_2\mathbf{1}_{\Lambda^\complement}$ with $h_1, h_2 \in C^2({{\mathbb T}}^d)$ and for all $t\in[0,T]$, the following the integral equation holds: $$\begin{split}
&{\langle}\rho_t, H{\rangle}- {\langle}\rho_0 , H{\rangle}- \int_0^t\! {\langle}\rho_s , \Delta H {\rangle}\, ds-\int_0^t\!\int_{{\partial}\Lambda}\!\rho_s(u^+)\sum_{j=1}^d{\partial}_{u_j} H(u^+){\langle}\vec{\zeta}(u),e_j{\rangle}\,dS(u)ds
\\
&
+\int_0^t\int_{{\partial}\Lambda}\!\rho_s(u^-)\sum_{j=1}^d{\partial}_{u_j} H(u^-){\langle}\vec{\zeta}(u),e_j{\rangle}\,dS(u)ds\\
&+\int_0^t\int_{{\partial}\Lambda}\!\alpha\,(\rho_s(u^-)-\rho_s(u^+))(H(u^+)-H(u^-))\Big(\sum_{j=1}^{d}|{\langle}\vec{\zeta}(u),e_j{\rangle}|\Big)\,dS(u)ds\;=\; 0\,.
\end{split}$$
The reader should note that the function $H$ is (possibly) discontinuous at the boundary ${\partial}\Lambda$. Note also that the expression $\sum_{j=1}^d{\partial}_{u_j} H(u^\pm){\langle}\vec{\zeta}(u),e_j{\rangle}$ appearing in the integral equation above is nothing but ${\partial}H (u^\pm)/{\partial}\vec{\zeta}$ due to linearity of the directional derivative.
\[beta>1\] A bounded function $\rho : [0,T]\times {{\mathbb T}}^d \to {{\mathbb R}}$ is said to be a weak solution of the heat equation with Neumann boundary conditions $$\label{edpbc}
\left\{
\begin{array}{ll}
{\displaystyle {\partial}_t \rho(t,u) \; =\; \Delta\rho(t,u)},& t\geq0, \, u\in {{\mathbb T}}^d,\smallskip \\
\displaystyle \frac{\partial\rho(t,u^{+})}{\partial\vec{\zeta}(u)}=\frac{\partial\rho(t,u^{-})}{\partial\vec{\zeta}(u)}=0, & t\geq0, \, u\in {\partial}\Lambda, \smallskip\\
{\displaystyle \rho(0,u) \;=\; \rho_0(u)}, & u\in{{\mathbb T}}^d \,,
\end{array}
\right.$$ if $\rho\in L^2([0,T], {{\mathcal H}}^1({{\mathbb T}}^d\backslash {\partial}\Lambda))$ and, for all functions $H=h_1\mathbf{1}_{\Lambda}+h_2\mathbf{1}_{\Lambda^\complement}$ with $h_1, h_2 \in C^2({{\mathbb T}}^d)$ and for all $t\in[0,T]$, the following integral equation holds: $$\label{eqint3}
\begin{split}
&{\langle}\rho_t, H{\rangle}\;-\; {\langle}\rho_0 , H{\rangle}- \!\int_0^t \!{\langle}\rho_s , \Delta H {\rangle}\, ds-\int_0^t\!\!\int_{{\partial}\Lambda}\!\rho_s(u^+)\sum_{j=1}^d{\partial}_{u_j} H(u^+){\langle}\vec{\zeta}(u),e_j{\rangle}\,dS(u)ds\\
&+\int_0^t\!\!\int_{{\partial}\Lambda}\rho_s(u^-)\sum_{j=1}^d{\partial}_{u_j} H(u^-){\langle}\vec{\zeta}(u),e_j{\rangle}\,dS(u)ds\;=\;0\,.
\end{split}$$
Since in Definitions \[beta=1\] and \[beta>1\] we impose $\rho \in L^2([0,T], {{\mathcal H}}^1({{\mathbb T}}^d\backslash{\partial}\Lambda))$, the integrals above are well-defined on the boundary due to the notion of trace in Sobolev spaces, see [@e] on the subject. We clarify that the notion of weak solutions above have been defined in the standard way of Analysis: the reader can check that a strong solution of , or is indeed a weak solution of the respective PDE.
Fix a measurable density profile $\rho_0:{{\mathbb T}}^d\rightarrow [0,1]$. For each $N\in{{\mathbb N}}$, let $\mu_N$ be a probability measure on $\Omega_N$. A sequence of probability measures $\{\mu_N \,: \,N\geq 1 \}$ is said to be *associated to a profile* $\rho_0 :{{\mathbb T}}^d \to [0,1]$ if, for every $\delta>0$ and every continuous function $H:{{\mathbb T}}^d\rightarrow {{\mathbb R}}$ the following limit holds: $$\label{profile}
\lim_{N\to\infty}
\mu_N \Bigg\{ \, \Bigg| \frac {1}{N^d}\!\! \sum_{x\in{{\mathbb T}}_N^d} H(x/N) \eta(x)
- \int H(u) \rho_0(u) du \Bigg| > \delta \Bigg\} \;=\; 0\,.$$
Below, we establish the main result of this paper, the hydrodynamic limit for the exclusion process with slow bonds, which depends on the regime of $\beta$.
\[t01\] Fix $\beta\in[0,\infty]$. Consider the exclusion process with slow bonds over ${\partial}\Lambda$ with rate $\alpha N^{-\beta}$ at each one of these slow bonds. Fix a Borel measurable initial profile $\rho_0 : {{\mathbb T}}^d \to [0,1]$ and consider a sequence of probability measures $\{\mu_N\}_{N\in{{\mathbb N}}}$ on $\Omega_N$ associated to $\rho_0$ in the sense of . Then, for each $t\in[0,T]$, $$\lim_{N\to\infty} {{\mathbb P}}^N_{\mu_N} \Bigg[\,\eta\,: \, \Bigg| \,\frac{1}{N^d} \sum_{x\in{{\mathbb T}}^d_N} H(x/N)\, \eta_t(x) - \int_{{{\mathbb T}}^d} H(u)\, \rho(t,u) du \,\Bigg| \,>\, \delta\, \Bigg] \;=\; 0 \, ,$$ for every $\delta>0$ and every function $H\in C({{\mathbb T}}^d)$ where:
- If $\beta \in [0,1)$, then $\rho$ is the unique weak solution of .
- If $\beta=1$, then $\rho$ is the unique weak solution of .
- If $\beta \in (1,\infty]$, then $\rho$ is the unique weak solution of .
The assumption that $\Lambda$ is simple and connected may be dropped, being imposed only for the sake of clarity. Otherwise, notation would be highly overloaded.
Scaling Limit and Proof’s Structure {#s3}
===================================
Let ${{\mathcal M}}$ be the space of positive Radon measures on ${{\mathbb T}}^d$ with total mass bounded by one, endowed with the weak topology. Let $\pi_t^N\in{{\mathcal M}}$ the empirical measure at time $t$ associated to $\eta_t$, it is a measure on ${{\mathbb T}}^d$ obtained rescaling space by $N$: $$\pi_t^N(du)\;=\;\pi_t^N(\eta_t, du)\;:=\;\frac{1}{N^d}\sum_{x\in{{\mathbb T}}^d_N}\eta_t(x)\delta_{x/N}(du)\,,$$ where $\delta_u$ denotes the Dirac measure concentrated on $u\in {{\mathbb T}}^d$. For a measurable function $H:{{\mathbb T}}^d\rightarrow{{\mathbb R}}$ which is $\pi$-integrable, denote by ${\langle}\pi_t^N, H{\rangle}$ the integral of $H$ with respect to $\pi_t^N$: $${\langle}\pi_t^N, H{\rangle}\; =\; \frac{1}{N^d}\sum_{x\in{{\mathbb T}}^d_N}H\left({\genfrac{}{}{}{1}{x}{N}}\right)\eta_t(x)\,.$$ Note that this notation ${\langle}\cdot,\cdot{\rangle}$ is also used as the inner product of $L^2({{\mathbb T}}^d)$. Fix once and for all a time horizon $T>0$. Let $D([0,T], {{\mathcal M}})$ be the space of ${{\mathcal M}}$-valued *càdlàg* trajectories $\pi:[0,T]\to{{\mathcal M}}$ endowed with the *Skorohod* topology. Then, the ${{\mathcal M}}$-valued process $\{\pi^N_t:t\ge 0\}$ is a random element of $D([0,T], {{\mathcal M}})$ determined by $\{\eta_t : t\ge 0\}$. For each probability measure $\mu_N$ on $\Omega_N$, denote by ${{\mathbb Q}}_{\mu_N}^{\beta,N}$ the distribution of $\{\pi^N_t:t\ge 0\}$ on the path space $D([0,T], {{\mathcal M}})$, when $\eta_0^N$ has distribution $\mu_N$.
Fix a continuous Borel measurable profile $\rho_0 : {{\mathbb T}}^d \to [0,1]$ and consider a sequence $\{\mu_N:N\geq1\}$ of measures on $\Omega_N$ associated to $\rho_0$. Let ${{\mathbb Q}}^\beta$ be the probability measure on $D([0,T], {{\mathcal M}})$ concentrated on the deterministic path $\pi(t,du) = \rho (t,u)du$, where:
- if $\beta \in [0,1)$, then $\rho$ is the unique weak solution of ,
- if $\beta=1$, then $\rho$ is the unique weak solution of ,
- if $\beta \in (1,\infty]$, then $\rho$ is the unique weak solution of .
\[s15\] For any $\beta\in[0,\infty]$, the sequence of probability measures ${{\mathbb Q}}_{\mu_N}^{\beta,N}$ converges weakly to ${{\mathbb Q}}^{\beta}$ as $N$ goes to infinity.
The proof of this result is divided into three parts. In the next section, we show that tightness of the sequence $\{{{\mathbb Q}}_{\mu_N}^{\beta,N}: N\geq1\}$. In Section \[s5\], we prove a suitable *Replacement Lemma* for each regime of $\beta$, which will be crucial in the task of characterizing limit points. In Section \[s6\] we characterize the limit points of the sequence for each regime of the parameter $\beta$. Finally, the uniqueness of weak solutions is presented in Section \[s7\] and this implies the uniqueness of limit points of the sequence $\{{{\mathbb Q}}_{\mu_N}^{\beta,N}: N\geq1\}$.
Finally, we note that Theorem \[t01\] is a consequence of Proposition \[s15\]. Actually, since ${{\mathbb Q}}_{\mu_N}^{\beta,N}$ weakly converges to ${{\mathbb Q}}^{\beta}$ for all continuous functions $H:{{\mathbb T}}^d\rightarrow {{\mathbb R}}$, it follows that the path $\{{\langle}\pi_t^N, H{\rangle}:\, 0\leq t\leq T\}$ converges in distribution to $\{{\langle}\pi_t, H{\rangle}:\, 0\leq t\leq T\}$. Since $\{{\langle}\pi_t, H{\rangle}:\, 0\leq t\leq T\}$ is a deterministic path, convergence in distribution is equivalent to convergence in probability. Therefore, $$\begin{aligned}
&\lim_{N\to\infty} {{\mathbb P}}^N_{\mu_N} \Bigg\{ \, \Bigg\vert \frac{1}{N^d} \sum_{x\in{{\mathbb T}}_N^d} H(x/N)\, \eta_t(x) - \int_{{{\mathbb T}}^d} H(u) \rho(t,u) du \Bigg\vert
> \delta \Bigg\}\\
&= \lim_{N\to\infty} {{\mathbb Q}}_{\mu_N}^{\beta,N}\big\{|{\langle}\pi_t^N, H{\rangle}-{\langle}\pi_t, H{\rangle}|>\delta\big\}\;=\; 0\,, \end{aligned}$$ for all $\delta>0$ and $0\leq t \leq T$. This gives the strategy of proof for the hydrodynamic limit. Next, we make some general observations.
Since particles in the exclusion process evolve independently as a nearest neighbor random walk, except for exclusion rule, the exclusion process with slow bonds over ${\partial}\Lambda$ is related to the random walk on $N^{-1}{{\mathbb T}}^d_N$ that describes the evolution of the system with a single particle. To be used throughout the paper we introduce the generator of the random walk described above, which is $$\label{bbLN}
{{\mathbb L}}_N H\big({\genfrac{}{}{}{1}{x}{N}}\big) = \sum_{j=1}^d \Big\{ \xi^N_{x,x+e_j} \, \Big[ H\big({\genfrac{}{}{}{1}{x+e_j}{N}}\big)
- H\big({\genfrac{}{}{}{1}{x}{N}}\big) \Big] + \xi^N_{x,x-e_j} \, \Big[H\big({\genfrac{}{}{}{1}{x-e_j}{N}}\big) - H\big({\genfrac{}{}{}{1}{x}{N}}\big) \Big] \Big\}$$ for every $H: N^{-1} {{\mathbb T}}^d_N \rightarrow \mathbb{R}$ and every $x \in {{\mathbb T}}^d_N$. Above, it is understood that $\xi_{x\pm e_j,x}=\xi_{x,x\pm e_j}$. By Dynkin’s formula (see A.1.5.1 in [@kl]), $$M^{N}_{t}(H)\;=\;{\langle}\pi^{N}_{t}, H{\rangle}- {\langle}\pi^{N}_{0}, H{\rangle}-\int_{0}^{t}N^{2} \mathscr{L}_N{\langle}\pi^{N}_{s},H{\rangle}ds$$ is a martingale with respect to the natural filtration ${{\mathcal F}}_t:=\sigma(\eta_s^N\,:\, s\leq t)$. By some elementary calculations, $$N^2\mathscr{L}_N{\langle}\pi_s^N,H{\rangle}\;=\;\frac{1}{N^{d-2}}\sum_{x\in{{\mathbb T}}^d_N}\eta_s(x){{\mathbb L}}_N H\Big(\frac{x}{N}\Big)\;=\;{\langle}\pi_s^N,N^2{{\mathbb L}}_N H{\rangle}\,,$$ hence the martingale can be rewritten as $$\label{M}
M^{N}_{t}(H)\;=\;{\langle}\pi^{N}_{t}, H{\rangle}- {\langle}\pi^{N}_{0}, H{\rangle}-\int_{0}^{t}{\langle}\pi^{N}_{s},N^2{{\mathbb L}}_N H{\rangle}ds\,.$$ Note that this observation stands for any jump rates. The particular form of jump rates for the SSEP with slow bonds over ${\partial}\Lambda$ will play a role when characterizing limit points and proving replacement lemmas.
Tightness {#s4}
=========
This section deals with the issue of tightness for the sequence $\{{{\mathbb Q}}_{\mu_N}^{\beta,N}: N\geq1\}$ of probability measures on $D([0,T], {{\mathcal M}})$.
\[tight\] For any fixed $\beta\in[0,\infty]$, the sequence of measures $\{{{\mathbb Q}}_{\mu_N}^{\beta,N}: N\geq1\}$ is tight in the Skorohod topology of $D([0,T], {{\mathcal M}})$.
In order to prove tightness of $\{\pi_t^N:0\leq t\leq T\}$, it is enough to show tightness of the real-valued process $\{{\langle}\pi_t^N,H{\rangle}:0\leq t \leq T\}$ for $H\in C({{\mathbb T}}^d)$. In fact, (cf. Proposition 1.7, chapter 4 of [@kl]) it is enough to show tightness of $\{{\langle}\pi_t^N,H{\rangle}:0\leq t \leq T\}$ in $D([0,T],{{\mathbb R}})$ for a dense set of functions in $C({{\mathbb T}}^d)$ with respect to the uniform topology.
For that purpose, fix $H\in C^2({{\mathbb T}}^d)$. Since the sum of tight processes is tight, in order to prove tightness of $\{{\langle}\pi_t^N, H{\rangle}: N\geq1\}$, it is enough to assure tightness of each term in . The quadratic variation of $M_t^N(H)$ is given by $$\label{quamar0}
{\langle}M^{N}(H){\rangle}_t=\!\int_{0}^{t}\sum_{j=1}^d\sum_{x\in{{\mathbb T}}_{N}^d} \frac{\xi^{N}_{x,x+e_j}}{N^{2d-2}}\Big[(\eta_{s}(x)-\eta_{s}(x+e_j))(H({\genfrac{}{}{}{1}{x+e_j}{N}})-H({\genfrac{}{}{}{1}{x}{N}}))\Big]^{2} ds,$$ implying that $$\label{quamar}
{\langle}M^{N}(H){\rangle}_t \;\leq\; \frac{\alpha t}{N^{d}} \, \sum_{j=1}^d\|{\partial}_{u_j}H\|^2_{\infty} \, ,$$ where $\|H\|_{\infty}:= \sup_{u\in {{\mathbb T}}^d}|H(u)|$, hence $M^{N}_{t}$ converges to zero as $N\rightarrow\infty$ in $L^{2}({{\mathbb P}}^\beta_{\mu_N})$. Therefore, by Doob’s inequality, for every $\delta>0$, $$\label{limmart}
\lim_{N\rightarrow\infty} {{\mathbb P}}^N_{\mu_N}\Big[\sup_{0\leq t\leq T}|M_t^N(H)|>\delta\Big]\;=\;0\,,$$ which implies tightness of the sequence of martingales $\{M_t^N(H)\,:\,N\geq1\}$. Next, we will prove tightness for the integral term in . Let $\Gamma_N$ be the set of vertices in ${{\mathbb T}}^d_N$ having some incident edge with exchange rate not equal to one, that is, $$\begin{aligned}
\label{GammaN}
\Gamma_N=\Big\{x\in {{\mathbb T}}_N^d: \text{ for some } j=1,\ldots,d, \quad\xi^N_{x,x+e_j}=\frac{\alpha}{N^\beta}\text{ or } \xi^N_{x,x-e_j}=\frac{\alpha}{N^\beta}\Big\}.\end{aligned}$$ The term ${\langle}\pi^{N}_{s},N^2{{\mathbb L}}_N H{\rangle}$ appearing inside the time integral in can be then written as $$\begin{aligned}
&\frac{1}{N^{d}}\sum_{j=1}^d\sum_{x\notin \Gamma_N}\eta_{s}(x)N^2\Big[H({\genfrac{}{}{}{1}{x+e_j}{N}})+H({\genfrac{}{}{}{1}{x-e_j}{N}})-2H({\genfrac{}{}{}{1}{x}{N}})\Big] \nonumber\\
&+ \frac{1}{N^{d-1}}\sum_{j=1}^d\sum_{x\in \Gamma_N}\!\eta_{s}(x)\Big[\xi^{N}_{x,x+e_j}N\big(H({\genfrac{}{}{}{1}{x+e_j}{N}})\!-\! H({\genfrac{}{}{}{1}{x}{N}})\big)\!+\!\xi^{N}_{x,x-e_j}
N\big(H({\genfrac{}{}{}{1}{x-e_j}{N}})\!-\!H({\genfrac{}{}{}{1}{x}{N}})\big)\!\Big]\label{qqq} \end{aligned}$$ since $\xi_{x,x+e_j}=\xi_{x+e_j,x}=1$ for every $x\notin{\Gamma_N}$. By a Taylor expansion on $H\in C^2({{\mathbb T}}^d)$, the absolute value of the summand in the first double sum above is bounded by $\|\Delta H\|_{\infty}$. Since there are ${{\mathcal O}}(N^{d-1})$ elements in $\Gamma_N$, and $\xi_{x,x+e_j}\leq \alpha$, the absolute value of summand in second double sum above is bounded by $\sum_{j=1}^d \alpha \Vert{\partial}_{u_j}H\Vert_\infty$. Therefore, there exists $C>0$, depending only on $H$, such that $|N^2{{\mathbb L}}_N{\langle}\pi_s^N, H{\rangle}|\leq C$, which yields $$\left|\int_s^t N^2{{\mathbb L}}_N{\langle}\pi_s^N, H{\rangle}dr\right|\;\leq\; C|t-s| \,.$$ By [@kl Proposition 4.1.6], last inequality implies tightness of the integral term, concluding the proof of the proposition.
Replacement Lemma and Energy Estimates {#s5}
======================================
This section gives a fundamental result that allow us to replace a mean occupation of a site by the mean density of particles in a small macroscopic box around this site. We start by introducing some tools to be used in the sequel.
Denote by $H_N(\mu_N|\nu_\theta)$ the relative entropy of $\mu_N$ with respect to the invariant state $\nu_\theta$. For a precise definition and properties of the entropy, we refer the reader to [@kl]. Assuming $0<\theta<1$, the formula in [@kl Theorem A1.8.3] assures the existence a finite constant $\kappa_0=\kappa_0(\theta)$ such that $$\label{cte}
H_N(\mu_N|\nu_\theta)\; \leq \; \kappa_0 N^d$$ for any probability measure $\mu_N$ on $\{0,1\}^{{{\mathbb T}}^d_N}$. Denote by ${{\mathfrak D}}_N$ the Dirichlet form of the process, which is the functional acting on functions $f:\{0,1\}^{{{\mathbb T}}^d_N}\to {{\mathbb R}}$ as $$\label{Dirichlet}
{{\mathfrak D}}_N(f) \,:=\,{\langle}f,- \mathscr{L}_N f{\rangle}_{\nu_\theta} =\sum_{j=1}^d\sum_{x\in {{\mathbb T}}^d_N}\!\frac{\xi^N_{x,x+e_j}}{2}\!\!\int{\!\left({f(\eta^{x,x+e_j})}-{f(\eta)}\right)^2\nu_{\theta}(d\eta)}\,.$$ In the sequence, we will make use of the functional ${{\mathfrak D}}_N(\sqrt{f})$, where $f$ is a probability density with respect to $\nu_\theta$.
Replacement Lemma for beta<1
-------------------------------
Below, we define the local density of particles, which corresponds a to the mean occupation in a box around a given site. Abusing of notation, we denote by ${\varepsilon}N-1$ the integer part of ${\varepsilon}N-1$. For $\beta\in[0,1)$, we define the local mean by $$\label{localmean}
\eta^{\varepsilon N}(x)\;=\;\frac{1}{({\varepsilon}N)^d}\sum_{j_1, j_2, \ldots, j_d=0}^{\varepsilon N-1}\eta\left(x+j_1e_1+\ldots+j_de_d\right)\,.$$ Note that the sum on the right hand side of above may contain sites in and out of $\Lambda$ in the sense that $x/N\in \Lambda$ or $x/N\in \Lambda^\complement$. By ${{\mathcal O}}(f(N))$ we will mean a function bounded in modulus by a constant times $f(N)$.
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(2.75,0.75) node [$\frac{x}{N}$]{}; (4.75,2.75) node [$\frac{y}{N}$]{};
\[r1\] Fix $\beta\in [0,1)$. Let $f$ be a density with respect to the invariant measure $\nu_\theta$, $\lambda_N:{{\mathbb T}}_N^d\to{{\mathbb R}}$ a function such that $\Vert \lambda_N\Vert_\infty\leq M<\infty$ and $\gamma>0$. Then, $$\begin{aligned}
&\int \gamma N \sum_{x\in \Gamma_N}\lambda_N(x)\big\{\eta(x)-\eta^{\varepsilon N}(x)\big\}f(\eta)\nu_\theta(d\eta)\\
&\leq\; \frac{\gamma^2 M^2 {{\mathcal O}}(N^{d})}{2}\Big(\frac{N^{\beta-1}}{\alpha}+d{\varepsilon}\Big)+N^2{{\mathfrak D}}_N(\sqrt{f}) \, .\end{aligned}$$
By the definition of local mean $\eta^{\varepsilon N}(x)$, $$\begin{aligned}
&\int \lambda_N(x)\Big\{\eta(x)-\eta^{\varepsilon N}(x)\Big\}f(\eta)\nu_\theta(d\eta)\;=\;\nonumber\\
&= \int\lambda_N(x)\frac{1}{\varepsilon^d N^d}\sum_{j_1, \ldots, j_d=0}^{\varepsilon N-1}\Big\{\eta(x)-\eta(x+j_1e_1+\ldots+j_de_d)
\Big\}f(\eta)\nu_\theta(d\eta)\,.\label{eq51}\end{aligned}$$ The next step is to write $\eta(x)-\eta(x+j_1e_1+\cdots+j_d e_d)$ as a telescopic sum: $$\eta(x)-\eta(x+j_1e_1+\ldots+j_d e_d)\; =\; \sum_{\ell=1}^{j_1+\cdots+j_d}
\eta(a_{\ell-1})-\eta(a_{\ell})\,,$$ where $a_0=x$, $a_{j_1+\cdots+j_\ell}=x+j_1e_1+\cdots+j_d e_d$, and $\Vert a_{\ell-1}-a_\ell\Vert_1=1$ for any $\ell=1,\ldots, j_1+\cdots+j_d$. Note that the path $a_0,a_1,\ldots,a_{j_1+\cdots+j_\ell}$ depends on the initial point $x$ and the final point $x+j_1e_1+\cdots+j_d e_d$. See Figure \[Fig2\] for an illustration and keep in mind that the length of this path is bounded by $d{\varepsilon}N$. Inserting the previous equality into , we get $$\begin{aligned}
& \int\lambda_N(x)\frac{1}{(\varepsilon N)^d}\sum_{j_1, \ldots, j_d=0}^{\varepsilon N-1}\Big\{\sum_{\ell=1}^{j_1+\cdots+j_d}
\eta(a_{\ell-1})-\eta(a_{\ell})\Big\}f(\eta) \, \nu_\theta(d\eta)\,. \end{aligned}$$ Rewriting the expression above as twice the half and performing the transformation $\eta\mapsto \eta^{a_{\ell-1},a_\ell}$ for which the probability measure $\nu_\theta$ is invariant, expression above becomes: $$\label{eq555}
\frac{1}{2(\varepsilon N)^d}\sum_{j_1, \ldots, j_d=0}^{\varepsilon N-1}\sum_{\ell=1}^{j_1+\cdots+j_d}\int\lambda_N(x) \left(\eta(a_{\ell-1})-\eta(a_\ell)\right)\left(f\left(\eta^{a_\ell, a_{\ell-1}}\right)-f\left(\eta\right)\right) \, d\nu_\theta \, .$$ Since $ab=\sqrt{c}a{\genfrac{}{}{}{1}{b}{\sqrt{c}}}\leq{\genfrac{}{}{}{1}{1}{2}}ca^2+{\genfrac{}{}{}{1}{1}{2}}{\genfrac{}{}{}{1}{b^2}{c}}$, which holds for any $c>0$, the previous expression is smaller or equal than $$\begin{split}
& \frac{1}{2(\varepsilon N)^d} \sum_{j_1, \ldots, j_d=0}^{\varepsilon N-1}\sum_{\ell=1}^{j_1+\cdots+j_d}\Bigg[\frac{\xi^N_{a_{\ell-1}, a_{\ell}}}{2A}\int\left(\sqrt{f\left(\eta^{a_\ell, a_{\ell-1}}\right)}-\sqrt{f\left(\eta\right)}\right)^2 d\nu_\theta\\
&+\frac{A}{2\xi^N_{a_{\ell-1}, a_\ell}}\int \lambda_N^2(x) \left(\eta(a_\ell)-\eta(a_{\ell-1})\right)^2\Big(\sqrt{f\left(\eta^{a_\ell, a_{\ell-1}}\right)}+\sqrt{f\left(\eta\right)}\Big)^2 d\nu_\theta\Bigg]\,.
\end{split}$$ Summing over $x\in \Gamma_N$, we can bound the last expression by $$\begin{split}
& \frac{1}{2(\varepsilon N)^d} \sum_{x\in\Gamma_N}\sum_{j_1, \ldots, j_d=0}^{\varepsilon N-1}\sum_{\ell=1}^{j_1+\cdots+j_d}\Bigg[\frac{\xi^N_{a_{\ell-1}, a_{\ell}}}{2A}\int\left(\sqrt{f\left(\eta^{a_\ell, a_{\ell-1}}\right)}-\sqrt{f\left(\eta\right)}\right)^2 d\nu_\theta\\
&+\sum_{x\in\Gamma_N}\frac{A}{2\xi^N_{a_{\ell-1}, a_\ell}}\int \lambda_N^2(x) \left(\eta(a_\ell)-\eta(a_{\ell-1})\right)^2\Big(\sqrt{f\left(\eta^{a_\ell, a_{\ell-1}}\right)}+\sqrt{f\left(\eta\right)}\Big)^2 d\nu_\theta\Bigg]\,.
\end{split}$$ Recalling , we can bound the first parcel in the sum above by $$\begin{aligned}
&\frac{1}{2(\varepsilon N)^d}\sum_{j_1, \ldots, j_d=0}^{\varepsilon N-1}\frac{1}{A}{{\mathfrak D}}_N(\sqrt{f}) \;=\; \frac{1}{2A}{{\mathfrak D}}_N(\sqrt{f}) \,.\end{aligned}$$ Since $f$ is a density and $|\lambda_N(x)|\leq M$, the second parcel is bounded by $$\begin{aligned}
\label{eeee}
\begin{split}
&\frac{1}{2(\varepsilon N)^d}\sum_{x\in\Gamma_N}\sum_{j_1, \ldots, j_d=0}^{\varepsilon N-1}\sum_{\ell=1}^{j_1+\cdots+j_d} \frac{A}{2}\cdot \frac{4M^2}{\xi^N_{a_{\ell-1}, a_\ell}}\\
&\leq\;\frac{1}{(\varepsilon N)^d}\sum_{j_1, \ldots, j_d=0}^{\varepsilon N-1}AM^2{{\mathcal O}}(N^{d-1})\Big(\frac{N^\beta}{\alpha}+d\varepsilon N\Big)\\
&=\; AM^2{{\mathcal O}}(N^{d-1})\Big(\frac{N^\beta}{\alpha}+d\varepsilon N\Big) \,.
\end{split}\end{aligned}$$ Up to here we have achieved that $$\begin{aligned}
&\int \sum_{x\in \Gamma_N}\lambda_N(x)\big\{\eta(x)-\eta^{\varepsilon N}(x)\big\}f(\eta)\nu_\theta(d\eta)\\
&\;\leq\; AM^2 {{\mathcal O}}(N^{d-1})\Big(\frac{N^\beta}{\alpha}+d\varepsilon N\Big)+\frac{1}{2A}{{\mathfrak D}}_N(\sqrt{f}) \, .\end{aligned}$$ We point out that the quantity of sites on $\Gamma_N$ is of order ${{\mathcal O}}(N^{d-1}$), which is a consequence of the fact that ${\partial}\Lambda$ is a smooth surface of dimension $d-1$. Then, multiplying the inequality above by $\gamma N$ gives us $$\begin{aligned}
&\int \gamma N\sum_{x\in \Gamma_N}\lambda_N(x)\big\{\eta(x)-\eta^{\varepsilon N}(x)\big\}f(\eta)\nu_\theta(d\eta)\\
& \;\leq\; A\gamma {{\mathcal O}}(N^{d}) M^2\Big[\frac{N^\beta}{\alpha}+d\varepsilon N\Big]+\frac{\gamma N}{2A}{{\mathfrak D}}_N(\sqrt{f}) \,. \end{aligned}$$ Now choosing $A=\gamma N^{-1}/2$ the proof ends.
Recall the definition of $\Gamma_N$ in .
\[replacement\] Fix $\beta\in [0,1)$. Let $\lambda_N:{{\mathbb T}}_N^d\to{{\mathbb R}}$ be a sequence of functions such that $\Vert \lambda_N\Vert_\infty\leq M<\infty$. Then, $$\overline{\lim_{\varepsilon\rightarrow0}} \varlimsup_{N\rightarrow\infty} {{\mathbb E}}^\beta_{\mu_N}\Big[\,\Big|\int_0^t \frac{1}{N^{d-1}} \sum_{x\in\Gamma_N} \lambda_N(x) \{\eta_s^{\varepsilon N}(x)-\eta_s(x)\}\,ds\,\Big|\,\Big]\;=\;0 \, .$$
Using the variational formula for entropy, for any $\gamma\in{{\mathbb R}}$ (which will be chosen large *a posteriori*), $$\begin{aligned}
&{{\mathbb E}}^\beta_{\mu_N}\Big[\,\Big|\int_0^t\frac{1}{N^{d-1}} \sum_{x\in\Gamma_N}\lambda_N(x)\{\eta_s(x)-\eta_s^{\varepsilon N}(x)\}ds\Big|\,\Big] \notag \\
&=\frac{1}{\gamma N^{d}}{{\mathbb E}}^\beta_{\mu_N}\Big[\gamma N\,\Big|\int_0^t\sum_{x\in\Gamma_N}\lambda_N(x)\{\eta_s(x)-\eta_s^{\varepsilon N}(x)\}ds\Big|\Big] \notag \\
& \label{ent} \leq \frac{H_N(\mu_N|\nu_\theta)}{ \gamma N^d} + \frac{1}{\gamma N^d} \log {{\mathbb E}}_{\nu_\theta}\Big[\exp\Big(\gamma N\Big|\int_0^t\sum_{x\in\Gamma_N}\lambda_N(x)\{\eta_s(x)-\eta_s^{\varepsilon N}(x)\}ds\Big|\Big)\Big] . \end{aligned}$$ By the estimate on the entropy, the first parcel of above is negligible as $N\rightarrow\infty$ since we will choose $\gamma$ arbitrarily large. Therefore, we can focus on the second parcel. Using that $e^{|x|}\leq e^x+e^{-x}$ and $$\label{A.2}
\varlimsup_{N\rightarrow\infty}\frac{1}{N^d}\log (a_N+b_N)\;=\;\max \Big\{\varlimsup_{N\rightarrow\infty}\frac{1}{N^d}\log a_N, \, \varlimsup_{N\rightarrow\infty}\frac{1}{N^d} \log b_N\Big\}$$ for any sequences $a_N,b_N>0$, one can see that the second parcel on the right hand side of is less than or equal to the sum of $$\label{eq57b}
\varlimsup_{N\rightarrow\infty}\frac{1}{ \gamma N^d}\log\Big\{ {{\mathbb E}}_{\nu_\theta}\Big[\exp\Big(\gamma N\int_0^t\sum_{x\in\Gamma_N}\lambda_N(x)\{\eta_s(x)-\eta_s^{\varepsilon N}(x)\}ds\Big)\Big]\Big\}$$ and $$\label{eq58b}
\varlimsup_{N\rightarrow\infty}\frac{1}{ \gamma N^d}\log\Big\{ {{\mathbb E}}_{\nu_\theta}\Big[\exp\Big(-\gamma N\int_0^t\sum_{x\in\Gamma_N}\lambda_N(x)\{\eta_s(x)-\eta_s^{\varepsilon N}(x)\}ds\Big)\Big]\Big\} \, .$$ We handle only , being analogous. By Feynman-Kac’s formula, see [@kl Appendix 1, Lemma 7.2], expression is bounded by $$\varlimsup_{N\rightarrow\infty}\frac{1}{\gamma N^d} \log\Big\{ \exp\Big(\int_0^t \Phi_N\, ds\Big)\Big\}\;=\; \varlimsup_{N\rightarrow\infty}\frac{t\,\Phi_N^1 }{\gamma N^{d}}\, ,$$ where $$\begin{aligned}
\Phi_N^1
&=\sup_{f \ \textrm{density}}\left\{\int\gamma N\sum_{x\in\Gamma_N}\lambda_N(x)\{\eta(x)-\eta^{\varepsilon N}(x)\}f(\eta)\nu_\theta(d\eta)-N^2{{\mathfrak D}}_N(\sqrt{f})\right\}\,.\end{aligned}$$ Applying Lemma \[r1\] finishes the proof.
Replacement Lemma for beta>1 {#subsec5.2}
-------------------------------
Here, some additional notation is required. The idea is actually very simple: the local mean shall be over a region avoiding slow bonds. Let $B_N[x,\ell]\subset {{\mathbb T}}^d_N$ be the discrete box centered on $x\in {{\mathbb T}}^d_N$ which edge has size $2\ell$, that is, $B_N[x,\ell]= \{y\in {{\mathbb T}}^d_N: \Vert y-x\Vert_\infty\leq \ell\}$, where we have written $\Vert \cdot \Vert_\infty$ for the supremum norm on ${{\mathbb T}}^d_N$, that is, $\Vert (x_1,\ldots,x_d)\Vert_\infty = \max\big\{|x_1|\wedge |N-x_1|,\ldots,|x_d|\wedge |N-x_d|\big\}$.
(1,2.2) to\[out=45,in=225\] (3,3.2) to\[out=45,in=0\] (3,0.2) to\[out=180,in=225\] (1,2.2); (2,0) rectangle (4,2);
(1,2) to\[out=45,in=225\] (3,3) to\[out=45,in=0\] (3,0) to\[out=180,in=225\] (1,2);
(1.25,1.75) node [$\Lambda$]{}; (4.25,3.25) node [$\Lambda^\complement$]{}; (0,-1) grid (5,4); (5.7,3.75) node\[above\][$N^{-1}{{\mathbb T}}^d_N$]{}; (3,1) circle (0.1); (2.75,0.75) node [$\frac{x}{N}$]{};
Let $\Lambda_N =\{x\in {{\mathbb T}}^d_N: \frac{x}{N}\in \Lambda\}$ the set of sites in $\frac{1}{N}{{\mathbb T}}^d_N$ belonging to $\Lambda$. We define now the region $C_N[x,\ell]\subset {{\mathbb T}}^d_N$ by $$\label{counterpart}
C_N[x,\ell]\;:=\; \begin{cases}
B_N[x,\ell] \cap \Lambda_N & \text{if }\frac{x}{N}\in \Lambda\,, \medskip \\
B_N[x,\ell] \cap \Lambda_N^\complement& \text{if }\frac{x}{N}\in \Lambda^\complement\,,
\end{cases}$$ see Figure \[Fig3\] for an illustration. For $\beta\in[1,\infty]$, we define the local density as the average over $C_N[x,\ell]$, that is, $$\label{localmean1}
\eta^{\varepsilon N}(x)\;:=\;\frac{1}{\#C_N[x,{\varepsilon}N]}\sum_{y\in C_N[x,{\varepsilon}N]}\eta(y)\,.$$
\[r2\] Fix $\beta\in [1,\infty]$. Let $f$ be a density with respect to the invariant measure $\nu_\theta$, let $\lambda_N:{{\mathbb T}}_N^d\to{{\mathbb R}}$ a function such that $\Vert \lambda_N\Vert_\infty\leq M<\infty$ and $\gamma>0$. Then, the following inequalities hold: $$\label{ineq5.10}
\begin{split}
&\int \gamma N\! \sum_{x\in \Gamma_N}\!\lambda_N(x)\big\{\eta(x)-\eta^{\varepsilon N}(x)\big\}f(\eta)\nu_\theta(d\eta)\;\leq\;{\genfrac{}{}{}{1}{1}{2}}\gamma^2 M^2 {{\mathcal O}}(N^{d})d{\varepsilon}+N^2{{\mathfrak D}}_N(\sqrt{f})
\end{split}$$ and $$\label{ineq5.11}
\begin{split}
&\int \gamma \sum_{x\in{{\mathbb T}}^d_N} \lambda_N(x)\{\eta(x)-\eta^{\varepsilon N}(x)\}f(\eta)\nu_\theta(d\eta)\;\leq\; {\genfrac{}{}{}{1}{1}{2}}\gamma^2 M^2 {{\mathcal O}}(N^{d-1})d{\varepsilon}+N^2{{\mathfrak D}}_N(\sqrt{f})\,.
\end{split}$$
Let us prove the inequality . As commented in the beginning of this subsection, the local average $\eta^{{\varepsilon}N}$ is taken over $C_N[x,{\varepsilon}N]$. Thus, we can write $$\begin{aligned}
&\int\lambda_N(x)\{\eta(x)-\eta^{\varepsilon N}(x)\}f(\eta)\nu_\theta(d\eta)\nonumber\\
&=\int \lambda_N(x)\Big\{\frac{1}{\#C_N[x,{\varepsilon}N]}\sum_{y\in C_N[x,{\varepsilon}N]}\big(\eta(x)-\eta(y)\big)\Big\}f(\eta)\nu_\theta(d\eta)\,.\label{eqtag}\end{aligned}$$ For each $y\in C[x,{\varepsilon}N]$, let $\gamma(x,y)$ be a polygonal path of minimal length connecting $x$ to $y$ which does not crosses ${\partial}\Lambda$. That is, $\gamma(x,y)$ is a sequence of sites $(a_0,\ldots,a_M)$ such that $x=a_0$, $y=a_M, \Vert a_i-a_{i+1}\Vert_1 =1$ and $\xi_{a_,a_{i+1}}=1$ for $i=0,\ldots,M-1$, and $\gamma(x,y)$ has minimal length, that is, $M=M(x,y)=\Vert x-y\Vert_1+1$. Now we repeat the steps in the proof of Lemma \[r1\], observing that in this case the sum will be over ${{\mathbb T}}^d_N$, obtaining that is bounded from above by $$\begin{aligned}
& \frac{1}{2\#C_N[x,{\varepsilon}N]}\sum_{x\in{{\mathbb T}}^d_N}\sum_{y\in C_N[x,{\varepsilon}N]}\sum_{\ell=1}^{M(x,y)-1}\Bigg[\frac{1}{2A} \int\Big(\sqrt{f(\eta^{a_\ell, a_{\ell-1}})}-\sqrt{f(\eta)}\Big)^2 \,d\nu_\theta\Bigg.\\
&+\frac{A}{2}\int\Big( \lambda_N(x)\Big)^2(\eta(a_\ell)-\eta(a_{\ell-1}))^2\Big(\sqrt{f(\eta^{a_\ell, a_{\ell-1}})}+\sqrt{f(\eta)}\Big)^2 \, d\nu_\theta\Bigg] \,.\end{aligned}$$ We can bound the first parcel in the sum above by $\frac{1}{2A}{{\mathfrak D}}_N(\sqrt{f})$ and the second parcel by $$\begin{aligned}
\begin{split}
&\frac{1}{2\#C_N[x,{\varepsilon}N]}\sum_{x\in{{\mathbb T}}^d_N}\sum_{y\in C_N[x,{\varepsilon}N]}\sum_{\ell=1}^{M(x,y)-1} \frac{4AM^2}{2}\\
& \leq\;\frac{1}{\#C_N[x,{\varepsilon}N]}\sum_{y\in C_N[x,{\varepsilon}N]}AM^2{{\mathcal O}}(N^{d})d\varepsilon N\;=\; AM^2{{\mathcal O}}(N^{d})d\varepsilon N\,.
\end{split}\end{aligned}$$ We hence have $$\begin{aligned}
\int \sum_{x\in {{\mathbb T}}^d_N}\lambda_N(x)\big\{\eta(x)-\eta^{\varepsilon N}(x)\big\}f(\eta)\nu_\theta(d\eta)\;\leq\; AM^2 {{\mathcal O}}(N^{d})d\varepsilon N+\frac{1}{2A}{{\mathfrak D}}_N(\sqrt{f}) \, .\end{aligned}$$ Then, multiplying the inequality above by $\gamma$ gives us $$\begin{aligned}
\int \gamma \sum_{x\in {{\mathbb T}}^d_N}\lambda_N(x)\big\{\eta(x)-\eta^{\varepsilon N}(x)\big\}f(\eta)\nu_\theta(d\eta)\;\leq\; A\gamma {{\mathcal O}}(N^{d}) M^2 d\varepsilon N+\frac{\gamma}{2A}{{\mathfrak D}}_N(\sqrt{f}) \,. \end{aligned}$$ Now choosing $A=\gamma N^{-2}/2$ the proof of ends. The proof of inequality similar to the proof of Lemma \[r1\], under the additional feature that rates of bonds over a path connecting two sites will be always equal to one, which facilitates the argument.
\[replacement2\] Fix $\beta\in [1,\infty]$. Let $\lambda_N:{{\mathbb T}}_N^d\to{{\mathbb R}}$ be a sequence of functions such that $\Vert \lambda_N\Vert_\infty\leq c<\infty$. Then, $$\varlimsup_{\varepsilon\rightarrow0} \varlimsup_{N\rightarrow\infty} {{\mathbb E}}^\beta_{\mu_N}\Big[\Big|\int_0^t \frac{1}{N^{d-1}}\sum_{x\in\Gamma_N} \lambda_N(x) \{\eta_s^{\varepsilon N}(x)-\eta_s(x)\}\,ds\Big|\Big]\;=\;0$$ and $$\varlimsup_{\varepsilon\rightarrow0} \varlimsup_{N\rightarrow\infty} {{\mathbb E}}^\beta_{\mu_N}\Big[\Big|\int_0^t\frac{1}{N^{d}} \sum_{x\in{{\mathbb T}}^d_N}\lambda_N(x)\{\eta_s^{\varepsilon N}(x)-\eta_s(x)\}\,ds\Big|\Big]\;=\;0 \, .$$
The proof is similar to the one of Lemma \[replacement\], being sufficient to show that expressions $$\begin{aligned}
\Phi_N^2\;&:=\;\sup_{f\, \textrm{density}} \Big\{\int\gamma N\sum_{x\in\Gamma_N}\lambda_N(x)\{\eta^{\varepsilon N}(x)-\eta(x)\}f(\eta)d\nu_\theta - N^{2}{{\mathfrak D}}_N(\sqrt{f})\Big\},\\
\Phi_N^3\;&:=\;\sup_{f\, \textrm{density}} \Big\{\int\gamma\sum_{x\in {{\mathbb T}}^d_N}\lambda_N(x)\{\eta^{\varepsilon N}(x)-\eta(x)\}f(\eta)d\nu_\theta - N^{2}{{\mathfrak D}}_N(\sqrt{f})\Big\}\end{aligned}$$ satisfy $$\begin{aligned}
&\lim_{N\to\infty}\frac{ t\Phi_N^2}{\gamma N^d}\;=\;0\qquad \text{and} \qquad
\lim_{N\to\infty}\frac{ t\Phi_N^3}{\gamma N^d}\;=\;0\,,\end{aligned}$$ which is a consequence of Lemma \[r2\], finishing the proof.
Energy Estimates
----------------
In this subsection, consider $\beta\in[1,\infty]$. Our goal here is to prove that any limit point ${{\mathbb Q}}^\beta_*$ of the sequence $\{{{\mathbb Q}}^{\beta, N}_{\mu_N}:N>1\}$ is concentrated on trajectories $\rho(t,u) du$ with *finite energy*, meaning that $\rho(t,u)$ belongs to a suitable Sobolev space.
This result plays a both role in the uniqueness of weak solutions of and in the characterization of limit points. The fact that ${{\mathbb Q}}^\beta_*$ is concentrated in trajectories with density with respect to the Lebesgue measure of the form $\rho(t,u) du$, with $0\leq \rho\leq 1$, is a consequence of maximum of one particle per site, see [@kl]. The issue here is to prove that the density $\rho(t,u)$ belongs to the Sobolev space ${L^2\big([0,T];\mathcal{H}^1({{\mathbb T}}^d\backslash \partial\Lambda)\big)}$, see Section \[s2\] for its definition.
Assume without loss of generality that the entire sequence $\{{{\mathbb Q}}^{\beta,N}_{\mu_N}: \, N\geq1\}$ weakly converges to ${{\mathbb Q}}^\beta_*$. Let $B[u,{\varepsilon}]:= \{r\in {{\mathbb T}}^d: \Vert r-u\Vert_\infty<{\varepsilon}\}$ and $$C[u,{\varepsilon}]\;:=\; \begin{cases}
B[u,{\varepsilon}] \cap \Lambda & \text{if }u\in \Lambda\,, \\\medskip
B[u,{\varepsilon}] \cap \Lambda^\complement& \text{if }u\in \Lambda^\complement\,,
\end{cases}$$ where we have written $\Vert \cdot \Vert_\infty$ for the supremum norm on the continuous torus ${{\mathbb T}}^d=[0,1)^d$, that is, $\Vert (u_1,\ldots,u_d)\Vert_\infty = \max\big\{|u_1|\wedge |1-u_1|,\ldots,|u_d|\wedge |1-u_d|\big\}$. See Figure \[Fig4\] for an illustration.
(1,2.2) to\[out=45,in=225\] (3,3.2) to\[out=45,in=0\] (3,0.2) to\[out=180,in=225\] (1,2.2); (2,0) rectangle (4,2);
(1,2) to\[out=45,in=225\] (3,3) to\[out=45,in=0\] (3,0) to\[out=180,in=225\] (1,2);
(1.25,1.75) node [$\Lambda$]{}; (4.25,3.25) node [$\Lambda^\complement$]{}; (0,-1) rectangle (5,4); (5.5,3.75) node\[above\][${{\mathbb T}}^d$]{}; (3,1) circle (0.08); (2.75,0.75) node [$u$]{}; (2,0) rectangle (4,2);
We define an approximation of the identity $\iota_{\varepsilon}$ in the continuous torus ${{\mathbb T}}^d$ by $$\label{approxidentity}
\iota_{\varepsilon}(u,v)\;:=\; \frac{1}{|C[u,{\varepsilon}]|} \mathbf{1}_{C[u, \varepsilon]}(v)\,,$$ where $|C[u,{\varepsilon}]|$ above denotes the Lebesgue measure of the set $C[u,{\varepsilon}]$. Recall that the convolution of a measure $\pi$ with $\iota_{\varepsilon}$ is defined by $$\label{convolution}
(\pi\ast\iota_{\varepsilon})(u)\;=\;\int_{{{\mathbb T}}^d}\iota_{\varepsilon}(u,v)\pi(dv)\quad \text{ for any }u\in {{\mathbb T}}^d\,.$$ Given a function $\rho$, the convolution $\rho\ast\iota_{\varepsilon}$ shall be understood as the convolution of the measure $\rho(v) dv$ with $\iota_{\varepsilon}$. An important remark now is the equality $$\begin{aligned}
\label{remark}
(\pi^N_t\ast\iota_{\varepsilon})\big({\genfrac{}{}{}{1}{x}{N}}\big)\;=\;\eta_t^{\varepsilon N}(x)+{{\mathcal O}}\big(({\varepsilon}N)^{1-d} \big)\,,\end{aligned}$$ where $\eta^{{\varepsilon}N}_t$ has been defined in , being the small error above due to the fact that sites on the boundary of $C_N[x,\ell]$ may or may not belong to $C[u,{\varepsilon}]$ when taking $u=x/N$ and $\ell={\varepsilon}N$. Given a function $H:{{\mathbb T}}^d\rightarrow {{\mathbb R}}$, let $$\label{eq5.16}
V_N(\varepsilon, j, H, \eta):=\frac{1}{ N^d}\!\! \sum_{x\in{{\mathbb T}}^d_N}\!\!H\big({\genfrac{}{}{}{1}{x}{N}}\big)\frac{\{\eta(x)-\eta(x+\varepsilon N e_j)\}}{{\varepsilon}}-\frac{2}{N^d}\!\sum_{x\in{{\mathbb T}}^d_N}\!\!\Big(H\big({\genfrac{}{}{}{1}{x}{N}}\big)\Big)^2.$$
\[lemma5.5\] Consider $H_1, \ldots, H_k$ functions in $C^{0,1}([0,T]\times {{\mathbb T}}^d)$ with compact support contained in $[0,T]\times ({{\mathbb T}}^d\backslash\partial\Lambda)$. Hence, for every $\varepsilon >0$ and $j=1,\ldots,d$, $$\label{5.8e}
\varlimsup_{\delta\rightarrow0} \varlimsup_{N\rightarrow\infty}{{\mathbb E}}_{\mu_N}^{\beta}\Big[\max_{1\leq i \leq k}\Big\{\int_0^T V_N(\varepsilon, j, H_i(s, \cdot), \eta_s^{\delta N})\,ds\Big\}\Big]\;\leq \;\kappa_0\,,$$ where $\kappa_0$ has been defined in .
Provided by Lemma \[replacement2\], it is enough to prove that $$\label{using}
\varlimsup_{N\rightarrow\infty}\,\,{{\mathbb E}}^\beta_{\mu_N}\Big[\max_{1\leq i \leq k}\Big\{\int_0^t V_N(\varepsilon, j, H_i(s,\cdot), \eta_s)\,ds\Big\}\Big]\;\leq\; \kappa_0\, .$$ By the entropy inequality, for each fixed $N$, the expectation above is smaller than $$\begin{aligned}
\frac{H(\mu^N|\nu_\theta)}{N^d}+\frac{1}{N^d}\log {{\mathbb E}}_{\nu_\theta}\Big[\exp \Big\{\max_{1\leq i \leq k} N^d\Big\{\int_0^T V_N(\varepsilon, j, H_i(s, \cdot), \eta_s)\,ds\Big\}\Big\}\Big]\,.\end{aligned}$$ Using , we bound the first parcel above by $\kappa_0$. Since $\exp\big\{\max_{1\leq i \leq k} a_j\big\}\leq \sum_{1\leq i \leq k}\exp\{a_j\}$ and by , we conclude that the limsup as $N\uparrow\infty$ of the second parcel above is less than or equal to $$\begin{aligned}
&\varlimsup_{N\rightarrow\infty} \frac{1}{N^d}\log {{\mathbb E}}_{\nu_\theta}\Big[\sum_{1\leq i \leq k}\exp \Big\{ N^d \int_0^T V_N(\varepsilon, j, H_i(s, \cdot), \eta_s)\,ds\Big\}\Big]\\
&=\max_{1\leq i \leq k} \varlimsup_{N\rightarrow\infty} \frac{1}{N^d}\log {{\mathbb E}}_{\nu_\theta}\Big[\exp \Big\{ N^d \int_0^T V_N(\varepsilon, j, H_i(s, \cdot), \eta_s)\,ds\Big\}\Big]\,.\end{aligned}$$ Thus, in order to conclude the proof, it is enough to show that the limsup above is non positive for each $i=1,\ldots,k$. By the Feynman-Kac formula (see [@kl p. 332, Lemma 7.2]) for each fixed $N$ and $d\geq 2$, $$\begin{aligned}
& \frac{1}{N^d}\log {{\mathbb E}}_{\nu_\theta}\Big[\exp \Big\{ N^d \int_0^T V_N(\varepsilon, j, H_i(s, \cdot), \eta_s)\,ds\Big\}\Big]\label{eq5.18} \\
&\leq \int_0^T \sup_f \Big\{\int V_N(\varepsilon, j, H_i(s, \cdot), \eta)f(\eta) d \nu_\theta -N^{2-d}{{\mathfrak D}}_N(\sqrt{f})\Big\}\,ds\,,\label{eq5.19}\end{aligned}$$ where the supremum above is taken over all probability densities $f$ with respect to $\nu_\theta$. By assumption, each of the functions $\{H_i : i=1,\ldots, k\}$ vanishes in a neighborhood of $\partial\Lambda$. Thus, we make following observation about the first sum in the RHS of : for small ${\varepsilon}$, non-zero summands are such that $x/N$ and $(x+{\varepsilon}Ne_j)N$ lay both in $\Lambda$ or both in $\Lambda^\complement$. Henceforth, in such a case, it is possible to find a path no slow bonds connecting $x$ and $x+{\varepsilon}Ne_j$. Keeping this in mind, we can repeat the arguments in the proof of Lemma \[r2\] to deduce that $$\begin{aligned}
& \int \frac{1}{N^d} \sum_{x\in{{\mathbb T}}^d_N}H\big({\genfrac{}{}{}{1}{x}{N}}\big)\frac{\{\eta(x)-\eta(x+\varepsilon N e_j)\}}{{\varepsilon}} f(\eta) d \nu_\theta\\
&\leq N^{2-d}{{\mathfrak D}}_N(\sqrt{f})+ \frac{2}{N^d}\sum_{x\in{{\mathbb T}}^d_N}\Big(H\big({\genfrac{}{}{}{1}{x}{N}}\big)\Big)^2\,.
\end{aligned}$$ Plugging this inequality into implies that has a nonpositive limsup, showing and therefore finishing the proof.
\[5.7\] $${{\mathbb E}}_{{{\mathbb Q}}^\beta_*}\left[\sup_{H}\left\{\int_0^T\!\!\int_{{{\mathbb T}}^d}(\partial_{u_j}H)(s,u)\rho(s,u)duds-2\int_0^T\!\!\int_{{{\mathbb T}}^d}\left(H(s,u)\right)^2duds\right\}\right]\;\leq\; \kappa_0\,,$$ where the supremum is carried over all functions $H\in C^{0,1}([0,T]\times{{\mathbb T}}^d)$ with compact support contained in $[0,T]\times ({{\mathbb T}}^d\backslash \partial\Lambda)$.
Consider a sequence $\{H_i:\,i\geq1\}$ dense in the subset of $C^2([0,t]\times{{\mathbb T}}^d)$ of functions with support contained in $[0,T]\times({{\mathbb T}}^d\backslash {\partial}\Lambda)$, being the density with respect to the norm $\Vert H\Vert_\infty+\Vert{\partial}_u H\Vert_\infty$. Recall we are assuming that $\{{{\mathbb Q}}_{\mu_N}^{\beta, N}: N\geq 1\}$ converges to ${{\mathbb Q}}^\beta_*$. Then, by and the Portmanteau Theorem, $$\begin{aligned}
\varlimsup_{\delta\rightarrow0}{{\mathbb E}}_{{{\mathbb Q}}_{*}^{\beta}}&\Big[\max_{1\leq i\leq k}\Big\{\frac{1}{\varepsilon}\int_0^T\int_{{{\mathbb T}}^d}H_i(s,u) )\{\rho_s^{\delta}(u)-\rho_s^{\delta}(u+\varepsilon e_j)\}\,duds\\
&-2\int_0^T\int_{{{\mathbb T}}^d}(H_i(s,u))^2\,duds\Big\}\Big]\;\leq\; \kappa_0,\end{aligned}$$ where $\rho_s^\delta(u)=(\rho_s\ast \iota_\delta)(u)$ as defined in . Letting $\delta\downarrow0$, the Lebesgue Differentiation Theorem assures that $\rho_s^\delta(u)$ converges almost surely to $\rho_s$. Then, performing a change of variables and letting $\varepsilon\downarrow0$, we obtain that $${{\mathbb E}}_{{{\mathbb Q}}_{*}^{\beta}}\Big[\max_{1\leq i\leq k}\Big\{\int_0^T\!\!\int_{{{\mathbb T}}^d} ({\partial}_{u_j}H_i(s,u)) \rho_s(u)\,duds-2\int_0^T\!\!\int_{{{\mathbb T}}^d}(H_i(s,u))^2\,duds\Big\}\Big]\leq \kappa_0.$$ Since the maximum increases to the supremum, we conclude the lemma by applying the Monotone Convergence Theorem to $\{H_i:\, i\geq1\}$, which is a dense sequence in the subset of functions $C^2([0,T]\times{{\mathbb T}}^d)$ with compact support contained in $[0,T]\times {{\mathbb (}}T^d \backslash {\partial}\Lambda)$.
\[Prop5.7\] The measure ${{\mathbb Q}}^\beta_*$ is concentrated on paths $\pi(t,u)=\rho(t,u)du$ such that $\rho\in {L^2\big([0,T];\mathcal{H}^1({{\mathbb T}}^d\backslash \partial\Lambda)\big)}$.
Denote by $\ell:C^2([0,T]\times{{\mathbb T}}^d)\rightarrow {{\mathbb R}}$ the linear functional defined by $$\ell(H)\;=\;\int_0^T\int_{{{\mathbb T}}^d} (\partial_{u_j}H)(s,u)\rho(s,u)\, du\,ds\,.$$ Since the set of functions $H\in C^2([0,T]\times{{\mathbb T}}^d)$ with support contained in $[0,T]\times({{\mathbb T}}^d\backslash\partial\Lambda)$ is dense in $L^2([0,T]\times{{\mathbb T}}^d)$ and since by Lemma \[5.7\] $\ell$ is a ${{\mathbb Q}}^\beta_*$-a.s. bounded functional in $C^2([0,T]\times{{\mathbb T}}^d)$, we can extend it to a ${{\mathbb Q}}^\beta_*$-a.s. bounded functional in $L^2([0,T]\times{{\mathbb T}}^d)$, which is a Hilbert space. Then, by the Riesz Representation Theorem, there exists a function $G\in L^2([0,T]\times{{\mathbb T}}^d)$ such that $$\ell(H)\;=\;-\int_0^T\int_{{{\mathbb T}}^d} H(s,u)G(s,u)\, du\,ds\,,$$ concluding the proof.
Characterization of limit points {#s6}
================================
Before going into the details of each regime $\beta\in[0,1)$, $\beta=1$ or $\beta\in(1,\infty]$, we make some useful considerations for all cases.
We will prove in this section that all limit points of the sequence $\{{{\mathbb Q}}^{\beta,N}_{\mu_N}: \, N\geq1\}$ are concentrated on trajectories of measures $\pi(t,du)=\rho(t,u)\,du$, whose density $\rho(t,u)$ with respect to the Lebesgue measure is the weak solution of the hydrodynamic equation , or for each corresponding value of $\beta$. Provided by tightness, let ${{\mathbb Q}}_*^\beta$ be a limit point of the sequence $\{{{\mathbb Q}}^{\beta,N}_{\mu_N}: \, N\geq1\}$ and assume, without loss of generality, that $\{{{\mathbb Q}}^{\beta,N}_{\mu_N}: \, N\geq1\}$ converges to ${{\mathbb Q}}^\beta_*$.
Since there is at most one particle per site, it is easy to show that ${{\mathbb Q}}_*^\beta$ is concentrated on trajectories $\pi(t,du)$ which are absolutely continuous with respect to the Lebesgue measure $\pi(t,du)=\rho(t,u)\,du$ and whose density $\rho(t,\cdot)$, is nonnegative and bounded by one. Recall the martingale $M_t^N(H)$ in .
\[bm\] If
a) $\beta\in[0,1)$ and $H\in C^2({{\mathbb T}}^d)$, or
b) $\beta\in [1,\infty]$ and $H\in C^2({{\mathbb T}}^d \backslash {\partial}\Lambda)$,
then, for all $\delta>0$, $$\label{quamarprop}
\lim_{N\rightarrow\infty} {{\mathbb P}}^N_{\mu_N}\Big[\sup_{0\leq t\leq T}|M_t^N(H)|>\delta\Big]\;=\;0\, .$$
Item a) has been already proved in . For item b), recalling note that $$\label{61}
{\langle}M^{N}(H){\rangle}_t \;\leq\; \frac{T}{N^{2d-2}}\sum_{j=1}^d\sum_{x\in{{\mathbb T}}_{N}^d} \xi^{N}_{x,x+e_j}\Big[H({\genfrac{}{}{}{1}{x+e_j}{N}})-H({\genfrac{}{}{}{1}{x}{N}})\Big]^2\, .$$ Since $H\in C^2({{\mathbb T}}^d \backslash {\partial}\Lambda)$, $H$ is differentiable with bounded derivative except over ${\partial}\Lambda$. Therefore, if the edge $x, \, x+e_j$ is not a slow bond, then $$\label{62}
\xi^{N}_{x,x+e_j}\Big[H({\genfrac{}{}{}{1}{x+e_j}{N}})-H({\genfrac{}{}{}{1}{x}{N}})\Big]^2 \;\leq\; \frac{1}{N^2}\Vert{\partial}_{u_j}H\Vert^2_\infty \, .$$ On the other hand, if the edge $x, \, x+e_j$ is a slow bond, then $$\begin{aligned}
\label{63}
\xi^{N}_{x,x+e_j}\Big[H({\genfrac{}{}{}{1}{x+e_j}{N}})-H_t({\genfrac{}{}{}{1}{x}{N}})\Big]^2 &\;\leq\; \frac{4\alpha \Vert H\Vert_{\infty}^2}{N^{\beta}} \,.\end{aligned}$$ Since the number of slow bonds is of order ${{\mathcal O}}(N^{d-1})$, plugging and into gives us ${\langle}M^{N}(H_t){\rangle}_t\leq {{\mathcal O}}(1/N^d)$. ’ Then, Doob’s inequality concludes the proof.
Characterization of limit points for beta em \[0,1). {#6.1}
----------------------------------------------------
\[6.1.1\] Let $H\in C^2({{\mathbb T}}^d)$. Then, for any $\delta>0$, $${{\mathbb Q}}^\beta_*\Big[\pi.:\, \sup_{0\leq t\leq T}\Big|{\langle}\pi_t, H{\rangle}\,-\, {\langle}\pi_0, H {\rangle}\,-\, \int_0^t \, {\langle}\pi_s ,\Delta H {\rangle}\,ds\Big| >\delta\Big]\;=\;0\, .$$
Since ${{\mathbb Q}}^{\beta, N}_{\mu_N}$ converges weakly to $Q^\beta_*$, by Portmanteau’s Theorem (see [@bili Theorem 2.1]), $$\begin{aligned}
&{{\mathbb Q}}^\beta_*\Big[\pi.:\, \sup_{0\leq t\leq T}\Big|{\langle}\pi_t, H{\rangle}\,-\, {\langle}\pi_0, H {\rangle}\,-\, \int_0^t \, {\langle}\pi_s ,\Delta H {\rangle}\,ds\Big| >\delta\Big] \nonumber\\
&\leq\varlimsup_{N\rightarrow\infty}{{\mathbb Q}}^{\beta, N}_{\mu_N}\Big[\pi.:\, \sup_{0\leq t\leq T}\Big|{\langle}\pi_t, H{\rangle}\,-\, {\langle}\pi_0, H {\rangle}\,-\, \int_0^t \, {\langle}\pi_s ,\Delta H {\rangle}\,ds\Big| >\delta\Big]\label{eq62}\end{aligned}$$ since the supremum above is a continuous function in the Skorohod metric, see Proposition \[A.3\]. Recall that ${{\mathbb Q}}^{\beta, N}_{\mu_N}$ is the probability measure induced by ${{\mathbb P}}^{\beta}_{\mu_N}$ via the empirical measure. With this in mind and then adding and subtracting ${\langle}\pi_s^N, N^2 {{\mathbb L}}_N H{\rangle}$, expression can be bounded from above by $$\begin{split}
&\varlimsup_{N\rightarrow\infty}{{\mathbb P}}^{\beta}_{\mu_N}\Big[\pi.:\, \sup_{0\leq t\leq T}\Big|{\langle}\pi^N_t, H{\rangle}\,-\, {\langle}\pi^N_0, H {\rangle}\,-\, \int_0^t \, {\langle}\pi^N_s ,N^2{{\mathbb L}}_N H {\rangle}\,ds\Big| >\delta/2\Big]\\
&+\varlimsup_{N\rightarrow\infty}{{\mathbb P}}^{\beta}_{\mu_N}\Big[\pi.:\, \sup_{0\leq t\leq T}\Big|\int_0^t \, {\langle}\pi^N_s ,\Delta H-N^2{{\mathbb L}}_N H {\rangle}\,ds\Big| >\delta/2\Big]\,.
\end{split}$$ By Lemma \[bm\], the first term above is null. Since there is at most one particle per site, the second term in last expression is bounded by $$\begin{split}
&\varlimsup_{N\rightarrow\infty}{{\mathbb P}}^{\beta}_{\mu_N}\Big[\frac{T}{N^d}\sum_{x\notin\Gamma_N}\Big|\Delta H\Big(\frac{x}{N}\Big)-N^2{{\mathbb L}}_N\Big(\frac{x}{N}\Big)\Big| >\delta/4\Big]\\
&+\varlimsup_{N\rightarrow\infty}{{\mathbb P}}^{\beta}_{\mu_N}\Big[\sup_{0\leq t \leq T}\Big|\int_0^t \frac{1}{N^d}\sum_{x\in\Gamma_N}\Big\{\Delta H\Big(\frac{x}{N}\Big)-N^2{{\mathbb L}}_N\Big(\frac{x}{N}\Big)\Big\}\eta_s(x) \, ds\Big| >\delta/4\Big] \,.
\end{split}$$ Outside $\Gamma_N$, the operator $N^2{{\mathbb L}}_N$ coincides with the discrete Laplacian. Since $H\in C^2({{\mathbb T}}^d)$, the first probability above vanishes for $N$ sufficiently large. Recall that the number of elements in $\Gamma_N$ is of order $N^{d-1}$. Applying the triangular inequality, the second expression in the previous sum becomes bounded by the sum of $$\label{eq6.2}
\varlimsup_{N\rightarrow\infty}{{\mathbb P}}^{\beta}_{\mu_N}\Big[{{\mathcal O}}(N^{-1})T\|\Delta H\|_{\infty}>\delta/8\Big]$$ and $$\label{eq6.3}
\varlimsup_{N\rightarrow\infty}{{\mathbb P}}^{\beta}_{\mu_N}\Big[\sup_{0\leq t \leq T}\Big| \int_0^t \frac{1}{N^{d-1}}\sum_{x\in\Gamma_N}N{{\mathbb L}}_N \Big(\frac{x}{N}\Big)\eta_s(x) \, ds\Big| >\delta/8\Big]\,.$$ For large $N$, the probability in vanishes. We deal now with . Let $x\in \Gamma_N$. By definition of $\Gamma_N$, some adjacent bond to $x$ is a *slow bond*. Thus, the opposite vertex to $x$ with respect to this bond is also in $\Gamma_N$, see Figure \[Fig5\].
(1,2) to\[out=45,in=225\] (3,3) to\[out=45,in=0\] (3,0) to\[out=180,in=225\] (1,2);
(1.25,1.75) node [$\Lambda$]{}; (4.25,3.25) node [$\Lambda^\complement$]{}; (0,-1) grid (5,4); (5.7,3.75) node\[above\][$N^{-1}{{\mathbb T}}^d_N$]{}; (3,1) circle (0.1); (2.75,0.7) node [$\frac{x}{N}$]{};
(4,1) circle (0.1); (3.8,0.7) node [$\frac{y}{N}$]{};
(3,0) circle (0.1); (2.75,-0.3) node [$\frac{z}{N}$]{};
Recall the definition of ${{\mathbb L}}_N$ in . Whenever $\{x,x-e_j\}$ neither $\{x,x+e_j\}$ are slow bonds, the expression $$\begin{aligned}
\xi^N_{x,x+e_j} \, \Big[ H\big({\genfrac{}{}{}{1}{x+e_j}{N}}\big)
- H\big({\genfrac{}{}{}{1}{x}{N}}\big) \Big] + \xi^N_{x,x-e_j} \, \Big[H\big({\genfrac{}{}{}{1}{x-e_j}{N}}\big) - H\big({\genfrac{}{}{}{1}{x}{N}}\big) \Big] \end{aligned}$$ is of order ${{\mathcal O}}(N^{-2})$ due to assumption $H\in C^2({{\mathbb T}}^d)$. Therefore, in we can disregard terms of this kind, reducing the proof that is null to prove that $$\label{eq6.5}
\varlimsup_{N\rightarrow\infty}{{\mathbb P}}^{\beta}_{\mu_N}\Big[\sup_{0\leq t \leq T}\Big| \int_0^t \frac{1}{N^{d-1}}\hspace{-0.3cm}\sum_{{\genfrac{}{}{0pt}{}{e=\{x,x+e_j\}}{e \text{ is a slow bond}}}}\!\!{\bf A}(e) \, ds\Big| >\delta/16\Big]\;=\;0\,,$$ where $$\begin{aligned}
{\bf A}(e)\;=\; & \Bigg[\alpha N^{1-\beta}\Big(H\big({\genfrac{}{}{}{1}{x+e_j}{N}}\big)-H\big({\genfrac{}{}{}{1}{x}{N}}\big)\Big)+\frac{H\big({\genfrac{}{}{}{1}{x-e_j}{N}}\big)-H\big({\genfrac{}{}{}{1}{x}{N}}\big)}{1/N}\Bigg]\eta_s(x)\\
+\;& \Bigg[\frac{H\big({\genfrac{}{}{}{1}{x+2e_j}{N}}\big)-H\big({\genfrac{}{}{}{1}{x+e_j}{N}}\big)}{1/N}+\alpha N^{1-\beta}\Big(H\big({\genfrac{}{}{}{1}{x}{N}}\big)-H\big({\genfrac{}{}{}{1}{x+e_j}{N}}\big)\Big)\Bigg]\eta_s(x+e_j)\,.\end{aligned}$$ Since $H$ is smooth, the terms inside parenthesis involving $N^{1-\beta}$ are of order ${{\mathcal O}}(N^{-\beta})$ and hence negligible. On the other hand, the remaining terms are close to plus or minus the derivative of $H$ at $x/N$. We have thus reduced the proof of to the proof of $$\label{eq6.6}
\varlimsup_{N\rightarrow\infty}{{\mathbb P}}^{\beta}_{\mu_N}\Big[\sup_{0\leq t \leq T}\Big| \int_0^t \frac{1}{N^{d-1}}\hspace{-0.3cm}\sum_{{\genfrac{}{}{0pt}{}{e=\{x,x+e_j\}}{e \text{ is a slow bond}}}}\!\!{\partial}_{u_j}H\big({\genfrac{}{}{}{1}{x}{N}}\big) \big(\eta_s(x+e_j)-\eta_s(x)\big) \, ds\Big| >\delta/32\Big]\;=\;0\,.$$ Let $t_0=0<t_1<\cdots<t_n=T$ be a partition of $[0,T]$ with mesh bounded by an arbitrary $\tilde{{\varepsilon}}>0$. Via the triangular inequality, if we prove that $$\sum_{k=0}^n\varlimsup_{N\to\infty} {{\mathbb P}}^{\beta}_{\mu_N} \Big[ \,\Big\vert
\int_0^{t_k} \frac{1}{N^{d-1}}\hspace{-0.3cm}\sum_{{\genfrac{}{}{0pt}{}{e=\{x,x+e_j\}}{e \text{ is a slow bond}}}}\!\!{\partial}_{u_j}H\big({\genfrac{}{}{}{1}{x}{N}}\big) \big(\eta_s(x+e_j)-\eta_s(x)\big) \, ds\,\Big\vert
\, > \, \delta \,\Big]$$ vanishes, then we will conclude that vanishes as well. Therefore, it is enough now to show that, for any $\delta>0$ and any $t\in[0,T]$, $$\varlimsup_{N\to\infty} {{\mathbb P}}^{\beta}_{\mu_N} \Big[ \,\Big\vert
\int_0^{t}
\frac{1}{N^{d-1}}\hspace{-0.3cm}\sum_{{\genfrac{}{}{0pt}{}{e=\{x,x+e_j\}}{e \text{ is a slow bond}}}}\!\!{\partial}_{u_j}H\big({\genfrac{}{}{}{1}{x}{N}}\big) \big(\eta_s(x+e_j)-\eta_s(x)\big) \, ds\,\Big\vert
\, > \, \delta \,\Big]\;=\;0\,.$$ Markov’s inequality then allows us to bound the expression above by $$\label{markovex}
\varlimsup_{N\to\infty} \delta^{-1}{{\mathbb E}}^\beta_{\mu_N}\Big[\, \Big| \int_0^{t}
\frac{1}{N^{d-1}}\hspace{-0.3cm}\sum_{{\genfrac{}{}{0pt}{}{e=\{x,x+e_j\}}{e \text{ is a slow bond}}}}\!\!{\partial}_{u_j}H\big({\genfrac{}{}{}{1}{x}{N}}\big) \big(\eta_s(x+e_j)-\eta_s(x)\big) \, ds\,\Big|\,\Big]\,.$$ Adding and subtracting $\eta^{{\varepsilon}N}_s(x)$ and $\eta^{{\varepsilon}N}_s(x+e_j)$, we bound from above by $$\begin{split}
&\varlimsup_{N\to\infty} \delta^{-1} {{\mathbb E}}^\beta_{\mu_N}\Big[\, \Big| \int_0^{t}
\frac{1}{N^{d-1}}\hspace{-0.3cm}\sum_{{\genfrac{}{}{0pt}{}{e=\{x,x+e_j\}}{e \text{ is a slow bond}}}}\!\!{\partial}_{u_j}H\big({\genfrac{}{}{}{1}{x}{N}}\big) \big(\eta_s(x+e_j)-\eta_s^{{\varepsilon}N}(x+e_j)\big) \, ds\,\Big|\,\Big]\\
&+\varlimsup_{N\to\infty} \delta^{-1}{{\mathbb E}}^\beta_{\mu_N}\Big[\, \Big| \int_0^{t}
\frac{1}{N^{d-1}}\hspace{-0.3cm}\sum_{{\genfrac{}{}{0pt}{}{e=\{x,x+e_j\}}{e \text{ is a slow bond}}}}\!\!{\partial}_{u_j}H\big({\genfrac{}{}{}{1}{x}{N}}\big) \big(\eta^{{\varepsilon}N}_s(x+e_j)-\eta_s^{{\varepsilon}N}(x)\big) \, ds\,\Big|\,\Big]\\
&+\varlimsup_{N\to\infty} \delta^{-1} {{\mathbb E}}^\beta_{\mu_N}\Big[\, \Big| \int_0^{t}
\frac{1}{N^{d-1}}\hspace{-0.3cm}\sum_{{\genfrac{}{}{0pt}{}{e=\{x,x+e_j\}}{e \text{ is a slow bond}}}}\!\!{\partial}_{u_j}H\big({\genfrac{}{}{}{1}{x}{N}}\big) \big(\eta^{{\varepsilon}N}_s(x)-\eta_s(x)\big) \, ds\,\Big|\,\Big] \,.
\end{split}$$ Since $|\{\eta_s^{\varepsilon N}(x+e_j)-\eta_s^{\varepsilon N}(x)\}|\leq \frac{2(\varepsilon N)^{d-1}}{(\varepsilon N)^d}=\frac{2}{\varepsilon N}$, $|\Gamma_N|$ is of order $N^{d-1}$ and $\Vert{\partial}_{u_j}H\Vert_\infty<\infty$, the second term above vanishes. For the remaining terms, we apply Lemma \[replacement\], finishing the proof.
Characterization of limit points for beta=1. {#6.2}
--------------------------------------------
This subsection is devoted to the proof of the next proposition. Keep in mind that Proposition \[Prop5.7\] allows us to write $\pi(t,u)=\rho(t,u)du$ when considering the measure ${{\mathbb Q}}^\beta_*$.
\[prop62one\] Let $H\in C^2({{\mathbb T}}^d\backslash \partial\Lambda)$. For all $\delta>0$, $$\label{6.4}
\begin{split}
&{{\mathbb Q}}^\beta_*\Big[\pi.: \sup_{0\leq t\leq T}\Big|{\langle}\rho_t, H{\rangle}- {\langle}\rho_0 , H{\rangle}- \int_0^t {\langle}\rho_s , \Delta H {\rangle}\, ds\\
&-\int_0^t\int_{{\partial}\Lambda}\rho_s(u^+)\sum_{j=1}^d{\partial}_{u_j} H(u^+){\langle}\vec{\zeta}(u),e_j{\rangle}\,dS(u)ds\\
&+\int_0^t \int_{{\partial}\Lambda}\rho_s(u^-)\sum_{j=1}^d{\partial}_{u_j} H(u^-){\langle}\vec{\zeta}(u),e_j{\rangle}\,dS(u)ds\\
&+\!\int_0^t\!\! \int_{{\partial}\Lambda}\!\!\!\alpha (\rho_s(u^-)-\rho_s(u^+))(H(u^+)-H(u^-))\sum_{j=1}^d|{\langle}\vec{\zeta}(u), e_j{\rangle}|\,dS(u)ds\Big| >\delta\,\Big]=0.
\end{split}$$
(1,2) to\[out=45,in=225\] (3,3) to\[out=45,in=0\] (3,0) to\[out=180,in=225\] (1,2);
(2.25,1.75) node [$\Lambda$]{}; (4.25,3.25) node [$\Lambda^\complement$]{}; (0,-1) grid (5,4); (4.5,4) node\[above\][$N^{-1}{{\mathbb T}}^d_N$]{}; (3,0.5) circle (0.1); (3.5,0.5) circle (0.1); (3.5,1) circle (0.1); (3.5,1.5) circle (0.1); (3.5,2) circle (0.1); (3.5,2.5) circle (0.1); (3.5,3) circle (0.1); (3,3) circle (0.1); (2.5,2.5) circle (0.1); (2,2.5) circle (0.1); (1.5,2.5) circle (0.1); (1,2) circle (0.1); (1,1.5) circle (0.1); (1.5,1) circle (0.1); (2,0.5) circle (0.1); (2.5,0.5) circle (0.1);
(1,2) to\[out=45,in=225\] (3,3) to\[out=45,in=0\] (3,0) to\[out=180,in=225\] (1,2);
(2.25,1.75) node [$\Lambda$]{}; (4.25,3.25) node [$\Lambda^\complement$]{}; (0,-1) grid (5,4); (4.5,4) node\[above\][$N^{-1}{{\mathbb T}}^d_N$]{}; (3,0.5) circle (0.1); (3.5,0.5) circle (0.1); (3.5,3) circle (0.1); (3,3) circle (0.1); (2.5,2.5) circle (0.1); (2,2.5) circle (0.1); (1.5,2.5) circle (0.1); (1,2) circle (0.1); (1,1.5) circle (0.1); (1.5,1) circle (0.1); (2,0.5) circle (0.1); (2.5,0.5) circle (0.1);
(0,-1) – (0,0) node\[above\][$e_2$]{}; (0,-1) – (1,-1) node\[right\][$e_1$]{};
Let us gather some ingredients for the proof of above. The first one is a suitable expression for $N{{\mathbb L}}_N $ over $\Gamma_N$. Define $$\begin{split}
{\Gamma_{N,-}}& \;=\; \Gamma_N \cap \big\{x\in{{\mathbb T}}^d_N\,: \, {\genfrac{}{}{}{1}{x}{N}}\in \Lambda \big\}\quad \text{ and }\\
{\Gamma_{N,+}}& \; =\;\Gamma_N \cap \big\{x\in{{\mathbb T}}^d_N\,: \, {\genfrac{}{}{}{1}{x}{N}}\in \Lambda^\complement \big\}
\end{split}$$ Such a notation has been chosen to agree with . Let us focus on ${\Gamma_{N,-}}$, being the analysis for ${\Gamma_{N,+}}$ completely analogous. It is convenient to consider the decomposition ${\Gamma_{N,-}}=\bigcup_{j=1}^d {\Gamma_{N,-}^j}$, where $$\begin{aligned}
&{\Gamma_{N,-}^j}\;=\; {\Gamma_{N,-}^{j,\text{\rm left}}}\cup {\Gamma_{N,-}^{j,\text{\rm right}}}\,, \qquad \text{with}
\end{aligned}$$ $$\begin{aligned}
&{\Gamma_{N,-}^{j,\text{\rm left}}}\;=\;\Big\{x\in{\Gamma_{N,-}}\,: \, \frac{x-e_j}{N}\in \Lambda^\complement \Big\}\;\text{ and }\;
{\Gamma_{N,-}^{j,\text{\rm right}}}\;=\; \Big\{x\in{\Gamma_{N,-}}\,: \, \frac{x+e_j}{N}\in \Lambda^\complement \Big\}\,,\end{aligned}$$ see Figure \[Fig6\] for an illustration. Note that ${\Gamma_{N,-}^{j,\text{\rm right}}}$ and ${\Gamma_{N,-}^{j,\text{\rm left}}}$ are not necessarily disjoint for a fixed $j$. Nevertheless, due to the smoothness of ${\partial}\Lambda$, the number of elements in the intersection of these two sets is of order ${{\mathcal O}}(N^{d-2})$, hence negligible to our purposes. We will henceforth assume that ${\Gamma_{N,-}^{j,\text{\rm right}}}$ and ${\Gamma_{N,-}^{j,\text{\rm left}}}$ are disjoint sets for all $j=1,\ldots, d$.
At first sight, the reader may imagine that ${\Gamma_{N,-}}$ is equal to ${\Gamma_{N,-}^{j,\text{\rm left}}}\cup {\Gamma_{N,-}^{j,\text{\rm right}}}$ for any $j$, or at least very to close to. This is false, as illustrated by Figure \[Fig6\]. Moreover, for $i\neq j$ and large $N$, the sets ${\Gamma_{N,-}^j}$ and ${\Gamma_{N,-}^i}$ in general are not disjoint with a no negligible intersection.
Define now $$N{{\mathbb L}}^j_N H({\genfrac{}{}{}{1}{x}{N}})=
N\xi^N_{x,x+e_j}\big(H({\genfrac{}{}{}{1}{x+e_j}{N}})-H({\genfrac{}{}{}{1}{x}{N}})\big)+N\xi^N_{x,x-e_j}\big(H({\genfrac{}{}{}{1}{x-e_j}{N}})-H({\genfrac{}{}{}{1}{x}{N}})\big)\,.$$ Then, by By Fubini’s Lemma, $$\begin{aligned}
&\sum_{x\in{\Gamma_{N,-}}}N{{\mathbb L}}_N H\big({\genfrac{}{}{}{1}{x}{N}}\big)\eta^{\varepsilon N}_s(x)\;=\; \sum_{x\in{\Gamma_{N,-}}}\sum_{j=1}^{d}N{{\mathbb L}}^j_N H\big({\genfrac{}{}{}{1}{x}{N}}\big)\eta^{\varepsilon N}_s(x)\nonumber \\
&=\sum_{j=1}^{d}\Big\{ \sum_{x\in{\Gamma_{N,-}^{j,\text{\rm right}}}}N{{\mathbb L}}^j_N H\big({\genfrac{}{}{}{1}{x}{N}}\big)\eta^{\varepsilon N}_s(x)+\sum_{x\in{\Gamma_{N,-}^{j,\text{\rm left}}}}N{{\mathbb L}}^j_N H\big({\genfrac{}{}{}{1}{x}{N}}\big)\eta^{\varepsilon N}_s(x)\Big\}\,.\label{sumnj}\end{aligned}$$ If $x\in {\Gamma_{N,-}^{j,\text{\rm right}}}$, then $\xi^N_{x,x+e_j}=\alpha/N$ and $\xi^N_{x,x-e_j}=1$, see Figure \[Fig5\]. In this case, $$N{{\mathbb L}}^j_N H\big({\genfrac{}{}{}{1}{x}{N}}\big)\;=\;\alpha \Big(H\big({\genfrac{}{}{}{1}{x+e_j}{N}}\big)-H\big({\genfrac{}{}{}{1}{x}{N}}\big)\Big)-{\partial}_{u_j}H\big({\genfrac{}{}{}{1}{x}{N}}\big)+{{\mathcal O}}(N^{-1})\,.$$ On the other hand, if $x\in {\Gamma_{N,-}^{j,\text{\rm left}}}$, then $\xi^N_{x,x-e_j}=\alpha/N$ and $\xi^N_{x,x+e_j}=1$. In this case, $$N{{\mathbb L}}^j_N H\big({\genfrac{}{}{}{1}{x}{N}}\big)\;=\;{\partial}_{u_j}H\big({\genfrac{}{}{}{1}{x}{N}}\big)+\alpha\Big(H\big({\genfrac{}{}{}{1}{x-e_j}{N}}\big)-H\big({\genfrac{}{}{}{1}{x}{N}}\big)\Big)+{{\mathcal O}}(N^{-1})\,.$$ Now, let ${{\bf u}}:{{\mathbb T}}^d\to{\partial}\Lambda$ be a function such that $$\label{uu}
\Vert {{\bf u}}(u) -u\Vert\;=\; \min_{v\in{\partial}\Lambda} \Vert v -u\Vert\,,$$ and ${{\bf u}}$ is continuous in a neighborhood of ${\partial}\Lambda$. That is, ${{\bf u}}$ maps $u\in {{\mathbb T}}^d$ to some of its closest points over ${\partial}\Lambda$ and ${{\bf u}}$ is continuous on the set $({\partial}\Lambda)^{\varepsilon}=\{u\in {{\mathbb T}}^d: \text{dist}(u,{\partial}\Lambda)< {\varepsilon}\}$ for some small ${\varepsilon}>0$. There are more than one function fulfilling , but any choice among them will be satisfactory for our purposes, once this function is continuous near ${\partial}\Lambda$. With this mind we can rewrite , achieving the formula $$\label{sumnj2}
\begin{split}
&\frac{1}{N^{d-1}}\sum_{x\in{\Gamma_{N,-}}}N{{\mathbb L}}_N H\big({\genfrac{}{}{}{1}{x}{N}}\big)\eta^{\varepsilon N}_s(x)\\
& \;=\;
\frac{1}{N^{d-1}}\sum_{j=1}^d \Bigg\{\sum_{x\in{\Gamma_{N,-}^{j,\text{\rm right}}}}\Big[\alpha \big(H({{\bf u}}^+)-H({{\bf u}}^-)\big)-{\partial}_{u_j}H({{\bf u}}^-)\Big]\eta^{\varepsilon N}_s(x)\\
&\hspace{2.5cm}+\sum_{x\in{\Gamma_{N,-}^{j,\text{\rm left}}}}\Big[{\partial}_{u_j}H({{\bf u}}^-)+\alpha\big(H({{\bf u}}^+)-H({{\bf u}}^-)\big)\Big]\eta^{\varepsilon N}_s(x)\Bigg\}\,.
\end{split}$$ plus a negligible error, where by $H({{\bf u}}^-)$ and $H({{\bf u}}^+)$ are the sided limits of $H$ at ${{\bf u}}$. The dependence of ${{\bf u}}$ on $x/N$ will be dropped to not overload notation. Defining $$\begin{aligned}
&{\Gamma_{N,+}^j}\;=\; {\Gamma_{N,+}^{j,\text{\rm left}}}\cup {\Gamma_{N,+}^{j,\text{\rm right}}}\,, \qquad \text{with}
\end{aligned}$$ $$\begin{aligned}
{\Gamma_{N,+}^{j,\text{\rm left}}}=\Big\{x\in{\Gamma_{N,+}}\,: \, \frac{x+e_j}{N}\in \Lambda \Big\}\;\text{ and }\;
{\Gamma_{N,+}^{j,\text{\rm right}}}= \Big\{x\in{\Gamma_{N,+}}\,: \, \frac{x-e_j}{N}\in \Lambda \Big\}\,\,,\end{aligned}$$ we similarly have $$\label{sumnj3}
\begin{split}
&\frac{1}{N^{d-1}}\sum_{x\in{\Gamma_{N,+}}}N{{\mathbb L}}_N H\big({\genfrac{}{}{}{1}{x}{N}}\big)\eta^{\varepsilon N}_s(x)\\
& \;=\;
\frac{1}{N^{d-1}}\sum_{j=1}^d \Bigg\{\sum_{x\in{\Gamma_{N,+}^{j,\text{\rm right}}}}\Big[{\partial}_{u_j}H({{\bf u}}^+)+\alpha \big(H({{\bf u}}^-)-H({{\bf u}}^+)\big)\Big]\eta^{\varepsilon N}_s(x)\\
&\hspace{2.5cm}+\sum_{x\in{\Gamma_{N,+}^{j,\text{\rm left}}}}\Big[\alpha\big(H({{\bf u}}^-)-H({{\bf u}}^+)\big)-{\partial}_{u_j}H({{\bf u}}^+)\Big]\eta^{\varepsilon N}_s(x)\Bigg\}\,.
\end{split}$$
The second ingredient is about convergence of sums over $\Gamma_N$ towards integrals over ${\partial}\Lambda$. Let us review some standard facts about integrals over surfaces. Consider a smooth compact manifold ${{\mathcal M}}\subset {{\mathbb R}}^d$ of dimension $(d-1)$. Assume that ${{\mathcal M}}$ is the graph of a function $f: R\subset {{\mathbb R}}^{d-1}\to {{\mathbb R}}$, that is, ${{\mathcal M}}=\{(x,f(x)): x\in R\}$. Then, given a smooth function $g:{{\mathcal M}}\to {{\mathbb R}}$, the surface integral of $g$ over ${{\mathcal M}}$ will be given by $$\label{eqGdS}
\begin{split}
&\int_{{{\mathcal M}}}g(u)\, dS(u) \;=\; \int_{R}g(x,f(x)) \frac{dx}{|\cos (\gamma(x,f(x)))|}\\
& =\; \int_{R}g\big(x_1,\ldots,x_{d-1},f(x_1,\ldots,x_{d-1})\big) \frac{dx_1\cdots dx_{d-1}}{|{\langle}\vec{\zeta}(x_1,\ldots,x_{d-1}),e_d{\rangle}|}\,,
\end{split}$$ where $\gamma(x,f(x))$ is defined as the angle between the normal exterior vector $\vec{\zeta}(u)=\vec{\zeta}(x_1,\ldots,x_{d-1})$ and $e_d$, the $d$-th element of the canonical basis of ${{\mathbb R}}^d$. Of course, a manifold in general is only locally a graph of a function as above. Nevertheless, the notion of partition of unity allows to use this local property to evaluate a surface integral. Recall the definition of ${{\bf u}}$ given in .
\[lemma65\] Let $g:\Lambda\backslash ({\partial}\Lambda)\subset {{\mathbb T}}^d \to{{\mathbb R}} $ be a function which is continuous near ${\partial}\Lambda$ with an extension to $\Lambda$ which is also continuous near ${\partial}\Lambda$. Then, $$\begin{aligned}
& \int_{{\partial}\Lambda}g(u^-)|{\langle}\vec{\zeta}(u),e_j{\rangle}|\,dS(u) \;=\; \lim_{N\to\infty}\frac{1}{N^{d-1}}\sum_{x\in{\Gamma_{N,-}^j}} g\big({\genfrac{}{}{}{1}{x}{N}}\big) \quad \text{ and} \label{eq617}\\
&\int_{{\partial}\Lambda}g(u^-){\langle}\vec{\zeta}(u),e_j{\rangle}\,dS(u) \;=\;\lim_{N\to\infty}\frac{1}{N^{d-1}}\Bigg[\sum_{x\in{\Gamma_{N,-}^{j,\text{\rm right}}}} g\big({\genfrac{}{}{}{1}{x}{N}}\big) - \sum_{x\in{\Gamma_{N,-}^{j,\text{\rm left}}}} g\big({\genfrac{}{}{}{1}{x}{N}}\big)\Bigg] \,.\label{eq617b}\end{aligned}$$ Analogously, if $g:\Lambda^\complement\subset {{\mathbb T}}^d \to{{\mathbb R}} $ is a function which is continuous near ${\partial}\Lambda$ with an extension to the closure of $\Lambda^\complement$ which is also continuous near ${\partial}\Lambda$, then $$\begin{aligned}
& \int_{{\partial}\Lambda}g(u^+)|{\langle}\vec{\zeta}(u),e_j{\rangle}|\,dS(u) \;=\; \lim_{N\to\infty}\frac{1}{N^{d-1}}\sum_{x\in{\Gamma_{N,+}^j}} g\big({\genfrac{}{}{}{1}{x}{N}}\big) \quad \text{ and} \label{eq617AA}\\
&\int_{{\partial}\Lambda}g(u^+){\langle}\vec{\zeta}(u),e_j{\rangle}\,dS(u) \;=\;\lim_{N\to\infty}\frac{1}{N^{d-1}}\Bigg[\sum_{x\in{\Gamma_{N,+}^{j,\text{\rm right}}}} g\big({\genfrac{}{}{}{1}{x}{N}}\big) - \sum_{x\in{\Gamma_{N,+}^{j,\text{\rm left}}}} g\big({\genfrac{}{}{}{1}{x}{N}}\big)\Bigg] \,.\label{eq617bAA}\end{aligned}$$
In view of the previous discussion, we claim that $$\label{eq123}
\lim_{N\to\infty}\frac{1}{N^{d-1}}\sum_{x\in{\Gamma_{N,-}^j}}\frac{h\big({\genfrac{}{}{}{1}{x}{N}}\big)}{ \big|\big{\langle}\vec{\zeta}\big({{\bf u}}({\genfrac{}{}{}{1}{x}{N}})\big),e_j\big{\rangle}\big|}\;=\; \int_{{\partial}\Lambda}h(u^-)\,dS(u)\,.$$ for any continuous function $h: \Lambda\to {{\mathbb R}}$ such that $h(u)=0$ on the set $\{u\in {\partial}\Lambda: {\langle}\vec{\zeta}(u),e_j{\rangle}=0\}$. This is due to the fact that the sum in the left hand side of is equal to a Riemann sum for the integral on the right hand side of modulus a small error. To see this, it is enough to note that if $x\in{\Gamma_{N,-}}$, then $x/N$ is at a distance less or equal than $1/N$ to ${\partial}\Lambda$, and recall that $\Lambda$ is compact, thus any continuous function over $\Lambda$ is uniformly continuous.
Consider now the function $h: \Lambda\to {{\mathbb R}}$ given by $$h(u)\;:=\; g(u) \, |{\langle}\vec{\zeta}\big({{\bf u}}(u)\big),e_j{\rangle}|\,.$$ Since ${{\bf u}}(u)=u$ for $u\in{\partial}\Lambda$, we have that $h(u)=0$ on the set $\{u\in {\partial}\Lambda: {\langle}\vec{\zeta}(u),e_j{\rangle}=0\}$. Then, considering this particular function $h$ in leads to . The limit can be derived from noticing that, for $N$ sufficiently large,
- if $x\in {\Gamma_{N,-}^{j,\text{\rm right}}}$, then $ {\langle}\vec{\zeta}\big({{\bf u}}(x/N)\big), e_j{\rangle}>0$ and
- if $x\in {\Gamma_{N,-}^{j,\text{\rm left}}}$, then ${\langle}\vec{\zeta}\big({{\bf u}}(x/N)\big), e_j{\rangle}<0 $,
see Figure \[Fig5\] for support. The proofs for and are analogous.
The fact that boundary integrals are not well-defined in the whole Skorohod space ${{\mathcal D}}([0,T ],{{\mathcal M}})$ forbids us to directly apply Portmanteau’s Theorem. To circumvent this technical obstacle, fix $\varepsilon > 0$ which will be taken small later. Adding and subtracting the convolution of $\rho(t,u)$ with the approximation of identity $\iota_{\varepsilon}$ defined in , we bound the probability in by the sum of $$\label{6.5}
\begin{split}
&{{\mathbb Q}}^\beta_*\Big[\pi.:\sup_{0\leq t\leq T}\Big|{\langle}\rho_t, H{\rangle}- {\langle}\rho_0, H {\rangle}-\int_0^t{\langle}\rho_s ,\Delta H{\rangle}\,ds \Big.\\
&-\int_0^t\int_{{\partial}\Lambda}(\rho_s\ast\iota_{\varepsilon})(u^+)\sum_{j=1}^{d}{\partial}_{u_j} H(u^+){\langle}\vec{\zeta}(u),e_j{\rangle}\,dS(u)ds\\
&+\int_0^t\int_{{\partial}\Lambda} (\rho_s\ast\iota_{\varepsilon})(u^-)\sum_{j=1}^{d}{\partial}_{u_j} H(u^-){\langle}\vec{\zeta}(u),e_j{\rangle}\,dS(u)ds\\
&+\int_0^t \int_{{\partial}\Lambda}\alpha ((\rho_s\ast\iota_{\varepsilon})(u^-)-(\rho_s\ast\iota_{\varepsilon})(u^+))\\
&\hspace{2cm}\times (H(u^+)-H(u^-))\sum_{j=1}^d|{\langle}\vec{\zeta}(u), e_j{\rangle}|\,dS(u)ds\Big|>\delta/2\Big]
\end{split}$$ and $$\label{second}
\begin{split}
&{{\mathbb Q}}^\beta_*\Big[\pi.:\, \sup_{0\leq t\leq T}\Big|\int_0^t\int_{{\partial}\Lambda}\Big((\rho_s\ast\iota_{\varepsilon})(u^+)-\rho_s(u^+)\Big)\sum_{j=1}^{d} H(u^+){\langle}\vec{\zeta}(u),e_j{\rangle}\,dS(u)ds\Big.\Big.\\
&-\int_0^t\int_{{\partial}\Lambda}\Big((\rho_s\ast\iota_{\varepsilon})(u^-)-\rho_s(u^-)\Big)\sum_{j=1}^{d}{\partial}_{u_j}H(u^-){\langle}\vec{\zeta}(u),e_j{\rangle}\,dS(u)ds\Big.\\
&-\int_0^t\int_{{\partial}\Lambda}\alpha \Big((\rho_s\ast\iota_{\varepsilon})(u^-)-\rho_s(u^-)\Big)(H(u^+)-H(u^-))\sum_{j=1}^d|{\langle}\vec{\zeta}(u), e_j{\rangle}|\,dS(u)ds\Big.\\
&+\!\int_0^t\!\!\int_{{\partial}\Lambda}\!\!\!\alpha \Big((\rho_s\ast\iota_{\varepsilon})(u^+)-\rho_s(u^+)\Big)\\
&\hspace{2cm}\times(H(u^+)-H(u^-))\sum_{j=1}^d|{\langle}\vec{\zeta}(u), e_j{\rangle}|\,dS(u)ds\Big|>\delta/2\Big].
\end{split}$$ where $\iota_{\varepsilon}$ and the convolution $\rho_s\ast\iota_{\varepsilon}$ were defined in . Adapting results of [@Adams Chapter III] to our context, the reader can check that functions in the Sobolev space ${L^2\big([0,T];\mathcal{H}^1({{\mathbb T}}^d\backslash \partial\Lambda)\big)}$ are continuous in ${{\mathbb T}}^d\backslash {\partial}\Lambda$. Thus, Lemma \[Prop5.7\] gives us that vanishes as ${\varepsilon}\to 0$. It remains to deal with . By Portmanteau’s Theorem, is bounded from above by $$\begin{aligned}
&\varlimsup_{N\rightarrow\infty}{{\mathbb Q}}^{\beta, N}_{\mu_N}\Big[\pi.:\sup_{0\leq t\leq T}\Big|{\langle}\pi_t, H{\rangle}- {\langle}\pi_0, H {\rangle}-\int_0^t{\langle}\pi_s ,\Delta H{\rangle}\,ds\Big.\\
&-\int_0^t\int_{{\partial}\Lambda}(\pi_s\ast\iota_{\varepsilon})(u^+)\sum_{j=1}^{d}{\partial}_{u_j} H(u^+){\langle}\vec{\zeta}(u),e_j{\rangle}\,dS(u)ds\Big.\\
&+\int_0^t\int_{{\partial}\Lambda}(\pi_s\ast\iota_{\varepsilon})(u^-)\sum_{j=1}^{d}{\partial}_{u_j} H(u^-){\langle}\vec{\zeta}(u),e_j{\rangle}\,dS(u)ds\Big.\\
&+\!\!\int_0^t\!\!\int_{{\partial}\Lambda}\hspace{-0.3cm}\alpha ((\pi_s\ast\iota_{\varepsilon})(u^-)\!-\!(\pi_s\ast\iota_{\varepsilon})(u^+))\\
&\hspace{2cm}\times(H(u^+)\!-\!H(u^-))\sum_{j=1}^d|{\langle}\vec{\zeta}(u), e_j{\rangle}|\,dS(u)ds\Big|>\delta/2\Big]\!,\end{aligned}$$ since the supremum above is a continuous function in the Skorohod metric. Now, recalling that ${{\mathbb Q}}^{\beta, N}_{\mu_N}$ is the probability induced by ${{\mathbb P}}^{\beta}_{\mu_N}$ via the empirical measure, adding and subtracting ${\langle}\pi_s^N, N^2 {{\mathbb L}}_N H{\rangle}$, adding and subtracting $\frac{1}{N^{d-1}}\sum_{x\in\Gamma_N}N{{\mathbb L}}_N H({\genfrac{}{}{}{1}{x}{N}})\eta_s^{{\varepsilon}N}(x)$, applying and the Lemma \[lemma65\], we can bound the previous expression by the sum of $$\label{eq614a}
\varlimsup_{N\rightarrow\infty}{{\mathbb P}}^{\beta}_{\mu_N}\Big[\sup_{0\leq t\leq T}\Big|{\langle}\pi^N_t, H{\rangle}-{\langle}\pi^N_0, H {\rangle}- \int_0^t {\langle}\pi^N_s ,N^2{{\mathbb L}}_N H {\rangle}\,ds\Big| >\delta/8\Big]\,,$$ $$\label{eq614aa}
\varlimsup_{N\rightarrow\infty}{{\mathbb P}}^{\beta}_{\mu_N}\Big[\sup_{0\leq t\leq T}\Big|\int_0^t \sum_{x\notin\Gamma_N}\Big(N^2{{\mathbb L}}_N H\big({\genfrac{}{}{}{1}{x}{N}}\big)-\Delta H\big({\genfrac{}{}{}{1}{x}{N}}\big)\Big)\eta_s(x) \,ds\Big| >\delta/8\Big]\,,$$ $$\label{eqmaA}
\varlimsup_{N\rightarrow\infty}{{\mathbb P}}^{\beta}_{\mu_N}\Big[\sup_{0\leq t \leq T} \Big| \frac{1}{N^{d-1}}\int_0^t \sum_{x\in\Gamma_N}N{{\mathbb L}}_N H\big({\genfrac{}{}{}{1}{x}{N}}\big)(\eta_s(x)-\eta^{\varepsilon N}_s(x))\,ds\Big|>\delta/8\Big]$$ and $$\label{eq6.15a}
\begin{split}
&\varlimsup_{N\rightarrow\infty}{{\mathbb P}}^{\beta}_{\mu_N}\Big[\sup_{0\leq t\leq T}\Big|\int_0^t \sum_{x\in\Gamma_N}N{{\mathbb L}}_N H\Big(\frac{x}{N}\Big)\eta_s^{{\varepsilon}N}(x) \,ds\\
&+\sum_{j=1}^{d}\int_0^t\frac{1}{N^{d-1}}\hspace{-0.2cm}\sum_{x\in {\Gamma_{N,-}^{j,\text{\rm right}}}}\hspace{-0.2cm}\eta_s^{\varepsilon N}(x){\partial}_{u_j}H({{\bf u}}^-)\,ds\\
&-\sum_{j=1}^{d}\int_0^t\frac{1}{N^{d-1}}\hspace{-0.2cm}\sum_{x\in {\Gamma_{N,-}^{j,\text{\rm left}}}}\hspace{-0.2cm}\eta_s^{\varepsilon N}(x){\partial}_{u_j}H({{\bf u}}^-)\,ds\\
&-\sum_{j=1}^{d}\int_0^t\vspace{-0.2cm}\frac{1}{N^{d-1}}\hspace{-0.2cm}\sum_{x\in {\Gamma_{N,+}^{j,\text{\rm right}}}}\hspace{-0.2cm}\eta_s^{\varepsilon N}(x){\partial}_{u_j}H({{\bf u}}^+)\,ds\\
&+\sum_{j=1}^{d}\int_0^t\vspace{-0.2cm}\frac{1}{N^{d-1}}\hspace{-0.2cm}\sum_{x\in {\Gamma_{N,+}^{j,\text{\rm left}}}}\hspace{-0.2cm}\eta_s^{\varepsilon N}(x){\partial}_{u_j}H({{\bf u}}^+)\,ds\\
& +\sum_{j=1}^d\int_0^t\!\!\frac{1}{N^{d-1}}\hspace{-0.2cm}\sum_{x\in{\Gamma_{N,-}^j}}\hspace{-0.2cm}\alpha\,\eta_s^{\varepsilon N}(x)(H({{\bf u}}^+)-H({{\bf u}}^-))\,ds \\
& -\sum_{j=1}^d\int_0^t\!\!\frac{1}{N^{d-1}}\hspace{-0.2cm}\sum_{x\in{\Gamma_{N,+}^j}}\hspace{-0.2cm}\alpha\,\eta_s^{\varepsilon N}(x)(H({{\bf u}}^+)-H({{\bf u}}^-))\,ds+ \text{err}(N) \Big|>\delta/8\Big]\,,
\end{split}$$ where $\text{err}(N)$ is a error that goes in modulus to zero as $N\to\infty$. Proposition \[bm\] tells us that is null. The approximation of the continuous Laplacian by the discrete Laplacian assures that is null. Since $N{{\mathbb L}}_N H$ is a sequence of uniformly bounded functions, Lemma \[replacement2\] allows we conclude that vanishes as ${\varepsilon}\searrow 0$. Finally, provided by formulas and and recalling the decomposition $\Gamma_N={\Gamma_{N,+}}\cup {\Gamma_{N,-}}$, we can see that, except for the error term, all terms inside the supremum in cancel. This concludes the proof.
Characterization of limit points for beta>1. {#6.3}
-----------------------------------------------
Let $H\in C^2({{\mathbb T}}^d\backslash \partial\Lambda)$. For all $\delta>0$, $$\label{6.11}
\begin{split}
{{\mathbb Q}}^\beta_*\Big[\pi.: \sup_{0\leq t\leq T}\Big|&{\langle}\rho_t, H{\rangle}- {\langle}\rho_0 , H{\rangle}- \int_0^t {\langle}\rho_s , \Delta H {\rangle}\, ds\Big.\\
&-\int_0^t\int_{{\partial}\Lambda}\rho_s(u^+)\sum_{j=1}^d{\partial}_{u_j} H(u^+){\langle}\vec{\zeta},e_j{\rangle}\,dS(u)ds\\
&+\int_0^t\int_{{\partial}\Lambda}\rho_s(u^-)\sum_{j=1}^d{\partial}_{u_j} H(u^-){\langle}\vec{\zeta},e_j{\rangle}\,dS(u)ds\Big| >\delta\,\Big]\;=\;0.
\end{split}$$
The proof of this proposition is similar, in fact, simpler than the one of Proposition \[prop62one\]. In this case, $$\label{sumnj2neumann}
\begin{split}
&\frac{1}{N^{d-1}}\sum_{x\in{\Gamma_{N,-}}}N{{\mathbb L}}_N H\big({\genfrac{}{}{}{1}{x}{N}}\big)\eta^{\varepsilon N}_s(x)\\
& \;=\;
\frac{1}{N^{d-1}}\sum_{j=1}^d \Bigg\{\sum_{x\in{\Gamma_{N,-}^{j,\text{\rm right}}}}\Big[\alpha N^{1-\beta} \big(H({{\bf u}}^+)-H({{\bf u}}^-)\big)-{\partial}_{u_j}H({{\bf u}}^-)\Big]\eta^{\varepsilon N}_s(x)\\
&\hspace{2cm}+\sum_{x\in{\Gamma_{N,-}^{j,\text{\rm left}}}}\Big[{\partial}_{u_j}H({{\bf u}}^-)+\alpha N^{1-\beta}\big(H({{\bf u}}^+)-H({{\bf u}}^-)\big)\Big]\eta^{\varepsilon N}_s(x)\Bigg\}\,.
\end{split}$$ and $$\label{sumnj3neumann}
\begin{split}
&\frac{1}{N^{d-1}}\sum_{x\in{\Gamma_{N,+}}}N{{\mathbb L}}_N H\big({\genfrac{}{}{}{1}{x}{N}}\big)\eta^{\varepsilon N}_s(x)\\
& \;=\;
\frac{1}{N^{d-1}}\sum_{j=1}^d \Bigg\{\sum_{x\in{\Gamma_{N,+}^{j,\text{\rm right}}}}\Big[{\partial}_{u_j}H({{\bf u}}^+)+\alpha N^{1-\beta} \big(H({{\bf u}}^-)-H({{\bf u}}^+)\big)\Big]\eta^{\varepsilon N}_s(x)\\
&\hspace{2cm}+\sum_{x\in{\Gamma_{N,+}^{j,\text{\rm left}}}}\Big[\alpha N^{1-\beta}\big(H({{\bf u}}^-)-H({{\bf u}}^+)\big)-{\partial}_{u_j}H({{\bf u}}^+)\Big]\eta^{\varepsilon N}_s(x)\Bigg\}\,.
\end{split}$$ Since $\beta\in (1,\infty]$, we conclude that all terms above involving $\alpha$ disappear in the limit as $N\to \infty$. Noting that there are no surface integrals in involving $\alpha$, it is a simple game to repeat the steps in the proof of Proposition \[prop62one\] to finally conclude .
Uniqueness of weak solutions {#s7}
============================
The hydrodynamic equation is the classical heat equation, which does not need any consideration about uniqueness of weak solutions. Thus, we only need to guarantee that weak solutions of and are unique.
Let us trace the strategy for the proof of uniqueness, which works for both and . Considering in each case $\beta=1$ or $\beta\in(1,\infty]$ a suitable set of test functions, we can annul all surface integrals. Being more precise, consider the following definitions:
\[operator1\] Let ${{{\mathfrak D}}^{{\mbox{\rm \scriptsize Rob}}}}\subset L^2({{\mathbb T}}^d)$ be the set of functions $H:{{\mathbb T}}^d\to {{\mathbb R}}$ such that $H(u)=h_1(u){\bf 1}_\Lambda(u)+h_2(u){\bf 1}_\Lambda^\complement(u)$, where
(i) $h_i\in C^2({{\mathbb T}}^d)$ for $i=\{1,2\}$.
(ii) ${\langle}\nabla h_1(u),\vec{\zeta}(u){\rangle}={\langle}\nabla h_2(u),\vec{\zeta}(u){\rangle}=\big(h_2(u)-h_1(u)\big){\displaystyle}\sum_{j=1}^d|{\langle}\vec{\zeta}(u), e_j{\rangle}|$ , $\forall u\in{\partial}\Lambda$.
Define the operator ${{{\mathfrak L}}^{{\mbox{\rm \scriptsize Rob}}}}: {{{\mathfrak D}}^{{\mbox{\rm \scriptsize Rob}}}}\rightarrow L^{2}({{\mathbb T}}^d)$ by $${{{\mathfrak L}}^{{\mbox{\rm \scriptsize Rob}}}}H(u)\,=\;\left\{\begin{array}{cl}
\Delta h_1(u), \,\, & \mbox{if}\,\,\,\,u\in\Lambda\,,\smallskip\\
\Delta h_2(u), \,\, & \mbox{if}\,\,\,\,u\in\Lambda^\complement \,.
\end{array}
\right.$$
\[operator\] Let ${{{\mathfrak D}}^{{\mbox{\rm \scriptsize Neu}}}}\subset L ^2({{\mathbb T}}^d)$ be the set of functions $H:{{\mathbb T}}^d\to {{\mathbb R}}$ such that $H(u)=h_1(u){\bf 1}_\Lambda(u)+h_2(u){\bf 1}_\Lambda^\complement(u)$, where:
(i) $h_i\in C^2({{\mathbb T}}^d)$ for $i=\{1,2\}$.
(ii) ${\langle}\nabla h_1(u),\vec{\zeta}(u){\rangle}={\langle}\nabla h_2(u),\vec{\zeta}(u){\rangle}=0$ , $\forall u\in{\partial}\Lambda$.
Define the operator ${{{\mathfrak L}}^{{\mbox{\rm \scriptsize Neu}}}}: {{{\mathfrak D}}^{{\mbox{\rm \scriptsize Neu}}}}\rightarrow L^{2}({{\mathbb T}}^d)$ by $${{{\mathfrak L}}^{{\mbox{\rm \scriptsize Neu}}}}H(u)\;=\;\left\{\begin{array}{cl}
\Delta h_1(u) \,\, & \mbox{if}\,\,\,\,u\in\Lambda\,,\smallskip\\
\Delta h_2(u) \,\, & \mbox{if}\,\,\,\,u\in\Lambda^\complement \,.
\end{array}
\right.$$
It is straightforward to check that, if $\rho$ is a weak solution of , then $$\label{71}
\begin{split}
&{\langle}\rho_t, H{\rangle}- {\langle}\rho_0 , H{\rangle}- \int_0^t\! {\langle}\rho_s , {{{\mathfrak L}}^{{\mbox{\rm \scriptsize Rob}}}}H {\rangle}\, ds\;=\; 0\,,\quad \forall H\in {{{\mathfrak D}}^{{\mbox{\rm \scriptsize Rob}}}}\,,\,\forall t\in[0,T]\,,
\end{split}$$ while, if $\rho$ is a weak solution of , then $$\label{72}
\begin{split}
&{\langle}\rho_t, H{\rangle}- {\langle}\rho_0 , H{\rangle}- \int_0^t\! {\langle}\rho_s , {{{\mathfrak L}}^{{\mbox{\rm \scriptsize Neu}}}}H {\rangle}\, ds\;=\; 0\,,\quad \forall H\in {{{\mathfrak D}}^{{\mbox{\rm \scriptsize Neu}}}}\,, \,\forall t\in[0,T]\,.
\end{split}$$ In both cases, if an orthonormal basis of $L^2({{\mathbb T}}^d)$ composed of eigenfunctions for the corresponding operator (associated to nonpositive eigenvalues) is available, this would easily lead to the proof of uniqueness, as we shall see later. However, this is not the case. So, to overcome this situation we extend the corresponding operator via a *Friedrichs extension* (see [@z] on the subject) to achieve the desired orthonormal basis.
Let us briefly explain the notion of Friedrichs extension. Let $X$ be a Hilbert space and denote by ${\langle}\cdot,\cdot{\rangle}$ and $\Vert\cdot\Vert$ its inner product and norm, respectively. Consider a linear, strongly monotone and symmetric operator $\mathcal{A}:{{\mathfrak D}}\subset X\rightarrow X$, where by *strongly monotone* we mean that there exists $c>0$ such that $${\langle}\mathcal{A} H, H{\rangle}\;\geq\; c \Vert H\Vert^2\,,\quad \forall \,H\in{{\mathfrak D}}\,.$$ Denote by ${\langle}\cdot, \cdot{\rangle}_{{{\mathcal E}}(\mathcal{A})}$ the so-called *energetic* inner product on ${{\mathfrak D}}$ associated to ${{\mathcal A}}$, which is defined by $${\langle}F,G{\rangle}_{{{\mathcal E}}(\mathcal{A})} \; :=\; {\langle}F,\,\mathcal{A} G{\rangle}\,.$$ Let ${\mathscr{H}_{\text{\rm Fried}}}$ be the set of all functions $F$ in $X$ for which there exists a sequence $\{F_n : n\ge 1\}$ in ${{\mathfrak D}}$ such that $F_n$ converges to $F$ in $X$ and $F_n$ is Cauchy for the inner product ${\langle}\cdot, \cdot {\rangle}_{{{\mathcal E}}({{\mathcal A}})}$. A sequence $\{F_n : n\ge 1\}$ with these properties will be called an *admissible sequence* for $F$. For $F$, $G$ in ${\mathscr{H}_{\text{\rm Fried}}}$, let $$\label{innerproduct}
{\langle}F,G{\rangle}_{\text{\rm Fried}}\; :=\; \lim_{n\to\infty} {\langle}F_n,G_n{\rangle}_{{{\mathcal E}}(\mathcal{A})}\,,$$ where $\{F_n : n\ge 1\}$, $\{G_n : n\ge 1\}$ are admissible sequences for $F$ and $G$, respectively. By [@z Proposition 5.3.3], the limit exists and does not depend on the admissible sequence chosen and, moreover, the space ${\mathscr{H}_{\text{\rm Fried}}}$ endowed with the scalar product ${\langle}\cdot, \cdot {\rangle}_{\text{\rm Fried}}$ is a real Hilbert space, usually called the *energetic space* associated to ${{\mathcal A}}$.
The Friedrichs extension $\mathcal{A}_{\text{\rm Fried}}:{{{\mathfrak D}}_{\text{\rm Fried}}}\to X$ of the operator $\mathcal{A}$ is then defined as follows. Let ${{{\mathfrak D}}_{\text{\rm Fried}}}$ be the set of vectors in $F\in{\mathscr{H}_{\text{\rm Fried}}}$ for which there exists a vector $f\in X$ such that $${\langle}F,G{\rangle}_{\text{\rm Fried}} \;=\; {\langle}f,G{\rangle}\,, \quad \forall G\in {\mathscr{H}_{\text{\rm Fried}}}\,.$$ and let ${\mathcal{A}_{\text{\rm Fried}}}F=f$. See the excellent book [@z] for why this operator $\mathcal{A}_{\text{\rm Fried}}:{{{\mathfrak D}}_{\text{\rm Fried}}}\to X$ is indeed an extension of ${{\mathcal A}}:{{\mathfrak D}}\to X$ and more details on the construction. The main result about Friedrichs extensions and eigenfunctions we cite here is the next one.
\[proper\] Let $\mathcal{A}: {{\mathfrak D}}\subseteq X\rightarrow X$ be a linear, symmetric and strongly monotone operator and let $\mathcal{A}_{\text{\rm Fried}}: {{\mathfrak D}}_{\text{\rm Fried}}\subseteq X\rightarrow X$ be its Friedrichs extension. Assume additionally that the embedding ${\mathscr{H}_{\text{\rm Fried}}}\hookrightarrow X$ is compact. Then,
- The eigenvalues of $-\mathcal{A}_{\text{\rm Fried}}$ form a countable set $0<c\leq \mu_1\le\mu_2\le \cdots$ with $\lim_{n\to\infty} \mu_n=\infty$, and all these eigenvalues have finite multiplicity.
- There exists a complete orthonormal basis of $X$ composed of eigenvectors of ${{\mathcal A}}_{\text{\rm Fried}}$.
Denote by ${{\mathbb I}}$ the identity operator. If ${{\mathfrak L}}: {{\mathfrak D}}\subseteq X\rightarrow X$ is a symmetric nonpositive operator, then ${{\mathbb I}} -{{\mathfrak L}}:{{\mathfrak D}}\rightarrow X$ is symmetric and strongly monotone with $c=1$. In fact, $${\langle}({{\mathbb I}}-{{\mathfrak L}}) H, H{\rangle}\;=\; \Vert H\Vert^2 + {\langle}-{{\mathfrak L}} H, H{\rangle}\;\geq\; \Vert H\Vert^2\,,\quad \forall\,H\in {{\mathfrak D}}\,.$$ Therefore, under the hypothesis that ${{\mathfrak L}}: {{\mathfrak D}}\subseteq X\rightarrow X$ is a symmetric and nonpositive linear operator, we may consider the Friedrichs extension of ${({{\mathbb I}} -{{\mathfrak L}})}$.
\[prop72\] Let ${{\mathfrak L}}: {{\mathfrak D}}\subseteq X\rightarrow X$ be a symmetric nonpositive operator. Denote by $({{\mathbb I}} -{{\mathfrak L}})_{\text{\rm Fried}}: {{\mathfrak D}}_{\text{\rm Fried}}\rightarrow X$ the Friedrichs extension of $({{\mathbb I}} -{{\mathfrak L}}):{{\mathfrak D}}\to X$ and by ${\mathscr{H}_{\text{\rm Fried}}}$ the corresponding energetic space. Assume that the embedding ${\mathscr{H}_{\text{\rm Fried}}}\hookrightarrow X$ is compact. Then, there exists at most one measurable function $\rho:[0,T] \to X$ such that $$\label{sup}
\sup_{t\in[0,T]}\Vert \rho_t\Vert\;<\;\infty$$ and $$\begin{split}
{\langle}\rho_t, H{\rangle}- {\langle}\rho_0 , H{\rangle}- \int_0^t {\langle}\rho_s , {{\mathfrak L}} H {\rangle}\, ds\;=\; 0\,,\quad \forall H\in {{\mathfrak D}}\,, \,\forall t\in[0,T]\,.
\end{split}$$ where $\rho_0$ is a fixed element of $X$.
Consider $\rho^1, \rho^2$ two solutions of above and write $\rho=\rho^1-\rho^2$. By linearity, $${\langle}\rho_t, H{\rangle}- \int_0^t {\langle}\rho_s , {{\mathfrak L}} H {\rangle}\, ds\;=\; 0\,,\quad \forall H\in {{\mathfrak D}}\,, \,\forall t\in[0,T]\,.$$ which is the same as $${\langle}\rho_t, H{\rangle}+ \int_0^t {\langle}\rho_s , ({{\mathbb I}}-{{\mathfrak L}}) H {\rangle}\, ds- \int_0^t {\langle}\rho_s , H {\rangle}\, ds\;=\; 0\,,\quad \forall H\in {{\mathfrak D}}\,, \,\forall t\in[0,T]\,.$$ Since ${{{\mathfrak D}}_{\text{\rm Fried}}}\subseteq {\mathscr{H}_{\text{\rm Fried}}}$, the last equation can be extended to $$\label{pth}
{\langle}\rho_t, H{\rangle}+ \int_0^t {\langle}\rho_s , ({{\mathbb I}}-{{\mathfrak L}})_{\text{\rm Fried}} H {\rangle}\, ds- \int_0^t {\langle}\rho_s , H {\rangle}\, ds\;=\; 0\,,\quad \forall H\in {{{\mathfrak D}}_{\text{\rm Fried}}}\,, \,\forall t\in[0,T]\,.$$ By Theorem \[proper\], the Friedrichs extension $({{\mathbb I}}-{{\mathfrak L}})_{\text{\rm Fried}} : {{\mathfrak D}}_{\text{\rm Fried}} \to X$ has eigenvalues $1\le \lambda_1 \le \lambda_2 \le \cdots$, all of them having finite multiplicity with $\lim_{n\to\infty}\lambda_n=\infty$, and there exists a complete orthonormal basis $\{\Psi_j\}_{i\in {{\mathbb N}}}$ of $L^2({{\mathbb T}}^d)$ composed of eigenfunctions. Denote $${{\mathfrak L}}_{\text{\rm Fried}}\;:=\;{{\mathbb I}}-({{\mathbb I}}-{{\mathfrak L}})_{\text{\rm Fried}}\,.$$ Thus, $\{\Psi_j\}_{j\in {{\mathbb N}}}$ is also a set of eigenfunctions for the operator ${{\mathfrak L}}_{\text{\rm Fried}}$ whose eigenvalues are given by $\mu_j=1-\lambda_j\leq 0$. Define $$R(t)\;=\;\sum_{j=1}^\infty\frac{1}{j^{2}(1-\mu_j)}{\langle}\rho_t,\Psi_j{\rangle}^{2}\quad \text{ for } t\in [0,T]\,.$$ Since $\rho$ satisfy , we have that $$\label{derivada}
\frac{d}{dt}{\langle}\rho_t,\Psi_j{\rangle}^{2}\;=\;2{\langle}\rho_t,\Psi_j{\rangle}{\langle}\rho_t,{{\mathfrak L}}_{\text{\rm Fried}}\Psi_j{\rangle}\;=\;2\mu_j{\langle}\rho_t,\Psi_j{\rangle}^2\,.$$ By and the Cauchy-Schwarz inequality, we have that $$\sum_{j=1}^\infty\frac{2|\mu_j|}{j^{2}(1-\mu_j)}{\langle}\rho_t,\Psi_j{\rangle}^{2}\;\leq\;\sum_{j=1}^\infty\frac{2|\mu_j|}{j^{2}(1-\mu_j)}\Big(\sup_{t\in [0,T]}\Vert \rho_t\Vert^{2}\Big)\;<\;\infty\,,$$ which together with implies that $$\frac{d}{dt}R(t)\;=\;\sum_{j=1}^\infty\frac{2\mu_j}{j^{2}(1-\mu_j)}{\langle}\rho_t,\Psi_j{\rangle}^{2}\;\leq\;0\,.$$ Since $R(t)\geq0$, $R(0)=0$, and $dR/dt\leq 0$, we conclude that $R(t)=0$ for all $t\in[0,T]$ and hence ${\langle}\rho_t,\Psi_j{\rangle}^2=0$ for any $t\in[0,T]$. Due to $\{\Psi_j\}_{j\in {{\mathbb N}}}$ be a complete orthonormal basis of $X$, we deduce that $\rho\equiv0$, finishing the proof.
In view of and , considering $X$ as the Hilbert space $L^2({{\mathbb T}}^d)$ and applying the last proposition, to achieve the uniqueness of weak solutions of and it is enough to assure that
1. The operators ${{\mathbb I}}-{{{\mathfrak L}}^{{\mbox{\rm \scriptsize Rob}}}}:{{{\mathfrak D}}^{{\mbox{\rm \scriptsize Rob}}}}\subseteq L^2({{\mathbb T}}^d)\to L^2({{\mathbb T}}^d)$ and ${{\mathbb I}}-{{{\mathfrak L}}^{{\mbox{\rm \scriptsize Neu}}}}:{{{\mathfrak D}}^{{\mbox{\rm \scriptsize Neu}}}}\subseteq L^2({{\mathbb T}}^d)\to L^2({{\mathbb T}}^d)$ are symmetric nonpositive linear operators.
2. Denoting by ${\mathscr{H}_{\text{\rm Fried}}^{{\mbox{\rm \scriptsize Rob}}}}$ and ${\mathscr{H}_{\text{\rm Fried}}^{{\mbox{\rm \scriptsize Neu}}}}$ their respective energetic spaces, the embeddings ${\mathscr{H}_{\text{\rm Fried}}^{{\mbox{\rm \scriptsize Rob}}}}\hookrightarrow L^2({{\mathbb T}}^d)$ and ${\mathscr{H}_{\text{\rm Fried}}^{{\mbox{\rm \scriptsize Neu}}}}\hookrightarrow L^2({{\mathbb T}}^d)$ are compact.
This is precisely what we are going to do in the next four propositions. Denote by ${\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{\zeta}}}}}}(u)=-\vec{\zeta}(u)$ the normal exterior vector to the region $\Lambda^\complement$ at $u\in {\partial}\Lambda$. Recall that ${\langle}\cdot,\cdot{\rangle}$ is used for both the inner products in $L^2({{\mathbb T}}^d)$ and in ${{\mathbb R}}^d$.
\[simetrico1\] The operator $-{{{\mathfrak L}}^{{\mbox{\rm \scriptsize Rob}}}}:{{{\mathfrak D}}^{{\mbox{\rm \scriptsize Rob}}}}\rightarrow L^2({{\mathbb T}}^d)$ is symmetric and nonnegative.
Let $H, G\in {{{\mathfrak D}}^{{\mbox{\rm \scriptsize Rob}}}}$. We can write $H=h_1{\bf 1}_\Lambda+h_2{\bf 1}_\Lambda^\complement$ and $ G=g_1{\bf 1}_\Lambda+g_2{\bf 1}_\Lambda^\complement$, where $h_1, h_2, g_1, g_2 \in C^2({{\mathbb T}}^d)$. By the third Green identity (see Appendix \[appendix\], Theorem \[evans\]), $$\int_{{{\mathbb T}}^d} \big(h\Delta g - g \Delta h\big) \,du\;=\;\int_{{\partial}\Lambda}\Big(h{\langle}\nabla g,\vec{\zeta}\,{\rangle}-g{\langle}\nabla h,\vec{\zeta}\,{\rangle}\Big) \,dS\,,$$ where $dS$ is an infinitesimal volume element of ${\partial}\Lambda$. Thus, $$\begin{aligned}
{\langle}H,-{{{\mathfrak L}}^{{\mbox{\rm \scriptsize Rob}}}}G{\rangle}=& {\langle}h_1{\bf 1}_\Lambda+h_2{\bf 1}_{\Lambda^\complement}, -\Delta g_1{\bf 1}_\Lambda-\Delta g_2{\bf 1}_{\Lambda^\complement} {\rangle}\\
=& -\int_{\Lambda}h_1\Delta g_1\, du - \int_{\Lambda^\complement} h_2\Delta g_2\, du\\
=& -\int_{\Lambda}g_1\Delta h_1\, du -\int_{{\partial}\Lambda}\Big(h_1{\langle}\nabla g_1,\vec{\zeta}\,{\rangle}-g_1{\langle}\nabla h_1,\vec{\zeta}\,{\rangle}\Big)\, dS\\
&-\int_{\Lambda^\complement} g_2\Delta h_2\, du-\int_{{\partial}\Lambda^\complement}\Big(h_2{\langle}\nabla g_2,{\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{\zeta}}}}}}\,{\rangle}-g_2{\langle}\nabla h_2,{\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{\zeta}}}}}}\,{\rangle}\Big)\, dS\\
=& -\int_{\Lambda}g_1\Delta h_1\, du -\int_{{\partial}\Lambda}\Big(h_1{\langle}\nabla g_1,\vec{\zeta}\,{\rangle}-g_1{\langle}\nabla h_1,\vec{\zeta}\,{\rangle}\Big)\, dS\\
&-\int_{\Lambda^\complement} g_2\Delta h_2\, du-\int_{{\partial}\Lambda^\complement}\Big(g_2{\langle}\nabla h_2,\vec{\zeta}\,{\rangle}-h_2{\langle}\nabla g_2,\vec{\zeta}\,{\rangle}\Big)\, dS\,.\end{aligned}$$ Using the boundary condition in the item (ii) of Definition \[operator1\] and ${\partial}\Lambda^\complement={\partial}\Lambda$, we conclude that the last expression above is equal to $$\begin{aligned}
&-\int_{\Lambda}g_1\Delta h_1\, du- \int_{\Lambda^\complement} g_2\Delta h_2\, du\\
&-\int_{{\partial}\Lambda}\!\!\Big((h_1\!-\!h_2)\sum_{j=1}^d|{\langle}\vec{\zeta}, e_j{\rangle}|(g_2\!-\!g_1)\!-\!(g_1-g_2)\sum_{j=1}^d|{\langle}\vec{\zeta}, e_j{\rangle}|(h_2-h_1)\Big)dS\\
&=-\int_{\Lambda}g_1\Delta h_1\, du- \int_{\Lambda^\complement} g_2\Delta h_2\, du\,.\end{aligned}$$ Then, $
{\langle}H,-{{{\mathfrak L}}^{{\mbox{\rm \scriptsize Rob}}}}G{\rangle}=-\int_{\Lambda}g_1\Delta h_1\, du - \int_{\Lambda^\complement} g_2\Delta g_2\, du=\, {\langle}-{{{\mathfrak L}}^{{\mbox{\rm \scriptsize Rob}}}}H, G{\rangle}$. For the nonnegativeness, note that $$\begin{aligned}
&{\langle}H, -{{{\mathfrak L}}^{{\mbox{\rm \scriptsize Rob}}}}H{\rangle}\;=\; - \int_{\Lambda}h_1 \Delta h_1\,du-\int_{\Lambda^\complement}h_2 \Delta h_2\,du\\
&=\int_{\Lambda}|\nabla h_1|^2\,du+\int_{\Lambda^\complement}|\nabla h_2|^2\, du-\int_{{\partial}\Lambda}\Big({\langle}\nabla h_1,\vec{\zeta}\,{\rangle}h_1+{\langle}\nabla h_2, \vec{\zeta}\,{\rangle}h_2\Big) \,dS\end{aligned}$$ where the second equality above holds by the second Green identity, see Appendix, Theorem \[evans\], and ${\partial}(\Lambda^{\complement})={\partial}\Lambda$. Since $\int_{\Lambda}|\nabla h_i|^2\,du\geq0$, for $i=1,2$, it is enough to check that $-\int_{{\partial}\Lambda}\Big({\langle}\nabla h_1, \vec{\zeta}\,{\rangle}h_1+{\langle}\nabla h_2, \vec{\zeta}\,{\rangle}h_2 \Big)\,dS\;\geq\;0$. In fact, $$\begin{aligned}
&-\int_{{\partial}\Lambda}\Big({\langle}\nabla h_1, \vec{\zeta}\,{\rangle}h_1+{\langle}\nabla h_2, {\reflectbox{\ensuremath{\vec{\reflectbox{\ensuremath{\zeta}}}}}}\,{\rangle}h_2\Big) \,dS\; =\;-\int_{{\partial}\Lambda}\Big({\langle}\nabla h_1, \vec{\zeta}\,{\rangle}h_1-{\langle}\nabla h_2, \vec{\zeta}\,{\rangle}h_2 \Big)\,dS\\
&=\!\int_{{\partial}\Lambda}\sum_{j=1}^d|{\langle}\vec{\zeta},e_j{\rangle}|\Big((h_2-h_1)h_2-(h_2-h_1)h_1\Big) dS\\
&=2\int_{{\partial}\Lambda}\sum_{j=1}^d|{\langle}\vec{\zeta},e_j{\rangle}|(h_2-h_1)^2\, dS\;\geq\; 0\,,\end{aligned}$$ where the second equality holds by item (ii) of Definition \[operator1\].
\[embedding1\] The embedding ${\mathscr{H}_{\text{\rm Fried}}^{{\mbox{\rm \scriptsize Rob}}}}\hookrightarrow L^2({{\mathbb T}}^d)$ is compact.
Let $\{ H_n\}$ be a bounded sequence in ${\mathscr{H}_{\text{\rm Fried}}^{{\mbox{\rm \scriptsize Rob}}}}$. Fix $\{ F_n\}$ a sequence in ${{{\mathfrak D}}^{{\mbox{\rm \scriptsize Rob}}}}$ such that $\Vert F_n-H_n\Vert \to 0$ when $n\rightarrow\infty$ and $\{ F_n\}$ is also bounded in ${\mathscr{H}_{\text{\rm Fried}}^{{\mbox{\rm \scriptsize Rob}}}}$. Thus, to show the compact embedding we need prove that $\{H_n\}$ have a convergent subsequence in $L^2({{\mathbb T}}^d)$. To get a convergent subsequence of $\{ H_n\}$, it is sufficient to find a convergent subsequence of $\{F_n\}$ in $L^2({{\mathbb T}}^d)$. Write $F_n=f_n\mathbf{1}_\Lambda+\tilde{f_n}\mathbf{1}_{\Lambda^\complement}$, with $f_n, \tilde{f_n} \in C^2({{\mathbb T}}^d)$. Then, $$\begin{aligned}
&{\langle}F_n,F_n{\rangle}_{{{\mathcal E}}({{\mathbb I}}-{{{\mathfrak L}}^{{\mbox{\rm \scriptsize Rob}}}})}={\langle}F_n,F_n{\rangle}+{\langle}F_n, -{{{\mathfrak L}}^{{\mbox{\rm \scriptsize Rob}}}}F_n{\rangle}\\
&={\langle}f_n\mathbf{1}_\Lambda+\tilde{f_n}\mathbf{1}_{\Lambda^\complement},f_n\mathbf{1}_\Lambda+\tilde{f_n}\mathbf{1}_{\Lambda^\complement}{\rangle}+ {\langle}f_n\mathbf{1}_\Lambda+\tilde{f_n}\mathbf{1}_{\Lambda^\complement},-\Delta f_n\mathbf{1}_{\Lambda}-\Delta \tilde{f_n}\mathbf{1}_{\Lambda^\complement} {\rangle}\,.\end{aligned}$$ Expanding the right hand side of above and using Green identity (see Appendix \[appendix\], Theorem \[evans\]), we get that $$\begin{aligned}
&\int_\Lambda f_n^2\,du + \int_{\Lambda^\complement}\tilde{f_n}^2\,du - \int_\Lambda f_n\Delta f_n\,du -\int_{\Lambda^\complement}\tilde{f_n}\Delta \tilde{f_n}\,du\\
&=\Vert f_n \mathbf{1}_{\Lambda}\Vert^2+\Vert \tilde{f_n} \mathbf{1}_{\Lambda^\complement}\Vert^2+\Vert \nabla f_n \mathbf{1}_{\Lambda}\Vert^2+\Vert \nabla \tilde{f_n} \mathbf{1}_{\Lambda^\complement}\Vert^2\\
&+2\int_{{\partial}\Lambda}\sum_{j=1}^d|{\langle}\vec{\zeta}, e_j{\rangle}|(f_n-\tilde{f_n})^2 dS\,.\end{aligned}$$ Under the hypotheses of boundedness of the sequence $\{F_n\}$ in the norm induced by ${\langle}\cdot,\cdot{\rangle}_{{{\mathcal E}}({{\mathbb I}}-{{{\mathfrak L}}^{{\mbox{\rm \scriptsize Rob}}}})}$, the sequences $\{\Vert f_n \mathbf{1}_{\Lambda}\Vert^2\}$, $\{\Vert \tilde{f_n} \mathbf{1}_{\Lambda^\complement}\Vert^2\}$, $\{\Vert \nabla f_n \mathbf{1}_{\Lambda}\Vert^2\}$ and $\{\Vert \nabla \tilde{f_n} \mathbf{1}_{\Lambda^\complement}\Vert^2\}$ are bounded. By the Rellich-Kondrachov compactness theorem (see [@e Theorem 5.7.1]), $\{f_n \mathbf{1}_{\Lambda}\}$, $\{\tilde{f}_n\mathbf{1}_{\Lambda^\complement}\}$ have a common convergent subsequence in $L^2({{\mathbb T}}^d)$. This implies that $\{F_n\}$ has a convergent subsequence.
\[simetrico\] The operator $-{{{\mathfrak L}}^{{\mbox{\rm \scriptsize Neu}}}}:{{{\mathfrak D}}^{{\mbox{\rm \scriptsize Neu}}}}\rightarrow L^2({{\mathbb T}}^d)$ is symmetric and nonnegative.
Let $H, G\in {{{\mathfrak D}}^{{\mbox{\rm \scriptsize Neu}}}}$. We can write $H=h_1{\bf 1}_\Lambda+h_2{\bf 1}_\Lambda^\complement$ and $ G=g_1{\bf 1}_\Lambda+g_2{\bf 1}_\Lambda^\complement$, where $h_1, h_2, g_1, g_2 \in C^2({{\mathbb T}}^d)$. By the third Green identity, see Appendix \[appendix\], Theorem \[evans\], we have that $$\int_{{{\mathbb T}}^d} h\Delta g \,du- g \Delta h \,du\;=\;\int_{{\partial}\Lambda}h{\langle}\nabla g,\vec{\zeta}\,{\rangle}-g{\langle}\nabla h,\vec{\zeta}\,{\rangle}\,dS\;=\;0\,,$$ where $dS$ is the infinitesimal volume element of ${\partial}\Lambda$. Thus, $$\begin{aligned}
& {\langle}H,-{{{\mathfrak L}}^{{\mbox{\rm \scriptsize Neu}}}}G{\rangle}={\langle}h_1{\bf 1}_\Lambda+h_2{\bf 1}_{\Lambda^\complement}, -\Delta g_1{\bf 1}_\Lambda-\Delta g_2{\bf 1}_{\Lambda^\complement} {\rangle}\\
&= -\!\int_{\Lambda}h_1\Delta g_1 du - \!\int_{\Lambda^\complement} h_2\Delta g_2 du= -\!\int_{\Lambda}g_1\Delta h_1 du -\! \int_{\Lambda^\complement} g_2\Delta g_2 du= {\langle}-{{{\mathfrak L}}^{{\mbox{\rm \scriptsize Neu}}}}H, G{\rangle}.\end{aligned}$$ For nonnegativeness, $$\begin{aligned}
{\langle}H, -{{\mathcal L}}_{\Lambda}H{\rangle}&= - \int_{\Lambda}h_1 \Delta h_1\,du- \int_{\Lambda^\complement}h_2 \Delta h_2\,du=\int_{\Lambda}|\nabla h_1|^2\,du+\int_{\Lambda^\complement}|\nabla h_2|^2\, du\geq0,\end{aligned}$$ where the second equality above holds due to the second Green identity, see Appendix \[appendix\], Theorem \[evans\].
\[embedding\] The embedding ${\mathscr{H}_{\text{\rm Fried}}^{{\mbox{\rm \scriptsize Neu}}}}\hookrightarrow L^2({{\mathbb T}}^d)$ is compact.
Let $\{ H_n\}$ be a bounded sequence in ${\mathscr{H}^{{\mbox{\rm \scriptsize Neu}}}}$. Fix a sequence $\{ F_n\}$ of functions in ${{{\mathfrak D}}^{{\mbox{\rm \scriptsize Neu}}}}$ such that $\Vert F_n-H_n\Vert \to 0$ when $n\rightarrow\infty$ and $\{ F_n\}$ is also bounded in ${\mathscr{H}_{\text{\rm Fried}}^{{\mbox{\rm \scriptsize Neu}}}}$. Thus, to show the compact embedding we need to prove that $\{H_n\}$ has a convergent subsequence in $L^2({{\mathbb T}}^d)$. To get a convergent subsequence of $\{ H_n\}$, it is sufficient to find a convergent subsequence of $\{F_n\}$ in $L^2({{\mathbb T}}^d)$. Write $F_n=f_n\mathbf{1}_\Lambda+\tilde{f_n}\mathbf{1}_{\Lambda^\complement}$, with $f_n\in C^2({{\mathbb T}}^d)$. Then, $$\begin{aligned}
&{\langle}F_n,F_n{\rangle}_{{{\mathcal E}}({{\mathbb I}}-{{{\mathfrak L}}^{{\mbox{\rm \scriptsize Neu}}}})}={\langle}F_n,F_n{\rangle}+{\langle}F_n, -{{{\mathfrak L}}^{{\mbox{\rm \scriptsize Neu}}}}F_n{\rangle}\\
&={\langle}f_n\mathbf{1}_\Lambda+\tilde{f_n}\mathbf{1}_{\Lambda^\complement},f_n\mathbf{1}_\Lambda+\tilde{f_n}\mathbf{1}_{\Lambda^\complement}{\rangle}+ {\langle}f_n\mathbf{1}_\Lambda+\tilde{f_n}\mathbf{1}_{\Lambda^\complement},-\Delta f_n\mathbf{1}_{\Lambda}-\Delta \tilde{f_n}\mathbf{1}_{\Lambda^\complement} {\rangle}.\end{aligned}$$ Expanding the right hand side and using Green identity, see Appendix \[appendix\], Theorem \[evans\], we get that $$\begin{aligned}
&\int_\Lambda f_n^2\,du + \int_{\Lambda^\complement}\tilde{f_n}^2\,du - \int_\Lambda f_n\Delta f_n\,du -\int_{\Lambda^\complement}\tilde{f_n}^2\Delta \tilde{f_n}\,du\\
&=\Vert f_n \mathbf{1}_{\Lambda}\Vert^2+\Vert \tilde{f_n} \mathbf{1}_{\Lambda^\complement}\Vert^2+\Vert \nabla f_n \mathbf{1}_{\Lambda}\Vert^2+\Vert \nabla \tilde{f_n} \mathbf{1}_{\Lambda^\complement}\Vert^2\,.\end{aligned}$$ Under the hypotheses of boundedness of the sequence $\{F_n\}$ in the norm induced by ${\langle}\cdot,\cdot{\rangle}_{{{\mathcal E}}({{\mathbb I}}-{{{\mathfrak L}}^{{\mbox{\rm \scriptsize Neu}}}})}$, the sequences $\{\Vert f_n \mathbf{1}_{\Lambda}\Vert^2\}$, $\{\Vert \tilde{f_n} \mathbf{1}_{\Lambda^\complement}\Vert^2\}$, $\{\Vert \nabla f_n \mathbf{1}_{\Lambda}\Vert^2\}$ and $\{\Vert \nabla \tilde{f_n} \mathbf{1}_{\Lambda^\complement}\Vert^2\}$ are bounded. By the Rellich-Kondrachov Compactness Theorem, $\{f_n \mathbf{1}_{\Lambda}\}$, $\{\tilde{f}_n\mathbf{1}_{\Lambda^\complement}\}$ have a common convergent subsequence in $L^2({{\mathbb T}}^d)$. This implies that $\{F_n\}$ has a convergent subsequence.
Auxiliary results {#appendix}
=================
\[A.3\] Let $G_1, \, G_2, \, G_3$ are continuous functions defined on the torus d-dimensional ${{\mathbb T}}^d$. Then, the application from $D([0,T],{{\mathcal M}})$ to ${{\mathbb R}}$ that associates to a trajectory $\{\pi_t:0\leq t\leq T\}$ the number $$\sup_{0\leq t \leq T}\Big|{\langle}\pi_t, G_1{\rangle}-{\langle}\pi_0, G_2{\rangle}-\int_0^t{\langle}\pi_s, G_3{\rangle}\, ds\Big|$$ is continuous in the Skorohod metric of $D([0,T],{{\mathcal M}})$.
\[evans\] Let $u,v\in C^2(\bar{U})$, where $U$ is a bounded open subset of ${{\mathbb R}}^n$, and ${\partial}U$ is $C^1$. Denote by $\cdot $ the inner product in ${{\mathbb R}}^n$, and by $\nu$ the normal exterior unitary vector to $U$ at ${\partial}U$. Then,
1. $\int_U \Delta u dx \;= \;\int _{{\partial}U}\frac{{\partial}u}{{\partial}\nu}\, dS$,
2. $\int_U \nabla v\cdot \nabla u \, dx\;=\;-\int_U u\Delta v\,dx+ \int_{{\partial}U}\frac{{\partial}u}{{\partial}\nu}u\, dS$,
3. $\int_U u\Delta v-v\Delta u\, dx\;=\; \int _{{\partial}U}u\frac{{\partial}u}{{\partial}\nu}-v\frac{{\partial}u}{{\partial}\nu}\, dS$.
Acknowledgements {#acknowledgements .unnumbered}
================
T.F. was supported through a project Jovem Cientista-9922/2015, FAPESB-Brazil. M.T. would like to thank CAPES for a PhD scholarship, which supported her research.
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author:
- Georgios Kofinas
- Genly Leon
- 'Emmanuel N. Saridakis'
title: 'Dynamical behavior in $f(T,T_G)$ cosmology'
---
Introduction {#Introduction}
============
Since the discovery of the universe late-times acceleration, a large amount of research has been devoted to its explanation. In principle, one can follow two main directions to achieve this. The first way is to modify the content of the universe introducing the dark energy concept, with its simpler candidates being a canonical scalar field, a phantom field or the combination of both fields in a unified model dubbed quintom (for reviews on dark energy see [@Copeland:2006wr; @Cai:2009zp] and references therein). The second direction that one can follow is to modify the gravitational sector itself (for a review see [@Capozziello:2011et] and references therein), acquiring a modified cosmological dynamics. However, note that apart from the interpretation, one can transform from one approach to the other, since the crucial issue is just the number of degrees of freedom beyond General Relativity and standard model particles (see [@Sahni:2006pa] for a review on such a unified point of view). Finally, note that the above scenarios, apart from late-times implications, can be also used for the description of the inflationary stage [@Nojiri:2003ft].
In the majority of modified gravitational theories, one suitably extends the curvature-based Einstein-Hilbert action of General Relativity. However, an interesting class of gravitational modification arises when one modifies the action of the equivalent formulation of General Relativity based on torsion. In particular, it is known that Einstein himself constructed the so-called “Teleparallel Equivalent of General Relativity” (TEGR) [@Unzicker:2005in; @Hayashi:1979qx; @Pereira; @Maluf:2013gaa] using the curvature-less Weitzenb[ö]{}ck connection instead of the torsion-less Levi-Civita one. The corresponding Lagrangian, namely the torsion scalar $T$, is constructed by contractions of the torsion tensor, in a similar way that the usual Einstein-Hilbert Lagrangian $R$ is constructed by contractions of the curvature (Riemann) tensor. Thus, inspired by the $f(R)$ modifications of the Einstein-Hilbert Lagrangian [@DeFelice:2010aj; @Nojiri:2010wj], one can construct the $f(T)$ modified gravity by extending $T$ to an arbitrary function [@Ferraro:2006jd; @Ben09; @Linder:2010py]. Note that although TEGR coincides with General Relativity at the level of equations, $f(T)$ does not coincide with $f(R)$, that is they represent different modification classes. Thus, the cosmological implications of $f(T)$ gravity are new and interesting [@Linder:2010py; @Chen:2010va; @Dent:2011zz; @Zheng:2010am; @Sharif001; @Li:2011rn; @Cai:2011tc; @Boehmer:2011gw; @Capozziello:2011hj; @Daouda:2011rt; @Geng:2011aj; @Wu:2011kh; @Gonzalez:2011dr; @Wei:2011aa; @Atazadeh:2011aa; @Farajollahi:2011af; @Karami:2012fu; @Iorio:2012cm; @Cardone:2012xq; @Capozziello:2012zj; @Jamil:2012ti; @Ong:2013qja; @Amoros:2013nxa; @Otalora:2013dsa; @Geng:2013uga; @Nesseris:2013jea; @Bamba:2013ooa; @Nashed:2014uta; @Harko:2014sja; @Harko:2014aja].
However, in curvature gravity, apart from the simple $f(R)$ modification one can construct more complicated extensions using higher-curvature corrections such as the Gauss-Bonnet term $G$ [@Wheeler:1985nh; @Antoniadis:1993jc; @Nojiri:2005vv] or functions of it [@Nojiri:2005jg; @DeFelice:2008wz], Lovelock combinations [@Lovelock:1971yv; @Deruelle:1989fj; @Charmousis:2008kc], and Weyl combinations [@Mannheim:1988dj; @Flanagan:2006ra; @Grumiller:2013mxa]. Inspired by these, in the recent work [@Kofinas:2014owa], the $f(T,T_G)$ gravitational modification was constructed, which is based on the old quadratic torsion scalar $T$, as well as on the new quartic torsion scalar $T_G$ that is the teleparallel equivalent of the Gauss-Bonnet term. Obviously, $f(T,T_G)$ theories cannot arise from the $f(T)$ ones, and additionally they are different from $f(R,G)$ class of curvature modified gravity. Thus, $f(T,T_G)$ is a novel class of gravitational modification.
The cosmological applications of $f(T,T_G)$ gravity proves to be very interesting [@Kofinas:2014owa]. Therefore, it is both interesting and necessary to perform a dynamical analysis, examining in a systematic way the allowed cosmological behaviors, focusing on the late-times stable solutions. The phase-space and stability analysis is a very powerful tool, since it reveals the global features of a given cosmological scenario, independently of the initial conditions and the specific evolution of the universe. In the present investigation we perform such a detailed phase-space analysis, and we extract the late-times, asymptotic solutions, calculating also the corresponding observable quantities, such as the deceleration parameter, the effective dark energy equation-of-state parameter, and the various density parameters.
The plan of the work is the following: In section \[fTgravity\] we briefly review the scenario of $f(T,T_G)$ gravity and in section \[fTcosmology\] we present its application in cosmology. In section \[Phasespaceanalysis\] we perform the detailed dynamical analysis for the simplest non-trivial model of $f(T,T_{G})$ gravity. In section \[implications\] we discuss the cosmological implications and the physical behavior of the scenario. Finally, in section \[Conclusions\] we summarize our results.
$f(T,T_G)$ gravity {#fTgravity}
==================
In this section we briefly review the $f(T,T_G)$ gravitational modification following [@Kofinas:2014owa]. In the whole manuscript we use the following notation: Greek indices run over the coordinate space-time, while Latin indices run over its tangent space.
In this framework the dynamical variable is the vierbein field $e_a(x^\mu)$. In terms of coordinates, it can be expressed in components as $e_a=e^{\,\,\, \mu}_a\partial_\mu$, while the dual vierbein is defined as $e^a=e^a_{\,\,\, \mu}d x^\mu$. Concerning the other field, that is the connection 1-forms $\omega^a_{\,\,\,
b}(x^\mu)$ which defines the parallel transportation, one uses the Weitzenb[ö]{}ck one, which in all coordinate frames is defined as $$\begin{aligned}
\omega_{\,\,\,\mu\nu}^{\lambda}=e_{a}^{\,\,\,\lambda}e^{a}_{\,\,\,\mu
, \nu } .
\label{Weinzdef}\end{aligned}$$ Due to its inhomogeneous transformation law it has tangent-space components $\omega_{\,\,\,bc}^{a}=0$, assuring the property of vanishing non-metricity. Additionally, for an orthonormal vierbein the metric tensor is given by the relation $$\label{metrdef}
g_{\mu\nu} =\eta_{ab}\, e^a_{\,\,\,\mu} \, e^b_{\,\,\,\nu},$$ where $\eta_{ab}=\text{diag}(-1,1,1,1)$ and indices $a,b,...$ are raised/lowered with $\eta_{ab}$.
One can now define the torsion tensor as $$T^{\lambda}_{\,\,\,\mu\nu}=
e_{a}^{\,\,\,\lambda}\left(\partial_\nu e^{a}_{\,\,\,\mu}-\partial_\mu
e^{a}_{\,\,\,\nu}\right),$$ while the Riemann tensor is zero by construction, due to the teleparallelism condition which is imposed with the use of the Weitzenb[ö]{}ck connection. Moreover, the contorsion tensor, which equals the difference between the Weitzenböck and Levi-Civita connections, is defined as $$\label{cotorsion}
\mathcal{K}^{\mu\nu}_{\:\:\:\:\rho}=-\frac{1}{2}\Big(T^{\mu\nu}_{
\:\:\:\:\rho}
-T^{\nu\mu}_{\:\:\:\:\:\rho}-T_{\rho}^{\:\:\mu\nu}\Big).$$
Since in this formulation all the information concerning the gravitational field is included in the torsion tensor $T^{\lambda}_{\,\,\,\mu\nu}$, one can use it in order to construct torsion invariants. The simplest invariants that one can build are quadratic in the torsion tensor. In particular, the combination $$\begin{aligned}
T&=&\frac{1}{4}T^{\mu\nu\lambda}T_{\mu\nu\lambda}+\frac{1}{2}T^{\mu\nu\lambda}
T_{\lambda\nu\mu}-T_{\nu}^{\,\,\,\nu\mu}T^{\lambda}_{\,\,\,\lambda\mu},
\label{Tquad}\end{aligned}$$ which can in general be defined in an arbitrary dimension $D$, is the “torsion scalar”, and if it is used as a Lagrangian and be varied in terms of the vierbein it gives rise to the Einstein field equations. That is why the gravitational theory characterized by the action $$\begin{aligned}
S=-\frac{1}{2\kappa^2}\int d^4 x \,e\,T \, ,
\label{teleaction}\end{aligned}$$ with $e=\det{(e^{a}_{\,\,\,\mu})}=\sqrt{|g|}$ and $\kappa^2\equiv 8\pi
G$ the gravitational constant, is called Teleparallel Equivalent of General Relativity. In these lines, one can be based on $T$ in order to construct modified gravitational theories extending the TEGR action to [@Linder:2010py; @Chen:2010va; @Dent:2011zz; @Zheng:2010am; @Sharif001; @Li:2011rn; @Cai:2011tc; @Boehmer:2011gw; @Capozziello:2011hj; @Daouda:2011rt; @Geng:2011aj; @Wu:2011kh; @Gonzalez:2011dr; @Wei:2011aa; @Atazadeh:2011aa; @Farajollahi:2011af; @Karami:2012fu; @Iorio:2012cm; @Cardone:2012xq; @Capozziello:2012zj; @Jamil:2012ti; @Ong:2013qja; @Amoros:2013nxa; @Otalora:2013dsa; @Geng:2013uga; @Nesseris:2013jea; @Bamba:2013ooa; @Nashed:2014uta; @Harko:2014sja; @Harko:2014aja] $$\begin{aligned}
S=\frac{1}{2\kappa^2}\int d^4 x \,e\,f(T) \, .
\label{teleaction}\end{aligned}$$ We stress here that although the field equations of TEGR are identical with those of General Relativity, $f(T)$ modification gives rise to different equations than $f(R)$ modification.
However, one can use the torsion tensor in order to construct higher-order torsion invariants, in a similar way that one uses the Riemann tensor in order to construct higher-order curvature invariants. In particular, in [@Kofinas:2014owa] the invariant $$\begin{aligned}
&&T_G=\left(\mathcal{K}^{\kappa}_{\,\,\,\varphi\pi}\mathcal{K}^{\varphi\lambda}_{\,\,\,\,\,
\,\, \rho }\mathcal{K}^{\mu}_{\,\,\,\,\chi\sigma}
\mathcal{K}^{\chi\nu}_{\,\,\,\,\,\,\,\tau}
-2\mathcal{K}^{\kappa\!\lambda}_{\,\,\,\,\,\,\pi}\mathcal{K}^{\mu}_{
\,\,\,\varphi\rho}
\mathcal{K}^{\varphi}_{\,\,\,\chi\sigma}\mathcal{K}^{\chi\nu}_{\,\,\,\,\,\,\tau}\right.
{\nonumber}\\
&&\left. \ \ \ \ \ \ \ \ \ \ \
+2\mathcal{K}^{\kappa\!\lambda}_{\,\,\,\,\,\,\pi}\mathcal{K}^{\mu}_{
\,\,\,\,\varphi\rho}
\mathcal{K}^{\varphi\nu}_{\,\,\,\,\,\,\chi}\mathcal{K}^{\chi}_{\,\,\,\,\sigma\tau}
+2\mathcal{K}^{\kappa\!\lambda}_{\,\,\,\,\,\,\pi}\mathcal{K}^{\mu}_{
\,\,\,\,\varphi\rho}
\mathcal{K}^{\varphi\nu}_{\,\,\,\,\,\,\,\sigma,\tau}\right)
\delta^{\pi\rho\sigma\tau}_{\kappa \lambda \mu \nu}
\label{TG}\end{aligned}$$ was constructed in an arbitrary dimension $D$, where the generalized $\delta^{\pi\rho\sigma\tau}_{\kappa \lambda \mu \nu}$ is the determinant of the Kronecker deltas. This invariant is just the Teleparallel Equivalent of the Gauss-Bonnet combination $
G=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\kappa\lambda}R^{\mu\nu\kappa\lambda}
$, and in four dimensions it reduces to a topological term. Thus, inspired by the $f(G)$ extensions of General Relativity [@Nojiri:2005jg; @DeFelice:2008wz], one can consider general functions $f(T_G)$ in the action too.
Taking the above into account, one can propose a new class of gravitational modifications as [@Kofinas:2014owa] $$\begin{aligned}
S =\frac{1}{2\kappa^{2}}\!\int d^{4}x\,e\,f(T,T_G)\,,
\label{fGBtelaction}\end{aligned}$$ which is also valid in higher dimensions. Since $T_G$ is quartic in the torsion tensor, $f(T,T_G)$ gravity is more general than the $f(T)$ class. Additionally, $f(T,T_G)$ gravity is obviously different from $f(R,G)$ one [@Nojiri:2005jg; @DeFelice:2008wz; @Davis:2007ta; @DeFelice:2009aj; @Jawad:2013wla]. Note that the usual Einstein-Gauss-Bonnet theory for $D>4$ arises in the special case $f(T,T_G)=-T+\alpha T_G$ (with $\alpha$ the Gauss-Bonnet coupling), while TEGR (that is GR) is obtained for $f(T,T_G)=-T$.
$f(T,T_G)$ cosmology {#fTcosmology}
====================
In order to investigate the cosmological implication of the above action (\[fGBtelaction\]), we consider a spatially flat cosmological ansatz $$ds^{2}=-dt^{2}+a^{2}(t)\delta_{\hat{i}\hat{j}}dx^{\hat{i}}dx^{\hat{j}}\,,
\label{metriccosmo}$$ where $a(t)$ is the scale factor. This metric arises from the diagonal vierbein $$\label{vierbeincosmo}
e^{a}_{\,\,\,\mu}=\text{diag}(1,a(t),a(t),a(t))$$ through (\[metrdef\]), while the dual vierbein is $e_{a}^{\,\,\,\mu}=\text{diag}(1,a^{-1}(t),
a^{-1}(t),a^{-1}(t))$, and its determinant $e=a(t)^{3}$. Thus, inserting the vierbein (\[vierbeincosmo\]) into relations (\[Tquad\]) and (\[TG\]), we find $$\begin{aligned}
\label{Tcosmo1}
&&T=6H^2\\
&&T_G= 24H^2\big(\dot{H}+H^2\big) ,
\label{TGcosmo1}\end{aligned}$$ where $H=\frac{\dot{a}}{a}$ is the Hubble parameter and dots denote differentiation with respect to $t$.
Finally, in order to acquire a realistic cosmology we additionally consider a matter action $S_{m}$, corresponding to an energy-momentum tensor $\Theta^{\mu\nu}$, focusing on the case of a perfect fluid of energy density $\rho_m$ and pressure $p_m$.
As it was showed in [@Kofinas:2014owa], variation of the total action $S+S_m$ gives in the case of FRW geometry the following Friedmann equations $$f-12H^{2}f_{T}-T_G f_{T_G}
+24H^{3}\dot{f_{T_G}}=2\kappa^{2}\rho_m
\label{Fr1}$$ $$f-4\big(3H^2+\dot{H}\big)f_T-4H\dot{f_T}-T_G
f_{T_G}+\frac{2}{3H}T_G\dot{f_{T_G}}+8H^2\ddot{f_{T_G}}
=-2\kappa^{2} p_m\,,
\label{Fr2}$$ where $\dot{f_{T}}=f_{TT}\dot{T}+f_{TT_{G}}\dot{T}_{G}$, $\dot{f_{T_{G}}}=f_{TT_{G}}\dot{T}+f_{T_{G}T_{G}}\dot{T}_{G}$, $\ddot{f_{T_{G}}}=f_{TTT_{G}}\dot{T}^{2}+2f_{TT_{G}T_{G}}\dot{T}
\dot{T}_{G}+f_{T_{G}T_{G}T_{G}}\dot{T}_{G}^{\,\,2}+
f_{TT_{G}}\ddot{T}+f_{T_{G}T_{G}}\ddot{T}_{G}$, with $f_{TT}$, $f_{TT_{G}}$,... denoting multiple partial differentiations of $f$ with respect to $T$, $T_{G}$. Here, the involved time-derivatives of $\dot{T}$, $\ddot{T}$, $\dot{T}_{G}$, $\ddot{T}_{G}$ are straightforwardly obtained using (\[Tcosmo1\]), (\[TGcosmo1\]).
Therefore, we can rewrite the Friedmann equations (\[Fr1\]) and (\[Fr2\]) in the usual form $$\begin{aligned}
\label{Fr1b}
H^2& =& \frac{\kappa^2}{3}\left(\rho_m + \rho_{DE} \right) \\
\label{Fr2b}
\dot{H}& =&-\frac{\kappa^2}{2}\left(\rho_m +p_m+\rho_{DE}+p_{DE}\right),\end{aligned}$$ defining the energy density and pressure of the effective dark energy sector as $$\label{rhode}
\rho_{DE}\equiv\frac{1}{2\kappa^2}\left( 6H^2
-f+12H^{2}f_{T}+T_G f_{T_G}-24H^{3}\dot{f_{T_G}}\right)$$ $$\label{pde}
p_{DE} \equiv \frac{1}{2\kappa^2}\left[\!
-2(2\dot{H}\!+\!3H^2)+f\!-\!4\big(\dot{H}\!+\!3H^2\big)f_T-4H\dot{f_T}-T_G
f_{T_G}+\frac{2}{3H}T_G\dot{f_{T_G}}+8H^2\ddot{f_{T_G}}
\right]\!.$$ The standard matter $\rho_{m}$ is conserved independently, i.e. $\dot{\rho}_{m}+3H(\rho_{m}+p_{m})=0$. One can easily verify that the dark energy density and pressure satisfy the usual evolution equation $$\begin{aligned}
\dot{\rho}_{DE} +3H(\rho_{DE}+p_{DE})=0,\end{aligned}$$ and we can also define the dark energy equation-of-state parameter as usual $$\begin{aligned}
w_{DE}\equiv \frac{p_{DE}}{\rho_{DE}}.\end{aligned}$$
Dynamical analysis {#Phasespaceanalysis}
==================
In order to perform the stability analysis of a given cosmological scenario, one first transforms it to its autonomous form $\label{eomscol}
\textbf{X}'=\textbf{f(X)}$ [@Perko; @Ellis; @Copeland:1997et; @Ferreira:1997au; @Chen:2008ft; @Cotsakis:2013zha; @Giambo':2009cc], where $\textbf{X}$ are some auxiliary variables presented as a column vector and primes denote derivatives with respect to $N=\ln a$. Then, one extracts the critical points $\bf{X_c}$ by imposing the condition $\bf{X}'=0$, and in order to determine their stability properties one expands around them with $\textbf{U}$ the column vector of the perturbations of the variables. Therefore, for each critical point the perturbation equations are expanded to first order as $\label{perturbation} \textbf{U}'={\bf{Q}}\cdot
\textbf{U}$, with the matrix ${\bf {Q}}$ containing the coefficients of the perturbation equations. The eigenvalues of ${\bf {Q}}$ determine the type and stability of the specific critical point.
In order to perform the above analysis, we need to specify the $f(T,T_{G})$ form. In usual $f(T)$ gravity one starts adding corrections of $T$-powers. However, in the scenario at hand, since $T_G$ contains quartic torsion terms it is of the same order with $T^2$. Therefore, $T$ and $\sqrt{T^{2}+\alpha_{2}T_{G}}$ are of the same order, and thus, one should use both in a modified theory. Hence, the simplest non-trivial model, which does not introduce a new mass scale into the problem and differs from General Relativity, is the one based on $$f(T,T_G)=-T+\alpha_1\sqrt{T^2+\alpha_2 T_G}\,.
\label{ansantz}$$ The couplings $\alpha_{1},\alpha_{2}$ are dimensionless and the model is expected to play an important role at late times. Indeed, this model, although simple, can lead to interesting cosmological behavior, revealing the advantages, the capabilities, and the new features of $f(T,T_{G})$ cosmology. We mention here that when $\alpha_2=0$ this scenario reduces to TEGR, that is to General Relativity, with just a rescaled Newton’s constant, whose dynamical analysis has been performed in detail in the literature [@Copeland:1997et; @Ferreira:1997au; @Chen:2008ft]. Thus, in the following we restrict our analysis to the case $\alpha_2\neq0$.
In this case, the cosmological equations are the Friedmann equations (\[Fr1b\]), (\[Fr2b\]), with the effective dark energy density and pressure (\[rhode\]) and (\[pde\]) becoming $$\label{rhodeb}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\kappa^2 \rho_{DE}=
\frac{\sqrt{3} \alpha_1 H^2 \left\{\alpha_2^2 \ddot H+9
\alpha_2 H \dot H+\big[(3-2 \alpha_2) \alpha_2+9\big] H^3\right\}}{D^{3/2}}$$ $$\begin{aligned}
\label{pdeb}
&&\!\!\!\!\!\!
\kappa^2 p_{DE}
=\frac{\alpha_1 \!\left\{\!(2
\alpha_2\!+\!3) \big[\alpha_2 (10 \alpha_2\!-\!51)\!-\!18\big] H^4 \!+\!\alpha_2 \big[4
\alpha_2 (5 \alpha_2\!-\!21)\!-\!90\big] H^2 \dot H \!-\!54 \alpha_2^2 \dot H^2 \right\}\!H\dot{H}}
{\sqrt{3} D^{5/2}}
\nonumber\\
&&\ \ \ \ \ \ \ \ -\frac{\alpha_1 \alpha_{2}^{2} H \dddot H}
{\sqrt{3} D^{3/2}}
+
\frac{\sqrt{3} \alpha_1 \alpha_2^3 H \ddot H^2}{D^{5/2}}-
\frac{2
\alpha_1 \alpha_2^2 \ddot H \left[2 (\alpha_2\!-\!3) H^2 \dot H+2 \alpha_2
\dot H^2+(6 \alpha_2\!+\!9) H^4\right]}{\sqrt{3} D^{5/2}}
\nonumber\\
&&\ \ \ \ \ \ \ \ + \frac{\sqrt{3}\alpha_1 (\alpha_2\!-\!3) (2
\alpha_2\!+\!3)^2 H^7 }
{ D^{5/2}} \,,\end{aligned}$$ where $D=3H^2+2\alpha_2(\dot{H}+H^2)$. In order to perform the dynamical analysis of this cosmological scenario, we introduce the following auxiliary variables $$\begin{aligned}
\label{xdefin}
&&x=\sqrt{\frac{D}{3H^2}}=\sqrt{1+\frac{2 \alpha_2}{3}\Big(1+\frac{\dot
H}{H^2}\Big)}\\
&&\Omega_m=\frac{\kappa ^2 \rho_m}{3 H^2}.\end{aligned}$$ Thus, the cosmological system is transformed to the following autonomous form $$\begin{aligned}
&x'=-\frac{x \left[3 \alpha_1 x^2-6
(1\!-\!\Omega_m)x+\alpha_1 (3\!-\!4 \alpha_2)\right]}{2 \alpha_1 \alpha_2}
\label{eqx}\\
&\Omega_m'= -\frac{\Omega_m \left(3
x^2+\alpha_2+3 \alpha_2 w_m-3\right)}{\alpha_2}\,, \label{eqm}\end{aligned}$$ where primes denote differentiation with respect to the new time variable $N$, so $f'= H^{-1} \dot f$. The above dynamical system is defined in the phase space $\left\{(x,\Omega_m)| x\in [0,\infty), \Omega_m\in
[0,\infty]\right\}$. One can now express the various observables in terms of the above auxiliary variables $\Omega_m$ and $x$ (note that $\Omega_m$ is an observable itself, that is the matter density parameter). In particular, the deceleration parameter $q\equiv -1-\dot{H}/H^2$ is given by $$q=\frac{3 \left(1-x^2\right)}{2 \alpha_2}.
\label{decc}$$ Similarly, the dark energy density parameter straightaway reads $$\Omega_{DE}\equiv \frac{\kappa^2\rho_{DE}}{3 H^2}= 1-\Omega_m.$$ The dark energy equation-of-state parameter $w_{DE}$ is given by the relation $2q=1+3(w_{m}\Omega_{m}+w_{DE}\Omega_{DE})$, and therefore $$w_{DE}= \frac{3 x^2+\alpha_2+3 \alpha_2 w_m \Omega_m-3}{3 \alpha_2
(\Omega_m-1)}\,,
\label{wdephasespace}$$ where $w_m\equiv \frac{p_m}{\rho_m}$ is the matter equation-of-state parameter. In the following, without loss of generality we assume dust matter ($w_m=0$), but the extension to general $w_m$ is straightforward.
Finite phase space analysis
---------------------------
We now proceed to the detailed phase-space analysis. The real and physically interesting (that is corresponding to an expanding universe) critical points of the autonomous system -, obtained by setting the left hand sides of these equations to zero, are presented in Table \[tab1\]. In the same table we provide their existence conditions. Their stability is extracted by examining the sign of the real part of the eigenvalues of the $2\times2$ matrix ${\bf {Q}}$ of the corresponding linearized perturbation equations. This procedure is shown in the Appendix \[appendixfinite\], and in Table \[tab1\] we summarize the stability results. Furthermore, for each critical point we calculate the values of the deceleration parameter $q$ and the dark energy equation-of-state parameter $w_{DE}$ given by (\[decc\]) and (\[wdephasespace\]), and we present the results in Table \[tab2\]. Finally, in the same Table we summarize the physical description of the solutions, which we analyze in the next section.
Phase space analysis at infinity
--------------------------------
Due to the fact that the dynamical system - is non-compact, there could be non-trivial dynamical features in the asymptotic regime too. Therefore, in order to complete the phase space analysis we must extend our investigation with the analysis at infinity using the Poincaré projection method [@PoincareProj; @Xu:2012jf].
We introduce the new coordinates $(r,\theta)$ defined by $$\begin{aligned}
&&x=\frac{r}{1-r}\cos\theta\label{Poincvariables1}
\\
&& \Omega_m=\frac{r}{1-r}\sin\theta,
\label{Poincvariables2}\end{aligned}$$ with $\theta\in
\left[0,\frac{\pi}{2}\right]$ and $r\in \left[0,1\right)$. Thus, the critical points at infinity, that is $x\rightarrow+\infty$ or $\Omega_m\rightarrow+\infty$ (that is $R^2\equiv x^2+\Omega_m^2\rightarrow
+\infty$), correspond to $r\rightarrow 1^-$. Moreover, the region of the plane $(r,\theta)$ that is corresponding to $0\leq x,\, 0\leq \Omega_m\leq 1$ is given by $$\label{rest_infty}
\left\{(r,\theta): 0\leq r\leq \frac{1}{2}, 0\leq \theta\leq
\frac{\pi}{2}\right\}\cup \left\{(r,\theta): \frac{1}{2}<r<1,
0\leq \theta \leq \arcsin \left(\frac{1-r}{r}\right)\right\}.$$ Using relations (\[Poincvariables1\]), (\[Poincvariables2\]) and substituting into (\[decc\]) and (\[wdephasespace\]), we obtain the deceleration and equation-of-state parameters as a function of the new variables, namely $$\begin{aligned}
&&q=\frac{3 \left(1-2r+r^2\sin^2\theta\right)}{2 \alpha_2(1-r)^2}
\label{q22}\\
&&w_{DE}=\frac{\alpha_2(1-r)^2-3\left(1-2r+r^2\sin^2\theta\right)}{
3\alpha_2(1-r)\left[r(\sin\theta+1)-1\right] }\, ,
\label{wdephase22}\end{aligned}$$ while $\Omega_{DE}$ is just $1-\Omega_m$, that is $$\begin{aligned}
\Omega_{DE}=\frac{1-r(1+\sin\theta)}{1-r}.
\label{Omegasde22}\end{aligned}$$
Cr. P. $\theta$ Stability $\Omega_{DE}$ $q$ $w_{DE}$
-------- ------------------------------------------ --------------------------- --------------- -------------------------------- --------------------------------
$Q_1$ $0$ saddle point $1$ $-\text{sgn}(\alpha_2) \infty$ $-\text{sgn}(\alpha_2) \infty$
$Q_2$ $\arctan\left(\frac{\alpha_1}{2}\right)$ unstable for $\alpha_2>0$ $-\infty$ $-\text{sgn}(\alpha_2) \infty$ $\text{sgn}(\alpha_2) \infty$
stable for $\alpha_2<0$
$Q_3$ $\frac{\pi}{2}$ see numerical elaboration $-\infty$ $\frac{3}{2\alpha_2}$ $0$
Applying the procedure described in the Appendix \[appendixinfin\], we conclude that there are three critical point at infinity. These critical points, along with their stability conditions are presented in Table \[tab3\]. In the same Table we include the corresponding values of the observables $\Omega_{DE}$, $q$ and $w_{DE}$, calculated using (\[q22\]), (\[wdephase22\]) and (\[Omegasde22\]). These points correspond to Big Rip, sudden or other forms of singularities [@Sami:2003xv; @Nojiri:2005sx; @Briscese:2006xu; @Bamba:2008ut; @Capozziello:2009hc; @Saridakis:2009jq], depending on whether the singularity is reached at finite or infinite time, and on their observable features.
Cosmological Implications {#implications}
=========================
In the previous section we performed the complete phase-space analysis of the physically interesting model with $f(T,T_G)=-T+\alpha_1\sqrt{T^2+\alpha_2
T_G}$, both at the finite region and at infinity. Thus, in the present section we discuss the corresponding cosmological behavior. As usual, the features of the solutions can be easily deduced by the values of the observables. In particular, $q<0$ ($q>0$) corresponds to acceleration (deceleration), $q=-1$ to de Sitter solution, $w_{DE}>-1$ ($w_{DE}<-1$) corresponds to quintessence-like (phantom-like) behavior, and $\Omega_{DE}=1$ implies a dark-energy dominated universe.
Point $P_1$ is stable for the conditions presented in Table \[tab1\], and thus it can attract the universe at late times. Since the dark energy and matter density parameters are of the same order, this point represents a dark energy - dark matter scaling solution, alleviating the coincidence problem (note that in order to handle the coincidence problem one should provide an explanation of why the present $\Omega_m$ and $\Omega_{DE}$ are of the same order, although they follow different evolution behaviors). However, it has the disadvantage that $w_{DE}$ is $0$ and the universe is not accelerating, as expected [@Kofinas:2005hc]. Although this picture is not favored by observations, it may simply imply that the today universe has not yet reached its asymptotic regime.
Point $P_2$ is stable for the conditions presented in Table \[tab1\], and therefore, it can be the late-times state of the universe. It corresponds to a dark energy dominated universe that can be accelerating. Interestingly enough, depending on the model parameters, the dark energy equation-of-state parameter can lie in the quintessence regime, it can be equal to the cosmological constant value $-1$, or it can even lie in the phantom regime. These features are a great advantage of the scenario at hand, since they are compatible with observations, and moreover they are obtained only due to the novel features of $f(T,T_G)$ gravity, without the explicit inclusion of a cosmological constant or a scalar field, either canonical or phantom one.
Point $P_3$ is stable for the conditions presented in Table \[tab1\], and therefore, it can attract the universe at late times. It has similar features with $P_2$, but for different parameter regions. Namely, it corresponds to a dark energy dominated universe that can be accelerating, where the dark energy equation-of-state parameter can lie in the quintessence or phantom regime, or it can be exactly $-1$. These features make also this point a good candidate for the description of Nature.
Point $P_4$ corresponds to a dark energy dominated universe that can be accelerating, where the dark energy equation-of-state parameter can lie in the quintessence or phantom regime, or it can be exactly $-1$. However, $P_{4}$ is not stable and thus it cannot attract the universe at late times.
Finally, the present scenario possesses three critical points at infinity, two of which can be stable. They correspond to Big Rip, sudden or other forms of singularities, depending on the parameter choice. We mention that as the universe moves towards these stable points the matter density parameter $\Omega_m$ will be larger than $1$. Although this is not theoretically excluded, growth-index observations could indeed exclude these regions (as it happens in $f(R)$ gravity [@DeFelice:2010aj]), and thus the corresponding parameter range that leads the universe to their basin of attraction should be excluded in the model at hand. Such a detailed investigation has not been performed in torsion-based gravity, and therefore it has to be done from the beginning. However, since it lies outside the scope of the present work, it is left for a future project.
In order to present the aforementioned behavior more transparently, we first evolve the autonomous system - numerically for the parameter choices $\alpha_1=-\sqrt{33}$ and $
\alpha_2=4$, assuming the matter to be dust ($w_m=0$). The corresponding phase-space behavior is depicted in Fig. \[fig1\]. For completeness we also present the projection in the “Poincaré plane” $(r,\theta)$, where we depict the behavior at both the finite and the infinite region. In this case the universe at late times is attracted by the dark-energy dominated de Sitter attractor $P_2$, where the effective dark energy behaves like a cosmological constant. At infinity, there is not any stable point, and thus the universe cannot result in any form of singularity.
In Fig. \[fig2\] we present the phase-space behavior of the autonomous system - for the choice $\alpha_1=\frac{1}{2}$ and $
\alpha_2=-\frac{1}{2}$ (assuming $w_m=0$), and its projection on the “Poincaré plane” $(r,\theta)$. In this case the attractor at the finite region is the phantom solution $P_2$. Additionally, the attractor in the infinite region is $Q_2$, that is a future singularity.
Finally, in Fig. \[fig5\] are present some orbits and the corresponding Poincaré projections for the choice $\alpha_1=3$ and $
\alpha_2=\frac{3}{2}$, with $w_m=0$. In this case, the universe is attracted by the quintessence solution $P_3$. Furthermore, in the infinite region the attractor is $Q_3$, that is a future singularity. [^1]
Conclusions {#Conclusions}
===========
In the present work we studied the dynamical behavior of the recently proposed scenario of $f(T,T_G)$ cosmology [@Kofinas:2014owa]. This class of modified gravity is based on the quadratic torsion scalar $T$, which is the Lagrangian of the teleparallel equivalent of General Relativity, as well as on the new quartic torsion scalar $T_G$, which is the teleparallel equivalent of the Gauss-Bonnet term. Obviously, $f(T,T_G)$ theories are more general and cannot be spanned by the simple $f(T)$ ones, and additionally they are different from $f(R,G)$ class of curvature modified gravity too.
Without loss of generality, as a simple, but non-trivial example, capable of revealing the advantages and the new features of the theory, we considered a model where $T$ and $T_G$ corrections are of the same order, and thus expected to play an important role at late times. We performed for a spatially flat universe the complete and detailed phase-space behavior, both in the finite and infinite regions, calculating additionally also the values of basic observables such is the various density parameters, the deceleration parameter and the dark energy equation-of-state parameter.
This scenario exhibits interesting cosmological behaviors. In particular, depending on the model parameters, the universe can result in a dark energy dominated accelerating solution and the dark energy equation-of-state parameter can lie in the quintessence regime, it can be equal to the cosmological constant value $-1$, or it can even lie in the phantom regime. Additionally, it can result in a dark energy - dark matter scaling solution, and thus it can alleviate the coincidence problem. Finally, under certain parameter choices the universe can result to Big Rip, sudden, or other form of singularities, as it is usual in many modified gravitational theories. Definitely, before the scenario at hand can be considered as a good candidate for the description of Nature, a detailed confrontation with observations should be performed. In particular, one should use data from local gravity experiments (Solar System observations), as well as type Ia Supernovae (SNIa), Baryon Acoustic Oscillations (BAO), and Cosmic Microwave Background (CMB) radiation data, in order to impose constraints on the model. These necessary investigations lie beyond the scope of the present work and are left for a future project.
GL was supported by COMISIÓN NACIONAL DE CIENCIAS Y TECNOLOGÍA through Proyecto FONDECYT DE POSTDOCTORADO 2014 grant 3140244 and by DI-PUCV grant 123.730/2013. The research of ENS is implemented within the framework of the Action “Supporting Postdoctoral Researchers” of the Operational Program “Education and Lifelong Learning” (Actions Beneficiary: General Secretariat for Research and Technology), and is co-financed by the European Social Fund (ESF) and the Greek State.
Stability of the finite critical points {#appendixfinite}
=======================================
For the critical points $(x_c,\Omega_{mc})$ of the autonomous system -, presented in Table \[tab1\], the coefficients of the perturbation equations form a $2\times2$ matrix ${\bf {Q}}$, which reads: $$\begin{aligned}
{\bf
{Q}}_{11}&=&\frac{\alpha_1 (4 \alpha_2-3)-9 \alpha_1 x^2-12 x (\Omega_m-1)}{2 \alpha_1 \alpha_2}
\nonumber\\
{\bf
{Q}}_{12}&=&-\frac{3 x^2}{\alpha_1 \alpha_2}
\nonumber\\
{\bf
{Q}}_{21}&=&-\frac{6 x \Omega_m}{\alpha_2}
\nonumber\\
{\bf
{Q}}_{22}&=&-\frac{\alpha_2+3 x^2-3}{\alpha_2}
\nonumber\end{aligned}$$ Thus, we can straightforwardly see that using the explicit critical points shown in Table \[tab1\], the matrix ${\bf {Q}}$ acquires a simple form that allows for an easy calculation of its eigenvalues. Hence, by examining the sign of the real parts of these eigenvalues, we can classify the corresponding critical point. In particular, if all eigenvalues of a critical point have positive real parts then this point is unstable, if they all have negative real parts then it is stable, and if they change sign then it is a saddle one. In the following we present the results for each separate point.
Point $P_1$ has the coordinates $$P_1: (x,\Omega_m)=\left( \sqrt{1-\frac{\alpha_2}{3}}, \frac{\alpha_1
\sqrt{9-3 \alpha_2} (6-5 \alpha_2)+6 (\alpha_2-3)}{6
(\alpha_2-3)}\right),$$ that is it exists for either $
\alpha_2=\frac{6}{5}$ or $\frac{6}{5}<\alpha_2<3,\, \alpha_1 \geq -2\sqrt{\frac{3(3-\alpha_2)}{(5\alpha_2-6)^2}}$ or $\alpha_2<\frac{5}{6},\, \alpha_1 \leq 2
\sqrt{\frac{3(3-\alpha_2)}{(5\alpha_2-6)^2}}$. The eigenvalues of the corresponding linearization matrix are $$\begin{aligned}
\left\{-\frac{\sqrt{\alpha_1 \left[(336-71 \alpha_2)
\alpha_2-288\right]+32
\sqrt{3} (3-\alpha_2)^{3/2}}}{4 \sqrt{\alpha_1} \alpha_2}-\frac{3}{4},
\right. \nonumber \\ \left. \frac{\sqrt{\alpha_1
\left[(336-71 \alpha_2) \alpha_2-288\right]+32 \sqrt{3}
(3-\alpha_2)^{3/2}}}{4
\sqrt{\alpha_1}
\alpha_2}-\frac{3}{4}\right\}.
\end{aligned}$$ Therefore, $P_1$ is a stable spiral for $$\alpha_2<3,\ \ -32 \sqrt{3}
\sqrt{\frac{(3-\alpha_2)^3}{\left(71 \alpha_2^2-336
\alpha_2+288\right)^2}}<\alpha_1<0,$$ or $$\alpha_1<0,\alpha_2\leq \frac{1}{71} \left(168-36 \sqrt{6}\right)\lesssim 1.12421.$$ Otherwise it is a saddle (we have excluded the parameter values that leads to non-hyperbolic critical points).
Point $P_2$ has the coordinates $$P_2: (x,\Omega_m)=\left(\frac{3-\sqrt{3 \alpha_1^2 (4
\alpha_2-3)+9}}{3 \alpha_1}, 0\right),$$ that is it exists for either $\alpha_2<\frac{3}{4},0<\alpha_1\leq \sqrt{\frac{3}{3-4 \alpha_2}}$ or $ \alpha_1\neq 0, \alpha_2=\frac{3}{4}$ or $
\alpha_2>\frac{3}{4}, \alpha_1<0$. The eigenvalues of the linearization matrix are $$\!\!\!
\left\{\frac{2 \sqrt{3 \alpha_1^2 (4 \alpha_2-3)+9}}{\alpha_1^2
\alpha_2}-\frac{6}{\alpha_1^2 \alpha_2}+\frac{6}{\alpha_2}-5, \,
\frac{\sqrt{3 \alpha_1^2 (4 \alpha_2-3)+9}}{\alpha_1^2
\alpha_2}-\frac{3}{\alpha_1^2 \alpha_2}+\frac{3}{\alpha_2}-4\right\}.$$ Hence, it is an stable node for either $$\alpha_2<0, \ \ 0<\alpha_1<2
\sqrt{\frac{3(3-\alpha_2)}{(5
\alpha_2-6)^2}}$$ or $$\frac{6}{5}<\alpha_2\leq
3,\ \ \alpha_1<-2 \sqrt{\frac{3(3- \alpha_2)}{(5 \alpha_2-6)^2}}$$ or $$\alpha_2>3,\ \ \alpha_1<0.$$ Additionally, it is unstable node for $$0<\alpha_2<\frac{3}{4},\ \
0<\alpha_1<\sqrt{\frac{3}{3-4 \alpha_2}}.$$ Finally, for the remaining parameter range in the hyperbolic domain, the point behaves as a saddle.
Point $P_3$ has the coordinates $$P_3: (x, \Omega_m)=\left( \frac{3+\sqrt{3 \alpha_1^2 (4
\alpha_2-3)+9}}{3 \alpha_1}, 0\right),$$ that is it exists for $\alpha_2<\frac{3}{4},\, 0<\alpha_1\leq \sqrt{\frac{3}{3-4 \alpha_2}}$ or $\alpha_2\geq \frac{3}{4},
\alpha_1>0$. The eigenvalues of the corresponding linearization matrix are $$\begin{aligned}
\left\{-\frac{2 \sqrt{3} \sqrt{4 \alpha_1^2 \alpha_2-3
\alpha_1^2+3}}{\alpha_1^2 \alpha_2}-\frac{6}{\alpha_1^2
\alpha_2}+\frac{6}{\alpha_2}-5, \,
-\frac{\sqrt{3} \sqrt{4 \alpha_1^2 \alpha_2-3 \alpha_1^2+3}}{\alpha_1^2
\alpha_2}-\frac{3}{\alpha_1^2 \alpha_2}+\frac{3}{\alpha_2}-4\right\}.\end{aligned}$$ That is, it is stable node for $$\alpha_1>0,\ \ \alpha_2\geq \frac{6}{5},$$ it is unstable node for $$\alpha_2<0, \ \ 0<\alpha_1<\frac{\sqrt{3}}{\sqrt{3-4 \alpha_2}},$$ otherwise it is a saddle with the exclusion of the parameter values that leads to non-hyperbolic critical point.
Point $P_4$ has the coordinates $P_4: (x,\Omega_m)=(0,0)$ and it exists always. The eigenvalues of the linearization matrix read $$\left\{2-\frac{3}{2
\alpha_2},\,\frac{3}{\alpha_2}-1\right\}.$$ Therefore, it is unstable node for $\frac{3}{4}<\alpha_2<3$, it is non-hyperbolic for $\alpha_2\in \left\{\frac{3}{4},3\right\}$, otherwise it is a saddle.
The above results are summarized in Table \[tab4\].
Stability of the critical points at infinity {#appendixinfin}
============================================
We introduce the new coordinates $(r,\theta)$ defined by $$\begin{aligned}
&&x=\frac{r}{1-r}\cos\theta\nonumber\\
&& \Omega_m=\frac{r}{1-r}\sin\theta,\end{aligned}$$ with $\theta\in
\left[0,\frac{\pi}{2}\right]$ and $r\in \left[0,1\right)$. The limit $r\rightarrow 1^-$ corresponds to $R^2\equiv x^2+\Omega_m^2\rightarrow
\infty.$ Note that the physical region of the plane $(r,\theta)$, that is corresponding to $0\leq x,\, 0\leq \Omega_m\leq 1$, is given by $$\label{rest_infty2}
\left\{(r,\theta): 0\leq r\leq \frac{1}{2}, 0\leq \theta\leq
\frac{\pi}{2}\right\}\cup \left\{(r,\theta): \frac{1}{2}<r<1,
0\leq \theta \leq \arcsin \left(\frac{1-r}{r}\right)\right\}.$$ The leading terms of the equations for $r'$ and $\theta'$ as $r\rightarrow
1^-$ are $$\begin{aligned}
&&r' \rightarrow \frac{3 \cos ^2(\theta ) [\alpha_1 (\cos (2 \theta )-3)-2
\sin
(2 \theta )]}{4 \alpha_1 \alpha_2 (1-r)}
\label{inftyr}\\
&&\theta'\rightarrow -\frac{3 \sin (\theta ) \cos ^2(\theta ) [\alpha_1 \cos
(\theta )-2 \sin (\theta )]}{2 \alpha_1 \alpha_2 (1-r)^2}
\label{inftytheta}.\end{aligned}$$ Hence, the fixed points at infinity (that is for $r\rightarrow 1^-$) are obtained by setting $\theta'=0$, and solving for $\theta$.
Let us denote a generic fixed point by $\theta=\theta^*$. The stability of this point is studied by analyzing first the stability of the angular coordinates from equation , and then deducing, from the sign of the equation , the stability on the radial direction [^2]. A fixed point $\theta=\theta^*$ is said to be stable if both $$\frac{d \theta'}{d\theta}\left|_{\theta=\theta^*}\right. <0, r'
\left|_{\theta=\theta^*} \right. >0.$$ The first condition implies stability of the angular coordinate $\theta$. The second condition implies that the $r$-values increase before reaching the limit value $r=1$ (that is before the boundary “at infinity” is reached) at the fixed point $\theta=\theta^*$. Similarly, a fixed point $\theta=\theta^*$ is said to be unstable if both $$\frac{d \theta'}{d\theta}\left|_{\theta=\theta^*}\right. >0, r'
\left|_{\theta=\theta^*}\right. <0.$$ Finally, it is a saddle point if either $$\frac{d \theta'}{d\theta}\left|_{\theta=\theta^*}\right. >0, r'
\left|_{\theta=\theta^*} \right. >0,$$ or $$\frac{d \theta'}{d\theta}\left|_{\theta=\theta^*}\right. <0, r'
\left|_{\theta=\theta^*} \right. <0.$$
In summary, the fixed points of the autonomous system at hand at infinity are the following:
- $Q_1: \theta^*=0, r^*=1, x=\infty, \Omega_m=0$.
Since from the definition (\[xdefin\]) we have $x=\sqrt{1+\frac{2
\alpha_2}{3}\left(1+\frac{\dot
H}{H^2}\right)}$, we deduce that the corresponding cosmological solution satisfies $\text{sign}(\alpha_2\dot H)\frac{|\dot H|}{H^2}\rightarrow \infty,$ or $H\rightarrow 0$ ($\dot H$ is bounded, with $\text{sign}(\alpha_2 \dot H)
>0$). Since $q=w_{DE}=-\text{sgn}(\alpha_2)\infty$, the point represents a super-accelerated phantom solution for $\alpha_2>0$, where eventually the universe ends in Big Rip, sudden or other forms of singularities (depending on whether the singularity is reached at finite or infinite time, what are its features etc.) [@Sami:2003xv; @Nojiri:2005sx; @Briscese:2006xu; @Bamba:2008ut; @Capozziello:2009hc; @Saridakis:2009jq]. For $\alpha_2<0$ it is a decelerating solution where the universe asymptotically stops expanding. Since $\left(\frac{d \theta'}{d
\theta},r'\right)|_{\theta=0}=\left(-\frac{3}{2\alpha_2},-\frac{3}{2\alpha_2}
\right)$, we conclude that $Q_1$ is always a saddle point.
- $Q_2: \theta^*=\arctan\left(\frac{\alpha_1}{2}\right), \alpha_1\neq 0, r^*=1,
\frac{\Omega_m}{x}\rightarrow \frac{\alpha_1}{2}, x\rightarrow \infty,
\Omega_m\rightarrow \infty$. Since $0\leq \theta\leq \frac{\pi}{2},$ then $\alpha_1> 0$. Since $\left(\frac{d \theta'}{d
\theta},r'\right)|_{\theta=\frac{\pi}{2}}=\left(\frac{6}{
(1+\alpha_1^2)\alpha_2},-\frac{12}{(1+\alpha_1^2)\alpha_2}\right)$, we deduce that $Q_2$ is unstable for $\alpha_2>0$ or stable for $\alpha_2<0$, as confirmed in Figure \[fig2\]. Since at this point $\Omega_m$ diverges, it corresponds to some form of future (respectively past) singularity for $\alpha_2<0$ (respectively $\alpha_2>0$) [@Sami:2003xv; @Nojiri:2005sx; @Briscese:2006xu; @Bamba:2008ut; @Capozziello:2009hc; @Saridakis:2009jq]. Its detailed classification for the various parameter regions lies beyond the scope of the present work.
- $Q_3: \theta^*=\frac{\pi}{2}, r^*=1, x=0, \Omega_m=\infty$. Since $\left(\frac{d \theta'}{d
\theta},r'\right)|_{\theta=\frac{\pi}{2}}=\left(0,0\right)$, we cannot rely on the linearization to examine the stability, and therefore, we need to resort to numerical examination (see Figures \[fig1\], \[fig2\]). Since at this point $\Omega_m$ diverges, it corresponds to some form of future, past or intermediate singularity [@Sami:2003xv; @Nojiri:2005sx; @Briscese:2006xu; @Bamba:2008ut; @Capozziello:2009hc; @Saridakis:2009jq].
The above results are summarized in Table \[tab5\].
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Cr. P. $\theta^*$ $\frac{d \theta'}{d\theta}\left|_{\theta=\theta^*}\right.$ $r' Stability
\left|_{\theta=\theta^*} \right.$
-------- ------------------------------------------ ------------------------------------------------------------ -------------------------------------- --------------------------- -- --
$Q_1$ $0$ $-\frac{3}{2\alpha_2}$ $-\frac{3}{2\alpha_2}$ saddle point
$Q_2$ $\arctan\left(\frac{\alpha_1}{2}\right)$ $\frac{6}{(1+\alpha_1^2)\alpha_2}$ $-\frac{12}{(1+\alpha_1^2)\alpha_2}$ unstable for $\alpha_2>0$
stable for $\alpha_2<0$
$Q_3$ $\frac{\pi}{2}$ $0$ $0$ see numerical elaboration
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
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[^1]: Figures \[fig2\] and \[fig5\] are suggestive that the stable manifold of $P_1$ acts as a separatrix, separating the phase-space solutions which are very likely to end at a future singularity “at infinity” from those ending at the finite region. Hence, the detailed examination of the stable manifold of $P_1$ may give information of the basin of attraction of the future singularities.
[^2]: The special functional form of the terms in the denominator depending on $r$ is irrelevant for the discussion, since they can be removed by choosing a different time scale. What is important is that the sign of these terms is positive, which implies that the arrow of time is preserved under the time rescaling.
|
---
author:
- |
Spencer Chang$^a$, Lawrence Hall$^{b}$ and Neal Weiner$^a$\
$^a$ Center for Cosmology and Particle Physics, Dept. of Physics, New York University, New York, NY 10003\
$^b$ Physics Dept. and Theoretical Physics Group, Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720\
bibliography:
- 'twin.bib'
title: A Supersymmetric Twin Higgs
---
Introduction
============
Supersymmetric extensions of the standard model (SM) tame the quadratic divergences associated with the Higgs boson mass, allowing perfectly natural theories for all energies up to the Planck scale. Yet at first sight they present a new puzzle: given all the scalars in the theory, why is it the Higgs boson that acquires a vev rather than a squark or slepton? Remarkably, radiative corrections to the supersymmetry breaking scalar masses provide a dynamical understanding for why the Higgs, and no other scalar, acquires a mass. As these mass parameters are scaled to the infrared, they are increased by gauge interactions and decreased by Yukawa interactions. Thus symmetry breaking is induced for the field that appears in the largest Yukawa coupling but has the weakest gauge interactions. This field is the Higgs boson, and the resulting Electroweak Symmetry Breaking (EWSB) is driven by the large value of the top quark Yukawa coupling.
This elegant, almost inevitable, breaking of electroweak symmetry, was seen as a key element of supersymmetric theories in the early 1980s, and, with the precision measurement of the weak mixing angle at the beginning of the 1990s, cemented supersymmetry as the leading extension of the Standard Model. However, the top quark radiative mechanism for electroweak symmetry breaking is considered by many to require fine-tuning — the problem is that it is simply too efficient, driving too large a Higgs vev so that the $W$ and $Z$ bosons are too heavy. The size of the negative Higgs boson mass squared, and therefore the size of the EWSB vev, is determined by the top quark Yukawa coupling, the top squark mass and by the number of decades of renormalization evolution. The top quark is so heavy that the radiative mechanism is extremely powerful: even if the top squark mass is near its experimental limit, scaling from the Planck scale drives too large a Higgs vev. If gravity mediation of supersymmetry breaking is replaced by gauge mediation at a much lower scale, the experimental limit on the scalar tau mass forces the top squark to be quite heavy, that again the vev is naturally too large.
The inability of LEP2 to discover a Higgs boson has compounded the problems for top-driven radiative EWSB. For the Higgs boson to be sufficiently heavy a new quartic Higgs interaction is required beyond that provided by the supersymmetric electroweak gauge interactions. In the simplest models this can only arise from radiative corrections from a top squark that is significantly heavier than its experimental bound. This further increases the efficiency of the heavy top radiative EWSB mechanism, leading to significant fine tuning for simple realistic theories. For an excellent discussion of the details, see [@Chacko:2005ra].
This problem of EWSB in supersymmetric theories has received considerably attention recently. One can seek an alternative scheme for mediating supersymmetry breaking to the standard model sector at low energies, via a low effective mediation scale and sizable $A_t$ term [@Choi:2005hd; @Kitano:2005wc]. The mass correction can also be reduced through conformal dynamics or by generating the top Yukawa at a low scale [@Kobayashi:2004pu; @Kobayashi:2005mg]. The Higgs could be a pseudo-Goldstone [@Birkedal:2004xi; @Berezhiani:2005pb; @Roy:2005hg; @Csaki:2005fc] or even composite [@Harnik:2003rs; @Chang:2004db; @Delgado:2005fq], cutting off the log or allowing larger quartics. The Higgs boson could be made heavy by adding additional gauge [@Batra:2003nj; @Maloney:2004rc] or superpotential [@Espinosa:1992hp] interactions. Furthermore, one might actually have a lighter Higgs, but evade the LEP Higgs bounds by new decay mechanisms [@Dermisek:2005ar; @Chang:2005ht; @Schuster:2005py; @Dermisek:2005gg].
In this paper we introduce a new mechanism that weakens the strength of top-quark radiative EWSB. It works even for gravity mediated supersymmetry breaking, and does not require the top squark to be light. It makes use of the twin Higgs mechanism [@Chacko:2005pe]. The standard model has a mirror or twin duplicate, that is guaranteed to have the same couplings by an interchange $Z_2$ parity. The Higgs potential, involving both our Higgs doublet $H$ and the twin Higgs doublet $H'$, possesses an approximate $SU(4)$ symmetry acting on ${\cal H} = (H,H')$. A large negative mass squared $-m^2 {\cal H} {\cal H}^\dagger$ leads to a large EWSB in one of the sectors, which by definition is the twin sector. This was proposed as a way to make progress on the Little Hierarchy Problem in non-supersymmetric theories. However, in the simplest model a hierarchy of vevs between our sector and the twin sector itself requires some tuning. Nevertheless, adding a twin of the SM does lead to a theory with significantly improved naturalness over the SM, while still preserving agreement with precision electroweak data [@Barbieri:2005ri]. The naturalness of the theories has been improved by enlarging the Higgs sector [@Chacko:2005vw], or extending the gauge group to $SU(2)_L \times SU(2)_R$ [@Chacko:2005un]. However, in these theories, the cutoff remains quite low.
Starting with the Minimal Supersymmetric Standard Model (MSSM), we show that adding a twin MSSM$'$ can solve the fine tuning problem of supersymmetry. There are now two $SU(4)$ Higgs scalars, ${\cal H}_u$ and ${\cal H}_d$, with mass terms $m_u^2 {\cal H}_u {\cal H}_u^\dagger + m_d^2 {\cal H}_d {\cal H}_d^\dagger + B^2({\cal H}_u {\cal H}_d + h.c.)$. If the determinant $m_u^2 m_d^2 - B^4$ is positive there is no EWSB. The top and mirror top radiative corrections reduce $m_u^2$ leading to a negative determinant inducing a large vev for the twin Higgs. The D term quartics explicitly break the approximate $SU(4)$ symmetry, so that a negative Higgs mass squared would also be generated for our sector. However, employing the “supersoft" mechanism [@Fox:2002bu] on the twin sector, guarantees that this negative Higgs mass squared in our sector is significantly reduced. Thus the success of top-quark radiative EWSB is restored: its full power is felt only in the twin sector, while in our sector it is still operative, but with a reduced strength. There is now no barrier to a heavy top squark, needed in some mediation schemes. In fact, a heavy top squark is now preferred as the simplest origin for an additional quartic interaction to give sufficient mass to the Higgs boson.
A Toy Model
===========
In non-supersymmetric theories the twin idea may be implemented by having the standard model, SM, an identical mirror, or twin, standard model, SM$'$, and a quartic interaction $H H^\dagger H' H'^\dagger$ such that the combined Higgs potential is approximately $SU(4)$ symmetric. There is a $Z_2$ symmetry that interchanges the two sectors, forcing $SU(4)$ invariance on the Higgs mass terms, but not on the quartics. A large vev for the twin Higgs breaks $SU(4) \rightarrow SU(3)$ so that our Higgs boson appears as a pseudo-Goldstone boson. There is an obvious barrier to implementing this in supersymmetry. If our sector is the MSSM and we add an identical twin sector, MSSM$'$, then supersymmetry forbids any quartic interaction coupling our Higgs to the twin Higgs, so that the Higgs quartics cannot be made $SU(4)$ invariant. A simple way to over come this difficulty is to add a gauge singlet N together with superpotential terms $\lambda N(H_u H_d + H_u' H_d')$. This leads to a Higgs quartic that couples the two sectors together, and, because it arises from terms in the superpotential that are quadratic in Higgs fields, the $Z_2$ interchange parity is sufficient to guarantee $SU(4)$ invariance. Hence one sees that the twin idea actually fits very well in supersymmetric theories: the $Z_2$ in the superpotential automatically generates an $SU(4)$ in the quartics, needing no separate assumption.
We begin by considering a toy model, that has the features expected from the twin supersymmetric theory described above, but is simplified in two respects. Firstly the electroweak gauge group of each sector will be taken to be $U(1)$, and secondly each sector will be taken to have only a single Higgs, rather than separate ones for the up and down sectors. Of course, these do not occur in any realistic supersymmetric model, but they allow a transparent illustration of our mechanism. Thus we have a $U(1)\otimes U(1)'$ gauge group, with Higgs field ${\cal H}=(h\ h')$ possessing an approximate global $SU(2)$ symmetry, rather than $SU(4)$, and we assume a scalar potential of the form V= -m\^2 [H]{} [H]{}\^+ ( [H]{} [H]{}\^)\^2 +m\^2 h h\^+ (h h\^)\^2 + (h’ h’\^)\^2. Each term in the potential has an important significance. The negative $SU(2)$ invariant mass term arises from the large radiative correction in the top sector, and will be the origin of EWSB, both in the twin sector and in our sector. We imagine that if this negative mass squared were in our sector alone it would lead to the $Z$ boson being to heavy, and our aim is to understand how the presence of the twin sector could reduce the natural value for the $Z$ boson mass. The quartic coupling $\lambda^2$ is the toy model version of the $SU(4)$ invariant Higgs quartic that comes from the interaction with the singlet superfield $N$. The quartic interactions proportional to $g^2$ and $g'^2$ are the toy analogue of the electroweak $D^2$ terms in our sector and the twin sector. The $Z_2$ parity sets $g' = g$, but we keep the prime in the twin case so that we can keep track of the origin of the two interactions. These gauge quartics explicitly break the global $SU(2)$ symmetry of the toy model, and we shall return to this point shortly. Finally, we have allowed for a small $SU(2)$ and $Z_2$ symmetry breaking mass term, $\delta m^2$.
Following the twin idea, suppose that the large negative mass squared leads to EWSB in the twin sector, then [ ]{} = . \[h’vev\] Upon integrating out the heavy $h'$, we are left with a low energy effective theory for $h$, with potential V=(m\^2- m\^2) h h\^+ ( g\^2 +g’\^2 )( h h\^)\^2. In the limit that the $SU(2)$ violating terms $\delta m^2$, $g^2$ and $g'^2$ go to zero, we have an exact $SU(2)$ symmetry and $h$ is a Goldstone boson. In the presence of the $SU(2)$-violating terms, the quartic interactions for $h$ are welcome since we need them to get the Higgs boson sufficiently heavy. However, the mass terms are problemmatic if they are two large. The contribution from $\delta m^2$ can be naturally small, because the $Z_2: h\leftrightarrow h'$ exchange symmetry enforces an accidental $SU(2)$ symmetry on the quadratic terms. However, the $SU(4)$-violating D-term quartics are inevitable in supersymmetry, so reducing or eliminating the contribution to $h h^\dagger$ proportional to $g'^2$ is the most significant challenge. This term arises because the $g'$ quartic leads to a shift in the vev away from the $SU(2)$ invariant value, as shown in eq. \[h’vev\]. How can this effect be eliminated?
Modifying the toy model {#modifying-the-toy-model .unnumbered}
-----------------------
The troublesome quartic interaction arising from the twin sector electroweak $D$ terms can be removed by the inclusion of a new superfield. The analysis below in the toy model corresponds to the inclusion of a “supersoft” supersymmetry breaking term [@Fox:2002bu] in the realistic theory, as discussed in the next section.
We extend the toy model by including a real singlet field $s$, with a potential V\_s = s\^2 + ( m\_s s + g’ h’ h’\^)\^2. Such a potential can arise naturally in the presence of D-term supersymmetry breaking [@Fox:2002bu]. When $h'$ acquires a vev, a tadpole for $s$ is induced so that it is convenient to redefine $s$ by s s- h’ h’\^. This new field is not canonically normalized, so if we want to study the masses of $s$ or $h'$, we should shift back to canonically normalized fields. However, for the purposes of studying vevs and determining the properties of the pseudo-Goldstone boson, this is adequate.
Notice that with such a potential, in the limit that $\delta m_s^2 \rightarrow 0$, such a redefinition removes the $SU(4)$ violating quartic $g'^2$. At leading order in $\delta m_s^2$ and $\delta m^2$, the potential for $h$ is now V= (- ) m\^2 h h\^+ ( + ) (h h\^)\^2. Therefore we have reduced the problem of removing the problematic $g'$ term to a problem of keeping certain supersymmetry breaking masses small.
In the realistic supersymmetric model, we will need two fields, a singlet $S$ and an $SU(2)$ triplet $T$, which will pick up radiative corrections to their masses. Consequently, the naturalness of the models will be related to the size of SUSY breaking masses for [*electroweak*]{}-charged superpartners (specifically winos), rather than for colored superpartners, such as gluinos.
A Supersymmetric Twin Higgs
===========================
Let us begin by taking the MSSM and creating a twin copy, the MSSM$'$. We will refer to these sectors as the visible and twin sectors, respectively. We will insist upon a $Z_2:\ MSSM\leftrightarrow MSSM'$ symmetry. The general spirit of the construction will be this: all supersymmetric couplings will respect the $Z_2$, and all soft supersymmetry breaking operators (terms that are log sensitive) will also respect the $Z_2$. In principle, we can include supersymmetric $\mu$-type masses which violate the $Z_2$, but they are unnecessary phenomenologically. The origin of the $Z_2$ breaking will be a hidden sector $D$-term, which will generate supersoft supersymmetry breaking terms only in the twin sector. We will return to this point in a moment.[^1]
In order to establish $SU(4)$ invariant quartics, we will employ the superpotential W=N [H]{}\_u [H]{}\_d \[eq:quartic\] where ${\cal H}_i=(H_i\ H'_i)$ and $N$ is a singlet superfield under both the MSSM and MSSM’ gauge groups. Notice that because the superpotential is bilinear in the Higgs fields, the $Z_2$ symmetry alone is sufficient to achieve an $SU(4)$ in the quartic induced in the potential from the $F_N^2$ term.
Of course, we still have the $SU(4)$ violating D-term quartics as in the toy model, which we must cancel in the twin sector. This is where we will employ the $D$-term supersymmetry breaking operators. We add a singlet $S'$ and a triplet $T'$ under the MSSM$'$ allowing us to expand our superpotential to W=N [H]{}\_u [H]{}\_d + W\^[’]{}\_Y S’ + W\^[’]{}\_[SU(2)]{} T’. To simplify our calculations, we will add a large soft mass for $N$, such that ${ \left\langle {N} \right\rangle }=0$ and we can decouple it. $W^D_\alpha$ is a spurion, reflecting the $D$-term of some hidden sector $U(1)$, so ${ \left\langle {W^D_\alpha} \right\rangle }= \theta_\alpha D$. The effect of these operators has been explored elsewhere [@Fox:2002bu]. They generate scalar masses $m_S$ and $m_T$, with related trilinears which we describe below. They also generate a Dirac mass between the fermion and the gaugino of size $m_S/2$ and $m_T/2$.
At this point we have not included comparable operators for the MSSM, which breaks the $Z_2$. However, because these are [*supersoft*]{} SUSY breaking operators, that is, they induce no corrections to the RG flow of the soft SUSY breaking masses, all contributions to the $SU(4)$ violations will be loop suppressed, with no log enhancement. (A triplet under the MSSM $SU(2)$ must be added in order to preserve the $Z_2$ of the gauge couplings, but no equivalent superpotential term need be added.)
Aside from these small supersoft effects, the remainder of the Higgs potential arises from $SU(4)$ (i.e., $Z_2$) preserving soft masses $m_{h_u}^2$, $m_{h_d}^2$and $B^2 {\cal H}_u {\cal H}_d + h.c.$. It is important to note that $m_{h_u}^2,m_{h_d}^2$ must both be positive as the quartic in eq. \[eq:quartic\] will not stabilize against breaking in these directions. The proper spectrum is $m_{h_d}^2 > B^2 > m_{h_u}^2$. However, this is a natural expectation in that the top/stop loops will drive down $m_{h_u}^2$, turning the determinant of the mass matrix negative at a low scale.
Let us go to the basis ${\cal H}=\sin\beta {\cal H}_u + \cos\beta {\cal H}_d$, $\bar{\cal H}=\sin\beta {\cal H}_d-\cos\beta {\cal H}_u$ where the mass matrix is diagonal and of the form Here we know that only the field $\cal H$ will acquire a vev, so we can set $\bar {\cal H}=0$. Then potential reads V=&-&m\^2 [H]{} [H]{}\^+ ([H]{}[H]{}\^)\^2 \^2 2 + (H H\^(2 ))\^2\
&+& ( m\_S S’ + g\_Y’ H’ H’\^(2 ) )\^2 + ( m\_T T’ + g’ H’ H’\^(2 ) )\^2\
&+&S’\^2 + T’\^2 where the $S'$ and $T'$ fields are in an abuse of notation the real parts of the respective scalars. As before, we can redefine S’ S’ - .5 in T’ T’ - These redefined fields will not acquire vevs, so we can set them to zero in the potential, leaving us with V=&-&m\^2 [H]{} [H]{}\^+ ([H]{}[H]{}\^)\^2 \^2 2 + (H H\^(2 ))\^2\
&+& (H’ H’\^(2 ))\^2 where $\gamma_Y = \delta_S^2/(m_S^2 + \delta_S^2)$ and $\gamma = \delta_T^2/(m_T^2 + \delta_T^2)$. To the extent that the corrections $\delta_{S,T}$ are small compared to $m_{S,T}$ these will be small numbers.
Now $H'$ will acquire a vev [ ]{}=f\^2 = That the vev is in the twin direction is a dynamical selection due to the smaller quartic, not a choice. Integrating out the $H'$ and taking $g',g_Y'=g,g_Y$ gives us V= -m\^2 + H\^4 \^2 (2 ) This is the principal result of this paper. We have an order one quartic, but with a tree level mass suppressed by small numbers $\gamma,\ \gamma_Y$.
Scales and limits
-----------------
Now we have established the basic tools for constructing a twin Higgs. But is it a twin Higgs theory, or is it a supersymmetric theory? The answer depends on the scales $m_{SUSY}$ and $f$.
In the limit that $m_{SUSY}/f \ll 1$, the theory is nearly a standard supersymmetric theory. Because we have invoked supersymmetry breaking masses to break the $SU(4)$, one has a supersymmetric model in the decoupling limit, with the Higgs fields $H,A$ with masses $O(f)$. As a consequence, the quadratic divergence associated with the Higgs self-coupling is not cancelled until the scale $f$. However, the more serious quadratic divergences, associated with the top Yukawa and gauge couplings, are cancelled by supersymmetry, and the logarithmic divergence cut off at $f$. All other SUSY masses will arise from higher scales and will, in general, be much larger than $m_h$.
Since SUSY is invoked in the breaking of $SU(4)$, and because the heavy Higgses appear at the scale $f$, it is clear that the scale of $SU(4)$ breaking cannot be much higher than the SUSY scale.[^2]
In the limit that $m_{SUSY}/f \gg 1$ we achieve a standard twin Higgs model, albeit one greatly improved over the model presented in [@Chacko:2005pe], because of the presence of a tree-level quartic coupling. In this case, it is the $SU(4)$ which is protecting the Higgs mass, and supersymmetry + $Z_2$ protecting the $SU(4)$.
Since SUSY is invoked to protect the $SU(4)$, it is clear that one cannot take the SUSY breaking scale arbitrarily high. In particular, we shall see that the winos, and the sleptons in general, must remain light in order for the cancellation of the twin quartic to occur.
Ultimately, we are directed towards the $m_{SUSY}\sim f$ region. In this region, there are no quadratic divergences above $f$, but the Higgs mass is cancelled up to one loop corrections by the $SU(4)$ breaking. In general, some fields (namely, squarks and gluinos) will typically appear above the scale $f$, while others (sleptons, winos) will typically appear below the scale $f$. The particular limits, however, require a more careful study of the radiative corrections to the theory.
Radiative corrections
---------------------
We now understand that it is possible to have a small soft mass for the Higgs field at the scale $f$, but we have translated the problem of the Higgs mass divergences into a problem of controlling the masses of $S$ and $T$. This is much easier because we lack the sizeable Yukawas to colored particles that are so problematic in the MSSM. Nonetheless, radiative corrections impose significant constraints on the spectrum.
Because $S$ has no gauge charges and no Yukawa couplings, it has no radiative corrections to its mass. However, $T$ receives corrections to its mass from a variety of sources. Loosely, we can group the radiative corrections into two parts: the log-enhanced RG flow, and the finite one-loop effects arising from the $Z_2$ breaking supersoft terms.
The RG flow is simply the usual contribution (see, e.g., [@Martin:1997ns]) = - M\_2\^2. where $M_2$ refers to the $Z_2$ preserving Majorana mass for the $SU(2)$-gauginos. There are also two-loop corrections to its mass from the soft masses of other fields charged under $SU(2)$ (see, e.g., [@Arkani-Hamed:1998kj]) = \_i T\_i m\_i\^2, where $T_i$ is the Dynkin index of the $i$ representation. Both of these terms should be properly evolved in a given model, as they depend on the value of $\alpha_2(\mu)$, which, in turn, depends on the particle content of the theory. However, given that new matter makes UV values of $\alpha$ larger, we can use the MSSM limit for the former as an estimate of the radiative correction assuming the masses are generated at a high scale, m\_T\^2 \~0.5 C\_2\^T M\_2\^2 In contrast, the effects of the two-loop running cannot be simply estimated without input as to the values of the scalar masses at the high scale as well. However, they are in general much smaller than the one loop contribution from gaugino masses.
Additionally, we must concern ourselves with the $Z_2$-violating corrections arising from the Dirac gaugino masses in the twin sector. In the limit that the Dirac masses are much larger than the Majorana masses, the radiative corrections are simply given by [@Fox:2002bu] m\^2\_ = (4) \[eq:ssoftcorr\] where $m_i$ is the supersoft mass of the scalar associated with the gauge group indexed by $i$ (as a warning, throughout the rest of the text, we will refer to these as $m_S, m_T, m_O$). The $\log(4)$ term arises from the ratio of the scalar and Dirac gaugino masses squared. When the Dirac and Majorana masses are comparable, there is unfortunately no simple formula, but the overall magnitude of the effect does not change.
Naturalness \[sec:naturalness\]
-------------------------------
As in the MSSM, many questions of naturalness arise due to the assumption of unification. If experimental limits on charginos or staus, for instance, indirectly imply scales for gluinos and squarks, there can be a significant effect on the naturalness of the theory. However, in the case of the supersymmetric twin Higgs, there are significant direct relationships between the wino Majorana mass, and the Dirac bino and wino masses.
In particular, we require that the soft mass for $T$ be small in order that the cancellations of the D-term quartic in the twin sector are complete. If this is not the case, one ends up with a correction to the Higgs mass $O(\delta m_T^2 m_{\cal H}^2/m_T^2)$. If one wishes to cancel the existing mass term of the Higgs to a few percent, and given that $\delta m_T \sim M_2$, one must have $m_T \approx 5-10 \times M_2$. Given limits on charginos from LEP2, this tells us $m_T { \mathop{}_{\textstyle \sim}^{\textstyle >} }500\ {{\rm GeV}}\sim 1\ {{\rm TeV}}$.
It is important to state that most, but not all, models will generate scalar masses at a similar scale to gaugino masses. Nonetheless, some models, e.g., low scale gaugino mediation [@Kaplan:1999ac; @Chacko:1999mi], could have the scalar masses considerably lower than the gaugino masses. Thus, while we believe that a lighter wino is a generic feature of these theories, it is by no means essential.
A similar statement can be made about the scalar, specifically slepton, masses. While they can, in principle, be quite heavy, with only two-loop running feeding into the $T$ mass, they are quite often generated from gauge interactions, and so we expect those masses to be comparable to the $T$ mass as well. A good estimate of the ratio of masses squared would be the ratio of the casimirs of the fundamental to the adjoint representation. Hence, a $T$ lighter than $400\ {{\rm GeV}}$ would likely be accompanied by left-handed sleptons in the $250\ {{\rm GeV}}$ range. Again, these limits are not absolute, but demonstrate that preserving the accidental $SU(4)$ is associated with light ($m<f$) fields, although such a statement is not obvious from the outset.
The Dirac masses in the twin sector are more model-independent in their effects. In fact, there are upper bounds on their size due to naturalness. The presence of a large supersoft $m_T$ will generate radiative corrections in the twin sector that are not cancelled in the visible sector. From eq. \[eq:ssoftcorr\], we see that there will be a correction roughly $\delta m_h^2 \sim {\rm few}\ \times 10^{-3} m_T^2$. For a natural completely natural Higgs mass ($O(100 {{\rm GeV}})$), one expects $m_T { \mathop{}_{\textstyle \sim}^{\textstyle <} }2 {{\rm TeV}}$, while for moderate tuning, $m_T { \mathop{}_{\textstyle \sim}^{\textstyle <} }5 {{\rm TeV}}$ would be reasonable.
The combination of these two requirements suggests that a light wino in the visible sector is most natural. While there is no absolute upper bound, the requirement of naturalness suggests an upper bound of $m_{\rm wino} { \mathop{}_{\textstyle \sim}^{\textstyle <} }400 {{\rm GeV}}$ in the most natural models. Sleptons are expected to be light, but, similarly, the limits are not absolute. Once unification is included, our expected parameter range will narrow somewhat.
To summarize, there is in general a correlation between the soft contributions to the $T'$ mass and the electroweak soft masses for the wino and sleptons. When considering experimental limits on the visible particles, requiring that this doesn’t upset the D-term cancellation gives a lower bound for the supersoft $T'$ mass. On the other hand, there is an upper bound on the same mass due to naturalness, as the $Z_2$ violation becomes too large. Therefore, these considerations will impact prospects of collider searches.
Unification \[sec:unification\]
===============================
The inclusion of an $SU(2)'$ adjoint in the twin sector necessitates one in the visible sector by $Z_2$. The natural consequence of this is to spoil unification. However, this is easily addressed by GUT-completing the adjoint with additional fields.
There are essentially two options, as outlined in [@Fox:2002bu]. The most obvious would be to GUT-complete into a [**24**]{} of $SU(5)$. This amounts to adding a total of five flavors to each sector, resulting in a landau pole below the GUT scale. It is difficult to continue to claim the quantitative successes of unification under such circumstances.
A more restrained approach is to GUT-complete into a [**24**]{} of $SU(3)^3$. This amounts to the addition of an [**8**]{} of $SU(3)$ color, as well as a vectorlike pair of $(1,2,\pm 1/2)$ fields, two pairs of $(1,1,\pm 1)$ fields, and four gauge singlets. This amounts to the addition of 3 flavors to the theory, which retains perturbativity in the theory, and thus the quantitative successes of unification.
The basic features of the spectrum have been laid out in [@Fox:2002bu] which we follow here. The $\beta$ functions are given by $(b_1,b_2,b_3)=(33/5,1,-3)$ in the MSSM. With the additional matter, we now have $(48/5,4,0)$. Since $\alpha_3$ is asymptotically flat at one loop, we can take $\alpha_i (M_{GUT}) = \alpha_3$. The supersoft mass parameters run due to gauge kinetic term renormalization, as well as the adjoint kinetic term renormalization. Hence &m\_S& = ()\^[1/2]{} M\
&m\_T& = M\
&m\_O& = ()\^[ ]{}M=M\^ M\_[GUT]{}\^ where $m_O$ is the supersoft mass of the new scalar color octet, and $M$ is the common supersoft scalar mass at the unification scale.[^3]
In terms of the unified mass term, the one loop scalar soft masses squared are (again, see [@Fox:2002bu]) &m\_r\^2& =\
&m\_l\^2& =\
&m\_c\^2&= ()\^ Because the color octet mass is so large, these corrections to the twin squarks will induce two-loop corrections to the twin Higgs mass, which are $Z_2$ violating. At leading order in $\alpha_3$, this is given by m\_h\^[’2]{} = - ()\^ (M\_3/m\_[t’]{})
Ultimately, the tension is between our desire to have a large $m_T$, and hence a robust cancellation of the $D$-term quartics and the consequent radiative correction to the twin Higgs mass squared. Because in this unified scenario, the two loop $Z_2$-violating contributions are larger than the one loop $Z_2$-violations, the upper bound on $m_T$ comes indirectly from an upper bound on $m_O$. Numerically, when one includes the relationship to $m_O$, one finds $\delta m_h^2/m_T^2 \approx 10^{-2}$, so a completely natural model with $m_h^2 \sim (100 {{\rm GeV}})^2$ would require $m_T \sim 1\ {{\rm TeV}}$. The upshot of this are slightly more stringent requirements on the wino and slepton masses in their relation to the radiative corrections to the soft mass of $T$.
Having assessed the consequences for naturalness on MSSM fields, where are the new exotic fields in the observable sector? They must have some mass, which is most simply understood by adding a supersymmetric $\mu$-term masses for these fields. A priori, this mass needn’t be the same as the mass in the twin sector, in that it will not lead to any sizeable $Z_2$ violating radiative corrections. Thus, in principle, the new fields in the visible sector can be quite heavy ($\sim {{\rm TeV}}$).
However, under the assumption that only supersoft violates the $Z_2$, we can make stronger statements. Because the supersymmetric masses interfere with the cancellation of the D-term in the twin sector, the masses for $S$ and $T$ should be small - at most of the order of the wino mass. Under the assumption of unification, we can calculate the spectrum in the visible sector. \_S &=&\
\_T &=&\
\_O &=& M\_[GUT]{}\^ \^
Numerically, we have $\mu_S:\mu_T:\mu_O \approx 1:3:21$. If we assume $Z_2$ symmetry in these masses, and require $\mu_T { \mathop{}_{\textstyle \sim}^{\textstyle <} }200 \,{{\rm GeV}}$, then we have $m_O { \mathop{}_{\textstyle \sim}^{\textstyle <} }1.5 \,{{\rm TeV}}$. Such a particle should be produced at the LHC. We will discuss the phenomenology in section \[sec:pheno\].
The additional $SU(2)\times U(1)$ fields which complete the $SU(3)^3$ adjoint (the so-called “bachelor” fields [@Fox:2002bu]), should have a mass related by unification to these $\mu$-terms. However, the RG-evolution of these masses depends on the particular couplings of the bachelor fields, specifically Yukawa-type couplings. As we shall see in section \[sec:cosmo\], such couplings may naturally enable a Froggatt-Nielsen theory of flavor [@Froggatt:1978nt]. Hence, we can say little about their specific spectrum except that they should be in the $100 \ {{\rm GeV}}$ range rather than the $1 \ {{\rm TeV}}$ range.
$\Lambda_{QCD}'$ {#lambda_qcd .unnumbered}
----------------
Under the assumption of unification, the new colored fields in both sectors can affect $\Lambda_{QCD}$. Under the assumption that the same sets of fields remain light, and so contribute to the running of the strong coupling (i.e., gluons, up, down and strange quarks), there is a simple relationship between $\Lambda_{QCD}'$ and $\Lambda_{QCD}$, specifically \_[QCD]{}’ = \_[QCD]{} \_i ()\^[-b\_i/b\_[light]{}]{} where $M_i$ ($M_i'$) is the common mass of fields in the observable (twin) sector, contributing a [*positive*]{} value $b_i$ to the $\beta$ function of QCD, while $b_{light}$ is the [*negative*]{} contribution to the $\beta$ function from the fields lighter than the strong coupling scale.
Assuming common squark masses, the twin strong couplings scale is simply \_[QCD]{}’ = \_[QCD]{} ([f v]{} )\^[2/9]{} ([m\_[q]{}’ m\_[q]{}]{} )\^[2/9]{} ([m\_[gluino]{}’ m\_[gluino]{}]{} )\^[2/9]{} ([m\_[ferm]{}’ m\_[ferm]{}]{} )\^[2/9]{} ([m\_[scal]{}’ m\_[scal]{}]{} )\^[1/18]{} ([m\_[pseudo]{}’ m\_[pseudo]{}]{} )\^[1/18]{} where $m_{ferm}$, $m_{scal}$ and $m_{pseudo}$ are the masses of the octet fields. If we want TeV squarks in the observable sector, we cannot have twin squarks much heavier than 3 TeV without generating unacceptably large two-loop corrections to the Higgs mass. The gluino and scalar octet masses could be a factor of ten larger, while the pseudoscalar, picking up its $Z_2$ violating mass difference from the radiative corrections, is likely of a similar ratio to the squarks. Thus, we expect the twin QCD scale to be roughly $\Lambda_{QCD}' \simeq (4 - 7)\times \Lambda_{QCD}$.
Cosmology \[sec:cosmo\]
=======================
As discussed previously [@Chacko:2005pe; @Barbieri:2005ri], the presence of a twin sector can have significant cosmological effects if the extra degrees of freedom were in thermal equilibrium at some earlier time in the universe. In Twin Higgs models, the presence of cross term quartic couplings that link twin sector Higgses to visible ones mediate processes that keep the twin sector in thermal equilibrium. Below the scale of the two electroweak breakings, one can integrate out the massive scalars to generate four fermion couplings between the two sectors of the following form: | |’ ’ These operators keep the sectors in thermal equilibrium to low temperatures; for example for an f scale of 500 GeV, processes that convert charm quarks into twin muons occur at a rate \^[-1]{} \~ T\^5 using $m_c = 1.4 \,{{\rm GeV}}, m_\mu' = 300 \,{{\rm MeV}}$ determines that this process decouples at a temperature of about 2 GeV. Other processes give similar decoupling temperatures.
If one insists on a low reheat temperature, anywhere from a few MeV to just above the QCD phase transition, cosmological difficulties are evaded. However, in more standard thermal histories, one is forced to address the issues raised by the additional thermal degrees of freedom. Of primary concern is additional relativistic energy at the BBN era as well as its effects on the CMBR. The first assumption would be that the $\gamma'$ is massless and the twin neutrinos are a factor of $f/v$ or $(f/v)^2$ heavier than in the Standard Model - assuming no other $Z_2$ violation is present in the theory. If this is the case, the crucial factor is the temperature of these light degrees of freedom, when the regular universe is at MeV temperatures. The additional relativistic energy density is given by ’ = \_[1]{} when normalized to the relativistic energy of one SM neutrino. Since current BBN limits restrict the number of additional neutrinos to be no more than one (for a recent discussion, see [@Dolgov:2003sg]), the bracketed terms are constrained to sum to less than one. There are two relevant scenarios for which we can determine the temperatures involved. First, if the twin electrons are below the decoupling temperature, when they annihilate, the $T_{\gamma'}$ will be enhanced relative to $T_{\nu'}$ as in the SM. In this case, we have = = ( )\^. where $T_d$ is the decoupling temperature and $g_*(T_d)$ and $g'_*(T_d)$ the number of relativistic degrees of freedom (including 7/8 for fermions) in the visible and twin sectors respectively at decoupling. The BBN constraints requires that $g'_*(T_d)/g_*(T_d) \lesssim 1/4.5$, which is a reasonably stringent constraint. The other relevant case is when only the $\gamma'$ and $\nu'$ are below the decoupling temperature. Then there is no relative heating up of the photons and we have = = ( )\^. In this case, the BBN constraint requires $g_*(T_d) \gtrsim 32,$ or equivalently $T_d \gtrsim T_{QCD}$, just above the QCD phase transition and consistent with the numbers given above.
To be consistent with the constraints in either case requires a smaller $g'_*(T_d)$ than that given by a spectrum scaled by $f/v$. In the first case, one requires at least the second generation to be heavier than the 1-2 GeV decoupling temperature (giving $\Delta N_\nu \sim 1$) whereas in the second case, the first two generations must be above the decoupling temperature. This required increase in the prime Yukawas can come from different realizations.
The first possibility is to just allow $Z_2$ breaking in the Yukawas for the first two generations as in [@Chacko:2005pe; @Barbieri:2005ri]. They can be increased to a sufficient level without introducing significant $Z_2$ breaking in the Higgs potential. This is potentially given by some high scale flavor physics, but appears to be an ad hoc assumption that runs counter to the $Z_2$ symmetry of the model. On the other hand, the supersoft $Z_2$ breaking already yields what appears to be hard $SU(4)$ breaking in the Higgs potentials at low energies. Given this, it is not surprising that other indirect effects of the supersoft $Z_2$ breaking can induce what appear to be hard breaking in other areas of the theory.
This can be trivially implemented using the large vevs of the adjoint fields in the context of a low-scale Froggatt-Nielsen model [@Froggatt:1978nt]. Most simply, if the Yukawas of the twin and visible sectors are generated by the presence of interactions of new fields in the $100\, {{\rm TeV}}$ range, then it is quite plausible to imagine that these Yukawas would depend on the vevs of the singlet fields as + + or + + In the former case, the singlet vev would generate larger Yukawas in the twin sector, while in the latter case the singlet vev would suppress Yukawas in the visible sector.[^4] It is also important to emphasize that the constraints do not require a significant change in the Yukawas, since raising the 2nd generation above a few GeV is at most an order of magnitude change in the Yukawas.
The particular model which achieves this is not important for our present discussion. However, the necessity of this (relatively) low scale of flavor suggests that, in spite of the large SUSY breaking masses in the theory, flavor violation may still be large enough to be observable. Similarly, even though the BBN constraints can be satisfied, improved measurements on relativistic energy density at BBN or in the CMB are expected to give deviations from that of the SM.
Phenomenology {#sec:pheno}
=============
The general twin Higgs phenomenology applies to this model, for instance there are new invisible Higgs decays into twin particles [@Chacko:2005pe; @Schabinger:2005ei]. This is mediated due to mixing and hence gives an $O((v/f)^2)$ branching ratio into invisible decays. However, since the model is also supersymmetric, there are new signatures of the model within the supersymmetry phenomenology. Furthermore, since this is the primary signal of new physics, one can hope to find features (or even smoking guns) that indirectly confirm this model at future colliders.
For Higgs physics, we expect a MSSM two Higgs doublet model in the decoupling limit. This is due to the fact that there is only a single Higgs doublet that is protected to be light by the Twin Higgs mechanism. Also, the SM Higgs mass limit suggests that $\tan \beta$ is reasonably large. So although this is not a smoking gun, if such Higgs physics is not observed, this model will be disfavored. Similar statements can be made within the top squark sector. They should be reasonably heavy so as to make the Higgs heavy enough, probably in a region that would be considered tuned normally. On the other hand, if there is gaugino unification, the gluino is probably lighter than expected, since the gaugino spectrum is light (see gaugino comments below).
However, there are interesting constraints on the electroweak sector within the supersymmetry breaking sector. As discussed in section \[sec:naturalness\], the cancellation of the D-term in the twin sector requires the additional masses for the electroweak adjoint scalars to be small, which implies that the common supersymmetry breaking masses for electroweak particles are also small. Within the gaugino/higgsino sector, this implies that they are light and should be below roughly $400 \, {{\rm GeV}}$. If these are light enough, there will be excellent prospects for their direct detection at LHC. This also applies to the slepton sector, not only giving hope for their direct production, but also suggesting that decay cascades of colored sparticles should end up with copious leptons.
While the scale of leptons and winos is closely related to the scale of electroweak symmetry breaking, ironically, the Higgsino mass parameter $\mu$ is essentially decoupled from it. Because the contribution from $\mu$ is cancelled off by the twin sector (under the assumption of $Z_2$ symmetry), it does not contribute significantly to the final scale for $m_Z$. In usual SUSY scenarios, $\mu$ often ends up quite large, decoupling the Higgsinos from the other electroweak fields. Here, this is not a requirement. Additionally, because we require all other $\mu$-type masses to be small, it is likely that Higgsinos will be light as well.
The model also predicts the existence of exotics in the form of the adjoints and, including unification, their GUT partners, the bachelors. One would also expect the electroweak adjoints to be light, however there can be a supersymmetry preserving mass (a soft $Z_2$ breaking) for the visible adjoints and bachelors that makes them heavier. So although their discovery prospects are not linked to naturalness, finding these exotics would be an intriguing hint of this model.
There is also potential new physics for Dark Matter and, in general, long lived particles in the model. Because of possibly light Higgsinos, annihilation in the early universe can be more efficient, bringing down the relic abundance without resorting to coannihilation or s-channel poles. There is both the possibility of twin nucleons comprising a large portion of the Dark Matter as well as the conventional LSP.[^5] There is also the possibility that adjoints or bachelors are long lived or even stable [@Fox:2002bu]. This is because these exotics are stable unless additional interactions for them are introduced. If these come from GUT suppressed operators in the Kähler potential, these fields can have lifetimes of seconds and thus will be stable on collider scales. However, for the adjoints, an alternative decay is possible if supersoft breaking couples to the visible sector. As long as this is suppressed compared to the twin sector, EWSB remains natural, while allowing the adjoints to decay promptly into SM particles. Therefore, there can be a large variety in the exotic sector in their appearance in events.
Some phenomenological aspects differ as the model interpolates between the twin Higgs and SUSY limits, i.e. as $f/m_{\tilde{t}}$ increases. One interesting signature of the twin Higgs limit is that the Higgs quartic coupling is smaller than expected, since the quartic Higgs coupling stops running above the mass of the $t'$ and not at $m_{\tilde{t}}$. Also, the radiative correction to the Higgs mass parameter is cut off by the $t'$ mass and not the stop’s, giving another potential handle on the twin sector. If this could be differentiated from an NMSSM or other MSSM extensions, this would be a curious hint of the new physics. On the other hand, this scenario would be very difficult to fit to a normal SUSY model. In the SUSY limit, one could interpret it as an MSSM/NMSSM model, but would have only hints of some UV structure that would be interpreted as a particular SUSY breaking scenario. In the intermediate case between SUSY and twin Higgs, the Higgses would be expected to mix more, leading to larger invisible decays for some of the Higgses, but this would probably be difficult to disentangle from a normal SUSY model.
In the best possible scenario, precision measurements of the SUSY/Higgs spectra and decays would give a compelling argument for this model. In this regard, it is fortunate that the naturalness constraints of the model suggest that SUSY phenomenology can be analyzed well at the LHC/ILC. This is because soft masses for electroweak particles are roughly bounded by 400 GeV and possibly lighter, giving many light sleptons, charginos and neutralinos. However, unfortunately the mass of the exotics are not guaranteed to be light, so this more distinctive signature is not guaranteed.
A Gauge Mediated Example {#a-gauge-mediated-example .unnumbered}
------------------------
To give a specific example spectrum, we can specialize to the case of a low scale gauge mediation model, with one set of messengers, under the assumption of $SU(3)^3$ unification. We choose $\Lambda = 60 \,{{\rm TeV}}, M_{mess}= 600 \,{{\rm TeV}}$ to determine the soft breaking masses and for the bachelors and adjoints we assume a unified $\mu$-term at the GUT scale of the value $\mu_{GUT} = 20 \,{{\rm GeV}}$. With these parameters, we get the spectrum in Table \[table:spectrum\]. To avoid complications in the Higgs sector, we do not attempt to break the spectrum down the electroweak gauginos and higgsinos down to physical neutralinos and charginos, but only give some mass entries to give an idea about the rough mass scales. From the point of view of naturalness, the large contributions to the scalar $T$ mass, suggest that the supersoft scale $m_T$ is quite large, which through unification suggests that the color supersoft scale $m_O$ gives too large a $Z_2$ violation to the twin top squark soft masses. So a realistic version of this spectrum would require that the coupling to the supersoft sector is not unified.
The phenomenology of this particular example is much like a normal gauge mediated scenario with the addition of the exotic adjoints and bachelors. There are decent prospects for discovering the electroweak charged adjoints given their low masses (via judicious choice of $\mu_{GUT}$). Unfortunately, unification pushes the colored adjoint to be very heavy and perhaps too difficult to discover. The normal SUSY spectrum is also quite reasonable and would be well explored at the LHC. Of course, the details of the decays of exotics could have an important impact on how much and how well the details of this model could be mapped out. In fact, it would be interesting to se if this can be done realistically, in this model or any other specific assumption of the SUSY breaking.
So far, we have unfortunately not determined a direct signal of the hidden twin sector in this model. This in fact was one of the motivations for the original Twin Higgs model, in demonstrating that the new physics that made EWSB natural did not have to be charged under the SM. In this model, the prospects are somewhat intermediate, where measurements within the SUSY phenomenology might give indirect signals of the hidden sector. The model is even better in that naturalness points to parameter space where the SUSY phenomenology is easily explorable at future colliders. So even though this is just one possible UV completion of the Twin Higgs, it does demonstrate that new physics signals may still exist in a Twin Higgs realization, maybe even enough to (indirectly) see the effects of a twin sector.
Scalars Mass (GeV) Gaugino Mass (GeV) Exotics Mass (GeV)
--------------- ------------ --------- ------------ ------------ ------------
$\tilde{e}_1$ 120 $M_1$ 120 $S,\Psi_S$ 20
$\tilde{e}_2$ 240 $M_2$ 240 $T$ 400
$\tilde{\nu}$ 230 $M_3$ 570 $\Psi_T$ 110
$\tilde{q}$ 930 $O$ 3900
$\Psi_O$ 3600
: Sample spectrum for a single messenger gauge mediation with $\Lambda = 60 \,{{\rm TeV}}, \mu_{GUT} = 20 \,{{\rm GeV}}$.\[table:spectrum\]
Discussion
==========
Supersymmetry, as a general idea, provides an elegant solution to the hierarchy problem. In practice, it is beset by a number of difficulties. The radiative electroweak symmetry breaking - an appealing aspect of the MSSM originally - is now too strong a force, inducing large values of $m_Z$ in the absence of significant fine tuning in the theory.
To this end, we have described a realization of the “twin Higgs" mechanism within the context of supersymmetry. Here, the visible sector is related to a “twin" sector by a $Z_2$ symmetry. The Higgs is a pseudo-Goldstone of an approximate $SU(4)$ symmetry, which arises as an accidental consequence of the $Z_2$, and the theory cancels the large Higgs mass terms without the inclusion of any new colored particles. Because the $Z_2$ is only broken by supersoft supersymmetry breaking operators, contributions to the Higgs mass are generally loop suppressed, but it maintains the usual $D$-term quartics of the MSSM.
The only model independent predictions involve some small mixing with the twin Higgs field, and the existence of an $SU(2)$-triplet. However, there is a great deal of phenomenology which is expected. GUT-completing the theory into trinification suggests the presence of a number of exotic fields, which should be light (100-300 [[GeV]{}]{}) by naturalness arguments. Such fields may lie at the end of new cascade decays, and may be stable on collider timescales. Similarly, we expect light winos, binos and sleptons, as their masses are indirectly tied to the effectiveness of the twin Higgs mechanism. The Higgsino mass, $\mu$ is essentially unrelated to the Higgs mass, and so can be much lighter than in many models.
The presence of light degrees of freedom (new photons and neutrinos) can be problematic for BBN and the CMBR. However, the $Z_2$ violating supersoft operators can yield apparently hard $Z_2$ violating terms in the low energy theory, such as larger Yukawas in the twin sector. Such small changes easily address the cosmological issues.
This proposal can be easily incorporated into most supersymmetric models, the most stringent requirements arise from the presence of new fields when the theory is unified. The most exciting phenomenological consequences arise from the “bachelor” fields - the unmarried GUT partners of the adjoints. A full discussion of their effects on cascades at the LHC, both with short and long bachelor lifetimes is warranted.
We would like to thank Gia Dvali, Jay Wacker for useful discussions. The work of S. Chang and N. Weiner was supported by NSF CAREER grant PHY-0449818. The work of L.J. Hall was supported by the US Department of Energy under contracts DE-AC03-76SF00098 and DE-FG03-91ER-40676 and by the National Science Foundation under grant PHY-00-98840.
[^1]: An alternative approach, employing an $SU(2)_L\times SU(2)_R$ gauge group with no twin sector, has been explored in [@martin].
[^2]: This could be possible if there is a linear term for $N$ of size $f^2$ in the superpotential. We do not consider this possibility due to the unknown mechanism which chooses this scale, but it is an interesting model in it’s own right as a solution to the Little Hierarchy.
[^3]: It is important to note that these expressions are only true under the assumption of no additional matter in the UV. In a gauge mediated scenario, for example, the additional messengers would modify the expressions above.
[^4]: Froggatt-Nielsen models are easily thought of as a mixing to a vectorlike fermion which is integrated out. For instance a superpotential $m \psi^c H + M \psi \psi^c + y \chi \chi^c \psi$ generates a low energy interaction $-y \frac{m}{M} \chi \chi^c H$. In our theory, if we simply allow either $m$ or $M$ to depend on ${ \left\langle {S'} \right\rangle }$, we trivially achieve either of the examples above. Note that the relevant new fields exist in trinification, by using the electroweak doublets in the ESPs as the $\psi, \psi^c$. Therefore, the unification story given in section \[sec:unification\] can still be maintained. In addition, for this choice, there is a GIM mechanism at work for the first two generations; thus the scale $M$ can be lowered to almost the weak scale.
[^5]: Due to the $\lambda N {\cal{H}}_u {\cal{H}}_d$ coupling, there is only a single R-parity and hence only one LSP.
|
---
abstract: 'Let $G$ be a finite group. We consider the set of the irreducible complex characters of $G$, namely $Irr(G)$, and the related degree set $cd(G)=\{\chi(1) : \chi\in Irr(G)\}$. Let $\rho(G)$ be the set of all primes which divide some character degree of $G$. In this paper we introduce the bipartite divisor graph for $cd(G)$ as an undirected bipartite graph with vertex set $\rho(G)\cup (cd(G)\setminus\{1\})$, such that an element $p$ of $\rho(G)$ is adjacent to an element $m$ of $cd(G)\setminus\{1\}$ if and only if $p$ divides $m$. We denote this graph simply by $B(G)$. Then by means of combinatorial properties of this graph, we discuss the structure of the group $G$. In particular, we consider the cases where $B(G)$ is a path or a cycle.'
address: 'Roghayeh Hafezieh, Department of Mathematics, Gebze Technical University, Gebze, Turkey'
author:
- Roghayeh Hafezieh
title: Bipartite divisor graph for the set of irreducible character degrees
---
Classification: Primary 05C25; secondary 05C75.
Keywords: bipartite divisor graph, irreducible character degree, path, cycle.
Introduction {#sec:introd}
============
Let $G$ be a finite group. It is well known that the set of irreducible characters of $G$, denoted by $Irr(G)$, can be used to obtain information about the structure of the group $G$. The value of each character at the identity is the degree of the character and by $cd(G)$ we mean the set of all irreducible character degrees of $G$. Let $\rho(G)$ be the set of all primes which divide some character degree of $G$. When studying problems on character degrees, it is useful to attach the following graphs which have been widely studied to the sets $\rho(G)$ and $cd(G)\setminus\{1\}$, respectively.
- Prime degree graph, namely $\Delta(G)$, which is an undirected graph whose set of vertices is $\rho(G)$; there is an edge between two different vertices $p$ and $q$ if $pq$ divides some degree in $cd(G)$.
- Common divisor degree graph, namely $\Gamma(G)$, which is an undirected graph whose set of vertices is $cd(G)\setminus\{1\}$; there is an edge between two different vertices $m$ and $n$ if $(m,n)\neq1$.
The notion of the bipartite divisor graph was first introduced by Iranmanesh and Praeger in [@IP] for a finite set of positive integers. As an application of this graph in group theory, in [@BDIP], the writers considered this graph for the set of conjugacy class sizes of a finite group and studied various properties of it. In particular they proved that the diameter of this graph is at most six, and classified those groups for which the bipartite divisor graphs of conjugacy class sizes have diameter exactly $6$. Moreover, they showed that if the graph is acyclic, then the diameter is at most five and they classified the groups for which the graph is a path of length five. Similarly, Taeri in [@T] considered the case that the bipartite divisor graph of the set of conjugacy class sizes is a cycle and (by using the structure of $F$-groups and the classification of finite simple groups) proved that for a finite nonabelian group $G$, the bipartite divisor graph of the conjugacy class sizes is a cycle if and only if it is a cycle of length $6$, and for an abelian group $A$ and $q\in\{4,8\}$, $G\simeq A\times SL_2(q)$. Inspired by these papers, in this work we consider the bipartite divisor graph for the set of irreducible character degrees of a finite group and define it as follows:
Let $G$ be a finite group. The bipartite divisor graph for the set of irreducible character degrees of $G$, is an undirected bipartite graph with vertex set $\rho(G)\cup (cd(G)\setminus\{1\})$, such that an element $p$ of $\rho(G)$ is adjacent to an element $m$ of $cd(G)\setminus\{1\}$ if and only if $p$ divides $m$.
Since classifying groups whose associated graphs have special graphical shapes is an important topic in this area, in this paper, we will discuss the cases where $B(G)$ is a path or a cycle for a group $G$.
In the second section, we suppose that $G$ is a solvable group. After finding the best upper bound for diameter of $B(G)$, we consider the case where $B(G)$ is a path of length $n$. We prove $n\leq 6$ and in Theorem \[thm:2\], which is the main theorem of this section, we give some group theoretical properties of such a group.
In the third section, we consider the case that $G$ is nonsolvable and $B(G)$ is a union of paths where by union of paths we mean that each connected components of $B(G)$ is a path, (so $B(G)$ is a path if there exists only one path in this union). Theorem \[thm:99\] is the main theorem of this section.
Finally in section four, we consider the case where $B(G)$ is a cycle. We prove that $B(G)$ is a cycle if and only if $G$ is solvable and $B(G)$ is a cycle of length four or six. By using these properties, we find a special subgroup of $G$ which explains the structure of the irreducible character degrees of $G$. Theorem \[thm:98\] is the main theorem of this section.
For positive integers $m$ and $n$, we denote the greatest common divisor of $m$ and $n$ by $(m,n)$; the number of connected components of a graph $\mathcal{G}$ by $n(\mathcal{G})$; the diameter of a graph $\mathcal{G}$ by $diam(\mathcal{G})$ (where by the diameter we mean the maximum distance between vertices in the same connected component of the graph). If $\alpha$ is a vertex of the graph $\mathcal{G}$, then $deg_{\mathcal{G}}(\alpha)$ is the number of vertices adjacent to $\alpha$ in $\mathcal{G}$. If the graph is well-understood, then we show it by $deg(\alpha)$. By length of a path or a cycle, we mean the number of edges in the path or in the cycle. Also, by $P_{n}$ and $C_{n}$ we mean a path of length $n$ and a cycle of length $n$, respectively. Let $G$ be a finite solvable group. As usual, we write $dl(G)$ and $h(G)$ to denote the derived length and Fitting height of $G$, respectively. Other notation throughout the paper is standard.
Solvable groups whose bipartite divisor graphs are paths {#sec:Path}
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We begin by giving the best upper bound for $diam(B(G))$.
For a finite solvable group $G$, $diam(B(G))\leq 7$ and this bound is best possible.
Since $G$ is solvable, by [@L Corollary 4.2, Theorem 7.2] we know that $diam(\Delta(G))$ and $diam(\Gamma(G))$ are both less than or equal to three. Now by [@IP Lemma 1] we have one of the following cases:
- $diam(B(G))=2 max\{diam(\Delta(G)),diam(\Gamma(G))\}\leq 2\times 3 =6$ or
- $diam(B(G))=2 diam(\Delta(G))+1= 2 diam(\Gamma(G))+1\leq (2\times 3)+1 =7$.
So in general, $diam(B(G))\leq 7$. Now let $G$ be the group as in [@1], then
$cd(G)=\{1, 3, 5, 3\times 5, 7\times 31\times 151, 2^{7}\times 7\times 31\times 151, 2^{12}\times 31\times 151, 2^{12}\times 3\times 31\times 151,$ $$2^{12}\times 7\times 31\times 151, 2^{13}\times 7\times 31\times 151, 2^{15}\times 3\times 31\times 151\}$$ It is easy to see that $diam(\Delta(G))=diam(\Gamma(G))=3$ and $diam(B(G))=7$.
\[prop:1\] Let $G$ be a finite solvable group. Assume that $B(G)$ is a path of length $n$. Then $n\leq 6$ and $dl(G)\leq 5$. In particular, if $B(G)$ equals $P_5$ or $P_6$, then $h(G)$ is less than or equal $3$ or $4$, respectively. (In general, we know that for a solvable group $h(G)\leq |cd(G)|$.)
Since $B(G)=P_{n}$, we can see that both $\Delta(G)$ and $\Gamma(G)$ are paths, (see [@IP Theorem 3]). Since $G$ is solvable, by [@L Theorem 4.1], we conclude that $\Delta(G)=P_m$ where $m\leq 3$. Furthermore [@L Theorem 4.5] implies that a path of length three cannot be the prime degree graph of a solvable group, so $m\leq 2$. On the other hand we know that $|diam(\Delta(G))-diam(\Gamma(G))|\leq 1$, so $n\leq 6$ which implies that $|cd(G)|\leq 5$. Now by [@7], we have $dl(G)\leq 5$. In the case that $B(G)$ equals $P_5$ or $P_6$, since $|cd(G)|\geq 4$, [@2] verifies that $h(G)\leq |cd(G)|-1$. Thus $h(G)\leq 3$ if $B(G)=P_5$ and $h(G)\leq 4$ if $B(G)=P_6$.
\[thm:2\] Let $G$ be a finite solvable group. Assume that $B(G)$ is a path of length $n$. Then we have the following cases:
- $G\simeq P\times A$, where $P$ is a $p$-group for some prime number $p$ and $A$ is an abelian group.
- There exist normal subgroups $N$ and $K$ of $G$ and a prime number $p$ with the following properties:
- $\frac{G}{N}$ is abelian.
- $\pi(G/K)\subseteq\rho(G)$.
- Either $p$ divides all the nontrivial irreducible character degrees of $N$, or there exists a unique nontrivial $\psi(1)\in cd(N)$ such that $[G:N]\psi(1)\in cd(G)$.
- $cd(G)=\{1,p^{\alpha},q^{\beta},p^{\alpha}q^{\beta}\}$, where $p$ and $q$ are distinct primes.
- There exists a prime $s$ such that $G$ has a normal $s$-complement.
- $G$ has an abelian normal subgroup $N$ such that $[G : N] = m\in cd(G)^{*}$.
Since $B(G)$ is a path of length $n$, Proposition \[prop:1\] implies that $n\leq 6$.
First suppose that $n\geq 4$. This implies that nonlinear irreducible character degrees of the solvable group $G$ are not all equal. We claim that there exists a normal subgroup $K>1$ of $G$ such that $\frac{G}{K}$ is nonabelian. If $G'$ is not a minimal normal subgroup of $G$, then $G$ has a nontrivial normal subgroup $N$ such that $\frac{G}{N}$ is nonabelian. Otherwise, if $G'$ is a minimal normal subgroup of $G$, then it cannot be unique since nonlinear irreducible character degrees of the solvable group $G$ are not all equal, [@5 Lemma 12.3]. So we can see that $G$ has a nontrivial normal subgroup $N$ such that $\frac{G}{N}$ is nonabelian. Let $K$ be maximal with respect to the property that $\frac{G}{K}$ is nonabelian. It is clear that $(\frac{G}{K})'$ is the unique minimal normal subgroup of $\frac{G}{K}$. Thus $\frac{G}{K}$ satisfies the hypothesis of [@5 Lemma 12.3]. So all nonlinear irreducible characters of $\frac{G}{K}$ have equal degree $f$ and we have the following cases:
$B(G)=P_6$.
In this case $|\rho(G)|$ is either $3$ or $4$. By [@L Theorem 4.5], $\Delta(G)$ cannot be a path of length $3$, so the case $|\rho(G)|=4$ is impossible. We may assume that $B(G): m-p-n-q-l-r-k$, where $p$, $q$, and $r$ are distinct prime numbers. We claim that $\frac{G}{K}$ is not an $s$-group for a prime $s$. If not, then $s\in\{p,q,r\}$. As $cd(G)$ does not contain a degree which is a power of $q$, it is clear that $s \neq q$. Thus either $s=p$ and $cd(\frac{G}{K})=\{1,f=m=p^{\alpha}\}$ for a positive integer $\alpha$ or $s=r$ and $cd(\frac{G}{K})=\{1,f=k=r^{\beta}\}$ for a positive integer $\beta$. Since the roles of $p$ and $r$ are the same, without loss of generality, we may assume that $s=p$ and there exists $\theta\in Irr(\frac{G}{K})$ such that $\theta(1)=p^{\alpha}=m$. Let $\chi\in Irr(G)$ with $\chi(1)=k$. Since $p$ does not divide $\chi(1)$, we have $\chi_{K}\in Irr(K)$. Now by Gallagher’s Theorem [@5], we conclude that $\chi\theta\in Irr(G)$, which is distinct from $\theta$. Thus $\chi(1)\theta(1)=mk\in cd(G)^{*}$, so either $mk=l$ or $mk=n$. Since $(n,k)=1$ and $(l,m)=1$, none of these cases are possible, so $\frac{G}{K}$ is not an $s$-group. Now [@5 Lemma 12.3] implies that $\frac{G}{K}$ is a Frobenius group with abelian Frobenius complement of order $f$ and Frobenius kernel $\frac{N}{K}=(\frac{G}{K})'$ which is an elementary abelian $s$-group for some prime $s$. Thus $[G/K:N/K] = [G:N] = f\in cd(\frac{G}{K})\subseteq cd(G) =\{1,m,n,k,l\}$. It is clear that $N$ is not abelian. First suppose that $f=m$ and let $\chi$ be a nonlinear irreducible character of $G$ with $\chi(1)=k$. If $s\notin \rho(G)$, then $(\chi(1),[G:K])=1$. This verifies that $\chi_{K}\in Irr(K)$ and by Gallagher’s Theorem [@5] we have $\chi(1)f=km\in cd(G)$ which is impossible. Therefore $\pi(G/K)\subseteq \rho(G)$. Let $\psi$ be a nonlinear irreducible character of $N$. If $s$ does not divide $\psi(1)$, then by [@5 Theorem 12.4] we conclude that $[G:N]\psi(1)\in cd(G)$. Now for any $1\neq \zeta(1)\in cd(N)$ which is different from $\psi(1)$, $[G:N]\psi(1)\notin cd(G)$. This implies that $s | \zeta(1)$ and $\psi(1)$ is a unique element in $cd(N)^{*}$ with respect to the property that $[G:N]\psi(1)\in cd(G)$. If $w$ is a nontrivial element of $cd(\frac{G}{N})$, then by Gallagher’s Theorem [@5] we can see that $kw\in cd(G)$ which is impossible. Thus $\frac{G}{N}$ is abelian and case $(ii)$ holds. The case $f=k$ is similar.
Now assume $f=n$. Suppose that $\psi\in Irr(N)$ with $\psi(1)\neq 1$. Since $(l,m)=1$, $(m,k)=1$ and $(n,k)=1$, we can see that $[G:N]\psi(1)\notin cd(G)$. Now by [@5 Theorem 12.4], we conclude that $s | \psi(1)$. Since $\psi\in Irr(N)$ was arbitrary, we deduce that $s$ divides all nontrivial irreducible character degrees of $N$. Furthermore, this implies that $\pi(\frac{G}{K})\subseteq \rho(G)$. As $\frac{G}{K}$ is Frobenius, we have $(s,[G:N])=1$. Since $f=n$, we have $s=r$. If $w$ is a nontrivial element of $cd(\frac{G}{N})$, then Gallagher’s Theorem [@5] implies that $kw\in cd(G)$, so $kw=l$. This forces $w$ to be a power of $q$ which is impossible since $cd(\frac{G}{N})\subseteq cd(G)$. Thus $\frac{G}{N}$ is abelian and case $(ii)$ holds. The case $f=l$ is similar.
$B(G)=P_5$.
We may assume that $B(G): p-m-q-l-r-n$, where $p$, $q$, and $r$ are distinct prime numbers. We claim that $\frac{G}{K}$ is not an $s$-group. If $\frac{G}{K}$ is an $s$-group for some prime $s$, then there exists $\eta\in Irr(\frac{G}{K})$ such that $\eta(1)=s^{\alpha}\in cd(G)^{*}=\{m,n,l\}$. According to the form of $B(G)$, we can see that $s=r$ and $\eta(1)=r^{\alpha}=n$, (since $n$ has only one prime divisor). Let $\chi\in Irr(G)$ with $\chi(1)=m$. Since $(m,n)=1$, $r$ does not divide $m$. Thus $\chi_{K}\in Irr(K)$. Now by Gallagher’s Theorem [@5], we conclude that $\chi(1)\eta(1)=mn\in cd(G)^{*}$, so $l=mn$. But in this case $l$ has three distinct prime divisors, which contradicts the form of $B(G)$. Hence $\frac{G}{K}$ is not an $s$-group. Now [@5 Lemma 12.3] implies that $\frac{G}{K}$ is a Frobenius group with abelian Frobenius complement of order $f$ and Frobenius kernel $\frac{N}{K}=(\frac{G}{K})'$ which is an elementary abelian $s$-group for some prime $s$. So $[G/K:N/K] = [G:N] = f\in cd(\frac{G}{K})\subseteq cd(G) =\{1,m,n,l\}$. Similar to the previous case, we can see that for $f=n$, case $(ii)$ occurs. So suppose that $f\neq n$. If $\psi\in Irr(N)$ with $\psi(1)\neq 1$, then $[G:N]\psi(1)\notin cd(G)$. Now [@5 Theorem 12.4] implies that $s | \psi(1)$. So either $f=l$ and $s=p$ or $f=m$ and $s=r$. Let $f=l$, $s=p$ and $\zeta$ be an irreducible constituent of $\psi^{G}$. Then $\zeta(1)=m$. Since $([G:N],\zeta(1))=1$, we have $\zeta_{N}\in Irr(N)$. Thus $\zeta_{N}=\psi$ and $\psi(1)=m$. Since any character degree of $\frac{G}{N}$ must divide the order of the group, $\frac{G}{N}$ is abelian and case $(ii)$ occurs. The case $f=m$ and $s=r$ is similar.
$B(G)=P_4$.
First suppose that $B(G): p-m-q-n-r$, where $p$, $q$, and $r$ are distinct prime numbers. Since $q$ divides every nonlinear character degree, $G$ has a normal $q$-complement, ( [@5 Corollary 12.2]). Thus case $(iv)$ occurs with $s=q$.
Now suppose that $cd(G)=\{1,m,n,l\}$. We may assume that $B(G): m-q-l-p-n$, where $p$ and $q$ are distinct prime numbers. Suppose $\frac{G}{K}$ is a Frobenius group with abelian Frobenius complement of order $f$ and Frobenius kernel $\frac{N}{K}=(\frac{G}{K})'$ which is an elementary abelian $s$-group for some prime $s$. If $f=m$ and $s\notin \rho(G)$, then by Gallagher’s Theorem [@5] we can conclude that $cd(G)=\{1,p^{\alpha},q^{\beta},p^{\alpha}q^{\beta}\}$, so case $(iii)$ occurs. If $f=m$ and $s\in \rho(G)$, then similar to the previous cases we can see that $s$ divides each nonlinear irreducible character degree of $N$ and $\frac{G}{N}$ is abelian. Thus case $(ii)$ occurs. The case $f=n$ is similar. Let $f=l$. Since $(s,f)=1$ and $\frac{N}{K}$ is nontrivial, we have $s\notin \rho(G)$. On the other hand, for $\psi\in Irr(N)$ with $\psi(1)\neq 1$, $[G:N]\psi(1)\notin cd(G)$. [@5 Theorem 12.4] implies that $s | \psi(1)$ and $s\in \rho(G)$ which is a contradiction. So $N$ is abelian. Since $N$ is normal and abelian, $\chi(1)$ divides $[G:N]=f$, for each $\chi\in Irr(G)$, which contradicts the form of $B(G)$. So $f\neq l$.
Now suppose $\frac{G}{K}$ is an $s$-group for a prime $s$. It is obvious that $f\neq l$. Without loss of generality we may assume that there exists $\eta\in Irr(\frac{G}{K})$ such that $\eta(1)=p^{\alpha}=n$. Let $\chi\in Irr(G)$ with $\chi(1)=m$. As $p$ does not divide $m$, we have $\chi_{K}\in Irr(K)$. Now by Gallagher’s Theorem [@5], we conclude that $\chi(1)\eta(1)=mn\in cd(G)^{*}$, so $l=mn$. Thus $cd(G)=\{1,p^{\alpha},q^{\beta},p^{\alpha}q^{\beta}\}$ and case $(iii)$ holds. It should be mentioned that such a group exists. We can see that for $G=S_3\rtimes A_4$ which is a solvable group we have $cd(G)=\{1,2,3,6\}$.
Now suppose that $B(G)$ is a path of length $n$ where $n\leq 3$. We have the following cases:
$B(G)=P_3$.
Assume $B(G): p-m-q-n$, where $p$ and $q$ are distinct prime numbers. Since $q$ divides every nonlinear character degree of $G$, we conclude that $G$ has a normal $q$-complement, ( [@5 Corollary 12.2]). Thus case $(iv)$ occurs with $s=q$.
$B(G)=P_2$.
Assume first that $cd(G)=\{1,m\}$ where $m$ is not a prime power. Then $m=p^{a}q^{b}$ for some primes $p\neq q$ and integers $a,b\geq 1$. Now by [@5 Theorem 12.5], $G$ has an abelian normal subgroup $N$ such that $[G:N]=m$, so case $(v)$ holds.
If $cd(G)=\{1,m,n\}$, then both $1<m<n$ are powers of a prime $s$ and so $G$ has a normal $s$-complement and case $(iv)$ holds.
$B(G)=P_1$.
Thus we have $cd(G)=\{1,p^{a}\}$ for some prime $p$ and integer $a\geq 1$. Now [@5 Theorem 12.5] implies that either $G\simeq P\times A$, where $P$ is a $p$-group, $A$ is abelian and case $(i)$ holds, or $G$ has an abelian normal subgroup of index $p^{a}$ and case $(v)$ occurs with $m=p^{a}$.
According to the proof of Theorem \[thm:2\], we have the following remarks.
Suppose that $G$ is a finite group and $q\in \rho(G)$ is an end point (a vertex of degree one) in $B(G)$. So $G$ is a finite group with exactly one irreducible character degree $m$ which is divisible by $q$. Let $Q\in Syl_{q}(G)$, then either $Q\lhd G$ or $U:= O_{q}(G)< Q$. In the second case, by [@LMNH Theorem A], we have the following properties:
- $U$ is abelian.
- $\frac{Q}{U}$ is cyclic.
- $|\frac{Q}{U}|=m_{q}$, which is the $q$-part of $m$.
- $\frac{Q}{U}$ is a $TI$-set in $\frac{G}{U}$.
Now suppose that $G$ is a finite group which is not nilpotent and $B(G)=P_n$ is a path of length $n$. It is clear that $B(G)$ has at least one prime as an end point if and only if $n=1$, $n=2$ with $cd(G)=\{1,m\}$, $n=3$, $n=4$ with $cd(G)=\{1,m,n\}$ or $n=5$. Since $G$ is not nilpotent, there exists a prime $q\in\rho(G)$ such that $O_{q}(G)$ has the above properties.
Suppose $G$ is a finite group such that $B(G)$ is $P_{6}$, $P_{5}$ or $P_{4}$ with $cd(G)=\{1,m,n\}$. There exist two distinct primes $p$ and $r$ such that they are not neighbors in the graph $\Delta(G)$. Now [@L Theorem 5.1] verifies that $l_{p}(G)\leq 2$ and $l_{r}(G)\leq 2$.
Nonsolvable groups whose bipartite divisor graphs are union of paths {#sec:nonsolvable}
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Let $G$ be a finite nonsolvable group. By [@L Theorem 6,4], we know that $\Delta(G)$ has at most three connected components. Since [@IP] implies that $n(B(G))=n(\Gamma(G))=n(\Delta(G))$, we conclude that $n(B(G))\leq 3$. In the rest of this section, we consider the case where each connected component of $B(G)$ is a path and we start by looking at simple groups.
For a nonabelian simple group $S$, $B(S)$ is disconnected and all the connected components are paths if and only if $S$ is isomorphic to one of the following groups:
- $PSL(2,2^{n})$ where $|\pi(2^{n}\pm 1)|\leq 2$;
- $PSL(2,p^{n})$ where $p$ is an odd prime and $|\pi(p^{n}\pm 1)|\leq 2$.
By [@IP] we know that $n(B(S))=n(\Gamma(S))=n(\Delta(S))$. Since all connected components of $B(S)$ are paths, so are the connected components of $\Delta(S)$. This implies that $\Delta(S)$ has no triangles. Thus by [@H Lemma 3.1], one of the following cases holds:
- $S\simeq PSL(2,2^{n})$ where $|\pi(2^{n}\pm 1)|\leq 2$ and so $|\pi(S)|\leq 5$;
- $S\simeq PSL(2,p^{n})$ where $p$ is an odd prime and $|\pi(p^{n}\pm 1)|\leq 2$ and so $|\pi(S)|\leq 4$.
Since $cd(PSL(2,2^{n}))=\{1,2^{n},2^{n}+1,2^{n}-1\}$, by [@L Theorem 6.4], we conclude that $B(S)$ has three connected components while $B(S)$ has two connected components in the case $(ii)$. Also it is clear that in both cases all the connected components are paths.
\[lem:22\] Let $G$ be a finite group. Assume that $B(G)$ is a union of paths. If $|\rho(G)|=5$, then $G\simeq PSL(2,2^{n})\times A$, where $A$ is an abelian group and $|\pi(2^{n}\pm 1)|=2$.
Since each connected component of $B(G)$ is a path, we can see that $\Delta(G)$ is triangle-free. We claim that $B(G)$ is disconnected. If $B(G)$ is connected, then $\Delta(G)$ is a path. Now [@H Theorem B] implies that $G\simeq H\times K$, where $H$ is isomorphic with $A_{5}$ or $PSL(2,8)$ and $K$ is a solvable group. This contradicts that $\Delta(G)$ is a path. Thus $n(\Delta(G))=n(B(G))>1$. By [@H Theorem B], we have $G\simeq PSL(2,2^{n})\times A$, where $A$ is an abelian group and $|\pi(2^{n}\pm 1)|=2$. So $B(G)$ is a graph with three connected components, one of them is a path of length one and the other two components are paths of length two.
\[thm:99\] Let $G$ be a finite nonsolvable group. Assume that $B(G)$ is a union of paths. Then $B(G)$ is disconnected and we have the following cases:
- If $n(B(G))=2$, then let $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ be the connected components of $B(G)$. We have $|\rho(G)|\in\{3,4\}$, $\mathcal{C}_{1}$ is a path of length one and $\mathcal{C}_{2}$ is isomorphic with $P_{n}$, where $n\in\{|\rho(G)|, |\rho(G)|+1\}$.
- If $n(B(G))=3$, then $G\simeq PSL(2,2^{n})\times A$, where $A$ is an abelian group and $n\geq 2$.
Since $B(G)$ is a union of paths, we deduce that $\Delta(G)$ is triangle-free. Now [@H Theorem A] implies that $|\rho(G)|\leq 5$. In Lemma \[lem:22\] we saw that if $|\rho(G)|=5$, then $G\simeq PSL(2,2^{n})\times A$, where $A$ is an abelian group and $|\pi(2^{n}\pm 1)|=2$. This implies that $n(B(G))=3$. On the other hand, since $G$ is a nonsolvable group, by [@L Theorem 6.4] we conclude that $n(B(G))=3$ if and only if $G\simeq PSL(2,2^{n})\times A$, where $A$ is an abelian group and $n\geq 2$. So we may assume that $|\rho(G)|\leq 4$ and $n(B(G))\leq 2$. It is obvious that $|\rho(G)|> 1$. Since $G$ is nonsolvable, it must have a nonabelian chief factor say $M/K\simeq S^{k}$, where $S$ is a nonabelian simple group and $k$ is a positive integer. Now it follows from $It\hat{o}-Michler$ and $Burnside's$ $p^{a}q^{b}$ theorems that $|\rho(S)|\geq 3$ and thus $|\rho(G)|\geq |\rho(M/K)|=|\rho(S)|\geq 3$. Now we have the following cases:
- $|\rho(G)|=3$. First suppose that $n(B(G))=2$. Thus $n(\Gamma(G))=2$. Now [@L Theorem 7.1] implies that one of the connected components of $\Gamma(G)$ is an isolated vertex and the other one has diameter at most two. This isolated vertex generates a connected component in $B(G)$ which is a path of length one. We denote this component by $\mathcal{C}_{1}$. Let $\mathcal{C}_{2}$ be the other component of $B(G)$. Since $G$ is nonsolvable, we can easily see that $\mathcal{C}_{2}$ is either a path of length $3$ or $4$. Now suppose that $B(G)$ is connected. Since $G$ is nonsolvable, we have $|cd(G)|>3$. Now by [@MM Corollary B], we can conclude that $|cd(G)|\geq 5$. This implies that $\Gamma(G)$ is a path of length $3$ which is impossible by [@HQ Theorem A]. Note that the case $\Gamma(G)=P_{4}$ is not possible as $|\rho(G)|=3$.
- $|\rho(G)|=4$. If $B(G)$ is connected, then $\Delta(G)$ is a path of length three which is impossible by [@LW Theorem B], so we may assume that $n(B(G))=2$. Similar to the previous case, we can see that $\mathcal{C}_{1}$ is a path of length one and $\mathcal{C}_{2}$ is either a path of length $4$ or $5$.
Let $G_{1}=M_{10}$ and $G_{2}=PSL(2,25)$. Since $cd(M_{10})=\{1,9,10,16\}$ and $cd(PSL(2,25))=\{1,13,24,25,26\}$, it is easy to see that $B(G_{i})$ has two connected components $\mathcal{C}_{i,1}$ and $\mathcal{C}_{i,2}$, for $i\in\{1,2\}$. $\mathcal{C}_{i,1}$ is a path of length one for each $i$, $\mathcal{C}_{1,2}$ is a path of length $3$ and $\mathcal{C}_{2,2}$ is a path of length $5$.
Groups whose bipartite divisor graphs are cycles {#sec:Cycle}
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\[rem:01\] Let $G$ be a finite group whose $B(G)$ is a cycle of length $n\geq 6$. Then both $\Delta(G)$ and $\Gamma(G)$ are cycles.
Suppose that $B(G)=C_{n}$ and $n\geq 6$. Let $\Phi\in\{\Delta(G),\Gamma(G)\}$. By graph theory we know that $\Phi$ is a cycle if and only if it is a connected graph such that every vertex in $\Phi$ has degree two. Since $n(B(G))=1$, [@IP Lemma 3.1] implies that $\Phi$ is a connected graph. Let $\alpha$ be a vertex of $\Phi$. It is clear that $deg_{B(G)}(\alpha)=2$. Since $n\geq 6$, one can see that $deg_{\Phi}(\alpha)=2$. Thus $\Phi$ is a cycle.
\[thm: 1\] Let $G$ be a finite group whose $B(G)$ is a cycle of length $n$. Then $n\in\{4,6\}$.
Since $B(G)$ is a cycle of length $n$, it is clear that $n\geq 4$. Furthermore, by [@IP Theorem 3] both $\Delta(G)$ and $\Gamma(G)$ are acyclic if and only if $B(G)=C_{4}$. In this case $\Delta(G)\simeq\Gamma(G)\simeq P_2$. On the other hand, if $n\geq 6$, then Lemma \[rem:01\] implies that both $\Delta(G)$ and $\Gamma(G)$ are cycles. So $\Delta(G)$ is either a cycle or a path (which is a tree). Now [@H Theorem C] implies that $\Delta(G)$ has at most four vertices. Thus $B(G)$ can be $C_4$, $C_6$ or $C_8$. We claim that $B(G)$ cannot be a cycle of length eight. Otherwise, if $B(G)=C_8$, then $\Delta(G)\simeq\Gamma(G)=C_4$. First suppose that $G$ is solvable. By the main theorem of [@LMe], we have $G\simeq H\times K$, where $\rho(H)=\{p,q\}$, $\rho(K)=\{r,s\}$ and both $\Delta(H)$ and $\Delta(K)$ are disconnected graphs. This implies that there exists $m,n\in cd(H)^{*}$ and $l,k\in cd(K)^{*}$ such that $m=p^{\alpha}$, $n=q^{\beta}$, $l=r^{\gamma}$ and $k=s^{\delta}$, for some positive integers $\alpha$, $\beta$, $\gamma$ and $\delta$. By the structure of $G$ it is clear that $\{1, m, n, l, k, ml, mk, nl, nk\}\subseteq cd(G)$, which contradicts the form of $B(G)$. So in the solvable case, $B(G)$ is not a cycle of length eight. On the other hand, [@LW] implies that a square (i.e. $C_{4}$) cannot be the prime degree graph of a nonsolvable group. Thus when $G$ is nonsolvable $B(G)$ is not $C_8$.
Let $G$ be a finite group and $B(G)$ be a cycle. Then $G$ is solvable and $dl(G)\leq |cd(G)|\leq 4$.
Let $B(G)=C_{n}$. By Theorem \[thm: 1\], we deduce that $n\in\{4,6\}$ and $\Gamma(G)$ is a complete graph. Since $\Gamma(G)$ is complete, [@bian] implies that $G$ is solvable. As $|cd(G)|\leq 4$ in the solvable group $G$, we conclude that $dl(G)\leq |cd(G)|\leq 4$ by [@3].
\[ex:1\] From [@LM] we know that for every pair of odd primes $p$ and $q$ such that $p$ is congruent to $1$ modula $3$ and $q$ is a prime divisor of $p+1$, there exists a solvable group $G$ such that $cd(G)=\{1,3q,p^{2}q,3p^{3}\}$. This gives an example of a solvable group $G$ whose bipartite divisor graph related to the set of character degrees, is a cycle of length $6$.
There are exactly $66$ groups of order $588$. Among these groups, there are exactly two nonabelian groups whose bipartite divisor graphs are cycles of length four. These groups have $\{1,6,12\}$ as their irreducible character degrees set.
Let $G$ be a nonabelian finite group, $P\in Syl_{p}(G)$ such that $|P|=p^{2}$, where $p\geq 7$, $p \neq 11$ and $[G:P]=12$. $B(G)$ is a cycle if and only if $cd(G)=\{1,6,12\}$.
It is easy to see that $P$ is a normal abelian Sylow $p$-subgroup of $G$, so $p$ does not appear in the vertex set of $B(G)$. Therefore, $B(G)$ is a cycle of length four with $\rho(G)=\{2,3\}$. We claim that $\frac{G}{P}$ is abelian. Suppose $\frac{G}{P}$ is nonabelian. This implies that $\frac{G}{P}$ is one of the groups $A_4$, $D_{12}$, or $T\simeq \mathbb{Z}_3\rtimes\mathbb{Z}_4$. In all the above cases we can see that there exists $\chi$ in $Irr(G)$ such that $\chi(1)$ is either $2$ or $3$. Thus $\chi(1)$ is a vertex of degree one in $B(G)$, which contradicts that $B(G)$ is a cycle. Therefore $\frac{G}{P}$ is abelian. This implies that $G$ is either $P\rtimes \mathbb{Z}_{12}$ or $P\rtimes (\mathbb{Z}_2\times\mathbb{Z}_6)$. Now by [@p Lemma 2.3], we have $cd(G)=\{\beta(1)[G:I_{G}(\lambda)] : \lambda\in Irr(P), \beta\in Irr(\frac{I_{G}(\lambda)}{P})\}$. Since $\frac{G}{P}$ is abelian, we conclude that $\frac{I_{G}(\lambda)}{P}$ is abelian and $cd(G)=\{[G:I_{G}(\lambda)] : \lambda\in Irr(P)\}$. As for each $\lambda\in Irr(P)$, $[G:I_{G}(\lambda)]$ divides $[G:P]=12$ and $B(G)$ is a cycle of length four, we conclude that $cd(G)=\{1,6,12\}$.
\[rem:21\] Let $G$ be a finite group with $B(G)=C_{n}$. First suppose that $n=4$ and $\pi(G)=\rho(G)=\{p,q\}$. Since $p$ divides every nonlinear irreducible character degree of $G$, [@5 Corollary 12.2] implies that $G$ has a normal $p$-complement $Q$. As $\pi(G)=\rho(G)=\{p,q\}$, $Q$ is the normal Sylow $q$-subgroup of $G$. Let $P$ be a Sylow $p$-subgroup of $G$. Similarly, we can see that $P$ is normal in $G$. Thus all the Sylow subgroups of $G$ are normal which implies that $G$ is nilpotent. Therefore $G$ is the direct product of its Sylow subgroups which contradicts the structure of $B(G)$. So in this case we always have $\rho(G)\subset \pi(G)$. But this is not always the case if $B(G)=C_{6}$, because as we can see in Example \[ex:1\], $G$ is a group generated by $P$, $\sigma$ and $\tau$, where $P$ is a Camina $p$-group of nilpotent class three and $\sigma$, $\tau$ are two commuting automorphisms of $P$ with orders $q$ and $3$, respectively. As it is explained in [@LM], we have $|G|= p^{7}3q$. So in this case $\pi(G)=\rho(G)$.
Suppose that $G$ is a finite group. The following theorem shows that if $B(G)$ is a cycle of length four, then $G$ has a normal abelian subgroup which explains the structure of $cd(G)$.
\[thm:98\] Let $G$ be a finite group. Assume that $B(G)$ is a cycle of length $4$. There exists a normal abelian subgroup $N$ of $G$ such that $cd(G)=\{[G:I_{G}(\lambda)] : \lambda\in Irr(N)\}$.
Suppose that $G$ is a finite group of order $p_1^{a_1}p_2^{a_2}...p_l^{a_l}$. Without loss of generality, we may assume that $p_1=p$, $p_2=q$ and $B(G): p-m-q-n-p$. Thus for each $p_i\in \pi(G)\setminus\{p,q\}$, the Sylow $p_i$-subgroup of $G$ is a normal abelian subgroup of $G$. Let $N=P_3\times ...\times P_l$. Since $B(G)$ is a cycle, we deduce from Remark \[rem:21\] that $G$ is not a $\{p,q\}$-group. Thus $N$ is a nontrivial normal abelian subgroup of $G$. Now $\frac{G}{N}$ is a $\{p,q\}$-group, so its bipartite divisor graph is not a cycle of length four. As $cd(\frac{G}{N})\subseteq cd(G)$, there exists no element of $cd(\frac{G}{N})$ which is a prime power. So for each nonlinear $\chi\in Irr(\frac{G}{N})$, $\chi(1)=p^{\alpha}q^{\beta}$, for some positive integers $\alpha$ and $\beta$. This implies that $\frac{G}{N}$ is the direct product of its Sylow subgroups which are nonabelian. But this contradicts the form of $cd(\frac{G}{N})$. Thus $\frac{G}{N}$ is abelian and $G=N\rtimes H$, where $H$ is the hall $\{p,q\}$-subgroup of $G$. Now by [@p Lemma 2.3], we have $cd(G)=\{\beta(1)[G:I_{G}(\lambda)] : \lambda\in Irr(N), \beta\in Irr(\frac{I_{G}(\lambda)}{N})\}$. Since $\frac{G}{N}$ is abelian, we conclude that $\frac{I_{G}(\lambda)}{N}$ is abelian and $cd(G)=\{[G:I_{G}(\lambda)] : \lambda\in Irr(N)\}$.
Let $G$ be a finite group of order $p_1^{a_1}p_2^{a_2}...p_l^{a_l}$. Assume that $B(G)$ is a cycle of length six and $\pi(G)\neq\rho(G)$. Let $N$ be as in the proof of Theorem \[thm:98\]. So $N$ is nontrivial. If $\frac{G}{N}$ is abelian, similar to the above proof, we can explain $cd(G)$ with respect to $N$. So suppose that $\frac{G}{N}$ is not abelian. Since $B(G)$ is a cycle, each nonlinear irreducible character of $\frac{G}{N}$ is divisible by exactly two primes. $B(\frac{G}{N})$ is a cycle if and only if it is a cycle of length six and this will occur if and only if $cd(\frac{G}{N})=cd(G)$. So we may assume that $cd(\frac{G}{N})\subset cd(G)$. This implies that $B(\frac{G}{N})$ is either a path of length two or four. According to the proof of Theorem \[thm:2\], we can see that $\frac{G}{N}$ has a normal $t$-complement for a prime $t\in\{p,q,r\}$ or it has a normal abelian subgroup whose index is the only nontrivial degree of $\frac{G}{N}$ (and so of $G$).
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abstract: 'We report results of $^{12}$CO ($J=1$-0) mapping observations of the Wolf-Rayet starburst galaxy Mrk 1259 which has optical evidence for the superwind seen from a nearly pole-on view. The CO emission is detected in the central 4 kpc region. The nuclear CO spectrum shows a blue-shifted ($\Delta V \simeq -27$ km s$^{-1}$) broad (FWHM $\simeq$ 114 km s$^{-1}$) component as well as the narrow one (FWHM $\simeq 68$ km s$^{-1}$). The off-nuclear CO spectra also show the single-peaked broad component (FWHM $\simeq$ 100 km s$^{-1}$). The single-peaked CO profiles of both the nuclear and off-nuclear regions may be explained if we introduce a CO gas disk with a velocity dispersion of $\sim 100$ km s$^{-1}$. If this gas disk would be extended up to a few kpc in radius, we may explain the wide line widths of the off-nuclear CO emission. Alternatively, we may attribute the off-nuclear CO emission to the gas associated with the superwind. However, if all the CO gas moves along the biconical surface of the superwind, the CO spectra would show double-peaked profiles. Hence, the single-peaked CO profiles of the off-nuclear regions may be explained by an idea that the morphology and/or velocity field of the molecular-gas superwind are more complex as suggested by hydrodynamical simulations.'
author:
- 'Youichi Ohyama, & Yoshiaki Taniguchi'
title: 'MOLECULAR-GAS SUPERWIND FROM THE FACE-ON WOLF-RAYET GALAXY MRK 1259'
---
INTRODUCTION
============
In starburst galaxies, a large number of massive stars (e.g., $\sim 10^{4-5}$) are formed within a short duration (Weedman et al. 1981; Balzano 1983; Taniguchi et al. 1988). Therefore, a burst of supernova explosions occurs inevitably $\sim 10^7$ years after the onset of the starburst. Since these numerous supernovae release a huge amount of kinetic energy into the circumnuclear gas, the circumnuclear gas is thermalized and then blow out into the direction perpendicular to the galactic disk as a “superwind” (Tomisaka & Ikeuchi 1988; Heckman, Armus, & Miley 1990; Suchkov et al. 1994). A bubble of the ionized gas sweeps up the circumnuclear molecular gas, leading to the formation of molecular-gas superwind as well as the ionized-gas one (Tomisaka & Ikeuchi 1988; Suchkov et al. 1994). Thus, in order to understand the whole physical processes of superwinds, it is important to investigate the nature of molecular-gas superwinds (e.g., Nakai et al. 1987; Aalto et al. 1994; Irwin & Sofue 1996).
In this [*Letter*]{}, we present new evidence for the molecular-gas superwind from the Wolf-Rayet starburst galaxy Mrk 1259, which shows the optical evidence for the superwind viewed from a nearly pole-on view (Ohyama, Taniguchi, & Terlevich 1997; hereafter Paper I). Mrk 1259 is a peculiar S0 galaxy (de Vaucouleurs et al. 1991; hereafter RC3) at a distance of 26.64 Mpc [^1]. The logarithmic major-to-minor diameter ratio, log $R_{\rm 25}=0.10\pm 0.08$ (RC3), gives a nominal inclination angle, $i=37\fdg 4^{+11.2}_{-20.1}$, and the galaxy appears to be elongated along the EW direction. If this elongation were attributed to the inclination, we would observe the rotational motion along the EW direction. However, our long slit optical spectrum along the EW direction which was analyzed in Paper I shows no hint on the rotational motion; $\Delta V\lesssim 50$ km s$^{-1}$, suggesting strongly that the galaxy is seen from an almost face-on view. Therefore the oval shape of Mrk 1259 may not be due to the inclination[^2].
OBSERVATIONS
============
The $^{12}$CO ($J=1$-0) observations of Mrk 1259 were made using the 45m radio telescope at Nobeyama Radio Observatory, equipped with an SIS100 receiver in February 1997. The observations were made in the midnight under little wind velocity. Pointing check was made in every 30 minutes and the pointing errors in our observations are smaller than a few arcsec. The half power beam width (HPBW) is 15 arcsec at 115 GHz, corresponding to $\simeq$ 2 kpc at the distance of Mrk 1259. We observed the radio center position \[$\alpha_{1950}=10^{\rm h}36^{\rm m}02\fs8$; $\delta_{1950}=-06\arcdeg54\arcmin37\arcsec$; Sramek & Weedman (1986)\] and four off-nuclear positions (15$^{\prime\prime}$N, 15$^{\prime\prime}$W, 15$^{\prime\prime}$S, and 15$^{\prime\prime}$E). The main beam efficiency ($\eta_{\rm mb}$) was 0.51 at 115 GHz. The backend was the 2048 channel wide-band acousto optical spectrometer operated with the band width of 250 MHz, covering a velocity range of 650 km s$^{-1}$. The final data were box-car averaged, resulting in a final velocity resolution of 5 km s$^{-1}$. The final spectra are shown in Figure 1.
RESULTS AND DISCUSSION
======================
In Table 1, we give a summary of our observational results. The integrated CO intensity was estimated by $I({\rm CO}) = \int T_{\rm A}^* \eta_{\rm mb}^{-1} dv$ K km s$^{-1}$ where $\eta_{\rm mb} = 0.51$. Using a galactic conversion factor, $N_{\rm H_{2}}/I_{\rm CO} = 3.6\times 10^{20}$ cm$^{-2}$ (K km s$^{-1}$)$^{-1}$ (Scoville et al. 1987), we estimate the molecular gas mass, $M_{\rm H_2} = 5.8 \times 10^6 I({\rm CO}) A$, in each position where $A$ is the projected area of a 15$^{\prime\prime}$ HPBW in units of kpc$^2$. For the off-nuclear regions, we also give total values of $I$(CO) and $M_{\rm H_2}$. The total molecular gas mass detected in our observations amounts to $1.2 \times 10^9 M_\odot$. Since we do not observe the entire disk of this galaxy, this mass is regarded as a lower limit.
The Nuclear CO Emission
-----------------------
The nuclear CO emission shows a single-peaked profile with the evident blueward asymmetry. Applying a two-component Gaussian profile fitting (see the midst panel of Figure 1), we obtain the blueshifted broad component with FWHM $\simeq$ 114 km s$^{-1}$ and the narrow one with FWHM $\simeq$ 68 km s$^{-1}$. The peak velocity of the broad component is blueshifted by 27 km s$^{-1}$ with respect to that of the narrow one (Table 1). Both the intensities are nearly the same. We also mention that the red wing cannot be seen in the nuclear CO profile.
Even though there is the broad CO emission component, its width is significantly narrower than those observed for typical starburst galaxies; e.g., FWHM(CO) $\simeq$ 200 - 250 km s$^{-1}$ for M82 (Young & Scoville 1984; Nakai et al. 1987), $\sim 350$ km s$^{-1}$ for NGC 1808 (Aalto et al. 1994), and $\sim 325$ km s$^{-1}$ for NGC 4945 (Dahlem et al. 1993). This difference can be attributed to the effect of viewing angles between Mrk 1259 and the other starburst galaxies. It is remembered that the CO line width is generally affected by the galactic rotation. Given a typical rotation velocity of a disk galaxy, $V_{\rm rot}
\sim 200$ km s$^{-1}$, the observed full widths would amount to 2$V_{\rm rot} \sim$ 400 km s$^{-1}$ if seen from the edge-on view. In fact, since we observe M82, NGC 1808, and NGC 4945 from highly inclined viewing angles, their line widths are considered to be broadened by the effect of galactic rotation. On the other hand, since Mrk 1259 appears to be a nearly face-on galaxy, the observed width is not affected by the galactic rotation. Irwin & Sofue (1996) suggested that one of the nearby superwind galaxies, NGC 3628, has a nuclear molecular gas disk with a velocity dispersion of $\sim$ 100 km s$^{-1}$. If Mrk 1259 has also such a nuclear gas disk, we can explain the velocity width of the nuclear CO emission. Therefore, it is suggested that the observed FWHM of Mrk 1259 is due mainly to the broadening by some dynamical effect of the starburst activity. The blueward asymmetry of the nuclear CO line profile suggests that the CO gas is affected significantly by the superwind.
The Off-Nuclear CO Emission
---------------------------
The detection of the off-nuclear CO emission from Mrk 1259 is very intriguing from the following two points. The first point is that the host galaxy of Mrk 1259 appears to be an S0 galaxy (RC3). It is often observed that early type galaxies such as S0 and elliptical galaxies tend to have less molecular gas (e.g., Young & Scoville 1991) although CO emission has been detected from a number of S0 galaxies (Thronson et al. 1989; Wiklind & Henkel 1989; Sage 1989; Sage & Wrobel 1989). It is also known that the molecular gas in (non-active) S0 galaxies tends to be concentrated in the region whose diameter is typically less than one tenth of the optical diameter (Taniguchi et al. 1994). If this is also the case for Mrk 1259, the molecular gas would be concentrated within the central $12\arcsec =0.1 D_{\rm 0}$ region where $D_{\rm 0}$ is the isophotal optical diameter (RC3). Therefore, the presence of the bright off-nuclear CO emission is one of very important characteristics of Mrk 1259.
The second point is that the line widths of the off-nuclear CO emission are comparable to that of the nuclear CO emission, FWHM $\sim 100$ km s$^{-1}$. If there were an inclined off-nuclear CO disk, we may explain the wide line width because of the velocity gradient in the disk. If this is the case, we would observe that the peak velocity at 15$^{\prime\prime}$E is significantly different from that at 15$^{\prime\prime}$W. However, since our observations show that the velocity field of the off-nuclear regions is almost symmetric, this possibility is rejected. The second possibility is that there are spatially extended starburst regions and a significant amount of molecular gas is associated with them. However, radio continuum (1.5 GHz and 5 GHz) images show that the starburst region of Mrk 1259 is concentrated in the central several arcsec region (R. A. Sramek 1997, private communication). Therefore, there is no observational evidence for active star forming regions in the off-nuclear regions. As described before, we are observing the disk of Mrk 1259 from nearly a face-on view and thus the CO line width would be as narrow as $\sim$ 10 km s$^{-1}$ if Mrk 1259 were a normal disk galaxy (Lewis 1984, 1987; Kamphuis & Sancisi 1993). If there were an extended molecular gas disk with a velocity dispersion of $\sim$ 100 km s$^{-1}$ up to a radius of a few kpc, we could explain the wide line width. However, the size of the nuclear gas disk in NGC 3628 is much smaller ($\simeq 230$ pc, or $\sim 0.01 D_{\rm 0}$) than the off-nuclear distance of Mrk 1259 ($\sim 2$ kpc, or $\sim 0.13 D_{\rm 0}$). Although we cannot rule out the possibility that Mrk 1259 has such a very extended molecular gas disk with a large velocity dispersion, we need further detailed molecular-line observations to confirm this possibility. The third possibility is that the off-nuclear CO gas is associated with the superwind (i.e., blown out from the nuclear region). Since the ionized-gas superwind is extended to $r \sim 3.3$ kpc (Paper I), this possibility seems to be quite high. In fact, such extended CO emission is detected in M82 at the scale of 600 pc (Nakai et al. 1987) and even at the larger scale ($\sim 2$ kpc; Sofue et al. 1992). We discuss this possibility in detail in the next section.
Biconical Superwind Model for Mrk 1259
--------------------------------------
Since the superwind of Mrk 1259 is observed from nearly the pole-on view, it is interesting to investigate both the velocity field and the geometry of the superwind. In order to perform this, we investigate the off-nuclear CO line profile using a simple biconical outflow model in which the superwind flows toward the polar directions symmetrically with its apex at the nucleus. Such a superwind geometry is expected theoretically by hydrodynamical numerical simulations (Suchkov et al. 1994) and indeed observed in M82 (e.g., Nakai et al. 1987). In our model, we assume that the molecular gas can only move along the cone surface. We assume that the axis of the cone lies along our line of sight. The full opening angle of the cone ($\theta$) is not well constrained by the observations because of its nearly face-on viewing angle. Therefore we take this as a free parameter although Paper I has suggested as $\theta \lesssim 90\arcdeg$. A mean tangential velocity of the ionized gas on the sky can be estimated as $V_{\rm t, ion}\simeq R_{\rm SW}/T_{\rm SW}\simeq (2.3$ kpc$)
/(5.5\times 10^6$ years)$\simeq 410$ km s$^{-1}$ where $R_{\rm SW}$ is the projected radius of the superwind and $T_{\rm SW}$ is the age of the superwind (Paper I). We note that the outflow velocity of the molecular gas is [*slower*]{} than that of the ionized gas because the molecular gas along the cone surface is [*dragged*]{} by the ionized gas, rather than directly [*pushed out*]{} (Suchkov et al. 1994). For example, the model A1 of Suchkov et al. (1994) shows that the velocity of the outflowing dense gas is slower by a factor of $\sim 5$ than that of the ionized gas at the age of 8.3 Myr. In fact, comparing the outflow velocity of the ionized gas (Heckathorn 1972) with that of the molecular gas (Nakai et al. 1987) of M82, we find that the outflow velocity of the molecular gas is slower by a factor of $\sim 3$ than that of the ionized-gas. Thus, the mean tangential velocity of the molecular gas on the sky can be $V_{\rm t, mol}=V_{\rm t, ion}/\epsilon$ where $\epsilon$ is the decelerating factor ($\epsilon \simeq 3 - 5$).
We examine if the model can explain the observed CO line profiles in the off-nuclear regions. No effect of radiative transfer is included in the model calculation. We assume that the size of the cone is large enough to cover the whole off-nuclear regions. For simplicity, we also assume that the velocity field has a power-law form; i.e., $V(r) \propto r^a$, with a boundary condition of $V_{\rm t, ion}$ ($r = 2.3$ kpc) = 410 km s$^{-1}$. The emissivity (strength of the CO emission per a unit area) is also assumed to have a power-law form; i.e., $I(r) \propto r^b$. Although the parameters $a$ and $b$ are not well constrained by the observations, we adopt $a = 1$ and $b = -1$ as representative values following the trend seen in M82 (Nakai et al. 1987). We calculate the model for the cases of $\theta = 60\arcdeg, 90\arcdeg, 120\arcdeg$, and $150\arcdeg$ and $\epsilon$ = 1, 2, 3, 4, 5, and 6. To explain the observed FWZI (Full Width at Zero Intensity) of the off-nuclear CO emission ($\sim 200$ km s$^{-1}$; see Figure 1), we find that only models with ($\theta = 90\arcdeg$ and $\epsilon \simeq 5 - 6$) and ($\theta = 120\arcdeg$ and $\epsilon \simeq 3 - 4$) are acceptable. Models with $\theta = 60\arcdeg$ and $150\arcdeg$ cannot reproduce the line width for any $\epsilon$. Therefore, we show our results only for the cases $\theta=90\arcdeg$ and $120\arcdeg$ and $\epsilon$ = 3, 4, 5, and 6 in Figure 2. Our simple model demonstrates that the CO line has always a double-peaked profile for any combinations of the parameters. The red peak corresponds to the recessing cone while the blue one corresponds to the advancing cone. On the other hand, our observations show that the off-nuclear CO lines have smooth profiles around at the systemic velocity.
We discuss why our simple model cannot reproduce the observed off-nuclear CO profiles. One possible idea is a “swirl”-like velocity field which is often found in the hydrodynamical numerical simulations (Tomisaka & Ikeuchi 1988; Suchkov et al. 1994). If the outflow actually shows such a complex geometry and/or velocity field, it is expected that some parts of the emission would contribute to the core emission and can explain the broad and smooth off-nuclear emission. In order to understand the molecular-gas superwind of Mrk 1259, detailed molecular-line observations with higher spatial resolution would be helpful.
We would like to thank the staff of Nobeyama Radio Observatory for their kind support for our observations. We thank Naomasa Nakai for useful discussion and encouragement and Takashi Murayama and Shingo Nishiura for kind assistance of the observations. We also thank R. A. Sramek for kindly providing us his VLA data. YO was supported by the Grant-in-Aid for JSPS Fellows by the Ministry of Education, Culture, Sports, and Science. This work was financially supported in part by Grant-in-Aids for the Scientific Research (No. 0704405) of the Japanese Ministry of Education, Culture, Sports, and Science.
[llccc]{} & & & Nucleus & Off-nucleus\
rms noise & $\delta T_{\rm A}^*$ (K) & & 0.018 & $\sim 0.015$\
Flux & $I$(CO)$^{a, b}$ (K km s$^{-1}$) & total & $28.0\pm 1.1$ & $43.8\pm 1.9$$^d$\
& & 15N & & $4.2\pm 1.0$\
& & 15E & & $18.6\pm 1.0$\
& & 15S & & $9.5\pm 1.0$\
& & 15W & & $11.5\pm 1.0$\
Mass & $M_{\rm H_2}^{b, c}$ ($M_\odot$) & & $(4.8\pm 0.2) \times 10^8$ & $(7.5\pm 0.3) \times 10^8$$^d$\
Line profile & & & &\
& FWHM$_{\rm narrow}$ (km s$^{-1}$) & & 68 &\
& $V_{\rm narrow}$ (km s$^{-1}$) & & 2178 &\
& FWHM$_{\rm broad}$ (km s$^{-1}$) & & 114 & $\sim 100$$^e$\
& $V_{\rm broad}$ (km s$^{-1}$) & & 2151 & $\sim 2170$$^e$\
& $I$(broad)/$I$(narrow) & & 1.0 &\
[^1]: Paper I adopted a distance toward Mrk 1259, $D = 33.5$ Mpc. However, $V_{\rm 3K}$ was misused instead of $V_{\rm GSR}$ in this estimate. In this [*Letter*]{}, using $V_{\rm GSR} = 1998$ km s$^{-1}$ (RC3), with a Hubble constant $H_0$ = 75 km s$^{-1}$ Mpc $^{-1}$, we adopt a distance $D$ = 26.64 Mpc. Therefore, the HeII$\lambda$4686 luminosity, the number of late WR (WRL) stars, the size of the superwind, and the average velocity of the superwind in Paper I should be read as $L$(HeII) = 7.0$\times 10^{39}$ erg s$^{-1}$, $N$(WRL) $\simeq$ 4100, $r$(superwind) $\simeq 3.3$ kpc, and the average wind velocity $\simeq$ 565 km s$^{-1}$, respectively.
[^2]: It seems no surprise even if an isolated galaxy shows some morphological peculiarity because any galaxy would experience some minor merger events in its life. It is also noted that minor mergers can cause nuclear starbursts (e.g., Hernquist & Mihos 1995; Taniguchi & Wada 1996).
|
=-9mm
TIFR/TH/96-57\
18 October 1996
0.5cm =.8cm
[**The $\Delta B =- \Delta Q$ transitions and $B_d\leftrightarrow \bar B_d$ oscillations**]{}\
\[7mm\] [**G.V. Dass$^a$ and K.V.L. Sarma**]{}$^{b,*}$\
\[3mm\] $^a$[*Department of Physics, Indian Institute of Technology, Powai, Mumbai, 400 076, India* ]{}\
0.3cm $^b$[*Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, 400 005, India* ]{}\
\[10mm\]
Abstract
=.8cm We estimate the product of the relative strength of the $\Delta B = -\Delta Q$ amplitude in the decay $B_d^0
\rightarrow D^{*} \ell \bar \nu_{\ell }$ and the width-difference parameter $y$. For this we have used the data on time-dependence of $B_{d}^0\bar B_{d}^0$ oscillations in $Z$ decays and the fraction of like-sign dilepton events at the $\Upsilon (4S)$.
PACS: 13.20.He, 11.30.Er.\
[*[Keywords:]{}*]{} Semileptonic decay rule; Neutral bottom mesons; Mixing; Oscillations.
$^*$ E-mail: [email protected]; fax: 091 22 215 2110
=.8cm There is now a considerable body of experimental evidence in favour of $B_d^0 \bar B_d^0$ oscillations. This comes from the studies of charge correlations in $Z\rightarrow b\bar b$ decays at the $Z$ resonance [@MIX] and also in hadron collisions wherein the bottom flavour is produced [@HAD]. The general strategy adopted by the various experimental groups (see e.g., Ref [@WU]) can be briefly stated as follows: (a) tag the bottom flavour at production (when $Z$ decays) by the charge of a high $p_T$ lepton, or by the “jet charge”, (b) in the opposite side hemisphere, look for $B^0$ meson decay events that contain, say, $D^{*\mp }$ and $\ell ^{\pm }$, (c) measure the displacement of the decay vertex from the production vertex, (d) estimate the parent $B^0$ momentum, and (e) determine the duration of propagation of the neutral beon. From such measurements it has been shown that the fraction of events arising from $B_d\bar B_d$ mixing has a sinusoidal time-dependence characteristic of oscillations.
In these analyses, it has been implicitly assumed that the quark model rule $\Delta B = \Delta Q$ is valid in the semileptonic decays of neutral $B$ mesons. Here we wish to examine whether any information on this rule can be gleaned from the oscillation data obtained at the $Z$ resonance. To this end we make use of the mass difference $\Delta m$ extracted from oscillation data and the like-sign dilepton fraction $\chi _d $ obtained in the decay of $\Upsilon (4S)$ state. We are able to extract the product of the relative strength $\rho $ of the $\Delta
B = -\Delta Q$ amplitude in the $ D^* \ell \nu $ mode of semileptonic $B_d^0$ decay and the width difference parameter y = ; = [\_2 + \_1 2]{} .
In regard to the determination of the initial flavour (also known as production tag) we consider, for illustration, the jet-charge technique (see, e.g., [@OP94]). Consider the jet produced by a $b$ quark coming from $Z$ decay. This jet can show up as having the normal jet charge ($Q_J=-1/3$) with a probability $b_n$, or an abnormal jet charge ($Q_J=+1/3$) with a probability $b_a$. The abnormal jet charge arises in the Standard Model due to the $b$ fragmenting into a $\bar
B_d$ or $\bar B_s$ which undergoes oscillation to a neutral meson with positive bottom flavour. Therefore the probabilities associated with jet production are b\_n=[Prob]{}(b J\^[-1/3]{}), b\_a=[Prob]{}(b J\^[+1/3]{}), |b\_n=[Prob]{}(|b J\^[+1/3]{}), |b\_a=[ Prob]{}(|b J\^[-1/3]{}), wherein the superscripts denote the jet charges. Requiring $CP$ invariance would imply the relations $b_n=\bar b_n$ and $b_a=\bar
b_a$.
As for the decay tag, we take the time $t=0$ when the primary decay $Z\rightarrow b\bar b$ (we ignore $Z$ decays involving multiple $b
\bar b$ pairs) occurs. Let the $b$ quark produce a $\bar B_d^0 $ meson which undergoes flavour oscillations during its propagation and decays semileptonically at time $t$. We shall, for definiteness, focus on the decay mode $B\rightarrow D^*(2010)\ell \nu $. The corresponding normal and abnormal decay rates are denoted by $$\begin{aligned}
\bar B_n(t) &\equiv & \Gamma (\bar B^0_d(t) \rightarrow D^{*+}~
\ell^-~\bar {\nu _{\ell }}),\\
\bar B_a(t) &\equiv & \Gamma (\bar B^0_d(t) \rightarrow D^{*-}~
\ell^+~ {\nu _{\ell }}),\end{aligned}$$ where $\bar B_d^0(t) $ denotes the state at time $t$ that evolved from a state that was pure $\bar B_d^0$ at $t=0$. In the Standard Model the abnormal rate arises from $B_d\bar B_d$ oscillations. Also, for an initial $\bar b$ producing a $B_d^0$ , the corresponding normal and abnormal decay rates are $$\begin{aligned}
B_n(t) &\equiv & \Gamma (B^0_d(t) \rightarrow D^{*-}~
\ell^+~{\nu _{\ell } }),\\
B_a(t) &\equiv & \Gamma (B^0_d(t) \rightarrow D^{*+}~
\ell^-~ \bar {\nu _{\ell } }).\end{aligned}$$ Again $CP$ invariance implies the conditions $B_n(t)=\bar B_n(t)$ and $B_a(t)=\bar B_a(t)$.
In what follows, we shall ignore second order $CP$ violations by neglecting terms that are bilinear in the differences $[B_n(t)-\bar
B_n(t)]$, $[B_a(t)-\bar B_a(t)]$, $(b_n-\bar b_n)$ and $(b_a-\bar
b_a)$.
The number of unmixed events in which a bottom jet is observed on one side and the particle pair ($D^*\ell $) is present on the other side, following the primary decay $Z\rightarrow b\bar b$, can be written (including the $\bar bb$ configuration) as $$\begin{aligned}
N_{{\rm unmixed}}
&=& (b_n B_n + b_aB_a)+(\bar b_n\bar B_n +\bar
b_a\bar B_a) \\
&\simeq & {1 \over 2}(b_n+\bar b_n)(B_n+\bar B_n )+ {1\over
2}(b_a+\bar b_a)(B_a+\bar B_a ).\label{app1}\end{aligned}$$ The last step neglects $CP$ violations of second-order. The number of mixed events which ought to have abnormal charge either at production or at decay (but not at both), is given by $$\begin{aligned}
N_{{\rm mixed}}&=&(b_nB_a+b_aB_n )+(\bar b_n\bar B_a + \bar b_a\bar
B_n)
\\
&\simeq & {1\over 2}(b_n+\bar b_n)(B_a+\bar B_a)+ {1\over 2}(b_a+
\bar b_a)(B_n+\bar B_n);\label{app2}\end{aligned}$$ again, the last step neglects $CP$ violations of second order.
The observable of interest is the charge-correlation function defined by C\_Q(t) = [ N\_[[unmixed]{} ]{} - N\_[[mixed]{}]{} N\_[[unmixed]{} ]{} + N\_[[mixed]{}]{} ]{} . This takes a factorized form when we substitute Eqs. (\[app1\]) and (\[app2\]) C\_Q(t) K [ (B\_n+|B\_n) - (B\_a+|B\_a) (B\_n +|B\_n) + (B\_a +|B\_a)]{} . The only assumption behind this simple relation is the neglect of second and higher order $CP$-violation effects. The time-independent constant $K$ refers to the jet production tag, while the remaining factor refers to the decay tag ($K$ is obtained from the decay tag factor by simply replacing $B_i $ by $b_i$ and $\bar B_i $ by $\bar
b_i$). Hence for convenience in studying the decay time distribution, we define C’\_Q(t) C\_Q(t) .
In the framework of the Standard Model, it is reasonable (and customary) to assume $y\simeq 0$ and that the mass-eigenstates and $CP$-eigenstates of the neutral beons are nearly the same (which means $|q/p|\simeq 1$ in the usual notation, $B_{1,2}= pB\pm q\bar B$). We thus get the standard probabilities per unit time $P_{u,m}(t)$ for finding, respectively, a $B_d$ or a $\bar B_d $ at time $t$ having started with an initial $B_d$ [@WU], $$\begin{aligned}
B_d\rightarrow B_d~:~~ P_u(t)={\Gamma \over 2}e^{-
\Gamma t}(1+\cos \Delta mt),
\\
B_d\rightarrow \bar B_d~:~~P_m(t)={\Gamma \over 2}e^{-
\Gamma t}(1-\cos \Delta mt).\end{aligned}$$ Starting with a $\bar B_d $ at time $t=0$ also leads to the same unmixed and mixed probabilities per unit time.
For describing the subsequent decays of the beons, we shall appeal to $CPT$ invariance to relate the semileptonic partial widths of $B_d$ and $\bar B_d$ mesons. If there is a single hadron in the final semileptonic channel, there is no strong phase due to final state interactions and the corresponding decay rates of $B_d$ and $\bar B_d$ into individual conjugate channels are equal. Thus the time-dependence describing the decays into exclusive single-hadron semileptonic channels will be B\_n=|B\_n = R P\_u(t), B\_a=|B\_a = R P\_m(t),with the same constant of proportionality $R$ in the normal and abnormal cases. In this way we are led to the function C’\_Q(t)= mt which may be fitted to the observed (proper) time dependence for extracting the mixing parameter $\Delta m$.
Now we examine how the above analysis gets modified when $\Delta
B=-\Delta Q$ amplitudes are present. Focusing on the specific channel $D^*(2010)\ell \nu $ , we define the ratios = , |= . $CPT$ invariance (the validity of which we assume throughout) implies the relation $\bar \rho =\rho ^*$ . On the other hand, $CP$ invariance implies $\bar \rho =\rho $. However when we allow for the presence of $\Delta B=-\Delta Q$ amplitudes, it is reasonable to allow also for possible $CP$ violations. We do this by retaining the terms which are linear in the $CP$ violating quantities $(\rho -\bar \rho )$ and $(|p|^2-|q|^2)$ (where $p$ and $q$ are the coefficients which define the propagation states as mixtures of flavour states).
We reevaluate the charge correlation function by neglecting terms that are second (and higher) order small in $\rho $ or/and $CP$ violations, to obtain C’\_Q(t).
Instead of fitting the data to the above function, we consider the corresponding time-integrated version = , a , and obtain information on $y{\rm Re}~\rho $. For the parameter $a$ we use the relation connecting it to the like-sign dilepton fraction $\chi _d$ [@OK], \_d= .From the experiments at $\Upsilon (4S)$ by ARGUS [@ARG94] and CLEO [@CLE93] collaborations, we have the average value [@PDG] \_d= 0.156 0.024 . By definition $\chi _d$ is $CP$ even and hence unaffected by first order $CP$ violations; it is also unaffected by terms which are linear in $\rho $ [@DS]. Therefore, substituting for $a$, we obtain = .A determination of the combination $y {\rm Re}~\rho $ is thus possible provided we know ${\cal C'}$. We observe that the experimental value for $x_d$ deduced from $B\bar B$ oscillations at $Z$ does give us ${\cal C'}$ through the relation =[1(1+x\_d\^2)]{} because $y=0$ and $\rho =0 $ are assumed in the experimental determinations of $x_d$.
We consider, mainly to illustrate our procedure, the oscillation data in $Z$ decays using the $D^*\ell / Q_J$ method. This method uses the jet charge $Q_J$ for production tag and hence deals with a bigger event sample than that using lepton tag. Also, it uses for the decay tag, the semi-exclusive channel $
B_d\rightarrow D^*(2010)\ell \nu X $ for which contamination from the semileptonic decays of $B_s$ and charged $B$ is expected to be minimal. Recently the OPAL group [@OP96] had reported a value for $\Delta m$ using this method; it is based on a sample of 1200 $D^{*\pm
}\ell ^{\mp } $ candidate events of which 778 $\pm $84 are expected to be from $B_d^0$ decays. Multiplying this $\Delta m$ by the average lifetime of the $B_d$ meson [@PDG], we obtain $$\begin{aligned}
x_d &=&(0.539 \pm 0.060 \pm 0.024)~{\rm ps}^{-1}~.~(1.56
\pm 0.06)~{\rm ps}~\\
&=& 0.84\pm 0.11.\end{aligned}$$ Assuming that the observed events are all due to the 3-body channel $B_d\rightarrow D^*(2010)\ell \nu $ , we are therefore led to conclude that $$\begin{aligned}
y {\rm Re}~\rho &=& {1\over 2}[(1-2\chi _d)(1+x_d^2)-1]\\
&=& 0.09\pm 0.07.\end{aligned}$$ A more accurate determination of this and other such quantities should be possible [@DS96] with the future dilepton data at the asymmetric $B$ Factories.
To conclude, we have shown that the magnitude of the product $y {\rm
Re}~\rho $ cannot exceed 0.21 at 90% CL. This limit depends on the dilepton data at $\Upsilon (4S)$ and the $B_d\bar B_d$ oscillation data from $Z$ decays in which the decaying neutral beon is tagged by the pair $D^*(2010)^{\mp }\ell ^{\pm }$.
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|
---
abstract: 'We measured stacked weak lensing cluster masses for a sample of 1323 galaxy clusters detected by the RedGOLD algorithm in the Canada-France-Hawaii Telescope Legacy Survey W1 and the Next Generation Virgo Cluster Survey at $0.2<z<0.5$, in the optical richness range $10<\lambda<70$. This is the most comprehensive lensing study of a $\sim 100\%$ complete and $\sim 80\%$ pure optical cluster catalog in this redshift range. We test different mass models, and our final model includes a basic halo model with a Navarro Frenk and White profile, as well as correction terms that take into account cluster miscentering, non-weak shear, the two-halo term, the contribution of the Brightest Cluster Galaxy, and an a posteriori correction for the intrinsic scatter in the mass–richness relation. With this model, we obtain a mass–richness relation of $\log{M_{\rm 200}/M_{\odot}}=(14.46\pm0.02)+(1.04\pm0.09)\log{(\lambda/40)}$ (statistical uncertainties). This result is consistent with other published lensing mass–richness relations. We give the coefficients of the scaling relations between the lensing mass and X-ray mass proxies, $L_X$ and $T_X$, and compare them with previous results. When compared to X-ray masses and mass proxies, our results are in agreement with most previous results and simulations, and consistent with the expected deviations from self-similarity.'
author:
- 'Carolina Parroni , Simona Mei , Thomas Erben, Ludovic Van Waerbeke, Anand Raichoor, Jes Ford, Rossella Licitra , Massimo Meneghetti, Hendrik Hildebrandt, Lance Miller, Patrick Côté, Giovanni Covone, Jean-Charles Cuillandre, Pierre-Alain Duc, Laura Ferrarese, Stephen D.J. Gwyn, Thomas H. Puzia'
title: 'Next Generation Virgo Cluster Survey. XXI. The weak lensing masses of the CFHTLS and NGVS RedGOLD galaxy clusters and calibration of the optical richness'
---
INTRODUCTION
============
Galaxy clusters are the largest and most massive gravitationally bound systems in the universe and their number and distribution permit us to probe the predictions of cosmological models. They are the densest environments where we can study galaxy formation and evolution, and their interaction with the intra-cluster medium [@voit2005]. For both these goals, an accurate estimate of the cluster mass is essential.
The cluster mass cannot be measured directly, but is inferred using several mass proxies. Galaxy clusters emit radiation at different wavelengths and their mass can be estimated using different tracers. Different mass proxies usually lead to mass estimations that are affected by different systematics.
From X-ray observations of the cluster gas, we can derive the gas temperature, which is related to its total mass [@sarazin1988], under the assumption of hydrostatic equilibrium. X-ray mass measurements are less subjected to projection and triaxiality effects, but the mass proxies are not reliable in systems undergoing mergers or in the central regions of clusters with strong AGN feedback [@allen2011].
The intracluster medium (ICM) can also be detected in the millimeter by the thermal Sunyaev–Zel’dovich effect [S-Z effect; @sz1972] and the S-Z flux is related to the total cluster mass. Unlike optical and X-ray surface brightness, the integrated S-Z flux is independent of distance, allowing for almost constant mass limit measurements at high redshifts. For the same reason, though, the method is also subjected to projection effects due to the overlap of all the groups and clusters along the line of sight [@voit2005].
In the optical and infrared bandpasses, we observe the starlight from cluster galaxies. If a cluster is in dynamical equilibrium, the velocity distribution of its galaxies is expected to be Gaussian and the velocity dispersion can be directly linked to its mass through the virial theorem. An advantage of this method is that, unlike X-ray and S-Z mass measurements, it is not affected by forms of non-thermal pressure such as magnetic fields, turbulence, and cosmic ray pressure. On the downside, it is sensitive to triaxiality and projection effects, the precision of the measurements is limited by the finite number of galaxies, and the assumption of dynamical and virial equilibrium is not always correct [@allen2011].
The total optical or infrared luminosity of a cluster is another indicator of its mass, given that light traces mass. @abell1958 defined a *richness* class to categorize clusters based on the number of member galaxies brighter than a given magnitude limit. The luminosity distribution function of cluster galaxies is also well described by the @schechter1976 profile, and the observation of the high-luminosity tip of this distribution allows us to better constrain cluster masses. @postman1996, for example, defined the richness parameter as the number of cluster galaxies brighter than the characteristic luminosity of the @schechter1976 profile, $L_{\rm *}$. Different definitions are possible and intrinsically related to the technique used to optically detect galaxy clusters.
@rykoff2014 built an optical cluster finder based on the red-sequence finding technique, redMaPPer and applied it to the Sloan Digital Sky Survey [SDSS; @york2000]. Their richness is computed using optimal filtering as a sum of probabilities and depends on three filters based on colors, positions, and luminosity [@rozo2009; @rykoff2012; @rykoff2014; @rykoff2016; @rozo2014].
In @licitra2016a [@licitra2016b], we introduced a simplified definition of cluster richness based on the redMaPPer richness measurement, within our detection and cluster selection algorithm RedGOLD. RedGOLD is based on a revised red-sequence technique.
RedGOLD richness quantifies the number of red, passive, early-type galaxies (ETGs) brighter than $0.2L_{\rm *}$, inside a scale radius, subtracting the scaled background. When compared to X-ray mass proxies, the RedGOLD richness leads to scatters in the X-ray temperature-richness relation similar to those obtained with redMaPPer [@rozo2014], which is very promising because RedGOLD was applied to a lower richness threshold (i.e. lower cluster mass).
The total cluster mass can also be derived by its strong and weak gravitational lensing of background sources. In the weak lensing regime, the gravitational potential of clusters of galaxies produces small distortions in the observed shape of the background field galaxies, creating the so-called shear field, which is proportional to the cluster mass.
Because the shear is small relative to the intrinsic ellipticity of the galaxies (due to their random shape and orientation), a statistical approach is required to measure it and the signal is averaged over a large number of background sources to increase the signal-to-noise ratio [S/N; @schneider2006]. Gravitational lensing does not require any assumptions about the dynamical state of the cluster and it is sensitive to the projected mass along the line of sight, making it a more reliable tool to determine total cluster masses [@meneghetti2010; @allen2011; @rasia2012].
In the future, as shown in @ascaso2016, optical and near-infrared (NIR) cluster surveys, such as Euclid[^1] [@laureijs2011], Large Synoptic Survey Telescope (LSST)[^2] and J-PAS [@benitez2014], will reach deeper than X-ray and S-Z surveys, such as e-Rosita [@merloni2012], SPTpol [@carlstrom2011] and ACTpol [@marriage2011]. It is thus important to understand the reliability of optical and NIR mass proxies because they will be the only mass proxy available for these new detections.
Several works in the literature have proven that the optical richness shows a good correlation with the cluster total masses derived from weak lensing [@johnston2007; @covone2014; @ford2015; @vanuitert2015; @melchior2016; @simet2016]. From these works, the typical uncertainty found in the cluster mass at a given richness is $\sim10-25\%$ including statistical and systematic errors, in the mass range $6 \times 10^{12}M_{\rm \odot} \lesssim M \lesssim 10^{15}M_{\rm \odot}$ and in the redshift range $0.1\lesssim z \lesssim 0.9$.
The aim of this work is to calibrate and evaluate the precision of the RedGOLD richness as a mass proxy and to compare it to stacked weak-lensing masses. We then compare our lensing-calibrated masses to X-ray mass proxies. Our approach mainly follows the one adopted by @johnston2007 and @ford2015, and we compare our results to @simet2016, @farahi2016 and @melchior2016.
The paper is organized as follows: in Section 2, we describe the shear data set and the photometric redshifts catalog; in Section 3, we briefly present the RedGOLD detection algorithm and the cluster catalogs; in Section 4, we describe the weak-lensing equations and our method; in Section 5, we present our results; in Section 6, we discuss our findings in comparison with other recent works; in Section 7, we present our conclusions.
Throughout this work, we assume a standard $\Lambda CDM$ model, with $\Omega_{\rm {m}}=0.3$, $\Omega_{\rm \Lambda}=0.7$ and $\rm{H_{\rm 0}}=70~km~s^{-1}~Mpc^{-1}$.
Magnitudes are given in the AB system [@oke1983; @sirianni2005].
DATA
====
For our analysis, we use our own data reduction [@raichoor2014] of the Canada-France-Hawaii Telescope Legacy Survey [CFHT-LS; @gwyn2012] Wide 1 (W1) field and of the Next Generation Virgo Cluster Survey [NGVS; @ferrarese2012]. We describe these two data sets below.
CFHTLenS and NGVSLenS {#shear_data}
---------------------
The CFHT-LS is a multi-color optical survey conducted between 2003 and 2008 using the CFHT optical multi-chip MegaPrime instrument . The survey consists of 171 pointing covering $\sim154~deg^{2}$ in four wide fields ranging from 25 to 72 $\rm deg^{2}$, with complete color coverage in the five filters $u^{*}g^{\prime}r^{\prime}i^{\prime}z^{\prime}$. All the pointings selected for this analysis were obtained under optimal seeing conditions with a seeing $<0.8^{\arcsec}$ in the primary lensing band $i^{\prime}$ [@erben2013]. The $5\sigma$ point source limiting magnitudes in a $2.0^{\arcsec}$ aperture in the five $u^{*}g^{\prime}r^{\prime}i^{\prime}z^{\prime}$ filters are $\sim25.2$, $\sim25.6$, $\sim24.9$, $\sim24.5$, $\sim23.5$ mag, respectively [@erben2013].
The NGVS [@ferrarese2012] is a multi-color optical imaging survey of the Virgo Cluster, also obtained with the CFHT MegaCam instrument. This survey covers $104~deg^{2}$ with 117 pointings in the four filters $u^{*}g^{\prime}i^{\prime}z^{\prime}$. Thirty-four of these pointings are also covered in the $r^{\prime}$ band. As for the CFHT-LS, the optimal seeing conditions were reserved to the $i^{\prime}$-band, which covers the entire survey with a seeing $<0.6^{\arcsec}$. The $5\sigma$ point source limiting magnitudes in a $2.0^{\arcsec}$ aperture in the five $u^{*}g^{\prime}r^{\prime}i^{\prime}z^{\prime}$ filters are $\sim25.6$, $\sim25.7$, $\sim24.7$, $\sim24.4$, $\sim23.6$ mag, respectively [@raichoor2014].
Both our CFHTLenS and NGVSLenS photometry and photometric redshift catalogs were derived using the dedicated data processing described in @raichoor2014. The preprocessed *Elixir*[^3] data, available at the Canadian Astronomical Data Centre (CADC)[^4]) were processed with an improved version of the THELI pipeline [@erben2005; @erben2009; @erben2013; @raichoor2014] to obtain co-added science images accompanied by weights, flag maps, sum frames, image masks, and sky-subtracted individual chips that are at the base of the shear and photometric analysis. We refer the reader to @erben2013 and @heymans2012 for a detailed description of the different THELI processing steps and a full systematic error analysis. @raichoor2014 modified the standard pipeline performing the zero-point calibration using the SDSS data, taking advantage of its internal photometric stability. The SDSS covers the entire NGVS field and 62 out of 72 pointings of the CFHT-LS W1 field ($\sim 60~deg^{2}$). @raichoor2014 constructed the photometric catalogs as described in @hildebrandt2012, adopting a [*global*]{} PSF homogenization to measure unbiased colors. Multicolor catalogs were obtained from PSF-homogenized images using SExtractor [@bertin1996] in dual-image mode, with the un-convolved $i^{\prime}$-band single-exposure as the detection image.
We restrict our analysis to the entire NGVS and the $\sim60~deg^2$ of the W1 field that were reprocessed by @raichoor2014, to have an homogeneously processed photometric catalog on a total of $\sim164~deg^2$.
For the shear analysis, as described in @miller2013, shape measurements were obtained applying the Bayesian *lens*fit algorithm to single-exposure $i^{\prime}$-band images with accurate PSF modeling, fitting PSF-convolved disc plus bulge galaxy models. The ellipticity of each galaxy is estimated from the mean likelihood of the model posterior probability, marginalized over model nuisance parameters of galaxy position, size, brightness, and bulge fraction. The code assigns to each galaxy an inverse variance weight $w_{\rm {lens}} \propto (\sigma_{\rm{e}}^{2}+\sigma_{\rm {pop}}^{2})^{-1}$, where $\sigma_{\rm {e}}^{2}$ is the variance of the ellipticity likelihood surface and $\sigma_{\rm {pop}}^{2}$ is the variance of the ellipticity distribution of the galaxy population. Calibration corrections consist of a multiplicative bias $m$, calculated using simulated images, and an additive bias $c$, estimated empirically from the data. As discussed in @miller2013, the former increases as the size and the S/N of a galaxy detection decrease, while the latter increases as the S/N of a galaxy detection increases and the size decreases.
Photometric Redshifts {#sec:photoz}
---------------------
The photometric redshift catalogs of the $\sim 60~deg^{2}$ of the CFHTLenS covered by the SDSS and of the entire NGVSLenS were obtained using the Bayesian software packages *LePhare* [@arnouts1999; @arnouts2002; @ilbert2006] and BPZ [@benitez2000; @benitez2004; @coe2006], as described in @raichoor2014. We used the re-calibrated SED template set of @capak2004.
Both [*LePhare*]{} and BPZ are designed for high-redshift studies, giving biased or low-quality photo-z’s estimations for objects with $i^{\prime}<20$ mag, which represent a non-negligible fraction of both samples. In order to improve the performance at low redshift, @hildebrandt2012 used an *ad hoc* modified prior for the CFHTLenS data. @raichoor2014 adopted a more systematic solution for our reprocessed CFHTLenS W1 field and for the NGVSLenS, building a new prior calibrated on observed data, using the SDSS Galaxy Main Sample spectroscopic survey [@york2000; @strauss2002; @ahn2014] to include bright sources.
To analyze the accuracy of the photometric redshift estimates, @raichoor2014 used several spectroscopic surveys covering the CFHTLenS and NGVSLenS: the SDSS Galaxy Main Sample, two spectroscopic programs at the Multiple Mirror Telescope (MMT; Peng, E. W. et al. 2016, in preparation) and at the Anglo-Australian Telescope [AAT; @zhang2015 2016, in preparation], the Virgo Dwarf Globular Cluster Survey (Guhathakurta, P. et al. 2016, in preparation), the DEEP2 Galaxy Redshift Survey over the Extended Groth Strip [DEEP2/EGS; @davis2003; @newman2013], the VIMOS Public Extragalactic Redshift Survey [VIPERS; @guzzo2014], and the F02 and F22 fields of the VIMOS VLT Deep Survey [VVDS; @lefevre2005; @lefevre2013].
As shown in @raichoor2014, when using all five filters, for $0.2<z_{\rm phot}<1$ and $i^{\prime}<23$ mag, we found a bias $\Delta z=\frac{z_{\rm phot}-z_{\rm spec}}{1+z_{\rm spec}}<0.02$ with scatter values in the range $0.02<\sigma<0.05$ and $<5\%$ of outliers. When using four bands, the quality of the measurements slightly decreases, due to the lack of the $r^{\prime}$-band to sample the 4000 [Å]{} break. In the range $0.3<z_{\rm phot}<0.8$ and $i^{\prime}>21$ mag, we obtained $-0.05<bias<0.02$, a scatter $\sigma\sim0.06$ and an outlier rate of $10-15\%$. Our photometric redshifts are not reliable for $z<0.2$ [@raichoor2014] and we excluded these low redshifts from our cluster detection in @licitra2016a [@licitra2016b] and our weak lensing analysis.
In this analysis, we use the photometric redshifts derived with BPZ, corresponding to $z_{\rm best}$, the peak of the redshift posterior distribution (hereafter, $z_{\rm phot}$).
CLUSTER CATALOGS
================
The RedGOLD Optical Cluster Catalogs {#redgold}
------------------------------------
### The RedGOLD Algorithm
The RedGOLD algorithm [@licitra2016a; @licitra2016b] is based on a modified red-sequence search algorithm. Because the inner regions of galaxy clusters host a large population of passive and bright ETGs, RedGOLD searches for passive ETG overdensities. To avoid the selection of dusty red star-forming galaxies, the algorithm selects galaxies on the red sequence both in the rest-frame $(U - B)$ and $(B - V)$, using red sequence rest-frame zero point, slope, and scatter from @mei2009, as well as an ETG spectral classification from *LePhare*. In order to select an overdensity detection as a cluster candidate, the algorithm also imposes that the ETG radial distribution follows an NFW [@NFW1996] surface density profile.
RedGOLD centers the cluster detection on the ETG with the highest number of red companions, weighted by luminosity. This is motivated by the fact that the brightest cluster members lying near the X-ray centroid are better tracers of the cluster centers compared to using only the BCG [@george2012]. The redshift of the cluster is the median photometric redshift of the passive ETGs.
Each detection is characterized by two parameters–the significance $\sigma_{\rm det}$ and the richness $\lambda$–which quantifies the number of bright red ETGs inside the cluster, using an iterative algorithm.
The entire galaxy sample is divided into overlapping photometric redshift slices. Each slice is then divided in overlapping circular cells, with a fixed comoving radius of $500~kpc$. The algorithm counts $N_{\rm gal}$, the number of red ETGs inside each cell, brighter than $0.2L_{\rm *}$, building the galaxy count distribution in each redshift slice. The background contribution is defined as $N_{\rm bkg}$, the mode of this distribution, with standard deviation $\sigma_{\rm bkg}$. The detection significance is then defined as $\sigma_{\rm det}=(N_{\rm gal}-N_{\rm bkg})/\sigma_{\rm bkg}$. Overdensities larger than $N_{\rm bkg}+\sigma_{\rm det}\times \sigma_{\rm bkg}$ are selected as preliminary detections. The uncertainties on the cluster photometric redshift range between 0.001 and 0.005, with an average of $0.003\pm0.002$. In this paper, we assume that these uncertainties are negligible for our analysis [see also @simet2016].
The algorithm then estimates the richness $\lambda$, counting $N_{\rm gal}$ inside a scale radius, initially set to $1~Mpc$. The radius is iteratively scaled with richness as in @rykoff2014, until the difference in richness between two successive iterations is less than $N_{\rm bkg}$.
RedGOLD is optimized to produce cluster catalogs with high completeness and purity. In @licitra2016a [@licitra2016b], the *completeness* is defined as the ratio between detected structures corresponding to true clusters and the total number of true clusters, and the *purity* is defined as the number of detections that correspond to real structures to the total number of detected objects.
Following the definition of a *true cluster* in the literature [e.g, @finoguenov2003; @lin2004; @evrard2008; @finoguenov2009; @mcgee2009; @mead2010; @george2011; @chiangoverzier2013; @gillis2013; @shankar2013], we define a *true cluster* as a dark matter halo more massive than $10^{14}\ \rm M_{\odot} $. In fact, numerical simulations show that 90$\%$ of the dark matter halos more massive than $10^{14}\ M_{\odot}$ are a very regular virialized cluster population up to redshift $z\sim1.5$ [e.g., @evrard2008; @chiangoverzier2013]. In order to validate the performance of our algorithm to find clusters with a total mass larger than $10^{14}\ \rm M_{\odot} $ and measure our obtained sample completeness and purity, we have applied RedGOLD to both galaxy mock catalogs and observations of X-ray detected clusters [@licitra2016a]. For details on the method and the performance of the algorithm when applied to simulations and observations, we refer the reader to @licitra2016a.
### The RedGOLD CFHT-LS W1 and NGVS Cluster Catalogs {#redgold_cat}
We use the CFHT-LS W1 and NGVS cluster catalogs from @licitra2016a and @licitra2016b, respectively. For both surveys, when using five bandpasses, in the published catalogs, we selected clusters more massive than $\approx 10^{14}M_{\rm \odot}$, the mass limit for which $\sim 90\%$ of dark matter halos at $z_{\rm phot}<1.5$ are virialized [@evrard2008]. In @licitra2016a [@licitra2016b], we calibrated the $\sigma_{\rm det}$ and $\lambda$ parameters to maximize the completeness and purity of the catalog of these type of objects.
@licitra2016a demonstrated that, when we considered only detections with $\sigma_{\rm det}\ge4$ and $\lambda\ge10$ at $z_{\rm phot}\le0.6$, and $\sigma_{\rm det}\ge4.5$ and $\lambda\ge10$ at $z_{\rm phot}\lesssim1$, we obtain catalogs with a completeness of $\sim100\%$ and $\sim70\%$, respectively, and a purity of $\sim80\%$ [see Figure 7 and 8 from @licitra2016a].
In both the CFHT-LS W1 and the NGVS, we masked areas around bright stars and nearby galaxies. We found that in only $\sim2\%$ of the cluster candidates (low richness structures at high redshift) are more than $10\%$ of their bright potential members masked [@licitra2016a]. Therefore, our richness estimates are not significantly affected by masking.
For the NGVS, as explained above, the five-band coverage was limited to only the $\sim30\%$ of the survey. The lack of the $r^{\prime}$-band in the remaining pointings, causes higher uncertainties on the determination of photometric redshifts for sources at $0.3<z_{\rm phot}<0.8$ but the global accuracy on the photometric redshifts remains high even for this sample, as shown in @raichoor2014. Because there are some fields in which the quality of the $r^{\prime}$-band is lower because of the lower depth and the lack of coverage of the intra-CCD regions, this adds to the difficulty of detecting the less-massive structures at intermediate and high redshifts, as well as the determination of the clusters center and richness.
To quantify this effect in the richness estimation, @licitra2016b compared the values recovered with a full band coverage $\lambda_{\rm r}$ to the ones obtained without the $r^{\prime}$-band $\lambda_{\rm wr}$, and measured $\Delta \lambda/\lambda_{\rm r} \equiv (\lambda_{\rm r}-\lambda_{\rm wr})/\lambda_{\rm r}$, in different redshift bins. Median values of $\Delta \lambda/\lambda_{\rm r}$ and their standard deviations are listed in Table 2 of @licitra2016b. At $z_{\rm phot}<0.5$ and $z_{\rm phot}>0.8$, the two estimates are in good agreement, with $\Delta \lambda/\lambda_{\rm r}<10\%$. This is due to the fact that the $(g-z)$ and $(i-z)$ colors straddle the 4000 [Å]{} break at $z_{\rm phot}<0.5$ and $z_{\rm phot}>0.8$, respectively. At $0.5<z_{\rm phot}<0.6$, $\lambda_{\rm wr}$ is systematically underestimated of $\sim40\%$ on average and, at $0.6<z_{\rm phot}<0.8$, it is systematically overestimated of $\sim20\%$ on average. The first systematic is due to the use of the $(g-z)$ color, which changes less steeply with redshift and has larger photometric errors, compared with $(r-i)$ and $(i-z)$ colors. The latter is caused by the use of the $(i-z)$ color only, without the additional cut in the $(r-z)$ or $(r-i)$ colors that allows us to reduce the contamination of dusty red galaxies on the red sequence [@licitra2016b].
To take this into account, we correct the $\lambda_{\rm wr}$ estimations using the average shifts given in Table 2 of @licitra2016b. As we will discuss later, since for this analysis we are only selecting clusters at $z_{\rm phot}<0.5$ (see below), so using four bands preserves the same level of completeness and purity as using the five-bands catalog.
 
For these reasons, we built two separate catalogs for the NGVS: the first for the $\sim20~deg^{2}$ covered by the $r^{\prime}$-band and the second for the entire NGVS using only four bandpasses. In this last catalog, we corrected for the average shift in $\lambda$ when applying our thresholds [@licitra2016b]. Hereafter, we define the NGVS catalog obtained on the area covered by the five bandpasses as NGVS5 and the catalog obtained with four bandpasses as NGVS4.
The CFHT-LS W1 published catalog includes 652 cluster candidate detections in an area of $\sim60~deg^{2}$. The NGVS published catalogs include 279 and 1505 detections, in the $\sim20~deg^{2}$ with the five band coverage and in the rest of the survey, respectively.
We select cluster subsamples from these catalogs for our weak lensing analysis. Knowing that the peak in the lensing efficiency is found at $z_{\rm phot}\sim0.4$ for source galaxies at $z_{\rm phot}\sim1$ [@hamana2004] and that shear measurements from ground-based telescopes are reliable for clusters with redshifts $0.2<z_{\rm phot}<0.5$ [@kasliwal2008], we select detections only in this redshift range $0.2<z_{\rm phot}<0.5$, where the lower limit is due to the fact that our photometric redshifts are not reliable for $z_{\rm phot}<0.2$, as noted in Section \[sec:photoz\] and presented in @raichoor2014 and @licitra2016a. We also discard clusters with richness $\lambda<10$ and $\lambda>70$. In fact, as shown in @licitra2016a at richness $\lambda<10$, our purity decreases for a given significance threshold. For our significance threshold of $\sigma_{\rm det} >4$, $\lambda<10$ implies a contamination of false detections larger than $\sim 20\%$. For $\lambda>70$, we have very few detections and there are not enough clusters to obtain an average profile from a statistically significant sample.
Our final selection for the weak lensing analysis includes 1323 clusters. Their richness and redshift distributions are shown in Figure \[fig:isto\]. Hereafter, we will define the catalogs to which we applied the thresholds in significance, richness, and redshift for the weak lensing analysis as [*selected*]{} catalogs. The published @licitra2016a [@licitra2016b] catalogs, to which we applied the thresholds in significance and richness, will be referred to as Licitra’s [*published*]{} catalogs. The @licitra2016a [@licitra2016b] catalogs, without any threshold, will be called [*complete*]{} catalogs.
The X-Ray Cluster Catalogs {#xray_cat}
--------------------------
@gozaliasl2014 analyzed the [*XMM-Newton*]{} observations in the $\sim 3~deg^{2}$ overlapping the CFHT-LS W1 field, as a part of the XMM-LSS survey [@pierre2007] [^5]. They presented a catalog of 129 X-ray groups, in a redshift range $0.04<z_{\rm phot}<1.23$, characterized by a rest frame $0.1-2.4~keV$ band luminosity range $10^{41}-10^{44}~ergs~s^{-1}$. They removed the contribution of AGN point sources from their flux estimates and applied a correction of $\sim 10\%$ for the removal of cool core flux based on the high-resolution [*Chandra*]{} data on COSMOS as shown in @leauthaud2010. They used a two-color red-sequence finder to identify group members and calculate the mean group photometric redshift. They inferred cluster’s $M_{\rm 200}$ masses using the $L_{\rm X}-M$ relation of @leauthaud2010, with a systematic uncertainty of $\sim 20\%$.
@mehrtens2012 presented the first data release of the XMM Cluster Survey, a serendipitous search for galaxy clusters in the [*XMM-Newton*]{} Science Archive data [^6]. The catalog consists of 503 optically confirmed clusters, in a redshift range $0.06<z_{\rm phot}<1.46$. Four hundred and two of these clusters have measured X-ray temperatures in the range $0.4<T_{\rm X}<14.7~keV$. They derived photometric redshifts with the red-sequence technique, using one color. They used a spherical $\beta$-profile model [@cavaliere1976] to fit the surface brightness profile and derive the bolometric (0.05 - 100 keV band) luminosity in units of $10^{44}~erg~s^{-1}$ within the radius $R_{\rm 200}$ and $R_{\rm 500}$.
In Section \[xray\], we will use these catalogs to compare our lensing masses with X-ray masses and calculate the scaling relations between lensing masses and X-ray temperature and luminosity. We analyze the two catalogs separately because the different treatment of the emission from the central regions of the clusters leads to different mass estimates. In Section \[xray2\] we will discuss these results.
WEAK LENSING ANALYSIS
=====================
In this section, we describe our weak lensing analysis. Our aim is to infer cluster masses by reconstructing the tangential shear radial profile $\gamma_{\rm t}(R)$, averaging in concentric annuli around the halo center, and fitting it to a known density profile. Here, $\gamma_{\rm t}(R)$ accounts for the distortion, due to the gravitational potential of the lens, of the shape of the background sources in the tangential direction with respect to the center of the lens. It is defined as: $$\gamma_{\rm t}=-Re\left[\gamma e^{-2i\phi}\right]$$ with $\gamma=\epsilon_{\rm 1}+i\epsilon_{\rm 2}=|\gamma|e^{2i\phi}$, where $\epsilon_{\rm 1}$ and $\epsilon_{\rm 2}$ are the ellipticity components of the galaxy and $\phi$ is the position angle of the galaxy respect to the center of the lens [@schneider2006].
As described in @wright2000, the tangential shear profile $\gamma_{\rm t}(R)$ is related to the surface density contrast by: $$\Delta\Sigma(R)=\left<\gamma_{\rm t}(R)\right>{\Sigma_{\rm c}}$$ where $R$ is the projected radius with respect to the center of the lens and: $$\label{sigmac}
\Sigma_{\rm c}=\frac{c^{2}}{4\pi G}\frac{D_{\rm s}}{D_{\rm l}D_{\rm ls}}$$ is the critical surface density. Here, $c$ is the speed of light and $D_{\rm s}$, $D_{\rm l}$, and $D_{\rm ls}$ are the angular diameter distances from the observer to the source, from the observer to the lens, and from the lens to the source, respectively.
To infer cluster masses, we fit the measured $\Delta \Sigma(R)$ profile, obtained as described in Section \[measured\_prof\], to the theoretical models introduced in Section \[model\_prof\].
Cluster Profile Measurement {#measured_prof}
---------------------------
To measure cluster masses, we need to fit the cluster radial profiles. This is possible individually only for the most massive clusters in our sample ($M_{\rm 200}> 4\times 10^{14}M_{\rm \odot}$ for a signal-to-noise ratio $S/N > 3$; they represent the $\sim 2\%$ of the sample), while the noise dominates for the others. In order to increase the S/N and measure average radial profiles for all the other detections, we stack galaxy clusters in five richness bins, from $\lambda =10$ to $\lambda =70$, in steps of 10 (20 for the last bin) in richness.
We select the background galaxy sample using the following criteria:
$$\begin{gathered}
z_{\rm phot, s}>z_{\rm phot, l}+3\times \sigma_{\rm z_{phot}}\left(i^\prime-mag_{\rm s} \right)\\\times \left(1+z_{\rm phot, s} \right)
\end{gathered}$$
where $z_{\rm phot, s}$ is the source redshift, $z_{\rm phot, l}$ is the lens redshift, and $\sigma_{\rm z_{phot}}\left(i^\prime-mag_{\rm s}\right)$ is the error on the photometric redshift as a function of the source $i^\prime$-band magnitude. This function was obtained by interpolating the values in Figure 9 of @raichoor2014, up to $i^\prime \sim 24.7~mag$. We tested different cuts in magnitude ($i^\prime \sim 24.7, ~24, ~23.5, ~23~mag$), and found consistent results in all cases. We can conclude that the inclusion of faint sources in the background sample does not introduce a bias in the total cluster mass estimation.
Following @ford2015, we then sort the background galaxies in 10 logarithmic radial bins from $0.09~Mpc$ from the center of the lens to $5~Mpc$. In fact, at radii closer than 0.09 $Mpc$, galaxy counts are dominated by cluster galaxies, and at larger radii, the scatter in the mass estimate can be $\geq 20\%$ because of the contribution of large-scale structure [@becker2011; @oguri2011].
In each radial bin, we perform a weighted average of the lensing signal as follows:
$$\label{eq:mean1}
\Delta\Sigma(R)=\frac{{\sum_{i=0}^{l}}{\sum^s_{j=0}}w_{ij}\Sigma_{{\rm c},ij}\gamma_{{\rm t},ij}}{{\sum^l_{i=0}}{\sum^s_{j=0}} w_{ij}}$$
where we sum over every *lens-source* pair (i.e. [*i–j*]{} indices up to the $l$ number of lenses and $s$ number of sources). The weights $w_{ij}=\Sigma^{-2}_{{\rm c},ij}w_{\rm lens}$ [@mandelbaum2005] quantify the quality of the shape measurements through the *lens*fit weights $w_{\rm lens}$ (defined in Section \[shear\_data\]) and down-weight source galaxies that are close in redshift to the lens through $\Sigma^{-2}_{{\rm c},ij}$, which is evaluated for every *lens-source* pair using $z_{\rm phot}$ to calculate the angular diameter distances that appear in Equation \[sigmac\].
We then need to correct the measured signal, applying the calibration corrections introduced in Section \[shear\_data\]. As shown in @heymans2012, the ellipticity estimated by [*lens*]{}fit can be related to the true ellipticity (i.e. the sum of the shear and of the galaxy intrinsic ellipticity) as $\epsilon_{\rm lens}=(1+m)[\gamma + \epsilon_{\rm int}] +c$, where $m$ and $c$ are the multiplicative and additive biases. While the latter can be simply added on single ellipticity measurements, the first needs to be applied as a weighted ensemble average correction:
$$1+K(R) \equiv \frac{\sum_{i=0}^{l}\sum_{j=0}^{s}w_{ij}[1+m_{ij}]}{\sum_{i=0}^{l}\sum_{j=0}^{s}w_{ij}}$$
This is done to avoid possible instabilities in case the term $(1+m)$ tends to zero. In this way, we also remove any correlation between the calibration correction and the intrinsic ellipticity [@miller2013]. The calibrated signal is written as:
$$\label{eq:mean}
\Delta\Sigma_{\rm cal}(R)=\frac{\Delta\Sigma(R)}{1+K(R)}$$
To estimate the errors on $\Delta\Sigma(R)$, we create a set of 100 bootstrap realizations for each richness bin, selecting the same number of clusters for each stack but taking them with replacements. We apply Equation \[eq:mean1\] to obtain $\Delta\Sigma(R)$ for each bootstrap sample.
Following @ford2015, we then calculate the covariance matrix: $$\begin{gathered}
C(R_{i},R_{j})=\\ \left[\frac{N}{N-1}\right]^{2}\frac{1}{N} \sum_{k=1}^{N} \left[ \Delta\Sigma_{k}(R_{i})-\overline{\Delta\Sigma}(R_{i})\right]\\ \times \left[ \Delta\Sigma_{k}(R_{j})-\overline{\Delta\Sigma}(R_{j})\right]
\label{eq:cov}
\end{gathered}$$ where $R_{i}$ and $R_{j}$ are the radial bins, $N$ is the number of bootstrap samples, and $\overline{\Delta\Sigma}(R_{i})$ is the average over all bootstrap realizations.
For each radial bin, we weight the shear using the *lens*fit weights as shown in Equation \[eq:mean1\], so these error bars also include the error on the shape measurements of the source galaxies. We calculate the covariance matrix to take into account the correlation between radial bins and the contribution to the stacked signal of clusters with different masses inside the same richness bin.
Cluster Profile Model {#model_prof}
---------------------
In order to fit the tangential shear profiles, we use a basic analytic model for the cluster profile, to which we progressively add additional terms to obtain our fiducial model, which we will call [*Final model*]{}. This procedure permits us to quantify how adding additional terms changes the final cluster profile model.
Our basic analytic model is the following (hereafter [*Basic Model*]{}):
$$\begin{gathered}
\label{deltasigmatot}
\Delta\Sigma(R)=p_{\rm cc}[ \Delta\Sigma_{\rm NFW}(R)+ \Delta\Sigma_{\rm nw}(R)]+ \\ (1-p_{\rm cc})\Delta\Sigma_{\rm sm}(R)+\Delta\Sigma_{\rm 2halo}(R)
\end{gathered}$$
Here, $\Delta\Sigma_{\rm NFW}$ is the surface density contrast calculated from an NFW density profile, assumed as the halo profile; $\Delta\Sigma_{\rm nw}$, $\Delta\Sigma_{\rm sm}$ and $\Delta\Sigma_{\rm 2halo}$ are correction terms that take into account, respectively, non-weak shear, cluster miscentering, and the contribution to the signal from large-scale structure; and $p_{\rm cc}$ is a free parameter related to the miscentering term, and represents the percentage of correctly centered clusters in each stack.
Each term and the free parameters of the [*Basic Model*]{} are described in detail in the following sections.
As shown by @gavazzi2007, the two contributions to the shear signal from the luminous and dark matter can be distinguished by fitting a two-component mass model, which takes into account the contribution from the stellar mass of the halo central galaxy $M_{\rm BCG}$. In order to model the BCG signal, we follow @johnston2007 and add a point mass term to Equation \[deltasigmatot\] (hereafter [*Two Component Model*]{}):\
$$\begin{gathered}
\label{deltasigmatot2}
\Delta\Sigma(R)=\frac{M_{\rm BCG}}{\pi R^{2}} + \\p_{\rm cc}[ \Delta\Sigma_{\rm NFW}(R)+ \Delta\Sigma_{\rm nw}(R)]+ \\ (1-p_{\rm cc})\Delta\Sigma_{\rm sm}(R)+\Delta\Sigma_{\rm 2halo}(R)
\end{gathered}$$
The BCG mass, $M_{\rm{BCG}}$, is either fixed at the value of the mean BCG stellar mass in each bin (hereafter $M_{\rm BCG}^{*}$), or left as a free parameter in the fit. We obtained $M_{\rm BCG}^{*}$ using our photometric and photometric redshift catalogs from @raichoor2014, and @bruzual2003 stellar population models with *LePhare*, in fixed redshift mode at the galaxy photometric redshift.
Previous works [@becker2007; @rozo2009] have also shown that, when fitting the model profile to the halo profile derived from the observations in richness bins, the intrinsic scatter between the dark matter halo mass and the richness biases mass measurements. Following their modeling, we assume that the mass $M_{\rm 200}$ has a log-normal distribution at fixed richness, with the variance in $\ln{M_{\rm 200}}$, $\sigma_{\rm \ln{M200}|\lambda}$, and we add $\sigma_{\rm \ln{M200}|\lambda}$ to our [*Basic Model*]{} (hereafter [*Added Scatter Model*]{}).
All the averages in the equations below are performed using the same weighting as in equation \[eq:mean1\].
### $\Delta\Sigma_{\rm NFW}$ Profile
For the cluster halo profile, we assume an NFW profile. Numerical simulations have shown that dark matter halos density profiles, resulting from the dissipationless collapse of density fluctuations, can be well-described by this profile:
\[nfw\] $$\begin{aligned}
\rho_{\rm NFW}(r)=\frac{\delta_{\rm c}\rho_{\rm c}}{(\frac{r}{r_{\rm s}})(1+\frac{r}{r_{\rm s}})^{2}} \tag{\ref{nfw}}\\
\rho_{\rm c}=\frac{3H(z)^{2}}{8 \pi G}\\
r_{\rm s}=\frac{r_{\rm 200}}{c}\\
\delta_{\rm c}=\frac{200}{3}\frac{c^{3}}{\ln{(1+c)}-\frac{c}{1+c}}
\end{aligned}$$
where $\rho_{\rm c}$ is the critical density of the universe; $c$ is the concentration parameter; $\delta_{\rm c}$ is a dimensionless parameter that depends only on the concentration; $r_{\rm s}$ is the scale radius of the cluster; and $r_{\rm 200}$ is the radius at which the density is 200 times the critical density of the universe and can be considered as an approximation of the virial radius of the halo. The mass $M_{\rm 200}$ is the mass of a sphere of radius $r_{\rm 200}$ and average density of $200\rho_{\rm c}$:
$$\label{mass}
M_{\rm 200}=M(r_{\rm 200})=\frac{4\pi}{3} r^{3}_{\rm 200} \times 200 \rho_{\rm c}$$
Simulations have also shown that there is a relation between $M_{\rm 200}$ and $c$ [e.g. @NFW1996; @bullock2001]. In order to take this into account, we use the @dutton2014 mass–concentration relation:
$$\log{c_{\rm 200}}=a+b\log{\left ( M_{\rm 200}/[10^{12}h^{-1}M_{\rm \odot}] \right )}$$
with $a=0.520+(0.905-0.520)\exp{(-0.617z^{1.21})}$ and $b=-0.101+0.026z$. This reduces the dimensionality of the model to one parameter, $r_{\rm 200}$, from which we can calculate the halo mass using Equation \[mass\].
Integrating the tridimensional NFW density profile along the line of sight, we can calculate the NFW surface density:
$$\Sigma_{\rm NFW}(R)=2\int^{\infty}_{\rm 0}{\rho_{\rm NFW}(R,z)dz}$$
Integrating again, we get $\overline{\Sigma}_{\rm NFW}(R)$, the average surface density inside a radius $R$: $$\overline{\Sigma}_{\rm NFW}(<R)=\frac{2}{R^{2}}\int^{R}_{\rm 0}{R'\Sigma_{\rm NFW}(R')dR'}$$ Finally, we can calculate the first term in Equation \[deltasigmatot\]:\
$$\Delta\Sigma_{\rm NFW}=\overline{\Sigma}_{\rm NFW}(<R)-\Sigma_{\rm NFW}(R)$$
### Miscentering Term
Because the NFW density profile is spherically symmetric, an error in the determination of the halo center would lead to systematic underestimation of the lens mass. In fact, the random stacking offset smooths the differential surface mass density profile [@george2012].
Following @licitra2016a, we use both simulations and X-ray observations to obtain a model of the distribution of the offsets between the RedGOLD center and the cluster true center. We apply RedGOLD to the lightcones of @henriques2012, and calculate the offsets between the centers estimated by the algorithm and the true centers from the simulations. We also match our RedGOLD detections to X-ray detections in the same areas [@gozaliasl2014] to measure our average offset between RedGOLD and X-ray cluster centers. We perform the match between the RedGOLD and the @gozaliasl2014 catalogs by imposing a maximum separation between centers of $1~Mpc$ and a maximum difference in redshift of $\Delta z =0.1$.
 
In both cases, we find that the distribution of the offsets on the plane perpendicular to the line of sight can be modeled as a Rayleigh distributions with modes of 23 and $13~arcsec$, respectively [Figure \[fig:offset\], on the left; see also @johnston2007; @george2012; @ford2015]. What is important is that the model (a Rayleigh distribution) is the same in both cases, even if the precise values of the mode are different. In fact, the mode of the Rayleigh distribution, from which its mean, median and variance can be derived, will be derived as a free parameter from our analysis. In Figure 2, on the right, we also show the offset distributions in kpc. A Rayleigh distribution is also consistent with the published center offset distribution predicted from cosmological simulations for X-ray detected clusters, including AGN feedback [@cui2016].
We assume that this distribution represents the general offset distribution for our entire RedGOLD sample $P(R_{\rm off})$, and model it following @johnston2007: $$P(R_{\rm off})=\frac{R_{\rm off}}{\sigma^{2}_{\rm off}}\exp \left[-\frac{1}{2}\left(\frac{R_{\rm off}}{\sigma_{\rm off}}\right)^{2}\right]$$ where $R_{\rm off}$ is the offset between the true and the estimated center, projected on the lens plane, and $\sigma_{\rm off}$ is the mode, or scale length, of the distribution. The surface density measured at the coordinates $(R, \theta)$, with $\theta$ the azimuthal angle, relative to the offset position, $R_{\rm off}$, is: $$\begin{gathered}
\Sigma_{\rm NFW}(R,\theta | R_{\rm off})=\\ \Sigma_{\rm NFW}\left(\sqrt{R^{2}+R^{2}_{\rm off}-2RR_{\rm off}\cos \theta}\right)
\end{gathered}$$ and the azimuthal averaged surface density around $R_{\rm off}$ is given by: $$\Sigma_{\rm NFW}(R|R_{\rm off})=\frac{1}{2\pi}\int^{2\pi}_{\rm 0}{\Sigma_{\rm NFW}(R,\theta | R_{\rm off})d\theta}$$
To model the effect of miscentering, we smooth the $\Sigma_{\rm NFW}(R|R_{\rm off})$ profile convolving it with $P(R_{\rm off})$: $$\Sigma_{\rm sm}(R)=\int^{\infty}_{\rm 0}{\Sigma_{\rm NFW}(R|R_{\rm off})P(R_{\rm off})dR_{\rm off}}$$ and obtain the stacked surface density profile $\Sigma_{\rm sm}(R)$ around the offset positions of our ensemble of clusters with offset distribution $P(R_{\rm off})$ [@yang2006; @johnston2007; @george2012].
Finally we can write the miscentering term as: $$\Delta\Sigma_{\rm sm}(R)=\overline{\Sigma}_{\rm sm}(<R)-\Sigma_{\rm sm}(R)$$ with $\overline{\Sigma}_{\rm sm}(<R)$ being, as before, the average surface density within the radius $R$.
The miscentering term adds two free parameters to our model, $\sigma_{\rm off}$ and $p_{\rm cc}$, which is the percentage of correctly centered clusters in the stack, already introduced in Equation \[deltasigmatot\].
### Non-weak Shear Term
The non-weak shear correction arises from the fact that what we actually measure is the reduced shear: $$g_{\rm t}=\frac{\gamma_{\rm t}}{1-k}$$ where $k\equiv\Sigma_{\rm NFW}/\Sigma_{\rm c}$ is the convergence. Usually in the weak lensing regime $g_{\rm t} \approx \gamma_{\rm t}$, if $\gamma_{\rm t} << 1$ and $k<<1$, but for relatively massive halos, this assumption may no longer hold at the innermost radial bins in which we want to measure the cluster profile.
As described in @johnston2007, we introduce the non-weak shear correction term, calculated in @mandelbaum2006. In the non-weak regime, the tangential ellipticity component, $\epsilon_{\rm t}$ is proportional to $g_{\rm t}$, instead of $\gamma_{\rm t}$. We can expand $\epsilon_{\rm t}$ in power series as: $$\begin{gathered}
\label{eq:nonlin}
\epsilon_{\rm t}=\sum_{\rm n=0}^{\infty}Ag_{\rm t}^{2n+1}\\=A\left( \frac{\gamma_{\rm t}}{1-k}\right)^{2n+1}=A\left( \frac{\Delta\Sigma\Sigma_{\rm c}^{-1}}{1-\Sigma\Sigma_{\rm c}^{-1}}\right)^{2n+1}
\end{gathered}$$ As shown in detail in appendix A of @mandelbaum2006, we can calculate the correction term from the expansion in power series to the second order of $\epsilon_{\rm t}$, in powers of $\Sigma_{\rm c}$. We obtain the following term, which we add in Equation \[deltasigmatot\]: $$\Delta\Sigma_{\rm nw}(R)=\Delta\Sigma_{\rm NFW}(R)\Sigma_{\rm NFW}(R)\frac{\left<\Sigma^{-3}_{\rm c}\right>}{\left<\Sigma^{-2}_{\rm c}\right>}$$
### Two-halo Term
On large scales, the lensing signal is dominated by nearby mass concentrations, halos, and filaments. @seljak2000 developed an analytic halo model in which all the matter in the universe is hosted in virialized halos described by a universal density profile. They computed analytically the power spectrum of dark matter and galaxies, and their cross-correlation based on the @press1974 model. They found that, ignoring the contribution from satellite galaxies, a cluster can be modeled by two contributions: the one-halo term and the two-halo term. The first represents the correlation between the central galaxy and the host dark matter halo and corresponds to $\Delta\Sigma_{\rm NFW}(R)$. The second accounts for the correlation between the cluster central galaxy and the host dark matter halo of another cluster.
On large scales, the two-halo power spectrum is proportional to the halo bias and the linear power spectrum, $P_{\rm 2halo}\propto b(M_{\rm 200, z})P_{\rm lin}(k)$. In order to calculate the surface density associated to the two-halo term, we integrate the galaxy-dark matter linear cross-correlation function $\xi_{\rm lin}(r)$, obtained by the Fourier transform of the linear power spectrum.
Following @johnston2007 and @ford2015, we can write the two-halo term as: $$\begin{gathered}
\label{bias}
\Delta\Sigma_{\rm 2halo}(R,b)= \\ b(M_{\rm 200},z)\Omega_{\rm m}\sigma^{2}_{\rm 8}D(z)^{2}\Delta\Sigma_{\rm l}(R)
\end{gathered}$$ where $b(M_{\rm 200},z)$ is the bias factor, $\Omega_{\rm m}$ is the matter density parameter, $\sigma^{2}_{\rm 8}$ is the amplitude of the power spectrum on scales of 8 $h^{-1}Mpc$, $D(z)$ is the growth factor and
$$\Delta\Sigma_{\rm l}(R)=\overline{\Sigma}_{\rm l}(<R)-\Sigma_{\rm l}(R)$$
where $$\begin{gathered}
\Sigma_{\rm l}(R,z)= \\ (1+z)^{3}\rho_{\rm c,0}\int^{\infty}_{\rm -\infty}{\xi_{\rm lin}\left((1+z)\sqrt{R^{2}+y^{2}}\right)}dy
\end{gathered}$$
The factor $(1+z)$ arises from the conversion from physical units to comoving units.
For the bias factor, we use the analytic formula calculated by @seljak2004, and for $P_{\rm lin}(k)$, we use tabulated values from CAMB [@lewis2000].
RESULTS
=======
Cluster Mass Estimation
-----------------------
### Fit the Profile Model to the Shear Profile {#subsec:fit}
We fit the shear profiles, obtained as described in Section \[measured\_prof\] with the density profile models of Section \[model\_prof\], progressively adding model parameters to quantify their impact on the final results.
We start from the [*Basic Model*]{} with an NFW surface density contrast and correction terms that take into account cluster miscentering, non-weak shear, and the two halo term. This model has three free parameters: the radius $r_{\rm 200}$, from which we calculate the mass $M_{\rm 200}$ from Equation \[mass\], and the miscentering parameters $p_{\rm cc}$, and $\sigma_{\rm off}$.
We then take into account the intrinsic scatter in the mass–richness relation through the [*Added Scatter Model*]{}, which has four free parameters: $\log{M_{200}}$, $p_{\rm cc}$, $\sigma_{\rm off}$, and $\sigma_{M|\lambda}$. For each bin, we use the mass–richness relation, calculated from the [*Basic Model*]{} to infer the mean mass of the stacked clusters, as a first approximation. We then randomly scatter the mass using a gaussian distribution with mean $\left<\ln{M_{\rm 200}}\right>$ and width $\sigma_{\rm \ln{M200}|\lambda}$.
Finally, we consider the [*Two Component Model*]{}, with four free parameters: $r_{\rm 200}$, $p_{\rm cc}$, $\sigma_{\rm off}$, and $\log{M_{\rm BCG}}$. When we fix the BCG mass to the mean stellar mass for each richness bin, $M_{\rm BCG}=M_{\rm BCG}^{*}$, the free parameters reduce to three.
The parameters used in each case are summarized in Table \[tab:priors\].
We perform the fit using Markov Chains Monte Carlo [MCMC; @metropolis1953]. This method is particularly useful when the fitting model has a large number of parameters, the posterior distribution of the parameters is unknown, or the calculation is computationally expensive. MCMC allows efficient sampling of the model likelihood by constructing a Markov chain that has the target posterior probability distribution as its stationary distribution. Each step of the chain is drawn from a model distribution and is accepted or rejected based on the criteria defined by the sampler algorithm.
To run our MCMC, we use *emcee*[^7] [@emcee], a Python implementation of the parallel Stretch Move by @goodman2010. In order to choose the starting values of the chain we first perform a minimization with the Python version of the Nelder–Mead algorithm, also known as downhill simplex [@nelder1965]. We used flat priors (i.e. a uniform distribution within a given range) for all parameters. Our initial priors, for the three different models, are shown in Table \[tab:priors\]. All parameters are constrained to be positive and inside a range chosen according to their physical meaning. To choose the range for the intrinsic scatter, we refer to the values calculated by @licitra2016a. They found $\sigma_{\rm \ln{M}|\lambda}=0.39\pm0.07$ using the X-ray catalog of @gozaliasl2014 and $\sigma_{\rm \ln{M}|\lambda}=0.30\pm0.13$ from @mehrtens2012.
MCMC produce a representative sampling of the likelihood distribution, from which we obtain the estimation of the error bars on the fitting parameters and of the confidence regions for each couple of parameters. We calculate the model likelihood using the bootstrap covariance matrix of Equation \[eq:cov\]: $$\begin{gathered}
ln\mathcal{L}=-\frac{1}{2}\left(\Delta\Sigma_{\rm data}- \Delta\Sigma_{\rm model}\right)^{T}C^{-1}\\ \left(\Delta\Sigma_{\rm data}- \Delta\Sigma_{\rm model} \right)
\end{gathered}$$ We use an ensemble of 100 walkers, a chain length of 1000 steps and a burn-in of 100 steps leading to a total of 90,000 points in the parameters space. In order to test the result of our chain, we check the acceptance fraction and the autocorrelation time, to be sure that we efficiently sample the posterior distribution and have enough independent samples.
### Fit Parameters {#results}
We perform the fit of the models to the observed profiles on each of the three samples, CFHT-LS W1, NGVS5, and NGVS4. We then combine the CFHTLS and NGVS5, and all the three samples together.
The profiles obtained using the [*Basic Model*]{} and the complete sample (CFHT-LS W1 + NGVS5 + NGVS4) are shown in black in Figure \[fig:profiles\], on the left. The error bars on the shear profiles are the square root of the diagonal elements of the covariance matrix.
![Shear profiles measured with the weak lensing [*selected*]{} CFHT-LS W1 + NGVS5, in red, with weak lensing selected NGVS4, in blue, and with the weak lensing [*selected*]{} CFHT-LS W1 + NGVS5 + NGVS4, in black. We notice that the addition of the four bands sample does not significantly change the profiles. The profiles measured using the three different samples are compatible within $1\sigma$ and the profiles obtained using CFHT-LS W1 + NGVS5 + NGVS4 have smaller error bars.[]{data-label="fig:profiles_comp"}](profiles_plot_comp.pdf)
The profiles measured using the CFHT-LS W1 + NGVS5 sample, the NGVS4 sample, and the complete sample are shown in Figure \[fig:profiles\_comp\]. They are consistent within $1\sigma$ and the error bars are smaller in the last case. We can conclude that the richness shifts applied to NGVS4 seem not to bias our results when this sample is added to the other two that are covered by five bands. Increasing the sample size, we notice a progressive improvement in the profiles that are recovered with a lower noise level.
Because the miscentering correction is the one that most affects the mass estimation, in Figure \[fig:profiles\], on the left, we show the fitted profiles (green lines), and the profiles that we would obtain with and without the miscentering term. The red lines represent the profiles we would obtain in the case in which all the clusters in the stack were perfectly centered ($p_{\rm cc}=1$), and the blue lines show the opposite case ($p_{\rm cc}=0$). An incorrect modeling of this effect leads to biased mass values [i.e. mass underestimation between 10 and $40\%$, @ford2015].
In Figure \[fig:profiles\], on the right, we show the lensing S/N maps. These maps were calculated using aperture mass statistics [@schneider1996; @schirmer2006; @dufan2014]. For each richness bin, we create a grid with a side of $1~Mpc$ and binning of $0.001~deg$, centered on the stacked clusters. In each cell, we evaluate the amount of tangential shear, filtered by a function that maximizes the S/N of an NFW profile, inside a circular aperture, following @schirmer2006. For stacked clusters, a $S/N\sim10$ is considered sufficient to recover the fitting parameters [@oguri_takada2011]. All richness bins have $S/N\geq10$. The highest richness bin shows the lowest S/N, being less populated than the others.
We show the results of our fits in Table \[tab:fit\], for the [*Basic Model*]{}, [*Added Scatter Model*]{}, and [*Two Component Model*]{}. The values of the radius, of the mass, and of the miscentering parameters for each richness bin are consistent within 1$\sigma$ for the three models. We found that the intrinsic scatter and BCG mass are not constrained by the data. The main effect of the addition of $\sigma_{M|\lambda}$ to the fit is to introduce more uncertainties and to increase the error on the estimated parameters. The inclusion of $M_{\rm BCG}$ in the model (either set as a free parameter or fixed to $M_{\rm BCG}^{*}$) has no impact on the estimated parameters, which are therefore the same as those obtained using the [*Basic Model*]{}. We can conclude that the contribution of the BCG mass is not significant in the radial range we are using to fit the shear profiles.
In Figure \[fig:corner\], we show an example of error bars and the confidence regions of the parameters, obtained using the python package *corner* by @corner. This example corresponds to the third richness bin, fitted with the [*Two Component Model*]{}. On the diagonal, we show the one-dimensional histograms of the parameter values, representing the marginalized posterior probability distributions. Under the diagonal, we show the two-dimensional histograms for each couple of parameters and the confidence levels corresponding to $0.5\sigma$, $1\sigma$, $1.5\sigma$ and $2\sigma$.
Mass–Richness Relation
----------------------
Using the mass measured for each richness bin, we perform a fit to a power law to infer the mass–richness relation for all three models, using the python orthogonal distance regression routine [@boggs1990] to take into account the errors in both $\log{\lambda}$ and $\log{M_{\rm 200}}$: $$\log{M_{\rm 200}}=\log{M_{\rm 0}}+\alpha\log{\lambda/\lambda_{\rm 0}}$$ with a pivot richness $\lambda_{\rm 0}=40$.
In Table \[tab:mr\_par\] and in Figure \[fig:massrich\], we show the results obtained fitting the three models. The slope and the normalization values are all consistent within $1 \sigma$, for the three models. We notice that the uncertainties in the fit of the [*Added Scatter Model*]{} are larger, due to the inclusion of the intrinsic scatter as a free parameter.
In order to also take into account the intrinsic scatter between richness and mass in the [*Basic*]{} and in the [*Two Component Models*]{}, we apply an a posteriori correction as in @ford2015. Using the mass–richness relation inferred from the [*Basic Model*]{} and from the [*Two Component Model*]{}, we calculate the mass of all the clusters in the sample, then we scatter those masses assuming a log-normal distribution centered on $\log{M_{\rm 200}}$ and with a width $\sigma_{\rm \ln{M}|\lambda}=0.39$, based on the scatter measured by @licitra2016a. We repeat this procedure, creating 1000 bootstrap realizations, choosing masses randomly with replacements from the entire sample. We then calculate the new mean mass values in each richness bin and average them over all bootstrap realizations. We then repeat the fit to infer the new mass–richness relation. This procedure is illustrated in Figure \[fig:scatter\], where we show the results from the fit to the [*Two Component Model*]{} (in black), the scattered masses (in light red), and the new mean masses and mass–richness relation (in red). Due to the shape of the halo mass function, the net effect of the intrinsic scatter correction is to lead to a slightly higher normalization value of the mass–richness relation. The introduction of the the intrinsic scatter between richness and mass does not significantly change our results obtained with the [*Basic*]{} or with the [*Two Component Model*]{}. In fact, the difference in normalization for the original models and their scattered versions is less than $1\%$.
Having verified the impact of each model term on the final results, we consider as our [*Final Model*]{} the model that takes into account all the parameters considered so far, the [*Two Component Model*]{} with the a posteriori intrinsic scatter correction. Our final mass–richness relation is then: $\log{M_{\rm 0}}=14.46\pm0.01$ and $\alpha=1.04\pm0.09$.
Our uncertainty on the mass–richness relation parameters above is statistical. We expect systematic biases to be of the same order as the statistical uncertainties, from previous work on the CFHT-LS survey. In fact, @miller2013 and @heymans2012 estimated that the residual bias in the CFHTLenS analysis (and as a consequence on the NGVSLenS, given that the survey characteristics and reduction techniques are the same) could reach maximal values around $3--5\%$ (see also @simet2016; @fenech2017), which is on the same order of magnitude of our statistical uncertainties ($\sim$ 5%).
We checked that our richness binning choice does not affect the recovered mass–richness relation. We performed the fit, discarding the lower (most contaminated) and the highest (less populated) bins, and found consistent results. We have also verified that our procedure does not significantly bias our results, compared to a joint fit [e.g. @viola2015; @simet2016]. We describe this test in Appendix \[appendix\].
Comparison with X-Ray Mass Proxies {#xray}
----------------------------------
To compare our mass estimates with X-ray mass proxies, we follow the same matching procedure as in @licitra2016a. We use the @gozaliasl2014 and @mehrtens2012 X-ray catalogs, and perform the match between their and our detections imposing a maximum separation of $1~Mpc$ and a maximum difference in redshift of 0.1. We include detections from both the [*published*]{} and the [*complete*]{} catalogs to broaden our sample, and have more statistics to perform the scaling relation fits. Results obtained with the [*complete*]{} catalogs might be affected by contamination biases, since for those, we estimated the purity to decrease to $\sim 60\%$ [Figure 8 and 9 of @licitra2016a].
Within all three fields, we recover 36(27) objects from the match of the [*complete*]{}([*published*]{}) catalog with @gozaliasl2014 (in this case, all objects are from the CFHT-LS W1 field), and 21(17) from objects from the match of the [*complete*]{}([*published*]{}) catalog with @mehrtens2012. As shown in @licitra2016a, RedGOLD recovers 38 clusters, up to $z\sim1$, in the $3~deg^2$ of the CFHT-LS W1 field, covered by @gozaliasl2014 catalog. The clusters detected by RedGOLD that do not have an X-ray counterpart seem to be, from visual inspection, small galaxy groups. It is possible that these systems have an X-ray emission below the X-ray detection limit, or that they are not relaxed systems and do not have any X-ray emission at all.
As explained in Section \[xray\_cat\], @gozaliasl2014 $M_{\rm 200}$ masses were estimated using the $M_{\rm X}-L$ relation of @leauthaud2010. We estimate @mehrtens2012 $M_{\rm 200}$ masses from the $r_{\rm 200}$ values given in their catalog, using Equation \[mass\]. Our masses $M_{\rm 200}^{lens}$ are calculated using our final mass–richness relation.
In Figure \[fig:xray\_res\], we show the normalized difference between the X-ray masses of @gozaliasl2014 and lensing masses $\left(M_{\rm 200}^{lens}-M_{\rm 200}^{X}\right)/M_{\rm 200}^{X}$ as a function of $M_{\rm 200}^{X}$, obtained using our [*Final Model*]{}. The ratio is measured with respect to $M_{\rm 200}^{X}$ since our sample is X-ray selected (i.e. we select the clusters in the X-ray catalog, and then compare their X-ray and lensing mass estimate).
![The weak lensing mass–richness relations obtained with the weak lensing [*selected*]{} CFHT-LS W1 + NGVS5 + NGVS4, using the [*Basic Model*]{} (black line and black dots), Added Scatter Model (blue line and blue squares), Two Component Model (red line and red triangles). See text for the description of the models.[]{data-label="fig:massrich"}](fit_mr.pdf)
![Effect of the a posteriori intrinsic scatter correction. Using the mass–richness relation inferred from the Two Component Model (in black), we calculated cluster masses for our selected sample. We scattered those masses, assuming a log-normal distribution centered on $\log{M_{\rm 200}}$ and with a width $\sigma_{\rm \ln{M}|\lambda}=0.39$, based on the value measured by @licitra2016a (in light red). We repeated this procedure, creating 1000 bootstrap realizations and calculated the new mean mass values in each richness bin, averaging over all realizations. We then repeated the fit to infer the new mass–richness relation (in red), which is shifted toward larger masses.[]{data-label="fig:scatter"}](scatter_04.pdf)
In the last four columns of Table \[tab:mr\_par\], we show the mean normalized difference and the mean ratio between lensing and X-ray masses, for the three models, obtained with @gozaliasl2014 and @mehrtens2012 catalogs. For all models, the mean differences obtained using $M_{\rm 200}^{X}$ from @gozaliasl2014 ($\sim0.1-0.2$) are higher than those obtained using @mehrtens2012 ($\sim -0.1-0.0$). However, uncertainties on the individual measurements are larger and the scatter in the difference are about an order of magnitude higher for the @mehrtens2012 sample. Because of the large uncertainty, we do not consider results obtained with the @mehrtens2012 catalogs reliable.
 
As explained in Section \[xray\_cat\], @gozaliasl2014 masses were calculated from the X-ray luminosity, after the excision of the AGN contribution and the correction for cool core flux removal. We find that this leads to X-ray mass estimates that are lower compared to masses derived with weak lensing than those calculated without core excision. Hereafter, we will use only the @gozaliasl2014 sample, given the larger uncertainty in our results obtained using the @mehrtens2012 sample, and the higher number of cluster matches. Core-excised X-ray temperatures are also known to better correlate with cluster masses [@pratt2009].
Using X-ray masses from the @gozaliasl2014 catalog and the lensing masses estimated from the mass–richness relation derived from our [*Final Model*]{}, applied on the [*complete*]{} catalogs, we find a mean normalized difference of $0.15\pm0.20$ ($\frac{M_{\rm 200}^{lens}}{M_{\rm 200}^{X}}=1.15\pm0.20$), considering the whole mass range. If we consider two different mass ranges, we find a mean normalized difference of $0.17\pm0.24$ for $M_{\rm 200}^{X}<10^{14}M_{\rm \odot}$, and a mean normalized difference of $0.14\pm0.18$ for $M_{\rm 200}^{X}\geq10^{14}M_{\rm \odot}$. This corresponds to $\sim15\%$ higher lensing masses in the whole mass range, and $\sim20\%$ and $\sim15\%$ higher lensing masses for $M_{\rm 200}^{X}<10^{14}M_{\rm \odot}$ and $M_{\rm 200}^{X}>10^{14}M_{\rm \odot}$, respectively.
To obtain scaling relations, we exclude the two clusters with mass $M_{\rm 200}^{X}< 2 \times10^{13}M_{\rm \odot}$ from the matched sample with @gozaliasl2014, because both our and the X-ray catalog are incomplete at these low masses. We also do not consider the two highest mass matches ($M_{\rm 200}^{X}>2 \times10^{14}M_{\rm \odot}$), because our catalog is incomplete in this mass range, given our low area coverage. All four excluded clusters were matches with the Licitra’s [*published*]{} catalog.
![We compare our derived weak lensing masses with the X-ray masses from @gozaliasl2014 catalog. The weak lensing masses have been derived from our fit of the mass–richness relation using our [*Final Model*]{}. The black dots are the RedGOLD detections from the [*published*]{} catalogs (RG PC) and the black squares are the detections from the [*complete*]{} catalogs (RG CC). The red lines show the fits obtained with the slope as a free parameter, and the green lines those obtained with the slope fixed at unity. In both cases, solid lines refer to the [*published*]{} catalogs, and the dashed lines to the [*complete*]{} catalogs. The black dotted line is the diagonal. See Section \[redgold\_cat\] for the catalog definitions.[]{data-label="fig:xray_scal1"}](MX-ML.pdf)
In Figure \[fig:xray\_scal1\], we plot the $M_{\rm 200}^{X}-M_{\rm 200}^{lens}$ relation, and in Figure \[fig:xray\_scal2\], the $L_{\rm X}-M_{\rm 200}^{lens}$ and the $T_{\rm X}-M_{\rm 200}^{lens}$ relations. In those plots, the black dots represent matches with the RedGOLD cluster detections in Licitra’s [*published*]{} catalogs, while the black squares represent all those with the [*complete*]{} catalogs (see Section \[redgold\]). This difference between our lensing masses and those calibrated with lensing masses from @leauthaud2010 includes different contributions, and it is not a straightforward difference between our lensing masses and X-ray selected lensing masses. In fact, both the @gozaliasl2014 selection in $L_X$ (when stacking clusters to derive the @leauthaud2010 lensing masses), our selection based on the @licitra2016a [@licitra2016b] richness, and differences in the shear calibration in our data and @leauthaud2010 contribute to this difference, and interpreting it precisely implies degeneracies on each contribution.
In Figure \[fig:xray\_scal1\], we show the relation between X-ray and lensing masses:
$$\begin{aligned}
\log{\left({M_{\rm 200}^{lens}}\right)} & =a+b \log{\left({M_{\rm 200}^{X}}\right)}
\end{aligned}$$
The black dotted line is the diagonal, the solid lines are the fit to the [*published*]{} catalogs, and the dashed lines are the fit to the [*complete*]{} catalogs. The red lines were obtained with the slope as a free parameter of the fit, and the green lines with the slope fixed at unity. For the [*published* ]{} catalogs, our threshold in richness and $\sigma_{\rm det}$ is meant to select clusters with $M_{\rm 200}>10^{14}M_{\rm \odot}$ with a completeness $\sim80\%$. Part of these detections have X-ray masses lower than our selection threshold of $M_{\rm 200}>10^{14}M_{\rm \odot}$; in fact, their X-ray masses are in the range $ 2 \times 10^{13}M_{\rm \odot}< M_{\rm 200}^{X}<10^{14}M_{\rm \odot}$. We expect to have a contamination of clusters with these lower masses, and our purity of $\sim 80\%$ is calculated for real clusters with $M_{\rm 200}^{X}>10^{13}M_{\rm \odot}$. However, our completeness decreases (<80%) in this mass range ($M_{\rm 200}^{X}<10^{14}M_{\rm \odot}$), as shown in @licitra2016a.
 
When fixing the slope at the unity, we obtain $a=0.20\pm0.03$($a=0.23\pm0.03$), and a scatter of $\sigma_M = 0.20$ dex ($\sigma_M = 0.17$ dex) for the [*complete*]{} ([*published*]{}) catalogs. In this case, the difference in $a$ for the two samples is negligible, $\sim 0.03 \pm0.06$ dex. The small shift in normalization ($\sim 0.2$ dex) compared to the diagonal is expected, because lensing mass estimates are generally higher than X-ray masses [@zhang2008; @rasia2012; @simet2015]. When leaving the slope as a free parameter, we find $a=-0.13\pm2.96$ and $b=1.02\pm0.21$, with a scatter of $\sigma_M = 0.20$ dex ($a=6.42\pm3.17$ and $b=0.56\pm0.23$, with a scatter of $\sigma_M = 0.15$ dex) for the [*complete*]{} ([*published*]{}) catalogs. The incompleteness when using the [*published*]{} catalogs appears to bias our fit slope, which becomes much shallower than the diagonal.
In Figure \[fig:xray\_scal2\], we show the mass–luminosity and mass–temperature relations. We apply a logarithmic linear fit, in the form:\
$$\begin{aligned}
\log{\left(\frac{M_{\rm 200} E(z)}{M_{\rm 0}}\right)} & =a+b \log{\left(\frac{L_{\rm X}}{L_{\rm 0} E(z)}\right)}\\
\log{\left(\frac{M_{\rm 200} E(z)}{M_{\rm 0}}\right)} & =a+b \log{\left(\frac{T_{\rm X}}{T_{\rm 0}}\right)}
\end{aligned}$$
where $E(z)=H(z)/H_{\rm 0}$, $M_{\rm 0}=8\times10^{13}\;h^{-1}M_{\rm \odot}$ for the $M_{\rm 200}-L_{\rm X}$, $M_{\rm 0}=6\times10^{13}\;h^{-1}M_{\rm \odot}$ for the $M_{\rm 200}-T_{\rm X}$, $L_{\rm 0}=5.6\times10^{42}\;h^{-2}erg/s$, and $T_{\rm 0}=1.5\;keV$.
For the mass–luminosity relation, we find $a=0.10\pm0.03$ and $b=0.61\pm0.12$, with a scatter $\sigma_{\rm logM_{\rm 200}|L_X}=0.20$ dex($a=0.16\pm0.03$ and $b= 0.43\pm0.12$, with a scatter $\sigma_{\rm logM_{\rm 200}|L_X}=0.15$ dex) for the [*complete*]{}([*published*]{}) catalogs. For the mass–temperature relation, we find $a=0.23\pm0.03$ and $b=1.46\pm0.28$, with a scatter $\sigma_{\rm logM_{\rm 200}|T_X}=0.20$ dex($a=0.28\pm0.03$ and $b=1.03\pm0.30$, with a scatter $\sigma_{\rm logM_{\rm 200}|T_X}=0.15$ dex), for the [*complete*]{}([*published*]{} catalogs). The relations obtained with the [*published*]{} catalogs show again shallower slopes. Our results are consistent with the expected deviations from self-similarity [@bohringer2011].
We summarize our results in Table \[tab:xray\].
DISCUSSION {#disc}
==========
Comparison to Previously Derived Mass–Richness Relations
--------------------------------------------------------
In this section, we discuss our results in the context of similar current studies.
As stated before and shown in @licitra2016a [@licitra2016b], our richness estimator $\lambda$ is defined in a similar way as the richness from redMaPPer [@rykoff2014]. The redMaPPer richness is defined as $\lambda_{\rm RM}=\sum p_{\rm mem}\theta_{\rm L}\theta_{\rm R}$, where $p_{\rm mem}$ is the probability that each galaxy in the vicinity of the cluster is a red-sequence member and $\theta_{\rm L}, \theta_{\rm R}$ are weights that depend on luminosity and radius. In this calculation, only galaxies brighter than $0.2L_{\rm *}$ and within a scale radius $R_{\rm \lambda}$ are considered. The radius is richness-dependent and it scales as $R_{\rm \lambda}=1.0(\lambda/100)^{0.2}h^{-1}Mpc$.
The RedGOLD richness is a simplified version of $\lambda_{\rm RM}$. We constrained the radial distribution of the red-sequence galaxies with an NFW profile and applied the same luminosity cut and radius scaling as in @rykoff2014 but did not apply a luminosity filter. Unlike the redMaPPer definition, our richness is not a sum of probabilities. Those choices were made to minimize the scatter in the mass–richness relation. For redshifts $z<0.3$, the difference $\frac{\lambda_{\rm RM} - \lambda}{\lambda}$ is only of $5--15\%$, while it increases to $40--60\%$ at $0.4<z<0.5$, where the redMaPPer richness is systematically higher [@licitra2016a]. This difference might be due to the different depths of the CFHTLenS and SDSS surveys. This means that we can compare our results with others obtained using the redMaPPer cluster sample.
@simet2016 performed a stacking analysis of the redMaPPer cluster sample, using shear measurements from the SDSS. Their sample is much larger than ours, consisting of 5,570 clusters, with a redshift range $0.1<z<0.3$, lower than the one used for this work, and a richness range $20\leq \lambda_{\rm RM} \leq 140$. With these data, they were able to characterize the different systematic errors arising in their analysis with great accuracy. For the mass–richness relation, they obtained the normalization $\log{(M_{\rm 0}\;[h^{-1}M_{\rm \odot}])}=14.34\pm0.04$ (the error includes both statistical and systematic error) and the slope $\alpha=1.33^{+0.9}_{\rm -0.1}$. To compare our results to theirs, we use our masses in units of $h^{-1}M_{\rm \odot}$ and we repeat our fits. Using our [*Final Model*]{}, we obtain $\log{M_{\rm 0}}=14.31\pm0.02$ and $\alpha=1.04\pm0.09$ (the errors are only statistical). Our normalization is consistent within $1 \sigma$ and our slope is consistent within $\sim2\sigma$ of Simet et al.’s. Comparing the masses at the pivot richness, $\lambda_{\rm 0}=40$, we obtain $2.04\times10^{14}h^{-1}M_{\rm \odot}\pm0.02$ compared to Simet et al.’s $2.21\times10^{14}h^{-1}M_{\rm \odot}\pm0.15$.
In another recent work, @farahi2016 inferred the mass–richness relation using the same sample of SDSS redMaPPer clusters ($0.1<z<0.3$ and $\lambda_{\rm RM}>20$), performing a stacking analysis and estimating the velocity dispersion of the dark matter halos from satellite-central galaxy pairs measurements. For the mass–richness relation, they found a normalization of $14.19\pm0.1$ and a slope of $1.31\pm0.19$ (the error includes both statistical and systematic error), using a pivot $\lambda_{\rm 0}=30$. Repeating the fit using their pivot richness, we obtain $\log{M_{\rm 0}}=14.18\pm0.02$ and $\alpha=1.04\pm0.09$, consistent within less than $1 \sigma$ in normalization and $1.5\sigma$ in slope, with their results. At the pivot richness $\lambda_{\rm 0}=30$ our mass is $1.51\times10^{14}M_{\rm \odot}\pm0.02$, consistent with their value of $1.56\times10^{14}M_{\rm \odot}\pm0.35$.
@melchior2016 calibrated the mass–richness relation and its evolution with redshift up to $z<0.8$, using 8000 RedMaPPer clusters in the Dark Energy Survey Science Verification [DES; @des2016] with $5\leq\lambda_{\rm RM}\leq180$. They found a normalization $M_{\rm 0}=2.35\pm0.34\times10^{14}M_{\rm \odot}$ and a slope $1.12\pm0.26$, using the pivot richness $\lambda_{\rm 0}=30$ and a mean redshift $z=0.5$. Their errors include both statistical and systematic errors. These results are consistent with ours within less than $1 \sigma$, both in normalization and slope, even if this sample has a larger average redshift, where we expect our richness definitions to be less similar.
Our normalization is in perfect agreement with all the works cited above ($<1\sigma$). On the other hand, there is a slight tension between our slope and those of @simet2016 and @farahi2016 ($1.5-2\sigma$), but not with @melchior2016 ($<1\sigma$). Our slope is also consistent with the first mass–richness relation inferred using the redMaPPer cluster sample from @rykoff2012, and with the @saro2015 richness-mass relation, inferred by cross-matching the SPT-SZ survey with the DES redMaPPer cluster sample. They found values of 1.08 (the error is not given) and $0.91\pm0.18$, respectively. @saro2015 value has been converted from the slope of the richness-mass relation to the slope of the mass–richness relation by @simet2016.
We cannot compare our results with the scaling relations obtained in @johnston2007, @covone2014, @ford2015, or @vanuitert2015 because their definition of richness is different.
We conclude that our fit of the mass–richness relation is in agreement with the other works cited above. These results confirm the efficiency of the RedGOLD richness estimator, and quantify the relation between the RedGOLD richness measurements and the total cluster masses obtained with weak lensing. Even without using a probability distribution, our richness is as efficient as the more sophisticated redMaPPer richness definition.
Weak Lensing vs X-Ray Masses {#xray2}
----------------------------
In Figure \[fig:xray\_scal2\], we compare our lensing mass versus X-ray mass proxies relations to those of other works in literature.
In the $L_{\rm X}-M_{\rm 200}^{lens}$ plot, we compare our results with those from @kettula2015 and @leauthaud2010. We remind the reader that the fit to the [*published*]{} catalogs (solid red line) shows a shallower slope because of our selection in mass, which, while it optimizes purity, leads to a bias in slope due to the lack of clusters detected at masses $M_{\rm 200}<10^{14}M_{\rm \odot}$ (see discussion in Section \[xray\]).
Because of the large uncertainties, the fit to both the [*complete*]{} and [*published*]{} catalogs (dashed red line) are consistent within $<1 \sigma$ and $<2 \sigma$, respectively, in normalization and slope, with results from @kettula2015, even if our normalizations are higher.
With respect to the $E(z) M_{\rm 200}$ derived from @leauthaud2010 (and, as a consequence, from @gozaliasl2014, because they use @leauthaud2010 to derive their mass relations), we are consistent within $<2.5 \sigma$ in normalization and within $<1 \sigma$ in slope for the [*complete*]{} catalogs. For the [*published*]{} catalogs, we are inconsistent in normalization (the normalization difference is $\sim 3.7\sigma$) but consistent in slope within $<1.5 \sigma$.
Both @kettula2015 and @leauthaud2010 did not apply the miscentering correction but, while the first performed their lensing analysis on single clusters, the latter stacked their low-mass clusters in very poorly populated bins. This procedure could have introduced a bias that led to more smoothed profiles and thus to lower mass estimates and to a lower normalization of the scaling relation.
In the $T_{\rm X}-M_{\rm 200}^{lens}$ plot, we compare our results with @kettula2015 and @mantz2016. Because their masses are derived at the overdensity $\Delta=500$, we convert their $M_{\rm 500}$ values to $M_{\rm 200}$, using $M_{\rm 200}=1.35M_{\rm 500}$ from @rines2016, derived considering that the mass–concentration relation weakly depends on mass [@bullock2001] and assuming an NFW profile with a fixed concentration $c=5$. We find that the normalization and slope of our fit to the [*complete*]{}([*published*]{}) catalogs are consistent with the @kettula2015 results within $<1\sigma$($ \lesssim2\sigma$), and with @mantz2016 results within $<1\sigma$($<1.5\sigma$) in normalization and slope.
In Table \[tab:xray2\], we show the differences in normalization, $\Delta a$, and in slope, $\Delta b$, between our results and those used for comparison for the mass–luminosity, and the mass–temperature relations.
Given that our results based on the RedGOLD [*complete*]{} catalogs are consistent with other results in the literature, we conclude that the thresholds that we apply in the RedGOLD [*published*]{} catalog introduces systematics in the fit of the cluster lower mass end.
Selecting samples based on lensing measurements, simulations predict that mass measurements from lensing are systematically lower than the cluster true total mass by $\sim 5--10\%$ (in the mass range $M^{sim}>5\times 10^{14}$) and those from X-ray proxies (in the mass range $10^{14}<M_{\rm 200}^{X}<5\times 10^{15}$) by $\sim 25--35\%$, with $\left<M^{sim}_{\rm {X}}/M^{sim}_{\rm {L}}\right> \sim 0.7-0.8$ [@meneghetti2010; @rasia2012]. When we compare our weak lensing mass measurements to X-ray @gozaliasl2014 cluster masses (Figure \[fig:xray\_res\] and Table \[tab:mr\_par\]), for X-ray selected clusters, for the [*Final Model*]{} we obtain $\sim15\%$ higher lensing masses in the whole mass range, and $\sim20$ and $\sim15\%$ higher lensing masses for $M_{\rm 200}^{X}<10^{14}M_{\rm \odot}$ and $M_{\rm 200}^{X}>10^{14}M_{\rm \odot}$, respectively.
As we mentioned before in Section \[xray\], and from Table \[tab:mr\_par\] and Figure \[fig:xray\_res\], the mean residuals and ratio values obtained using @mehrtens2012 catalog are lower, with $\left<M_{\rm {L}}/M_{\rm {X}}\right> \sim -0.1-0.0$, which means that non core-excised temperature led to overestimated X-ray masses, as expected [@pratt2009].
Previously published [*XMM-Newton*]{} X-ray to lensing mass ratios are obtained with a selection on lensing, and show values of $\left<M_{\rm {X}}/M_{\rm {L}}\right> \sim 0.91-0.99$ [@zhang2008] and $\sim 0.72-0.96$ [@simet2015, using observations from @piffaretti2011; @hajian2013]. Given that we measure the bias on the lensing mass given an X-ray selection, we cannot compare our measurements directly with those obtained by the measure of the bias in the X-ray mass given the lensing mass. However, the trend is similar and consistent with simulation. Our uncertainty on $\left<M_{\rm {L}}/M_{\rm {X}}\right>$ ($\sigma_{\rm \left<M_{\rm {L}}/M_{\rm {X}}\right> } \sim15-20\%$) is also similar to those cited in these works ($\sigma_{\rm \left<M_{\rm {X}}/M_{\rm {L}}\right> } \sim3-20\%$).
We remind the reader, however, that even if our results are consistent with previous work, the scaling relations, difference and ratios that we obtain between our lensing masses and those in @gozaliasl2014 depend on the @gozaliasl2014 selection in $L_X$ (when stacking clusters to derive the @leauthaud2010 lensing masses). Our selection based on the @licitra2016a [@licitra2016b] richness and differences in the shear calibration in our data and @leauthaud2010 contribute to this difference; interpreting them precisely implies understanding the degeneracies on each contribution.
It is also known that [*XMM-Newton*]{} and *Chandra* have different instrument calibrations that lead to different temperature estimations, with *Chandra* X-ray temperatures being higher and leading to higher cluster mass estimation [@israel2014; @vonderlinden2014; @schellenberger2015]. Applying the correction from @schellenberger2015, to convert [*XMM-Newton*]{} masses to *Chandra* masses, we find $\left<M_{\rm {L}}/M_{\rm {X}}\right>_{\rm Chandra}=0.99\pm0.17$, using the lensing masses from our [*Final Model*]{}.
SUMMARY AND CONCLUSIONS
=======================
We measure weak lensing galaxy cluster masses for optically detected cluster candidates stacked by richness. We fit the weak lensing mass versus richness relation and compare our findings to X-ray detected mass proxies in the area.
Our cluster sample was obtained with the RedGOLD [@licitra2016a] optical cluster finder algorithm. The algorithm is based on a revised red-sequence technique and searches for passive ETG overdensities. RedGOLD is optimized to detect massive clusters ( $M_{\rm 200}>10^{14}M_{\rm \odot}$) with both high completeness and purity. We use the RedGOLD cluster catalogs from @licitra2016a [@licitra2016b] for the CFHT-LS W1 and NGVS surveys. The catalogs give the detection significance and an optical richness estimate that corresponds to a proxy for the cluster mass.
For our weak lensing analysis, we use a sample of 1323 published clusters, selected with a threshold in significance of $\sigma_{\rm det}\ge4$ and in richness $\lambda\ge10$ at redshift $0.2\le z\le0.5$, for which our published catalogs are $\sim 100\%$ complete and $\sim 80\%$ pure [@licitra2016a]. In order to compare our lensing masses to X-ray mass proxies, we considered both the [*published*]{} and [*complete*]{} Licitra et al.’s catalogs, as defined in Section \[redgold\_cat\].
Our photometric and photometric redshift catalogs were obtained with a modified version of the THELI pipeline [@erben2005; @erben2009; @erben2013; @raichoor2014], and weak lensing shear measurements with the shear measurement pipeline described in @erben2013, @heymans2012, and @miller2013.
We calculate our cluster mean shear radial profiles by averaging the tangential shear in logarithmic radial bins in stacked cluster detections binned by their richness. We apply lens-source pairs weights that depend on the lensing efficiency and on the quality of background galaxy shape measurements.
We obtain the average cluster masses in each richness bin by fitting the measured shear profiles using three models: (1) a basic halo model ([*Basic Model*]{}), with an NFW surface density contrast and correction terms that take into account cluster miscentering, non-weak shear, and the second halo term; (2) a model that includes the intrinsic scatter in the mass–richness relation ([*Added Scatter Model*]{}); and (3) a model that includes the contribution of the BCG stellar mass ([*Two Component Model*]{}). In the [*Basic*]{} and in the [*Two Component Models*]{}, we apply an a posteriori correction to take into account the intrinsic scatter in the mass–richness relation.
We find that our [*Final Model*]{} is the [*Two Component Model*]{}, which, with the inclusion of the a posteriori correction for the intrinsic scatter in the mass–richness relation, is more complete in taking into account the systematics, and more reliable in the obtained results.
Our main results are:
- We test different cluster profile models and fitting techniques. We find that the intrinsic scatter in the mass–richness relation and the BCG mass are not constrained by the data. While the miscentering correction is necessary to avoid a bias in the measured halo masses, the inclusion of the BCG mass does not affect the results.
- Comparing weak lensing masses to RedGOLD optical richness, we calibrate our optical richness with the lensing masses, fitting the power law $\log{M_{\rm 200}}=\log{M_{\rm 0}}+\alpha\log{\lambda/\lambda_{\rm 0}}$. For our [*Final Model*]{}, we obtain $\log{M_{\rm 0}}=14.46\pm0.02$ and $\alpha=1.04\pm0.09$, with a pivot richness $\lambda_{0}=40$.
Even if our sample is one order of magnitude smaller than the SDSS and DES redMaPPer cluster samples used in @simet2016, @farahi2016 and @melchior2016, our results are consistent with theirs within $1-2 \sigma$. This confirms that our cluster selection is not biased toward a different cluster selection when compared to the SDSS and DES redMaPPer cluster samples, as we expect.
- Using our mass–richness relation and X-ray masses from @gozaliasl2014, we infer scaling relations between lensing masses and X-ray proxies.
For the lensing mass versus X-ray luminosity relation $\log{\left(\frac{M_{\rm 200} E(z)}{M_{\rm 0}}\right)} =a+b \log{\left(\frac{L_{\rm X}}{L_{\rm 0} E(z)}\right)}$, we find $a=(0.10\pm0.03)$ and $b=(0.61\pm0.12)$, with $M_{\rm 0}=8 \times 10^{13} h^{-1}M_{\rm \odot}$ and $L_{\rm 0}=5.6\times 10^{42} h^{-2}erg/s$.
For the lensing mass versus X-ray temperature relation $\log{\left(\frac{M_{\rm 200} E(z)}{M_{\rm 0}}\right)}=a+b \log{\left(\frac{T_{\rm X}}{T_{\rm 0}}\right)}$, we obtain $a=(0.23\pm0.03)$ and $b=(1.47\pm0.28)$, with $M_{\rm 0}=6 \times 10^{13} h^{-1}M_{\rm \odot}$ and $T_{\rm 0}=1.5 KeV$.
Our results are consistent with those of @kettula2015 and @mantz2016, within $<1\sigma$. Our normalization is consistent within $<2.5\sigma$, and our slope within $1\sigma$, of the results of @leauthaud2010 (and therefore with @gozaliasl2014). They are also consistent with expected deviations from self-similarity [@bohringer2011].
- We find a scatter of $0.20$ dex, for all three relations, consistent with redMaPPer scatters, confirming the @licitra2016a [@licitra2016b] results that the RedGOLD optical richness is an efficient mass proxy. This is very promising because our mass range is lower than that probed by redMaPPer, and the scatter does not increase as expected to these lower mass ranges.
In order to increase the accuracy of the weak lensing mass estimates, it will be important to increase the number density of background sources to achieve a higher S/N in the shear profile measurements in the future. This will be possible with ground- and space-based large-scale surveys such as the LSST[^8], *Euclid*[^9] and WFIRST[^10]. Also, the next generation radio surveys such as SKA[^11] will allow us to extend weak lensing measurements to the radio band, giving access to even larger scales. Cluster samples will then be an order of magnitude bigger than the one used for this work, allowing us to constrain cluster masses and their scaling relations with even higher accuracy (e.g. @sartoris2016, @ascaso2016).\
This work is based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/IRFU, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This research used the facilities of the Canadian Astronomy Data Centre, operated by the National Research Council of Canada with the support of the Canadian Space Agency. CFHTLenS data processing was made possible thanks to significant computing support from the NSERC Research Tools and Instruments grant program. R.L., S.M., and A.Ra. acknowledge the support of the French Agence Nationale de la Recherche (ANR) under the reference ANR10- BLANC-0506-01-Projet VIRAGE (PI: S.Mei). S.M. acknowledges financial support from the Institut Universitaire de France (IUF), of which she is senior member. H.H. is supported by the DFG Emmy Noether grant Hi 1495/2-1. We thank the Observatory of Paris and the University of Paris D. Diderot for hosting T.E. under their visitor programs.
[*Facilities:*]{} .
JOINT FIT TEST {#appendix}
==============
In order to check that individually fitting the profile of each richness bin does not introduce a bias in the determination of the mass–richness relation parameters, we tested a *joint fit* [e.g. @viola2015; @simet2016]. This method consists of the simultaneous fitting of the profiles associated whit all richness bins. In this case, the fitting parameters will be directly the normalization and slope of the mass–richness relation, and the likelihood of the model will be the sum of the likelihoods of all shear profiles.
Also for the *joint fit*, we changed the free parameters to understand how each free parameter could change the results. We tested different models, each with different free parameters:
Model 1
: has four parameters: $\log M_{0}$, $\alpha$, $p_{\rm cc}$, and $\sigma_{\rm off}$, which are the normalization and slope of the mass–richness relation, and the miscentering parameters. The BCG mass is fixed at $M_{\rm BCG}^{*}$.
Model 2
: has five parameters: $\log M_{0}$, $\alpha$, $p_{\rm cc}$, $\sigma_{\rm off}$, and $\sigma_{M|\lambda}$, which are the parameters of Model 1 with the addition of the intrinsic scatter of the mass–richness relation. The BCG mass is fixed at $M_{\rm BCG}^{*}$.
Model 3
: has five parameters: $\log M_{0}$, $\alpha$, $p_{\rm cc}$, $\sigma_{\rm off}$, and $aMbcg$, which are the parameters of Model 1 with the addition of a constant that multiplies $M_{\rm BCG}^{*}$, so that $M_{\rm BCG}=aMbcg\times M_{\rm BCG}^{*}$.
Model 4
: has five parameters: $\log M_{0}$, $\alpha$, $p_{\rm cc}$, $\sigma_{\rm off}$, and $aCM$, which are the parameters of Model 1 with the addition of the amplitude of the mass-concentration relation used [i.e. @dutton2014]. The BCG mass is fixed at $M_{\rm BCG}^{*}$.
Model 5
: has seven parameters: $\log M_{0}$, $\alpha$, $p_{\rm cc}$, $\sigma_{\rm off}$, $\sigma_{M|\lambda}$, $aMbcg$, and $aCM$.
In Table \[tab:priors\_joint\] we find the priors on the parameters. In Table \[tab:res\_joint\], we find the results of the MCMC for the different models.
We found that the results from the different models are consistent with each other within $<1.5\sigma$; except one, the $\log M_{0}$ obtained with Model 4, which is only consistent with those from Models 1 and 3 within $2.2\sigma$ . The normalization and slope of the mass–richness relation are well-constrained in all models. Here, $aMbcg$ is not constrained, and the inclusion of this parameter in the fit does not affect the other parameters. This result is consistent with what was found in Section \[results\], from the comparison of the [*Basic*]{} and [*Two Component Models*]{}. The amplitude of mass–concentration relation is constrained, but it is slightly degenerate with the miscentering parameters that are less well-constrained in the models that include $aCM$. Moreover, for these models, the normalization and slope of the mass–richness relation have lower values compared to the models without $aCM$. Here, $\sigma_{M|\lambda}$ is constrained but it has a lower value than expected from @licitra2016a [@licitra2016b].
When comparing the results from the *joint fit* to the results from our [*Final Model*]{}, we find consistent results ($<1-2\sigma$), confirming that the two approaches are consistent and equivalent.
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[^1]: http://euclid- ec.org
[^2]: http://www.lsst.org
[^3]: http://www.cfht.hawaii.edu/Instruments/Elixir/
[^4]: http://www4.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/cadc/
[^5]: https://heasarc.gsfc.nasa.gov/W3Browse/all/cfhtlsgxmm.html
[^6]: https://heasarc.gsfc.nasa.gov/W3Browse/all/xcs.html
[^7]: https://github.com/dfm/emcee
[^8]: https://www.lsst.org/
[^9]: http://euclid- ec.org
[^10]: http://wfirst.gsfc.nasa.gov
[^11]: http://www.skatelescope.org
|
---
author:
- 'Anisur Rahaman Molla[^1]'
- 'William K. Moses Jr.[^2]'
bibliography:
- 'references.bib'
title: 'Dispersion of Mobile Robots: The Power of Randomness'
---
Introduction {#sec:intro}
============
Technical Preliminaries {#sec:prelims}
=======================
Local Leader Election {#sec:local-leader-election}
=====================
Dispersion on Rooted Rings {#sec:rooted-ring}
==========================
Dispersion on Rooted Trees {#sec:rooted-tree}
==========================
Dispersion on Rooted Graphs {#sec:rooted-graph}
===========================
Dispersion on Arbitrary Graphs (without Termination) {#sec:arb-graph}
====================================================
Extending Algorithms to Arbitrary $k$ {#sec:arb-k}
=====================================
Lower Bound on Memory {#sec:lower-bounds}
=====================
Conclusion and Future Work {#sec:conc}
==========================
[^1]: Research supported in part by DST Inspire Faculty research grant DST/INSPIRE/04/2015/002801. ORCID ID: 0000-0002-1537-3462
[^2]: Research supported in part by a grant of his postdoctoral fellowship hosts from the Israeli Ministry of Science. ORCID ID: 0000-0002-4533-7593
|
---
abstract: 'We calculate the dependence of the Casimir force on the isotopic composition of the interacting objects. This dependence arises from the subtle influence of the nuclear masses on the electronic properties of the bodies. We discuss the relevance of these results to current experiments utilizing the iso-electronic effect to search at very short separations for new weak forces suggested by various unification theories.'
address:
- '$^{1}$Physics Department, Wabash College, Crawfordsville, IN 47933-0352'
- '$^{2}$Physics Department, Purdue University, West Lafayette, IN 47907-1396'
author:
- 'Dennis E. Krause$^{1,2}$ and Ephraim Fischbach$^{2}$'
title: Isotopic Dependence of the Casimir Force
---
The Casimir effect [@Casimir] has been the subject of intense study recently, both experimentally and theoretically [@Bordag; @Review; @Books; @Lamoreaux; @Mohideen; @experiments; @Harris; @Ederth; @Chan; @Bressi]. Not only is it of interest in its own right, as a novel and fundamental quantum effect, but the increasing precision of experimental tests of the Casimir effect has led to its use in searching for new macroscopic forces acting over sub-millimeter scales [@Bordag; @1998; @Most; @and; @Novello; @2001; @Fischbach; @2001a]. This follows from the fact that the Casimir force becomes the dominant background to any new macroscopic force over short distance scales after electrostatic and magnetic effects have been eliminated. If we consider, for example, the Casimir force per unit area $P_{C}(d)$ between two perfectly conducting parallel plates at temperature $T = 0$ separated by a distance $d$, then [@Casimir] $$P_{C}(d) = -\frac{\pi^{2}\hbar c}{240}\frac{1}{d^{4}}
= -\frac{0.013}{(d/\mbox{$\mu$m})^{2}}\,\mbox{dyn/cm$^{2}$}.
\label{Casimir pressure}$$ It is instructive to compare $P_{C}(d)$ to the gravitational force: For two Cu plates having dimensions 1 cm$^{2} \times 1$ mm a distance $d$ apart, the Casimir force exceeds the Newtonian gravitational force when $d \lesssim 14$ $\mu$m. (This distance becomes slightly larger at room temperature [@Krause].) However, this is the very distance scale over which recent extra-dimensional theories suggest the possibility of new short-range gravitational forces [@Arkani; @Randall; @Randall; @Review]. If follows that detecting such forces requires understanding the (presumably larger) Casimir background.
For Casimir experiments using real (rather than idealized) conductors, the effects of finite conductivity, finite temperature, and surface roughness become important, and considerable progress has been made in recent years in dealing with these effects [@Bordag; @Review; @KlimMost; @2001; @Bezerra; @Lambrecht]. Still, the inherent difficulty in calculating the Casimir force for real materials to high precision limits the use of Casimir force experiments to constrain new forces. The object of the present paper is to implement a recent proposal [@Fischbach; @2001a; @Krause; @Fischbach; @2001b] which aims to sidestep these problems when searching for new very short-ranged macroscopic forces. The approach relies on the fact that the Casimir force depends primarily on the [*electronic*]{} properties of the interacting bodies, while the proposed new forces, including those arising from new spatial dimensions, depend on the test bodies’ [*nuclear*]{}, as well as on their electronic, properties. One should therefore be able to set limits on new forces at sub-micron separations by measuring the [*differences*]{} in forces between test bodies composed of different isotopes of the same element (the iso-electronic effect), since the Casimir force should be independent of isotope to a good approximation. Any observed differences between the test bodies could then be attributed to new physics after other effects (differences in sample preparation, etc.) have been accounted for. Furthermore, this method does not require a detailed calculation of the Casimir force, if this force is known to be the same for the isotopes being compared. However, before one can extract reliable limits on new forces from an experiment based on the iso-electronic effect, one must be confident that any small isotopic dependence of the Casimir force will produce a force difference that is less than the force resolution of the experiment. In what follows we present the first calculation of the isotopic dependence of the Casimir force. This calculation is of interest for two reasons: First, it is directly relevant to an experiment currently underway [@Decca] to search for new short-range forces by comparing the Casimir forces for two isotopes of the same element. Secondly, our results reveal new characteristics of the Casimir force, the dependence on lattice spacing and nuclear masses, which may be utilized in future applications.
To understand how the isotopic dependence of the Casimir force comes about we note that for two infinitely thick parallel plates composed of real dielectrics at $T \neq 0$, the expression for the Casimir force $F_{C}(d,T)$ is more complicated than Eq. (\[Casimir pressure\]), and is given by the Lifshitz formula [@Lifshitz; @1956; @Lifshitz; @1961; @Lifshitz; @1980; @KlimMost; @2001]: $$\begin{aligned}
F_{C}(d,T) & = & -\frac{k_{B}TA}{\pi c^{3}}\sum_{l=0}^{\infty}\!^\prime\,
\xi_{l}^{3}\int^{\infty}_{1}p^{2}dp
\nonumber\\
& &
\times \left\{\left[\left(\frac{K(i\xi_{l},T) + \varepsilon(i\xi_{l},T)p}
{K(i\xi_{l},T) - \varepsilon(i\xi_{l},T)p}\right)^{2}
e^{2d(\xi_{l}/c)p} - 1\right]^{-1}\right.
\nonumber\\
& & \nonumber\\
& & \phantom{sp}\mbox{} + \left.\left[\left(\frac{K(i\xi_{l},T) + p}
{K(i\xi_{l},T) - p}\right)^{2}
e^{2d(\xi_{l}/c)p} - 1\right]^{-1}\right\}.
\label{Lifshitz formula}\end{aligned}$$ Here $A$ is the area of the plates, $k_{B}$ is Boltzmann’s constant, $\omega = i\xi_{l}$, $\xi_{l} = 2\pi k_{B}Tl/\hbar$, and $$K(i\xi_{l},T) = \left[p^{2} - 1 + \varepsilon(i\xi_{l},T)\right]^{1/2},$$ where $\varepsilon(\omega,T)$ is the frequency and temperature dependent dielectric constant. The prime on the summation sign indicates that the $l~=~0$ term should be multiplied by 1/2. The dielectric properties of the interacting media determine the Casimir force acting between real metallic plates. In practice, one obtains $\varepsilon(\omega,T)$ from a Drude or plasma model for low frequencies and from tables of optical data for higher frequencies.
We note from Eq. (\[Lifshitz formula\]) that the Casimir force depends on the temperature $T$, as well as on $d$, and that this $T$-dependence between real metals enters in two different ways [@Bezerra]. First, the quantum electromagnetic field is at temperature $T$ and this contributes a thermal pressure from real photons on the plates which becomes significant for larger plate separations. When $d \gtrsim \hbar c/2k_{B}T$, the energy spacing between the field modes decreases allowing thermal energy to more easily excite higher modes. \[Note that for a room temperature experiment ($T = 300$ K), $\hbar c/2k_{B}T = 3.8\,
\mbox{$\mu$m}$, while for an experiment operating at liquid helium temperatures ($T = 4$ K), $\hbar c/2k_{B}T = 0.29\,$ mm.\] If $d
\ll
\hbar c/2k_{B}T$, there is a sufficiently large gap between the ground state and first excited states to prevent significant thermal excitation of higher modes in which case $F(d,T)$ reduces to [@Lifshitz; @1956; @Lifshitz; @1961; @Lifshitz; @1980; @KlimMost; @2001] $$\begin{aligned}
F_{T}(d) & = & -\frac{\hbar A}{2\pi^{2} c^{3}}
\int^{\infty}_{0}d\xi\, \xi^{3}\int^{\infty}_{1}p^{2}dp
\nonumber\\
& &
\times \left\{\left[\left(\frac{K(i\xi,T) + \varepsilon(i\xi,T)p}
{K(i\xi,T) - \varepsilon(i\xi,T)p}\right)^{2}
e^{2d(\xi/c)p} - 1\right]^{-1}\right.
\nonumber\\
& & \nonumber\\
& & \mbox{} + \left.\left[\left(\frac{K(i\xi,T) + p}
{K(i\xi,T) - p}\right)^{2}
e^{2d(\xi/c)p} - 1\right]^{-1}\right\}.
\label{Lifshitz T = 0 formula}\end{aligned}$$ As was observed recently [@Bezerra], we can see from Eq. (\[Lifshitz T = 0 formula\]), that there remains another temperature dependence to the Casimir force. The dielectric constant $\varepsilon(\omega,T)$ is also temperature dependent as discussed below, even when one can neglect the temperature fluctuations of the field. Therefore, following Ref. [@Bezerra], we define $F(d,T= 0) \equiv F_{T}(d)$, were the subscript denotes this implicit $T$-dependence. Thus, it is important to check that the tabulated data for the dielectric constant are appropriate for the temperature at which the experiment is conducted. This has not been a problem in previous experiments since they have all been conducted at room temperature at which most of the tabulated optical data are obtained. However, these data may be inappropriate for experiments conducted at low temperatures.
For the case of two infinite plates, we see that the isotopic dependence of the Casimir force must enter through $\varepsilon(i\xi,T)$, which depends mainly upon the electronic properties of the material. Optical data for the isotopes of interest, at temperatures relevant for an experiment, are difficult to find in the literature. Furthermore, experience from room temperature Casimir force experiments indicates that, ideally, one should obtain optical data directly from the actual samples used, since there is sufficient variation from sample to sample.
In the absence of relevant experimental data, we can estimate the isotopic dependence of the Casimir force from theoretical considerations. As we have discussed, the Casimir force is determined through the Lifshitz formula by the dielectric constant $\varepsilon(\omega,T)$. For metals which can be described by the plasma model, $\varepsilon(\omega,T)$ is characterized by a single parameter, the plasma frequency $\omega_{p}$ which depends, in turn, on the lattice constant $a$. In the presence of an anharmonic potential, the lattice spacing $a$ will be different for two isotopes of the same element, since the zero point motion of the isotopes at $T = 0$ depends on the respective isotopic masses. Thus, the isotopic dependence of the Casimir force arises from the dependence of the lattice constant on mass, and this dependence affects the dielectric constant, and eventually the Casimir force, through the Lifshitz formula.
To quantify the preceding discussion, we consider a Casimir force experiment utilizing conductors which can be described by the plasma model. Although this is a simple model of metals, it is sufficiently reliable for our present purposes. In the plasma model the dielectric constant $\varepsilon(\omega = i\xi)$ is given by $$\varepsilon(i\xi) = 1 + \frac{\omega_{p}^{2}}{\xi^{2}},
\label{plasma model}$$ where the plasma frequency $\omega_{p}$ is $$\omega_{p}^{2} = \frac{4\pi Ne^{2}}{m_{\rm eff}V}.
\label{plasma frequency}$$ Here $N/V$ is the number of conduction electrons/volume and $m_{\rm eff}$ is the effective electron mass. If Eq. (\[plasma model\]) is substituted into Eq. (\[Lifshitz T = 0 formula\]), one finds in the limits $d \gg 2\pi c/\omega_{p}$ and $T \ll
\hbar c/2k_{B}d$ [@Hargreaves; @Bordag; @Review], $$F(d) \simeq -\frac{\pi^{2}}{240}\frac{\hbar c A}{d^{4}}
\left(1 - \frac{16}{3}\frac{c}{\omega_{p}d}\right).
\label{Plasma Casimir}$$ In the simplest case, let $N_{\rm val}/V$ be the number of [*valence*]{} electrons/volume and let $m_{\rm eff} =
m_{e}$, the free electron mass, in which case, Eq. (\[plasma frequency\]) reduces to $$\omega_{p}^{2} = \frac{4\pi N_{\rm val}e^{2}}{m_{e}V}.
\label{simple plasma frequency}$$ From Eq. (\[simple plasma frequency\]), we see that in this case, all of the isotopic dependence must arise from $V$, the volume per atom, which is proportional to $a^{3}$.
The isotopic dependence of the lattice spacing has been a topic of interest for some time [@London; @Plekhanov]. It is well-known that the temperature dependence of $a$, which leads to thermal expansion of solids, arises from anharmonic terms in the interatomic potential [@Kittel]. For example, in a one-dimensional lattice with a typical interatomic distance given by $x$, let an atom’s potential energy be approximated by $V(u) \simeq (1/2)ku^{2} - (1/6)bu^{3}$, where $u = x - x_{0}$, $x_{0}$ is the equilibrium separation, $k$ is the effective spring constant, and $b$ characterizes the anharmonic contribution. At temperature $T$, the thermal average displacement depends on the anharmonic term so that the lattice constant in this model becomes temperature dependent and proportional to $b$ [@Kittel]: $$a(T)= x_{0} + \langle u\rangle \simeq x_{0} + \frac{b}{2k^{2}}k_{B}T,$$ where $k_{B}$ is Boltzmann’s constant. This temperature dependence of the lattice spacing affects the plasma frequency, which leads to a temperature-dependent dielectric constant $\varepsilon(\omega,T)$ as mentioned earlier. If one replaces thermal vibrations, which allow atoms to sense the anharmoniticity of the potential, with quantum zero-point motion ($kT
\rightarrow \hbar\omega/2$), one finds that the lattice spacing in this model becomes dependent on the atomic mass $M$, since the vibrational frequency $\omega$ is proportional to $1/\sqrt{M}$ [@Ramdas]: $$a(T = 0) \simeq x_{0} + \frac{b}{4k^{2}}\hbar\omega .$$ It follows that the temperature and isotopic dependence of $\varepsilon(\omega,T)$ are linked through the anharmonic term of the interatomic potential. At finite temperatures, both thermal and zero-point motions contribute, although the former dominate at higher temperatures. Hence the isotopic dependence of $a$ is most significant at temperatures much less than the Debye temperature. For a current review of theoretical estimates and experimental results for the isotopic dependence of the lattice constant, see Plekhanov [@Plekhanov]. With the exception of nickel, which is one of the metals being considered for an experiment utilizing the iso-electronic effect [@Fischbach; @2001a], most of this effort has focused on materials which would not be appropriate for Casimir force experiments. Nonetheless, one finds (Table \[isotope table\]) that $\Delta a/a \sim 10^{-4}$ for those elements that have been studied, and this agrees with theoretical estimates.
If we assume that the entire isotopic dependence of the Casimir force is dominated by the isotopic dependence of the lattice constant through Eq. (\[simple plasma frequency\]), and that the Casimir force is given by Eq. (\[Plasma Casimir\]), then we find that a variation of the lattice constant $\Delta
a$ leads to a relative difference in the Casimir force for two different isotopes, $$\frac{\Delta F_{21}}{F} \simeq
\left(\frac{16}{3}\frac{c}{\omega_{p}d}\right)\frac{\Delta\omega_{p}}{\omega_{p}}
=
-\left(\frac{8c}{\omega_{p}d}\right)
\frac{\Delta a_{21}}{a},$$ where $\Delta F_{21} = F_{2} - F_{1}$, $\Delta a_{21} = a_{2} - a_{1}$, and we have used $\Delta\omega_{p}/\omega_{p} = -(1/2) \Delta V/V = -(3/2)\Delta a/a$. Since Eq. (\[Plasma Casimir\]) is valid when $2\pi c/\omega_{p}d
\ll 1$, and experimentally one finds (e.g., for nickel) $\Delta a/a \sim 10^{-4}$, one expects $$\frac{\Delta F_{21}}{F} \ll 10^{-4},$$ under these conditions. This is several orders of magnitude below the current resolution of Casimir force experiments ($\Delta
F/F \sim 10^{-2}$). However, since this problem remains unexplored experimentally, it may be possible to find situations in which the isotopic $\Delta F/F$ is large enough to be observed.
To summarize, we have shown that for metals which can be described by the plasma model, the relative difference in the Casimir force between plates composed of different isotopes separated by $d \gg 2\pi c/\omega_{p}$ is negligible in any current experiment [@Decca] utilizing force differences to extract limits on new forces. For metals which are not well described by the plasma model, and for experiments with shorter separations, further analysis will be needed to ascertain how well our conclusions hold, particularly when more realistic models of the dielectric constants are used for the actual experimental samples. Additionally, other effects of an isotopic mass difference should be explored. These include the isotopic dependence of $m_{\rm eff}$, and the possibility that the dielectric constant can depend on isotopic mass in other ways besides the lattice constant. However, these effects are not likely to alter the principal conclusion of our analysis, that current searches for new short-range forces using the iso-electronic effect [@Decca] can ignore the isotopic dependence of the Casimir force.
The authors thank G. Carugno, R. Decca, A. Lambrecht, D. López, V. M. Mostepanenko, A. W. Overhauser, A. K. Ramdas, S. Reynaud, G. Ruoso, and S. Rodriguez for helpful discussions. This work was supported in part by the U. S. Department of Energy under contract No. DE-AC02-76ER071428.
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Isotopes $\Delta a/a$ Reference
------------------------------ --------------------------------------- -----------------
$^{58}$Ni, $^{64}$Ni $1.4 \times 10^{-4}$ \[$T$ = 78 K\] [@Kogan; @1962]
$5.7 \times 10^{-5}$ \[$T$ = 300 K\] [@Kogan; @1962]
$^{12}$C, $^{13}$C (Diamond) $-1.5 \times 10^{-4}$ \[$T$ = 298 K\] [@Holloway]
$^{6}$Li, $^{7}$Li $-2 \times 10^{-4}$ \[$T$ = 293 K\] [@Covington]
$^{20}$Ne, $^{22}$Ne $-1.9 \times 10^{-3}$ \[$T$ = 3 K\] [@Batchelder]
$-1.6 \times 10^{-3}$ \[$T$ = 24 K\] [@Batchelder]
$^{70}$Ge, $^{76}$Ge $-5.3 \times 10^{-5}$ \[$T$ = 30 K\] [@Sozontov]
$-2.2 \times 10^{-5}$ \[$T$ = 300 K\] [@Sozontov]
: Experimental values of $\Delta a/a$ for several isotopes.[]{data-label="isotope table"}
|
---
abstract: 'Weak gravitational lensing changes the angular power spectra of the cosmic microwave background (CMB) temperature and polarization in a characteristic way containing valuable information for cosmological parameter estimation and weak lensing reconstructions. So far, analytical expressions for the lensed CMB power spectra assume the probability density function (PDF) of the lensing excursion angle to be Gaussian. However, coherent light deflection by nonlinear structures at low redshifts causes deviations from a pure Gaussian PDF. Working in the flat-sky limit we develop a method for computing the lensed CMB power spectra which takes these non-Gaussian features into account. Our method does not assume any specific PDF but uses instead an expansion of the characteristic function of the lensing excursion angle into its moments. Measuring these in the CMB lensing deflection field obtained from the Millennium Simulation we show that the change in the lensed power spectra is only at the 0.1%–0.4% level on very small scales $(\Delta \theta \lesssim 4'', \, l\gtrsim 2500)$ and demonstrate that the assumption of a Gaussian lensing excursion angle PDF is well applicable.'
author:
- |
Philipp M. Merkel[^1] and Bj[ö]{}rn Malte Schäfer\
Institut f[ü]{}r Theoretische Astrophysik, Zentrum f[ü]{}r Astronomie, Universit[ä]{}t Heidelberg, Albert-Ueberle-Stra[ß]{}e 2, 69120 Heidelberg, Germany\
Astronomisches Recheninstitut, Zentrum f[ü]{}r Astronomie, Universit[ä]{}t Heidelberg, M[ö]{}nchhofstra[ß]{}e 12, 69120 Heidelberg, Germany
bibliography:
- 'bibtex/aamnem.bib'
- 'bibtex/references.bib'
title: Gravitational lensing of the cosmic microwave background by nonlinear structures
---
\[firstpage\]
cosmology: large-scale structure, gravitational lensing, methods: analytical, numerical
Introduction
============
Cosmological parameter analysis as well as investigation of inflationary models require a precise knowledge of the fluctuations of the cosmic microwave background. Therefore supreme technical efforts are made to equip current and future CMB experiments with instruments with ever greater resolution including sensitivity on CMB polarization. In order to interpret these low-noise data correctly a profound understanding of the physics of the primary CMB is not longer sufficient but requires an equivally good understanding of CMB foregrounds.
One of the most important of these is weak gravitational lensing: on their way from the last-scattering-surface to today’s observer CMB photons are deflected by the intervening large-scale structure. Although each single deflection is small, their cumulative effect changes the statistics of the CMB fluctuations and accordingly their power spectra observed on today’s sky [see @2001PhR...340..291B; @2006PhR...429....1L]. At present, the lensing effect has been detected in the temperature data of the WMAP satellite at moderate significance [@2007PhRvD..76d3510S; @2008PhRvD..78d3520H]. Apart from being a CMB foreground, the lensing pattern itself is a valuable source of cosmological information [@2000PhRvD..62d3517G; @2002ApJ...574..566H; @2003PhRvD..67d3001H; @2003PhRvD..67h3002O; @2004NewA....9..687A; @2006PhRvD..73d5021L; @2006JCAP...10..013P; @2009PhRvD..79f5033D].
@1996ApJ...463....1S [@1998PhRvD..58b3003Z; @2005PhRvD..71j3010C] developed efficient and accurate methods for computing the lensed CMB power spectra starting from the corresponding correlation functions. These methods all work under the assumption of a Gaussian distributed lensing excursion angle, which measures the difference between the deflection of two nearby light rays. Since the deflection angle is given by the gradient of the lensing potential one commonly accounts for the impact of nonlinear structures on the lensed CMB spectra semi-analytically by applying nonlinear but Gaussian corrections from the HALOFIT model of @2003MNRAS.341.1311S to the lensing potential power spectrum. This approach takes the additional power on small scales due to nonlinear evolution into account but ignores that coherent deflection by nonlinear structures at low redshifts alters the Gaussian character of the lensing excursion angle distribution. In this work we investigate how exactly this non-Gaussianity influences the lensed CMB power spectra.
This paper is structured as follows. In Section \[sec:formalism\] we develop the formalism for computing the lensed CMB temperature power spectrum in case of a non-Gaussian lensing excursion angle PDF. We relegate the discussion of the power spectra involving CMB polarization to Appendix \[sec:polarization\]. Section \[sec:numerics\] is devoted to our numerical results. Here we quantify the non-Gaussianity present in the lensing excursion angle PDF which we compute from the CMB deflection field of the Millennium Simulation (MS). With this deflection field we then derive the lensed CMB power spectra and compare our results to power spectra resulting from using a pure Gaussian lensing excursion angle PDF. In Section \[sect\_summary\] we summarize our results and give an outlook on future investigations. Finally, in Appendix \[sec:explicit\_expressions\] we give explicit expressions for the lensed CMB correlation functions used in our numerical computations.
Throughout this work we assume a spatially flat $\Lambda\textnormal{CDM}$ cosmology with adiabatic Gaussian initial perturbations. The relevant parameter values are: $\Omega_m=0.25$, $\Omega_\Lambda=0.75$, $H_0 = 73 \, \textnormal{km} \, \textnormal{s}^{-1} \, \textnormal{Mpc}^{-1}$, $\Omega_b=0.045$, $n_s=1.0$, $\sigma_8 = 0.9$ and $r = 0.0$ (no primordial gravitational waves present). These parameters are equal to those of the Millennium Simulation [@2006Natur.440.1137S].
Formalism {#sec:formalism}
=========
Lensed CMB temperature power spectrum {#subsec:lensed_CMB_temp_power_spectrum}
-------------------------------------
Working in the flat-sky limit the 2D lensed temperature field is given by the remapping $$\tilde{\Theta} ( \mathbf x ) = \Theta ( \mathbf x + \balpha (\mathbf x ) )$$ mediated by the lensing deflection angle $\balpha$, i.e. the gradient of the lensing potential $\psi$: $\balpha = \nabla \psi$. In the approximation of instantaneous recombination the CMB can be described by a single source plane at conformal distance $\chi = \chi^*$. In the absence of anisotropic stress the lensing potential is then given by the line of sight projection of the physical peculiar gravitational potential $\phi$: $$\psi \left(\mathbf{\hat{n}}\right) = \frac{2}{c^2} \int\limits_{0}^{\chi^*} {\mathrm{d}}\chi \;
\frac{\chi^* - \chi}{\chi^*\chi} \phi \left( \chi \mathbf{\hat{n}} ; \chi \right) ,$$ where the underlying geometry is flat [@2001PhR...340..291B; @2006PhR...429....1L]. Introducing the Fourier transform of the temperature field via $$\Theta(\mathbf x ) = \int \frac{{\mathrm{d}}^2 l}{2\pi} \Theta (\mathbf l ) e^{\mathrm i \mathbf l \cdot \mathbf x},$$ the spectrum for a statistically homogeneous and isotropic field reads $$\left\langle \Theta (\mathbf l) \Theta^* ( \mathbf l' ) \right\rangle = C_l^{{\Theta\Theta}} \delta ( \mathbf l - \mathbf l' ).$$ Then, ignoring the weak large-scale correlation between CMB and lensing potential due to the integrated Sachs-Wolfe effect, the lensed correlation function of the CMB temperature fluctuations is given by $$\tilde{\xi} ( r ) = \left\langle \tilde{\Theta} ( \mathbf x ) \tilde{\Theta} ( \mathbf x' ) \right\rangle
= \int \frac{{\mathrm{d}}^2 l}{(2\pi)^2} C_l^{{\Theta\Theta}}
e^{\mathrm i \mathbf l \cdot \mathbf x}
\left\langle e^{\mathrm i \mathbf l \cdot \left[ \balpha (\mathbf x) - \balpha (\mathbf x')\right]} \right\rangle
\label{eq:lensed_correlation_function}$$ where $r = \left|\mathbf x - \mathbf x'\right|$. It is worth noting that the lensed correlation function only depends on the *relative* displacement, the so-called lensing excursion angle, $\bdelta\balpha(\mathbf r) \equiv \balpha (\mathbf x) - \balpha (\mathbf x')$ and that this dependence is given by the *characteristic function* of the lensing excursion angle: $$\varphi_{\bdelta\balpha} (\mathbf l ) \equiv \left\langle e^{\mathrm i \mathbf l \cdot \mathbf \bdelta \balpha} \right\rangle =
\int {\mathrm{d}}\left(\bdelta\balpha\right) \; p\left(\bdelta\balpha\right) e^{\mathrm i \mathbf l \cdot \mathbf \bdelta \balpha}.$$ The last equality reveals that the characteristic function is the Fourier transform of the PDF $p\left(\bdelta \balpha\right)$. Hence, it carries the same information as the PDF itself. From equation [(\[eq:lensed\_correlation\_function\])]{} the lensed power spectrum is readily obtained by $$\tilde C_l^{{\Theta\Theta}} = 2\pi \int {\mathrm{d}}r \; r \tilde \xi (r) J_0(lr)$$ where $J_n(z)$ denotes the $n$-th order Bessel function [@1965hmfw.book.....A].
Lensing excursion angle {#subsec:lensing_ex_angle}
-----------------------
In linear theory the lensing potential is a Gaussian field and so is its gradient, the lensing deflection angle. Accordingly, the lensing excursion angle is a Gaussian variate and therefore the characteristic function of the lensing excursion angle is just given in terms of the variance $$\left\langle e^{\mathrm i \mathbf l \cdot \left[ \balpha (\mathbf x) - \balpha (\mathbf x')\right]} \right\rangle
= \exp \left( -\frac{1}{2}\left\langle \left[ \mathbf l \cdot \bdelta\balpha \right]^2\right\rangle \right)
= \exp \left( -\frac{1}{2} l^2 \left[\sigma^2(r) + \cos2(\phi_l - \phi_r) {C_{\mathrm{gl,2}}}(r) \right] \right)
\label{eq:gaussian_char_fct}$$ where $\phi_{\mathbf l, \mathbf r}$ denotes the angle between $\mathbf l$, $\mathbf r$ and the $x$-axis. We have defined $\sigma^2(r) = \frac{1}{2}\bigl\langle \bdelta\balpha^2 \bigr\rangle = \frac{1}{2}\left({C_{\mathrm{gl}}}(0) - {C_{\mathrm{gl}}}(r)\right)$. ${C_{\mathrm{gl}}}$ and ${C_{\mathrm{gl,2}}}$ are given in terms of the power spectrum of the lensing potential $C^{\psi\psi}_l$ [@2005PhRvD..71j3010C]: $${C_{\mathrm{gl}}}(r) = \frac{1}{2\pi} \int {\mathrm{d}}l \; l^3 C_l^{\psi\psi} J_0(lr) \qquad \textnormal{and} \qquad {C_{\mathrm{gl,2}}}(r) = \frac{1}{2\pi} \int {\mathrm{d}}l \; l^3 C_l^{\psi\psi} J_2(lr).
\label{eq:def_of_cgl_and_cgl2}$$
Inserting equation [(\[eq:def\_of\_cgl\_and\_cgl2\])]{} into equation [(\[eq:lensed\_correlation\_function\])]{} and performing a perturbative expansion in ${C_{\mathrm{gl,2}}}$ up to second order one recovers the expressions derived by @2005PhRvD..71j3010C. However, @2001MNRAS.327..169H and @2005MNRAS.356..829H showed in numerical weak lensing ray-tracing experiments that the lensing excursion angle is not Gaussian distributed. Its PDF has indeed a Gaussian core but also exponential wings. These wings are a consequence of coherent scattering by individual massive haloes with mass larger than $10^{14}M_{\sun}/h$. Since coherent deflection demands a sufficiently small intrinsic separation of the light rays the exponential wings appear prominent in the excursion angle PDFs obtained from light rays with intrinsic separation of a few arcminutes and are negligible in those of separations larger than one degree [@2005MNRAS.356..829H]. The contributions from coherent scattering broaden the PDFs, i.e. larger excursion angles are more probable than in case of a pure Gaussian PDF. For example, for intrinsic separations smaller than two arcminutes an excursion angle of one arcminute is about ten times more probable than for a Gaussian PDF.
Non-Gaussian probability density function {#subsec:non_gaussian_pdf}
-----------------------------------------
In order to investigate how the non-Gaussian features of the lensing excursion angle PDF affects the lensed CMB temperature spectrum one has to use (in principle) all moments of the lensing excursion angle for computing the lensed correlation function (\[eq:lensed\_correlation\_function\]), which is now denoted with a hat to distinguish it from the correlation function derived under the Gaussian assumption (denoted with a tilde) $$\hat\xi(r) = \left\langle \tilde \Theta (\mathbf x ) \tilde \Theta(\mathbf x') \right\rangle
= \int \frac{{\mathrm{d}}^2l}{(2\pi)^2} C_l^{{\Theta\Theta}} e^{\mathrm i \mathbf l \cdot \mathbf r}
\left\langle \sum_{n=0}^{\infty} \frac{\left( \mathrm i \mathbf l \cdot \bdelta \balpha\right)^n}{n!}\right\rangle
= \int \frac{{\mathrm{d}}^2l}{(2\pi)^2} C_l^{{\Theta\Theta}} e^{\mathrm i \mathbf l \cdot \mathbf r}
\sum_{n=0}^{\infty} \frac{\mathrm i^n}{n!} \sum_{k=0}^{n} {n \choose k} l_x^kl_y^{n-k}
\left\langle \delta\alpha_x^k \delta\alpha_y^{n-k}\right\rangle.$$ For the last equality we used the binomial law $$\left( \mathbf l \cdot \bdelta\balpha \right)^n = \left( l_x\delta\alpha_x + l_y \delta\alpha_y \right)^n
= \sum_{k=0}^{n} {n \choose k} \bigl(l_x \delta\alpha_x\bigr)^k \bigl(l_y\delta\alpha_y\bigr)^{n-k}
\quad \textnormal{with} \quad {n \choose k} = \frac{n!}{k!\, (n-k)!}.$$ Following @1997PhRvD..55.7368K one should choose the local coordinate system, in which to define the correlation function, aligned with the great circle connecting the two points where the temperature fluctuations are measured. In the flat-sky limit this choice of coordinate system corresponds to evaluating the two-point correlator at the origin of the flat coordinate system and at a point on the $x$-axis at distance $r$. One of the great advantages of this coordinate system is that here the correlation tensor of the lensing excursion angle is diagonal, i.e $\left\langle \alpha_i (\mathbf x) \alpha_{j}(\mathbf x') \right\rangle \propto \delta_{ij}$. In case of a Gaussian distribution it then follows immediately by virtue of Wick’s theorem that different moments of different components are statistically independent. In Section \[subsec:corr\_coeff\] we will show that it is reasonable to assign this property also to the (non-Gaussian) PDF of the lensing excursion angle. Hence, introducing polar coordinates $\mathbf l = (l\cos\phi, l\sin\phi)$, the lensed correlation function reads $$\hat\xi(r) = \frac{1}{(2\pi)^2} \int \int {\mathrm{d}}\phi \: l{\mathrm{d}}l \; C^{{\Theta\Theta}}_l e^{\mathrm i l r \cos \phi}
\sum_{n=0}^{\infty} \frac{\mathrm i^n}{n!} \sum_{k=0}^{n} {n \choose k} l^n\cos^k\phi \sin^{n-k}\phi
\left\langle \delta\alpha_x^k \right\rangle \left\langle \delta\alpha_y^{n-k}\right\rangle.$$ A further simplification can be obtained by demanding that components of the lensing excursion angle are distributed symmetrically about zero. This assumption is very natural since otherwise there would be a preferred direction along the coordinate axes. For a symmetric PDF all odd moments vanish, hence, $$\begin{aligned}
\hat\xi(r)
& = &
\frac{1}{(2\pi)^2} \int \int {\mathrm{d}}\phi \: l{\mathrm{d}}l \; C^{{\Theta\Theta}}_l e^{\mathrm i l r \cos \phi}
\sum_{n=0}^\infty \sum_{k=0}^{n} \sum_{q=0}^{n-k} \frac{(-1)^{n+q}}{(2n)!} {2n \choose 2k} {n-k \choose q}l^{2n} \cos^{2(k+q)} \phi
\left\langle \delta\alpha_x^{2k} \right\rangle \left\langle \delta\alpha_y^{2(n-k)}\right\rangle
\nonumber\\
& = &
\frac{1}{2\pi} \int l{\mathrm{d}}l \; C^{{\Theta\Theta}}_l
\sum_{n=0}^\infty \sum_{k=0}^{n} \sum_{q=0}^{n-k} \sum_{r=0}^{2(k+q)}
\frac{(-1)^{n+k+r}}{(2n)!4^{k+q}} {2n \choose 2k} {n-k \choose q} {2(k+q) \choose r}
l^{2n} J_{2(k+q-r)}(lr)
\left\langle \delta\alpha_x^{2k} \right\rangle \left\langle \delta\alpha_y^{2(n-k)}\right\rangle.
\label{eq:lensed_corr_fct_non_linear}\end{aligned}$$ To get the second line we used the fact that $$\int {\mathrm{d}}\phi \; e^{\mathrm i z \cos \phi} \cos^n \phi = 2\pi (-\mathrm i )^n \frac{{\mathrm{d}}^n}{{\mathrm{d}}z^n} \: J_0(z)$$ and that $$\frac{{\mathrm{d}}^n}{{\mathrm{d}}z^n} \: J_0 (z) = \frac{1}{2^n} \sum_{k=0}^n(-1)^k { n \choose k} J_{-n+2k}(z).$$ The expression for the lensed CMB temperature correlation function given in equation [(\[eq:lensed\_corr\_fct\_non\_linear\])]{} is exact for any PDF of the lensing excursion angle that is symmetric about zero and whose moments of different components are uncorrelated, which we have shown to be valid in the deflection field obtained from the Millennium Simulation (cf. Section \[subsec:moments\]).
Numerics {#sec:numerics}
========
Truncation {#subsec:truncation}
----------
For explicitly computing the lensed CMB temperature power spectrum via equation [(\[eq:lensed\_corr\_fct\_non\_linear\])]{} one has to truncate the expansion of the characteristic function at a certain order $n$. To find a reasonable value for $n$ we resorted to the approximation of a purely Gaussian distributed lensing excursion angle and verified the performance of the series expansion in comparison to the lensing method of @2005PhRvD..71j3010C described in Section \[subsec:lensing\_ex\_angle\] and numerically implemented in CAMB[^2]. Successively increasing the order $n$ taken into account in the series expansion we determined that $n$ for which both lensing methods work equivally well. We confirmed that in the case of a pure Gaussian lensing excursion angle PDF for $n=3$ almost perfect agreement between both lensing methods can be achieved. The deviations are largest on very small scales but do not exceed $\mathcal O ( 10^{-4})$. Thus, for the numerical implementation of our lensing method we truncated the series expansion in equation [(\[eq:lensed\_corr\_fct\_non\_linear\])]{} at $n=3$, i.e. we included the sixth moments in the series expansion. Explicit expressions for $n=3$ are given in Appendix \[sec:explicit\_expressions\].
Moments {#subsec:moments}
-------
Aiming at the influence of the non-Gaussian features in the lensing excursion angle PDF on the lensed CMB power spectra we cannot compute the moments needed for its calculation analytically using the power spectrum approach developed by @1994ApJ...436..509S [@1996ApJ...463....1S] (cf. Section \[subsec:lensing\_ex\_angle\]). This approach is well suited for describing gravitational light deflection by the intervening large scale structure but since it is based on linear perturbation theory it does not account for coherent lensing scatter by individual massive haloes, which gives rise to the exponential wings in the lensing excursion angle PDF (cf. Section \[subsec:lensing\_ex\_angle\]). Therefore, we either have to resort to numerical simulations or to use the halo model of large scale structure reviewed by @2002PhR...372....1C.
In this work the moments needed for the computation of the lensed CMB power spectra are obtained from the lensing deflection field constructed by @2008MNRAS.388.1618C which is based on the Millennium Simulation (MS) [@2006Natur.440.1137S]. Being an all-sky map the deflection field is given as angular gradient of the lensing potential $$\balpha(\mathbf{\hat{n}}) = \left( \mathbf{\hat{e}}_\theta \frac{\partial}{\partial \theta}
+ \mathbf{\hat{e}}_\phi \frac{1}{\sin\phi}\frac{\partial}{\partial \phi} \right) \psi(\mathbf{\hat{n}})
= \alpha_\theta (\mathbf{\hat{n}}) \mathbf{\hat{e}}_\theta + \alpha_\phi(\mathbf{\hat{n}})\mathbf{\hat{e}}_\phi.
\label{eq:spherical_gradient_of_lensing_potential}$$ Since the non-Gaussian features of the lensing excursion angle PDF are only substantial for intrinsic light ray separations smaller than one degree (cf. Section \[subsec:lensing\_ex\_angle\]), i.e on scales where the curvature of the sky is negligible, we approximate $$\bdelta \balpha(r) = \balpha(\mathbf x) - \balpha (\mathbf x') \approx \bar{\alpha} (\mathbf{\hat{n}}) - \bar{\alpha} (\mathbf{\hat{n}}') = \bdelta \bar{\balpha} (\beta)
\quad \textnormal{with} \quad r \approx \beta \quad \textnormal{and} \quad \mathbf{\hat{n}} \cdot \mathbf{\hat{n}}' = \cos \beta,
\label{eq:approx_flat_full_sky}$$ where the bars indicate the basis defined by the geodesic connecting $\mathbf{\hat{n}}$ and $\mathbf{\hat{n}}'$. The error of this approximation is slightly increasing with the light ray separation and finally reaches $\sim 2\%$ for two rays intrinsically separated by one degree.
Correlation coefficients {#subsec:corr_coeff}
------------------------
The expression for the lensed correlation function [(\[eq:lensed\_corr\_fct\_non\_linear\])]{} assumes that different moments of different components of the excursion angle are statistically independent. To show that this is indeed the case we compute the correlation coefficient defined by $$\rho(X,Y) = \frac{\left\langle (X-\left\langle X \right\rangle )(Y - \left\langle Y \right\rangle ) \right\rangle }
{\sqrt{\left\langle (X - \left\langle X \right\rangle )^2 \right\rangle } \sqrt{\left\langle (Y - \left\langle Y \right\rangle )^2 \right\rangle }},
\qquad X,Y \; \in \left\lbrace \left. \delta\alpha_{\theta,\phi}^n \right|n = 1,...,5 \right\rbrace$$ for all relevant combinations of $X$ and $Y$. All $\rho(X,Y)$ are compatible with zero at the level of $10^{-3}$, justifying the assumption of statistical independence.
Non-Gaussianity {#subsec:non_gaussianity}
---------------
Equation [(\[eq:lensed\_corr\_fct\_non\_linear\])]{} reveals that all information about the non-Gaussianity of the lensing excursion angle PDF is carried by its moments. It is therefore natural to quantify the amount of non-Gaussianity of the excursion angle PDF via comparing its moments with those of a Gaussian PDF. The moments of the latter can all be expressed in terms of the variance: $$\left\langle X^{2n} \right\rangle = (2n-1)!!\left\langle X^2 \right\rangle^n \quad \textnormal{with} \quad (2n-1)!! = (2n-1) \cdot (2n-2) \cdot ... \cdot 5 \cdot 3 \cdot 1.$$ Thus $$\kappa \equiv \frac{\left\langle X^4 \right\rangle}{\left\langle X^2 \right\rangle^2} \qquad \textnormal{and} \qquad
\eta \equiv \frac{\left\langle X^6 \right\rangle }{\left\langle X^2 \right\rangle^3}
\label{eq:kappa_eta}$$ or rather their deviation from 3 and 15 carry information about the amount of non-Gaussianity present in the fourth and sixth moment of any symmetric PDF. Consequently, in order to ensure to capture only the effects on the lensed power spectrum which arise from the non-Gaussianity of the lensing excursion angle PDF it is recommended not to use directly the moments of the excursion angle components but to use instead $\kappa_{\theta,\phi}$ and $\eta_{\theta,\phi}$ defined by $$\kappa_{\theta,\phi}(\beta) \equiv \frac{\left\langle \delta\bar\alpha_{\theta,\phi}^4 (\beta)\right\rangle}{\left\langle \delta\bar\alpha_{\theta,\phi}^2(\beta) \right\rangle^2}
\qquad \textnormal{and} \qquad
\eta_{\theta,\phi}(\beta) \equiv \frac{\left\langle \delta\bar\alpha_{\theta,\phi}^6(\beta) \right\rangle }{\left\langle \delta\bar\alpha_{\theta,\phi}^2(\beta) \right\rangle^3}
\label{eq:kappa_eta_for_l_e_a}$$ in analogy to equation [(\[eq:kappa\_eta\])]{}. Thus we used $$\left\langle \delta{\alpha}_{x}^2(r) \right\rangle = \sigma^2 (r) + {C_{\mathrm{gl,2}}}(r), \quad
\left\langle \delta{\alpha}_{x}^4(r) \right\rangle = \kappa_{\theta}(r) \left\langle \delta{\alpha}_{x}^2(r) \right\rangle^2 \quad \textnormal{and} \quad
\left\langle \delta{\alpha}_{x}^6(r) \right\rangle = \eta_{\theta}(r) \left\langle \delta{\alpha}_{x}^2(r) \right\rangle^3
\label{eq:actually_used_moments_x}$$ and $$\left\langle \delta{\alpha}_{y}^2(r) \right\rangle = \sigma^2 (r) - {C_{\mathrm{gl,2}}}(r), \quad
\left\langle \delta{\alpha}_{y}^4(r) \right\rangle = \kappa_{\phi}(r) \left\langle \delta{\alpha}_{y}^2(r) \right\rangle^2
\quad \textnormal{and} \quad
\left\langle \delta{\alpha}_{y}^6(r) \right\rangle = \eta_{\phi}(r) \left\langle \delta{\alpha}_{y}^2(r) \right\rangle^3,
\label{eq:actually_used_moments_y}$$ respectively, for our actual computation of the lensed correlation function accounting for the excess of small scale power of the lensing potential due to nonlinear growth via semianalytical corrections from the HALOFIT model [cf. @2008MNRAS.388.1618C]. For computing $\kappa_{\theta,\phi}$ and $\eta_{\theta,\phi}$ from the CMB deflection field of the MS we sampled pairs of pixels with fixed angular separation and computed their differences in the local coordinate system defined by the connecting geodesic. The rotation angle needed for the transion from the coordinate system used by the MS (cf. equation \[eq:spherical\_gradient\_of\_lensing\_potential\]) to the geodesic basis can be readily obtained from identities of spherical triangles. From these samples we estimated the first three even moments of the lensing excursion angle components and computed $\kappa_{\theta,\phi}$ and $\eta_{\theta,\phi}$. They are shown in Figure \[fig:kurtosen\]. Note that the sampling errors are negligible and are not shown. The non-Gaussian features are more pronounced in the distribution of the $\phi$-component of the lensing excursion angle. The deviations from the Gaussian expectation are larger in case of $\eta_{\theta,\phi}$ than for $\kappa_{\theta,\phi}$ as expected since the PDFs are broadened due the coherent scattering by nonlinear structures (cf. Section \[subsec:lensing\_ex\_angle\]). Furthermore Figure \[fig:kurtosen\] reveals that there is still a small amount of non-Gaussianity for light ray separations larger than one degree. The Gaussian expectation is not reached until $\beta \sim $ 20. For such large light ray separations, however, the approximation $ \bdelta\bar{\balpha}(\beta) \approx \bdelta\balpha(r)$ given in equation [(\[eq:approx\_flat\_full\_sky\])]{} is not longer valid. Since the error from extending this approximation up to separations larger than one degree would fairly exceed the one resulting from neglecting the small amount of non-Gaussianity present for [$\beta > $ 1]{} the latter is omitted in the remainder of this work.
![Quantifying the non-Gaussianity of the PDFs of the lensing excursion angle components, obtained from the MS deflection field, by the deviation of the fourth (left panel) and sixth (right panel) moment, respectively, from the Gaussian case, i.e. $\kappa_{\theta,\phi} \equiv \left\langle \delta\bar\alpha_{\theta,\phi}^4 \right\rangle/\left\langle \delta\bar\alpha_{\theta,\phi}^2 \right\rangle^2$ and $\eta_{\theta,\phi} \equiv \left\langle \delta\bar\alpha_{\theta,\phi}^6 \right\rangle / \left\langle \delta\bar\alpha_{\theta,\phi}^2 \right\rangle^3$. The deviations from the Gaussian expectation ($\kappa_{\theta,\phi}\equiv3$, $\eta_{\theta,\phi}\equiv15$) decrease rapidly with increasing intrinsic light ray separation $\beta$. For separations larger than one degree there are only small amounts of non-Gaussianity left.[]{data-label="fig:kurtosen"}](kurtosen.eps "fig:"){height="6.0cm" width="8.6cm"} ![Quantifying the non-Gaussianity of the PDFs of the lensing excursion angle components, obtained from the MS deflection field, by the deviation of the fourth (left panel) and sixth (right panel) moment, respectively, from the Gaussian case, i.e. $\kappa_{\theta,\phi} \equiv \left\langle \delta\bar\alpha_{\theta,\phi}^4 \right\rangle/\left\langle \delta\bar\alpha_{\theta,\phi}^2 \right\rangle^2$ and $\eta_{\theta,\phi} \equiv \left\langle \delta\bar\alpha_{\theta,\phi}^6 \right\rangle / \left\langle \delta\bar\alpha_{\theta,\phi}^2 \right\rangle^3$. The deviations from the Gaussian expectation ($\kappa_{\theta,\phi}\equiv3$, $\eta_{\theta,\phi}\equiv15$) decrease rapidly with increasing intrinsic light ray separation $\beta$. For separations larger than one degree there are only small amounts of non-Gaussianity left.[]{data-label="fig:kurtosen"}](hexatosen.eps "fig:"){height="6.0cm" width="8.6cm"}
Results {#subsec:results}
-------
As seen in Section \[subsec:moments\], for intrinsic light ray separations up to one degree the lensing excursion angle can be well approximated by the difference of two deflection angles defined in the geodesic basis. In the regime of larger separations the non-Gaussianity of the lensing excursion angle PDFs is weak and we confirmed that in this regime the Gaussian approximation is very good. Thus, it is natural to establish the following computation scheme for the lensed correlation function: For intrinsic light ray separations less than one degree one uses formula [(\[eq:lensed\_corr\_fct\_non\_linear\])]{} together with the moments computed via equations [(\[eq:actually\_used\_moments\_x\])]{} and [(\[eq:actually\_used\_moments\_y\])]{}. For larger separations one uses equation [(\[eq:lensed\_correlation\_function\])]{} together with equation [(\[eq:gaussian\_char\_fct\])]{}, which is valid for Gaussian distributed excursion angles. The same computation scheme can be applied to the lensed CMB polarization power spectra by using the corresponding equations given in the appendix.
The influence of a non-Gaussian PDF on the lensed CMB power spectra can now be quantified by computing two sets of power spectra $\hat{C}_l^{XY}$ and $\hat{C}_{l,\mathrm{ref}}^{XY}$. For both sets the computation scheme for the lensed correlation functions described above is used but in case of the reference spectra we use Gaussian moments on all scales, i.e. setting $\kappa_{\theta,\phi}(r) \equiv 3$ and $\eta_{\theta,\phi}(r) \equiv 15$. Figure \[fig:lensed\_power\_spectra\] shows both the lensed and reference CMB power spectra. In the lower half of each panel the ratio between the corresponding spectra is depicted.
![ Lensed CMB power spectra computed via the method described in the text accounting for the non-Gaussian features of the lensing excursion angle PDF discussed in Section \[subsec:lensing\_ex\_angle\] (blue dashed curves). The reference power spectra (solid red curves) were computed by the same method but assume a Gaussian PDF of the lensing excursion angle. The ratio of the corresponding spectra is shown in the lower part of each panel (solid green lines). []{data-label="fig:lensed_power_spectra"}](power_spec_two_methods_TT.eps "fig:"){height="5.5cm" width="8.6cm"} ![ Lensed CMB power spectra computed via the method described in the text accounting for the non-Gaussian features of the lensing excursion angle PDF discussed in Section \[subsec:lensing\_ex\_angle\] (blue dashed curves). The reference power spectra (solid red curves) were computed by the same method but assume a Gaussian PDF of the lensing excursion angle. The ratio of the corresponding spectra is shown in the lower part of each panel (solid green lines). []{data-label="fig:lensed_power_spectra"}](power_spec_two_methods_EE.eps "fig:"){height="5.5cm" width="8.6cm"}\
![ Lensed CMB power spectra computed via the method described in the text accounting for the non-Gaussian features of the lensing excursion angle PDF discussed in Section \[subsec:lensing\_ex\_angle\] (blue dashed curves). The reference power spectra (solid red curves) were computed by the same method but assume a Gaussian PDF of the lensing excursion angle. The ratio of the corresponding spectra is shown in the lower part of each panel (solid green lines). []{data-label="fig:lensed_power_spectra"}](power_spec_two_methods_TE.eps "fig:"){height="5.5cm" width="8.6cm"} ![ Lensed CMB power spectra computed via the method described in the text accounting for the non-Gaussian features of the lensing excursion angle PDF discussed in Section \[subsec:lensing\_ex\_angle\] (blue dashed curves). The reference power spectra (solid red curves) were computed by the same method but assume a Gaussian PDF of the lensing excursion angle. The ratio of the corresponding spectra is shown in the lower part of each panel (solid green lines). []{data-label="fig:lensed_power_spectra"}](power_spec_two_methods_BB.eps "fig:"){height="5.5cm" width="8.6cm"}
These ratios show that the influence of the non-Gaussianity in the lensing excursion angle distribution function is marginal. The differences caused by the exponential wings of the PDF are just several per mile on very small scales $(\Delta \theta \lesssim 4', \, l \ga 2500) $. On large and intermediate scales all lensed power spectra but the $B$-modes are almost unaffected. The lensed $B$-modes, however, are altered on all scales, reflecting the fact that in the cosmological model assumed in this work primordial gravitational waves are absent and thus polarization of the $B$-type is completely lensing induced.
Taking into account that neglecting the curvature of the sky, as we did in this work, already causes an error in the lensed CMB power spectra at the 0.3% –1.0% level [@2005PhRvD..71j3010C] we conclude that the influence of the non-Gaussian part of the lensing excursion angle PDF on the lensed CMB spectra is far from being observable and can be safely neglected. Furthermore, it is very likely that uncertainties in the recombination history and contamination from various late-time secondaries affect the lensed power spectra in an even stronger way [@2006PhR...429....1L].
Our results have been anticipated by the purely numerical work of @2009MNRAS.396..668C. Using the MS they constructed lensed CMB temperature and polarization maps and computed the corresponding power spectra. They agree well with the ones obtained from CAMB including nonlinear corrections to the lensing potential from the HALOFIT model. Hence, @2009MNRAS.396..668C increased the variance of the lensing excursion angle but did not drop the assumption of a Gaussian PDF [cf. @2005PhRvD..71h3008L].
Our analytical approach, however, takes the non-Gaussianity of the lensing excursion angle PDF explicitly into account revealing that its influence on the lensed CMB spectra is weak and therefore the assumption of a pure Gaussian lensing excursion angle PDF is well applicable. The weakness of the impact of the non-Gaussian features is due to the fact that they are only prominent on small scales (see Figure \[fig:kurtosen\]), while lensing smooths the CMB signal over a large angular range.
Summary {#sect_summary}
=======
The topic of this paper is a derivation of the fluctuation statistics of the lensed CMB temperature and polarisation, taking non-Gaussian features of the lensing deflection angle distribution into account.
1. [Starting point of including non-Gaussian distributions of the deflection angle is the expansion of the characteristic function, which enters the computation of the lensed CMB spectra, into a a series in terms of its moments. For computing lensed CMB spectra with non-Gaussian deflection angle fields we provide a set of analytical expressions, and expand the expressions up to the sixth-order moment of the deflection angle distribution.]{}
2. [Simulated deflection maps [provided by @2009MNRAS.396..668C] show considerable amounts of non-Gaussianity on small scales below a degree, which we quantified with the fourth and sixth moment. On angular scales of an arcminute, they exceed the Gaussian expectation by a factor of 1.5 and 3, respectively, and drop close to their fiducial, Gaussian values close to one degree.]{}
3. [We show that deviations in the angular spectra relative to those derived assuming Gaussian statistics are most important on small angular scales, but remain below the percent level, confirming that the Gaussian approximation is very good. In particular we confirm that errors introduced into the spectra by non-Gaussian deflection angle statistics are smaller than those caused by other secondary anisotropies.]{}
4. [One can give a simple physical argument, why the impact of the non-Gaussian lensing excursion angle PDF is so weak: On the subdegree scale, where the non-Gaussianity is considerable, the CMB spectra are almost featureless. As lensing cannot generate features in a featureless CMB [see @2000PhRvD..62d3007H; @2006PhR...429....1L] the influence of the non-Gaussianity of the lensing excursion angle, which itself is only substantial on subdegree scales, is very weak.]{}
5. [Estimates of cosmological parameters and weak lensing reconstructions are not seriously impeded by non-Gaussianities in the deflection angle distribution, given the small differences relative to spectra derived with a Gaussian approximation. Furthermore, the differences in the spectra involving the temperature and $E$-type polarization are substantial only at high multipole order where the signal-to-noise ratio is small and where the spectra do not possess strong parameter constraining power due to Silk-damping.]{}
The formalism presented here can be applied to investigating primordial non-Gaussianities by CMB lensing e.g. in terms of the $f_\mathrm{NL}$-model or the $\chi^2$-model, for which the higher-order moments are directly calculable. It would be interesting to see if lensed CMB spectra are significantly distorted in these cosmological models, because they inherently are able to provided stronger non-Gaussian features on larger angular scales compared to non-Gaussianities generated by nonlinear structure formation.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Matthias Bartelmann for valuable comments, and Carmelita Carbone for providing the lensing deflection map. For some of our numerical results we used routines of the HEALPix package [@2005ApJ...622..759G]. We are grateful to Martin Reinecke for his help with these routines. BMS’s work is supported by the German Research Foundation (DFG) within the framework of the excellence initiative through the Heidelberg Graduate School of Fundamental Physics.
Polarization {#sec:polarization}
============
Correlation functions and power spectra {#subsec:pol_corr_fct_and_pow_spec}
---------------------------------------
The polarization of the CMB is described by the Stokes parameters $Q$ and $U$ constituting the spin-2 polarization field $P = Q +\mathrm i U$, whose decomposition in gradient-like $E$- and curl-like $B$-modes reads (in the limit of a flat sky) $$P(\mathbf x ) = - \int \frac{{\mathrm{d}}^2 l}{2\pi} (E(\mathbf l) - \mathrm i B (\mathbf l) ) e^{-2\mathrm i \phi} e^{\mathrm i \mathbf l \cdot \mathbf x}$$ [@2005PhRvD..71j3010C]. Defining the correlation functions involving $P$ in the same local coordinate system as before, i.e. with the $x$-axis adapted to the vector connecting $\mathbf x$ and $\mathbf x'$, we have $$\xi_+(r) \equiv \left\langle P^*(\mathbf x)
P(\mathbf x') \right\rangle,
\quad
\xi_-(r) \equiv \left\langle P(\mathbf x)
P(\mathbf x') \right\rangle
\quad \textnormal{and} \quad
\xi_\times(r) \equiv \left\langle \Theta(\mathbf x)
P(\mathbf x') \right\rangle$$ [@2005PhRvD..71j3010C]. The corresponding power spectra are then calculated via $$C_l^{EE} + C_l^{BB} = 2\pi \int r{\mathrm{d}}r \; J_0(lr) \xi_+(r),
\quad
C_l^{EE} - C_l^{BB} = 2\pi \int r{\mathrm{d}}r \; J_4(lr) \xi_-(r)
\quad \textnormal{and} \quad
C_l^{\Theta E} = 2\pi \int r{\mathrm{d}}r \; J_2(lr) \xi_\times(r)$$ [@2006PhR...429....1L].
Lensed correlation functions {#subsec:pol_lensed_corr_fct}
----------------------------
The lensed correlation functions involving the polarization field valid for a general distribution function of the lensing excursion angle can be derived in complete analogy to Section \[subsec:non\_gaussian\_pdf\]. The expression for $\hat\xi_+(r)$ is identical with equation [(\[eq:lensed\_corr\_fct\_non\_linear\])]{} if one replaces $C^{{\Theta\Theta}}_l$ by $C^{EE}_l + C^{BB}_l$: $$\begin{aligned}
\hat\xi_+(r) &=& \int \frac{{\mathrm{d}}^2l}{(2\pi)^2} \left(C^{EE}_l + C^{BB}_l\right) e^{\mathrm i \mathbf l \cdot \mathbf r}
\left\langle \sum_{n=0}^{\infty} \frac{\left( \mathrm i \mathbf l \cdot \bdelta \balpha\right)^n}{n!}\right\rangle
\nonumber\\
&=& \frac{1}{2\pi} \int l{\mathrm{d}}l \; \left(C^{EE}_l + C^{BB}_l\right)
\sum_{n=0}^\infty \sum_{k=0}^{n} \sum_{q=0}^{n-k} \sum_{r=0}^{2(k+q)}
\frac{(-1)^{n+k+r}}{(2n)!4^{k+q}} {2n \choose 2k} {n-k \choose q} {2(k+q) \choose r}
J_{2(k+q-r)}(lr)
\left\langle \delta\alpha_x^{2k} \right\rangle \left\langle \delta\alpha_y^{2(n-k)}\right\rangle.
\label{eq:xi_plus_nonlinear}\end{aligned}$$ Some additional effort, however, has to be put in the computation of $\hat\xi_-(r)$ and $\hat\xi_\times(r)$, since here the additional factors of $e^{-2\mathrm i \phi}$ in the spin 2 polarization do not cancel. After a somewhat lengthy but straightforward calculation we find $$\begin{aligned}
\hat\xi_-(r)
& = &
\int \frac{{\mathrm{d}}^2 l }{(2\pi)^2} \left( C_l^{EE} - C_l^{BB} \right) \cos4\phi \:
e^{\mathrm i \mathbf l \cdot \mathbf r}
\left\langle \sum_{n=0}^{\infty} \frac{\left( \mathrm i \mathbf l \cdot \bdelta \balpha\right)^n}{n!}\right\rangle
\nonumber\\
& = &
\frac{1}{2\pi} \int l {\mathrm{d}}l \; \left( C_l^{EE} - C_l^{BB} \right)
\sum_{n=0}^{\infty} \sum_{k=0}^{n} \sum_{q=0}^{n-k} \frac{(-1)^{n+k}}{(2n)!4^{k+q}} {2n\choose 2k}{n-k \choose q}
\left\langle \delta\alpha_x^{2k} \right\rangle \left\langle \delta\alpha_y^{2(n-k)}\right\rangle
\nonumber\\
&&
\qquad \cdot \: \left[
8 \sum_{r=0}^{2(k+q+2)}(-1)^r {2(k+q+2) \choose r} J_{2(k+q+2-r)}(lr)
-8 \sum_{r=0}^{2(k+q+1)}(-1)^r {2(k+q+1) \choose r} J_{2(k+q+1-r)}(lr)
\right.
\nonumber\\
&&
\left.
\qquad \qquad \:+\sum_{r=0}^{2(k+q)}(-1)^r {2(k+q) \choose r} J_{2(k+q-r)}(lr)
\right]
\label{eq:xi_minus_nonlinear}\end{aligned}$$ and $$\begin{aligned}
\hat\xi_\times (r)
& = &
- \int \frac{{\mathrm{d}}^2 l }{(2\pi)^2} C_l^{\Theta E} \cos2\phi \:
e^{\mathrm i \mathbf l \cdot \mathbf r}
\left\langle \sum_{n=0}^{\infty} \frac{\left( \mathrm i \mathbf l \cdot \bdelta \balpha\right)^n}{n!}\right\rangle
\\
& = &
\frac{1}{2\pi} \int l{\mathrm{d}}l\; C_l^{\Theta E} \sum_{n=0}^\infty \sum_{k=0}^{n} \sum_{q=0}^{n-k}
\frac{(-1)^{k+n}}{(2n)! 4^{k+q}} {2n \choose 2k} {n-k \choose q}
\left\langle \delta\alpha_x^{2k} \right\rangle \left\langle \delta\alpha_y^{2(n-k)}\right\rangle \nonumber\\
& &
\qquad \cdot \: \left[ 2 \sum_{r=0}^{2(k+q+1)} (-1)^r {2(k+q+1) \choose r} J_{2(k+q+1-r)}(lr)
+ \sum_{r=0}^{2(k+q)} (-1)^r {2(k+q) \choose r} J_{2(k+q-r)}(lr) \right].
\label{eq:xi_X_nonlinear}\end{aligned}$$
Explicit expressions {#sec:explicit_expressions}
====================
Here we give explicit expressions for the lensed correlation functions used for our numerical calculations. Truncating at $n = 3$ in the equations [(\[eq:lensed\_corr\_fct\_non\_linear\])]{}, [(\[eq:xi\_plus\_nonlinear\])]{}, [(\[eq:xi\_minus\_nonlinear\])]{} and [(\[eq:xi\_X\_nonlinear\])]{}, we obtain: $$\begin{aligned}
\hat\xi (r)
& = &
\frac{1}{2\pi}\int l\mathrm dl\; C_{l}^{\Theta\Theta} \biggl( J_0(lr)
-\frac{1}{4}l^2 \left[\left({\left\langle \delta\alpha_x^{2}\right\rangle}+{\left\langle \delta\alpha_y^{2}\right\rangle} \right)J_0(lr)
- \left({\left\langle \delta\alpha_x^{2}\right\rangle} -{\left\langle \delta\alpha_y^{2}\right\rangle} \right)J_2(lr)\right]\biggr.
\nonumber\\
& & \qquad + \frac{1}{24}l^4 \left[\frac{1}{8}\left({\left\langle \delta\alpha_x^{4}\right\rangle} + {\left\langle \delta\alpha_y^{4}\right\rangle}\right)\left(3J_0(lr)+J_4(lr)\right)
-\frac{1}{2}\left({\left\langle \delta\alpha_x^{4}\right\rangle}-{\left\langle \delta\alpha_y^{4}\right\rangle}\right)J_2(lr) \right.
\nonumber\\
& &
\qquad\qquad\qquad\quad \left.+\frac{3}{4} {\left\langle \delta\alpha_x^{2}\right\rangle}{\left\langle \delta\alpha_y^{2}\right\rangle}\left(J_0(lr)-J_4(lr)\right)\right]
\nonumber\\
& &
\qquad - \frac{1}{720}l^6 \left[\frac{1}{16} \left({\left\langle \delta\alpha_x^{6}\right\rangle}+{\left\langle \delta\alpha_y^{6}\right\rangle}\right) \left(5J_0(lr) +3J_4(lr)\right)
+ \frac{1}{32}\left({\left\langle \delta\alpha_x^{6}\right\rangle}-{\left\langle \delta\alpha_y^{6}\right\rangle}\right)\left(-15J_2(lr)-J_6(lr)\right)\right.
\nonumber\\
& &
\qquad\qquad\qquad\quad
+\frac{15}{32}{\left\langle \delta\alpha_x^{2}\right\rangle}{\left\langle \delta\alpha_y^{4}\right\rangle}\left(2J_0(lr)+J_2(lr)-2J_4(lr)-J_6(lr)\right)
\nonumber\\
& &
\qquad\qquad\qquad\qquad
\left.+\frac{15}{32}{\left\langle \delta\alpha_x^{4}\right\rangle}{\left\langle \delta\alpha_y^{2}\right\rangle}\left(2J_0(lr)-J_2(lr)-2J_4(lr)+J_6(lr)\right)\right]
\biggl. + \; ... \;\biggr),\end{aligned}$$ $$\begin{aligned}
\hat\xi_+(r)
& = &
\frac{1}{2\pi}\int l\mathrm dl\; \left(C^{EE}_l+C^{BB}_l\right) \biggl( J_0(lr)
-\frac{1}{4}l^2 \left[\left({\left\langle \delta\alpha_x^{2}\right\rangle}+{\left\langle \delta\alpha_y^{2}\right\rangle} \right)J_0(lr)
- \left({\left\langle \delta\alpha_x^{2}\right\rangle} -{\left\langle \delta\alpha_y^{2}\right\rangle} \right)J_2(lr)\right]\biggr.
\nonumber\\
& &
\qquad + \frac{1}{24}l^4 \left[\frac{1}{8}\left({\left\langle \delta\alpha_x^{4}\right\rangle} + {\left\langle \delta\alpha_y^{4}\right\rangle}\right)\left(3J_0(lr)+J_4(lr)\right)
-\frac{1}{2}\left({\left\langle \delta\alpha_x^{4}\right\rangle}-{\left\langle \delta\alpha_y^{4}\right\rangle}\right)J_2(lr) \right.
\nonumber\\
& &
\qquad\qquad\qquad\quad \left.+\frac{3}{4} {\left\langle \delta\alpha_x^{2}\right\rangle}{\left\langle \delta\alpha_y^{2}\right\rangle}\left(J_0(lr)-J_4(lr)\right)\right]
\nonumber\\
& &
\qquad - \frac{1}{720}l^6 \left[\frac{1}{16} \left({\left\langle \delta\alpha_x^{6}\right\rangle}+{\left\langle \delta\alpha_y^{6}\right\rangle}\right) \left(5J_0(lr) +3J_4(lr)\right)
+ \frac{1}{32}\left({\left\langle \delta\alpha_x^{6}\right\rangle}-{\left\langle \delta\alpha_y^{6}\right\rangle}\right)\left(-15J_2(lr)-J_6(lr)\right)\right.
\nonumber\\
& &
\qquad\qquad\qquad\quad
+\frac{15}{32}{\left\langle \delta\alpha_x^{2}\right\rangle}{\left\langle \delta\alpha_y^{4}\right\rangle}\left(2J_0(lr)+J_2(lr)-2J_4(lr)-J_6(lr)\right)
\nonumber\\
& &
\qquad\qquad\qquad\qquad
\left.+\frac{15}{32}{\left\langle \delta\alpha_x^{4}\right\rangle}{\left\langle \delta\alpha_y^{2}\right\rangle}\left(2J_0(lr)-J_2(lr)-2J_4(lr)+J_6(lr)\right)\right]
\biggl. + \; ... \;\biggr),\end{aligned}$$ $$\begin{aligned}
\hat\xi_-(r)
& = &
\frac{1}{2\pi}\int l\mathrm dl\; \left(C^{EE}_l - C^{BB}_l\right) \biggl( J_4(lr)
-\frac{1}{4}l^2 \left[\left({\left\langle \delta\alpha_x^{2}\right\rangle}+{\left\langle \delta\alpha_y^{2}\right\rangle} \right)J_4(lr)
- \frac{1}{2}\left({\left\langle \delta\alpha_x^{2}\right\rangle} -{\left\langle \delta\alpha_y^{2}\right\rangle} \right)\left(J_2(lr)+J_6(lr)\right)\right]\biggr.
\nonumber\\
& &
\qquad + \frac{1}{24}l^4 \left[\frac{1}{16}\left({\left\langle \delta\alpha_x^{4}\right\rangle} +
{\left\langle \delta\alpha_y^{4}\right\rangle}\right)\left(J_0(lr)+6J_4(lr)+J_8(lr)\right)
-\frac{1}{4}\left({\left\langle \delta\alpha_x^{4}\right\rangle}-{\left\langle \delta\alpha_y^{4}\right\rangle}\right)\left(J_2(lr)+J_6(lr)\right) \right.
\nonumber\\
& &
\qquad\qquad\qquad\quad \left.-\frac{3}{8} {\left\langle \delta\alpha_x^{2}\right\rangle}{\left\langle \delta\alpha_y^{2}\right\rangle}\left(J_0(lr)-2J_4(lr)+J_8(lr)\right)\right]
\nonumber\\
& &
\qquad - \frac{1}{720}l^6 \left[\frac{1}{32} \left({\left\langle \delta\alpha_x^{6}\right\rangle}+{\left\langle \delta\alpha_y^{6}\right\rangle}\right) \left(3J_0(lr)
+10J_4(lr)+3J_8(lr)\right)\right.
\nonumber\\
& &
\qquad\qquad\qquad\quad -\frac{1}{64}\left({\left\langle \delta\alpha_x^{6}\right\rangle}-{\left\langle \delta\alpha_y^{6}\right\rangle}\right)
\left(16J_2(lr)+15J_6(lr)+J_{10}(lr)\right)
\nonumber\\
& &
\qquad\qquad\qquad\qquad\quad
+\frac{15}{64}{\left\langle \delta\alpha_x^{2}\right\rangle}{\left\langle \delta\alpha_y^{4}\right\rangle}\left(-2J_0(lr)+4J_4(lr)+J_6(lr)-2J_8(lr)-J_{10}(lr)\right)
\nonumber\\
& &
\qquad\qquad\qquad\qquad\qquad
\left.+\frac{15}{64}{\left\langle \delta\alpha_x^{4}\right\rangle}{\left\langle \delta\alpha_y^{2}\right\rangle}\left(-2J_0(lr)+4J_4(lr)-J_6(lr)-2J_8(lr)+J_{10}(lr)\right)\right]
\biggl. + \; ... \;\biggr)\end{aligned}$$ and $$\begin{aligned}
\hat\xi_\times(r)
& = &
\frac{1}{2\pi}\int l\mathrm dl\; C^{\Theta E}_l \biggl( J_2(lr)
- \frac{1}{4}l^2 \left[\left({\left\langle \delta\alpha_x^{2}\right\rangle}+{\left\langle \delta\alpha_y^{2}\right\rangle} \right)J_2(lr)
- \frac{1}{2}\left({\left\langle \delta\alpha_x^{2}\right\rangle} -{\left\langle \delta\alpha_y^{2}\right\rangle} \right)\left(J_0(lr)+J_4(lr)\right)\right]\biggr.
\nonumber\\
& &
\qquad - \frac{1}{24}l^4 \left[-\frac{1}{16}\left({\left\langle \delta\alpha_x^{4}\right\rangle} +
{\left\langle \delta\alpha_y^{4}\right\rangle}\right)\left(7J_2(lr)+J_6(lr)\right)
+\frac{1}{4}\left({\left\langle \delta\alpha_x^{4}\right\rangle}-{\left\langle \delta\alpha_y^{4}\right\rangle}\right)\left(J_0(lr)+J_4(lr)\right) \right.
\nonumber\\
&&
\qquad\qquad\qquad\quad \left.+\frac{3}{8} {\left\langle \delta\alpha_x^{2}\right\rangle}{\left\langle \delta\alpha_y^{2}\right\rangle}\left(-J_2(lr)+J_6(lr)\right)\right]
\nonumber\\
&&
\qquad + \frac{1}{720}l^6 \left[-\frac{1}{32} \left({\left\langle \delta\alpha_x^{6}\right\rangle}+{\left\langle \delta\alpha_y^{6}\right\rangle}\right) \left(13J_2(lr)
+3J_6(lr)\right)\right.
\nonumber\\
&&
\qquad\qquad\qquad\quad +\frac{1}{64}\left({\left\langle \delta\alpha_x^{6}\right\rangle}-{\left\langle \delta\alpha_y^{6}\right\rangle}\right)
\left(15J_0(lr)+16J_4(lr)+J_{8}(lr)\right)
\nonumber\\
&&
\qquad\qquad\qquad\qquad\quad
+\frac{15}{64}{\left\langle \delta\alpha_x^{2}\right\rangle}{\left\langle \delta\alpha_y^{4}\right\rangle}\left(-J_0(lr)-2J_2(lr)+2J_6(lr)+J_8(lr)\right)
\nonumber\\
&& \qquad\qquad\qquad\qquad\qquad
\left.+\frac{15}{64}{\left\langle \delta\alpha_x^{4}\right\rangle}{\left\langle \delta\alpha_y^{2}\right\rangle}\left(J_0(lr)-2J_2(lr)+2J_6(lr)-J_8(lr)\right)\right]
\biggl. + \; ... \;\biggr).\end{aligned}$$
\[lastpage\]
[^1]: e-mail: [email protected]
[^2]: http://camb.info/
|
---
abstract: 'This paper presents self-contained proofs of the strong subadditivity inequality for von Neumann’s quantum entropy, $S(\rho)$, and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps. Moreover, the approach presented here, which is based on Klein’s inequality and Lieb’s theorem that the function $A \raw \tr \, e^{K + \log A}$ is concave, allows one to obtain conditions for equality. In the case of strong subadditivity, which states that $S(\rho_{123}) + S(\rho_2) \leq S(\rho_{12})+S(\rho_{23})$ where the subscripts denote subsystems of a composite system, equality holds if and only if $\log \rho_{123} = \log \rho_{12} - \log \rho_2 + \log \rho_{23}$. Using the fact that the Holevo bound on the accessible information in a quantum ensemble can be obtained as a consequence of the monotonicity of relative entropy, we show that equality can be attained for that bound only when the states in the ensemble commute. The paper concludes with an Appendix giving a short description of Epstein’s elegant proof of Lieb’s theorem.'
author:
- |
Mary Beth Ruskai\
Department of Mathematics\
University of Massachusetts Lowell\
Lowell, MA 01854 USA\
[MaryBeth\[email protected]]{}
date: |
\
\
[PACS Numbers: 03.65.-w, 03.67.-a, 03.67.Lx ]{}\
[MR classification: 81P68, 94A17, 82B10]{}
title: |
Inequalities for Quantum Entropy:\
A Review with Conditions for Equality
---
Introduction
============
Quantum Entropy
---------------
Quantum information science [@NC] is the study of the information carrying and processing properties of quantum mechanical systems. Recent work in this area has generated renewed interest in the properties of the quantum mechanical entropy. It is interesting to note that von Neumann [@vN27; @vNbk] introduced the notion of mixed state, represented by a density matrix $\rho$ (a positive semi-definite operator with $\tr \rho = 1$), into quantum theory defined its entropy as as $ S(\rho) \equiv - \tr(\rho \log \rho)$ in 1927, well before the corresponding classical quantity was introduced in Shannon’s seminal work [@Shan] on “The Mathematical Theory of Communication” in 1948. (Admittedly, von Neumann’s motivation was the extension of the classical theory of statistical mechanics, developed by Gibbs, Boltzman, et al to the quantum domain rather than the development of a theory of quantum communication.) Many fundamental properties of the quantum entropy were proved in a remarkable, but little-known, 1936 paper of Delbrück and Molèiere [@DM]. For further discussion of the history of quantum entropy, see [@OP; @Pz.vN; @W] and the introductory remarks in [@RuSt].
One important class of inequalities relates the entropy of subsystems to that of a composite system, whose Hilbert space is a tensor product is ${\cal H}_{12} = {\cal H}_1 \ot {\cal H}_2$ of the Hilbert spaces for the subsystems. When the state of the composite system is described by the density matrix $\rho_{12}$, the states of the subsystems are given by the reduced density matrices, e.g., $\rho_1 = T_2(\rho_{12})$, obtained by taking the partial trace. The subadditivity inequality $$\begin{aligned}
\label{eq:subadd}
S(\rho_{12}) \leq S(\rho_1)+S(\rho_2)\end{aligned}$$ was proved in [@DM] and [@LanRob]. (It should not be confused with the concavity $$\begin{aligned}
\label{eq:concav}
S(x \rho' + (1-x) \rho'') \geq x S(\rho') + (1-x) S(\rho'')\end{aligned}$$ which can actually be obtained [*from*]{} subadditivity by considering block matrices [@LbBull; @SSA; @W]). In the more complex situation in which the composite system is composed of three subsystems the following stronger inequality, known as strong subadditivity (SSA), holds. $$\begin{aligned}
\label{eq:ssa}
S(\rho_{123}) + S(\rho_2) \leq S(\rho_{12}) + S(\rho_{23})\end{aligned}$$ This inequality was conjectured by Lanford and Robinson in [@LanRob] and proved in [@SSA.PRL; @SSA]. In this paper, we review its proof in a form that easily yields the following condition for equality.
\[thm:ssa.eq\] Equality holds in strong subadditivity [*(\[eq:ssa\])*]{} if and only if $$\begin{aligned}
\label{ssa.equal}
\log \rho_{123} - \log \rho_{12} = \log \rho_{23} - \log \rho_2.\end{aligned}$$
We have suppressed implicit tensor products with the identity so that, e.g., $\log \rho_{12}$ means $(\log \rho_{12}) \ot I_3$. Rewriting (\[ssa.equal\]) as $ \log \rho_{123} + \log \rho_2 = \log \rho_{12} + \log \rho_{23}$, multiplying by $\rho_{123}$ and taking the trace immediately establishes the sufficiency of this equality condition. In Section \[sect:SSA\], we will also show that it is also necessary.
Relative entropy {#sect:relent}
----------------
The SSA inequality can be restated as a property of the [*quantum relative entropy*]{} which is defined as $$\begin{aligned}
\label{eq:relent}
H(\rho,\gamma) \equiv \tr \, \rho
\big( \log \rho - \log \gamma \big).\end{aligned}$$ It is usually assumed that $\rho, \gamma$ are density matrices, although (\[eq:relent\]) is well-defined for any pair of positive semi-definite matrices for which $\ker(\gamma) \subset \ker(\rho)$. Strong subadditivity can now be restated as $$\begin{aligned}
\label{eq:mono.SSA}
H(\rho_{12},\rho_{2}) \leq H(\rho_{123},\rho_{23})\end{aligned}$$ where we again write, e.g., $\rho_{23}$ for $I_1 \ot \rho_{23}$. More generally, the relative entropy is monotone under completely positive, trace-preserving maps (also known as “quantum operations” [@NC] and “stochastic maps” [@AU; @KR1] and discussed in more detail in section \[sect:Lind.rep\]), i.e., $$\begin{aligned}
\label{eq:mono}
H[\Phi(\rho),\Phi(\gamma)] \leq H(\rho,\gamma).\end{aligned}$$ This monotonicity implies (\[eq:mono.SSA\]) when $\Phi = T_3$ is the partial trace operation; perhaps surprisingly, the converse is also true [@Lind75]. This, and other connections between strong subadditivity and relative entropy are discussed in Section \[sect:relation\] .
The approach to SSA presented here can also be used to obtain conditions for equality in properties of relative entropy, including its joint convexity and monotonicity. The explicit statements are postponed to later sections. Since the monotonicity can be used to give a simple proof of the celebrated Holevo bound [@Hv0; @NC] on accessible information, we show how our results can be used to recover the equality conditions in that bound. As discussed in section \[sect:Petz\], Petz [@OP; @Pz.eq] has also obtained several equality conditions in different, but equivalent, forms. However, Theorem \[thm:eqal.gen.mono\], which applies to the most general form of monotonicity, appears to be new.
Lieb’s convex trace functions
-----------------------------
One of the most frequently cited approaches to strong subadditivity is to present it as a consequence of the concavity of a quantity known as the Wigner-Yanase-Dyson entropy [@WY]. This property, conjectured by Bauman [@B], is equivalent to the joint concavity in $A$ and $B$ of the map $$\begin{aligned}
\label{eq:WYD}
(A,B) \raw \tr \, A^s K^{\dg} B^{(1-s)} K ~~~ {\rm for} ~~ A, B > 0,
~~~ 0 < s < 1\end{aligned}$$ (where $\dg$ is used to denote the adjoint). Lieb’s proof [@LbWYD] of the concavity of the WYD function (\[eq:WYD\]) and his realization of a connection between SSA and Bauman’s concavity conjecture was a crucial breakthrough. However, concavity of the WYD function was only one of several concave trace functions studied in [@LbWYD]; the following result was also established by Lieb.
\[exp.conc\] For any fixed self-adjoint matrix $K$, the function $A \mapsto F(A) = \tr \, e^{K + \log A}$ is concave in $A > 0$.
This result played a fundamental role in the original proof [@SSA.PRL; @SSA] of SSA and the closely related property of joint concavity of the relative entropy [@SSA.PRL; @SSA; @Lind74]. Although SSA is a deep theorem, a complete proof is not as forbidding as is sometimes implied. Therefore, for completeness, we include Epstein’s elegant proof [@Ep] of Theorem \[exp.conc\] in Appendix A, and then follow the original strategies of Lieb and Ruskai [@SSA] to show how it implies SSA.
Overview
--------
Although this paper grew out of questions about the conditions for equality in strong subadditivity and related inequalities, it seems useful to present these conditions within a more comprehensive exposition. For simplicity, we confine our discussion to finite dimensions, and assume that, unless otherwise stated, the density matrices under consideration are strictly positive.
The remainder of the paper is structured as follows. In Section \[sect:reform\] we discuss some consequences and interpretations of the SSA equality condition. In Section \[sec:tools\] we summarize some mathematical results needed for the proofs in the sections that follow. Section \[sect:SSA\], which might be regarded as the heart of the paper, presents the proof of strong subadditivity in a form which easily yields the equality conditions. (A reader primarily interested in this proof can proceed directly to Section \[sect:SSA\] with a willingness to accept the results of section \[sec:tools\].) Section \[sect:other\] presents proofs with equality conditions for the monotonicity of the relative entropy under partial traces, the joint convexity of the relative entropy; and the general monotonicity under stochastic maps. This section also contains a discussion of the connection between these properties, SSA and their proofs. Section \[sect:gen.mono\] contains the proof of the equality conditions for monotonicity of relative entropy. Section \[sect:Holv\] consider bounds, most notably the Holevo bound, on the accessible information that can be extracted from an ensemble of quantum states, and the conditions under which they can be attained. The paper concludes with some additional historical comments in Section \[sect:conc\].
Implications of the equality conditions for SSA {#sect:reform}
===============================================
Classical conditions {#sect:class}
--------------------
To describe the corresponding classical inequalities, let the subsystems $A,B$ and $C$ correspond to classical random variables. One can recover the classical Shannon entropy $ - \sum_a p(a) \log p(a) $ from the von Neumann entropy by taking $\rho$ to be a diagonal matrix with elements $p(a)$ on the diagonal. Employing a slight abuse of notation, we write $S[p(a)]$ for this quantity. Then the classical strong subadditivity inequality can be stated as $$\begin{aligned}
\label{eq:SSA.class}
S[p(a,b,c)] + S[p(b)] \leq S[p(a,b)] + S[p(b,c)] .\end{aligned}$$ The classical relative entropy of the distribution $q(a)$ with respect to $p(a)$ is $H[p(a),q(a)] = \sum_a p(a) \log \frac{p(a)}{q(a)}$. It is well-known (see, e.g.,[@Kull]) that the convexity of the function $f(x) = x \log x$ implies that $H[p(a),q(a)] \geq 0$ and its strict convexity implies that equality holds if and only if $p(a) = q(a) ~~ \forall~a$. (The generalization of this result to quantum situations is discussed in section \[sect:klein\].)
The classical form (\[eq:SSA.class\]) of SSA is equivalent to $H[p(a,b,c), q(a,b,c)] \geq 0$ when the second distribution is $q(a,b,c) = p(a,b) [p(b)]^{-1} p(b,c)$, Thus, equality holds in (\[eq:SSA.class\]) if and only if $$\begin{aligned}
\label{eq:SSA.class.prod}
p(a,b,c) = p(a,b) [p(b)]^{-1} p(b,c) ~~~~ \forall~a,b,c\end{aligned}$$ which can be rewritten as $$\begin{aligned}
\label{eq:SSA.class.eqal}
\log p(a,b,c) - \log p(a,b) = \log p(b,c) - \log p(b)
~~~~ \forall~a,b,c.\end{aligned}$$ which is identical to what one would obtain from Theorem \[thm:ssa.eq\]. Using $p(c|b)$ to denote the classical conditional probability distribution, (\[eq:SSA.class.eqal\]) can be rewritten as $$\begin{aligned}
\label{eq:classical_ssa}
p(c|a,b) = p(c|b),\end{aligned}$$ which is precisely the condition that the sequence $A \rightarrow B \rightarrow C$ forms a Markov chain.
Special cases of SSA equality {#sect:equal}
-----------------------------
Some insight into equality condition (\[ssa.equal\]) may be obtained by looking at special cases in which it is satisfied. The most obvious is when $\rho_{123}$ is a tensor product of its three reduced density matrices. However, it is readily verified that (\[ssa.equal\]) also holds when either $\rho_{123} = \rho_1 \ot
\rho_{23}$ [*or*]{} $\rho_{123} = \rho_{12} \ot \rho_{3}$. One can generalize this slightly further. If the subsystem $2$ can be partitioned further into two subsystems $2'$ and $2''$, then one can verify equality holds if $\rho_{123} = \rho_{12'} \otimes \rho_{2''3}$, where $\rho_{12'}$ and $\rho_{2''3}$ are states of the composite systems $1,2'$ and $2'',3$ respectively.
However, such a decomposition into tensor products is not necessary; indeed, we have already seen that equality also holds for the case of classical Markov processes. Moreover, by comparison to (\[eq:classical\_ssa\]) it is natural to regard (\[ssa.equal\]) as a kind of quantum Markov condition. Thus, the conditions in Theorem \[thm:ssa.eq\] can also be viewed as a natural non-commutative analogue of the conditions for equality in classical SSA. Another way of regarding (\[ssa.equal\]) is as a concise statement of a subtle intertwining condition discussed below. Unfortunately, we have not found explicit examples which satisfy it other than the two classes discussed above, that is, a partial decomposition into tensor products or a classical Markov chain.
Petz’s conditions {#sect:Petz}
-----------------
Using a completely different approach, Petz [@OP; @Pz.eq] gave conditions for equality in (\[eq:mono\]) when $\Phi$ can be identified with a mapping of an algebra onto a subalgebra, a situation which includes (\[eq:mono.SSA\]). In that case Petz’s conditions become $$\begin{aligned}
\label{eq:mono.Pz.eqal}
\rho_{12}^{it} \rho_2^{-it} = \rho_{123}^{it} \rho_{23}^{-it}.\end{aligned}$$ Taking the derivative of both sides of (\[eq:mono.Pz.eqal\]) at $t = 0$ yields (\[ssa.equal\]). Although (\[eq:mono.Pz.eqal\]) appears stronger than (\[ssa.equal\]), it is not since, as noted above, (\[ssa.equal\]) is sufficient for equality in (\[eq:mono.SSA\]). Moreover, since (\[ssa.equal\]) implies $$\begin{aligned}
e^{i t \, \log ( \rho_{123})} & = &
e^{i t \, \left[ \log \rho_{12} - \log \rho_2 + \log \rho_{23}\right]}\end{aligned}$$ our results can be combined with those of Petz to see that equality holds in SSA $\iff$ (\[ssa.equal\]) $\iff$ (\[eq:mono.Pz.eqal\]) and that any of these conditions suffices to imply $$\begin{aligned}
e^{i t \, \left[ \log \rho_{12} - \log \rho_2 + \log \rho_{23}\right]}
= e^{i t \, \log ( \rho_{12})} e^{-i t \, \log ( \rho_{2})}
e^{i t \, \log ( \rho_{23})} .\end{aligned}$$ Note that one can also relate Petz’s conditions to those for equality in classical SSA by rewriting (\[eq:SSA.class.prod\]) as $ p(a,b,c)[p(b,c)]^{-1} = p(a,b) [p(b)]^{-1}$ and then raising to the $it$ power.
Fundamental mathematical tools {#sec:tools}
==============================
Klein’s inequality {#sect:klein}
------------------
The fact that the relative entropy is positive, i.e., $H(\rho,\gamma) \geq 0$ when $\tr \, \rho = \tr \, \gamma$ is an immediate consequence of the following fundamental convexity result due to Klein [@Kl; @NC; @W].
\[thm:klein\] [*(Klein’s Inequality)*]{} For $A, B > 0$ $$\begin{aligned}
\label{ineq:klein}
\tr \, A \big( \log A - \log B \big) \geq \tr (A - B),\end{aligned}$$ with equality if and only if $A = B$.
The closely related Peierls-Bogoliubov inequality [@OP; @W] is sometimes used instead of Klein’s inequality. However, the equality conditions in Theorem \[thm:klein\] play a critical role in the sections that follow.
Lieb’s golden corollary
-----------------------
The proofs in Section \[sect:SSA\] do not use Theorem \[exp.conc\] directly, but a related result generalizing the following inequality, which we will also need.
\[thm:gold\] [*(Golden-Thompson-Symanzik)*]{} For self-adjoint matrices $A$ and $B$ $\tr \, e^{A+B} \leq \tr \, e^A e^B$ with equality if and only if $A$ and $B$ commute.
Although this inequality is extremely well-known, the conditions for equality do not appear explicitly in such standard references as [@HJ2; @Sim; @W]. However, one method of proof is based on the observation that $ \tr \, [e^{A/2^k} e^{B/2^k}]^{2^k}$ is monotone decreasing in $k$, yielding $e^{A+B}$ in the limit as $k \raw \infty$. The equality conditions then follow easily from those for the Schwarz inequality for the Hilbert-Schmidt inner product $\tr C^{\dg} D$. Indeed, $k = 1$ yields $$\begin{aligned}
\tr \Big( e^{A/2} e^{B/2} \Big) \Big( e^{A/2} e^{B/2} \Big) \leq
\Big[ \tr \, e^{B/2} e^A e^{B/2} \Big]^{1/2}
\Big[ \tr \, e^{A/2} e^B e^{A/2} \Big]^{1/2} = \tr \, e^A e^B\end{aligned}$$ with $C = e^{B/2} e^{A/2}$ and $D = e^{A/2} e^{B/2}$. The equality condition that $C$ is a multiple of $D$ implies $e^{B/2} e^{A/2} = e^{A/2} e^{B/2}$ which holds if and only if $A$ and $B$ commute. One reference [@OP] that does discuss equality does so by making the interesting observation that (as shown in [@Pz.gt]) Theorem \[thm:gold\] and its equality conditions, can be derived as a consequence of the monotonicity of relative entropy, Theorem \[eq:mono\].
The natural extension to three matrices $\tr \, e^{A + B + C} \leq |\tr e^A e^B e^C|$, fails; see, for example, Problem 20 on pages 512–513 of [@HJ2]. Therefore, the following result of Lieb [@LbWYD] is particularly noteworthy.
[*(Lieb)*]{} \[cor:trip.gold\] For any $R, S, T > 0$ $$\begin{aligned}
\label{eq:trip.gold}
\tr \, e^{\log R - \log S + \log T} \leq
\tr \, \int_0^\infty R \frac{1}{S + u I} T \frac{1}{S + u I} du.\end{aligned}$$
One might expect that equality holds if and only if $R,S, T$ commute. Although this is sufficient, it is not necessary. One easily checks that both sides of (\[eq:trip.gold\]) equal $\tr \, \rho_1 \ot \rho_{23}$ when $R = \rho_1 \ot \rho_2 \ot I_3, S = I_1 \ot \rho_2 \ot I_3,
T = I_1 \ot \rho_{23}$, even when $T$ does not commute with $R$ or $S$. Lieb’s proof of (\[eq:trip.gold\]) begins with the easily-established fact [@Rock] that if $F(A)$ is concave and homogeneous in the sense $F(xA) = x F(A)$ , then $$\begin{aligned}
\label{conc.deriv.ineq}
\lim_{x \raw 0} \frac{F(A + xB) - F(A)}{x} \geq F(B) .\end{aligned}$$ Applying this to the functions in Theorem \[exp.conc\] with $A = S, B = T, K = \log R - \log S$ yields $$\begin{aligned}
\label{eq:intermediate_Lieb}
\tr e^{\log R-\log S+\log T} \leq \lim_{x\rightarrow 0}
\frac{\tr e^{\log R-\log S+\log(S+xT)}-\tr R}{x}.\end{aligned}$$ To complete the proof, we need the well-known integral representation $$\begin{aligned}
\label{eq:int_repn}
\log(S + xT) - \log S = \int_0^\infty
\frac{1}{S + u I} \, xT \, \frac{1}{S + xT + u I} \, du ~.\end{aligned}$$ Substituting (\[eq:int\_repn\]) into (\[eq:intermediate\_Lieb\]) and noting that $$\begin{aligned}
\tr \, e^{\log R + x \int_0^\infty
\frac{1}{S + u I} \, T \, \frac{1}{S + xT + u I} \, du} =
\tr \, R + x \, \tr \, R \int_0^\infty \!
\frac{1}{S + u I} \, T \, \frac{1}{S + u I} \, du + O(x^2)\end{aligned}$$ yields the desired result.
Purification {#sect:pure}
------------
Araki and Lieb [@AkLb; @LbBull] observed that one could obtain useful new entropy inequalities by applying what is now known as the “purification process” to known inequalities. Any density $\rho_1$ can be extended to a pure state density matrix $\rho_{12}$ on a tensor product space; moreover, $S(\rho_1) = S(\rho_2)$. Applying this to the subadditivity inequality (\[eq:subadd\]), i.e., $S(\rho_{12}) \leq S(\rho_1)+S(\rho_2)$, yields the equivalent result $S(\rho_3) \leq S(\rho_{23})+S(\rho_2)$ which can be combined with (\[eq:subadd\]) to give the triangle inequality [@AkLb; @LbBull] $$\begin{aligned}
\label{eq:tri}
| S(\rho_1) - S(\rho_2)| \leq S(\rho_{12}) \leq S(\rho_1)+S(\rho_2).\end{aligned}$$ By purifying $\rho_{123}$ to $\rho_{1234}$ one can similarly show that SSA (\[eq:ssa\]) is equivalent to $$\begin{aligned}
\label{eq:ssa.alt}
S(\rho_4) + S(\rho_2) \leq S(\rho_{12}) + S(\rho_{14}).\end{aligned}$$
Lindblad’s representation of stochastic maps {#sect:Lind.rep}
--------------------------------------------
Stochastic maps arise naturally in quantum information as a description of the effect on a subsystem $A$ interacting with the environment in the pure state $\gamma_B = |\psi_B \kb \psi_B|$ via the unitary operation $U_{AB}$, $$\begin{aligned}
\rho_A \raw
\tr_B \Big( U_{AB} \, \rho_A \otimes \gamma_B \, U_{AB}^{\dagger} \Big).\end{aligned}$$ Lindblad [@Lind75] used Stinespring’s representation to show that any completely positive trace-preserving map $\Phi$ which maps an algebra into itself can be represented as if it arose in this way. That is, given such a map $\Phi$ one can always find an auxiliary system, ${\cal H}_B$, a density matrix $\gamma_B$ on ${\cal H}_B$, and a unitary map $U_{AB}$ on the combined system ${\cal H}_A \ot {\cal H}_B$ (where $A$ denotes the original system) such that $$\begin{aligned}
\label{eq:Lind.rep}
\Phi(\rho) = \tr_B
\Big( U_{AB} \, \rho \ot \gamma_B U_{AB}^{\dagger} \Big)\end{aligned}$$ where $\tr_B$ denotes the partial trace over the auxiliary system.
Using the Kraus representation $\Phi(\rho) = \sum_k F_k \rho
F_k^{\dg}$ (and noting that the requirement that $\Phi$ be trace-preserving is equivalent to $\sum_k F_k^{\dg}F_k = I$), one can give a construction equivalent to Lindblad’s by initially defining $U_{AB}$ as $$\begin{aligned}
\label{eq:unit.def}
U_{AB}|\psi \ket \ot |\beta \ket \equiv \sum_k F_k|\psi \ket \ot
|k \ket,\end{aligned}$$ where $|\beta \ket$ is a fixed normalized state of the auxiliary system, and $\{ |k \ket \}$ is some orthonormal basis for the auxiliary system. Then $U_{AB}$ is a partial isometry from ${\cal H}_A \ot |\beta \kb \beta|$ to ${\cal H}_A \ot {\cal H}_B$ which can be extended to a unitary operator on all of ${\cal H}_A \ot {\cal H}_B$. This yields (\[eq:Lind.rep\]) with $\gamma_B = |\beta \kb \beta|$ a pure state.
However, $U_{AB}$ can also be extended to ${\cal H}_A \ot {\cal H}_B$ in other ways. In particular, it can be extended, instead, to the partial isometry for which $U_{AB}^{\dg} U_{AB}$ is the projection onto ${\cal H}_A \ot |\beta \kb \beta|$ so that $U_{AB} = 0$ on the orthogonal complement of ${\cal H}_A \ot |\beta \kb \beta|$. We describe this in more detail when $\Phi$ requires at most $m$ Kraus operators $F_k$, in which case one can choose the auxiliary system to be ${\bf C}^m$. One can also choose $| k \ket = |e_k \ket$, and $| \beta \ket = | e_1 \ket$ with $ |e_k \ket$ the standard basis of column vectors with elements $c_j = \delta_{jk}$. Then (\[eq:unit.def\]) depends only on the first column of $U_{AB}$ which we denote $V$ and regard as a map from ${\cal H}$ to ${\cal H} \ot {\bf C}^m$. In block form $$\begin{aligned}
\label{eq:Vrep}
V \rho V^{\dg} & = & U_{AB} \, \rho \ot |e_1 \kb e_1 | U_{AB}^{\dagger}
\\ & ~ & \nonumber \\ & = &
\pmatrix{ F_1 \cr F_2 \cr ~ \cr \vdots \cr ~ \cr F_m} \rho
\pmatrix{ F_1^{\dg} & F_2^{\dg} & \ldots & F_m^{\dg} }
= \pmatrix{ F_1 \rho F_1^{\dg} & F_1 \rho F_2^{\dg} & \ldots &
F_1 \rho F_m^{\dg} \cr F_2 \rho F_1^{\dg} & F_2 \rho F_2^{\dg}
& \ldots & F_2 \rho F_m^{\dg} \cr ~ & ~ & ~ & ~ \cr
\vdots & \vdots & ~ & \vdots \cr ~ & ~ & ~ & ~ \cr
F_m \rho F_1^{\dg} & ~ & \ldots & F_m \rho F_m^{\dg}} \nonumber\end{aligned}$$ from which it easily follows that $\tr_B (V \rho V^{\dg}) = \sum_k F_k \rho F_k^{\dg} = \Phi(\rho)$. The requirement that $\Phi$ be trace-preserving gives $V^{\dg} V = \sum_k F_k^{\dg} F_k = I$ which again implies that $V$ is a partial isometry. Moreover, $V \rho V^{\dg}$ has the same non-zero eigenvalues as $ (V \sqrt{\rho})^{\dg} (V \sqrt{\rho}) = \rho $ so that $S[V \rho V^{\dg}] = S(\rho)$.
This construction can be readily extended to situations in which $\Phi$ maps operators acting on one Hilbert space ${\cal H}_A$ to those acting on another space ${\cal H}_{A'}$, e.g., $\Phi : {\cal B}({\cal H}_A) \mapsto {\cal B}({\cal H}_{A'})$. In this case, the Kraus operators $F_k : {\cal H}_A \mapsto {\cal H}_{A'}$, and $U_{AB}$ is a partial isometry from ${\cal H}_A \ot |\beta \kb \beta|$ to a subspace of $ {\cal H}_{A'} \ot {\cal H}_B$. Alternatively, $V$ can be defined as a partial isometry from ${\cal H}_A$ to $ {\cal H}_{A'} \ot {\bf C}^m$.
Measurements and their representations {#sect:POVM}
--------------------------------------
A von Neumann or [*projective measurement*]{} is a partition of the identity $I = \sum_b E_b$ into mutually orthogonal projections, i.e., $E_b E_c = \delta_{bc} E_b$. A positive operator valued measurement (POVM) is a set of positive semi-definite operators $E_b$ such that $\sum_b E_b = I$, i.e., the orthogonality condition is dropped. It is well-known that a general POVM can be represented as a projective measurement on a tensor product space [@NC].
In fact, by noting that the map $\rho \mapsto \sum_b \sqrt{E_b} \, \rho \sqrt{E_b}$ is completely positive and trace-preserving with Kraus operators $F_b = \sqrt{E_b}$ one use the construction above. Write $V = \sum_b \sqrt{E_b} \ot | b \ket$ where $| b \ket$ is an orthonormal basis for ${\bf C}^M$ and $M$ is the number of measurements in the POVM, i.e., $b = 1 \ldots M$. Then $V \rho V^{\dg} =
\sum_{b,c} \sqrt{E_b} \, \rho \, \sqrt{E_c} \ot | b \kb c|$. Now, if $F_b = I \ot |b \kb b|$, then $\{ F_b \}$ is a projective measurement on ${\cal H} \ot {\bf C}^M$ and $\tr\, F_b \, (V \rho V^{\dg}) = \tr \, E_b \rho$.
Adjoint maps {#sect:adj}
------------
It is sometimes useful to consider the adjoint, which we denote $\wh{\Phi}$, of a stochastic map $\Phi$ with respect to the Hilbert-Schmidt inner product $\bra A, B \ket = \tr \, A^{\dg} B$. When $\Phi$ acts on $n \times n$ matrices, this adjoint (or dual) is fully defined by the requirement $$\begin{aligned}
\label{eq:HSadj}
\tr \, [\Phi(A)]^{\dg} B = \tr \, A^{\dg} \wh{\Phi}(B).\end{aligned}$$ for all $n \times n$ matrices, $A,B$. Indeed, when $\Phi(\rho) = \sum_k F_k \rho F_k^{\dg}$, the adjoint is given by $\wh{\Phi}(\rho) = \sum_k F_k^{\dg} \rho F_k$. Moreover, $\Phi$ is trace-preserving if and only if $\wh{\Phi}$ is unital, i.e, $\wh{\Phi}(I) = I$. When $\Phi$ is the partial trace, $T_2$, its adjoint takes $A \mapsto A \ot I_2$.
Subadditivity proofs {#sect:SSA}
====================
To understand the proof of strong subadditivity, it is instructive to first understand how Klein’s inequality can be used to prove two weaker inequalities. First, we consider the subadditivity inequality (\[eq:subadd\]). Substituting $A = \rho_{12}$ and $B = \rho_1 \ot \rho_2$ into Klein’s inequality (\[ineq:klein\]) yields $$\begin{aligned}
- S( \rho_{12} ) + S(\rho_1) + S(\rho_2) \geq
\tr \big( \rho_{12} - \rho_1 \ot \rho_2 \big) = 0, \end{aligned}$$ which is equivalent to subadditivity. Furthermore, the well-known conditions for equality in subadditivity follow from the conditions for equality in Klein’s inequality, namely that equality holds if and only if $\rho_{12}$ is a tensor product, that is, $\rho_{12} = \rho_1 \ot \rho_2$.
A second, more powerful subadditivity inequality was obtained by Araki and Lieb [@AkLb], $$\begin{aligned}
\label{eq:Arak.Lb}
S( \rho_{123} ) \leq S(\rho_{12} ) + S(\rho_{23} )\end{aligned}$$ under the constraint $\tr \rho_{123} = 1$. To prove this, choose $A = \rho_{123}$ and $B = e^{\log \rho_{12} + \log \rho_{23}}$ in Klein’s inequality to obtain $$\begin{aligned}
- S( \rho_{123} ) + S(\rho_{12} ) + S(\rho_{23} ) & \geq & 1 -
\tr e^{\log \rho_{12} + \log \rho_{23}}.\end{aligned}$$ Applying Theorem \[thm:gold\], to the right-hand side gives $$\begin{aligned}
- S( \rho_{123} ) + S(\rho_{12} ) + S(\rho_{23} ) & \geq &
1 - \tr_{123} \rho_{12} \rho_{23} \\
& = & 1 - \tr_2 (\rho_2)^2 \\
& \geq & 1-\tr_2 \rho_2 = 0,\end{aligned}$$ where the last line follows from $(\rho_2)^2 \leq \rho_2$ (which is the [*only*]{} place the normalization condition $\tr \rho_{123} = 1$ is needed).
The strategy for proving SSA is similar to that above, but with Theorem \[thm:gold\] replaced by Theorem \[cor:trip.gold\]. Let $A = \rho_{123}$ and choose $B$ so that $\log B = \log \rho_{12} - \log \rho_2 + \log \rho_{23}$. Then Klein’s inequality implies $$\begin{aligned}
\lefteqn{ - S( \rho_{123} ) + S(\rho_{12} ) - S(\rho_2) + S(\rho_{23} ) }
~~~~~~~~~~~~~~~~~~~~~~ \nonumber \\
& \geq &
\tr \left( \rho_{123} - e^{ \log \rho_{12} - \log \rho_2 + \log
\rho_{23}} \right).\end{aligned}$$ Applying Lieb’s result (\[eq:trip.gold\]) to the right-hand side above, we obtain $$\begin{aligned}
\lefteqn{ - S( \rho_{123} ) + S(\rho_{12} ) - S(\rho_2) + S(\rho_{23} ) }
~~~~~~~~~~~~~~~~~~~~~~ \nonumber \\
& \geq & \tr \left( \rho_{123} - \int_0^\infty \rho_{12} \frac{1}{\rho_2
+ u I}
\rho_{23} \frac{1}{\rho_2 + u I} du \right) \\ & = &
\tr_{123} \, \rho_{123} -
\tr_2 \int_0^\infty \rho_{2} \frac{1}{\rho_2 + u I}
\rho_{2} \frac{1}{\rho_2 + u I} du\\
& = & \left( \tr_{123} \, \rho_{123} - \tr_2 \rho_2 \right) = 0.\end{aligned}$$ This proves SSA. Moreover, this approach allows us to easily determine the conditions for equality, and thus complete the proof of Theorem \[thm:ssa.eq\]. The first inequality in the derivation above is satisfied with equality if and only if $A = B$ which is just the condition (\[ssa.equal\]). Although the conditions for equality in (\[eq:trip.gold\]) are more difficult to analyze, this is not necessary here. When $A = B$, it immediately follows that $\tr \, A =
\tr \, B$ so that the second inequality in the above derivation automatically becomes an equality when (\[ssa.equal\]) holds.
Inequalities for relative entropy {#sect:other}
==================================
Monotonicity under partial trace {#sect:mono}
--------------------------------
We now show how the same strategy can be applied to obtain a proof with equality conditions for the monotonicity of relative entropy under partial trace.
\[thm:mon.tr\] When $\rho_{12},\gamma_{12} > 0$ and $\tr \rho_{12} = \tr \gamma_{12}$ $$\begin{aligned}
\label{eq:mon.tr}
H(\rho_2,\gamma_2) \leq H(\rho_{12},\gamma_{12})\end{aligned}$$ with equality if and only if $ \log \rho_{12} - \log \gamma_{12} = \log \gamma_2 + \log \rho_{2} $.
This condition should be interpreted as $ \log \rho_{12} - \log \gamma_{12} = I_1 \ot
\Big[ \log \gamma_2 - \log \rho_{2} \Big] $. Since, as noted in section \[sect:adj\], when $\Phi = T_1$, the action $\wh{\Phi}$ is precisely $I_1 \ot $, the equality condition can be written as $ \log \rho_{12} - \log \gamma_{12} = \wh{T}_1
\Big[ \log T_1(\gamma_{12}) - \log T_1(\rho_{12}) \Big]$ which is a special case of the more general form (\[eq:eqal.gen.mono\]) developed later.
SSA can be regarded as a special case of this monotonicity result via the correspondence $\rho_{12} \raw \rho_{123}, ~\gamma_{12} \raw \rho_{12}$, and Petz’s form of the equality condition becomes $ \rho_{2}^{it} \gamma_2^{-it} = \rho_{12}^{it} \gamma_{12}^{-it}$. It is interesting to note that in [@SSA], Lieb and Ruskai actually obtained equation (\[eq:mon.tr\]) [*from*]{} SSA using the convexity of the conditional entropy $S(\rho_1) - S(\rho_{12})$ and the inequality (\[conc.deriv.ineq\]).
[**Proof:**]{} Let $A = \rho_{12}$, $\log B = \log \gamma_{12} - \log \gamma_2 + \log \rho_{2}$. Then Klein’s inequality and (\[eq:trip.gold\]) imply $$\begin{aligned}
H(\rho_{12},\gamma_{12}) - H(\rho_2,\gamma_2) & \geq & \tr_{12} \left(
\rho_{12}-
e^{\log \gamma_{12} - \log \gamma_2 + \log \rho_{2}} \right) \\
& \geq & \tr_{12} \left( \rho_{12} - \int_0^\infty \gamma_{12}
\frac{1}{\gamma_2 + uI} \rho_{2} \frac{1}{\gamma_2 + u I} du \right) \\ & = &
\tr_{12} \, \rho_{12} - \tr_{ 2}\int_0^\infty \gamma_{2}
\frac{1}{\gamma_2 + u I}
\rho_{2} \frac{1}{\gamma_2 + u I} du \\ & = &
\tr_{12} \, \rho_{12} - \tr_2 \rho_{2} = 0.
\end{aligned}$$ The equality condition is again precisely the condition $A =B$.
Joint convexity of the relative entropy {#sect:convexity}
---------------------------------------
The joint convexity of relative entropy can be obtained directly from Theorem \[thm:mon.tr\] by choosing $\rho_{12}$ (and similarly $\gamma_{12}$) to be a block diagonal matrix with blocks $\lambda_k
\rho^{(k)}$ (and $\lambda_k \gamma^{(k)}$). We can interpret the partial trace as a sum over blocks so that $\rho \equiv \rho_2 =
\sum_k \lambda_k \rho^{(k)}$. However, it is worth giving a direct proof of the joint convexity since it demonstrates the central role of Theorem \[exp.conc\].
\[thm:relent.conv\] The relative entropy is jointly convex in its arguments, i.e., if $\rho = \sum_k \lambda_k \rho^{(k)}$ and $\gamma = \sum_k \lambda_k \gamma^{(k)}$, then $$\begin{aligned}
\label{eq:relent.conv}
H(\rho, \gamma) \leq
\sum_k \lambda_k \, H \left( \rho^{(k)} \, , \, \gamma^{(k)} \right)\end{aligned}$$ with equality if and only if $\log \rho - \log \gamma = \log \rho^{(k)} - \log \gamma^{(k)}$ for all $k$.
[**Proof:**]{} Let $A = \rho^{(k)}$ and $\log B = \log \rho - \log \gamma+ \log \gamma^{(k)} $ with $\rho = \sum_k \lambda_k \rho^{(k)}$ and $\gamma = \sum_k \lambda_k \gamma^{(k)}$. Then Klein’s inequality implies $$\begin{aligned}
H \left( \rho^{(k)} \, , \, \gamma^{(k)} \right) -
\tr \, \rho^{(k)} \big[ \log \rho - \log \gamma \big] \geq
\tr \left( \rho - e^{\log \rho - \log \gamma + \log \gamma^{(k)}}
\right)\end{aligned}$$ Multiplying this by $\lambda_k$ with $\lambda_k > 0 $ and $\sum_k \lambda_k = 1$ yields, after summation, $$\begin{aligned}
\lefteqn{ \sum_k \lambda_k \, H \left( \rho^{(k)} \, , \, \gamma^{(k)}
\right)
- H(\rho,\gamma) } ~~~~~~~~~~~~ \\
& \geq & \tr \left( \rho - \sum_k \lambda_k
e^{\log \rho - \log \gamma + \log \gamma^{(k)}} \right) \\
& \geq & \tr \left( \rho -
e^{\log \rho - \log \gamma + \log \sum_k \lambda_k \gamma^{(k)}}
\right) \\
& = & \tr \left( \rho - e^{\log \rho} \right) = 0\end{aligned}$$ where the second inequality is precisely the concavity of $C \raw F(C) = \tr e^{K + \log C}$ with $K = \log \rho - \log \gamma$ and $ C = \sum_k \lambda_k \gamma^{(k)}$.
Relationships among inequalities {#sect:relation}
--------------------------------
We make some additional remarks about connections between SSA and various properties of relative entropy. To facilitate the discussion, we will use MONO to denote the general monotonicity inequality (\[eq:mono\]), MPT to denote the special case of monotonicity under partial traces, i.e., Theorem \[thm:mon.tr\], and JC to denote the joint convexity, Theorem \[thm:relent.conv\]. Using the restatement of SSA in the form (\[eq:mono.SSA\]), it is easy to see that MONO $\Raw$ MPT $\Raw$ SSA. Before theorem \[thm:relent.conv\], we showed that MPT $\Raw$ JC. Similarly, by choosing $\rho_{123}$ to be block diagonal with blocks $\rho_{123}^{k}$ one can show that SSA implies that the map $\rho_{12} \mapsto S(\rho_1) - S(\rho_{12} )$ is convex. In [@SSA] it was observed that applying the convexity inequality (\[conc.deriv.ineq\]) to this map (with $A + xB = \rho_{12} + x \gamma_{12}$), yields (\[eq:mon.tr\]). This shows that SSA $\Raw$ MPT so that we have the chain of implications $$\begin{aligned}
\label{eq:chain}
\hbox{MONO} \Raw \hbox{MPT} \iff SSA \Raw \hbox{JC}.\end{aligned}$$ One can show that JC $\Raw$ MPT by using Uhlmann’s observation [@Uhl1] that the partial trace can be written as a convex combination of unitary transformations.
One can also show directly that JC $\Raw$ SSA by using the purification process described in section \[sect:pure\] to show that SSA is equivalent to $$\begin{aligned}
\label{eq:SSA.alt}
\rho_4 + \rho_2 \leq \rho_{12} + \rho_{14} .\end{aligned}$$ Moreover, if $\rho_{124}$ is pure, then $\rho_4 = \rho_{12}$ and $\rho_2 = \rho_{14}$ so that equality holds in (\[eq:SSA.alt\]). Since the extreme points of the convex set of density matrices are pure states, the inequality (\[eq:SSA.alt\]) then follows from the joint convexity, Theorem \[thm:relent.conv\]. Thus we have $$\begin{aligned}
\label{eq:chain2}
\hbox{MONO} \Raw \hbox{MPT} \iff SSA \iff \hbox{JC}.\end{aligned}$$ Lindblad [@Lind75] completed this circuit by showing that MPT $\Raw$ MONO.
Using the representation described in Section \[sect:Lind.rep\], with $V$ the partial isometry from ${\cal H}$ to ${\cal H} \ot {\bf
C}^m$ as in (\[eq:Vrep\]), one finds $$\begin{aligned}
H\big[\Phi(\rho), \Phi(\gamma) \big] & = &
H \Big[ \, \tr_B \big( V \rho V^{\dg} \big) \, ,
\, \tr_B \big( V \gamma V^{\dg} \big) \, \Big]
\nonumber \\ \label{eq:mono.via.pt} & \leq &
H \Big[\,V \rho \, V^{\dg} \, , V \gamma \,V^{\dg} \, \Big] \\
& = & H(\rho,\gamma)\end{aligned}$$ since $\tr V \rho V^{\dg} \log (V \gamma V^{\dg}) = \tr \, \rho \log \gamma$ for a partial isometry $V$.
Equality in monotonicity under stochastic maps {#sect:gen.mono}
==============================================
Conditions for equality in the general monotonicity inequality (\[eq:mono\]) may be more subtle since it is not always possible to achieve equality. Indeed, it was noted in [@LesRu] that $ \sup_{\rho \neq \gamma} \frac{H[\Phi(\rho) ,\Phi(\gamma)] }
{H(\rho,\gamma) } $ can be strictly less than 1. Using the reformulation (\[eq:mono.via.pt\]) above, we prove the following result.
\[thm:eqal.gen.mono\] Equality holds in [*(\[eq:mono\])*]{}, $H\big[\Phi(\rho), \Phi(\gamma) \big] \leq H(\rho,\gamma)$, if and only if $$\begin{aligned}
\label{eq:eqal.gen.mono}
\log \rho - \log \gamma = \wh{\Phi}
\left[ \log \Phi(\rho) - \log \Phi(\gamma) \right] \label{eq:cond.d}\end{aligned}$$ where $\widehat{\Phi}$ denotes the adjoint of $\Phi$ with respect to the Hilbert-Schmidt inner product as defined in (\[eq:HSadj\]).
To verify sufficiency, multiply (\[eq:eqal.gen.mono\]) by $\rho$ and take the trace to obtain $$\begin{aligned}
H[\rho,\gamma] & = & \tr \, \rho \, \wh{\Phi}
\left[ \log \Phi(\rho) - \log \Phi(\gamma) \right] \\
& = & \tr \, \Phi(\rho)
\left[ \log \Phi(\rho) - \log \Phi(\gamma) \right] \\
& = & H\big[\Phi(\rho), \Phi(\gamma) \big] .\end{aligned}$$ It is tempting to follow our previous strategy and choose $A = \rho$, $\log B = \log \gamma + \widehat{\Phi}
\big[ \log \Phi(\rho) - \log \Phi(\gamma) \big]$. However, we have been unable to verify that $ \tr e^ {\log \gamma +
\widehat{\Phi} [ \log \Phi(\rho) - \log \Phi(\gamma) ]} \leq 1$ as required by this approach.
Instead, we use the representation (\[eq:Lind.rep\]) or (\[eq:Vrep\]). Rather than applying the equality conditions in Theorem \[thm:mon.tr\] directly to (\[eq:mono.via.pt\]), it is useful to repeat the argument for an appropriate choice of $A$ and $B$.
Choose $A = V \rho V^{\dg} $, $\log B = \log (V \gamma V^{\dg})
+ \log \tr_2 \big( V \rho V^{\dg} \big)
- \log \tr_2 \big( V \gamma V^{\dg} \big)$ where $V$ is again the partial isometry as in (\[eq:Vrep\]) of Section \[sect:Lind.rep\]. $B$ is defined so that the last two terms in $\log B$ are extended from ${\cal H}$ to ${\cal H} \ot {\bf C}^m$ so that $\ker(B) \subset \ker(A)$. The condition for equality in (\[eq:mono.via.pt\]) is then $$\begin{aligned}
\label{eq:eqal.mono.form.a}
\log ( V \rho V^{\dg} )
- \log (V \gamma V^{\dg} ) & = &
\log \tr_2 (V \rho V^{\dg} )
- \log \tr_2 (V \gamma V^{\dg} )
\\
& = & \log \Phi(\rho) - \log \Phi(\gamma) \nonumber\end{aligned}$$ We can put this into a more useful form by noting that for a partial isometry $V$ $$\begin{aligned}
\log \, ( V \rho V^{\dg} ) - \log \, (V \gamma V^{\dg} ) =
V \Big[ \log \rho - \log \gamma \Big] V^{\dg}\end{aligned}$$ from which it follows that (\[eq:eqal.mono.form.a\]) is equivalent to $$\begin{aligned}
\label{eq:cond.a}
V \Big[ \log \rho - \log \gamma \Big] V^{\dg}
= \log \Phi(\rho) - \log \Phi(\gamma) .\end{aligned}$$ Multiplying by $V^{\dg} $ on the left and $V$ on the right and using that $V^{\dg}V = I$, one sees that (\[eq:cond.a\]) implies $$\begin{aligned}
\label{eq:cond.b}
\log \rho - \log \gamma
= V^{\dg} \Big[ \log \Phi(\rho) - \log \Phi(\gamma) \Big] V .\end{aligned}$$ Taking the partial trace $\tr_2$ over the auxiliary space in (\[eq:cond.b\]) yields (\[eq:eqal.gen.mono\]) since $\wh{\Phi}(P) = \sum_k F_k^{\dg} P F_k = V^{\dg} P V$ for all $P$ in ${\cal H}$.
Another useful necessary condition for equality in (\[eq:mono\]) can be obtained by multiplying both sides of (\[eq:cond.a\]) by the projection $ VV^{\dg}$. Since $ V^{\dg}V = I$, one finds $$\begin{aligned}
\label{eq:imp.proj}
VV^{\dg} \Big[ \log \Phi(\rho) - \log \Phi(\gamma) \Big] & = &
V \Big[ \log \rho - \log \gamma \Big] V^{\dg} \nonumber \\
& = & \Big[ \log \Phi(\rho) - \log \Phi(\gamma) \Big] VV^{\dg}\end{aligned}$$ i.e., the projection $VV^{\dg}$ commutes with $\big[ \log \Phi(\rho) - \log \Phi(\gamma) \big] $. Taking the partial trace and noting that $\Phi(I) = \tr_2 VV^{\dg} $ we can summarize this discussion in the following
\[cor:eqal.mon.nec2\] If equality holds in [*(\[eq:mono\])*]{}, then $$\begin{aligned}
\label{eq:cond.c}
\Phi \big( \log \rho - \log \gamma \big) =
\Phi(I) \left[ \log \Phi(\rho) - \log \Phi(\gamma) \right]
= \left[ \log \Phi(\rho) - \log \Phi(\gamma) \right] \Phi(I).\end{aligned}$$ Moreover, $ \log \Phi(\rho) - \log \Phi(\gamma) $ commutes with the projection $VV^{\dg} = \sum_{k, \ell} |k \kb \ell | F_k F_{\ell}^{\dg}$ where $\{ F_k\}$ is a set of Kraus operators for $\Phi$, i.e., $\Phi(\rho) = \sum_k F_k \rho F_k^{\dg}$ and $| k \ket$ is an orthonormal basis for the auxiliary space ${\cal H}_2$.
The results of this section also hold in the more general situation when $\Phi : {\cal B}({\cal H}_A) \mapsto {\cal B}({\cal H}_A^{\prime})$ maps operators on one Hilbert space to those on another, in which case $F_k : {\cal H}_A) \mapsto {\cal H}_A^{\prime}$.
The Holevo bound {#sect:Holv}
================
Background
----------
One reason for studying conditions for equality is that other results, such as Holevo’s celebrated bound [@Hv0] on the accessible information, can be obtained rather easily from SSA or some form of the monotonicity of relative entropy. However, obtaining the corresponding conditions for equality is not as straightforward as one might hope because of the need to introduce an auxiliary system. Although Holevo’s bound is quite general, it is often applied in situations where $\wt{\rho}_j = \Phi(\rho_j)$ is the output of a noisy quantum channel $\Phi$ with input $\rho_j$. We use the tilde $\wt{}$ as a reminder of this, as well as to ensure a distinction from other density matrices which arise.
For any fixed POVM and density matrix $\gamma$, $p(b) = \tr \, (\gamma E_b)$ defines a classical probability distribution whose entropy we denote $S[\tr \, \gamma E_b]$. The Holevo bound states that for any ensemble of density matrices ${\cal E} = \{ \pi_j \wt{\rho}_j \}$ with average density matrix $\wt{\rho} = \sum_j \pi_j \wt{\rho}_j$ the accessible information in the ensemble satisfies $$\begin{aligned}
\label{eq:acc.inf}
I(\cale, \calm) & \equiv &
S[\tr \, \wt{\rho} E_b] - \sum_j \pi_j \, S[\tr \, \wt{\rho}_j E_b]
\\ & \leq & S(\wt{\rho}) - \sum_j \pi_j S(\wt{\rho}_j) \label{eq:Holv}\end{aligned}$$ for any POVM $\calm = \{ E_b \}$.. If all of the $\wt{\rho}_j$ commute, then it is easy to see that equality can be achieved by choosing the $E_b$ to be the spectral projections which simultaneously diagonalize the density matrices $\wt{\rho}_j$. We wish to show that this condition is also necessary, i.e., equality can only be achieved in (\[eq:Holv\]) if all the $\wt{\rho}_j$ commute.
It is known [@KR2; @YO] that (\[eq:Holv\]) can be obtained from (\[eq:mono\]). First, observe that $$\begin{aligned}
\label{eq:Holv.equiv}
S(\wt{\rho}) - \sum_j \pi_j S(\wt{\rho}_j) =
\sum_j \pi_j H(\wt{\rho}_j, \wt{\rho})\end{aligned}$$ Now let $ \Omega_{\calm}$ be the map $\Omega_{\calm}(A)
= \sum_b |b \kb b| \, \tr (A E_b)$ where $\calm = \{ E_b \}$. Then $\Omega_{\calm}$ is a stochastic map of the special type known as a Q-C channel and the Holevo bound (\[eq:Holv\]) follows immediately from (\[eq:Holv.equiv\]) and $$\begin{aligned}
\label{eq:Holv.Omega}
H[\Omega_{\calm}(\wt{\rho}_j), \Omega_{\calm}(\wt{\rho}) ]
\leq H(\wt{\rho}_j,\wt{\rho}).\end{aligned}$$
Equality conditions {#sect:eqal.Holv}
-------------------
We will henceforth assume that $\{ \pi_j ,\wt{\rho}_j\}$ is a fixed ensemble and seek conditions under which we can find a POVM satisfying the equality requirements. Since $\wh{\Omega}_{\calm} (D) = \sum_b E_b \bra b , D b \ket$, applying Theorem \[thm:eqal.gen.mono\] yields conditions for equality in (\[eq:Holv.Omega\]). For equality in (\[eq:Holv\]) these conditions must hold for every $j$ and reduce to $$\begin{aligned}
\label{eq:cond.g}
\log\wt{\rho}_j- \log \wt{\rho} = \sum_b \, E_b ~
\log \frac{\tr E_b \wt{\rho}_j }{\tr E_b \wt{\rho}}
~~ \forall ~ j\end{aligned}$$ where this should be interpreted as a condition on $\ker(\wt{\rho}_j)^\perp$ in which case all terms are well-defined. (Indeed, since the condition arises from the use of Klein’s inequality and the requirement $A = B$, the operators in $B$ must be defined to be zero on $\ker(A)$, which reduces to $\ker(\wt{\rho}_j)$ in the situation considered here.) If the POVM $\{ E_b \}$ consists of a set of mutually orthogonal projections, then it is immediate that the operators $Z_j \equiv \log\wt{\rho}_j- \log \wt{\rho} $ commute, since (\[eq:cond.g\]) can be regarded as the spectral decomposition of $Z_j$. To show that the $\wt{\rho}_j$ themselves commute, observe that $$\begin{aligned}
1 = \tr \, \wt{\rho}_j & = &
\tr \, e^{log \wt{\rho} + \, [\log \wt{\rho}_j - \log \wt{\rho}]} \\
& \leq & \tr \, \wt{\rho} ~ e^{\log \wt{\rho}_j - \log \wt{\rho}} \\
& = & \tr \, \wt{\rho} ~ e^{\sum_b \, E_b
\log \frac{\tr E_b \wt{\rho}_j }{\tr E_b \wt{\rho} } } \\
& = & \tr \, \wt{\rho} \, \sum_b \, E_b
\frac{\tr E_b \wt{\rho}_j }{\tr E_b \wt{\rho} } \\
& = & \sum_b \tr E_b \wt{\rho}_j = 1\end{aligned}$$ where we have used Theorem \[thm:gold\] with $A = \log \wt{\rho}, B = \log \wt{\rho}_j - \log \wt{\rho}$, and the fact that for orthogonal projections $e^{\sum_b a_b E_b} = \sum_b e^{a_b} E_b$. The conditions for equality in Theorem \[thm:gold\] then imply that $\log \wt{\rho}_j$ and $\log \wt{\rho}$ commute for all $j$. Hence $\wt{\rho}_j$ and $\wt{\rho}_k$ also commute for all $j,k$ when the POVM consists of mutually orthogonal projections.
Using King’s observation in the next section, one can reduce the general case to that of projective measurements. However, we prefer to use the equality conditions to show directly that the elements of the POVM must be orthogonal. Moreover, the commutativity condition involving $VV^{\dg}$ is reminiscent of the more sophisticated Connes cocyle approach used by Petz, and thus of some interest.
Since the Kraus operators for the Q-C map $\Omega_{\calm}$ can be chosen as $F_{kb} = |b \kb k | \sqrt{E_b}$ where $|b \ket$ and $|k \ket$ are orthonormal bases, one finds $$\begin{aligned}
\label{eq:proj.cond}
VV^{\dg} = \sum_{b,c} \sum_{k, \ell}
| b \kb c| \, \bra k \sqrt{E_b} \sqrt{E_c} \, \ell \ket
= \sum_{b,c} | b \kb c| \, \bra \phi \sqrt{E_b} \sqrt{E_c} \, \phi \ket
.\end{aligned}$$ By (\[eq:imp.proj\]), this must commute for all $j$ with $\log
\Omega_{\calm}(\wt{\rho}_j) -
\log \Omega_{\calm}(\wt{\rho}_j) $ which can be written in the form $ \sum_b z_{bj}|b \kb b| $ with $z_{bj} = \log \frac{\tr E_b\wt{\rho}_j }{\tr E_b \wt{\rho}} $. A diagonal operator of the form $ \sum_b z_b|b \kb b|$ with all $z_b \neq 0$ will commute with the projection in (\[eq:proj.cond\]) if and only if all off-diagonal terms are zero. This will hold if the POVM is a projective measurement, since then $\sqrt{E_b} \sqrt{E_c} = E_b E_c = E_b \delta_{bc}$. To see that this is necessary, note that the possibility that the vector $\phi$ is orthogonal to all $E_b$ is precluded by the condition that $\sum_b E_b = I$. Moreover, since the orthonormal basis $|k \ket$ is arbitrary, $\phi$ can be chosen to be arbitrary. The restriction that (\[eq:cond.g\]) hold only on $\ker(\wt{\rho}_j)^\perp$ may permit some $z_{bj} = 0$; however, for each $b$ there will always be at least one $j$ for which $z_{bj} \neq 0$, and this suffices.
One can obtain an alternate form of the equality conditions from Corollary \[cor:eqal.mon.nec2\]. Since $\Phi(I) = \sum_b |b \kb b| \tr E_b $, another necessary condition for equality in (\[eq:Holv\]) is $$\begin{aligned}
\label{eq:cond.e}
\tr E_b \left[ \log \wt{\rho}_j - \log \wt{\rho} \right] =
\tr E_b \Big( \log \tr E_b \wt{\rho}_j -
\log \tr E_b \wt{\rho} \Big)~~~~\forall ~j, ~b\end{aligned}$$ Inserting this in (\[eq:cond.g\]) yields the requirement $$\begin{aligned}
\label{eq:cond.f}
\log\wt{\rho}_j- \log \wt{\rho} & = & \sum_b \frac{1}{\tr E_b}
E_b \tr \, E_b \, \big[ \log\wt{\rho}_j- \log \wt{\rho} \big]\end{aligned}$$ which can be rewritten as $$\begin{aligned}
\label{eq:cond.Z}
Z_j = \sum_b \frac{ | E_b \ket }{\tr E_b} \, \bra E_b, Z_j \ket
~~~\forall j\end{aligned}$$ where $Z_j = \log\wt{\rho}_j- \log \wt{\rho} $ and the bra-ket now refer to the Hilbert-Schmidt inner product. This implies that $ \sum_b \frac{ | E_b \kb E_b | }{\tr E_b}$ projects onto the span($\{ Z_j \}$). However, this alone is not sufficient to imply that the $E_b$ form a projective measurement.
Other approaches
----------------
Chris King has observed [@King3] that when the POVM is a projective measurement of the form $E_b = |b \kb b|$, one can obtain the Holevo bound from the joint convexity of relative entropy. Let $\beta(\wt{\rho}) = \sum_b |b \kb b| \tr E_b \wt{\rho}$. Then applying Theorem \[thm:relent.conv\] to $H[\wt{\rho}, \beta(\wt{\rho})]$ yields $$\begin{aligned}
\label{eq:Holv.JC}
& ~ & -S(\wt{\rho}) + S(\tr E_b \wt{\rho}) \leq \sum_j \pi_j
\Big[ - S(\wt{\rho}_j) + S(\tr E_b \wt{\rho}_j) \Big] \\
{\rm or} & ~ & \nonumber \\
& ~ & S(\tr E_b \wt{\rho}) - \sum_j \pi_j S(\tr E_b \wt{\rho}_j) \leq
S(\wt{\rho}) - \sum_j \pi_jS(\wt{\rho}_j) \nonumber\end{aligned}$$ with equality if and only if $$\begin{aligned}
\label{eq:JC.Hv.cond.a}
\log \wt{\rho} - \sum_b |b \kb b| \log \tr E_b \wt{\rho} =
\log \wt{\rho}_j - \sum_b |b \kb b| \log \tr E_b \wt{\rho}_j
~~~\forall ~ j.\end{aligned}$$ This is equivalent to (\[eq:cond.g\]) when $E_b = |b \kb b|$, and the argument can be extended to more general projective measurements.
King also pointed out that if $\{ E_b \}$ is an arbitrary POVM, the construction in Section \[sect:POVM\] can be used to show that (\[eq:Holv\]) and (\[eq:cond.g\]) are equivalent to the equalities obtained when $\wt{\rho}_j$ is replaced by $V \wt{\rho}_j V^{\dg}$ and $E_b$ by $F_b$. Since the $\{ F_b \}$ form a projective measurement, we can conclude from the argument above that equality implies that all $V\wt{\rho}_jV^{\dg}$ commute, which implies that all $\wt{\rho}_j$ also commute since $V^{\dg} V = I$.
It should be noted that Petz was able to use his equality conditions to find the conditions for equality in the Holevo bound and this is sketched in [@OPW]. Indeed, Petz’s analogue of (\[eq:JC.Hv.cond.a\]) is $\wt{\rho}^{it} D^{-it} = \wt{\rho}_j^{it} D_j^{-it} ~~ \forall ~j$ where $D, D_j$ denotes the diagonal parts of $\wt{\rho}, \wt{\rho}_j$ respectively. Then $$\begin{aligned}
\label{eq:Petz.Hv}
\wt{\rho}_j^{it} = \wt{\rho}^{it} D^{-it} D_j^{it} .\end{aligned}$$ Since (\[eq:Petz.Hv\]) holds for all real $t$, as well as all $j$, it also implies $\wt{\rho}_j^{\, -it} = \wt{\rho}^{\,-it} D^{it} D_j^{-it}$.; However, taking the adjoint of (\[eq:Petz.Hv\]) yields $\wt{\rho}_j^{\,-it} = D_j^{-it} D^{it} \wt{\rho}^{\,-it} $. Therefore, $\wt{\rho}^{\, -it}$ commutes with the diagonal matrix $D^{it} D_j^{-it} = D_j^{-it} D^{it}$ and must also be diagonal. This gives a simultaneous diagonalization of all $\wt{\rho}_j^{it} $ which means that all $\wt{\rho}_j$ commute.
Holevo’s original longer derivation [@Hv0] of the bound (\[eq:Holv\]) also concluded that commutativity was necessary and sufficient for equality. Some simplifications of this argument were given by Fuchs [@Fuchs] in his thesis.
Another bound on accessible information {#sect:alt.bnd}
---------------------------------------
When $\rho$ is a density matrix, the mapping $A \mapsto \rho^{-1/2} A \rho^{-1/2}$ and its inverse gives a duality between ensembles and POVM’s. Hall [@Hall] observed that this duality can be used to give another upper bound on the accessible information (\[eq:acc.inf\]) in terms of the POVM and average density $\rho$, i.e., $$\begin{aligned}
\label{eq:alt.bnd}
I(\cale, \calm) & \leq & S(\rho) - \sum_b \tau_b \,
S\left( \ts{\frac{1}{\tau_b}} \sqrt{\rho} \, E_b
\sqrt{\rho} \right) \\
& = & \sum_b \tau_b \, H\left( \ts{\frac{1}{\tau_b}}
\sqrt{\rho} \, E_b \sqrt{\rho} \, , \, \rho \right)\end{aligned}$$ where $\tau_b = \tr\, E_b \rho$. This inequality can be obtained from the monoticity of relative entropy under the Q-C map $\Omega_{\cale}(A) =
\sum_j |j \kb j| \pi_j \rho^{-1/2} \rho_j \rho^{-1/2}$ applied to $H\left( \ts{\frac{1}{\tau_b}}
\sqrt{\rho} \, E_b \sqrt{\rho} \, , \, \rho \right)$ as in (\[eq:Holv.Omega\]); or as in [@KR2] where an equivalent bound was given. The argument in Section \[sect:eqal.Holv\] can then be used to show that equality can be achieved in (\[eq:alt.bnd\]) if and only if all $ \sqrt{\rho} \, E_b \sqrt{\rho} $ commute. Hall [@Hall] also found this condition and noted that it implies that $\rho$ commutes with every $E_b$ in the POVM.
One is often interested in (\[eq:Holv\]) and (\[eq:alt.bnd\]) when one wants to optimize the accessible information after using a noisy quantum channel, $\Phi$. It was observed in [@KR2] that, since $\tr \Phi(\rho_j) E_b = \tr \rho_j \wh{\Phi}(E_b)$, one can regard the noise as either acting to transform pure inputs $\rho_j$ to mixed state outputs $\Phi(\rho_j)$ [*or*]{} as acting through the adjoint $\wh{\Phi}$ on the POVM with uncorrupted outputs. In the first case, one can bound the right side of (\[eq:alt.bnd\]) by choosing the $E_b$ to be the spectral projections of the average output state $\Phi(\rho)$ to yield $I[\Phi(\cale), \calm] \leq S[\Phi(\rho)]$ which is weaker than the corresponding Holevo bound. Moreover, since the optimal choice for $\Phi(\rho_j)$ need not be in the image of $\Phi$, it not necessarily achievable even though the commutativity condition holds. Hall [@Hall] discussed other situations in which the bound can not be achieved despite the fact that all $ \sqrt{\rho} \, E_b \sqrt{\rho} $ commute.
Viewing the noise as acting on the POVM, King and Ruskai [@KR2] defined $$\begin{aligned}
\label{eq:UEP}
U_{EP}(\Phi) = \sup_{\rho,\calm} \left[ S(\rho) - \sum_b \tau_b \,
S\left( \ts{\frac{1}{\tau_b}}
\sqrt{\rho} \, \wh{\Phi}(E_b) \sqrt{\rho} \right) \right]\end{aligned}$$ with $\tau_b = \tr \rho \wh{\Phi}(E_b) = \tr \Phi(\rho) E_b$. If the supremum in (\[eq:UEP\]) is achieved with an average density and POVM for which $\sqrt{\rho} \, \wh{\Phi}(E_b) \sqrt{\rho}$ do not commute, then $U_{EP}(\Phi)$ is strictly greater than the accessible information. The questions of whether or not (\[eq:UEP\]) can actually exceed the optimal accessible information, and how it might then be interpreted are under investigation.
Concluding remarks {#sect:conc}
==================
The proof presented here for each inequality, SSA, Theorem \[thm:mon.tr\], Theorem \[thm:relent.conv\] and the general monotonicity (\[eq:mono\]), is quite short — only half a page using results from Section \[sec:tools\] which require less than one additional page [*and*]{} Theorem \[exp.conc\] . However, as shown in the Appendix, even this result does [*not*]{} require a long argument if one is permitted to use some powerful tools of complex analysis.
It is certainly not unusual to find that complex analysis can extremely be useful, even when the functions of interest are real-valued. Indeed, Lieb’s original proof of the concavity of WYD entropy used a complex interpolation argument. In his influential book [@Sim] on Trace Ideals, Simon (extracting ideas from Uhlmann [@Uhl2]) gave a longer“elementary” proof using the Schwarz inequality, perhaps inadvertently reinforcing the notion that any complete proof of SSA is long and forbidding. Similar ideas are implicit in Ando [@Ando] who restates the result in terms of tensor product spaces and block matrices. Uhlmann [@Uhl2] again demonstrated the power of complex interpolation by using it to prove the monotonicity of relative entropy under completely positive trace-preserving maps. SSA then follows immediately as a special case. However, Uhlmann’s approach, which has been extended by Petz [@Pz.mono; @OP], was developed within the framework of the relative modular operator formalism developed by Araki [@Ak; @BR; @OP] for much more general situations. Recently, Lesniewski and Ruskai [@LesRu] observed that within this relative modular operator framework, monotonicity can be established directly using an argument based on the Schwarz inequality.
The approach of this review is similar to that of Wehrl [@W] in that we view Theorem \[exp.conc\] as the “essential ingredient”. Indeed, Uhlmann [@Uhl1; @W], using a completely different approach, had independently recognized that Theorem \[exp.conc\] would imply SSA. However, Wehrl’s otherwise excellent review stated (at the end of section III.B) that “Unfortunately, the proof of \[this\] is not easy at all.” Later (in section III.C) Werhl again states that “... the proof is surprisingly complicated. I want to indicate only that the concavity of $\tr \, e^{K + \log A}$ can be obtained from Lieb’s theorem \[on concavity of the WYD entropy\] through a sequence of lemmas.” Although aware that Epstein’s approach [@Ep], which was developed shortly after Lieb announced his results, permitted a “direct” proof of Theorem \[exp.conc\], Wehrl does not seem to have fully appreciated it. The utility of Epstein’s technique may have been underestimated, in part, because he presented his results in a form which applied to the full collection of convex trace functions studied in [@LbWYD]. Checking Epstein’s hypotheses for the WYD function requires some non-trivial mapping theorems. This may have obscured the elegance of the argument in Appendix A.
It is worth noting that if the concavity of WYD entropy is regarded as the key result, it is not necessary to use the long sequence of lemmas Wehrl refers to in order prove SSA. Lindblad [@Lind74] gave a direct proof of the joint convexity, Theorem \[thm:relent.conv\], directly by differentiating the WYD function. Once this is done, SSA follows via the purification argument sketched after equation (\[eq:SSA.alt\]) or, alternatively, the variant of Uhlmann’s argument described in [@Sim; @W]. Combining this with Lieb’s original complex interpolation proof of the concavity of the WYD function, yields another “short” proof of SSA, albeit one which does not appear to be well-suited to establishing conditions for equality.
Finally, we mention that Carlen and Lieb [@CarLb] obtained another proof of SSA by using Epstein’s technique to prove some Minkowski type inequalities for $L_p$ trace norms. Using a different approach, King [@King1; @King2] recently proved several additivity results for the minimal entropy and Holevo capacity of a noisy channel by using $L_p$ inequalities in which Epstein’s technique provided a critical estimate. This suggests that connections with $L_p$ inequalities, as advocated by Amosov, Holevo and Werner [@AHW], may be a promising avenue for studying entropy and capacity in quantum information. Despite the results mentioned above, many open conjectures remain; see [@AHW; @CarLb; @King1; @King2; @WH] for further details.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work of M.B. Ruskai was partially supported by the National Security Agency (NSA) and Advanced Research and Development Activity (ARDA) under Army Research Office (ARO) contract number DAAG55-98-1-0374 and by the National Science Foundation under Grant numbers DMS-9706981 and DMS-0074566. This paper is the result of questions first posed to the author by M. Nielsen and it is a pleasure to acknowledge the value of extensive discussions with him. Some of these discussions took place, and parts of this paper were written, when MBR and MN were visiting the Institute for Theoretical Physics at the University of California, Santa Barbara and thereby also partly supported by the National Science Foundation under Grant PHY-9907949.
The author is extremely grateful to C. King for careful readings of earlier versions of the manuscript leading to many useful comments, and for communicating (and allowing to be included) several critical observations needed to obtain the equality conditions for the Holevo bound in Section \[sect:Holv\]. It is also a pleasure to thank C. Fuchs for explaining the history of the equality conditions in the Holevo bound, E. Lieb to clarifying remarks about Bauman’s responsibility for the actual conjecture of the concavity of the WYD function, D. Petz for discussions about his approach, and P. Shor for asking a question which stimulated the analysis in Section \[sect:alt.bnd\], and to M.J.W. Hall for bringing Reference [@Hall] to my attention.
Epstein’s proof of concavity of $A \raw \tr e^{K + \log A}$
===========================================================
Let $f(x) = \tr e^{K + \log (A + xB)}$ with $A > 0$ strictly positive and $K, B$ self-adjoint. For sufficiently small $x$, the function $f(x)$ is well-defined and the concavity of $F(A)$ in Theorem \[exp.conc\] follows immediately if $f^{\prime\prime}(0) < 0$ for all choices of $B = B^*$.
Instead of dealing with $f$ directly, Epstein considered the function $g(x) = x f(x^{-1})$ which is well-defined for $|x| > \mu^{-1} \equiv \| A^{-1}\| \, \| B \| $ and can be analytically continued to the upper half plane so that $$\begin{aligned}
g(z) = \tr \, e^{K + \log (zA + B)}.\end{aligned}$$ There are a number of equivalent (when meaningful) ways of defining functions of matrices. For the purposes needed here it is natural to assume that the spectrum $\sigma(A)$ of the operator $A$ is contained in the domain of an analytic function $F(z)$ and that $$\begin{aligned}
F(A) = \frac{1}{2 \pi i} \oint \frac{F(z)}{zI - A} dz.\end{aligned}$$
One can then use the spectral mapping theorem $\sigma [F(A)] \subset F[\sigma(A)]$ for an appropriate sequence of functions to verify that $$\begin{aligned}
\Im \, z > 0 & \Raw & \Im \, \omega(zA + B) > 0 \\
& \Raw & \pi > \Im \, \omega[\log(zA + B)] > 0 \\
& \Raw & \pi > \Im \, \omega[K + \log(zA + B)] > 0 \\
& \Raw & \Im \, \omega \left[e^{K + \log (zA + B)} \right] > 0 \\
& \Raw & \Im \, \tr \, e^{K + \log (zA + B)} > 0\end{aligned}$$ where $\Im $ denotes the imaginary part of a complex number and $\omega$ is used to denote an arbitrary element of the spectrum of the indicated operator. Thus, $g(z)$ maps the upper half plane into the upper half plane. Functions with this property have been studied extensively under various names, including, “operator monotone”, “Herglotz” or “Pick”. (See, for example, [@Ando; @D; @OP]). It then follows that $g$ has an integral representation of the form $$\begin{aligned}
g(z) = a + bz +
\int_{-\mu}^{\mu} \frac{1}{t - z} \, dm(t)\end{aligned}$$ for some positive measure $\mu(t)$. This yields (via the change of variables $s = t^{-1}$) $$\begin{aligned}
f(x) & = & ax + b + \int_{-\mu}^{\mu} \frac{x^2}{tx - 1} \, dm(t)\end{aligned}$$ Differentiation under the integral sign can then be used to establish that $f^{\prime\prime}(0) < 0$ as desired by observing $ \frac{x^2}{tx - 1} = t^{-2} [ (xt + 1) + (xt -1)^{-1} ].$
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|
---
author:
- |
L. E. Barr, P. Karlsen, S. M. Hornett, I. R. Hooper, M. Mrnka,\
C. R. Lawrence, D. B. Phillips, E. Hendry
bibliography:
- 'LEB\_THz\_NF\_Supp.bib'
title: 'Millimetre-wave total internal reflection based computational imaging - Supplementary Information'
---
Introduction
============
In this supplementary material further details on the experimental design will be discussed, along with a deeper analysis of the images that are presented in the accompanying paper. Future works are explored, including thickness and optical parameter extraction from images, and imaging through opaque media.
Parameters of the system
========================
Details of experimental setup
-----------------------------
Since imaging in the mm-wave range is still a challenge we employ all-optical modulation (known as photomodulation), using a visible or near-IR light source to modulate the low frequency radiation source in a photo-active medium (such as silicon). We use light from a 4.7 W, 623 $\pm$5 nm LED, which travels up through the transparent prism to the silicon wafer. This allows photoexcitation of the silicon wafer while keeping the top surface of the silicon free for samples. Using a digital micromirror device (DMD) from Vialux, we pattern the optical beam in order to locally modulate the photoconductivity in the silicon wafer. A light guide is used for homogeneous illumination of the DMD, and projection of the patterned light to our modulator occurs using a series of achromatic lenses (focal lengths 10 cm and 7.5 cm) and mirrors. We generate masks of spatially varying photoconductivity, which are imprinted on to the reflected mm-wave beam, providing the spatial information needed to construct an image using a single detector. We use a set of Hadamard masks, similar to [@Pratt1969; @Sloane1976a], which, when implemented correctly, can significantly boost the signal compared to conventional raster scanning [@Stantchev2016]. As shown below, this approach provides good signal to noise in images while avoiding the complex image reconstruction calculations associated with compressive imaging techniques [@Stantchev2016], allowing us to both measure and reconstruct the image in a few seconds. More details on the image reconstruction are given in section 3.1.
The majority of images shown in this paper are defined by 64 $\times$ 64 pixels, requiring $2 \times 64^2 = 8192$ masks to be generated, and the mm-wave signal collected for each one. We collect the measurements at a frame rate of 2 kHz, so each image is recorded in 4 seconds. Images presented below are an average of 100 sequential image measurements. Images with smaller numbers of pixels or fewer averages can be collected in shorter times, and are presented in section 3.3 for comparison.
Wafer thickness
---------------
![Simulated modulation assuming 140 GHz radiation incident on a wafer with a effective charge carrier lifetime and photoexcited by a 623 nm pump source of intensity 200 Wm$^{-2}$. The peaks in modulation are periodic with the wafer thickness, and suggest a Fabry-Perot cavity resonance in the wafer plays an important role on the modulation.[]{data-label="fig_sup_wafthick"}](Sup_Fig1_WafThick.png){width="10cm"}
We mention in the main text that the thickness of the wafer is very important when designing the imaging system. Fig. S\[fig\_sup\_wafthick\] shows the modulation calculated for an incident wave of 140 GHz at various angles for various wafer thicknesses. Two clear peaks are shown with a separation of around at the critical angle, as shown by the white arrow. The refractive index of silicon is 3.42, therefore the wavelength of the mm-wave beam inside the silicon is . The peaks are separated by almost one half-wavelength, and shift to higher frequencies as the angle of the incident beam is increased. These are both behaviours that are characteristic of a Fabry-Perot resonance inside the silicon.
Frequency and incident angle
----------------------------
![(a) Calculated modulation and (b) penetration length of evanescent fields at the silicon-air boundary as a function of both angle of incidence and frequency. The white dashed lines represent the critical angle above which total internal reflection occurs. The red circles highlight the set of parameters chosen for the experimental measurements.[]{data-label="fig_penetration"}](Fig3_Penetration.png){width="8cm"}
In Fig. S\[fig\_penetration\] (a) we plot the modulation as a function of both incident angle and frequency. We see that the modulation for all frequencies with incident angles below $\theta_{\textrm{crit}}$ (white dotted line) is small, but for larger angles the modulation is highly frequency dependent. We observe a peak in the modulation just above 140 GHz, which varies with the thickness of the silicon wafer, and is attributed to a resonant Fabry-Perot cavity mode inside the wafer (see Fig. S\[fig\_sup\_wafthick\]. Near this resonant feature, and for TE polarisation with an angle of incidence around 82$^{\circ}$, we predict a modulation depth of $>99\%$ with an optical illumination intensity of just 220 Wm$^{-2}$, achievable with a commercial LED source.
While matching the frequency to the silicon wafer thickness is beneficial to achieve the largest signals, one must also consider the thickness of the sample. When there is a dielectric medium above the silicon with a refractive index lower than that of the prism, the penetration length of the evanescent fields (L) will determine the depth of material that can be probed. We plot
$$\label{eqn_pen}
L = \frac{\lambda}{4\pi\sqrt{n_{\textrm{silicon}} \sin \theta_{\textrm{silicon}}^2 - n_{\textrm{sample}}^2}}$$
as a function of frequency and incident angle in the prism in Fig. S\[fig\_penetration\] (b), assuming a sample index of 1. $\theta_{\textrm{silicon}}$ is the angle of refraction in the silicon, and is related to the angle of incidence in the prism, $\theta_{\textrm{prism}}$ via Snell’s Law: $\theta_{\textrm{silicon}} = \arcsin \left( \frac{ n_{\textrm{prism}} \sin \theta_{\textrm{prism}}}{n_{\textrm{silicon}}}\right)$. For all frequencies and all angles below $\theta_{\textrm{crit}}$ there are no evanescent fields at the silicon-air boundary. Above $\theta_{\textrm{crit}}$ the penetration length is larger for lower frequencies and drops off towards higher frequencies. Thus, if we are interested in probing only the surface of a low-index sample, higher frequencies are more suitable. However, if one would like to probe the bulk of a sample, or even determine the thickness of layers, lower frequencies mean longer penetration of the fields into a sample. Hence, in practice, there is a trade off between the incident angles and frequencies which give the largest signals, and those that probe the bulk of the material in the half-space above the wafer. In the main text, we image tissue layers of a few hundred micron thickness, with refractive indices that are close to or higher than that of the prism. In this case the mm-wave beam is totally internally reflected from the sample-air boundary on the side furthest from the silicon, and instead the absorption and scattering of propagating waves inside the sample provide the contrast in the image. The red circles in Figs. S\[fig\_penetration\] (a) and (b) represent the parameters selected for the experimental measurements - a frequency of 140 GHz (chosen from a limited selection of available sources and the proximity to the wafer cavity resonance) and incident angle of 49$^{\circ}$ (corresponding to a penetration depth of around in air). The experimentally determined modulation for this geometry is 69.9% which is higher than the theoretical predictions in Fig. S\[fig\_penetration\] (a) of around 40%.
Measuring the effective charge carrier lifetime
-----------------------------------------------
![Plot of the normalised transmission of a 60 GHz signal through the thick wafer used in the experiment. The point where the intensity of the signal falls to $1/e$ of the original is marked, and corresponds to an effective charge carrier lifetime of .[]{data-label="fig_sup_LT"}](Sup_Fig2_LT.png){width="10cm"}
In order to measure the effective charge carrier lifetime of the wafer used for the photomodulator, the transmission through the wafer at 60 GHz was measured as a function of time as a photoexciting light source was modulated. Note that the lifetime of the charge carriers is a property of the wafer and will not change as a function of frequency. The wafer was oriented at a $45^{\circ}$ angle to a 1 ms TTL modulated photoexciting source (a collimated 4.8 W SOLIS-623C LED from Thorlabs with an output wavelength of 623 nm). The 60 GHz source was an Anritsu Vectorstar MS4647B Vector Network Analyser in CW mode with a Flann 25240 25 dB standard gain horn antenna, which was also oriented at $45^{\circ}$ to the wafer, and orthogonal to the photoexciting light. The magnitude of the signal transmitted through the wafer was detected using a Sage Millimeter SFD-503753-15SF-P1 waveguide detector connected to an oscilloscope triggered from the modulated light source. The waveform of both switch on and switch off events were recorded for a range of photoexciting intensities. Fig. S\[fig\_sup\_LT\] shows the transmission through the wafer as a function of time (blue dots) and an exponential curve fitted to the data. The photoexciting source is switched on at time = 0, and the time elapsed when the signal has fallen to $1/e$ of the original is taken to be the lifetime of the wafer, as marked by the black dashed lines.
Effect of charge carrier distribution
-------------------------------------
![Simulated modulation of a thick wafer with an effective lifetime of . The red line assumes a uniform homogeneous charge carrier distribution, each of the other lines are calculated assuming different combinations of bulk and surface lifetimes that each lead to an effective lifetime of. The inset shows a zoomed-in section that illustrates the small deviation in modulation for each wafer.[]{data-label="fig_sup_CarrDist"}](Sup_Fig3_CarrDist.png){width="10cm"}
Throughout the analysis in the main text we assume that the modulation from the silicon wafer could be predicted reasonably well by assuming that the distribution of charge carriers is uniform across the wafer thickness. Under this assumption the modulation depth is proportional to the excess charge carrier density, $\Delta n$, which is dependent only on the effective lifetime of the carriers, $\tau_{\textrm{eff}}$, the excitation function, $G$ (proportional to the intensity of the incident light) and the thickness of the silicon wafer, $d$, according to $\Delta n = \tau_\textrm{eff}G/d$.
For modulators in a transmission geometry this approximation is very good, as the wave will travel through the entire wafer and be subject to the total amount of charge carriers. However in the total internal reflection geometry this approximation is not necessarily valid. Instead it may be more appropriate to separate the recombination of charge carriers into bulk, $\tau_{\mathrm{b}}$ and surface $\tau_{\mathrm{s}}$ contributions, which are related to the effective charge carrier lifetime via $1/\tau_{\mathrm{eff}} = 1/\tau_{\mathrm{b}} + 1/\tau_{\mathrm{s}}$. However for comparison to the experiment we would need accurate measurements of bulk and surface lifetimes, and this is not a straightforward measurement to perform. Instead we can measure the effective lifetime (see figure \[fig\_sup\_LT\]) and use this value in the simulation. It is still pertinent to check that the assumption of effective lifetime is valid.
Fig. S\[fig\_sup\_CarrDist\] shows the calculated modulation for wafers of various bulk,$\tau_{\mathrm{b}}$, and surface, $\tau_{\mathrm{s}}$, charge carrier lifetime. The distribution of charge carriers in silicon was found using a diffusion calculation, assuming illumination from one side. More details can be found in reference [@Hooper2019]. The red line represents a wafer with homogeneous charge carrier distribution, and assumes an effective charge carrier lifetime of . The following lines show the modulation expected from wafers that each have an effective charge carrier lifetime of , but which have different bulk and surface lifetimes. The charge carrier distribution will be most inhomogeneous in wafers with large bulk lifetimes and high surface recombination rates, which correspond to low surface lifetimes. This is the case where we would expect the maximum deviation in modulation from the homogeneous distribution. However for all lifetime combinations investigated here (which cover the typical values found in undoped silicon wafers), the difference in modulation between homogeneous and non-homogeneous distribution does not exceed 5% . We can therefore say that our approximation still leaves us with a reasonably valid model for the modulation, although we must be aware that there is a possibility the simulation will predict modulations slightly higher than those measured in experiment. It is also worth keeping in mind that if the wafer thickness was reduced this approximation becomes worse as the surface recombination rate becomes more dominant in determining the effective charge carrier lifetime.
Parameters for imaging
======================
Orthonormal single-pixel computational imaging
----------------------------------------------
Without photomodulation of the silicon wafer, the total internal reflection geometry results in the entire underside of the sample being immersed in a mm-wave field when the source is active. Inhomogeneities in the complex refractive index of the sample result in a spatial variation in the level of scattering of the evanescent field. Therefore, the intensity of totally internally reflected mm-wave radiation spatially varies in a sample dependent manner. In order to reconstruct an image of this spatial variation using only a single-element detector, we sequentially project a series of $N$ patterns, each of which probes a different subset of the spatial information of the sample. In each case, the detector records the totally internally reflected signal $a_n$, which is proportional to the level of spatial overlap (i.e. dot product) between the scattering profile of the sample and the $n^{th}$ displayed 2D pattern $\mathbf{H}_n$. If these patterns are chosen to be orthonormal, then the resulting image $\mathbf{I}$ can be reconstructed from a sum of the patterns, each weighted by the measured overlap with the sample: $$\label{Eqn:reconstrction}
\mathbf{I} = \sum_{n=1}^{N}a_n\mathbf{H}_n,$$ where $\mathbf{I}$ represents a 2D image containing $N$ pixels. In this case the patterns are drawn from the orthonormal Hadamard basis, similar to [@Pratt1969; @Sloane1976a], which, significantly boosts the signal compared to a set of patterns equivalent to conventional raster scanning, as shown in [@Stantchev2016]. In practice, use of the fully sampled Hadamard basis requires the display of 2$N$ patterns to ensure that the patterns are orthogonal, as described in [@Gibson2017].
Number of pixels
----------------
![A comparison between images with $32 \times 32$ pixels (left panels) and $64 \times 64$ pixels (right panels), and a larger (top panels) and smaller (bottom panels) field of view.[]{data-label="fig_sup_pix"}](Sup_Fig1_pixels.png){width="8cm"}
When taking mm-wave images in the near-field we are able to select the field of view and the number of pixels of the image. Fig. S\[fig\_sup\_pix\] shows four images taken of the same porcine tissue sample with a larger (top row) and smaller (bottom row) field of view, and either 32 $\times$ 32 (left) or 64 $\times$ 64 (right) pixels. The darker areas are filaments of fat running through an area of protein.
Increasing the number of pixels does improve the clarity and resolution of the images, however this also leads to an increase in imaging time by a factor equal to twice the number of additional pixels. Reducing the field of view while maintaining the same number of pixels does not affect the time taken to collect images, and does allow finer structures in the sample to be seen more clearly. However reducing the field of view is done by decreasing the area of the digital micromirror device (DMD) that is used, and therefore reduces the portion of the mm-wave beam that is modulated and leads to an increase in the background signal. In extreme cases it will also limit the achievable resolution, as there needs to be as many micromirrors as pixels in the selected DMD region. Therefore when considering any application, careful thought should be given to the required speed and acceptable resolution, in order to decide on the most appropriate number of pixels.
Measurement time
----------------
![A selection of images taken of the same porcine tissue sample (optical image at top) taken with different imaging parameters. Rows contain images with the same frame rate, and columns contain images with the same averages. In the very top row the images have $64 \times 64$ pixels, all others have $32 \times 32$ pixels.[]{data-label="fig_sup_ave"}](Sup_Fig2_ave.png){width="16cm"}
Another parameter of the imaging system that can be tuned is the time over which the mm-wave signal is collected for each frame. Fig. S\[fig\_sup\_ave\] shows an array of images taken of the same porcine tissue sample at various frame rates and averages, all with 32 $\times$ 32 pixels apart from the top row which has 64 $\times$ 64 pixels, as indicated.
The signal to noise ratio is drastically improved by either increasing the number of averages taken, or decreasing the frame rate and increasing the signal collection time accordingly. We can investigate which of these methods is best for reducing the noise, as this depends on the nature of the noise itself. Taking only a few averages of signals collected over a long time with a low frame rate will effectively eliminate noise that is quickly varying. However if the noise in the system is varying slowly with time, it is more effectively minimised by increasing the number of averages taken for each frame.
In this system, when comparing two images that have equivalent total collection times (e.g. 20 averages at 2 kHz and 5 averages at 0.5 kHz), it can be seen that the high frame rate/high averaging option gives a signal to noise ratio that is slightly better than the low frame rate/low averaging option, although they are both very similar.
Polarisation
------------
![(a) Calculated reflection at 140 GHz from a thick silicon wafer with a charge carrier lifetime on a half-space of prism material (TPX polymer), as a function of incident angle for both TE and TM polarisations. In the dark state there is no photoexcitation, and in the bright state a 623 nm optical beam is incident on the wafer with an intensity of 220 W/m$^2$. (b) Comparison between images taken with the same imaging parameters but with different polarisations, represented by the arrows above each image.[]{data-label="fig_sup_pol"}](Sup_Fig3_pol.png){width="9cm"}
One other parameter at our disposal is the polarisation of the mm-wave beam we are using. The calculation presented in Fig. S\[fig\_sup\_pol\] (a) shows that, for an air sample, much larger modulation is expected for TE polarisation than for TM. This should lead to greater contrast in images, and improved signal-to-noise ratios.
To illustrate this, Fig. S\[fig\_sup\_pol\] (b) shows two images of porcine tissue taken with identical measurement parameters, but with opposite linear polarisations. It is clear that the contrast and signal-to-noise ratio are greatly improved for the TE polarisation compared to the TM.
Analysis of images
==================
In this section we will analyse some possible sources of artefacts arising from the imaging process, such as diffraction of mm-wave beam from the masks and the use of Hadamard masks. We will also discuss in more detail the interpretation of the information held in each image, including the extraction of material parameters and thickness of samples. Finally, we will present evidence of imaging objects hidden behind a layer of biological tissue.
Diffraction from mask
---------------------
![(a) shows the measured angular response of the detector. (b), (c) and (d) show the calculated diffraction pattern of a 140 GHz beam that is travelling through the prism material and reflected from a Hadamard mask shown in the inset. Coarse structured masks lead to some diffracted orders falling inside the detection range of the detector, but most masks have fine structures and the diffracted beams cannot be detected in the set-up.[]{data-label="fig_sup_ft"}](Sup_Fig4_FT.png){width="12cm"}
A fundamental part of our imaging technique requires periodic structures in the conductivity of the silicon to be created. For some of the masks in the Hadamard set, these can be of a similar size to the wavelength. Therefore for these masks there will be some diffraction of the mm-wave beam that is reflected off this structured surface. It could be suggested that this will lead to artefacts in the images, and so is worth looking at in more detail.
Fig. S\[fig\_sup\_ft\] (a) shows how the mm-wave signal measured by the detector changes as a function of angle. For the measurement, the source detector and mm-wave lens were positioned as they were in the imaging system. The source was modulated at 1 kHz and the signal from the detector recorded with a lock-in amplifier for each angle as the source was rotated about the exit aperture. It is seen from Fig. S\[fig\_sup\_ft\] (a) that the signal has significantly reduced to less than 20% at angles greater than $\pm 10^{\circ}$ from normal incidence.
Figs. S\[fig\_sup\_ft\] (b) - (d) show how this angular response compares to typical angles of diffraction from a selection of Hadamard masks, shown in the inset in each plot. To obtain these, a 2-dimensional fast Fourier transform was taken of the electric field at the surface of each mask, assuming that it is projected over an area of 25 mm$^2$, and for a frequency of 140 GHz. A slice is taken through the centre of the 2-dimensional plot for comparison. Only masks with 1-dimensional structure have been presented here for simplicity, but this technique can be applied to all masks, as diffraction along the y-axis has the same behaviour as diffraction along the x-axis.
Panel (b) in Fig. S\[fig\_sup\_ft\] shows the Fourier transform of a mask with relatively coarse structure, which is one of very few that will produce diffracted orders within the acceptance angle of the detector. In this case, the first diffracted order will be detected when imaging. However, as the masks become finer and more sub-wavelength in structure, such as those in panels (c) and (d), the beams are diffracted at higher angles and will not be recorded by the detector. We can say that the majority of masks have a structure that is sufficiently sub-wavelength that the diffracted orders will not interfere with collected signals.
It is also worthwhile considering the fact that the total power that can be diffracted into any of the higher orders is very limited. Ignoring the diffusion of charge carriers, and the less than 100% modulation, the masks can be approximated as a square-profile amplitude grating. The diffraction efficiency depends on the duty cycle of the grating, which will vary for different masks, and is highest for a 50% duty cycle grating. According to calculations performed here [@Harvey2019], at most 10.1% of the total power can be diffracted into the first order mode, and will decrease for duty cycles other than 50% and for less than 100% modulation, and further due to the smearing of the square profile due to charge carrier diffusion. Taking into account the angles and efficiency of diffraction, for the majority of the masks used in the imaging process, collection of diffracted beams is not a likely cause of artefacts in the images that we collect. The masks that may be affected by diffraction are those with very coarse structure, and if this was a significant effect we would clearly see a coarse structure over the images, and we do not.
Simulation of imaging
---------------------
![Results of simulation of imaging a Siemens star. The original object is in panel (a), (b) shows an image taken with a $64 \times 64$ raster scan, (d) shows an image taken with a $32 \times 32$ pixel Hadamard scan, and (c) shows an image taken with a $64 \times 64$ pixel Hadamard scan.[]{data-label="fig_sup_simim"}](Sup_Fig3_simim_2.png){width="12cm"}
Another possible cause of unusual features in the images could be artefacts arising from the use of Hadamard masks themselves. In order to investigate this, a simulation of the imaging was created based on the analytical methods described in [@Hooper2019].
A transfer matrix code is used to calculate the reflection from an infinitely large area of several different stacks that are possible in the imaging system, i.e. combinations of illuminated or dark silicon, and protein or fat samples. These values are assigned to each pixel of a mask, and for mask pixels that cover regions with both fat and protein, linear interpolation between reflection coefficients is used. An arbitrary binary pattern of fat and protein regions is drawn. This is then overlaid with a Hadamard matrix, and the material in each pixel is used to select the reflection coefficients for dark and illuminated silicon. The reflection coefficients for each pixel are added and normalised to find the total reflected signal from that mask, which is then used in the reconstruction in the same way as in the experiment to find the value of $\delta r$. $\delta r$ is the difference in reflection when the modulator is on (illuminated) and off (dark), similar to the modulation but with a sample in place. This is repeated for all masks in the set, and to obtain a true comparison with experiment, the matrices are separated into positive (i.e. in the Hadamard matrix +1s become 1s and -1s become 0s) and negative masks (i.e. +1s become 0s and -1s become 1s) in the simulation as well.
This model is a very simple approximation of what is happening in the imaging. It does not take into account any effects due to diffraction of the mm-wave beam by the mask or the sample, and any resonances in the plane of the sample will not be reproduced, as each pixel is assumed to be infinitely large and not interacting with the neighbouring pixels. However it is still a useful tool for interpreting images and predicting the effects of sample index and thickness.
![Plots of simulated $\delta r$ as a function of sample thickness and incident angle of the mm-wave beam. All results are calculated at 140 GHz mm-wave frequency for a thick silicon wafer with a charge carrier lifetime of , and an incident photoexcitation wavelength of 623 nm and power of 220 Wm$^{-2}$ when illuminated. The samples in each column have the same real permittivity, and in each row the imaginary permittivity is the same.[]{data-label="fig_lossmod"}](Sup_Fig_LossMod.png){width="12cm"}
Fig. S\[fig\_sup\_simim\] shows the result of a simulation of a fat and protein sample. The thickness of the sample is , and the material parameters for fat and protein are the same as in the main text, along with all other experimental parameters. The object being imaged, (a), is a Siemens star made of arms of fat (cream) and protein (pink). The image in panel (b) was taken using a simulated raster scan, (c) using a lower resolution simulated Hadamard scan, and (d) using a simulated Hadamard scan with the same resolution as (b). It is clear that the magnitude of signals collected is much larger when using Hadamard masks over Raster scanning, and also that the resolution is reduced when the number of pixels is reduced. However, in all images there are no features or artefacts that arise from the use of Hadamard masks.
Sample losses and critical coupling
-----------------------------------
It is explained in the main text of the paper that a change in sign of an image is due to a change in sign of the $\delta r$. This is proposed to be due to meeting the condition for critical coupling, where absorption of the mm-wave beam will be most efficient. This is achieved when the damping coefficients for radiative and non-radiative loss mechanisms are equal [@Herminghaus1994], and is due to a Fabry-Perot resonance in the thickness of the sample. This condition, rather counter-intuitively, can be reached by reducing the losses in the silicon wafer, and lead to a situation where more of the mm-wave is absorbed when the silicon is not illuminated. If this is a correct interpretation, we expect that the losses in the sample will play an important role in reaching this condition. We propose that it will also depend on a resonance determined by the thickness and index of the sample, which provides the radiative loss component. In this section we explore the parameter space, and look at how the $\delta r$ changes with real and imaginary parts of the permittivity of the sample.
Fig. S\[fig\_lossmod\] shows the $\delta r$ as a function of the incident angle and sample thickness for a range of real and imaginary permittivities. The sample that is closest to the experimental example in the main text is (c). Comparing (a) and (b), we see that changing the real part of the permittivity changes the sample thickness at which the resonant condition is met, and shifts the resonant peak in $\delta r$ to lower thicknesses for higher indices, as well as reducing the angular dispersion. These are both features of a Fabry-Perot type cavity resonance in the sample. However, due to the low losses in the sample in (a) and (b), we do not observe any negative $\delta r$ below very high incident angles. Increasing the loss, as in (c) and (d), introduces a range of thicknesses where the $\delta r$ is negative, followed by a range of sample thicknesses that give positive $\delta r$. Again the real part of the permittivity effects the thickness and angle where this is observed. Finally, in (e) and (f), the losses are increased again, and the magnitude of the positive $\delta r$ drops in general, while there is still a region where negative $\delta r$ can be observed.
Variation in tissue properties
------------------------------
![Calculated $\delta r$ as a function sample thickness and incident angle for protein (panels (a) and (c)) and fat (panels (b) and (d)) samples, with permittivities taken from reference [@Gabriel1996] (panels(a) and (b)), [@Bowman2016] (panel (C)) and [@Ashworth2009] (panels (d)). All results are calculated at 140 GHz mm-wave frequency for a thick silicon wafer with a charge carrier lifetime of , and an incident photoexcitation wavelength of 623 nm and power of 220 W/m$^2$ when illuminated.[]{data-label="fig_fatprotein"}](Sup_Fig5_FatProtein3.png){width="12cm"}
There are varying values for the permittivity of biological tissues reported in the literature. For the analysis conducted in this work we use values reported by Gabriel et al [@Gabriel1996], where the complex permittivities are $\epsilon_{\mathrm{fat}} = 2.89 + 0.64i$ and $\epsilon_{\mathrm{protein}} = 8.63 + 11.20i$. Fig. S\[fig\_fatprotein\] (a) and (b) show the $\delta r$ calculated as a function of incident angle and sample thickness for layers of protein and fat respectively, assuming these permittivities. However, there are wide ranges of values reported for the complex permittivities of fat and protein in the literature [@Gabriel1996; @Bowman2016; @Ashworth2009; @Sun2009; @Sun2011]. As an example, Bowman et al [@Bowman2016] report a different value for the permittivity of protein, $\epsilon_{\mathrm{protein}} = 8.28 + 5.10i$. Note in particular the discrepancy between the losses in values reported by both references. Fig. S\[fig\_fatprotein\] (c) shows the calculated $\delta r$ as a function of sample thickness and incident angle for protein, assuming the material properties from Bowman et al [@Bowman2016]. The effect of the lower losses given by Bowman et al is evident from the general increase in the magnitude of the $\delta r$, as well as the stronger dependence on the sample thickness, as there is less absorption of the mm-wave beam.
It is also worth noting that in reference [@Ashworth2009], Ashworth et al state a real part of the permittivity of fat that is just below that of our prism ($\epsilon_{\mathrm{prism}} = 2.5 > \epsilon_{\mathrm{fat}} = 2.29 + 1.05i$). In this case, the contrast mechanism for the imaging changes slightly, as the mm-wave is now reflected from the silicon-sample boundary at high incident angles. Fig. S\[fig\_fatprotein\] (d) shows the $\delta r$ calculated using this permittivity. By comparison with Fig. S\[fig\_fatprotein\] (b) we see that the magnitude of the $\delta r$ decreases, as the imaginary part of the permittivity given by Ashworth et al is higher. In addition, a band of large negative $\delta r$ at high angles is seen. This is indicative of evanescently decaying fields inside the sample, which are very sensitive to the material parameters, but not to the thickness.
Extracting parameters
---------------------
![(a) Calculated $\delta r$ as a function sample thickness and incident angle, for a sample with permittivity $\epsilon = 3.5 + 0.525i$. (b) Calculated $\delta r$ as a function of the real part of the sample permittivity and incident angle, for a thick sample with imaginary permittivity $\epsilon_i = 0.525$. (c) and (d) Calculated (blue line, blue circles show the points chosen for fitting) $\delta r$ for a sample with material parameters listed in blue inset box, and the $\delta r$ calculated using the fitted material parameters in the yellow inset box (yellow circles). The grey shaded region shows the range of angles below the critical angle.[]{data-label="fig_fitting"}](Sup_Fig5_Fitting2.png){width="12cm"}
As mentioned in the main text, the ability to extract the permittivity and thickness of a sample from an image would open the door to many applications. Fig. S\[fig\_lossmod\] also gives us another insight, as we can see that the real and imaginary parts of the sample permittivity change the $\delta r$ in independent ways. In other words, in any of the plots in Fig. S\[fig\_lossmod\], any vertical slice of data taken will not be identical to the same slice in any other, with particularly large contrast at higher angles. This suggests that, if we were able to collect enough information, it is possible to determine the complex permittivity of a sample. Furthermore, Fig. S\[fig\_fitting\] (a) and (b) show that the permittivity and thickness of the sample affect the $\delta r$ in non-identical ways. We can show that it is in fact possible to extract the complex permittivity and thickness of a sample from the $\delta r$.
Figs. S\[fig\_fitting\] (c) and (d) show the results of two such extractions from simulated data, for two different samples. The inset blue box contains the expected parameters, and the blue circles on the plot correspond to the $\delta r$ calculated from that sample at 6 different angles. The $\delta r$ was then used in a non-linear least-squares fitting algorithm, and the parameters that we found as a result of the fitting are in the yellow box inset in each plot. The fitted $\delta r$ is also shown by the yellow circles.
In both cases the fit is very good, and we are able to extract accurate values for the thickness, and real and imaginary parts of the permittivity simultaneously. While six data points have been given to this fitting routine, in theory only three points are needed to extract the three parameters. Adding more data reduces the chances of finding a local minimum, which is always possible. In addition, if fitting to experimental data, noise can introduce some uncertainty and means local minima are less easy to avoid. However more complex and robust fitting routines can be used to help avoid this, such as genetic algorithms, simulated annealing or swarm optimisation.
Imaging through tissue
----------------------
![Visible (bottom row) and mm-wave (top row) images of the same sample of porcine tissue. (a) shows the sample in isolation, and in (b) a rigid strip of metal was placed on the far side of the sample, covering regions of both fat and protein. The metal is visible in the mm-wave image through fatty tissue but not through the protein due to the high mm-wave absorption. Both 64 $\times$ 64 pixel mm-wave images were taken at 140 GHz using TE polarisation, and are averages of 100 images that took 4.1 seconds each to collect.[]{data-label="fig_metal"}](Fig5_HamRuler.png){width="10cm"}
We present one final demonstration of near-field imaging using a TIR photomodulator, in which we image a metal object on the far side of a thin porcine tissue sample. This is shown in Fig. S\[fig\_metal\], using the same experimental parameters as described in the main text. Fig. S\[fig\_metal\] (a) shows the porcine tissue with only air on the top side, whereas in Fig. S\[fig\_metal\] (b) a rigid metal strip (shaded region in optical image) has been placed behind the tissue, on the opposite side to the modulator. In the region where the metal strip is behind the protein it cannot be observed, as the mm-wave is heavily absorbed within the protein, and the TIR wave is not strongly reflected due to the large real permittivity of protein. However the metal is clearly visible as an increase in the reflected mm-wave in the fatty tissue region, seen by comparison to Fig. S\[fig\_metal\] (a). This demonstrates that it is possible to see through a thin layer of fat and observe objects behind it using a TIR mm-wave imaging technique.
|
---
abstract: 'The purpose of this study was to test some validation simulations for a 3D finite element model of contracting muscle. The model was based on continuum theory for fibre-reinforced composite materials. Here we simulated contractions for an idealized medial gastrocnemius muscle in man, using the model. Simulations were performed to test the force-length relation of the whole muscle, to evaluate the changes in internal fascicle geometry during contractions, and to assess the importance of material formulations for the aponeurosis and tendon. The simulation results were compared to previously published experimental values. The force-length profile for the whole muscle showed a realistic profile. As the muscle contracted the fascicles curved into S-shaped trajectories and curled around 3D paths, both of which matched previous experimental findings. As the fascicles shortened they increased in their cross-sectional area, but this increase was asymmetric with the smaller increase occurring within the fascicle-plane: the Poisson’s ratio in this plane matched that previously shown from ultrasound imaging. The distribution of strains in the aponeurosis and tendon was shown to be a function of their material properties. This study demonstrated that the model could replicate realistic patterns of whole muscle-force, and changes to the internal muscle geometry, and so will be useful for testing mechanisms that affect the structural changes within contracting muscle.'
author:
- |
Hadi Rahemi$^{*,1}$, Nilima Nigam$^2$, James M. Wakeling$^1$\
[1. Dept. of Biomechanics and Kinesiology, Simon Fraser University]{}\
[2. Dept. of Mathematics, Simon Fraser University]{}
bibliography:
- 'refjbio.bib'
title: 'Structural Changes of Active Skeletal Muscles: Modelling, Validation and Numerical Experiments'
---
[**Keywords:**]{} Skeletal muscle model, finite element method, model validation, fascicle pennation, fascicle curvature, connective tissue properties
Introduction
============
Forces developed by contracting skeletal muscle depend on the structure and geometry of the contracting fascicles, and their interaction with the surrounding connective tissues. Recent studies have highlighted the complexity of the internal structure of the muscles in 3D, and the changes to this structure during contraction e.g. [@Rana20133d]. However, relatively little is known about the mechanisms that relate the structure to function. It is likely that regional variations in muscle structure, tissue properties and activation patterns all contribute to the force output from the muscle. In order to understand such effects it is necessary to use a muscle model that can incorporate these complexities. An efficient way, in terms of both time and cost, to test these effects would be with a 3D finite element simulation platform based on a realistic mathematical model of muscle.
Muscle models and their related simulations have evolved over the last decade to incorporate 3D structural and architectural parameters such as fascicle orientations and connective tissue properties e.g.,[@Oomens2003; @Blemker2005; @Bol2008]. Features such as fascicle activation patterns, structural changes (for instance changes in fascicle curvature and orientation) under isometric and dynamic contractions and their effects on the force and power generated by the whole muscle have been investigated in a number of previous works e.g.[@rahemi2014regionalizing; @Carrasco1999]. While recent developments in imaging and signal processing techniques are enhancing our ability to measure detailed structure [@namburete2011computational; @Rana20133d] and activation profiles e.g.[@hodson2013myoelectric; @kinugasa2011unique; @staudenmann2009heterogeneity] in a muscle, all the intended parameters may be hard or impossible to collect in a single experiment. Therefore, there is a need to use mathematical models to get insight into muscle function where large number of parameters can be manipulated or measured during a simulation of muscle contraction.
Here we present the results of 3D finite element simulations of a skeletal muscle model that has been developed specifically to investigate the relation between the muscle’s internal structure and activation patterns and its force output [@rahemi2014regionalizing]. The model has the ability to include detailed 3D architecture and regionalized submaximal activity in different groups of fascicles. It integrates the effects of different tendon and aponeurosis properties on the force transfer within the muscle-tendon unit from its origin to insertion. Furthermore, we have previously shown that this mathematical modelling framework can predict the deformations of the internal structure within the muscle, and the force vector developed by the whole muscle, while the activity patterns within the muscle can be varied and regionalized [@rahemi2014regionalizing].
The main purpose of the current work is to present the validity of this modelling framework using different sets of experimental data. A validated computational model of muscle can be used to test mechanisms and investigate the effect of parameters that are difficult or impossible to measure. The second purpose of this work is to demonstrate some of the effects of the tendon and aponeurosis properties on the structural properties of the muscle during contraction.
Methods
=======
A 3D finite-element model of MG muscle was developed based on the continuum theory for fibre-reinforced composite materials. The tissues were transversely isotropic and were constrained to have nearly incompressible behaviour. The mathematical framework for this work has been previously described [@rahemi2014regionalizing]. The computational model was validated by comparing the force-length properties of the whole muscle to experimental measures, and also by comparing the shape, orientation and curvature of the modelled muscle fascicles to similar measures that have recently been made available through ultrasound imaging studies.
A unipennate muscle belly was modelled with dimensions similar to the medial gastrocnemius in man. The model coordinate system had the Z-axis running proximal-distal along the line of action of the muscle, the Y-axis ran from the deep to the superficial direction and the X-axis ran across the medial-lateral width of the muscle. This model had the same constitutive law and geometries that we have previously used [@rahemi2014regionalizing]. However, for this study the activation patterns and structural parameters along with mathematical boundary and initial condition were altered. The end planes of aponeuroses were defined as the transverse planes where the aponeuroses would join onto the external tendons, and mark the proximal and distal ends of the muscle belly. Some simulations were run for isometric contractions of the muscle belly where the end planes of the aponeuroses were fixed. Other simulations were run for the whole muscle-tendon unit with the external tendons included: for these, the proximal and distal ends of the muscle-tendon unit were fixed during contraction.
Simulations in this study were done using a set of C++ libraries for finite element modelling [@dealii]. Each simulation was run with an increasing and uniform level of activation across all fascicles. The simulations were terminated when the nonlinear iterations did not converge within specified tolerences within given number of steps; this point depended on the initial state and boundary conditions for each simulation. Where groups of simulations are compared together, they were compared up to the highest activation level that was commonly achieved across the set. Each simulation took approximately 10 minutes to run \[on a standalone 8-core (16 thread) computer\], and this time included that for mesh initialization, matrix setup, iterative solving and result output.
Simulation vs. Experiments - Validation of a muscle model
---------------------------------------------------------
Two sets of simulations were carried out on a muscle belly geometry (see Figure 1 in [@rahemi2014regionalizing]). Initially the muscle belly was a parallelepiped with 65 mm initial fascicle length, 15 degree pennation angle, and each aponeurosis was a rectangular cuboid of 210 $\times$ 55 $\times$ 3 mm. The initial stretch values for both the muscle and aponeuroses fascicles were set to one. This stretch corresponds to the optimal length for the muscle fascicles. A set of simulations was run to map the force-length relation for the muscle belly, and a second set of simulations was run to test the trajectories of the muscle fascicles and the strains within the tissues during contraction.
### Force-length test for isometric contractions of a muscle belly
The model of the muscle belly was adjusted to different lengths by fixing one end at its aponeurosis end plane, and passively displacing the other aponeusosis end plane to a new position. When the length of the muscle belly reached the desired length, both end planes for the aponeuroses were fixed to maintain the muscle belly at an isometric length, and the activation level in the muscle fascicles was then ramped up. The range over which the muscle belly length changed was selected so that pre-activation fascicle stretch in the fascicles was between 0.75 and 1.35. This is close to the range for stretches in the medial gastrocnemius (MG) that have been reported when the ankle is passively moved from 30 degrees plantarflexion to 15 degrees dorsiflexion [@Maganaris1998]. To achieve this, the muscle belly was shortened about 6% for the lower bound of the fascicle stretch range. However, lengthening of the belly was selected to surpass the natural range so the force-stretch curve could be plotted for a longer range. The simulations at different lengths reached a common activity level of 30%. The magnitude of the passive and total belly forces were computed along with the muscle fascicle lengths at which those forces were developed. The active muscle force was taken as the difference between the total force and the passive force for a set of common muscle fascicle lengths.
### Internal structural changes during isometric contractions of the muscle belly
Both end planes of the aponeuroses for the initial geometry were fixed and the activation was uniformly ramped up. Geometrical properties of fascicles both in 2D (fascicle curvature) mid-longitudinal and transverse planes (Figs. \[fig:planes\], \[fig:curvemap\]) and 3D (fascicle path, along-fascicle and transverse strains) were measured at different activity levels (Fig. \[fig:3dpath\] and Table \[tab:strains\]). Undeformed fascicles (Figs. \[fig:planes\], \[fig:3dpath\]) were chosen as groups of points that fit along lines that connect the two aponeuroses and have 15 degrees inclination (pennation) in the initial geometry. These fascicles were then tracked throughout all simulations to measure the structural deformations at the fascicle level. The mean pennation and curvature of the fascicles along with the along-fascicle (longitudinal) and transverse strains were extracted from the deformed fascicle data after the contractions had been simulated. The extent of fascicle curvature across the whole muscle belly in its mid-longitudinal plane was quantified by its root-mean-square (RMS) value for each activity level (% MVC). Fascicle sheets were defined as the 3D faces that run longitudinally through muscle and contain fascicles that were originally in the same YZ-plane of the undeformed geometry. Figure \[fig:planes\]B shows the intersection of these sheets with the mid-transverse plane.
The effect of tendon and aponeurosis properties on structural changes of the muscle tendon unit
-----------------------------------------------------------------------------------------------
Proximal and distal tendons were attached to the geometry of the muscle belly, where the distal tendon mimics the Achilles tendon. Both tendons had the same thickness and width as aponeuroses, but had lengths of 20 and 160 mm for the proximal and distal tendons, respectively. Initial tests showed that considerable rotations of the muscle belly during contraction as the aponeuroses end planes aligned along the line-of-action of the whole muscle tendon unit (Fig. \[fig:unsupported\]). To minimize this rotation, the deep aponeurosis (that was attached to the distal tendon) was constrained to not move any more in a deep direction during contraction. The free end of proximal tendon was fixed and the free end of the distal tendon was pulled about 0.2% of the total muscle-tendon unit length as an initialization step to settle the system into a initially stable structure. It was then fixed to keep the muscle-tendon unit isometric. Two situations were investigated: (1) the tendon and aponeurosis had the same material properties that were equal to the tendon properties, and (2) the tendon and aponeurosis had distinctive material properties which are shown below. These simulations achieved a common activation level of 10%, and the patterns of aponeurosis and tendon strains were compared for the two material formulations.
Constitutive equations for the tendon and aponeurosis. The mathematical formulation and implementation of these properties can be found in [@rahemi2014regionalizing]. We denote by $\lambda$ and $\sigma$ are along-fascicle stretch and stress, respectively, and $I_1$ is the first invariant of right Cauchy-Green deformation tensor.s
For the tendon, the along-fascicle stress-stretch (in Pa) is given by $$\label{equ:tend_stress} \sigma_{Tend}(\lambda)=\begin{cases}
10^4 \times 1.904\times(\lambda ^{68.8} -1), & 1 \leq \lambda \leq 1.07 \\
10^4 \times 1.904\times(6758\times(\lambda-1.07)+104.1), & 1.07< \lambda.
\end{cases}$$ The tendon base (matrix) material strain energy (Pa) is given by $$\label{equ:base_tend}
\Psi_{Tend}= 10^4 \times 2.857\times(I_1-3).$$ For the Aponeurosis, the along-fascicle stress-stretch properties are given by $$\label{equ:apol_stress}
\sigma_{Apo\ }(\lambda)=\begin{cases}
10^6 \times 3.053\times(\lambda ^{124.6} -1), & 1 \leq \lambda \leq 1.025 \\
10^6 \times 3.053\times(17375 \times(\lambda-1.025)+20.7), & 1.025< \lambda.
\end{cases}$$ while the base material of the aponeurosis is given by $$\label{equ:base_tend}
\Psi_{Apo\ }= 10^4 \times 57.84\times e^{579.6\times(I_1-3)} .$$
Results
=======
The force-length properties for the contracting muscle belly are shown in Figure \[fig:fl\] along with selected data from experimental studies on human muscle. As the muscle was activated, the stretch in the connective tissues allowed the fascicles to shorten, and so the fascicle lengths were different between the active and passive states. Plots shown in Figure \[fig:fl\] are all for equivalent fascicle lengths, and so the active force was calculated by subtracting the passive force at a slightly longer belly length away from the total force for a contracting muscle. The total and active muscle belly force showed a peak for fascicle stretch of 1, however, the overall shapes of the active and passive plots for the muscle belly were different from the plots for purely muscle fascicles due to the effects from the aponeurosis, muscle structure and pennation.
This modelling framework has previously shown [@rahemi2014regionalizing] that the belly force and fascicle pennation becomes larger when the activation state of the muscle belly increases. In the current study the pennation also increased when the belly was passively shortened, and decreased when the belly was passively lengthened. The range of pennation for passive and 30% active belly were 11.6-19.3 degrees and 13.4-21.2 degrees, respectively, as the belly length was reduced.
The muscle fascicles in the MG belly, changed from their initially straight configuration to a curved state during contraction. The fascicles showed an S-shaped profile in the mid-longitudinal plane (Fig. \[fig:planes\]) with the fascicles intersecting with the aponeurosis at a lower angle than their mean orientation would predict. These curvatures profiles match those that we have previously seen experimentally using ultrasound-based imaging [@namburete2011computational], and both are shown in Fig. \[fig:curvemap\]). The magnitude of the fascicle curvatures increased as the contraction level increased, and the increases in curvature matched the increases experimentally observed in contracting MG (Figs. \[fig:curvemap\], \[fig:RMS\]).
Strain measures for muscle tissue in the centre of the muscle belly are shown for an isometric contraction at 40% in Table \[tab:strains\] along with experimentally measured values [@wakeling2014transverse]. The transverse strains in the fascicle (mid-longitudinal:YZ) plane were much smaller than the strains normal to this plane. The Poisson’s ratio in the fascicle plane was calculated as the magnitude of the ratio between transverse and along-fascicle strains in this plane and was 0.089.
The fascicle sheets bulged in both medial and lateral directions when the muscle belly contracted (Figs. \[fig:planes\], \[fig:3dpath\]), and the bulge increased as the activity level rose. The path of the fascicles in 3D showed them running along the fascicle sheets as they bulged, and thus formed a part of a helix (demonstrated by their varying azimuthal angle along their length (Fig. \[fig:3dpath\]).
When the whole muscle-tendon unit was simulated (with the external tendons included), the muscle belly showed substantial rotations as the aponeurosis end planes aligned to be closer to the line-of-action of the muscle (Fig. \[fig:unsupported\]). Subsequent simulations of the MTU constrained the deep aponeurosis to not displace any deeper, and this forced the bulging of the muscle belly to be in the superficial direction. This was to emulate a simplified set of constraints that occur on the MG within the intact leg. The final simulations (Fig. \[fig:multi\]) showed that when a stiffer aponeurosis was used instead of adopting tendon properties, the strains in aponeurosis were smaller. Also the strains in the muscle tissue were more uniform when a stiffer material for the aponeurosis was used.
Discussion
==========
Validating a mathematical framework and numerical implementation of it for human muscle is a challenge, due in part to the fact that muscle forces cannot be directly measured in vivo. In this study we have compared the force output from a computational 3D FEM model with the forces estimated from studies of ankle joint flexion-extension experiments. The general pattern of the force-length relationship generated by the model matches those from the experimental studies. Experimental measures can identify the overall shape of the muscle with MRI [@gilles2006anatomical] and even the internal trajectories of the muscle fascicles using diffusion-tensor MRI [@heemskerk2009quantitative; @heemskerk2011vivo; @infantolino2012arrangement]. While this information is very important, the relatively long scan times of MR imaging preclude such measurements for active contractions [@rana2011vivo]. However, the aim of the presented muscle model is to understand the mechanisms occurring during muscle contractions. It is therefore important to validate the muscle model in its contracted state. For this study we have used ultrasound-based measures [@namburete2011computational; @Rana20133d] of the internal structure during contraction (fascicle orientations, curvatures, and strains) to validate the model.
The model in this study has a simplistic initial geometry that has the overall dimensions and mean fascicle pennation of the MG in man, but without the details of the geometry or internal structure. Furthermore, all the muscle fascicles within the model had the same material properties and thus represented the same fibre-types. Additionally, the activation was uniform across all fascicles: again these are gross simplifications compared to the physiological complexities and variations that occur within muscles *in-vivo*. Nonetheless, the emergent features from the model showed a remarkable similarity to the experimental measures that are available for comparison, giving confidence that the model can identify general features and consequences of the muscle structure that were not a result of idiosyncrasies or muscle-specific details of geometry, structure or activation.
Intramuscular pressure develops within muscles during contraction [@Sjogaard1986; @sejersted1984intramuscular; @Maton2006], and the fascicles curve around the regions of higher pressure. Previous modelling studies [@Van_Leeuwen1992; @vanLeeuwen1995] have shown how the curvatures in both the muscle fascicles and aponeurosis must balance the intramuscular pressure, and indeed our current model shows curvatures developing in both these structures. However, in these previous studies the curvatures of the muscle fascicles were constrained to be constant along their lengths, whereas this was not a constraint in the current model. The muscle fascicles in the current model started straight in their initial configuration, but developed S-shaped profiles when quantified in the mid-longitudinal plane. Both the S-shaped profiles and the increases in curvature that occurred with increasing activity and muscle force mirror those that we have previously imaged for the MG using B-mode ultrasound [@namburete2011computational; @Rana20133d]. A consequence of the S-shaped trajectories is that the angle at which the fascicles insert onto the aponeurosis can be reduced, allowing for a greater component of traction in the line of action of the whole muscle along the direction of the aponeuroses.
When tracked in 3D, the muscle fascicles followed curved paths on their fascicle sheets indicating that change in architecture is not simply due to a bulge of the sheets. The active configuration of these fascicles indicate that S-shaped fascicles in 2D curvature maps (Fig. \[fig:curvemap\]) are not only the result of projecting the fascicles on a 2D plane [rana2014curve]{} but comes from curling of the fascicles in a helical path. These 3D helical paths are curved around the centre of the muscle (Fig. \[fig:3dpath\]) where the intramuscular pressure is higher.
It is generally assumed that muscle fascicles are isovolumetric [@Baskin1967], and isovolumetric assumptions dictate the relation between longitudinal and transverse strains. Poisson’s ratio is the absolute value of ratio of the transverse to the longitudinal strain, and should be 0.5 for small strains in an incompressible and elastic material. The simulations in this study showed that as the activation increased, the transverse strain (in the mid-longitudinal plane) was lower than expected, resulting in a Poisson’s ratio of 0.089, however this was compensated for by greater transverse strains in the orthogonal direction (Table \[tab:strains\]). The muscle fascicles were represented as transversely isotropic materials in this model [@rahemi2014regionalizing], and so the asymmetry in their transverse bulging must reflect asymmetries in the transverse stresses acting on the fascicles. Being a unipennate model, there would have been a larger compressive force in the mid-longitudinal plane that was bounded by the aponeuroses that were being squeezed together by the pennate fascicles, than in the medial-lateral direction where there was no aponeurosis bounding the muscle. Indeed, the model has showed muscle belly bulging to its sides, but decreasing in its thickness between the aponeuroses during contraction [@rahemi2014regionalizing], in a similar manner to the decreases in thickness observed for the MG in vivo [@randhawa2013-1]. Recently we have quantified transverse bulging of the muscle fascicles in the MG from B-mode ultrasound images [@wakeling2014transverse], showing a Poisson’s ratio of 0.09; this matches the simulated results and provides confidence that emergent features of the model explain realistic features of muscle contraction.
When the model was evaluated with external tendons, there was a need to constrain displacements of the geometry since the unconstrained simulation (Fig. \[fig:unsupported\]) showed a large displacement of the muscle in the Y-direction. This illustrates that a range of additional boundary constraints may need to be applied to finite element models of muscle-tendon units in order to result in more realistic deformation.
In the case that the aponeurosis and tendon were given the same material properties a pattern of non-uniform strains resulted in the aponeurosis. This non-uniformity in strain is similar to that observed in previous experiments [@finni2003nonuniform; @Muramatsu2001], but our modelling study shows this can be an emergent feature of the muscle, and not necessarily due to differnces between active and inactive motor units in submaximally activated muscle, as previously suggested [@finni2003nonuniform]. The aponeurosis strains were smaller than the tendon strains for both formulations (Table \[tab:materials\]) of material property (Fig.\[fig:multi\]). Although there is an obvious jump in strain between the tendon and aponeurosis when a stiffer material is used for the aponeurosis, the difference in strains was less than 2%. A benefit of such a material distribution would be that a more uniform distribution of strains occurs in the fascicles, and this would allow the fascicles to produce more consistent forces along their length.
The simulated results from this finite element model match the general patterns from experimental and imaging results. Whole muscle force is partly shaped by the internal geometry of the muscle fascicles, and their interactions with the aponeuroses [@rahemi2014regionalizing], and so cannot be explained entirely by modelling a muscle as a scaled-up muscle fibre [@wakeling2011modelling]. As the fascicles shorten they must increase in cross-sectional area in order to maintain their volume, but asymmetric bulging occurs due to asymmetries in the compressive stress acting on the fascicles during contraction. The fascicles curve and adopt S-shaped profiles that align their traction to be closer to the aponeurosis direction, and they curl across fascicle sheets that in turn bulge around the intramuscular pressure that develops during contraction. Material properties of the aponeuroses affect the strains in the fascicles and thus their force generating potenitial. The muscle model that we have validated in this study will provide a useful tool for understanding the mechanisms that relate muscle structure to its contractile function.
Acknowledgement {#acknowledgement .unnumbered}
===============
We gratefully acknowledge funding from Natural Sciences and Engineering Research Council of Canada (Nilima Nigam and James M. Wakeling) and the Canada Research Chairs Program (Nilima Nigam).
Figures {#figures .unnumbered}
=======
Figures and captions.\
\
\
\
\
\
\
{width="\textwidth"}
**\[fig:planes\] Figure .**[ Geometry of the muscle fascicles within the muscle belly (A), shown for their mid-transverse (B) and mid-longitudinal (C) planes. The frames with black fascicle lines are in relaxed state and the frames with red fascicle lines belong to muscle fascicles at a 40% activity level. The active fascicles show a decrease in thickness and an increase in width in the longitudinal and transverse sections, respectively. Note that the fascicles in the longitudinal section (fascicle plane) are mostly curved to S-shapes in the active state.]{}
{width="\textwidth"}
**\[fig:curvemap\] Figure .**[ Intensity map showing the magnitude of the fascicle curvature for 30 and 60% activity. Mid-longitudinal plane fascicle curvature map after contraction had been simulated (A). Curvature map for a similar fascicle plane measured in human MG [@namburete2011computational] (B).]{}
{width="\textwidth"}
**\[fig:3dpath\] Figure .**[ 3D paths of three fascicles crossing the mid-transverse plane. Each fascicle is plotted for 0 (green), 30 (blue) and 60% (red) activity levels. The arrows show the normals to a medial/lateral fascicle at 30% activity and are coloured by their azimuthal angle where the azimuthal angle is the angle between the projection of the fascicle path in the XY-plane with the X-axis. The change in azimuthal angle from 80 ( yellow ) to 99 degrees ( red ) shows that the fascicle sheets curve away from the centre of the muscle belly.]{}
{width="\textwidth"}
**\[fig:unsupported\] Figure .**[ Displacement of whole muscle-tendon unit when activated without deep or superficial constraints. ]{}
{width="\textwidth"}
**\[fig:fl\] Figure .**[ Measured (gray) and modelled (black) force-length properties of human calf muscles. The simulations reached a 30% activation, and the forces have been normalize to achieve a maximum active force of 1. The black lines without symbols show the active (solid line) and passive (dashed line) force-length properties that were input for the fascicles [@rahemi2014regionalizing]. The black lines with symbols show normalized active (inverted triangles), passive (squares) and total (circles) forces for the whole muscle belly. The normalized passive (stars) and active (diamonds) human MG force was measured from twitch contractions [@Hoffman2012]. The active human soleus (triangles) forces were measured from tetanic contractions [@maganaris2001vivo].]{}
{width="\textwidth"}
**\[fig:RMS\] Figure .**[ The root-mean-square curvatures of the fascicles in mid-longituninal plane increased with activation for both simulation (black) and experimental (gray line with symbols)[@namburete2011computational] results. The lower gray lines show the change in mean RMS curvature ($\pm$S.D.) from the experimental study.]{}
[\
\
]{} **\[fig:multi\] Figure .**[ Total strain in the muscle tendon unit tissue at a 10% activity level for two material conditions: equal material properties for aponeurosis and tendon (A), tissue specific properties (Table \[tab:materials\]) for aponeurosis and tendon (B). ]{}
Tables {#sec:tabs .unnumbered}
======
Tables and captions.\
\
\
\
\
\
\
**\[tab:strains\] Table .**[ Along-fascicle and transverse strains for fascicles in the middle of the muscle belly for 40% activity (Fig.\[fig:planes\]). The Poisson’s ratio in the mid-longitudinal plane is calculated as the magnitude of the ratio of the transverse (cross-fascicle) to the along-fascicle strain. The last row shows the measured Poisson’s ratio from 2D ultrasound images in the mid-longitudinal plane of the MG during dynamic contractions [@wakeling2014transverse].\
\
]{}
|
---
abstract: 'We study ruled real hypersurfaces whose shape operators have constant squared norm in nonflat complex space forms. In particular, we prove the nonexistence of such hypersurfaces in the projective case. We also show that biharmonic ruled real hypersurfaces in nonflat complex space forms are minimal, which provides their classification due to a known result of Lohnherr and Reckziegel.'
address: 'Department of Mathematics, University of Santiago de Compostela, Spain.'
author:
- 'Olga Pérez-Barral'
title: |
Some problems on ruled hypersurfaces\
in nonflat complex space forms
---
[^1]
Introduction
============
A ruled real hypersurface in a nonflat complex space form, that is, in a complex projective or hyperbolic space, $\mathbb{C}P^{n}$ or $\mathbb{C}H^{n}$, is a submanifold of real codimension one which is foliated by totally geodesic complex hypersurfaces of $\mathbb{C}P^{n}$ or $\mathbb{C}H^{n}$. Ruled hypersurfaces in nonflat complex space forms constitute a very large class of real hypersurfaces. It becomes then an interesting problem to classify these objects under some additional geometric properties. For example, Lohnherr and Reckziegel classified ruled minimal hypersurfaces in nonflat complex space forms into three classes [@LR]: Kimura type hypersurfaces in $\mathbb{C}P^{n}$ or $\mathbb{C}H^{n}$, bisectors in $\mathbb{C}H^{n}$ and Lohnherr hypersurfaces in $\mathbb{C}H^{n}$. Moreover, they proved that Lohnherr hypersurfaces in $\mathbb{C}H^{n}$ are the only complete ruled hypersurfaces with constant principal curvatures in nonflat complex space forms [@LR].
Another important notion in the context of real hypersurfaces is that of Hopf hypersurface, which is defined as a real hypersurface whose Reeb vector field is an eigenvector of the shape operator at every point. Ruled hypersurfaces in nonflat complex space forms are never Hopf; indeed, the smallest tangent distribution invariant under the shape operator and containing the Reeb vector field has rank two. In particular, the minimal ruled hypersurfaces mentioned above have an additional property, which has been introduced in [@DDV:annali]: they are strongly 2-Hopf, that is, the smallest distribution invariant under the shape operator and containing the Reeb vector field is integrable and has rank two, and the principal curvatures associated with the principal directions defining such distribution are constant along its integral submanifolds. This concept is also important since it characterizes, at least in the complex projective and hyperbolic planes $\mathbb{C}P^{2}$ and $\mathbb{C}H^{2}$, the real hypersurfaces of cohomogeneity one that can be constructed as union of principal orbits of a polar action of cohomogeneity two on the ambient space.
Motivated by some recent results concerning the classification of ruled hypersurfaces in nonflat complex space forms having constant mean curvature [@holi] or having constant scalar curvature [@Maeda_2], we firstly focus on studying ruled hypersurfaces in $\mathbb{C}P^{n}$ or $\mathbb{C}H^{n}$ whose shape operators have constant squared norm, proving that there are no such hypersurfaces in $\mathbb{C}P^{n}$, whereas any possible example in $\mathbb{C}H^{n}$ must be strongly 2-Hopf.
Let $M$ be a ruled real hypersurface in a nonflat complex space form whose shape operator has constant norm. Then, $M$ is a strongly 2-Hopf real hypersurface in a complex hyperbolic space. In particular, there are no ruled hypersurfaces in complex projective spaces whose shape operator has constant norm.
We note that, in the hyperbolic case, the Lohnherr hypersurfaces (as homogeneous ruled hypersurfaces) are examples of ruled real hypersurfaces with shape operator of constant norm. The problem of deciding whether these are the only such examples remains open.
A hypersurface is said to be biharmonic if its defining isometric immersion is a biharmonic map, that is, a smooth map which is a critical point of the so-called bienergy functional (see, for example, [@ou] for more information on biharmonic hypersurfaces). It is well known that any minimal hypersurface is biharmonic and there exist some known results and conjectures claiming that the converse is, under certain circumstances, also true. For example, Chen conjectured that any minimal hypersurface in the Euclidean space is biharmonic. In the context of Riemannian manifolds of nonpositive curvature, it has been proved that both compact biharmonic hypersurfaces and biharmonic hypersurfaces with constant mean curvature are exactly the minimal ones [@jiang; @ou]. However, if one removes these conditions one cannot ensure (in principle) minimality. Indeed, deciding whether, in general, biharmonicity implies minimality in ambient spaces of nonpositive curvature is the content of the generalized Chen’s conjecture, proposed by Caddeo, Montaldo and Oniciuc in [@caddeo]. Ou and Tang have constructed some counterexamples which prove that this conjecture is not true [@outang]. However, it is still one of the main motivations for studying biharmonic hypersurfaces in the setting of Riemannian manifolds of nonpositive curvature due to the incompleteness of the examples provided by these two authors. Then, it becomes interesting to study biharmonic hypersurfaces satisfying other conditions, such as ruled ones, particularly in complex hyperbolic spaces, which are negatively curved. We point out that a general study of biharmonic submanifolds of arbitrary codimension in complex space forms has been developed in [@fetcu].
It has been recently proved (see [@toru]) that biharmonic ruled hypersurfaces in complex projective spaces are minimal. Our second goal in this article is to extend this result to the entire context of nonflat complex space forms. In particular, we prove the following result.
Let $M$ be a biharmonic ruled real hypersurface in a nonflat complex space form. Then, $M$ is minimal, and therefore an open part of one of the following hypersurfaces:
1. a Kimura type hypersurface in a complex projective or hyperbolic space, or
2. a bisector in a complex hyperbolic space, or
3. a Lohnherr hypersurface in a complex hyperbolic space.
Preliminaries {#sec:preliminaries}
=============
In this section we introduce the notation and terminology that we are going to use throughout this article. For more information, we refer to [@CR] and [@NR].
Let $\bar{M}$ be a Riemannian manifold and $M$ a smooth hypersurface of $\bar{M}$. Since the arguments that follow are local, we can assume that $M$ is embedded and take a unit normal vector field $\xi$ on $M$. We denote by $\langle\,\cdot\,,\,\cdot\,\rangle$ the metric of $\bar{M}$, by $\bar{\nabla}$ its Levi-Civita connection and by $\bar{R}$ its curvature tensor. Let $\nabla$ be the Levi-Civita connection of $M$. Then, the relation between $\nabla$ and $\bar{\nabla}$ is given by the Gauss formula $\bar{\nabla}_{X}Y=\nabla_{X}Y+{\ensuremath{I\!I}}(X,Y)$, where ${\ensuremath{I\!I}}$ denotes the second fundamental form of $M$.
The shape operator $S$ of $M$ is the endomorphism of the tangent bundle of $M$ given by $SX=-(\bar{\nabla}_{X}\xi)^{\top}$, where $X$ is a tangent vector to $M$ and $(\cdot)^\top$ denotes the orthogonal projection onto the tangent space to $M$. As the shape operator is a self-adjoint endomorphism with respect to the induced metric on $M$, it can be diagonalized with real eigenvalues and orthogonal eigenspaces. Each eigenvalue $\lambda$ is called a principal curvature and its corresponding eigenspace, $T_{\lambda}$, is a principal curvature space. In this paper we will need some second order equations of submanifold geometry. In particular, we will use the Codazzi equation $$\langle \bar{R}(X,Y)Z,\xi\rangle = \langle(\nabla_{X}S)Y,Z\rangle-\langle(\nabla_{Y}S)X,Z\rangle,$$ as well as the Gauss equation $$\langle\bar{R}(X,Y)Z,W\rangle = \langle R(X,Y)Z,W\rangle + \langle SX,Z\rangle\langle SY,W\rangle -\langle SX,W\rangle\langle SY,Z\rangle,$$ where $X,Y,Z,W\in\Gamma(TM)$ are sections of $TM$.
In what follows, we will assume that the ambient manifold $\bar{M}$ is a complex space form of complex dimension $n$ and constant holomorphic sectional curvature $c\in\mathbb{R}$, $\bar{M}^{n}(c)$. It is known that its curvature tensor $\bar{R}$ is given by $$\begin{aligned}
\langle\bar{R}(X,Y)Z,W\rangle={}&
\frac{c}{4}\Bigl(
\langle Y,Z\rangle\langle X,W\rangle
-\langle X,Z\rangle\langle Y,W\rangle\\[-1ex]
&\phantom{\frac{c}{4}\Bigl(}
+\langle JY,Z\rangle\langle JX,W\rangle
-\langle JX,Z\rangle\langle JY,W\rangle
-2\langle JX,Y\rangle\langle JZ,W\rangle
\Bigr),
\end{aligned}$$ where $J$ denotes the complex structure of $\bar{M}^{n}(c)$. Moreover, since $\bar{M}^{n}(c)$ is a Kähler manifold, $\bar{\nabla}J=0$.
Now let $M$ be a real hypersurface of $\bar{M}^{n}(c)$, that is, a real submanifold of real codimension one. The tangent vector field $J\xi$ is usually called the Hopf or Reeb vector field of $M$. We define the integer-valued function $h$ on $M$ as the number of the principal curvature spaces onto which $J\xi$ has nontrivial projection.
A real hypersurface $M$ of $\bar{M}^{n}(c)$ is said to be ruled if it is foliated by totally geodesic complex hypersurfaces of $\bar{M}^{n}(c)$. Equivalently, $M$ is ruled if the orthogonal distribution to the Hopf vector field $J\xi$ is integrable and its leaves are totally geodesic submanifolds of $\bar{M}^{n}(c)$. Locally, ruled hypersurfaces are embedded, but globally, they may have self-intersections and singularities. See [@Maeda] and [@CR Section 8.5.1] for more information on ruled real hypersurfaces.
We finally recall the notion of strongly 2-Hopf real hypersurface [@DDV:annali]. A real hypersurface $M$ in $\bar{M}^{n}(c)$ is said to be strongly 2-Hopf if the following conditions hold:
1. The smallest $S$-invariant distribution $\mathcal{D}$ of $M$ that contains the Hopf vector field $J\xi$ has rank 2.
2. $\mathcal{D}$ is integrable.
3. The spectrum of $S|_{\mathcal{D}}$ is constant along the integral submanifolds of $\mathcal{D}$.
Notice that the first condition is equivalent to $h=2$ and that a hypersurface satisfying both (1) and (2) is said to be a 2-Hopf hypersurface [@CR Section 8.5.1].
Proof of the Main Theorems
==========================
Let $M$ be a ruled real hypersurface in a nonflat complex space form $\bar{M}^{n}(c)$, $c\neq0$, with (locally defined) unit normal vector field $\xi$.
We briefly recall some facts from [@holi Section 3]. It is known that there exists an open and dense subset ${\ensuremath{\mathcal{U}}}$ of $M$ where $h=2$. ${\ensuremath{\mathcal{U}}}$ has exactly two nonzero principal curvature functions $\alpha$ and $\beta$, both of multiplicity one at every point, and $J\xi=aU+bV$ for some unit vector fields $U\in\Gamma(T_\alpha)$ and $V\in \Gamma(T_\beta)$ and nonvanishing smooth functions $a$, $b\colon {\ensuremath{\mathcal{U}}}\to\mathbb{R}$ satisfying $a^2+b^2=1$ and $$\label{eq:1}
a^2=\frac{\alpha}{\alpha-\beta}, \qquad b^2=\frac{\beta}{\beta-\alpha}.$$ From now on, we will work in the open and dense subset ${\ensuremath{\mathcal{U}}}$ of $M$.
With this notation, it can be proved [@DD:indiana Lemma 3.1] that there exists a unit vector field $A\in\Gamma(T_0)$ such that $$\begin{aligned}
\label{eq:alg}
J\xi&=aU+bV, &JU&=-bA-a\xi, &JV&=aA-b\xi, &JA&=bU-aV.\end{aligned}$$
Taking these expressions into account, we obtain the following.
\[prop:Levi-Civita\] Let $M$ be a ruled hypersurface in a nonflat complex space form. Then, its Levi-Civita connection satisfies the following equations: $$\begin{aligned}
&\langle\nabla_{U}U,V\rangle=\frac{V(\alpha)}{\alpha-\beta},
&&\langle\nabla_{V}V,U\rangle=-\frac{U(\beta)}{\alpha-\beta},\\
&\langle\nabla_{U}U,A\rangle=\frac{4A(\alpha)-3abc}{4\alpha},
&&\langle\nabla_{V}V,A\rangle=\frac{4A(\beta)+3abc}{4\beta},\\
&\langle\nabla_{U}V,A\rangle=\frac{3c}{4(\alpha-\beta)}+\alpha-\frac{aA(\alpha)}{b\alpha},
&&\langle\nabla_{V}U,A\rangle=\frac{3c}{4(\alpha-\beta)}-\beta-\frac{bA(\beta)}{a\beta},\\
&\langle\nabla_{A}U,V\rangle=\frac{ac\beta-4a\alpha\beta^{2}-4b\alpha A(\beta)}{4a\beta(\alpha-\beta)},
&&\nabla_{A}A=0.
\end{aligned}$$ Moreover, for any unit vector field orthogonal to $A$ in the 0-principal curvature distribution, $W\in\Gamma(T_{0}\ominus\mathbb{R}A)$, the following relations hold: $$\begin{aligned}
&\langle\nabla_{U}U,W\rangle=\frac{W(\alpha)}{\alpha},
&&\langle\nabla_{V}V,W\rangle=\frac{W(\beta)}{\beta},
&&\langle\nabla_{W}U,V\rangle=\frac{b\alpha W(\beta)}{a\beta(\beta-\alpha)}.\end{aligned}$$ In addition, $$\begin{aligned}
&U(\beta)=-\frac{\beta^{2}
U(\alpha)+2ab\alpha(\alpha-\beta)V(\beta)}{3\alpha\beta},
&&V(\alpha)=\frac{2ab\beta(\alpha-\beta)U(\alpha)-\alpha^{2}V(\beta)}{3\alpha\beta},\\
&A(\alpha)=\frac{b(\alpha-\beta)(a\beta(4\alpha\beta-c)+2b\alpha A(\beta))}{2\beta^{2}},
&&W(\alpha)=-\frac{\alpha}{\beta}W(\beta).\end{aligned}$$
Since $U$ and $V$ are orthogonal eigenvectors of the shape operator $S$ associated with the eigenvalues $\alpha$ and $\beta$, respectively, we have $$\begin{aligned}
\langle (\nabla_{U}S)V, U\rangle &= \langle \nabla_{U}(SV)-S\nabla_{U}V, U\rangle = \langle \nabla_{U}(\beta V), U\rangle - \alpha\langle \nabla_{U}V, U\rangle\\
&=\langle U(\beta)V+\beta\nabla_{U}V, U\rangle -\alpha \langle\nabla_{U}V,U \rangle= -(\alpha-\beta)\langle \nabla_{U}V, U\rangle.
\end{aligned}$$ As $U$ has constant length, $\langle \nabla_{V}U,U\rangle=0$, and thus, proceeding as before, $$\begin{aligned}
\langle (\nabla_{V}S)U, U\rangle &= \langle \nabla_{V}(SU)-S\nabla_{V}U, U\rangle = \langle \nabla_{V}(\alpha U), U\rangle - \alpha\langle \nabla_{V}U, U\rangle\\
&=\langle V(\alpha)U+\alpha\nabla_{V}U,U\rangle-\alpha\langle \nabla_{V}U, U\rangle=V(\alpha).
\end{aligned}$$ Using the expression for the curvature tensor of a complex space form and the relations , we obtain that $\langle\bar{R}(U,V)U,\xi\rangle=0$. Then, using the previous relations to apply the Codazzi equation to the triple $(U,V,U)$, we get $$\label{eq:cod_uvu}
0=V(\alpha)+(\alpha-\beta)\langle \nabla_{U}V,U\rangle,$$ which gives the first relation in the statement. Analogously, the Codazzi equation applied to the triple $(V,U,V)$ yields $$\label{eq:cod_vuv}
0=U(\beta)-(\alpha-\beta)\langle\nabla_{V}U,V\rangle,$$ which is equivalent to the second relation in the statement.
Since $\bar{\nabla}J=0$, using the definition of the shape operator and the relations $J\xi=aU+bV$ and $\langle\nabla_{U}U,U\rangle=0$, we obtain $$\begin{aligned}
U(a)&= U\langle J\xi,U\rangle = \langle \bar{\nabla}_{U}J\xi,U\rangle+\langle J\xi,\bar{\nabla}_{U} U\rangle = \langle J\bar{\nabla}_{U}\xi,U\rangle+\langle J\xi,\nabla_U U\rangle\\
&= \langle SU,JU \rangle + \langle aU+bV,\nabla_{U}U\rangle = b\langle V,\nabla_{U}U \rangle = -b\langle \nabla_{U}V,U \rangle.
\end{aligned}$$ By multiplying this expression by $2a$ and taking into account that $$2aU(a)=U(a^{2})=U\left(\frac{\alpha}{\alpha-\beta}\right)=\frac{\alpha U(\beta)-\beta U(\alpha)}{(\alpha-\beta)^{2}},$$ we get, using , $$\label{eq:deriv_ua}
0=\beta U(\alpha)-\alpha U(\beta)+2ab(\alpha-\beta)V(\alpha).$$ Analogously, expanding the relation $V(a)=V\langle J\xi,U\rangle$, we deduce, inserting , that $$\label{eq:deriv_va}
0=\beta V(\alpha)-\alpha V(\beta)-2ab(\beta-\alpha)U(\beta).$$
Equations and constitute a linear system in the unknowns $U(\beta)$ and $V(\alpha)$. After some calculations using , we get that the determinant of the matrix of this system vanishes if and only in $\alpha\beta$ does, which cannot occur in $\mathcal{U}$. Then, there exists a unique solution given by $$\begin{aligned}
\label{eq:ubeta,valpha}
&U(\beta)=-\frac{2ab\alpha(\alpha-\beta)V(\beta)+\beta^{2}U(\alpha)}{3\alpha\beta}, &&V(\alpha)=\frac{2ab\beta(\alpha-\beta)U(\alpha)-\alpha^{2}V(\beta)}{3\alpha\beta}.
\end{aligned}$$
Now, proceeding as above, the Codazzi equation applied to the triples $(U,A,U)$, $(V,A,V)$, $(A,U,A)$ and $(A,V,A)$ yields $$\begin{aligned}
\label{eq:cod_A}
&\langle\nabla_{U}U,A\rangle=\frac{4A(\alpha)-3abc}{4\alpha}, &&\langle \nabla_{V}V,A\rangle=\frac{3abc+4A(\beta)}{4\beta},\\\nonumber
&\langle\nabla_{A}A,U\rangle=0,
&&\langle\nabla_{A}A,V\rangle=0.
\end{aligned}$$
Since $\bar{\nabla}J=0$ and $J\xi=aU+bV$, expanding the relations $0=U\langle J\xi,A\rangle$ and $0=V\langle J\xi,A\rangle$, inserting the expressions for $\langle\nabla_{U}A,U\rangle$ and $\langle \nabla_{V}A,V\rangle$ that follow from , and using , we have $$\begin{aligned}
\label{eq:deriv_A}
&\langle\nabla_{U}V,A\rangle= \frac{3c}{4(\alpha-\beta)}+\alpha-\frac{aA(\alpha)}{b\alpha}, &&\langle \nabla_{V}A,U \rangle =\frac{bA(\beta)}{a\beta}+\beta-\frac{3c}{4(\alpha-\beta)}.
\end{aligned}$$
Now, the Codazzi equation applied to the triple $(V,A,U)$ yields, after inserting the expression for $\langle\nabla_{V}A,U\rangle$ given in , $$\begin{aligned}
0&=\langle \bar{R}(V,A)U,\xi\rangle-\langle(\nabla_V S)A,U\rangle+\langle(\nabla_A S)V,U\rangle\\ &=\frac{c(2\alpha+\beta)}{4(\alpha-\beta)}+\alpha\langle\nabla_{V}A,U\rangle-(\alpha-\beta)\langle\nabla_{A}V,U\rangle =-\frac{c}{4}+\alpha\beta+(\beta-\alpha)\langle\nabla_{A}V,U\rangle+\frac{b\alpha A(\beta)}{a\beta},
\end{aligned}$$ from where $$\label{eq:cod_vau}
\langle\nabla_{A}V,U\rangle=\frac{4b\alpha A(\beta)+4a\alpha\beta^{2}-ac\beta}{4a\beta(\alpha-\beta)}.$$ Similarly, applying the Codazzi equation to the triple $(U,V,A)$ and using the expressions for $\langle\nabla_{U}V,A\rangle$ and $\langle\nabla_{V}U,A\rangle$ given by , we obtain $$\begin{aligned}
0&=\langle \bar{R}(U,V)A,\xi\rangle-\langle (\nabla_U S)V,A\rangle+\langle(\nabla_V S)U,A\rangle\\&=-\frac{c}{4}-\beta\langle\nabla_{U}A,V\rangle+\alpha\langle\nabla_{V}A,U\rangle=\frac{c}{2}-2\alpha\beta+\frac{a\beta A(\alpha)}{b\alpha}-\frac{b\alpha A(\beta)}{a\beta},
\end{aligned}$$ from where, using , we get the following relation between $A(\alpha)$ and $A(\beta)$: $$\label{eq:a_alpha}
A(\alpha)=\frac{b(\alpha-\beta)(a\beta(4\alpha\beta-c)+2b\alpha A(\beta))}{2\beta^{2}}.$$
Finally, let $W\in\Gamma(T_{0}\ominus\mathbb{R}A)$ be an arbitrary unit vector field orthogonal to $A$ in the 0-principal curvature distribution. Using the expressions for $J\xi, JU$ and $JA$ given in , the Codazzi equation applied to the triple $(A,U,W)$ yields $\langle\nabla_{A}U,W\rangle=0$, from where we deduce, using , that $\nabla_{A}U$ is proportional to $V$. In particular, since $T_{0}\ominus\mathbb{R}A$ is a complex distribution, $\langle\nabla_{A}U,JW\rangle=0$. Expanding the relation $0=A\langle JU,W\rangle$, and taking the previous fact into account, as well as the expression for $JU$ given in , one gets $$b\langle\nabla_{A}A,W\rangle=\langle\nabla_{A}U,JW\rangle=0.$$ Then, $\langle\nabla_{A}A,W\rangle=0$ and, in fact, using , we obtain that $\nabla_{A}A=0$.
Proceeding as above, the Codazzi equation applied to the triples $(U,W,U)$ and $(V,W,V)$ yields $$\begin{aligned}
\label{eq:cod_w}
&\langle \nabla_{U}U,W\rangle=\frac{W(\alpha)}{\alpha}, &&\langle \nabla_{V}V,W\rangle=\frac{W(\beta)}{\beta}.
\end{aligned}$$
Expanding the relations $0=U\langle J\xi,W\rangle$ and $0=V\langle J\xi,W\rangle$ and inserting the expressions for $\langle\nabla_{U}U,W\rangle$ and $\langle\nabla_{V}V,W\rangle$ given by , we have $$\begin{aligned}
\label{eqs_w}
&\langle\nabla_{U}V,W\rangle=-\frac{aW(\alpha)}{b\alpha}, &&\langle\nabla_{V}U,W\rangle=-\frac{bW(\beta)}{a\beta}.
\end{aligned}$$
After some calculations using and , the Codazzi equation applied to the triple $(V,W,U)$ yields $$\label{eq:cod_vwu}
\langle\nabla_{W}V,U\rangle=\frac{b\alpha W(\beta)}{a\beta(\alpha-\beta)}$$ and, analogously, the Codazzi equation applied to the triple $(U,V,W)$ reads $$\label{eq:cod_uvw}
0=\frac{a\beta W(\alpha)}{b\alpha}-\frac{b\alpha W(\beta)}{a\beta},$$ from where we get, after using , the last formula in the statement.
Ruled hypersurfaces whose shape operator has constant norm
----------------------------------------------------------
We firstly focus on the proof of Theorem 1.
Let $k=\alpha^{2}+\beta^{2}$ denote the squared norm of the shape operator of $M$. Since by hypothesis $k$ is constant, $X(k)=0$ for each $X\in TM$. Thus, $\alpha X(\alpha)+\beta X(\beta)=0$, from where $$\label{eq:X_beta}
X(\beta)=-\frac{\alpha X(\alpha)}{\beta}, \ \text{ for each } X\in TM.$$ Taking this fact into account, one can rewrite some of the relations given in Proposition \[prop:Levi-Civita\] in an easier way.
\[prop:Levi-Civita-k\] Suppose that $\alpha\neq-\beta$ on an open subset of ${\ensuremath{\mathcal{U}}}$. Then, with the previous notations, the Levi-Civita connection of such open subset satisfies the following equations: $$\begin{aligned}
&\nabla_U U=-\frac{ab(8\alpha\beta^{2}+c(3\alpha+\beta))}{4\alpha(\alpha+\beta)}A,
&\nabla_U V=\frac{c(3\alpha+\beta)+4\alpha(\alpha^{2}+\beta^{2})}{4(\alpha^{2}-\beta^{2})}A,
\\
&\nabla_V V=\frac{ab(8\alpha^{2}\beta+c(\alpha+3\beta))}{4\beta(\alpha+\beta)}A,
&\nabla_V U=\frac{c(\alpha+3\beta)+4\beta(\alpha^{2}+\beta^{2})}{4(\alpha^{2}-\beta^{2})}A.
\end{aligned}$$ Moreover: $$\label{eq:deriv_functions}
U(\alpha)=V(\alpha)=W(\alpha)=0, \qquad \text{and}\qquad A(\alpha)=\frac{ab\beta(c-4\alpha\beta)}{2(\alpha+\beta)},$$ for any $W\in\Gamma(T_{0}\ominus\mathbb{R}A)$.
First of all, in order to prove that $U(\alpha)=V(\alpha)=0$, we rewrite the expressions for $U(\beta)$ and $V(\alpha)$ given in Proposition \[prop:Levi-Civita\] using the relation . Some calculations using show that such equations are equivalent to: $$\begin{aligned}
\beta(3\alpha^{2}-\beta^{2})\ U(\alpha)+2ab\alpha^{2}(\alpha-\beta)\ V(\alpha)&=0,\\
-2ab\beta^{2}(\alpha-\beta)\ U(\alpha)+\alpha(3\beta^{2}-\alpha^{2})\ V(\alpha)&=0,\end{aligned}$$ which constitute a homogeneous linear system in the unknowns $U(\alpha)$ and $V(\alpha)$. The determinant of the matrix of such system can be easily deduced to be, using , $-3\alpha\beta(\alpha^{2}-\beta^{2})^{2}$, which cannot vanish since $\alpha\beta\neq0$ and $\alpha\neq\pm\beta$ on an open subset of $\mathcal{U}$. Then, we conclude that $U(\alpha)=V(\alpha)=0$.
Again, using , we can rewrite the expression for $A(\alpha)$ given in Proposition \[prop:Levi-Civita\]. Some calculations using lead us to conclude that $2(\alpha+\beta)A(\alpha)=ab\beta(c-4\alpha\beta)$, which is equivalent to the last formula in the statement.
Given $W\in\Gamma(T_{0}\ominus\mathbb{R}A)$, $W(\alpha)=-\alpha W(\beta)/\beta$ by Proposition \[prop:Levi-Civita\] and $W(\beta)=-\alpha W(\alpha)/\beta$ by . Then $W(\alpha)(1-\alpha^{2}/\beta^{2})=0$, from where we deduce, since $\alpha\neq\pm\beta$, that $W(\alpha)=0$.
Inserting the expressions for $U(\alpha),\ V(\alpha),\ W(\alpha)$ and $A(\alpha)$ that we have just obtained into the relations given in Proposition \[prop:Levi-Civita\] one gets, after some calculations involving , the formulas for $\nabla_{U}U,\ \nabla_{U}V,\ \nabla_{V}U,\ \nabla_{V}V$ in the statement.
We can now conclude the proof of Theorem 1. First of all notice that, if $\alpha=-\beta$ on $\mathcal{U}$, then $0=X(k)=X(2\alpha^{2})=4\alpha X(\alpha)$, which implies that $X(\alpha)=0$ for any $X\in T\mathcal{U}$. Since $X\in TM$ is arbitrary, we deduce that both $\alpha$ and $\beta$ have to be constant on $\mathcal{U}$, and by the density of $\mathcal{U}$, also on $M$.
Suppose now that there exists a point $p\in\mathcal{U}$ such that $\alpha(p)\neq-\beta(p)$. Then, in an open neighborhood of $p$, $\alpha\neq-\beta$. Taking the expressions given by into account, the definition of the Lie bracket of $M$ yields $$\label{eq:[u,v]}
[U,V](\alpha)=U(V(\alpha))-V(U(\alpha))=0.$$ On the other hand, using the fact that the Levi-Civita connection of $M$ is torsion-free, inserting the expressions for $\nabla_{U}V$ and $\nabla_{V}U$ given in Proposition \[prop:Levi-Civita-k\], we obtain $$\label{eq:torsion_free}
[U,V](\alpha)=(\nabla_{U}V)(\alpha)-(\nabla_{V}U)(\alpha)=\frac{c+2(\alpha^{2}+\beta^{2})}{2(\alpha+\beta)}A(\alpha).$$ Then, either $A(\alpha)=0$ or $c+2(\alpha^{2}+\beta^{2})=c+2k=0$. If $A(\alpha)=0$ on an open subset, since $U(\alpha)=V(\alpha)=W(\alpha)=0$ for any $W\in\Gamma(T_{0}\ominus\mathbb{R}A)$, both $\alpha$ and $\beta$ must be constant, and thus, $M$ has constant principal curvatures on such open subset. Suppose now that $2k+c=0$ or, equivalently, that $k=-c/2$ on an open subset of $\mathcal{U}$.
In the projective case, since $c>0$, the equation $\alpha^{2}+\beta^{2}=-c/2$ has no solution and, on the other hand, there is no ruled hypersurface in the complex projective case having constant principal curvatures [@LR Remark 5]. Therefore, there is no example of ruled hypersurface in $\mathbb{C}P^{n}$ whose shape operator has constant norm.
In the hyperbolic case, let $\mathcal{D}:=\mathrm{span}\{U,V\}$ be the smallest $S$-invariant distribution of ${\ensuremath{\mathcal{U}}}$ that contains $J\xi$, which clearly has rank 2. On the one hand, if an open subset of $M$ has constant principal curvatures, then it is an open part of a Lohnherr hypersurface [@LR Remark 5], which is known to be strongly $2$-Hopf. This can be checked directly from Proposition 3.1: taking into account that it has constant principal curvatures $\alpha=\sqrt{-c}/2$, $\beta=-\sqrt{-c}/2$ and $0$, it follows that $[U,V]=\nabla_{U}V-\nabla_{V}U=0$, hence ${\ensuremath{\mathcal{D}}}$ is integrable, and moreover ${\ensuremath{\mathcal{D}}}(\alpha)={\ensuremath{\mathcal{D}}}(\beta)=0$. On the other hand, if an open subset of $M$ satisfies $k=-c/2$, it follows from that ${\ensuremath{\mathcal{D}}}$ is integrable and, by virtue of and the assumption that $\alpha^2+\beta^2$ is constant, again ${\ensuremath{\mathcal{D}}}(\alpha)={\ensuremath{\mathcal{D}}}(\beta)=0$. Thus, in any case, $M$ is a strongly 2-Hopf hypersurface, which concludes the proof.
Ruled biharmonic hypersurfaces
------------------------------
In this subsection we prove Theorem 2. Recall that a submanifold of a Riemannian manifold is said to be biharmonic if its defining isometric immersion is a critical point of the bienergy functional [@ou]. Moreover, they can be characterized as those submanifolds having vanishing bitension field. In the particular case of codimension one isometric immersions, that is, in the setting of biharmonic hypersurfaces, there exists an explicit formula which completely characterizes them.
[@ou Theorem 2.1] Let $\bar{M}$ be a Riemannian manifold and $M\subset\bar{M}$ a hypersurface with unit normal vector field $\xi$. $M$ is biharmonic if, and only if, it satisfies the following relations $$\label{eq:biharmonic}
\begin{cases}
&\Delta H-H|S|^{2}+H\overline{\operatorname{Ric}}(\xi,\xi)=0,\\
&2S(\nabla H)+H\nabla H-2H(\overline{\operatorname{Ric}}(\xi))^{\top}=0,
\end{cases}$$ where $H=\operatorname{Tr}(S)$ is the mean curvature of the hypersurface, $\nabla$ denotes the gradient, $\Delta$ is the Laplace-Beltrami operator of $M$, and $\overline{\operatorname{Ric}}$ denotes both the (0,2) and the (1,1) Ricci tensors of $\bar{M}$.
In our context, $\bar{M}=\bar{M}^{n}(c)$ is a complex space form of constant holomorphic sectional curvature $c\neq 0$. In this case, the Ricci tensor of $\bar{M}$ satisfies $\overline{\operatorname{Ric}}(\xi,\xi)=c(n+1)/2$ and $(\overline{\operatorname{Ric}}(\xi))^{\top}=0$, which follows immediately from the formula for the curvature tensor of a complex space form. Thus, in our case, equations can be rewritten as follows (cf. [@fetcu Proposition 2.1]) : $$\label{eq:biharmonic2}
\begin{cases}
& \Delta H-H|S|^{2}+\frac{1}{2}Hc(n+1)=0,\\
& 2S(\nabla H)+H\nabla H=0.
\end{cases}$$
We assume from now on that $M$ is a biharmonic ruled real hypersurface in a nonflat complex space form $\bar{M}^{n}(c)$. We will use the notations introduced in Section 3 for ruled real hypersurfaces. Thus, the mean curvature function of $M$ is $H=\alpha+\beta$.
As in the previous section, according to the discussion before Proposition \[prop:Levi-Civita\], there is an open and dense subset ${\ensuremath{\mathcal{U}}}$ of $M$ where $h=2$. Proposition \[prop:Levi-Civita\] and relations hold in this open subset. In what follows, we will work in terms of an orthonormal basis of eigenvectors $\{U,V,A,W_{4},\dots,W_{2n-1}\}$, where $W_{i}\in\Gamma(T_0\ominus\mathbb{R}A)$, $i\in\{4,\dots,2n-1\}$.
Suppose that the mean curvature, $H=\alpha+\beta$, is not zero on an open subset of ${\ensuremath{\mathcal{U}}}$. We will work on this open subset of $M$ from now on. Since $M$ is a biharmonic hypersurface, it satisfies equations . With respect to the orthonormal eigenbasis $\{U,V,A,W_{4},\dots,W_{2n-1}\}$, we have $$\nabla H=U(H)U+V(H)V+A(H)A+\sum\limits_{i=4}^{2n-1}W_{i}(H) W_{i}.$$ On the other hand, since $U, V, A$ and $W_{i}$, for $i\in\{4,\dots,2n-1\}$, are orthogonal eigenvectors of the shape operator $S$ of $M$ associated with eigenvalues $\alpha, \beta$ and 0, respectively, we have $$S(\nabla H)=\alpha U(H)U+\beta V(H)V.$$ Thus, inserting these relations into the second equation in , we obtain $$0=2S(\nabla H)+H\nabla H=(2\alpha+H)U(H)U+(2\beta+H)V(H)V+HA(H)A+\sum\limits_{i=4}^{2n-1}HW_{i}(H) W_{i}.$$ Since $H\neq0$ by assumption, one can deduce that $A(H)=0$ and $W_{i}(H)=0$ for $i\in\{4,\dots,2n-1\}$. Moreover, one of the following conditions holds on an open subset:
1. $U(H)=V(H)=0$.
2. $\alpha=\beta=-H/2$.
3. $U(H)=0$ and $2\beta+H=0$.
4. $V(H)=0$ and $2\alpha+H=0$.
Neither case (1) nor case (2) are possible. Indeed, if $U(H)=V(H)=0$ on an open subset, then such subset has constant mean curvature and, since it is ruled, $H=0$ [@holi], which gives a contradiction. Analogously, since $M$ is ruled, $\alpha\neq\beta$ on any open subset.
Suppose that $U(H)=0$ and $2\beta+H=0$ or, equivalently, $H=2\alpha/3$ (case (4) is analogous). Then, both $\alpha$ and $\beta$ can be expressed as $\alpha=3H/2$ and $\beta=-H/2$, respectively. Inserting these expressions in the formula for $A(\alpha)$ given in Proposition \[prop:Levi-Civita\], one gets $$\frac{3}{2}A(H)=A(\alpha)=2b(a(c+3H^{2})-3bA(H)).$$ Moreover, $A(H)\neq0$ on $\mathcal{U}$, from where $ab(3H^{2}+c)=0$. Since $a$ and $b$ are not zero in $\mathcal{U}$, $M$ has constant mean curvature on $\mathcal{U}$ and, as $M$ is a ruled, it must be minimal (see [@holi]), which concludes the proof.
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[^1]: The author acknowledges support by projects MTM2016-75897-P (AEI/FEDER, Spain) and ED431C 2019/10 (Xunta de Galicia, Spain), and by a research grant under the Ramón y Cajal project RYC-2017-22490 (AEI/FSE, Spain).
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---
abstract: 'We have found the Quasi Normal Mode (QNM) frequencies of a class of static spherically symmetric spacetimes having a [*[smeared]{}*]{} matter distribution, parameterized by $\Theta$ - an inherent length scale. Here our main focus is on the QNMs for the odd parity perturbation in this background geometry. The results presented here for diffused mass distribution reveal significant changes in the QNM spectrum. This could be relevant for future generation (cosmological) observations, specifically to distinguish the signals of GW from a non-singular source in contrast to a singular geometry. We also provide numerical estimates for the $\Theta$-corrected QNM spectrum applicable to typical globular cluster like spherical galaxies having a Gaussian spread in their mass distribution. We find that the $\Theta$-correction to the GW signal due to smeared distribution is accessible to present day observational precision.'
author:
- |
Kumar Das $^{(a)}$[^1], Souvik Pramanik $^{(b)}$[^2] Subir Ghosh $^{(a)}$[^3]\
\[10pt\] *$^{(a)}$ S. N. Bose National Centre For Basic Sciences, JD Block, Sector III, Salt-Lake,\
*Kolkata-700098, India\
*$^{(b)}$ Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B.T. Road,\
*Kolkata - 700 018, India\
*$^{(c)}$ Department of Physical Science, Indian Institute of Science Education and Research Kolkata,\
*Mohanpur, India.******
title: Quasinormal mode spectra for odd parity perturbations in spacetimes with smeared matter sources
---
Introduction {#introduc}
============
The recent detection of Gravitational Waves (GWs), from binary Black Hole (BH) mergers and Neutron stars by LIGO and Virgo collaboration [@Abbott:2016blz; @Abbott:2016nmj; @Abbott:2017vtc; @TheLIGOScientific:2017qsa], has provided us with a new window to study and understand physical processes at extreme conditions, where the role of gravity by far dominates the other known forms of interactions in nature. Since the metric to describe the gravitational collapse of a binary merger is unknown we fall back to numerical relativity simulations from which we gain a fairly reasonable understanding of such realistic phenomena. For the present work we are assuming that these collapse events are generally consistent with the theoretical predictions of general relativity (GR) [@Will:2014kxa], although later works [@kon] have shown that there are significant deviations. These observations now firmly suggest studying the possibility of having alternative sources that can account for such deviations in GW characteristic frequencies. Although an ultra-compact system of binaries is probably too exotic a system for producing strong GW signal to be detected at large distances, more common objects nevertheless can also produce GWs with frequencies that could be highly relevant for the next generation space-based GW detectors.
The popularly known GWs as observed by LIGO/Virgo collaboration are actually the Quasi Normal Modes or QNMs [@Berti:2005ys; @Macedo:2016wgh; @Kokkotas:1999bd; @kon1]. The waveforms of these GW signals consist of three parts: 1) inspiral, 2) merger and 3) ring-down. The ring-down phase shows characteristic frequencies of oscillation corresponding to damped resonances of the remnant BH. These damped oscillations or QNMs encode information about the BH source. Applying the linear perturbation results, the ring-down portion of the signal may be used to discriminate between BHs and other possible sources. The damped modes in turn possess a complex frequency whose real part corresponds to the oscillation frequency and whose imaginary part gives the lifetime. It is important to note that the QNM spectrum of a BH is completely characterized by the BH parameters, and does not depend on the initial conditions of the perturbations.
In this work our aim is to investigate the QNM frequencies for a spherically symmetric geometry having a smeared matter source. An interesting approach was pioneered by Nicolini, Smailagic and Spallucci [@nicsp]. These authors introduced a (spherically symmetric) smeared source in the matter sector and solved the Einstein equation thereby obtaining a generalized form of Schwarzschild (black hole) metric that successfully cured the black hole singularity problem. It was also tentatively proposed to identify the smearing scale with Planck length so that the metric can play a role in the context of quantum gravity. (For an exhaustive review see [@Nicolini:2008aj].) Various aspects of this generalized black hole have been studied: its thermodynamics [@rabin], effect of the smearing on AdS/CFT correspondence [@Pramanik:2015eka], among others. As mentioned above, the conventional Dirac delta source term for matter is replaced by a new type of matter source with the energy density given by a Gaussian distribution function. The resulting geometry will be helpful to understand the dynamics of objects, which have approximately a Gaussian mass profile. From astrophysical point of view, such a Gaussian profile can be applied to study the dynamics of GWs for elliptical galaxies (*e.g.* globular clusters having dense matter core in the center [@hogg:1965hg]). Besides that, this distribution is also relevant for the Dark matter profiles within galaxies (e.g. Press-Schechter mass distribution that is extensively used in the context of dark matter distribution profile [@Press:1973iz]). Therefore, the mathematical formulation of this study with smeared matter source will be particularly interesting for astrophysical objects, where the length scales would be $\mathcal{O}(Ly)$ $( 1Ly\sim3\times 10^{-7} Mpc )$. In this paper, we can tentatively identify this length scale with $\Theta$ - the smearing parameter.
Let us point out the proper perspective of our work in view of the recent works of Liang [@Liang:2018nmr; @Liang:2018uyk] who has also made an exhaustive study of smearing effect on QNM. The results of Liang are valid up to $3$’rd order in WKB. On the other hand we have used the framework of [@Konoplya:2003ii] yielding results valid up to $6$’th order in WKB. We explicitly demonstrate that there are appreciable modifications when the latter are taken in to account.
The organization of our paper is as follows — in Sec. \[methqnms\] we have briefly reviewed the basic aspects of QNMs and an elementary method to obtain them for static spherically symmetric Schwarschild spacetime. Sec. \[gwgauss\] deals with the gravitational perturbation of a spherically symmetric QG-inspired spacetime. Here the analysis is made in four segments. In Sec \[gwgauss1\] we have computed the equations for the odd parity gravitational perturbation of this QG-inspired spacetime. Then in Sec. \[gwgauss2\] and in Sec. \[gwgauss3\] we have obtained the QNM frequencies for this spacetime using the Ferrari-Mashoon formula [@Ferrari:1984zz] and also the WKB 6th order formula [@Konoplya:2003ii]. Here we discuss our results for the new QNMs by comparing them with the standard Schwarzschild QNMs and also with the QNMs for this QG-induced spactime, obtained earlier with 3rd order WKB method. Finally, in Sec. \[gwgauss4\] we discuss the relevance of the results from observational perspetives and in Sec. \[conc\] we conclude.
QNM and determination of QNM spectrum: a brief review {#methqnms}
======================================================
[**(a) Quasi-normal mode:**]{}
A black hole posses characteristic frequencies which arise from perturbations in it’s spacetime geometry. Such perturbations of the BH geometry can originate in many different ways. For example a certain mass falling along the geodesic of the Schwarzschild spacetime can be considered as a perturbation on the background Schwarzschild geometry. In the presence of such distortion of the BH equilibrium, the BH system undergoes damped oscillations with complex frequencies. These frequencies are called quasi-normal modes (QNMs). Here the term ‘quasi’ is referring to the fact that the frequencies are complex, thus they show damping. While the conventional normal modes of compact classical linear oscillating systems are non-dissipative, for black hole QNMs [@Berti:2005ys], the dissipations cannot be neglected, as the event horizon imposes necessary loss of energy. The real part of this QNM frequency corresponds to the oscillation frequency, whereas the imaginary part corresponds to the damping rate. From astrophysical point of view, QNMs dominate an exponentially decaying ringdown phase at intermediate times in the GW signal from a perturbed BH [@Kokkotas:1999bd]. Moreover, they also govern the ringdown phase of gravitational systems produced by the merger of a pair of black holes [@Pretorius:2005gq; @Campanelli:2005dd]. Since these QNMs are independent of the initial perturbation, we can infer crucial information regarding the fingerprints (*e.g* mass, charge and angular momentum of a BH [@Echeverria:1989hg]) of its source, the BH geometry.
Let us briefly discuss the equations governing the perturbation around a stationary spherically symmetric geometry *i.e.* Schwarzschild spacetime. This spacetime is represented by the metric around a fixed spherically symmetric center-of-mass M $$\begin{aligned}
ds^2 = -\bigg(1-\frac{2M}{r}\bigg)dt^2+\bigg(1-\frac{2M}{r} \bigg)^{-1}dr^2 + r^2 d\Omega^2
\label{schwarsc}\end{aligned}$$ where, $d\Omega^2 = d\theta^2 +\sin^2\theta d\phi^2$.
Now we consider a small non-spherical perturbation $h_{\mu\nu}$ such that the new perturbed metric is, $$\begin{aligned}
g_{\mu\nu} = \bar{\bf g}_{\mu\nu} + h_{\mu\nu} \qquad\text{where,} \qquad \frac{|h_{\mu\nu}|}{|\bar{\bf g}_{\mu\nu}|}<<1 .\end{aligned}$$ Here we denote the static generic background metric by $\bar{\bf g}_{\mu\nu}$. The inverse metric is then $$\begin{aligned}
g^{\mu\nu} = \bar{\bf g}^{\mu\nu} - h^{\mu\nu} + \mathcal{O}(h^2).\end{aligned}$$ The perturbed Christoffel symbols are given by $$\begin{aligned}
%\Gamma^{\alpha}_{\mu\nu}& = \frac 1 2 (\bar{\bf g}^{\alpha\sigma} - h^{\alpha\sigma})(g_{\sigma\nu,\mu}+g_{\sigma\mu,\nu}-g_{\mu\nu,\sigma} ) \nonumber \\
\Gamma^{\alpha}_{\mu\nu}& = \bar{\Gamma}^{\alpha}_{\mu\nu} + \frac 1 2 \bar{\bf g}^{\alpha\sigma} (h_{\sigma\nu,\mu}+h_{\sigma\mu,\nu}-h_{\mu\nu,\sigma} - 2h_{\sigma\kappa}\bar {\Gamma}^{\kappa}_{\mu\nu}) + \mathcal{O}(h^2) \nonumber \\
& \simeq \bar {\Gamma}^{\alpha}_{\mu\nu} + \delta\Gamma^{\alpha}_{\mu\nu},\end{aligned}$$ where the $\bar{\bf \Gamma}$ is the Christoffel symbol for the unperturbed metric $\bar{\bf g}_{\mu\nu}$ and the small perturbation $\delta\Gamma$’s is given by, $$\begin{aligned}
\delta\Gamma^{\alpha}_{\mu\nu} = \frac{\bar{\bf g}^{\alpha\beta}}{2}\big(\nabla_{\nu}h_{\mu\beta} + \nabla_{\mu}h_{\nu\beta} -
\nabla_{\beta}h_{\mu\nu} \big).
\label{chris_eqn}\end{aligned}$$ Using the definition of covariant derivative $\nabla_{\mu}$ (with $\nabla_{\mu}$ being with repect to $\bar{\bf g}_{\mu\nu}$) for the perturbed Christoffel symbol given in eqn. , the vacuum Einstein field equation can be put into a more convenient form as $$\begin{aligned}
\nabla_{\beta}\delta\Gamma_{\mu\nu}^{\beta} - \nabla_{\nu}\delta\Gamma_{\mu\beta}^{\beta} = 0.
\label{mod_eqn}\end{aligned}$$ Finally, putting the expression for $\delta\Gamma$ into eqn. and employing gauge freedom, we get the second order differential equation for $h_{\mu\nu}$, $$\begin{aligned}
\Box h_{\mu\nu} - 2\bar{R}^{\rho}_{\,\sigma\mu\nu}h_{\rho}^{\,\sigma} = 0\end{aligned}$$ in the TT (transverse traceless) gauge, where $$\begin{aligned}
\nabla^{\mu}h_{\mu\nu}=0 \quad \text{and} \quad h_{~\mu}^{\mu} = \bar{\bf g}^{\mu\nu}h_{\mu\nu} = h = 0.\end{aligned}$$
Now a generic perturbation, $h_{\mu\nu}$ of the spherically symmetric metric can be broken up into odd ($h_{\mu\nu}^{\text{odd}}$) and even ($h_{\mu\nu}^{\text{even}}$) parity components according to their transformation properties under parity *i.e.* $(\theta,\phi)\to(\pi-\theta,\pi+\phi)$ [@Chandrasekhar:1985kt; @Regge:1957td; @Zerilli:1971wd; @Rezzolla:2003ua]. Here, we will focus on the odd-parity perturbations $h_{\mu\nu}^{\text{odd}}$, also known as the axial perturbations. (We will comment about the even-parity perturbations towards the end.) Its components are simplified by using the residual freedom to choose a proper gauge (*e.g.* see [@Regge:1957td]) which eliminates all the highest derivatives in the angles $(\theta,\phi)$. Finally, the true gauge invariant axial perturbations are described by the functions $h_0(t,r)$ and $h_1 (t,r)$. The gravitational odd parity perturbations for this spherically symmetric spacetime are now described by the Regge-Wheeler equation $$\begin{aligned}
\frac{\partial^2Q(t,r)}{\partial t^2} - \frac{\partial^2Q(t,r)}{\partial r_{\star}^2} + V_{\text{axial}}(r)Q(t,r) = 0
\label{swaschrl}\end{aligned}$$ where $Q(t,r)$ is the gauge-invariant odd-parity variable, also known as Regge-Wheeler variable, and it is defined as $$\begin{aligned}
Q(t,r)= \bigg(1-\frac{2M}{r} \bigg)\frac{h_1(t,r)}{r}\end{aligned}$$ with $h_1(t,r)$ being an unknown function from the perturbation [^4] and the so called tortoise co-ordinate ($r_{\star}$) is defined as, $$\begin{aligned}
\frac{dr_{\star}}{dr} = \frac{1}{1- \frac{2M}{r}} .
\label{star_old}\end{aligned}$$ Integrating eqn. one obtains for $r_{\star}$ $$\begin{aligned}
r_{\star} &= r + 2M \ln\bigg({\frac{r}{2M}-1}\bigg) .\end{aligned}$$ Since $r_{\star}\to \infty$ as $r\to\infty$ and $r_{\star}\to-\infty$ as $r\to2M$, so tortoise co-ordinate will be helpful in this context for it does not suffer from coordinate singularity near the event horizon at $r=2M$ (since $r_{\star}$ is pushed to $-\infty$ at horizon).
Now extracting the time dependence in $ Q(t,r)$ as $Q(t,r) \sim e^{i\omega t} Q(r)$, eqn. takes the form $$\begin{aligned}
\frac{\partial^2 Q(r)}{\partial r_{\star}^2} + \big( \omega^2 - V_{\text{axial}}(r)\big)Q(r) = 0 .
\label{schr}\end{aligned}$$ The function $V_{\text{axial}}(r)$ is given by $$\begin{aligned}
V_{\text{axial}}(r) &= \bigg(1-\frac{2M}{r}\bigg)\bigg[\frac{l(l+1)}{r^2}-\frac{6M}{r^3} \bigg] .
\label{sc_po} \end{aligned}$$ The solutions of the eqn. define the QNMs of the black hole with QNM mode frequencies $\omega$. Below we describe how to compute this frequency.
[**(b) Method for computing the QNM spectrum to $6$’th order in WKB approximation:**]{}
There are various methods to determine the QNM spectrum of a black hole spacetime. Note that for Schwarzschild and Kerr black holes, there exist the method of Leaver [@Leaver:1985ax], who constructed exact eigensolutions of the radiative boundary-value problem of Chandrasekhar. Later Detweiler [@chandra] developed a stable numerical method in order to determine the quasinormal frequencies with an arbitrary precision. However, to the best of our knowledge, no such stable numerical method exists for the QG-inspired spherically symmetric BH space-time [@Nicolini:2008aj], that is capable of evaluating the QNMs with arbitrary precision. Therefore, to find the QNMs we need to address the problem using approximations. One of the easiest semi-analytic way to determine the QNMs in frequency domain is the approximation of the effective potential by the P$\ddot{\text o}$schel-Teller potential. This method was suggested by Ferrari and Mashoon [@Ferrari:1984zz].[^5] In this approach [@Ferrari:1984zz], the main problem of finding the QNMs $\omega$ for $V^{\text{axial}}(r)$ is reduced to finding the bound state of an inverse potential given by the profile $$\begin{aligned}
V_{PT} (r_{\star})= \frac{V_0}{\cosh^2{\alpha(r_{\star}-\tilde{r}_{\star})}}.
\label{pt_pt}\end{aligned}$$ This is the P$\ddot{\text o}$schel-Teller potential, where $\tilde{r}_{\star}$ is the point of extremum of the potential. Here $V_0=V_{PT}(\tilde{r}_{\star})$ is the height and $\alpha= \frac{1}{2V_0}\,\frac{d^2 V_{PT}}{dr_{\star}^2}\bigg|_{r_{\star}=\tilde{r}_{\star}} $ is the curvature of the potential at the extremum. The bound state frequencies $\Omega(V_0,\alpha)$ of this potential are exactly known to be $$\begin{aligned}
\Omega(V_0,\alpha) = \alpha \bigg[-\bigg(n+\frac 1 2 \bigg) + \bigg(\frac 1 4 + \frac{V_0}{\alpha^2} \bigg)^{1/2} \bigg] .
\label{bo_pt}\end{aligned}$$ The proper QNM frequencies of the original potential in eqn. are then obtained from eqn. by the parameter replacement $(V_0,\alpha) \to(V_0,i\alpha)$ [@Ferrari:1984zz], and are given by the expression $$\begin{aligned}
\omega = \Omega(i\alpha) = \pm\sqrt{\bigg(V_0 - \frac{\alpha^2}{4} \bigg)} - i \alpha \bigg(n+\frac1 2 \bigg) .
\label{mash_form} \end{aligned}$$
However, this approach is just the first order approximation of the standard WKB method. To obtain the QNMs of more complicated potentials, WKB method is a convenient procedure which offers good accuracy. This method was originally suggested in [@Schutz:1985zz], and developed to the 3rd order beyond the eikonal approximation in [@Iyer:1986np]. It should also be noted that there has been considerable development in the accuracy in the Ferrari-Mashoon procedure and results up to 6th order in WKB are provided in [@Konoplya:2003ii] (see [@Konoplya:2010kv] for an usage of the 6th order WKB formula to the scattering problem). Below we provide the results of the 6th order WKB formula that will be exploited subsequently.
In WKB method one starts with the Schrodinger like wave equation $$\begin{aligned}
\Psi''(x) + (\omega^2 - V(x))\Psi(x) = 0 \end{aligned}$$ where the potential $V(x)$ approaches a constant at $x\to\pm\infty$ and at some intermediate value $x_0$, it rises to a maximum. For the present problem we identify $x\equiv r_{\star}$ and $\Psi(x)\equiv Q(r(r_{\star}))$. This problem is now analogous to the quantum mechanical scattering problem from the peak of a potential barrier, where the turning points divide the potential into three regions. The solutions in those regions are then matched at the boundaries to obtain the energy spectrum. However for the higher order WKB extension, it turns out that an explicit match of the interior solutions to WKB solutions in the exterior regions to the same order is not necessary (see [@Konoplya:2010kv] for details). The result with 6th order WKB formula then has the form $$\begin{aligned}
\omega^2 = V_0 - i \sqrt{-V_2} \bigg(n+\frac1 2 \bigg) + \sum_{i=2}^6 A_i \qquad n=0,1,2,...
\label{6wkb}\end{aligned}$$ where $A_i$’s represent i-th order correction in the WKB formula *e.g.*, $$\begin{aligned}
A_2 = ~&(-11 V_{3}^2 + 9 V_2 V_4 - 30 V_{3}^2 n + 18 V_2 V_4 n - 30 V_{3}^2 n^2 + 18 V_2 V_4 n^2)/(144 V_{2}^2) \\
\frac{i A_3}{\sqrt{-2 V_2}} = ~& (-155 V_{3}^4 + 342 V_2 V_{3}^2 V_4 - 63 V_{2}^2 V_{4}^2 - 156 V_{2}^2 V_3 V_5 + 36 V_{2}^3 V_6 - 545 V_{3}^4 n \nonumber \\
& + 1134 V_2 V_{3}^2 V_4 n - 177 V_{2}^2 V_{4}^2 n - 480 V_{2}^2 V_3 V_5 n + 96 V_{2}^3 V_6 n - 705 V_{3}^4 n^2 +
\nonumber \\
& 1350 V_2 V_{3}^2 V_4 n^2 - 153 V_{2}^2 V_{4}^2 n^2 - 504 V_{2}^2 V_3 V_5 n^2 + 72 V_{2}^3 V_6 n^2 - 470 V_{3}^4 n^3
\nonumber \\
& + 900 V_2 V_{3}^2 V_4 n^3 - 102 V_{2}^2 V_{4}^2 n^3 - 336 V_{2}^2 V_3 V_5 n^3 + 48 V_{2}^3 V_6 n^3)/(6912 V_{2}^5) .\end{aligned}$$ Other correction terms $A_4,A_5, A_6$ are given in [@Konoplya:2003ii]. Here $V_0 (\tilde{r}_{\star})$ is the value of the effective potential in its maximum ($r=\tilde{r}_{\star})$ and $V_i(\tilde{r}_{\star})$, is the i-th derivative of $V$ with respect to tortoise coordinate in the maximum.
Spacetime for a smeared (Gaussian) matter distribution {#gwgauss}
======================================================
In this section, we consider the metric of a spherically symmetric spacetime geometry with a Gaussian distributed matter source (this kind of matter distribution, motivated by Quantum Gravity perspective, has an astrophysical interest as well, as mentioned in Sec.\[introduc\]). Our aim is to compute the QNM frequencies for this smeared system (to $6$’th order in WKB scheme). The first task is to find the form of the potential for odd parity perturbations of this background geometry.
Perturbation of the spacetime {#gwgauss1}
------------------------------
Let us now start with the QG-inspired spherically symmetric spacetime. The metric of such spacetime is given by [@Nicolini:2008aj] $$\begin{aligned}
ds^2 = -\bigg(1-\frac{4M}{r\sqrt\pi}\gamma(3/2, {r^2}/{4\Theta})\bigg)dt^2+\bigg(1-\frac{4M}{r\sqrt\pi}\gamma(3/2, r^2/{4\Theta}) \bigg)^{-1}dr^2 + r^2 d\Omega^2
\label{qg_sphe}\end{aligned}$$ where, $d\Omega^2 = d\theta^2 + \sin^2\theta d\phi^2$, $\sqrt{\Theta}$ is some minimal length scale which removes the singularity of the usual Schwarzschild spacetime, and $$\begin{aligned}
\gamma(3/2, r^2/{4\Theta})=\int_{0}^{r^2/{4\Theta}}\sqrt t\,e^{t}\,dt \end{aligned}$$ is the lower incomplete Gamma function. If we now expand the incomplete Gamma function of eqn. in the limit $r^2>>4\Theta$, the metric takes the following form [@Nicolini:2008aj] $$\begin{aligned}
ds^2 = -\bigg(1- \frac{2M}{r} + \frac{2M}{\sqrt{\pi\Theta}}e^{-r^2/{4\Theta}}\bigg)dt^2+\bigg(1- \frac{2M}{r} + \frac{2M}{\sqrt{\pi\Theta}}e^{-r^2/{4\Theta}}\bigg)^{-1}dr^2 + r^2 d\Omega .
\label{mod_sph}\end{aligned}$$
The perturbed Einstein equation in vacuum is then $$\begin{aligned}
R_{\mu\nu} = \delta R_{\mu\nu} = 0 \qquad \text{as,}\qquad \bar{R}_{\mu\nu}=0 . \end{aligned}$$ This equation has ten components. It turns out that only three of them (corresponding to the components $\mathbf{\delta R_{r \phi}}, \mathbf{\delta R_{t \phi}}$ and $\mathbf{\delta R_{\theta \phi}}$) survive and they are respectively given below in explicit form
$$\begin{aligned}
\bigg( 1-\frac{2M}{r}+\frac{2M}{\sqrt{\pi\Theta}}e^{-r^2/{4\Theta}} \bigg) \bigg[\partial^{2}_{rr}h_0 - \partial_{r}\partial_{t}h_1
+ \frac{2}{r} \partial_{t}h_1 \bigg] - \frac{l(l+1)}{r^2}h_0 \nonumber \\
+ \frac{2}{r}\bigg(\frac{2M}{r^2}-\frac{Mr}{\Theta\sqrt{\pi\Theta}}e^{-r^2/{4\Theta}}\bigg)h_0 = 0 ,\end{aligned}$$
$$\begin{aligned}
\bigg(1-\frac{2M}{r}+\frac{2M}{\sqrt{\pi\Theta}}e^{-r^2/{4\Theta}} \bigg)^{-1} \bigg[\partial^{2}_{tt}h_1 - \partial_{r}\partial_{t}h_0
+ \frac{2}{r} \partial_{t}h_0 \bigg] + \frac{(l+2)(l-1)}{r^2}h_1 = 0 ,\end{aligned}$$
$$\begin{aligned}
\partial_r \bigg[\bigg( 1-\frac{2M}{r}+\frac{2M}{\sqrt{\pi\Theta}}e^{-r^2/{4\Theta}} \bigg)h_1\bigg] +
\frac{\partial_t h_0}{\bigg( 1-\frac{2M}{r} + \frac{2M}{\sqrt{\pi\Theta}}e^{-r^2/{4\Theta}} \bigg)} = 0
\label{eq_tehi}\end{aligned}$$
where $h_0,h_1$ have been introduced earlier in Sec. \[methqnms\](a).
Let us now introduce a $\Theta$-dependent generalized Regge-Wheeler variable $Q_{\Theta}$ as, $$\begin{aligned}
Q_{\Theta}(t,r) = \bigg(1- \frac{2M}{r} + \frac{2M}{\sqrt{\pi\Theta}}e^{-r^2/{4\Theta}}\bigg) \frac{h_1(t,r)}{r} .
\label{rw_va}\end{aligned}$$ With the help of eqn. we can eliminate $h_0(t,r)$ and thus the final equation for the axial perturbation assumes the following simple form, $$\begin{aligned}
\frac{\partial^{2}}{\partial t^2}Q_{\Theta}(t,r) - \frac{\partial^{2}}{\partial r_{\Theta}^2}Q_{\Theta}(t,r) + V^{\Theta}_{\text{axial}}(r)Q_{\Theta}(t,r) = 0,
\label{rg_eq}\end{aligned}$$ where $Q_{\Theta}(t,r)$ is given in eqn. and the potential function is $$\begin{aligned}
V^{\Theta}_{\text{axial}}(r) = \bigg(1-\frac{2M}{r}+\frac{2M}{\sqrt{\pi\Theta}}e^{-r^2/{4\Theta}}\bigg)\bigg[\frac{l(l+1)}{r^2}-\frac{6M}{r^3}
&+ \frac{M}{\Theta\sqrt{\pi\Theta}}e^{-r^2/{4\Theta}} \nonumber \\
&+ \frac{4M}{\sqrt{\pi\Theta}\,r^2}e^{-r^2/{4\theta}}\bigg] \label{new_pot}.\end{aligned}$$ We have defined the new co-ordinate $r_{\Theta}$, in analogy with the definition of eqn. , as $$\begin{aligned}
\frac{dr_{\Theta}}{dr} = \frac{1}{(1- \frac{2M}{r} + \frac{2M}{\sqrt{\pi\Theta}}e^{-r^2/{4\Theta}})} .
\label{star_newr}\end{aligned}$$ Likewise eqn. we write $ Q_{\Theta}(t,r)$ as $Q_{\Theta}(t,r) \sim e^{i\omega^{\Theta} t} Q_{\Theta}(r)$. So, eqn. now becomes $$\begin{aligned}
\frac{\partial^2 Q_{\Theta}(r)}{\partial r_{\Theta}^2} + \big[(\omega^{\Theta})^2 -V^{\Theta}_{\text{axial}}(r)\big]Q_{\Theta}(r) = 0 .
\label{rweq}\end{aligned}$$
Thus, we have obtained the form of the potential for the odd parity perturbation of this spacetime. Here $(\omega^{\Theta})^2$ plays the role of energy whose numerical estimate is what we are interested in.
QNM due to smeared matter distribution {#gwgauss2}
--------------------------------------
In this section we will evaluate the QNM frequencies for the potential given in eqn. . Following the line of discussion made in Sec. \[methqnms\], we will first determine the QNMs by employing the Ferrari-Mashoon method (*i.e.* first order WKB) and subsequently we will also use the 6th order WKB formula for computing QNMs with much better precision.
To apply Ferrari-Mashoon method, we need to see whether this potential can also be mapped to the so-called P$\ddot{\text{o}}$schl-Teller potential of eqn. . Let us write the modified potential for axial perturbation after incorporating the $\Theta$-correction as, $$\begin{aligned}
V^{\Theta}_{\text{axial}}(r) &= V_{\text{axial}} (r) + V_{\text{axial}}^{\text{extra}}(r) \label{fu_po}
%\text{where,} ~~ V^{\text{axial}}_{\text{Sch}} (r) = & \bigg(1-\frac{2M}{r}\bigg)\bigg(\frac{l(l+1)}{r^2}-\frac{6M}{r^3}\bigg) \label{po_un}\end{aligned}$$ where $V_{\text{axial}}$ is given in eqn. and $$\begin{aligned}
V_{\text{axial}}^{\text{extra}}(r) = & \frac{2M}{\sqrt{\pi\Theta}}e^{-r^2/{4\Theta}}\bigg[ \frac{l(l+1)}{r^2}-\frac{6M}{r^3}
+\bigg( \frac{1}{2\Theta} + \frac{2}{r^2} \bigg) \bigg(1-\frac{2M}{r}\bigg) \bigg] .
\label{qg_part}\end{aligned}$$ Just by inspection of eqn. we can see that at large distance from the horizon at $r=2M$, the correction terms fall exponentially fast. Therefore, this effective potential is not expected to deviate much from $V_{PT}$. To see that, we find out the minimum of the $\Theta$-corrected effective potential in eqn. .
We assume that the extremum of the new potential $V^{\Theta}_{\text{axial}}(r)$ is perturbatively shifted to the point, $$\begin{aligned}
r_{0}^{\Theta} \simeq ~~ r_0 + \frac{2M}{\sqrt{\pi\Theta}}\,e^{-r_{0}^2/{4\Theta}}\,r^{\prime}
\label{minima_new}\end{aligned}$$ where $r^{\prime}$ is so far unknown and $r_0$ is the minimum of the effective potential of eqn. , given by $$\begin{aligned}
\frac{r_0}{M} = \frac{\sqrt{9(l^2+l+3)^2 -96 l(l+1)}}{2l(l+1)} + \frac{3(l^2 +l+3)}{2l(l+1)} .\end{aligned}$$ Now taking the derivative of eqn. and using eqn. , we solve for $r^{\prime}$ to get the new extremum located at $$\begin{aligned}
r_{0}^{\Theta} &\simeq \,\, r_0 + \frac{2M}{\sqrt{\pi\Theta}}\,e^{-r_{0}^2/{4\Theta}}\times\frac{1}{\Delta}\big[r_{0}^2 A +
r_{0}^4 B \big] \\
\text{where,}\quad A &= -2M\big[r_{0}^4 +12r_{0}^2\Theta +60\Theta^2\big], \nonumber\\
B &= \big[r_{0}^4 + 2(l^2 + l + 2)\Theta(r_{0}^2 + 4\Theta) \big], \nonumber \\
\Delta &= 24[ l(1 + l) r_{0}^2 - 4 (l^2 + l + 3) M r_0 + 40 M^2 ] \Theta^2 . \nonumber \end{aligned}$$
We note that for the $\Theta$-corrected metric of eqn. , the co-ordinate $r_{\Theta}$ is a function of the parameter $\Theta$. Integrating eqn. numerically, we found that the behaviour of this new coordinate $r_{\Theta}$ is not markedly different from $r_{\star}$, since $\frac{dr_{\star}}{dr}\sim\frac{dr_{\Theta}}{dr}$. Therefore, the maximum of the potential $V^{\text{axial}}_0$ and the curvature parameter $\alpha^{\text{new}}$ are then given by, $V^{\Theta}_0 = V_{\text{axial}}^{\Theta}(r) \big| _{r=r_{0}^{\Theta}}$ and $\alpha^{\Theta} = \frac{1}{2V^{\Theta}_0}\,\frac{d^2 V_{\text{axial}}^{\Theta}(r)}{dr_{\Theta}^2}\bigg|_{r=r_{0}^{\Theta}}$ where $d^2/dr_{\Theta}^2$ can be found by using the new transformation rule of eqn. .
![Plot showing the comparison of $V_{\text{axial}}$ (red) and $V_{PT}$ (dashed blue) as a function of the new coordinate $r_{\Theta}$ for $\Theta=0.3$, and $0.4$ respectively. The figure shows that the asymptotic behavior of both potentials are very close to each other.[]{data-label="qnm_pot"}](potsm03.pdf "fig:"){width="8.1cm"} ![Plot showing the comparison of $V_{\text{axial}}$ (red) and $V_{PT}$ (dashed blue) as a function of the new coordinate $r_{\Theta}$ for $\Theta=0.3$, and $0.4$ respectively. The figure shows that the asymptotic behavior of both potentials are very close to each other.[]{data-label="qnm_pot"}](potsm04.pdf "fig:"){width="8.1cm"}
In Fig. \[qnm\_pot\] we plot variation of the P$\ddot{\text{o}}$schel-Teller potential $V_{PT}$ and the potential for the axial perturbation $V_{\text{axial}}$ as a function of the co-ordinate $r_{\Theta}$. We have taken two different values of $\Theta$ to show that the form of the axial perturbation potential $V_{\text{axial}}^{\Theta}$ does not differ drastically from the form of $V_{PT}$ with respect to the new transformed co-ordinate $r_{\Theta}$. Since the QNMs of $V_{\text{axial}}^{\Theta}$ are related with the asymptotical behaviour of the potential, therefore one may legitimately use eqn. to get the QNMs for this potential.
As mentioned in Sec. \[methqnms\], the semi-analytic method due to Ferrari-Mashoon is one of the easiest tool to estimate the QNMs. However, the WKB treatment [@Schutz:1985zz] for computing QNMs offers much better accuracy. Also, in the context of QNM determination, the 3rd order WKB formula [@Iyer:1986np] was frequently used in the literature (see refs. [@Konoplya:2002ky; @Kokkotas:1993ef; @Andersson:1996xw; @Onozawa:1995vu]). Later it was shown that the WKB formula, when extended to the 6th order, gives the relative error which is about two orders less than that of the 3rd WKB order [@Konoplya:2004ip]. Therefore, in this work we will also compute the QNMs for the potential $V_{\text{axial}}^{\Theta}$ using the 6th order WKB formula given by eqn. . A previous study of QNMs for different perturbations of the $\Theta$-corrected metric was done by Liang [@Liang:2018nmr; @Liang:2018uyk] using 3rd order WKB formula. But we will shortly see that with the 6th order WKB formula, the associated QNMs will show non-trivial modifications in their numerical estimates. Finally, we will make a comparison of the results with various orders of the WKB method.
Results of QNM {#gwgauss3}
--------------
In fig. \[mfig\] we plot (colored dashed plots) the variation of Re$[\omega^{\Theta}]$ as a function of $\Theta$ using Ferrari-Mashoon method (*i.e.* 1st order WKB). The associated frequencies Re$[\omega]$ (corresponding to the normal case with potential of eqn. ) in this graph are also shown by colored continuous plots. Since Re$[\omega]$ being independent of $\Theta$ for all values $l$, so they appear to be parallel straight lines for the same set of $l$ values. Here, the difference (Re$[\omega^{\Theta}]-$Re$[\omega]$) is found to increase for higher $l$ values (though shown for $l$ upto 2, but this is true for $l>2$ also, as explicitely checked by us). But this simple observation does not remain strictly valid when the 6th order WKB formula eqn. is exploited. Therefore, while eqn. captures the relevant changes in QNMs as a first hint, it is better to rely on the 6th order WKB formula for numerical precision.
![Plot showing the variation of the real part of QNM frequency $\omega^{\Theta}$, with increasing value of the parameter $\Theta$. For the above plot we have considered $l=2,~3$. []{data-label="mfig"}](mashfig.pdf){width="10.2cm"}
![Plot showing the variation of the real part of QNM frequency $\omega^{\Theta}$, with increasing value of the parameter $\Theta$. For the above plot we have considered $l=3,~n=0$ and different orders of the WKB formula. []{data-label="freq_plot"}](wkbgraph6th.pdf){width="10.2cm"}
------------------------------------------------------------------------------------ -- -- -- -- --
**l & **n & **Gravitational & $\Theta$ & **Gravitational & **Gravitational\
& & **modes & & **modes & **modes\
& & (Schwarzschild) & & (Smeared matter) & (Smeared matter)\
& & & & 3rd order WKB & 6th order WKB\
& & & & &\
2 & 0 & 0.373616 - i0.0888891 & 0.1 & 0.373163 - i0.089221 & 0.374579 - i0.088672\
& & & 0.2 & 0.371753 - i0.089350 & 0.470351 - i0.044726\
& & & 0.3 & 0.360579 - i0.066717 & 0.309889 - i0.123928\
& 1 & 0.346297 - i0.273480 & 0.1 & 0.346028 - i0.274930 & 0.352931 - i0.268630\
& & & 0.2 & 0.339165 - i0.275108 & 0.918186 - i0.002729\
& & & 0.3 & 0.234593 - i0.185924 & 0.235278 - i0.629412\
3 & 0 & 0.599444 - i0.092703 & 0.1 & 0.599265 - i0.092732 & 0.599440 - i0.092735\
& & & 0.2 & 0.598163 - i0.091950 & 0.600413 - i0.090236\
& & & 0.3 & 0.594996 - i0.081382 & 0.596246 - i0.086833\
& 1 & 0.582642 - i0.281291 & 0.1 & 0.582362 - i0.281424 & 0.582510 - i0.281662\
& & & 0.2 & 0.576212 - i0.278026 & 0.573718 - i0.276666\
& & & 0.3 & 0.549907 - i0.239466 & 0.581571 - i0.261865\
& 2 & 0.551594 - i0.479047 & 0.1 & 0.553235 - i0.476731 & 0.550833 - i0.481339\
& & & 0.2 & 0.534658 - i0.469451 & 0.467130 - i0.566279\
& & & 0.3 & 0.442211 - i0.393935 & 0.648012 - i0.366030\
****************
------------------------------------------------------------------------------------ -- -- -- -- --
: Comparison between the QNM frequencies for the gravitational perturbation of Schwarzschild spacetime and spherically symmetric spacetime with smeared matter source. []{data-label="tab_gwv"}
The following Table (Tab.\[tab\_gwv\]) shows the values of QNM frequencies for the odd parity gravitational perturbation of the $\Theta$-corrected space-time calculated using 6th order WKB formula.
![Plot showing the variation of Re$[\omega^{\Theta}]$ as a function of the parameter $\Theta$ for $l=2, n=0$ and $l=2, n=1$ using the 6th order WKB formula. []{data-label="wkbm_plot"}](wkb6thl2n01.pdf){width="10.2cm"}
In fig. \[freq\_plot\], we plot the real part of QNM frequencies as a function of the parameter $\Theta$ for various orders of the WKB approximation. It is clear from the graph that why the higher order WKB gives significant alteration in the value of Re$[\omega^{\Theta}]$. For $0.2>\Theta >0.01$, the 3rd order WKB result for the QNM frequency is nearly a constant. But this is not the correct picture if we go to 5th or 6th order approximations. In fact, Re$[\omega^{\Theta}]$ begins to decrese much earlier for $\Theta\gtrsim0.12$. This change is significant. Fig. \[wkbm\_plot\] shows another plot for Re$[\omega^{\Theta}]$ for $l=2\,(n=0,1) $ modes computed using 6th order WKB formula. Let us compare our results which is valid up to 6th order in WKB with that of Liang [@Liang:2018nmr] that is valid up to 3rd order in WKB. Liang has considered $\Theta $ to be lower than $0.25$ approximately. It is straightforward to check that for $\Theta > 0.25 $ there appears oscillations in the value of QNM frequency which seems to be spurious indicating that the 3rd order WKB perturbation scheme used by Liang is reliable up to $\Theta \sim 0.25$. On the other hand, from Fig. \[freq\_plot\] it is clear that in 4th, 5th or 6th order in WKB such oscillations appear earlier at around $\Theta \sim 0.16$. Thus 6th order computational results restrict the value of $\Theta$ to lower values where the results are reliable. Naively it might seem that higher than 6th order results in WKB might restrict $\Theta$ further but as has been noted in [@Konoplya:2003ii] orders of WKB higher than six are not feasible in the WKB framework.
Observational aspects of $\Theta $-correction in QNM {#gwgauss4}
-----------------------------------------------------
So far, we have depicted the change in the QNM frequency spectrum when the source has smeared matter distribution. Here, we discuss the relevance of this result in the context of observational aspects. As an example, let us consider the fundamental GW mode (for $l=2, n=0$) of Schwarzschild geometry. The associated real part of the frequency $\omega_{re}= 0.373616$ from Tab. \[tab\_gwv\] can be expressed in Hz unit as $$\begin{aligned}
f= \frac{\omega_{re}}{2\pi M} \times \frac{c^3}{G} = \bigg(\frac{\omega_{re}}{2\pi}\times\frac{c^3}{GM_{\odot}}\bigg)
\times\frac{M_{\odot}}{M}
\label{conobs}\end{aligned}$$ where $M_{\odot}$ is the solar mass. Using this formula with $M=1~M_{\odot}$, the frequency $f$ for the fundamental mode turns out to be $12$ kHz.
Now there exist compact spherical star clusters (*e.g* globular clusters) that approximately follow a Gaussian matter distribution. A typical order of magnitude estimate for the mass ($ \tilde M $) of such a cluster is $10^5 M_{\odot}$. With $M=\tilde{M}$, it can be shown from eqn. that if the corresponding smeared distribution has a spread $\sqrt{\Theta}\sim 10^7 $ km (which matches with a $\Theta \sim 0.173$ within the range $\Theta \sim 0.16 - 0.19$ of Tab. \[tab\_gwv\]), then it yields a signal having frequency $f\sim13 $ kHz. As a first clue, this small change in frequency is significant to infer the nature of the source, that is to say, whether a GW detected with this frequency is associated to a point mass or a diffused mass pattern.
Conclusion {#conc}
==========
In this work, we have studied the QNM frequency spectrum for the static spherically symmetric spacetime having a [*[smeared]{}*]{} (Gaussian type) matter distribution. This type of matter distribution, involving a length scale, can be motivated from astrophysical perspectives (in the context of star clusters). Hence our result can be relevant for those scenarios depending on a proper choice of the length of smearing scale. Also, such a length scale is crucial to identify the character of the source density. As a demonstration with astrophysical objects, we found that the resulting frequency change due to smearing is of $\mathcal {O}(Hz)$ and hence within the current limits of the terrestrial GW detectors. Originally such metrics with smeared matter distribution was motivated by quantum gravity with the smearing length tentatively identified with Planck length. However, in that case, due to the smallness of Planck length scale such quantum gravity motivated corrections will be difficult to observe.
In summary, our analysis here focuses on the gravitational perturbations of a background geometry, that are odd multipoles under parity transformation. The gravitational perturbations do posses an even parity component as well. For the case of conventional spherically symmetric Schwarschild geometry with a delta-function source, there is a special property for the perturbation spectrum that ensures that the QNM spectra for odd and even parity perturbations are equal. In technical terms, one says that the QNM spectrum of odd parity perturbations is iso-spectral with the QNM spectrum of even parity perturbations [@Chandrasekhar:1985kt]. However, it is not clear whether the same would hold true for spacetime with a smeared matter distribution, that has been studied here. With Gaussian distributed matter density, the potential for even-parity perturbation would have a new ($\Theta$-dependent) scale. This feature may restrict the validity of the iso-spectral character between perturbations of opposite parity. This requires an explicit computation of the QNM spectrum of even-parity perturbations that we have left as a future work.
[**[Acknowledgement:]{}**]{} It is a pleasure to thank Professor Roman Konoplya for helpful suggestions and informing us about the relevant references for this work and specially for sending us the necessary code for computation. Also we thank the referee for constructive comments.
[unsrt]{}
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[^1]: [email protected]
[^2]: [email protected]
[^3]: [email protected]
[^4]: The other component $h_0(t,r)$ can be removed by using the $\delta R_{\theta\phi}$ components of eqn. ($\because \delta R_{\mu\nu} = \nabla_{\beta}\delta\Gamma_{\mu\nu}^{\beta} - \nabla_{\nu}\delta\Gamma_{\mu\beta}^{\beta} = 0$) [@Regge:1957td]
[^5]: For details and review of the other methods see for example [@Berti:2005ys; @Macedo:2016wgh].
|
\
[Department of Applied Mathematical and Physical Sciences, National Technical University of Athens, Greece]{}\
[[email protected]]{}\
**Abstract**
Using an elementary identity, we prove that for infinitely many polynomials $P(x)\in \mathbb{Z}[X]$ of fourth degree, the equation $\prod\limits_{k=1}^{n}P(k)=y^2$ has finitely many solutions in $\mathbb{Z}$. We also give an example of a quartic polynomial for which the product of it’s first consecutive values is infinitely often a perfect square. =12.875pt
Introduction
============
Over the last few years, there has been a growing interest in identifying if certain product sequences contain perfect squares. In 2008 Javier Cilleruelo [@1] proved that the product $(1^2+1)(2^2+1)\cdots (n^2+1)$ is a square only for $n=3$. Soon after, Jin-Hui Fang [@2] achieved to prove that both of the products $\prod\limits_{k=1}^{n}(4k^2+1)$ and $\prod\limits_{k=1}^{n}\big(2k(k-1)+1\big)$ are never squares. There are not many similar results for quadratic polynomials. However, in a recent paper [@3] two certain cases of quartic polynomials were settled. In this paper we will prove using elementary arguments that there is actually an infinite collection of quartic polynomials $P(x)$ such that the product $\displaystyle\prod\limits_{k=1}^{n}P(k)$ is a square finitely often. At the end of the paper we discuss some cases that can be handled by this method. We begin with a polynomial identity which is the key ingredient throughout this article.
Let $f(x)=x^2+ax+ b$ be a quadratic polynomial. For every $x\in \mathbb{R}$ the following formula is valid: $$f\big(f(x) + x\big) = f(x)f(x + 1)$$.
We can verify this just by doing elementary manipulations but we will prove the lemma using a clever observation. Since $f(x)$ is a polynomial of second degree, Taylor’s formula gives $f\big(f(x) + x\big)=f(x)+\frac{f'(x)f(x)}{1!}+f^2(x)$. This is equal to $f(x)\big(1+f'(x)+f(x) \big)$. But $1+f'(x)+f(x)=1+2x+a+x^2+ax+b=(x+1)^2+a(x+1)+b=f(x+1)$. Hence we have: $f\big(f(x) + x\big) = f(x)f(x + 1)$.
This simple formula will play a key role in the proof of the main theorem. For convenience of notation we set $f\big(f(k) + k\big)=P(k)=f(k)f(k+1)$. It can be seen that $P(k)=k^4+2(a+1)k^3+\big((a+1)^2+2b+a)\big)k^2+(a+1)(2b+a)k+b^2+ab+b$. In the proof of the main theorem, we require $a$ and $b$ to obey a certain restriction. Under this restriction, we are able to prove that equation (1) has finitely many solutions.
Main Results
============
Let $a, b, m\in \mathbb{Z}$ and $a+b+1=m^2$. Then the diophantine equation $$\displaystyle\prod\limits_{k=1}^{n} P(k)=y^2\label{1}$$ has finitely many solutions.
Using lemma 1 we can rewrite equation (\[1\]) as $f(1)f(2)f(2)f(3)\cdots f(n)f(n+1)=y^2$ which reduces to $f(1)f(n+1)\displaystyle\prod\limits_{k=2}^{n}(f(k))^2=y^2$. Since $f(1)=a+b+1=m^2$ we conclude that $f(n+1)=\frac{y^2 }{m^2\prod\limits_{k=2}^{n}(f(k))^2}$. It becomes clear that equation (\[1\]) is satisfied whenever $f(n+1)$ is a perfect square. It remains to prove that among the values of $f(k)$ occur finitely many squares. Write$$k^2+ak+b=z^2 \label{2}$$ for some $z\in \mathbb{Z}$. This means that for sufficiently large $k$, $k^2<z^2<(k+2a)^2$ if $a> 0$ or, $(k+2a)^2<z^2<k^2$ if $a< 0$. (If $a=0$ then equation (\[2\]) transposes to $(z-x)(z+x)=b$ which clearly has finitely many solutions). Both of the inequalities yield $z=k+c$ for some $c\in \mathbb{Z}$ with $|c|<|2a|$. So, (\[2\]) becomes $k^2+ak+b=(k+c)^2$ which has finitely many solutions as the reader may easily verify.
It suffices to choose some nice values for a and b in order to demonstrate the theorem. Choosing $(a, b) = (-1, 1)$ we have $f(k) = k^2-k + 1$ hence the following:
$\displaystyle\prod\limits_{k=1}^{n}(k^4+k^2+1)$ is a square only for $n=1$.
If $(a, b)=(-1, 1)$ then $f(1)=1^2$. Repeating the previous arguments, it suffices to show that $k^2-k+1=y^2$ has one solution. Indeed, if $k^2-k+1=y^2$ then we must have $k^2\le y^2<(k+1)^2$ which yields $y=k$ and so $k=1$. The claim follows.
Arguing as in the previous section, we may present an example which shows that equation (\[1\]) has infinitely many solutions. Choosing $(a, b)=(-4, 2)$ we have $f(k)=k^2-4k+2$ and $P(k)=\big(k(k-3)\big)^2-2$. We can prove that the product $\displaystyle\prod\limits_{k=4}^{n}\Big(\big(k(k-3)\big)^2-2\Big)$ is a square infinitely often. Here we start with $k=3$ to omit any trivial case in which the product has negative factors. The product is a square if $f(4)f(n+1)=2(n-1)^2-4=y^2$. It is a routine matter to prove that both $y$ and $n-1$ must be even. Thus, equation can be written as $(\frac{y}{2})^2-2(\frac{n-1}{2})^2=-1$ which is a special case of the negative Pell equation $X^2-2Y^2=-1$. This equation has the fundamental solution $(1, 1)$ and all it’s positive solutions can be found by taking odd powers of $1+\sqrt 2$. The positive solutions are $(X_n,Y_n)$ where $X_n+Y_n\sqrt 2=(1+\sqrt 2)^{2n-1}$. The next solution is $(X_2, Y_2)=(7, 5)$ which gives $n=11$. As an example we can verify that $\displaystyle\prod\limits_{k=4}^{11}\Big(\big(k(k-3)\big)^2-2\Big)=246988938224^2$
[99]{} J. Cilleruelo, Squares in $(1^2+1) \cdots (n^2+1)$, J. Number Theory 128 (2008) 2488-2491. J.-H. Fang, Neither $\prod\limits_{k=1}^{n}(4k^2+1)$ nor $\prod\limits_{k=1}^{n}(2k(k-1)+1)$ is a perfect square, Integers 9 (2009) 177-180 Erhan Gürel. “On the Occurrence of Perfect Squares Among Values of Certain Polynomial Products.” The American Mathematical Monthly 123.6 (2016): 597-99.
|
---
abstract: |
In a recently published paper Ciufolini reports on the so far performed tests aimed at the detection of the general relativistic gravitomagnetic Lense-Thirring effect in the gravitational field of the Earth by means of the analysis of the laser-ranged data of the existing LAGEOS and LAGEOS II geodetic satellites. In this paper we will critically discuss his claims by showing that the total error, mainly due to the systematic bias due to the mismodelling in the static and time-varying parts of the multipolar expansion of the Newtonian terrestrial gravitational potential, is larger than that claimed by Ciufolini. E.g., the systematic error due to the mismodelling in the static part of the geopotential in the tests performed with the EGM96 Earth gravity model and the combination involving the nodes of LAGEOS and LAGEOS II and the perigee of LAGEOS II realistically amounts to more than 80$\%$ (1-$\sigma$): the claimed total uncertainty, including also the non-gravitational perturbations which especially affect the perigee of LAGEOS II, is, instead, 20-25$\%$. The claimed accuracy in the more precise tests performed with the 2nd generation CHAMP-only EIGEN2 Earth gravity model and a combination involving the nodes of LAGEOS and LAGEOS II over 10 years is 18$\%$. With numerical simulations we will show that, instead, it is $\leq
51\%$ (1-$\sigma$) if the impact of the secular variations of the even zonal harmonics over a so long observational time span ($\sim
14\%$) is accounted for.
---
\#1[ \#1 ]{} \#1\#2[\_[\#1]{}\^[\#2]{}]{} \#1[eq. (\[\#1\])]{} \#1\#2[(\[\#1\])-(\[\#2\])]{} \#1[Eq. (\[\#1\])]{} \#1\#2[(\[\#1\])-(\[\#2\])]{}
\#1\#2 \#1\#2 \#1\#2
|[$$\begin{aligned}
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\def\ear{\end{aligned}$$]{} \#1\#2[[\#1\#2]{}]{} \#1 \#1[\[\#1\]]{}
2[$\mathcal{O}(c^{-2})$]{}
\#1
[****]{}\
\
\
[Lorenzo Iorio]{}\
[*Dipartimento Interateneo di Fisica dell’ Universit${\rm
\grave{a}}$ di Bari\
Via Amendola 173, 70126\
Bari, Italy\
e-mail: [email protected]*]{}
Introduction
============
Recent years have seen increasing efforts aimed to directly detecting various phenomena connected to the general relativistic gravitomagnetic field of the rotating Earth. It should be noted that, according to K. Nordtvedt , the multidecadal analysis of the Moon’orbit by means of the Lunar Laser Ranging (LLR) technique yields a comprehensive test of the various parts of order $\mathcal{O}(c^{-2})$ of the post-Newtonian equation of motion. The existence of gravitomagnetism as predicted by the Einstein’s General Theory of Relativity would, then, be indirectly inferred from the high accuracy of the lunar orbital reconstruction. In the same arguments are applied to the radial motion of the LAGEOS satellite.
The extraordinarily sophisticated and expensive Gravity Probe B (GP-B) mission has been launched in April 2004; it is aimed at the detection of the gravitomagnetic precession of the spins of four superconducting gyroscopes carried onboard at a claimed accuracy of 1$\%$ or better.
The Lense-Thirring effect on the orbital motion of a test particle could be measured by analyzing the orbital data of certain Earth artificial satellites with the Satellite Laser Ranging (SLR) technique . Up to now, the only performed tests are due to Ciufolini and coworkers.
In this paper we will analyze the latest results presented in from a critical point of view in order to show that the claimed accuracies are optimistic.
The Lense-Thirring effect on the orbit of a test particle and the strategy to measure it
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The gravitomagnetic field of a spinning mass of proper angular momentum $J$ induces tiny secular precessions on the longitude of the ascending node $\Omega$ and the argument of pericentre[^1] $\omega$ of a test particle \_[LT]{}=,\_[LT]{}=-,where $G$ is the Newtonian constant of gravitation, $c$ is the speed of light in vacuum, $a,e$ and $i$ are the semimajor axis, the eccentricity and the inclination, respectively, of the test particle’s orbit. In the terrestrial space environment the gravitomagnetic precessions are very small: for the geodetic SLR LAGEOS satellites, whose orbital parameters are listed in Table \[para\], they amount to a few tens of milliarcseconds per year (mas yr$^{-1}$ in the following)
[lllll]{}
Satellite & $a$ km) & $e$ & $i^{\circ}$ & $\dot\Omega_{\rm LT}$ (mas yr$^{-1}$)\
LAGEOS & 12270 & 0.0045 & 110 & 31\
LAGEOS II & 12163 & 0.0135 & 52.64 & 31.5\
The extraction of the Lense–Thirring precessions from the orbit data analysis is very difficult due to a host of competing classical orbital perturbations of gravitational and non-gravitational origin which have various temporal signatures and are often quite larger than the relativistic signal of interest. The most insidious ones are the perturbations which have the same temporal signature of the Lense-Thirring precessions[^2], i.e. secular trends. Indeed, whatever the length of the adopted observational time span $T_{\rm
obs}$ is, they cannot be fitted and removed from the time series without removing the relativistic signal as well. Then, it is of the utmost importance to assess as more accurately and reliably as possible their aliasing impact on the measurement of the Lense-Thirring effect.
It turns out that the perigees of the LAGEOS-like satellites are severely affected by the non-gravitational perturbations, contrary to the nodes. Moreover, since the non–conservative forces depend on the structure, the shape and the rotational status of the satellite their correct modelling is not a trivial task and, as we will see later, introduces large uncertainties in the correct assessment of the error budget in some of the performed gravitomagnetic tests.
The gravitational error
-----------------------
The even ($\ell=2,4,6...$) zonal ($m=0$) harmonic coefficients $J_{\ell}$ of the multipolar expansion of the Earth’s gravitational potential, called geopotential, induce secular precessions[^3] on the node and the perigee of any near-Earth artificial satellite which, of course, depend only on its orbital configuration and are independent of its physical structure. Such aliasing effects are many orders of magnitude larger than the Lense-Thirring precessions; the precision with which the even zonal harmonics are known in the currently available Earth gravity models would yield errors amounting to a significant fraction of the Lense-Thirring precessions or even larger.
Even more dangerous are the perturbations induced by the secular variations of the low degree even zonal harmonics $\dot
J_{\ell},\ell=2,4,6$ . Indeed, such perturbations grow quadratically in time if the shifts in mas are considered and linearly in time if the rates in mas yr$^{-1}$ are considered. Their impact on the orbital elements of the LAGEOS satellites have been worked out in . It turns out that, by using the results of , the errors induced by $\dot J_2$ would amount to 8$\%$, 14$\%$ and $5.4\%$ for the nodes of LAGEOS and LAGEOS II and the perigee of LAGEOS II, respectively, over an observational time span $T_{\rm obs}$ of just one year at 1$-\sigma$ level. This clearly shows that it would be impossible to analyze single orbital elements.
The time-dependent periodic perturbations are less dangerous because if their periods are shorter than the adopted observational time span they can be fitted and removed from the time series. The most insidious tidal perturbation is that induced by the even zonal constituent which has a period of 18.6 years and whose nominal impact on the orbital elements of the LAGEOS satellites amounts to thousands of mas . However, it turns out that it does not affect the observables which have been adopted for the performed Lense-Thirring tests because its main component is of degree $\ell=2$ and order $m=0$.
### The linear combination approach
The problem of reducing the impact of the mismodeling in the even zonal harmonics of the geopotential with the currently existing satellites can be coped in the following way .
Let us suppose we have at our disposal N (N$>1$) time series of the residuals of those Keplerian orbital elements which are affected by the geopotential with secular precessions, i.e. the node and the perigee: let them be $\psi^{\rm A},$ A=LAGEOS, LAGEOS II, etc. Let us write explicitly down the expressions of the observed residuals of the rates of those elements $\delta\dot\psi^{\rm A}_{\rm obs}$ in terms of the Lense-Thirring effect $\dot\psi_{\rm LT}^{\rm A}$, of N-1 mismodelled classical secular precessions $\dot\psi_{.\ell}^{\rm A}\delta J_{\ell}$ induced by those even zonal harmonics whose impact on the measurement of the gravitomagnetic effect is to be reduced and of the remaining mismodelled phenomena $\Delta$ which affect the chosen orbital element \_[obs]{}\^[A]{}=\_[LT]{}\^[A]{}\_[LT ]{}+\_[.]{}\^[A ]{}J\_+\^[A]{}, The parameter[^4] $\mu_{\rm LT}$ is equal to 1 in the General Theory of Relativity and 0 in Newtonian mechanics. The coefficients $\dot\psi_{.\ell}^{\rm A}$ are defined as \_[.]{}=and have been explicitly worked out for the node and the perigee up to degree $\ell=20$ in ; they depend on some physical parameters of the central mass ($GM$ and the mean equatorial radius $R$) and on the satellite’s semimajor axis $a$, the eccentricity $e$ and the inclination $i$. We can think about as an algebraic nonhomogeneuous linear system of N equations in N unknowns which are $\mu_{\rm LT}$ and the N-1 $\delta J_{\ell}$: solving it with respect to $\mu_{\rm LT}$ allows to obtain a linear combination of orbital residuals which is independent of the chosen N-1 even zonal harmonics. In general, the orbital elements employed are the nodes and the perigees and the even zonal harmonics cancelled are the first N-1 low-degree ones.
This approach is, in principle, very efficient in reducing the impact of the systematic error of gravitational origin because all the classical precessions induced by the static and time-dependent parts of the chosen N-1 $J_{\ell}$ do not affect the combination for the Lense-Thirring effect. Moreover, it is flexible because it can be applied to all satellites independently of their orbital configuration, contrary to the butterfly configuration in which the cancellation of the even zonal harmonics can be achieved only for supplementary orbital planes and identical orbital parameters. Apart from the first orbital element which enters the combination with 1, the other elements are weighted by multiplicative coefficients $c_i(a,e,i)\neq 1$ which are built up with $\dot\psi_{.\ell}$ and, then, depend on the orbital elements of the considered satellites. Their magnitude is very important with respect to the non-gravitational perturbations, which in general are not cancelled out by the outlined method, and to the other time-dependent perturbations of gravitational origin with $\ell\neq 2,4,6,..,m\neq 0$. Values smaller than 1 for the $c_i$ coefficients are, in general, preferable because they reduce the impact of such uncancelled perturbations. It is important to note that the order with which the orbital elements enter the combination is important: indeed, while the systematic error due to the even zonal harmonics of the geopotential remains unchanged if the orbital elements of a combination are exchanged, the coefficients $c_i$ do change and, consequently, also the non-gravitational error. The best results are obtained by choosing the highest altitude satellite as first one and by inserting the other satellites in order of decreasing altitudes.
This method was explicitly adopted for the first time in with the nodes of the LAGEOS satellites and the perigee of LAGEOS II. The obtained combination is \^[LAGEOS ]{}\_[obs ]{}+c\_1\^[LAGEOS II]{}\_[obs ]{}+c\_2\^[LAGEOS II]{}\_[obs ]{}\~\_[LT]{}60.2,where $c_1=0.304$, $c_2=-0.350$ and 60.2 is the slope, in mas yr$^{-1}$, of the expected gravitomagnetic linear trend. is insensitive to the first two even zonal harmonics $J_2$ and $J_4$. It has been used in when the level of accuracy of the JGM3 and EGM96 Earth gravity models, available at that time, made it necessary to consider a combination of observables which is independent of errors in both $J_2$ and $J_4$.
In view of the great improvements in the Earth gravity field modelling with the CHAMP and, especially, GRACE missions an extensive search for alternative combinations has been subsequently performed . In the following combination has been proposed[^5] \^[LAGEOS ]{}\_[obs ]{}+k\_1\^[LAGEOS II]{}\_[obs]{}\~\_[LT]{}48.2,where $k_1= 0.546$ and 48.2 is the slope, in mas yr$^{-1}$, of the expected gravitomagnetic linear trend. It has been adopted for the tests performed in with the 2nd generation CHAMP-only EIGEN2 Earth gravity model and the 1st generation GRACE-only GGM01S Earth gravity model. allows to cancel out the first even zonal harmonic $J_2$.
The performed Lense-Thirring tests with the LAGEOS satellites
=============================================================
The only performed tests aimed at the detection of the Lense-Thirring precessions of in the gravitational field of the Earth with the existing LAGEOS satellites have been performed, up to now, by Ciufolini and coworkers. They have used the node-node-perigee combination of and the node-node combination of .
In it is claimed that “\[...\] [*the Lense-Thirring effect exists and its experimental value,*]{} \[...\][*, fully agrees with the prediction of general relativity.*]{}" in regard to both the tests with the EGM96 and EIGEN-2 Earth gravity models. In this Section we will disprove such statements.
The main objections to the results presented in these works can be summarized as follows
- Ciufolini has not performed tests by varying the length of the adopted observational time span, running backward and forward the initial epoch of the analysis, varying the secular rates of the even zonal harmonics in order to check their impact over different time spans, using different Earth gravity models in order to obtain a scatter plot of the obtained results.
- The total error budget has been underestimated, especially the systematic error of gravitational origin. E.g., the impact of the secular variations of the even zonal harmonics of the geopotential, which may become a very limiting factor over time spans many years long as those used, has not been addressed. Almost always 1$-\sigma$ results have been presented without any explicit indication of this fact.
The node-node-perigee tests
---------------------------
The combination of has been analyzed by using the EGM96 Earth gravity model over 4 years in and over 7.3 years in . The claimed total error budget amounts to 20-25$\%$ over 4 years and to 20$\%$ over 7.3 years.
### The gravitational error
The impact of the remaining uncancelled even zonal harmonics of the geopotential $J_6, J_8, J_{10},...$ on has been estimated by Ciufolini and coworkers with the full covariance matrix of EGM96 in a root-sum-square calculation. In and, six years later, in it is claimed to be $\lesssim 13\%$. Apart from the fact that this is a $1-\sigma$ level estimate, in , as later acknowledged in a number of papers , the use of the full covariance matrix of EGM96 has been questioned. Indeed, it has been noted that in the EGM96 solution the recovered even zonal harmonics are strongly reciprocally correlated; it seems, e.g., that the 13$\%$ value for the systematic error due to geopotential is due to a lucky correlation between $J_6$ and $J_8$ which are not cancelled by . The point is that, according to , nothing would assure that the covariance matrix of EGM96, which is based on a multi–year average that spans the 1970, 1980 and early 1990 decades, would reflect the true correlations between the even zonal harmonics during the particular time intervals of a few years adopted in the analyses by Ciufolini and coworkers. Then, a more conservative, although pessimistic, approach would be to consider the sum of the absolute values of the errors due to the single even zonal as representative of the systematic error induced by our uncertainty in the terrestrial gravitational field according to EGM96 . In this case we would get a conservative upper bound of 83$\%$ (1-$\sigma$). If a root-sum-square calculation is performed by neglecting the correlations between the even zonals a 45$\%$ 1-$\sigma$ error is obtained .
### The non-gravitational error
Another important class of systematic errors is given by the non–gravitational perturbations which affect especially the perigee of LAGEOS II. The main problem is that it turned out that their interaction with the structure of LAGEOS II changes in time due to unpredictable modifications in the physical properties of the LAGEOS II surface (orbital perturbations of radiative origin, e.g. the solar radiation pressure and the Earth albedo) and in the evolution of the spin dynamics of LAGEOS II (orbital perturbations of thermal origin induced by the interaction of the electromagnetic radiation of solar and terrestrial origin with the physical structure of the satellites, in particular with their corner–cube retroreflectors). Moreover, such tiny but insidious effects were not entirely modelled in the GEODYN II software at the time of the analysis of [@science98; @ciucaz04], so that it is not easy to correctly and reliably assess their impact on the total error budget of the measurement performed during that particular time span. According to the evaluations in [@luc02], the systematic error due to the non–gravitational perturbations over a time span of 7 years amounts to almost 28$\%$. However, according to [@ries], their impact on the measurement of the Lense–Thirring effect with the nodes of LAGEOS and LAGEOS II and the perigee of LAGEOS II is, in general, quite difficult to be reliably assessed. So, by adding quadratically the gravitational and non–gravitational errors of [@luc02] we obtain for the systematic uncertainty $\delta\mu_{\rm LT}^{\rm systematic}\sim
54\%$ if we assume a 45$\%$ error due to geopotential. The sum of the absolute values of the errors due to gepotential added quadratically with the non–gravitational perturbations would yield a total systematic error of $\delta\mu_{\rm LT}^{\rm
systematic}\sim$ 88$\%$. It must be noted that the latter estimate is rather similar to those released in [@ries]. Note also that they are 1-$\sigma$ evaluations. Moreover, it should be considered that the perigee of LAGEOS II is also sensitive to the eclipses effect on certain non–gravitational perturbations. Such features are, generally, not accounted for in all such estimates. An attempt can be found in [@ves99] in which the impact of the eclipses on the effect of the direct solar radiation pressure on the LAGEOS–LAGEOS II Lense–Thirring measurement has been evaluated: it should amount to almost 10$\%$ over an observational time span of 4 years.
The node-node tests
-------------------
In this Section we will deal with the node-node combination of . Such observable only cancels out the gravitational bias of the first even zonal harmonic $J_2$, but has the great advantage of discarding the perigee of LAGEOS II and its insidious non-gravitational perturbations.
### The gravitational error
In the node-node combination of has been analyzed with the 2nd generation CHAMP-only EIGEN2 and the 1st generation GRACE-only GGM01S Earth gravity models over a time span of almost 10 years.
In the impact of the static part of the geopotential, according to the CHAMP-only EIGEN2 Earth gravity model, is evaluated as $18\%$ (1-$\sigma$ root-sum-square covariance calculation), $22\%$ (1-$\sigma$ root-sum-square calculation) and $37\%$ (1-$\sigma$ upper bound). Ciufolini reports 18$\%$ obtained in a root-sum-square fashion with the full covariance matrix of EIGEN2 for which the same remarks as for EGM96 holds. Moreover, he does not consider the fact that EIGEN2 is only based on six months of data and that the released sigmas of the even zonal harmonics of low degree, which are the most relevant in this kind of analyses with the LAGEOS satellites, are rather optimistic, as explicitly pointed out in and acknowledged in . In regard to the GGM01S model, the covariance matrix was not publicly released. Ciufolini correctly presents a 19$\%$ which is the 1$-\sigma$ upper bound obtained in . However, GGM01S is only based on 111 days of data.
In our opinion the author’s conclusion “We conclude, using the Earth gravity model EIGEN-2S, that the Lense-Thirring effect exists and its experimental value, $\mu=0.98\pm 0.18$, fully agrees with the prediction of general relativity" is optimisitc. Indeed, he claims that in his 18$\%$ total error budget all the error sources are included. Ciufolini neglects the impact of the time-dependent gravitational perturbations on . Indeed, they may turn out to be a serious limiting factor mainly due to the secular variations of the even zonal harmonics[^6] $\dot J_{\ell}$. Indeed, allows to cancel out $\dot J_2$, but is sensitive to $\dot J_4$, $\dot J_6$,..., as pointed out in . The uncertainties in the $\dot J_{\ell}$ are still quite large: see Table 1 of . From it the values of Table \[jdots\] can be inferred.
[llll]{}
& $\ell=2$ & $\ell=4$ & $\ell=6$\
$\dot J_{\ell}$ & -2.113 & -0.6992 & -0.3594\
$\sigma_{\dot J_{\ell}}$ & 0.0810 & 0.2029 & 0.1765\
On the other hand, their impact on the measurement grows linearly in time[^7]. Indeed, the mismodelled shift, in mas, of due to the secular variations of the uncancelled even zonal harmonics can be written as \_[=4]{}(\_[.]{}\^[LAGEOS]{}+k\_1 \_[.]{}\^[LAGEOS II ]{})T\^2\_[obs]{},where the coefficients $\dot\Omega_{.\ell}$ are $\partial \dot\Omega_{\rm class}/\partial
J_{\ell}$ and have explicitly been calculated up to degree $\ell=20$ in . It must be divided by the gravitomagnetic shift, in mas, of over the same observational time span (\_[LT]{}\^[LAGEOS]{}+k\_1 \_[LT]{}\^[LAGEOS II]{} ) T\_[obs]{}=48.2 [mas yr\^[-1]{}]{} T\_[obs]{}. By assuming, e.g., $\sigma_{\dot J_4}=0.6\times 10^{-11}$ yr$^{-1}$ and $\sigma_{\dot J_6}=0.5\times 10^{-11}$ yr$^{-1}$ [@cox], it turns out that the percent error on the combination grows linearly with $T_{\rm obs}$ and would amount to $1\%$ over one year at $1-\sigma$ level. This means that, over 10 years, their impact is $\sim 10\%$ (1-$\sigma$). In Section \[smerd\] we will quantitatively support this evaluation.
### The impact of the secular variations of the even zonal harmonics: a quantitative estimate
Here we describe a numerical experiment aimed at a quantitative evaluation of the impact of $\dot J_{\ell}$.
The first step consists in simulating the time series of $\delta\Omega^{\rm LAGEOS}+k_1\delta\Omega^{\rm LAGEOS II}$ in order to obtain the qualitative and quantitative features of Figure 4 of . It refers to EIGEN2 and shows the raw residual time series with a straight line which fits it. The post-fit residuals amounts to 12 mas. In our model, called Input Model (IM), we include
- LT$\equiv S_{\rm LT}t$ with $S_{\rm LT}=48.2$ mas yr$^{-1}$. Lense-Thirring trend as predicted by the General Theory of Relativity in order to simulate the fact that the residuals of LAGEOS and LAGEOS II have been built up by dealing with the gravitomagnetic force as a totally unmodelled feature
- ZONDOT$\equiv \sum_{\ell=4}^6\{r\}\left(\dot\Omega_{.\ell}^{\rm
LAGEOS}+c_1 \dot\Omega_{.\ell}^{\rm LAGEOS\ II} \right)\left(\rp{
\dot J_{\ell} }{2}\right)t^2$. Quadratic term due to the $\dot
J_{\ell}$ according to Table \[jdots\]. The numbers $\{r\}$ are randomly generated from a normal distribution with mean zero, variance one and standard deviation one. Note that EIGEN2 does not solve for $\dot J_{\ell}$. In there are no details about the values included in the dynamical force models of the orbital processor; thus we treat the secular rates of the even zonal harmonics as unmodelled features fully absorbed by the residuals.
- ZONALS$\equiv p\left(\rp{x}{100}\right)S_{\rm LT}t$. Linear trend with a slope of $x\%$ of the Lense-Thirring signal. For EIGEN2 $x=37$ (sum of the absolute values of the individual errors) is assumed. The number $p$ is randomly generated as for the $\{r\}$
- TIDE$\equiv\sum\{a_c\}\sigma_{A_c}\cos\left[\left(\rp{2\pi}{P}\right)t+\{f_c\}\right]+
\sum\{a_s\}\sigma_{A_s}\sin\left[\left(\rp{2\pi}{P}\right)t+\{f_s\}\right]$. Set of various tidal perturbations of known periods $P$. For the impact of such kind of perturbations on the orbits of the LAGEOS satellites see . The sets of numbers $\{a_c\},\{a_s\},\{f_c\},\{f_s\}$ are randomly generated as $p$ and the $\{r\}$
- NOISE. White gaussian noise with variable amplitude which simulates the observational errors of the laser-ranged measurement
The full IM used in our analysis is thus We include in our model the possibility of varying the length of the time series $T_{\rm obs
}$, the temporal step $\Delta t$ which simulates the orbital arc length, the amplitude of the noise and of the mismodelling in the perturbations and the initial phases of the sinusoidal terms in order to simulate different initial conditions and uncertainties in the dynamical force models of the orbital processors. More precisely, the magnitude of the mismodelling in the various effects is randomly varied within the currently accepted ranges (1-$\sigma$) by using random numbers generated from a normal distribution with mean zero, variance one and standard deviations one. The same also holds for ZONALS because it is impossible to know, a priori, the sign of the slope of the residual trend due to the even zonal harmonics. The so built IM represents the basis of our subsequent analyses.
With the so built IM we perform a set of 5000 runs by randomly varying the initial phases, the noise and the mismodelling amplitudes within the accepted intervals in order to simulate a wide range of initial conditions and measurement errors which can occur in the real world. The length of the time series is keep fixed at $T_{\rm obs}=9.5$ years. In every run we fit the IM with a straight line only (LF) and record the obtained slope $\mu$ with the related error $\delta\mu$. Then, we calculate the averages of $\mu$ and $\delta\mu$ and the standard deviation of the mean $\Sigma$. For EIGEN2, i.e. $x=37$, we obtain ||\~30%.Also the post-fit residuals are calculated. Figure \[simul\] shows the complete IM, its straight line fit compared with the nominal Lense-Thirring trend and the post-fit residuals for a given set of randomly chosen initial conditions. The RMS post-fit amounts to 12.7 mas.
![\[simul\] Simulated time series, straight line fit and post-fit residuals for $T_{\rm obs}=9.5$ years and $\Delta t=15$ days. The RMS of the post-fit residuals is 12.7 mas. The slope of the trend simulating the impact of the mismodelled even zonal harmonics has been fixed to 37$\%$ of the Lense-Thirring effect, according to EIGEN2. ](simul.eps){width="13cm" height="11cm"}
The averaged RMS post-fit is 11 mas. This shows that our procedure represents a realistic starting point for our analyses.
A first interesting result is that the departure of the measured slope $\mu$ from the nominal gravitomagnetic slope $\mu_{\rm LT}$, which is included in IM, amounts to $\sim 30\%$ on average, while for Ciufolini is $2\%$ only ($\mu=0.98$).
In order to evaluate the impact of the secular rates of the even zonal harmonics in every run we also fit the IM with a quadratic polynomial (QF) and compare the so obtained slope $\mu_{\rm QF}$ with the slope obtained in LF $\mu_{\rm LF}$. Note that procedure is analogous to that adopted in for the periodic perturbations. On average, the difference between the two slopes, i.e. the systematic error due to $\dot J_4,\dot J_6$, amounts to \~14%of the Lense-Thirring effect. This result holds for $1-\sigma$. It is important to note that Figure \[simul\] has been obtained by assuming that the combined residuals of the LAGEOS satellites absorb the quadratic signature of $\dot J_{\ell}$ according to Table \[jdots\] at 1-$\sigma$, i.e. IM also includes ZONDOT. Nonetheless, it is difficult to discern the parabolic signal which, instead, is present and does affect the recovery of the slope. This means that a simple visual inspection of the plots of the combination of cannot be considered conclusive about the effect of $\dot J_4,\dot J_6$.
Another important point is that the combination of cannot be used to reliably constrain the zonals’ rates by measuring a $\dot J_4^{\rm eff}$. Indeed, it turns out that the errors $\delta Q$ in the quadratic parameters of QF are always larger than the estimated values themselves $Q$ and their mean $<\delta Q/Q>$ over a given set of 5000 runs amounts to $\sim
260\%$ with a standard deviation of the mean of $68\%$. Moreover, these figures change for different sets of 5000 runs.
If we repeat the same numerical experiments for $T_{\rm obs}=9.5$ years with $x=18$ (1-$\sigma$ root-mean-square full covariance calculation) and $x=22$ (1-$\sigma$ root-mean-square variance calculation) the situation does not substantially change \~20%, \~14%.
Then, a more conservative 1-$\sigma$ estimate of the total systematic error of the measurement of the Lense-Thirring effect with the combination of and the EIGEN2 Earth gravity model is \_[LT]{}\^[total error]{}51%.
While the forthcoming solutions from CHAMP and, especially, GRACE will be able to improve the static part of the terrestrial gravitational potential, i.e. the $J_{\ell}$, it is not so for their secular rates $\dot J_{\ell}$. This fact sets for the systematic error of gravitational origin a sort of threshold below which it will not be possible to go unless much more accurate determinations of $\dot J_4,\dot J_6$ will be available.
Conclusions
===========
In this paper we have performed a detailed critical analysis of the reliability and robustness of the so far performed tests aimed to the detection of the Lense-Thirring effect in the gravitational field of the Earth with the existing or proposed LAGEOS satellites.
We can summarize our conclusions as follows
- In regard to the node-node-perigee LAGEOS-LAGEOS II combination, the claimed $20-25\%$ total accuracy obtained with the EGM96 Earth gravity model, still presented in , is not realistic because of the impact of the non-gravitational perturbations on the perigee of LAGEOS II and the mismodelling in the even zonal harmonics of the geopotential whose $1-\sigma$ upper bound is 83$\%$.
- In regard to the node-node LAGEOS-LAGEOS II combination of , extensive numerical tests have been performed in order to quantitatively assess the impact of the uncancelled secular variations of the even zonal harmonics on the proposed measurement of the Lense-Thirring effect. A simulated time series curve has been fitted with a straight line and a quadratic polynomial and the so obtained slopes have been compared. This procedure has been repeated over 5000 runs performed by randomly varying the initial phases, the noise and the mismodelling level within the currently accepted ranges of the simulated signal. It turns out that the bias due to $\dot J_4$ and $\dot J_6$ over 9.5 years amounts to $\sim 14\%$ on average. This yields an upper bound of the total systematic error of the performed tests with EIGEN2 of $\sim 51\%$.
- Alternative combinations involving the use of existing laser-ranged targets other than the LAGEOS satellites should be analyzed. The most promising combination is, in principle, the one that involves the nodes of LAGEOS, LAGEOS II, Ajisai and Jason-1 . It cancels out the first three even zonal harmonics $J_2, J_4, J_6$ along with their temporal variations at the price of introducing the relatively huge non-gravitational perturbations on Jason-1 which, however, should have a time-dependent periodic signature with short periodicities. According to the recently released combined CHAMP+GRACE+terrestrial gravimetry/altimetry EIGEN-CG01C Earth gravity model , the systematic error due to the remaining even zonal harmonics would amount to 0.7 $\%$ (root-sum-square calculation) and[^8] 1.6$\%$ (upper bound) at 1$-\sigma$ . The forthcoming more accurate and robust Earth gravity model solutions from GRACE should especially improve the higher degree even zonal harmonics, so that it might happen that the difference between the node-only LAGEOS-LAGEOS II and the node-only LAGEOS-LAGEOS II-Ajisai-Jason-1 combination will further enforce to the detriment of the former one, at least in regard to the gravitational error. The possibility of getting long time series of the Jason’s node should seriously be investigated with real data tests. Moreover, Jason-1 is also affected by the orbital maneuvers but they are mainly in-plane.
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[^1]: In their original paper Lense and Thirring use the longitude of pericentre $\varpi=\Omega+\omega$.
[^2]: Also the perturbations which grow quadratically in time are, of course, very dangerous. Those induced by the secular variations of the even zonal harmonics of the Earth’s geopotential fall in this category, as we will see in detail in Section \[zonrat\]. Time-dependent periodic perturbations with periods longer than the observational time span may also be insidious because they would resemble superimposed linear trends .
[^3]: Also the subtle non–gravitational Yarkovsky-Rubincam force, which is due to the interaction of the Earth’s electromagnetic IR radiation with the physical structure of the LAGEOS satellites, induces secular effects on their nodes and perigees .
[^4]: It can be expressed in terms of the PPN $\gamma$ parameter as $\mu_{\rm LT}=(1+\gamma)/2$.
[^5]: The possibility of using only the nodes of the LAGEOS satellites in view of the improvements in the Earth gravity models from GRACE has been propsed for the first time in , although without quantitative details. In it seems that Ciufolini refers to it as a proper own result with his reference \[6\] which includes of the present work and an announced paper; is not concerned with because it deals with and its analysis by means of EGM96. Moreover, Iorio retains the e-mails in which he passed to Ciufolini, with whom he was long in contact, the combination of along with the estimates of the systematic error obtained with EIGEN2 and is disposed to make them publicly available on request.
[^6]: The problem of the secular variations of the even zonal harmonics in post-Newtonian tests of gravity with LAGEOS satellites has been quantitatively addressed for the first time in . In regard to the measurement with , it has been, perhaps, misunderstood in .
[^7]: For a possible alternative combination which would cancel out the first three even zonal harmonics along with their temporal variations see .
[^8]: The 1-$\sigma$ errors for the node-node LAGEOS-LAGEOS II combination of are $5\%$ (root-sum-square calculation with the variance matrix) and $6\%$ (upper bound), according to EIGEN-CG01C, while they are $3\%$ and $4\%$, according to the 2nd generation GRACE-only EIGEN-GRACE02S model.
|
---
abstract: 'We analyze a random lozenge tiling model of a large regular hexagon, whose underlying weight structure is periodic of period $2$ in both the horizontal and vertical directions. This is a determinantal point process whose correlation kernel is expressed in terms of non-Hermitian matrix valued orthogonal polynomials. We obtain the limiting densities of the lozenges in the disordered flower-shaped region. The starting point of our analysis is a double contour formula (obtained by Duits and Kuijlaars) which involves the solution of a $4 \times 4$ Riemann-Hilbert problem. Our method generalizes the existing techniques to a model involving matrix valued orthogonal polynomials.'
author:
- Christophe Charlier
title: |
Doubly periodic lozenge tilings of a hexagon\
and matrix valued orthogonal polynomials
---
Introduction
============
A lozenge tiling of a hexagon is a collection of three different types of lozenges ($\tikz[scale=.25]{ \draw (0,0) {
--++(1,1)--++(0,1)--++(-1,-1)
--++(0,-1)
}; }$, $\tikz[scale=.25] { \draw (0,0) {--++(1,0)--++(0,1)--++(-1,0)--++(0,-1)
}; }$ and $\tikz[scale=.25] { \draw (0,0) {--++(1,1)--++(1,0)--++(-1,-1)--++(-1,0)
};}$) which cover this hexagon without overlaps, see Figure \[fig: non-intersecting paths\] (left). There are finitely many such tilings; hence by assigning to each tiling $\mathcal{T}$ a non-negative weight $\mathrm{W}(\mathcal{T})$, we define a probability measure on the tilings by $$\begin{aligned}
\label{prob over tilings}
\mathbb{P}(\mathcal{T}) = \frac{\mathrm{W}(\mathcal{T})}{\sum_{\mathcal{T}'}\mathrm{W}(\mathcal{T'})},\end{aligned}$$ where the sum is taken over all the tilings (and is assumed to be non-zero). Uniform random tilings of a hexagon (i.e. when $\mathrm{W}(\mathcal{T})=1$ for all $\mathcal{T}$) is a well-studied model. As the size of the hexagon tends to infinity (while the size of the lozenges is kept fixed), the local statistical properties of this model are described by universal processes [@Jptrf; @BKMM; @Gorin; @J17]. We also refer to [@CKP; @KO; @KOS] for important early results and to [@BG; @K] for general references on tiling models. Uniform lozenge tilings of more complicated domains (non-necessarily convex) have also been widely studied in recent years [@Petrov1; @Petrov2; @BuGo2; @AvMJ].
at (0,0) ; (0,0)–(2,0)–(4,2)–(4,4)–(2,4)–(0,2)–(0,0); (0,0.5)–(1,0.5)–(1,0)–(1.5,0.5)–(2.5,0.5); (0,1)–(0.5,1)–(1,1.5)–(1.5,1.5)–(1.5,1)–(2,1)–(3,2)–(3.5,2)–(4,2.5); (0,1)–(0.5,1.5)–(1,1.5)–(2.5,3)–(4,3); (0,1.5)–(0.5,2)–(1,2)–(2.5,3.5)–(4,3.5); (0.5,2)–(2,3.5)–(2.5,3.5); (2.5,3)–(2.5,4); (1.5,1.5)–(3,3)–(3,4); (1.5,0)–(3,1.5)–(3,2.5)–(3.5,3)–(3.5,4); (3.5,1.5)–(3.5,2.5)–(4,3); (0.5,0)–(0.5,1); (0.5,1.5)–(0.5,2.5); (1,1)–(1,2); (1,2.5)–(1,3); (1.5,0.5)–(1.5,1.5); (1.5,2)–(1.5,2.5); (1.5,3)–(1.5,3.5); (2,0.5)–(2,1); (2,1.5)–(2,2); (2,2.5)–(2,3); (2,3.5)–(2,4); (2.5,1)–(2.5,1.5); (2.5,2)–(2.5,2.5); (2,2.5)–(3.5,2.5); (0.5,0.5)–(1,1); (1,0.5)–(1.5,1)–(1,1); (1.5,1)–(2.5,2)–(3,2); (1,2.5)–(1.5,2.5); (1.5,3)–(2,3); (1.5,2)–(2,2); (2,1.5)–(2.5,2); (2,1.5)–(2.5,1.5); (2.5,1)–(3,1); (3,1.5)–(3.5,1.5);
at (0,0) ; (0,0)–(2,0)–(4,2)–(4,4)–(2,4)–(0,2)–(0,0); (0,0.5)–(1,0.5)–(1,0)–(1.5,0.5)–(2.5,0.5); (0,1)–(0.5,1)–(1,1.5)–(1.5,1.5)–(1.5,1)–(2,1)–(3,2)–(3.5,2)–(4,2.5); (0,1)–(0.5,1.5)–(1,1.5)–(2.5,3)–(4,3); (0,1.5)–(0.5,2)–(1,2)–(2.5,3.5)–(4,3.5); (0.5,2)–(2,3.5)–(2.5,3.5); (2.5,3)–(2.5,4); (1.5,1.5)–(3,3)–(3,4); (1.5,0)–(3,1.5)–(3,2.5)–(3.5,3)–(3.5,4); (3.5,1.5)–(3.5,2.5)–(4,3); (0.5,0)–(0.5,1); (0.5,1.5)–(0.5,2.5); (1,1)–(1,2); (1,2.5)–(1,3); (1.5,0.5)–(1.5,1.5); (1.5,2)–(1.5,2.5); (1.5,3)–(1.5,3.5); (2,0.5)–(2,1); (2,1.5)–(2,2); (2,2.5)–(2,3); (2,3.5)–(2,4); (2.5,1)–(2.5,1.5); (2.5,2)–(2.5,2.5); (2,2.5)–(3.5,2.5); (0.5,0.5)–(1,1); (1,0.5)–(1.5,1)–(1,1); (1.5,1)–(2.5,2)–(3,2); (1,2.5)–(1.5,2.5); (1.5,3)–(2,3); (1.5,2)–(2,2); (2,1.5)–(2.5,2); (2,1.5)–(2.5,1.5); (2.5,1)–(3,1); (3,1.5)–(3.5,1.5);
(0,0.25) circle (0.7mm); (0,0.75) circle (0.7mm); (0,1.25) circle (0.7mm); (0,1.75) circle (0.7mm);
(0.5,0.25) circle (0.7mm); (0.5,0.75) circle (0.7mm); (0.5,1.75) circle (0.7mm); (0.5,2.25) circle (0.7mm);
(1,0.25) circle (0.7mm); (1,1.25) circle (0.7mm); (1,1.75) circle (0.7mm); (1,2.75) circle (0.7mm);
(1.5,0.75) circle (0.7mm); (1.5,1.25) circle (0.7mm); (1.5,2.25) circle (0.7mm); (1.5,3.25) circle (0.7mm);
(2,0.75) circle (0.7mm); (2,1.75) circle (0.7mm); (2,2.75) circle (0.7mm); (2,3.75) circle (0.7mm);
(2.5,1.25) circle (0.7mm); (2.5,2.25) circle (0.7mm); (2.5,3.25) circle (0.7mm); (2.5,3.75) circle (0.7mm);
(3,1.75) circle (0.7mm); (3,2.25) circle (0.7mm); (3,3.25) circle (0.7mm); (3,3.75) circle (0.7mm);
(3.5,1.75) circle (0.7mm); (3.5,2.25) circle (0.7mm); (3.5,3.25) circle (0.7mm); (3.5,3.75) circle (0.7mm);
(4,2.25) circle (0.7mm); (4,2.75) circle (0.7mm); (4,3.25) circle (0.7mm); (4,3.75) circle (0.7mm);
(0,0.25)–(1,0.25)–(1.5,0.75)–(2,0.75)–(3,1.75)–(3.5,1.75)–(4,2.25); (0,0.75)–(0.5,0.75)–(1,1.25)–(1.5,1.25)–(2.5,2.25)–(3.5,2.25)–(4,2.75); (0,1.25)–(0.5,1.75)–(1,1.75)–(2.5,3.25)–(4,3.25); (0,1.75)–(2,3.75)–(4,3.75);
In this work, we consider the regular hexagon of (large) size $n$ $$\begin{aligned}
\label{def of Hn}
\mathcal{H}_{n} := \{(x,y)\in \mathbb{R}^{2}: 0 \leq x \leq 2n, 0 \leq y \leq 2n, -n \leq x-y \leq n\}, \qquad n \in \mathbb{N}_{\geq 1},\end{aligned}$$ but we deviate from the uniform measure and study instead measures with periodic weightings. To explain what this means, we first briefly recall a well-known one-to-one correspondence between tilings of a hexagon and certain non-intersecting paths. This bijection can be written down explicitly, but is best understood informally. The paths are obtained by drawing lines on top of two types of lozenges $$\begin{aligned}
\label{non-intersecting paths}
\tikz[scale=.5]{ \draw (0,0) {
--++(1,1)--++(0,1)--++(-1,-1)
--++(0,-1)
}; \draw[very thick] (0,.5)--(1,1.5); \filldraw (0,0.5) circle(3pt); \filldraw (1,1.5) circle(3pt); } \quad \tikz[scale=.5] { \draw (0,0) {--++(1,0)--++(0,1)--++(-1,0)--++(0,-1)
}; \draw[very thick] (0,.5)--(1,.5); \filldraw (0,0.5) circle(3pt); \filldraw (1,0.5) circle(3pt); }\quad \mbox{and} \quad \tikz[scale=.5] { \draw (0,0) {--++(1,1)--++(1,0)--++(-1,-1)--++(-1,0)
};},\end{aligned}$$ as shown in Figure \[fig: non-intersecting paths\] (right). The paths associated to the tilings of $\mathcal{H}_{n}$ lie on a graph $\mathcal{G}_{n}$ which depends only on the size of the hexagon, see Figure \[fig: lattice\] (left). To each edge $\mathfrak{e}$ of $\mathcal{G}_{n}$, we assign a non-negative weight $w_{\mathfrak{e}}$. The weight of a path $\mathfrak{p}$ is then defined as $w_{\mathfrak{p}}=\prod_{\mathfrak{e} \in \mathfrak{p}} w_{\mathfrak{e}}$, and the weight of a tiling $\mathcal{T}$ as $\mathrm{W}(\mathcal{T})=\prod_{\mathfrak{p} \in \mathcal{T}} w_{\mathfrak{p}}$. Provided that at least one tiling has a positive weight, this defines a probability measure on the set of tiling by . If each edge is assigned the same weight, then we recover the uniform measure over the tiling. We say that a lozenge tiling model has $p \times q$ periodic weightings if the weight structure on the edges is periodic of period $p$ in the vertical direction, and periodic of period $q$ in the horizontal direction, see Figure \[fig: lattice\] (right) for an illustration with $p=2$ and $q=3$. Thus a $p \times q$ periodic weighting is completely determined by $2pq$ edge weights. Note that all paths share the same number of horizontal edges, and also the same number of oblique edges; hence lozenge tiling models with $1 \times 1$ periodic weightings are all equivalent to the uniform measure.
at (0,0) ; (0,0) circle (0.4mm); (0.5,0) circle (0.4mm); (1,0) circle (0.4mm); (1.5,0) circle (0.4mm); (2,0) circle (0.4mm);
(0,0.5) circle (0.4mm); (0.5,0.5) circle (0.4mm); (1,0.5) circle (0.4mm); (1.5,0.5) circle (0.4mm); (2,0.5) circle (0.4mm); (2.5,0.5) circle (0.4mm);
(0,1) circle (0.4mm); (0.5,1) circle (0.4mm); (1,1) circle (0.4mm); (1.5,1) circle (0.4mm); (2,1) circle (0.4mm); (2.5,1) circle (0.4mm); (3,1) circle (0.4mm);
(0,1.5) circle (0.4mm); (0.5,1.5) circle (0.4mm); (1,1.5) circle (0.4mm); (1.5,1.5) circle (0.4mm); (2,1.5) circle (0.4mm); (2.5,1.5) circle (0.4mm); (3,1.5) circle (0.4mm); (3.5,1.5) circle (0.4mm);
(0.5,2) circle (0.4mm); (1,2) circle (0.4mm); (1.5,2) circle (0.4mm); (2,2) circle (0.4mm); (2.5,2) circle (0.4mm); (3,2) circle (0.4mm); (3.5,2) circle (0.4mm); (4,2) circle (0.4mm);
(1,2.5) circle (0.4mm); (1.5,2.5) circle (0.4mm); (2,2.5) circle (0.4mm); (2.5,2.5) circle (0.4mm); (3,2.5) circle (0.4mm); (3.5,2.5) circle (0.4mm); (4,2.5) circle (0.4mm);
(1.5,3) circle (0.4mm); (2,3) circle (0.4mm); (2.5,3) circle (0.4mm); (3,3) circle (0.4mm); (3.5,3) circle (0.4mm); (4,3) circle (0.4mm);
(2,3.5) circle (0.4mm); (2.5,3.5) circle (0.4mm); (3,3.5) circle (0.4mm); (3.5,3.5) circle (0.4mm); (4,3.5) circle (0.4mm);
(0,0)–(0.5,0); (0.5,0)–(1,0); (1,0)–(1.5,0); (1.5,0)–(2,0);
(0,0.5)–(0.5,0.5); (0.5,0.5)–(1,0.5); (1,0.5)–(1.5,0.5); (1.5,0.5)–(2,0.5); (2,0.5)–(2.5,0.5);
(0,1)–(0.5,1); (0.5,1)–(1,1); (1,1)–(1.5,1); (1.5,1)–(2,1); (2,1)–(2.5,1); (2.5,1)–(3,1);
(0,1.5)–(0.5,1.5); (0.5,1.5)–(1,1.5); (1,1.5)–(1.5,1.5); (1.5,1.5)–(2,1.5); (2,1.5)–(2.5,1.5); (2.5,1.5)–(3,1.5); (3,1.5)–(3.5,1.5);
(0.5,2)–(1,2); (1,2)–(1.5,2); (1.5,2)–(2,2); (2,2)–(2.5,2); (2.5,2)–(3,2); (3,2)–(3.5,2); (3.5,2)–(4,2);
(1,2.5)–(1.5,2.5); (1.5,2.5)–(2,2.5); (2,2.5)–(2.5,2.5); (2.5,2.5)–(3,2.5); (3,2.5)–(3.5,2.5); (3.5,2.5)–(4,2.5);
(1.5,3)–(2,3); (2,3)–(2.5,3); (2.5,3)–(3,3); (3,3)–(3.5,3); (3.5,3)–(4,3);
(2,3.5)–(2.5,3.5); (2.5,3.5)–(3,3.5); (3,3.5)–(3.5,3.5); (3.5,3.5)–(4,3.5);
(0,0)–(0.5,0.5); (0.5,0)–(1,0.5); (1,0)–(1.5,0.5); (1.5,0)–(2,0.5); (2,0)–(2.5,0.5);
(0,0.5)–(0.5,1); (0.5,0.5)–(1,1); (1,0.5)–(1.5,1); (1.5,0.5)–(2,1); (2,0.5)–(2.5,1); (2.5,0.5)–(3,1);
(0,1)–(0.5,1.5); (0.5,1)–(1,1.5); (1,1)–(1.5,1.5); (1.5,1)–(2,1.5); (2,1)–(2.5,1.5); (2.5,1)–(3,1.5); (3,1)–(3.5,1.5);
(0,1.5)–(0.5,2); (0.5,1.5)–(1,2); (1,1.5)–(1.5,2); (1.5,1.5)–(2,2); (2,1.5)–(2.5,2); (2.5,1.5)–(3,2); (3,1.5)–(3.5,2); (3.5,1.5)–(4,2);
(0.5,2)–(1,2.5); (1,2)–(1.5,2.5); (1.5,2)–(2,2.5); (2,2)–(2.5,2.5); (2.5,2)–(3,2.5); (3,2)–(3.5,2.5); (3.5,2)–(4,2.5);
(1,2.5)–(1.5,3); (1.5,2.5)–(2,3); (2,2.5)–(2.5,3); (2.5,2.5)–(3,3); (3,2.5)–(3.5,3); (3.5,2.5)–(4,3);
(1.5,3)–(2,3.5); (2,3)–(2.5,3.5); (2.5,3)–(3,3.5); (3,3)–(3.5,3.5); (3.5,3)–(4,3.5);
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By putting points on the paths as shown in , each tiling of the hexagon gives rise to a point configuration, see also Figure \[fig: non-intersecting paths\] (right). Thus the probability measure on tilings can be viewed as a discrete point process [@Borodin; @Soshnikov]. For lozenge tiling models with $p \times q$ periodic weightings, it follows from the Lindström-Gessel-Viennot theorem [@GV; @L] combined with the Eynard-Mehta theorem [@EM] that this point process is determinantal. Therefore, to understand the fine asymptotic structure (as $n \to + \infty$) it suffices to analyze the asymptotic behavior of the correlation kernel. However, until recently [@DK; @CDKL], the existing techniques were not appropriate for such analyses.
The main result of [@DK] is a double contour formula for the correlation kernels of various tiling models with periodic weightings (including lozenge tiling models of a hexagon as considered here). In this formula, the integrand is expressed in terms of the solution (denoted $Y$) to a $2p \times 2p$ Riemann-Hilbert (RH) problem. This RH problem is related to certain orthogonal polynomials (OPs), which are non-standard in two aspects:
- the OPs and the weight are $p \times p$ matrix valued,
- the orthogonality conditions are non-hermitian.
The size of the RH problem, the size of the weight, and the size of the OPs all depend on $p$, but quite interestingly not on $q$.
Lozenge tiling models of the hexagon with $p \times q$ periodic weightings are rather unexplored up to now. To the best of our knowledge, the model considered in [@CDKL] is the only one (other than the uniform measure) prior to the present work for which results on fine asymptotics exist. The model considered in [@CDKL] is $1 \times 2$ periodic and uses the formula of [@DK] as the starting point of the analysis. The techniques of [@CDKL] combine the Deift/Zhou steepest descent method [@DZ] of $Y$ (of size $2 \times 2$) with a non-standard saddle point analysis of the double contour integral. However, since $p=1$, the associated OPs are scalar (this fact was extensively used in the proof) and it is not clear how to generalize these techniques to the case $p \geq 2$.
The aim of this paper is precisely to make progress in this direction by studying a lozenge tiling model with $2 \times 2$ periodic weightings. Our model presents one simply connected liquid region (which has the shape of a flower with $6$ petals), $6$ frozen regions, and $6$ staircase regions (also called semi-frozen regions). The starting point of our analysis is the double contour formula from [@DK] which expresses the kernel in terms of $2 \times 2$ matrix valued OPs. Our first main result is a new expression for the kernel in terms of scalar OPs (which are orthogonal with respect to another, scalar, weight). This formula allows for a much simpler analysis than the original formula from [@DK]. Our second main result concerns the limiting densities of the different lozenges in the liquid region. The model and the results are presented in more detail in Sections \[Section: model\] and \[Section: main results\].
#### An expression for the kernel in terms of scalar OPs. {#an-expression-for-the-kernel-in-terms-of-scalar-ops. .unnumbered}
The eigenvalues and eigenvectors of the $2 \times 2$ orthogonality weight play an important role in the first step of the analysis. They are naturally defined on a $2$-sheeted Riemann surface $\mathcal{M}$, which turns out to be of genus $0$. This fact is crucial to obtain the new formula for the kernel in terms of scalar OPs. We expect that ideas similar to the ones presented here can be applied to other tiling models with periodic weightings, as long as the corresponding Riemann surface $\mathcal{M}$ is of genus $0$.
Lozenge tiling models of large hexagons with periodic weightings can feature all of the three possible types of phases known in random tiling models: the solid, liquid and gas phases. A solid region (also called frozen region) is filled with one type of lozenges. In the liquid and gas phases, all three types of lozenges coexist. The difference between these two phases is reflected in the correlations between two lozenges: in the liquid region, the correlation decay is polynomial with the distance between the lozenges, while in the gas region the decay is exponential. It is known that there is no gas phase for the uniform measure (corresponding to $p=q=1$). In fact, it turns out that the smallest periods that lead to the presence of a gas phase are either $p=2,q=3$ or $p=3, q=2$. For models that present gas phases, we expect $\mathcal{M}$ to have genus at least $1$, and then new techniques are required. This is left for future works.
#### Related works. {#related-works. .unnumbered}
Random lozenge tilings of the regular hexagon is a particular example of a tiling model. We briefly review here other tiling models with periodic weightings that have been studied in the literature and for which more results are known. We also discuss the related techniques and explain why they cannot be applied in our case.
The Aztec diamond is a well-studied tiling model [@JPS; @CEP; @CKP; @J05]. It consists of covering the region $\{(x,y):|x|+|y| \leq n+1\}$ with $2 \times 1$ or $1 \times 2$ rectangles (called dominos), where $n > 0$ is an integer which parametrizes the size of the covered region. Uniform domino tilings of the Aztec diamond features $4$ solid regions and one liquid region. The associated discrete point process is determinantal, and turns out to belong to the class of Schur processes (introduced in [@OR1]), for which there exists a double contour integral for the kernel that is suitable for an asymptotic analysis as $n \to + \infty$. Another important Schur process is the infinite hexagon with $1 \times k$ periodic weightings. The infinite hexagon is a non-regular hexagon whose vertical side is first sent to infinity either from above or from below, see e.g. [@BeD Figure 14] for an illustration. For more examples of other interesting tiling models that fall in the Schur process class, see e.g. [@BG]. Uniform lozenge tilings of the finite hexagon (such as $\mathcal{H}_{n}$) do not belong to the Schur class, but have been studied using other techniques based on some connections with Hahn polynomials [@Jptrf]: the limiting kernel in the bulk scaling regime has been established in [@BKMM] using a discrete RH problem, and in [@Gorin] using the approach developed in [@BO]. The doubly periodic Aztec diamond exhibits all three phases. It still defines a determinantal point process, but it falls outside of the Schur process class. However, Chhita and Young found in [@CY2014] a formula for the correlation kernel by performing an explicit inversion of the Kasteleyn matrix. This formula was further simplified in [@CJ] and then used in [@CJ; @BCJ] to obtain fine asymptotic results on the fluctuations of the liquid-gas boundary as $n \to + \infty$. This same model was analyzed soon afterward in [@DK] via a different (and more general) method based on matrix valued orthogonal polynomials and a related RH problem. For the doubly periodic Aztec diamond, this RH problem is surprisingly simple in the sense that it can be solved explicitly for finite $n$. The analyses of [@CY2014; @CJ; @DK] rely on the rather special integrable structure of the doubly periodic Aztec diamond. However, the approach of [@DK] applies to a much wider range of tiling models. Berggren and Duits [@BeD] have recently identified a whole class of tiling/path models for which it is possible to simplify significantly the formula of [@DK]. Quite remarkably, their final expression for the kernel does not involve any RH problem or OPs, which simplifies considerably the saddle point analysis. Using the results from [@BeD], Berggren in [@Berggren] recently studied the $2 \times k$ periodic Aztec diamond, for an arbitrary $k$. The class of models for which the formula from [@BeD] applies roughly consists of the models with an infinite number of paths whose (possibly matrix valued) orthogonality weight has a Wiener-Hopf type factorization. This class of models contains the Schur class, but also (among others) the doubly periodic Aztec diamond and doubly periodic lozenge tilings of an infinite hexagon.
However, lozenge tiling models of the finite hexagon cannot be represented as models with infinitely many paths (as opposed to the Aztec diamond and the infinite hexagon). In particular, they do not belong to the class of models studied in [@BeD] and thus the simplified formula from [@BeD] cannot be used. This fact makes lozenge tiling models of the finite hexagon much harder to analyze asymptotically (see also the comment in [@BeD beginning of Section 6]).
#### The figures.
In addition to being in bijection with non-intersecting paths, lozenge tilings of the hexagon are also in bijection with *dimer coverings*, which are perfect matchings of a certain bipartite graph. We refer to [@J17] for more details on the correspondence with dimers (see also [@Petrov1 Figure 1] for an illustration). The bijection with dimers is not used explicitly in this paper, but we do use it to generate the pictures via the shuffling algorithm [@ProppShuffling].
#### Acknowledgment.
The content of Theorem \[thm: correlation kernel final scalar expression\] and Section \[section: reducing the size\] is based on an unpublished idea of Arno Kuijlaars. I am very grateful to him for allowing me and even encouraging me to use his idea. Thanks to him, the length and technicalities of the paper have been considerably reduced (compared to an earlier draft), and the results are stronger. I also thank Arno Kuijlaars, Maurice Duits and Jonatan Lenells for interesting discussions, and for a careful reading of the introduction. This work is supported by the European Research Council, Grant Agreement No. 682537.
Model and background {#Section: model}
====================
In this section, we present a lozenge tiling model with $2 \times 2$ periodic weightings. We also introduce the necessary material to invoke the double contour formula from [@DK] for the kernel. In particular, we present the relevant $2 \times 2$ matrix valued OPs and the associated $4 \times 4$ RH problem.
### Affine transformation for certain figures of lozenge tilings {#affine-transformation-for-certain-figures-of-lozenge-tilings .unnumbered}
For the presentation of the model and the results, it is convenient to define the hexagon and the lozenges as in –. However, for the purpose of presenting certain figures of lozenge tilings, it is more pleasant to modify the hexagon and the lozenges by the following simple transformation: $$\begin{aligned}
\label{affine transformation on lozenges}
\tikz[scale=.5,baseline={([yshift=-0.5ex]current bounding box.center)}]{ \draw (0,0)--(0,1)--(1,2)--(1,1)--(0,0); \node at (1.6,0.9) {$\to$}; \draw[cm={1,0,0,1,(2.2,0)}] (0,0)--(0,1)--(0.866025,1.5)--(0.866025,0.5)--(0,0); }, \quad \tikz[scale=.5,baseline={([yshift=-0.5ex]current bounding box.center)}] { \draw (0,0)--(0,1)--(1,1)--(1,0)--(0,0); \node at (1.6,0.4) {$\to$}; \draw[cm={1,0,0,1,(2.2,0)}] (0,0)--(0,1)--(0.866025,0.5)--(0.866025,-0.5)--(0,0);} \quad \mbox{ and } \quad \tikz[scale=.5,baseline={([yshift=-0.5ex]current bounding box.center)}] { \draw (0,0)--(1,1)--(2,1)--(1,0)--(0,0); \node at (2.4,0.0) {$\to$}; \draw[cm={1,0,0,1,(3,0)}] (0,0)--(0.866025,0.5)--(1.73205,0)--(0.866025,-0.5)--(0,0);},\end{aligned}$$ so that $\mathcal{H}_{n}$ is mapped by this transformation to a hexagon whose $6$ sides are of equal length. Above the definition of $\mathcal{H}_{n}$, we used the standard terminology and called $\mathcal{H}_{n}$ “the regular hexagon"; note however that $\mathcal{H}_{n}$ becomes truly regular only after applying the transformation . In the figures, we will assign the colors red, green and yellow for the three lozenges in , from left to right, respectively.
Definition of the model
-----------------------
The regular hexagon $\mathcal{H}_{n}$ has corners located at $(0,0)$, $(0,n)$, $(n,2n)$, $(2n,2n)$, $(2n,n)$ and $(n,0)$. We normalize the lozenges such that they cover each a surface of area $1$, and the vertices of the lozenges have integer coordinates. We recall that each lozenge tiling of $\mathcal{H}_{n}$ gives rise, through , to a system of $n$ non-intersecting paths. These paths live on the graph $\mathcal{G}_{n}$, which is illustrated in Figure \[fig: lattice\] (left) for $n=4$. The vertices of $\mathcal{G}_{n}$ form a subset of $\mathbb{Z}\times (\mathbb{Z} + \frac{1}{2})$, and the bottom left vertex has coordinates $(0,\frac{1}{2})$. We denote the paths by $$\begin{aligned}
\label{def of the paths}
\mathfrak{p}_{j} : \{0,1,\ldots,2n\} \to \frac{1}{2}+\mathbb{Z}, \qquad j = 0,\ldots,n-1,\end{aligned}$$ and they satisfy the initial positions $\mathfrak{p}_{j}(0) = j + \frac{1}{2}$ and ending positions $\mathfrak{p}_{j}(2n) = n + j + \frac{1}{2}$. The lozenge tiling model we consider has $2 \times 2$ periodic weightings and depends on a parameter $\alpha \in (0,1]$. The weightings are defined on the $2 \times 2$ bottom left block of the lattice as shown in Figure \[fig: 2x2 periodic weightings\] (left), and is then extended by periodicity as shown in Figure \[fig: 2x2 periodic weightings\] (right). More formally, if $\mathfrak{e} = \big( (x_{1},y_{1}+\frac{1}{2}),(x_{2},y_{2}+\frac{1}{2}) \big)$ is an edge of $\mathcal{G}_{n}$, then
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$$\begin{aligned}
\label{def weightings}
w_{\mathfrak{e}} =\begin{cases}
\alpha^{2}, & \mbox{if } x_{1} \mbox{ is odd, } y_{1}=y_{2}, \mbox{ and } y_{1} \mbox{ is even,} \\
\alpha, & \mbox{if } x_{1}+y_{1} \mbox{ is odd, and } y_{2} = y_{1}+1, \\
1 & \mbox{otherwise}.
\end{cases}\end{aligned}$$
For any values of $\alpha \in (0,1]$, the weightings are such that $\mathrm{W}(\mathcal{T)} > 0$ for all $\mathcal{T}$, and thus we have a well-defined probability measure via . On the other hand, if $\alpha = 0$, then several edges have weights $0$, and it is easy to see (e.g. from Figure \[fig: 2x2 periodic weightings\] (right)) that $\mathrm{W}(\mathcal{T}) = 0$ for all $\mathcal{T}$. So in this case, does not induce a probability measure, and this explains why we excluded $\alpha = 0$ in the definition of the model. If $\alpha = 1$, all tilings have the same weight, and we recover the uniform distribution. Proposition \[prop: frozen\] states that for $\alpha < 1$, there is a particular tiling $\mathcal{T}_{\max}$ that is more likely to appear than any other tiling. $\mathcal{T}_{\max}$ is illustrated in Figure \[fig: alpha 0 vs alpha 1\] (left) for $n=60$.
\[prop: frozen\] Let $\alpha \in (0,1)$ and let $n \geq 1$ be an integer. There exists a unique tiling $\mathcal{T}_{\max}$ of $\mathcal{H}_{n}$ such that $\mathrm{W}(\mathcal{T}) \leq \alpha \mathrm{W}(\mathcal{T}_{\max})$ for all $\mathcal{T} \neq \mathcal{T}_{\max}$. Furthermore, $$\begin{aligned}
\mathrm{W}(\mathcal{T}_{\max}) = \begin{cases}
\alpha^{\frac{n^{2}}{4}}, & \mbox{if } n \mbox{ is even}, \\
\alpha^{\frac{n^{2}-1}{4}}, & \mbox{if } n \mbox{ is odd.}
\end{cases}\end{aligned}$$
See Subsection \[section: model is frozen as alpha to 0\].
![\[fig: alpha 0 vs alpha 1\]Two tilings taken at random accordingly to the measure induced by , for $n = 60$ and $\alpha = 5\times 10^{-4}$ (left), and $n = 100$ and $\alpha = 1$ (right).](a00005size60 "fig:"){width="5cm"} ![\[fig: alpha 0 vs alpha 1\]Two tilings taken at random accordingly to the measure induced by , for $n = 60$ and $\alpha = 5\times 10^{-4}$ (left), and $n = 100$ and $\alpha = 1$ (right).](7a1 "fig:"){width="5cm"}
It follows from Proposition \[prop: frozen\] that, as $\alpha \to 0$, the randomness disappears because the tiling $\mathcal{T}_{\max}$ becomes significantly more likely than any other tiling. Therefore, our model interpolates between the uniform measure over the tilings (for $\alpha = 1$) and a particular totally frozen tiling $\mathcal{T}_{\max}$ (as $\alpha \to 0$), see Figures \[fig: alpha 0 vs alpha 1\] and \[fig: alpha between 0 and 1\]. Intriguingly, these figures show similarities with the rectangle-triangle tiling of the hexagon obtained by Keating and Sridhar in [@KeaSri Figure 18].
![\[fig: alpha between 0 and 1\]Three tilings taken at random accordingly to the measure induced by with $n=100$ and $\alpha = 0.01$ (left), $\alpha = 0.05$ (middle), $\alpha = 0.2$ (right).](2a001 "fig:"){width="4.5cm"} ![\[fig: alpha between 0 and 1\]Three tilings taken at random accordingly to the measure induced by with $n=100$ and $\alpha = 0.01$ (left), $\alpha = 0.05$ (middle), $\alpha = 0.2$ (right).](3a005 "fig:"){width="4.5cm"} ![\[fig: alpha between 0 and 1\]Three tilings taken at random accordingly to the measure induced by with $n=100$ and $\alpha = 0.01$ (left), $\alpha = 0.05$ (middle), $\alpha = 0.2$ (right).](5a02 "fig:"){width="4.5cm"}
Several tiling models in the literature (e.g. those considered in [@BGR] and [@CDKL]) are defined by weightings on the lozenges, instead of weightings on the edges. To ease possible comparisons with these models, we give an alternative definition of our model. The weight $\mathrm{W}(\mathcal{T})$ of a tiling $\mathcal{T}$ can alternatively be defined as $$\begin{aligned}
\mathrm{W}(\mathcal{T}) = \prod_{{\tikz[scale=.18] \draw (0,0) {--++(1,0)--++(0,1)--++(-1,0)--++(0,-1)
};} \; \in \mathcal{T}} w(\tikz[scale=.27] { \draw (0,0) {--++(1,0)--++(0,1)--++(-1,0)--++(0,-1)
};}) \prod_{\tikz[scale=.18] {\draw (0,0) {
--++(1,1)--++(0,1)--++(-1,-1)
--++(0,-1)
};} \; \raisebox{0.11cm}{\scriptsize $\in \hspace{-0.08cm}\mathcal{T}$}} w\Big(\raisebox{-0.15cm}{\tikz[scale=.27] \draw (0,0) {
--++(1,1)--++(0,1)--++(-1,-1)
--++(0,-1)
}; } \hspace{-0.05cm}\Big),\end{aligned}$$ where $w$ is the weight function over the lozenges given by $$\begin{aligned}
& w\bigg( \tikz[scale=.3,baseline=(current bounding box.center)]{\draw (0,0) {--++(1,0)--++(0,1)--++(-1,0)--++(0,-1)
}; \draw[fill] (0,0) circle(5pt); \node[below] (i) at (0,-.2) {\tiny{$(i,j)$}};} \bigg)=
\begin{cases}
\alpha^{2}, & \mbox{if }i \text{ is odd and }j \mbox{ is even,}\\
1, & \text{otherwise},
\end{cases} \label{weight horizontal}
\\
& w\bigg( \hspace{-0.15cm}\raisebox{-0.5cm}{
\tikz[scale=.27] {
\draw (0,0) {
--++(1,1)--++(0,1)--++(-1,-1)
--++(0,-1)
};
\draw[fill] (0,0) circle(5pt);
\node[below] (i) at (0,-.2) {\tiny{$(i,j)$}};
}
} \hspace{-0.1cm} \bigg)=
\begin{cases}
\alpha, & \mbox{if }i+j \text{ is odd,}\\
1, & \text{otherwise},
\end{cases} \label{weight oblique}\end{aligned}$$ where $\alpha \in (0,1]$. The above lozenge weightings depend only on the parity of $i$ and $j$, and thus are periodic of period $2$ in both directions. By using the correspondence between lozenge tilings and non-intersecting paths, it is straightforward to verify that the weightings – define the same measure as the weightings .
Matrix valued orthogonal polynomials {#subsection: MVOPs}
------------------------------------
It will be convenient for us to define $\mathcal{G}_{\infty}$ as the graph whose vertex set is $\mathbb{Z}\times (\mathbb{Z} + \frac{1}{2})$, and whose edges are of the form $\mathfrak{e}= \big( (x_{1},y_{1}+\frac{1}{2}),(x_{2},y_{2}+\frac{1}{2}) \big)$ with $x_{2} = x_{1}+1$ and $y_{2}-y_{1} \in \{0,1\}$. The weighting was defined on the edges of $\mathcal{G}_{n}$, but it can be straightforwardly extended to the edges of $\mathcal{G}_{\infty}$. We follow the notations of [@DK equation (4.3)], and denote $T_{x,x+1}(y_{1},y_{2})$ for the weight associated to the edge $\mathfrak{e} = \big( (x_{1},y_{1}+\frac{1}{2}),(x_{2},y_{2}+\frac{1}{2}) \big)$ of $\mathcal{G}_{\infty}$. This weight can be obtained from and only depends on the parity of $x$. If $x$ is even, it is given by $$\begin{aligned}
\label{Transition x even}
T_{x,x+1}(y_{1},y_{2}) = \begin{cases}
1 & \mbox{if }y_{2}=y_{1}, \\
1 & \mbox{if }y_{2} = y_{1}+1 \mbox{ and } y_{1} \mbox{ is even}, \\
\alpha & \mbox{if } y_{2} = y_{1}+1 \mbox{ and } y_{1} \mbox{ is odd}, \\
0 & \mbox{otherwise},
\end{cases}\end{aligned}$$ while if $x$ is odd, we have $$\begin{aligned}
\label{Transition x odd}
T_{x,x+1}(y_{1},y_{2}) = \begin{cases}
\alpha^{2} & \mbox{if }y_{2}=y_{1}, \mbox{ and } y_{1} \mbox{ is even}, \\
1 & \mbox{if }y_{2}=y_{1}, \mbox{ and } y_{1} \mbox{ is odd}, \\
\alpha & \mbox{if }y_{2} = y_{1}+1 \mbox{ and } y_{1} \mbox{ is even}, \\
1 & \mbox{if } y_{2} = y_{1}+1 \mbox{ and } y_{1} \mbox{ is odd}, \\
0 & \mbox{otherwise}.
\end{cases}\end{aligned}$$ For each $x \in \mathbb{Z}$, $T_{x,x+1}$ is periodic of period $2$, namely $T_{x,x+1}(y_{1}+2,y_{2}+2) = T_{x,x+1}(y_{1},y_{2})$ for all $y_{1},y_{2} \in \mathbb{Z}$. The weightings $T_{x,x+1}$ can be represented as two $2 \times 2$ block Toeplitz matrices (one for $x$ even, and one for $x$ odd) that are infinite in both directions. These two infinite matrices can be encoded in two $2 \times 2$ matrix symbols $A_{x,x+1}(z)$, whose entries $(A_{x,x+1}(z))_{i+1,j+1}$, $0 \leq i,j \leq 1$, are given by $$\begin{aligned}
(A_{x,x+1}(z))_{i+1,j+1} = T_{x,x+1}(i,j) + zT_{x,x+1}(i,j+2).\end{aligned}$$ More explicitly, this gives $$\begin{aligned}
\label{1-transition symbol}
A_{x,x+1}(z) = \begin{cases}
\begin{pmatrix}
1 & 1 \\
\alpha z & 1
\end{pmatrix}, & \mbox{if }x \mbox{ is even}, \\[0.4cm]
\begin{pmatrix}
\alpha^{2} & \alpha \\
z & 1
\end{pmatrix}, & \mbox{if }x \mbox{ is odd},
\end{cases}\end{aligned}$$ and we can retrieve the entries of $T_{x,x+1}$ from its symbol by $$\begin{aligned}
\begin{pmatrix}
T_{x,x+1}(2y_{1},2y_{2}) & T_{x,x+1}(2y_{1},2y_{2}+1) \\
T_{x,x+1}(2y_{1}+1,2y_{2}) & T_{x,x+1}(2y_{1}+1,2y_{2}+1)
\end{pmatrix} = \frac{1}{2\pi i}\int_{\gamma} A_{x,x+1}(z) z^{y_{1}-y_{2}} \frac{dz}{z},\end{aligned}$$ where $\gamma$ is any close contour going around $0$ once in the positive direction. The symbol associated to $\mathcal{G}_{n}$ is then obtained by taking the following product (see [@DK equation (4.9)]): $$\begin{aligned}
A_{0,2n}(z) = \prod_{x=0}^{2n-1}A_{x,x+1}(z) = A(z)^{n},\end{aligned}$$ where $$\begin{aligned}
\label{def of A}
A(z) := \begin{pmatrix}
1 & 1 \\
\alpha z & 1
\end{pmatrix} \begin{pmatrix}
\alpha^{2} & \alpha \\
z & 1
\end{pmatrix} = \begin{pmatrix}
\alpha^{2}+z & 1+\alpha \\
(1+\alpha^{3})z & 1+\alpha^{2}z
\end{pmatrix}.\end{aligned}$$ In order to limit the length and technicalities of the paper, from now we take the size of the hexagon even, i.e. $n = 2N$ where $N$ is a positive integer. This is made for convenience; the case of odd integer $n$ could also be analyzed in a similar way, but then a discussion on the partity of $n$ is needed. Since $n=2N$, following [@DK equation (4.12)], the relevant orthogonality weight to consider is $$\begin{aligned}
\frac{A(z)^{2N}}{z^{2N}}. \label{def of weight}\end{aligned}$$ We consider two families $\{P_{j}\}_{j\geq 0}$ and $\{Q_{j}\}_{j\geq 0}$ of $2 \times 2$ matrix valued OPs defined by $$\begin{aligned}
& P_{j}(z) = z^{j}I_{2}+\bigO(z^{j-1}), \qquad \mbox{as } z \to \infty \label{cond for Pn} \\
& \frac{1}{2\pi i}\int_{\gamma} P_{j}(z) \frac{A(z)^{2N}}{z^{2N}}z^{k}dz = 0_{2}, & & k = 0,\ldots,j-1, \nonumber \end{aligned}$$ and $$\begin{aligned}
& \frac{1}{2\pi i}\int_{\gamma} Q_{j}(z) \frac{A(z)^{2N}}{z^{2N}}z^{j}dz = -I_{2}, \label{cond for Qn} \\
& \frac{1}{2\pi i}\int_{\gamma} Q_{j}(z) \frac{A(z)^{2N}}{z^{2N}}z^{k}dz = 0_{2}, & & k = 0,\ldots,j-1, \nonumber\end{aligned}$$ where $0_{2}$ denotes the $2\times 2$ zero matrix, $I_{2}$ is the identity matrix, and $\gamma$ is, as before, a close contour surrounding $0$ once in the positive direction. Since the weight is not hermitian, there is no guarantee that the above OPs exist for every $j$. However, it follows from [@DK Lemma 4.8 and equation (4.32)] that $P_{N}$ and $Q_{N-1}$ exist.
The $4 \times 4$ Riemann-Hilbert problem for $Y$
------------------------------------------------
Riemann-Hilbert problems for scalar orthogonal polynomials have been introduced by Fokas, Its and Kitaev in [@FIK]. Here, we need the generalization of this result for matrix valued OPs, which can be found in [@CassaManas2012; @Delvaux; @GrunIglesia]. Consider the $4 \times 4$ matrix valued function $Y(z) = Y(z;\alpha,N)$ defined by $$\label{Y definition}
Y(z) = \begin{pmatrix}
P_{N}(z) & \displaystyle \frac{1}{2\pi i} \int_{\gamma} P_{N}(s)\frac{A^{2N}(s)}{s^{2N}}\frac{ds}{s-z} \\
Q_{N-1}(z) & \displaystyle \frac{1}{2\pi i} \int_{\gamma} Q_{N-1}(s)\frac{A^{2N}(s)}{s^{2N}}\frac{ds}{s-z}
\end{pmatrix}, \qquad z \in \mathbb{C}\setminus \gamma.$$ The matrix $Y$ is characterized as the unique solution to the following RH problem.
### RH problem for $Y$ {#rh-problem-for-y .unnumbered}
- $Y : \mathbb{C}\setminus \gamma \to \mathbb{C}^{4\times 4}$ is analytic.
- The limits of $Y(z)$ as $z$ approaches $\gamma$ from inside and outside exist, are continuous on $\gamma$ and are denoted by $Y_+$ and $Y_-$ respectively. Furthermore, they are related by $$\label{jump relations of Y}
Y_{+}(z) = Y_{-}(z) \begin{pmatrix}
I_{2} & \frac{A^{2N}(z)}{z^{2N}} \\ 0_{2} & I_{2}
\end{pmatrix}, \hspace{0.5cm} \mbox{ for } z \in \gamma.$$
- As $z \to \infty$, we have $Y(z) = \left(I_{4} + \bigO(z^{-1})\right) \begin{pmatrix}
z^{N}I_{2} & 0_{2} \\ 0_{2} & z^{-N}I_{2}
\end{pmatrix}$.
Double contour formula from [@DK] for the kernel
------------------------------------------------
As mentioned in the introduction, the point process obtained by putting points on the paths, as shown in , is determinantal. We let $K$ denote the associated kernel. By definition of determinantal point processes, for integers $k \geq 1$, and $x_{1},\ldots,x_{k},y_{1},\ldots,y_{k}$ with $(x_{i},y_{i}) \neq (x_{j},y_{j})$ if $i \neq j$ we have $$\begin{aligned}
\label{def of determinantal}
\mathbb{P}\bigg[ \begin{matrix}
\mathfrak{p}_{0},\ldots,\mathfrak{p}_{2N-1} \mbox{ go through each of the points} \\
(x_{1},y_{1}+\frac{1}{2}),\ldots,(x_{k},y_{k}+\frac{1}{2})
\end{matrix} \bigg] = \det \big[ K(x_{i},y_{i},x_{j},y_{j}) \big]_{i,j=1}^{k}. \end{aligned}$$ The following proposition follows after specifying the general result [@DK Theorem 4.7] to our situation.[^1]
*(from [@DK])* Let $\alpha \in (0,1]$. For integers $x_{1},x_{2} \in \{1,2,\ldots,4N-1\}$ and $y_{1},y_{2} \in \mathbb{Z}$, we have $$\begin{aligned}
\big[ K(x_{1},2y_{1}+j,x_{2},2y_{2}+i) \big]_{i,j=0}^{1} = - \frac{\chi_{x_{1}>x_{2}}}{2\pi i}\int_{\gamma} A_{x_{2},x_{1}}(z)z^{y_{2}-y_{1}} \frac{dz}{z} \nonumber \\
+ \frac{1}{(2\pi i)^{2}}\int_{\gamma}\int_{\gamma}\frac{A_{x_{2},4N}(w)}{w^{2N-y_{2}}}\mathcal{R}^{Y}(w,z)\frac{A_{0,x_{1}}(z)}{z^{y_{1}+1}} dzdw \label{kernel general}\end{aligned}$$ where, $A_{a,b}$ is defined by $$\begin{aligned}
A_{a,b}(z) = \prod_{x=a}^{b-1}A_{x,x+1}(z), \qquad b>a,\end{aligned}$$ and $\mathcal{R}^{Y}$ is given by $$\begin{aligned}
\label{def of mathcal R}
\mathcal{R}^{Y}(w,z) = \frac{1}{z-w} \begin{pmatrix}
0_{2} & I_{2}
\end{pmatrix}Y^{-1}(w)Y(z)\begin{pmatrix}
I_{2} \\ 0_{2}
\end{pmatrix}.\end{aligned}$$
As particular cases of the above, we obtain the following formulas.
Let $\alpha \in (0,1]$. For integers $x \in \{1,\ldots,2N-1\}$ and $y \in \mathbb{Z}$, we have $$\begin{aligned}
& \big[ K(2x,2y+j,2x,2y+i) \big]_{i,j=0}^{1} = \frac{1}{(2\pi i)^{2}}\int_{\gamma}\int_{\gamma}\frac{A(w)^{2N-x}}{w^{2N-y}}\mathcal{R}^{Y}(w,z)\frac{A(z)^{x}}{z^{y+1}} dzdw \label{kernel diag even}\end{aligned}$$ and $$\begin{aligned}
& \big[ K(2x+1,2y+j,2x+1,2y+i) \big]_{i,j=0}^{1} = \label{kernel diag odd} \\
& \frac{1}{(2\pi i)^{2}}\int_{\gamma}\int_{\gamma}\begin{pmatrix}
\alpha^{2} & \alpha \\
w & 1
\end{pmatrix}\frac{A(w)^{2N-x-1}}{w^{2N-y}}\mathcal{R}^{Y}(w,z)\frac{A(z)^{x}}{z^{y+1}}\begin{pmatrix}
1 & 1 \\
\alpha z & 1
\end{pmatrix} dzdw. \nonumber\end{aligned}$$
This simply follows from $$\begin{aligned}
& A_{0,2x}(z) = A(z)^{x}, & & A_{2x,4N}(w) = A(w)^{2N-x}, \\
& A_{0,2x+1}(z) = A(z)^{x}\begin{pmatrix}
1 & 1 \\
\alpha z & 1
\end{pmatrix}, & & A_{2x+1,4N}(w) = \begin{pmatrix}
\alpha^{2} & \alpha \\
w & 1
\end{pmatrix}A(w)^{2N-x-1},\end{aligned}$$ where we have used and .
From [@DK Lemma 4.6], $\mathcal{R}^{Y}(w,z)$ is the unique bivariate polynomial of degree $\leq N-1$ in both variables $w$ and $z$ which satisfies the following reproducing property $$\label{reproducing property Y}
\frac{1}{2\pi i} \int_{\gamma} P(w) \frac{A^{2N}(w)}{w^{2N}}\mathcal{R}^{Y}(w,z)dw = P(z),$$ for every $2 \times 2$ matrix valued polynomial $P$ of degree $\leq N-1$. Because it satisfies , $\mathcal{R}^{Y}(w,z)$ is called a reproducing kernel.
Statement of results {#Section: main results}
====================
The new double contour formula for the kernel in terms of scalar OPs is stated in Theorem \[thm: correlation kernel final scalar expression\]. In this formula, the integrand is associated to a phase function, which in our case is defined on a two-sheeted Riemann surface $\mathcal{R}_{\alpha}$. The restriction of this phase function on the first and second sheet are denoted by $\Phi$ and $\Psi$, respectively. The saddle points are the solutions $\zeta \in \mathbb{C}$ for which either $\Phi'(\zeta) = 0$ or $\Psi'(\zeta) = 0$. In the liquid region, Proposition \[prop:saddle\] states that there is a unique saddle, denoted $s$, lying in the upper half plane. This saddle plays an important role in our analysis, and some of its properties are stated in Propositions \[prop:hightemp\] and \[prop: s on the Riemann surface\]. The limiting densities for the lozenges in the liquid region are given explicitly in terms of $s$ in Theorem \[thm:main\].
\[rem:alpha is not 1\] If $\alpha = 1$, our model reduces to the uniform measure and the kernel can be expressed in terms of scalar-valued OPs. However, our approach is based on the formulas –, and even though these formulas are still valid for $\alpha = 1$, this case requires a special attention (because of a different branch cut structure in the analysis). Since the limiting densities for the lozenges in this case are already well-known [@CLP], from now we will assume that $\alpha \in (0,1)$ to avoid unnecessary discussions.
New formula for the kernel in terms of scalar OPs {#subsection: new formula for the kernel}
-------------------------------------------------
We define the scalar weight $W$ by $$\label{def of W in intro}
W(\zeta) = \bigg( \frac{(\zeta-\alpha c)(\zeta-\alpha c^{-1})}{\zeta (\zeta-c)(\zeta-c^{-1})} \bigg)^{2N}, \qquad \mbox{ where } \qquad c = \sqrt{\frac{\alpha}{1-\alpha + \alpha^{2}}},$$ and consider the following $2 \times 2$ RH problem.
### RH problem for $U$ {#rh-problem-for-u .unnumbered}
- $U : \mathbb{C}\setminus \gamma_{\mathbb{C}} \to \mathbb{C}^{2\times 2}$ is analytic, where $\gamma_{\mathbb{C}}$ is a closed curve surrounding $c$ and $c^{-1}$ once in the positive direction, but not surrounding $0$.
- The limits of $U(\zeta)$ as $\zeta$ approaches $\gamma_{\mathbb{C}}$ from inside and outside exist, are continuous on $\gamma_{\mathbb{C}}$ and are denoted by $U_+$ and $U_-$ respectively. Furthermore, they are related by $$\label{jump relations of U}
U_{+}(\zeta) = U_{-}(\zeta) \begin{pmatrix}
1 & W(\zeta) \\ 0 & 1
\end{pmatrix}, \hspace{0.5cm} \mbox{ for } \zeta \in \gamma_{\mathbb{C}}.$$
- As $\zeta \to \infty$, we have $U(\zeta) = \left(I_{2} + \bigO(\zeta^{-1})\right) \begin{pmatrix}
\zeta^{2N} & 0 \\ 0 & \zeta^{-2N}
\end{pmatrix}$.
It is known [@FIK] that the solution $U$ to the above RH problem is unique (provided it exists), and can be expressed in terms of scalar-valued orthogonal polynomials as follows $$\begin{aligned}
U(\zeta) = \begin{pmatrix}
p_{2N}(\zeta) & \frac{1}{2\pi i} \int_{\gamma_{\mathbb{C}}} \frac{p_{2N}(\xi)W(\xi)}{\xi-\zeta}d\xi \\
q_{2N-1}(\zeta) & \frac{1}{2\pi i} \int_{\gamma_{\mathbb{C}}} \frac{q_{2N-1}(\xi)W(\xi)}{\xi-\zeta}d\xi
\end{pmatrix}, \qquad \zeta \in \mathbb{C}\setminus \gamma_{\mathbb{C}},\end{aligned}$$ where $p_{2N}$ and $q_{2N-1}$ are polynomials of degree $2N$ and $2N-1$ respectively, satisfying the following conditions $$\begin{aligned}
& p_{2N}(\zeta) = \zeta^{2N}+\bigO(\zeta^{2N-1}), \qquad \mbox{as } \zeta \to \infty, \label{cond for pn} \\
& \frac{1}{2\pi i}\int_{\gamma_{\mathbb{C}}} p_{2N}(\zeta) W(\zeta)\zeta^{k}d\zeta = 0, & & k = 0,\ldots,2N-1, \nonumber \end{aligned}$$ and $$\begin{aligned}
& \frac{1}{2\pi i}\int_{\gamma_{\mathbb{C}}} q_{2N-1}(\zeta) W(\zeta)\zeta^{2N-1}d\zeta = -1, \label{cond for qn} \\
& \frac{1}{2\pi i}\int_{\gamma_{\mathbb{C}}} q_{2N-1}(\zeta) W(\zeta)\zeta^{k}d\zeta = 0, & & k = 0,\ldots,2N-2. \nonumber\end{aligned}$$ The reproducing kernel $\mathcal{R}^{U}$ is defined by $$\label{reproducing kernel in terms of U intro}
\mathcal{R}^{U}(\omega,\zeta) = \frac{1}{\zeta-\omega} \begin{pmatrix}
0 & 1
\end{pmatrix} U^{-1}(\omega)U(\zeta) \begin{pmatrix}
1 \\ 0
\end{pmatrix}.$$ Now, we state our first main result.
\[thm: correlation kernel final scalar expression\]
For $x \in \{1,\ldots,2N-1\}$, $y \in \mathbb{Z}$ and $\epsilon_{x} \in \{0,1\}$, we have $$\begin{aligned}
& \big[ K(2x+\epsilon_{x},2y+j,2x+\epsilon_{x},2y+i) \big]_{i,j=0}^{1} = \frac{1}{(2\pi i)^{2}}\int_{\gamma_{\mathbb{C}}}\int_{\gamma_{\mathbb{C}}} H_{K}(\omega,\zeta;\epsilon_{x}) \label{new formula for the kernel in the thm} \\
& W(\omega) \mathcal{R}^{U}(\omega,\zeta) \frac{\omega^{N+x-y}}{\zeta^{N+x-y}} \frac{(\omega-c)^{y}(\omega-c^{-1})^{y}}{(\zeta-c)^{y}(\zeta-c^{-1})^{y}} \frac{(\zeta - \alpha c)^{x} (\zeta - \alpha c^{-1})^{x}}{(\omega - \alpha c)^{x} (\omega - \alpha c^{-1})^{x}}d\zeta d\omega, \nonumber\end{aligned}$$ where $\gamma_{\mathbb{C}}$ is a closed curve surrounding $c$ and $c^{-1}$ once in the positive direction that does not go around $0$, and where $H_{K}(\omega,\zeta;0)$ and $H_{K}(\omega,\zeta;1)$ are given by $$\begin{aligned}
& H_{K}(\omega,\zeta;0) = \begin{pmatrix}
\frac{1}{\zeta - c} & \frac{c(1-\alpha)}{\alpha (\zeta -c)(\zeta - c^{-1})} \\
\frac{\alpha}{(1-\alpha)c^{2} \omega}\frac{\omega-c}{\zeta -c} & \frac{\omega - c}{c \omega (\zeta -c)(\zeta -c^{-1})}
\end{pmatrix}, \label{HK epsx 0} \\
& H_{K}(\omega,\zeta;1) = \begin{pmatrix}
\frac{c (\zeta - \alpha c)}{\zeta(\zeta -c)(\omega - \alpha c)} & \frac{(1-\alpha)c(\zeta - \alpha c)}{(\zeta -c)(\zeta - c^{-1})(\omega - \alpha c)} \\
\frac{(\zeta - \alpha c)(\omega -c)}{(1-\alpha)\zeta(\zeta -c)(\omega-\alpha c)} & \frac{(\zeta - \alpha c)(\omega -c)}{(\zeta -c)(\zeta -c^{-1})(\omega - \alpha c)}
\end{pmatrix}. \label{HK epsx 1}\end{aligned}$$
Theorem \[thm: correlation kernel final scalar expression\] is proved in Section \[section: reducing the size\]. It is based on an unpublished idea of A. Kuijlaars that matrix valued orthogonal polynomials in a genus zero situation can be reduced to scalar orthogonality. In our case, the scalar orthogonality appears in – and a main part of the proof of Theorem \[thm: correlation kernel final scalar expression\] consists of relating the matrix valued reproducing kernel $\mathcal{R}^Y$ from to the scalar reproducing kernel $\mathcal{R}^U$ from .
The rational function $\mathcal{Q}$ {#sec:zeta}
-----------------------------------
The function $\mathcal{Q}$ is a meromorphic function that appears in the equilibrium problem associated to the varying weight $W$. Its explicit expression is obtained after solving a non-linear system of 5 equations with 5 unknowns. Here, we just state the formula for $\mathcal{Q}$, and refer to Section \[section: g-function\] for a more constructive approach. We define $\mathcal{Q}$ as follows $$\begin{aligned}
\label{def of Q in statement of results}
\mathcal{Q}(\zeta) = \frac{(\zeta-r_{1})^{2}(\zeta-r_{2})^{2}(\zeta-r_{3})^{2}(\zeta-r_{+})(\zeta-r_{-})}{4\zeta^{2} (\zeta-\alpha c)^{2} (\zeta-\alpha c^{-1})^{2} (\zeta-c)^{2} (\zeta-c^{-1})^{2}},\end{aligned}$$ where $c$ is given by , $r_{1}$, $r_{2}$ and $r_{3}$ are given by $$\begin{aligned}
\label{def of r1 r2 r3}
& r_{1} = - \sqrt{\alpha}, & & r_{2} = \sqrt{\alpha} \frac{\alpha c + \sqrt{\alpha}}{c + \sqrt{\alpha}}, & & r_{3} = \sqrt{\alpha} \frac{c + \sqrt{\alpha}}{\alpha c + \sqrt{\alpha}},\end{aligned}$$ and $r_{+}$ and $r_{-}$ are given by $$\begin{aligned}
& r_{+} = c \, \bigg( \frac{1+\alpha}{2} + i\sqrt{3} \frac{1-\alpha}{2} \bigg), & & r_{-} = c \, \bigg( \frac{1+\alpha}{2} - i\sqrt{3} \frac{1-\alpha}{2} \bigg). \label{def of r+ r-}\end{aligned}$$ The zero $r_{+}$ of $\mathcal{Q}$ lies in the upper half plane, $r_{-} = \overline{r_{+}}$, and the other zeros and poles of $\mathcal{Q}$ are real. Furthermore, for all $\alpha \in (0,1)$, they are ordered as follows: $$\begin{aligned}
\label{ordering of the zeros and the poles}
r_{1} < 0 < \alpha c < r_{2} < \alpha c^{-1} < c < r_{3} < c^{-1}.\end{aligned}$$
Lozenge probabilities
---------------------
The densities for the three types of lozenges at a point $(x,y)$, $x,y \in \{0,1,\ldots,4N\}$, are denoted by $$\begin{aligned}
\label{def of mathcal P1 P2 P3}
\mathcal{P}_{1}(x,y) = \mathbb P\Bigg(\tikz[scale=.3,baseline=(current bounding box.center)] {\draw (0,-1) {
--++(1,1)--++(0,1)--++(-1,-1)
--++(0,-1)
}; \filldraw (0,-1) circle(5pt); \draw (0,-1) node[below] {$(x,y)$}} \Bigg), \qquad \mathcal{P}_{2}(x,y) = \mathbb P\Bigg(\tikz[scale=.3,baseline=(current bounding box.center)] {\draw (0,0) {--++(1,0)--++(0,1)--++(-1,0)--++(0,-1)
}; \filldraw (0,0) circle(5pt); \draw (0,0) node[below] {$(x,y)$} }\Bigg), \qquad \mathcal{P}_{3}(x,y) = \mathbb P\Bigg(\tikz[scale=.3,baseline=(current bounding box.center)] {\draw (0,0) {--++(1,1)--++(1,0)--++(-1,-1)--++(-1,0)
}; \filldraw (1,0) circle(5pt); \draw (1,0) node[below] {$(x,y)$}} \Bigg),\end{aligned}$$ and satisfy $\sum_{j=1}^{3}\mathcal{P}_{j}(x,y) = 1$. Because our model is $2\times 2$ periodic, $\mathcal{P}_{1}(x,y)$, $\mathcal{P}_{2}(x,y)$ and $\mathcal{P}_{3}(x,y)$ depend crucially on the parity of $x$ and $y$, and it is convenient to consider the following matrices $$\begin{aligned}
\label{def of P1 P2 P3}
P_{j}(x,y) = \begin{pmatrix}
\mathcal{P}_{j}(2x,2y+1) & \mathcal{P}_{j}(2x+1,2y+1) \\
\mathcal{P}_{j}(2x,2y) & \mathcal{P}_{j}(2x+1,2y)
\end{pmatrix}, \qquad j=1,2,3,\end{aligned}$$ where $x,y \in \{0,1,...,2N-1\}$. Let $\{(x_{N},y_{N})\}_{N\geq 1}$ be a sequence satisfying $$\begin{aligned}
\label{good sequence}
& \begin{cases}
\frac{x_{N}}{N} = 1+\xi + o(1), \\
\frac{y_{N}}{N} = 1+\eta + o(1),
\end{cases} \qquad \mbox{as } N \to + \infty,\end{aligned}$$ where the point $(\xi,\eta)$ lies in the hexagon $$\label{eq:Hhexagon}
\mathcal H= \left\{(\xi,\eta) \mid -1\leq \xi \leq 1, \ -1 \leq \eta \leq 1,\ -1\leq \eta-\xi \leq 1 \right\}.$$ In Theorem \[thm:main\], we give explicit expressions for $$\begin{aligned}
\label{limits intro in intro}
\lim_{N \to + \infty} P_{j}(x_{N},y_{N}), \qquad j=1,2,3,\end{aligned}$$ in case $(\xi,\eta)$ belongs to the liquid region $\mathcal{L}_{\alpha} \subset \mathcal{H}$.
Saddle points and the liquid region {#subsection: saddle points and liquid region}
-----------------------------------
For each $(\xi,\eta)\in \mathcal{H}$, there are in total $8$ saddles for the double contour integral , which are the solutions to the algebraic equation $$\begin{aligned}
\label{eq:saddlepointeq}
\left[\frac{\xi-\eta}{2}\frac{1}{\zeta} - \frac{\xi}{2}\left( \frac{1}{\zeta-\alpha c} + \frac{1}{\zeta - \alpha c^{-1}} \right) + \frac{\eta}{2}\left( \frac{1}{\zeta - c} + \frac{1}{\zeta-c^{-1}} \right)\right]^{2} = \mathcal{Q}(\zeta),\end{aligned}$$ where $\mathcal{Q}(\zeta)$ is given by . Following the previous works [@BF; @Duits1; @Ok2; @Petrov1; @CDKL], we define the liquid region as the subset of $\mathcal{H}$ for which there exists a saddle lying in the upper half-plane $\mathbb{C}^{+} = \{\zeta \in \mathbb{C}: {\text{\upshape Im\,}}\zeta >0\}$. Proposition \[prop:saddle\] states that there is a unique such saddle (whenever it exists), which is denoted by $s(\xi,\eta;\alpha)$. This saddle plays a particular role in the analysis of Section \[section: saddle point analysis\] and appears in the final formulas for the limiting densities .
\[prop:saddle\] Let $(\xi, \eta) \in \mathcal{H}^{\mathrm{o}}$ (the interior set of $\mathcal{H}$). Then there exists at most one solution $\zeta=s(\xi, \eta; \alpha)$ to in $\mathbb C^+=\{\zeta \in \mathbb C \mid {\text{\upshape Im\,}}\zeta>0\}$.
We define the liquid region $\mathcal L_\alpha \subset \mathcal H$ by $$\label{eq:liquidLalpha}
\mathcal L_\alpha= \left\{ (\xi,\eta) \in \mathcal{H}^{\mathrm{o}} \mid
\mbox{there exists a solution } \zeta = s(\xi, \eta; \alpha) \in \mathbb{C}^+ \mbox{ to \eqref{eq:saddlepointeq}} \right\}$$ and we define the map $s: \mathcal L_\alpha \to\mathbb{C}^+$ by $(\xi,\eta) \mapsto s(\xi,\eta; \alpha)$.
at (0,0) [![\[fig: mapping s\] On the left, we draw for $\alpha = 0.4$ the parts of the lines $\xi = 0$ (red), $\eta = \frac{\xi}{2}$ (dashed red), $\eta = \xi$ (blue), $\eta = - \xi$ (dashed blue), $\eta = 0$ (green) and $\eta = 2 \xi$ (dashed green) that are in the liquid region. On the right, we draw the corresponding location of $s(\xi,\eta;\alpha)$ in the upper half plane. The black dots are, from left to right, $0$, $\alpha c$, $\alpha c^{-1}$, $c$ and $c^{-1}$.](alpha_04.pdf "fig:"){width="4.05cm"}]{}; (0,-2.05)–(0,2.04); (-1.78,0)–(1.77,0); (30:2.05)–(-150:2.05); (120:1.78)–(-60:1.77); (150:2.05)–(-30:2.05); (60:1.78)–(-120:1.77);
at (0,0) ;
(\[shift=(0:3.6cm)\]2.1,0) arc (0:156:3.6cm); (\[shift=(155:3.6cm)\]2.1,0) arc (155:180:3.6cm);
(\[shift=(0:2.4cm)\]-3.13,0) arc (0:35:2.4cm); (\[shift=(35:2.4cm)\]-3.13,0) arc (35:180:2.4cm);
(\[shift=(95:1.45cm)\]-1.05,0) arc (95:180:1.45cm); (\[shift=(0:1.45cm)\]-1.05,0) arc (0:95:1.45cm);
at (2.1,0) [ $\bullet$]{}; at (-0.38,0) [ $\bullet$]{}; at (-1.05,0) [ $\bullet$]{}; at (-2.035,0) [ $\bullet$]{}; at (-3.13,0) [ $\bullet$]{};
It is clear from and that $(0,0) \in \mathcal{L}_{\alpha}$ and $s(0,0;\alpha) = r_{+}$ for all $\alpha \in (0,1)$. We now describe some properties of $(\xi,\eta) \mapsto s(\xi,\eta;\alpha)$. Consider the following three circles: $$\begin{aligned}
& \gamma_{0} = \{\zeta \in \mathbb{C}:|\zeta|= R_{0}\}, \qquad \gamma_{\alpha} = \{\zeta \in \mathbb{C}:|\zeta - \alpha c^{-1}|= R_{\alpha}\}, \qquad \gamma_{1} = \{\zeta \in \mathbb{C}:|\zeta - c^{-1}|= R_{1}\}\end{aligned}$$ where $R_{0} = \sqrt{\alpha}$, $R_{\alpha} = (1-\alpha)\sqrt{\alpha}$ and $R_{1} := \frac{1-\alpha}{\sqrt{\alpha}}$ (see also Figure \[fig: crit traj alpha 04\]). It is a direct computation to verify that $$\begin{aligned}
\label{r2 r+ r- on the circle}
r_{+},r_{-},r_{2} \in \gamma_{1}, \qquad r_{+},r_{-},r_{3} \in \gamma_{\alpha} \qquad \mbox{ and } \qquad r_{+},r_{-},r_{1} \in \gamma_{0}.\end{aligned}$$ In particular, we can write $$\begin{aligned}
r_{\pm} = c^{-1} + R_{1}e^{\pm i \theta_{1}} = \alpha c^{-1} + R_{\alpha} e^{\pm i \theta_{\alpha}} = R_{0} e^{\pm i \theta_{0}},\end{aligned}$$ for certain angles $\theta_{1} \in (\frac{2\pi}{3},\pi)$, $\theta_{\alpha} \in (\frac{\pi}{3},\frac{2\pi}{3})$ and $\theta_{0} \in (0,\frac{\pi}{3})$. We also define $$\begin{aligned}
& \Sigma_{1} = \{\zeta \in \mathbb{C} : |\zeta - c^{-1}| = R_{1}, \; \arg z \in (-\theta_{1},\theta_{1}) \} \subset \gamma_{1}, \label{def of Sigma 1} \\
& \Sigma_{\alpha} = \{\zeta \in \mathbb{C} : |\zeta - \alpha c^{-1}| = R_{\alpha}, \; \arg z \in (-\pi,-\theta_{\alpha})\cup (\theta_{\alpha},\pi] \} \subset \gamma_{\alpha}, \label{def of Sigma alpha} \\
& \Sigma_{0} = \{\zeta \in \mathbb{C} : |\zeta| = R_{0}, \; \arg z \in (-\theta_{0},\theta_{0}) \} \subset \gamma_{0}. \label{def of Sigma 0}\end{aligned}$$ The following proposition is illustrated in Figure \[fig: mapping s\].
For a given set $A$, the notation $\overline{A}$ stands for the closure of $A$.
\[prop:hightemp\] The map $(\xi,\eta) \mapsto s(\xi,\eta;\alpha)$ satisfies $s(-\xi,-\eta;\alpha) = s(\xi,\eta;\alpha)$, and
1. \[item a in prop mapping s\] it maps $\{\xi = 0\}\cap \mathcal{L}_{\alpha}$ onto $\overline{\Sigma_{1}} \cap \mathbb{C}^{+}$,
2. \[item b in prop mapping s\] it maps $\{\eta = \frac{\xi}{2}\}\cap \mathcal{L}_{\alpha}$ onto $(\gamma_{1}\setminus \Sigma_{1}) \cap \mathbb{C}^{+}$,
3. \[item c in prop mapping s\] it maps $\{\eta = \xi\}\cap \mathcal{L}_{\alpha}$ onto $\overline{\Sigma_{0}} \cap \mathbb{C}^{+}$,
4. \[item d in prop mapping s\] it maps $\{\eta = -\xi\}\cap \mathcal{L}_{\alpha}$ onto $(\gamma_{0}\setminus \Sigma_{0}) \cap \mathbb{C}^{+}$,
5. \[item e in prop mapping s\] it maps $\{\eta = 0\}\cap \mathcal{L}_{\alpha}$ onto $\overline{\Sigma_{\alpha}} \cap \mathbb{C}^{+}$,
6. \[item f in prop mapping s\] it maps $\{\eta = 2\xi \}\cap \mathcal{L}_{\alpha}$ onto $(\gamma_{\alpha}\setminus \Sigma_{\alpha}) \cap \mathbb{C}^{+}$.
By definition, the saddles lie in the complex plane. We show here that they can be naturally projected on a Riemann surface. Define $\mathcal{Q}(\zeta)^{1/2}$ with a branch cut joining $r_{-}$ to $r_{+}$ along $\Sigma_{1}$, such that $\mathcal{Q}(\zeta)^{1/2}\sim \frac{1}{2\zeta}$ as $\zeta \to \infty$, and denote the associated Riemann surface by $\mathcal{R}_{\alpha}$: $$\begin{aligned}
\mathcal{R}_{\alpha} := \{(\zeta,w) \in \mathbb{C}^{2} : w^{2}=\mathcal{Q}(\zeta) \}.\end{aligned}$$ This is a two-sheeted covering of the $\zeta$-plane glued along $\Sigma_{1}$, and the sheets are ordered such that $w = \mathcal{Q}(\zeta)^{1/2}$ on the first sheet and $w = -\mathcal{Q}(\zeta)^{1/2}$ on the second sheet. For each solution $\zeta$ to , there exists a $w$ satisfying $w^{2} = \mathcal{Q}(\zeta)$, and such that $$\begin{aligned}
\label{eq:saddlepointeq sqrt}
\frac{\xi-\eta}{2}\frac{1}{\zeta} - \frac{\xi}{2}\left( \frac{1}{\zeta-\alpha c} + \frac{1}{\zeta - \alpha c^{-1}} \right) + \frac{\eta}{2}\left( \frac{1}{\zeta - c} + \frac{1}{\zeta-c^{-1}} \right) = w.\end{aligned}$$
The map $(\xi,\eta)\mapsto w(\xi,\eta;\alpha)$ is defined by $w(\xi,\eta;\alpha)^{2} = \mathcal{Q}(s(\xi,\eta;\alpha))$, such that holds with $\zeta = s(\xi,\eta;\alpha)$ and $w = w(\xi,\eta;\alpha)$.
\[prop: s on the Riemann surface\] The map $(\xi,\eta) \mapsto \big( s(\xi,\eta;\alpha),w(\xi,\eta;\alpha) \big)$ is a diffeomorphism from $\mathcal{L}_{\alpha}$ to $$\begin{aligned}
\label{calRplusdef}
\mathcal{R}_{\alpha}^+ := \{(\zeta,w) \in \mathcal R_{\alpha} \mid {\text{\upshape Im\,}}\zeta > 0 \}.\end{aligned}$$ It maps the left half $\mathcal L_\alpha^{l}=\left\{ (\xi,\eta) \in \mathcal L_\alpha \mid \xi < 0 \right\}$ to the upper half-plane of the first sheet of $\mathcal{R}_{\alpha}$, and it maps $\mathcal L_\alpha^{r}=\left\{ (\xi,\eta) \in \mathcal L_\alpha \mid \xi > 0 \right\}$ to the upper half-plane of the second sheet. Moreover, its inverse $(s,w)\mapsto (\xi,\eta)=(\xi(s,w;\alpha),\eta(s,w;\alpha))$ is explicitly given by $$\begin{aligned}
\label{inverse of the diffeomorphism}
\begin{pmatrix}
\xi \\ \eta
\end{pmatrix} = \begin{pmatrix}
{\text{\upshape Re\,}}\left( \frac{-(s - \alpha)(s +\alpha)(s -c)(s - \frac{1}{c})}{(s - \alpha c)(s - \frac{\alpha}{c})(s - 1)(s + 1)} \right) & 1 \\[0.2cm]
{\text{\upshape Im\,}}\left( \frac{-(s - \alpha)(s +\alpha)(s -c)(s - \frac{1}{c})}{(s - \alpha c)(s - \frac{\alpha}{c})(s - 1)(s + 1)} \right) & 0
\end{pmatrix}^{-1} \begin{pmatrix}
{\text{\upshape Re\,}}\left( \frac{2s (s -c)(s - \frac{1}{c})}{(s - 1)(s + 1)}w \right) \\[0.2cm]
{\text{\upshape Im\,}}\left( \frac{2s (s -c)(s - \frac{1}{c})}{(s - 1)(s + 1)}w \right)
\end{pmatrix}.\end{aligned}$$
#### Description of the liquid region. {#description-of-the-liquid-region. .unnumbered}
After clearing the denominator in , we get $$\begin{gathered}
\label{eq:saddleequationhigh}
(\zeta -r_{1})^{2}(\zeta -r_{2})^{2}(\zeta -r_{3})^{2}(\zeta -r_{+})(\zeta -r_{-}) = \\
\Big[ (\zeta-1)(\zeta +1)(\zeta-\alpha c)(\zeta-\tfrac{\alpha}{c})\eta - (\zeta-\alpha)(\zeta+\alpha)(\zeta-c)(\zeta-\tfrac{1}{c})\xi \Big]^{2}. \end{gathered}$$ Since is invariant under the map $(\xi,\eta) \mapsto (-\xi,-\eta)$, we conclude that $\mathcal{L}_{\alpha}$ is symmetric with respect to the origin. Also, this equation has real coefficients, so $s(\xi,\eta;\alpha)$ and $\overline{s(\xi,\eta;\alpha)}$ are both solutions whenever $(\xi,\eta) \in \mathcal{L}_{\alpha}$. At the boundary $\partial \mathcal L_\alpha$ of the liquid region, $s(\xi,\eta;\alpha)$ and $\overline{s(\xi,\eta;\alpha)}$ coalesce in the real line, so $\partial \mathcal L_\alpha$ is part of the zero set of the discriminant of (whose expression is too long to be written down). The curve $\partial \mathcal{L}_{\alpha}$ is tangent to the hexagon at $12$ points and possesses $6$ cusp points. The tangent points can be obtained by letting $s \to s_{\star}\in\{ 0,\alpha c, \alpha c^{-1}, c,c^{-1},\infty\}$ in , and the cusp points by letting $s \to s_{\star} \in \{r_{1},r_{2},r_{3}\}$ in (see also Figure \[fig: mapping s\]). Figure \[fig: flower\] illustrates $\partial \mathcal{L}_{\alpha}$ for different values of $\alpha$ (and has been generated using ). Denote $\mathcal{F}_{\alpha,j}$, $j=1,\ldots,6$ for the regions shown in Figure \[fig: frozen regions\] (left). They are disjoint from each other and from $\mathcal{L}_{\alpha}$, and are symmetric under $(\xi,\eta)\mapsto (-\xi,-\eta)$. As we will see, these regions are frozen (or semi-frozen).
![\[fig: flower\]The curve $\partial \mathcal{L}_{\alpha}$ with $\alpha=0.04$, $\alpha=0.2$ and $\alpha=0.4$ (from left to right).](alpha_004.pdf "fig:"){width="3.5cm"} ![\[fig: flower\]The curve $\partial \mathcal{L}_{\alpha}$ with $\alpha=0.04$, $\alpha=0.2$ and $\alpha=0.4$ (from left to right).](alpha_02.pdf "fig:"){width="3.5cm"} ![\[fig: flower\]The curve $\partial \mathcal{L}_{\alpha}$ with $\alpha=0.04$, $\alpha=0.2$ and $\alpha=0.4$ (from left to right).](alpha_04.pdf "fig:"){width="3.5cm"}
From Propositions \[prop:hightemp\] and \[prop: s on the Riemann surface\], we already infer the following:
\[limits s sbar as xi eta\] $$\begin{aligned}
& s,\overline{s} \to s_{\star} \in (0,\alpha c), & & \mbox{as } (\xi,\eta) \to (\xi^{\star},\eta^{\star}) \in \partial \mathcal{L}_{\alpha} \cap \partial \mathcal{F}_{1,\alpha}, \\
& s,\overline{s} \to s_{\star} \in (\alpha c^{-1},c), & & \mbox{as } (\xi,\eta) \to (\xi^{\star},\eta^{\star}) \in \partial \mathcal{L}_{\alpha} \cap \partial \mathcal{F}_{2,\alpha}, \\
& s,\overline{s} \to s_{\star} \in (c^{-1},+\infty), & & \mbox{as } (\xi,\eta) \to (\xi^{\star},\eta^{\star}) \in \partial \mathcal{L}_{\alpha} \cap \partial \mathcal{F}_{3,\alpha}, \\
& s,\overline{s} \to s_{\star} \in (\alpha c,\alpha c^{-1}), & & \mbox{as } (\xi,\eta) \to (\xi^{\star},\eta^{\star}) \in \partial \mathcal{L}_{\alpha} \cap \partial \mathcal{F}_{4,\alpha}, \\
& s,\overline{s} \to s_{\star} \in (-\infty,0), & & \mbox{as } (\xi,\eta) \to (\xi^{\star},\eta^{\star}) \in \partial \mathcal{L}_{\alpha} \cap \partial \mathcal{F}_{5,\alpha}, \\
& s,\overline{s} \to s_{\star} \in (c,c^{-1}), & & \mbox{as } (\xi,\eta) \to (\xi^{\star},\eta^{\star}) \in \partial \mathcal{L}_{\alpha} \cap \partial \mathcal{F}_{6,\alpha}.\end{aligned}$$
Limiting densities in the liquid region
---------------------------------------
Theorem \[thm:main\] states that the limits are expressed in terms of the angles shown in Figure \[fig: the four triangles\].
\[thm:main\] Let $\{(x_{N},y_{N}\}_{N \geq 1}$ be a sequence satisfying with $(\xi,\eta) \in \mathcal{L}_{\alpha}$. We obtain the following limits: $$\begin{aligned}
& \lim_{N \to \infty} P_{1}(x_{N},y_{N}) = \frac{1}{\pi}\begin{pmatrix}
\arg(s-\alpha c) - \arg(s) & \arg(s-\alpha c^{-1}) \\
\arg (s-\alpha c) & \arg(s-\alpha c^{-1}) - \arg s
\end{pmatrix}, \label{P1 limit main result} \\
& \lim_{N \to \infty} P_{2}(x_{N},y_{N}) = \frac{1}{\pi}\begin{pmatrix}
\arg(s-c^{-1}) - \arg(s-\alpha c) & \arg(s-c^{-1}) - \arg(s-\alpha c^{-1}) \\
\arg(s-c)-\arg(s-\alpha c) & \arg(s-c) - \arg(s-\alpha c^{-1})
\end{pmatrix}, \label{P2 limit main result} \\
& \lim_{N \to \infty} P_{3}(x_{N},y_{N}) = \frac{1}{\pi} \begin{pmatrix}
\pi-\arg(s-c^{-1})+\arg(s) & \pi-\arg(s-c^{-1}) \\
\pi-\arg(s-c) & \pi-\arg(s-c)+\arg(s)
\end{pmatrix}. \label{P3 limit main result}\end{aligned}$$ These limits can equivalently be stated as follows: $$\begin{aligned}
& \lim_{N \to \infty} P_{1}(x_{N},y_{N}) = \frac{1}{\pi}\begin{pmatrix}
\phi_{1,11} & \phi_{1,12} \\
\phi_{1,21} & \phi_{1,22}
\end{pmatrix}, \\
& \lim_{N \to \infty} P_{2}(x_{N},y_{N}) = \frac{1}{\pi}\begin{pmatrix}
\phi_{2,11} & \phi_{2,12} \\
\phi_{2,21} & \phi_{2,22}
\end{pmatrix}, \\
& \lim_{N \to \infty} P_{3}(x_{N},y_{N}) = \frac{1}{\pi} \begin{pmatrix}
\phi_{3,11}^{(l)}+\phi_{3,11}^{(r)} & \phi_{3,12} \\
\phi_{3,21} & \phi_{3,22}^{(l)}+\phi_{3,22}^{(r)}
\end{pmatrix},\end{aligned}$$ where $\phi_{k,ij}$, $1\leq i,j,k \leq 2$, and $\phi_{3,11}^{(l)}$, $\phi_{3,11}^{(r)}$, $\phi_{3,12}$, $\phi_{3,21}$, $\phi_{3,22}^{(l)}$ and $\phi_{3,22}^{(r)}$ are the angles represented in Figure \[fig: the four triangles\].
at (0,0) ; at (2.1,0) [ $\bullet$]{}; at (-0.38,0) [ $\bullet$]{}; at (-1.05,0) [ $\bullet$]{}; at (-2.035,0) [ $\bullet$]{}; at (-3.13,0) [ $\bullet$]{}; at (-1.8,2) [ $\bullet$]{};
(Start) at (-3.13,0); (End) at (2.1,0); (Middle) at (-2.035,0); (S) at (-1.8,2);
(S)–(Start)–(End)–(S)–(Middle);
(End,Start,S);
(Middle,S,End);
(Start,S,Middle);
(S,End,Start);
at (-2.45,1.6) [$\phi_{1,11}$]{}; at (-1.4,1.2) [$\phi_{2,11}$]{}; at (-3.3,0.4) [$\phi_{3,11}^{(l)}$]{}; at (0.8,0.3) [$\phi_{3,11}^{(r)}$]{};
at (0,0) ; at (2.1,0) [ $\bullet$]{}; at (-0.38,0) [ $\bullet$]{}; at (-1.05,0) [ $\bullet$]{}; at (-2.035,0) [ $\bullet$]{}; at (-3.13,0) [ $\bullet$]{}; at (-1.8,2) [ $\bullet$]{};
(Start) at (-1.05,0); (End) at (2.1,0); (S) at (-1.8,2);
(S)–(Start)–(End)–(S);
(End,Start,S);
(Start,S,End);
(S,End,Start);
at (-0.7,0.5) [$\phi_{1,12}$]{}; at (-1.2,1.95) [$\phi_{2,12}$]{}; at (1,0.3) [$\phi_{3,12}$]{};
at (0,0) ; at (2.1,0) [ $\bullet$]{}; at (-0.38,0) [ $\bullet$]{}; at (-1.05,0) [ $\bullet$]{}; at (-2.035,0) [ $\bullet$]{}; at (-3.13,0) [ $\bullet$]{}; at (-1.8,2) [ $\bullet$]{};
(Start) at (-2.035,0); (End) at (-0.38,0); (S) at (-1.8,2);
(S)–(Start)–(End)–(S);
(End,Start,S);
(Start,S,End);
(S,End,Start);
at (-1.6,0.55) [$\phi_{1,21}$]{}; at (-1.2,1.95) [$\phi_{2,21}$]{}; at (-0.1,0.3) [$\phi_{3,21}$]{};
at (0,0) ; at (2.1,0) [ $\bullet$]{}; at (-0.38,0) [ $\bullet$]{}; at (-1.05,0) [ $\bullet$]{}; at (-2.035,0) [ $\bullet$]{}; at (-3.13,0) [ $\bullet$]{}; at (-1.8,2) [ $\bullet$]{};
(Start) at (-3.13,0); (End) at (-0.38,0); (Middle) at (-1.05,0); (S) at (-1.8,2);
(S)–(Start)–(End)–(S)–(Middle);
(End,Start,S);
(Middle,S,End);
(Start,S,Middle);
(S,End,Start);
(-4,-0.2)–(-4,5.5); (-10.5,2.5)–(2.5,2.5);
at (-1.9,1.2) [$\phi_{1,22}$]{}; at (-1.2,1.7) [$\phi_{2,22}$]{}; at (-2.2,0.4) [$\phi_{3,22}^{(l)}$]{}; at (-0.2,0.4) [$\phi_{3,22}^{(r)}$]{};
By combining with Theorem \[thm:main\], we obtain the following immediate corollary.
\[coro: frozen\] Let $\{(x_{N},y_{N}\}_{N \geq 1}$ be a sequence satisfying with $(\xi,\eta) \in \mathcal{L}_{\alpha}$. We have $$\begin{aligned}
& \lim_{N \to \infty} P_{j}(x_{N},y_{N}) \to \left\{ \begin{pmatrix}
1 & 1 \\ 1 & 1
\end{pmatrix}, \begin{pmatrix}
0 & 0 \\ 0 & 0
\end{pmatrix},\begin{pmatrix}
0 & 0 \\ 0 & 0
\end{pmatrix} \right\} & & \mbox{as } (\xi,\eta) \to (\xi^{\star},\eta^{\star}) \in \partial \mathcal{L}_{\alpha} \cap \partial \mathcal{F}_{1,\alpha}, \\
& \lim_{N \to \infty} P_{j}(x_{N},y_{N}) \to \left\{ \begin{pmatrix}
0 & 0 \\ 0 & 0
\end{pmatrix},\begin{pmatrix}
1 & 1 \\ 1 & 1
\end{pmatrix},\begin{pmatrix}
0 & 0 \\ 0 & 0
\end{pmatrix} \right\} & & \mbox{as } (\xi,\eta) \to (\xi^{\star},\eta^{\star}) \in \partial \mathcal{L}_{\alpha} \cap \partial \mathcal{F}_{2,\alpha}, \\
& \lim_{N \to \infty} P_{j}(x_{N},y_{N}) \to \left\{ \begin{pmatrix}
0 & 0 \\ 0 & 0
\end{pmatrix},\begin{pmatrix}
0 & 0 \\ 0 & 0
\end{pmatrix},\begin{pmatrix}
1 & 1 \\ 1 & 1
\end{pmatrix} \right\} & & \mbox{as } (\xi,\eta) \to (\xi^{\star},\eta^{\star}) \in \partial \mathcal{L}_{\alpha} \cap \partial \mathcal{F}_{3,\alpha}, \\
& \lim_{N \to \infty} P_{j}(x_{N},y_{N}) \to \left\{ \begin{pmatrix}
0 & 1 \\ 0 & 1
\end{pmatrix},\begin{pmatrix}
1 & 0 \\ 1 & 0
\end{pmatrix},\begin{pmatrix}
0 & 0 \\ 0 & 0
\end{pmatrix} \right\} & & \mbox{as } (\xi,\eta) \to (\xi^{\star},\eta^{\star}) \in \partial \mathcal{L}_{\alpha} \cap \partial \mathcal{F}_{4,\alpha}, \\
& \lim_{N \to \infty} P_{j}(x_{N},y_{N}) \to \left\{ \begin{pmatrix}
0 & 1 \\ 1 & 0
\end{pmatrix},\begin{pmatrix}
0 & 0 \\ 0 & 0
\end{pmatrix},\begin{pmatrix}
1 & 0 \\ 0 & 1
\end{pmatrix} \right\} & & \mbox{as } (\xi,\eta) \to (\xi^{\star},\eta^{\star}) \in \partial \mathcal{L}_{\alpha} \cap \partial \mathcal{F}_{5,\alpha}, \\
& \lim_{N \to \infty} P_{j}(x_{N},y_{N}) \to \left\{ \begin{pmatrix}
0 & 0 \\ 0 & 0
\end{pmatrix},\begin{pmatrix}
1 & 1 \\ 0 & 0
\end{pmatrix},\begin{pmatrix}
0 & 0 \\ 1 & 1
\end{pmatrix} \right\} & & \mbox{as } (\xi,\eta) \to (\xi^{\star},\eta^{\star}) \in \partial \mathcal{L}_{\alpha} \cap \partial \mathcal{F}_{6,\alpha},\end{aligned}$$ where the three matrices inside each brackets correspond, from left to right, to $j=1,2,3$.
at (0,0) [![\[fig: frozen regions\]The six frozen regions for $\alpha = 0.3$, and a tiling of size $n=48$.](oblique_alpha_03.pdf "fig:"){width="7cm"}]{}; at (-3.3,0.5) [$\mathcal{F}_{1,\alpha}$]{}; at (3.3,-0.5) [$\mathcal{F}_{1,\alpha}$]{}; at (-2.95,-3.65) [$\mathcal{F}_{2,\alpha}$]{}; at (2.95,3.65) [$\mathcal{F}_{2,\alpha}$]{}; at (0,3.65) [$\mathcal{F}_{3,\alpha}$]{}; at (0,-3.65) [$\mathcal{F}_{3,\alpha}$]{}; at (-3.7,-1.6) [$\mathcal{F}_{4,\alpha}$]{}; at (3.7,1.6) [$\mathcal{F}_{4,\alpha}$]{}; at (-2,1.8) [$\mathcal{F}_{5,\alpha}$]{}; at (2,-1.8) [$\mathcal{F}_{5,\alpha}$]{}; at (-1.6,-3.65) [$\mathcal{F}_{6,\alpha}$]{}; at (1.6,3.65) [$\mathcal{F}_{6,\alpha}$]{};
at (0,0) [![\[fig: frozen regions\]The six frozen regions for $\alpha = 0.3$, and a tiling of size $n=48$.](size48_a_03.jpg "fig:"){width="7cm"}]{}; at (0,-1.2) [.]{};
From Figure \[fig: frozen regions\] (right), it transpires that the regions $\mathcal{F}_{j,\alpha}$, $j=1,2,3$ are frozen, and that $\mathcal{F}_{j,\alpha}$, $j=4,5,6$ are semi-frozen. More precisely, let $(x,y) \in \{0,\ldots,2N-1\}$ be such that $(\xi,\eta)=(\frac{x}{N}-1,\frac{y}{N}-1) \in \mathcal{F}_{j,\alpha}$, $j \in \{1,\ldots,6\}$. In Figure \[fig: frozen regions\] (right), we observe that $$\begin{aligned}
\tikz[scale=.3,baseline=(current bounding box.center)] {\draw (0,0) {
--++(1,1)--++(0,1)--++(-1,-1)
--++(0,-1)
}; \draw (1,0) {
--++(1,1)--++(0,1)--++(-1,-1)
--++(0,-1)
}; \draw (0,1) {
--++(1,1)--++(0,1)--++(-1,-1)
--++(0,-1)
}; \draw (1,1) {
--++(1,1)--++(0,1)--++(-1,-1)
--++(0,-1)
}; \filldraw (0,0) circle(5pt); \draw (0,0) node[below] {$(2x,2y)$}; } \; , \quad \tikz[scale=.3,baseline=(current bounding box.center)] {\draw (0,0) {--++(1,0)--++(0,1)--++(-1,0)--++(0,-1)
}; \draw (1,0) {--++(1,0)--++(0,1)--++(-1,0)--++(0,-1)
}; \draw (0,1) {--++(1,0)--++(0,1)--++(-1,0)--++(0,-1)
}; \draw (1,1) {--++(1,0)--++(0,1)--++(-1,0)--++(0,-1)
}; \filldraw (0,0) circle(5pt); \draw (0,0) node[below] {$(2x,2y)$}; } \; , \quad \tikz[scale=.3,baseline=(current bounding box.center)] {\draw (0,0) {--++(1,1)--++(1,0)--++(-1,-1)--++(-1,0)
}; \draw (1,0) {--++(1,1)--++(1,0)--++(-1,-1)--++(-1,0)
}; \draw (0,1) {--++(1,1)--++(1,0)--++(-1,-1)--++(-1,0)
}; \draw (1,1) {--++(1,1)--++(1,0)--++(-1,-1)--++(-1,0)
}; \filldraw (1,0) circle(5pt); \draw (1,0) node[below] {$(2x,2y)$};} \; , \quad \tikz[scale=.3,baseline=(current bounding box.center)] {\draw (0,0) {--++(1,0)--++(0,1)--++(-1,0)--++(0,-1)
}; \draw (1,0) {
--++(1,1)--++(0,1)--++(-1,-1)
--++(0,-1)
}; \draw (0,1) {--++(1,0)--++(0,1)--++(-1,0)--++(0,-1)
}; \draw (1,1) {
--++(1,1)--++(0,1)--++(-1,-1)
--++(0,-1)
}; \filldraw (0,0) circle(5pt); \draw (0,0) node[below] {$(2x,2y)$};} \; , \quad \tikz[scale=.3,baseline=(current bounding box.center)] {\draw (0,0) {
--++(1,1)--++(0,1)--++(-1,-1)
--++(0,-1)
}; \draw (0,0) {--++(1,1)--++(1,0)--++(-1,-1)--++(-1,0)
}; \draw (-1,1) {--++(1,1)--++(1,0)--++(-1,-1)--++(-1,0)
}; \draw (1,1) {
--++(1,1)--++(0,1)--++(-1,-1)
--++(0,-1)
}; \filldraw (0,0) circle(5pt); \draw (0,0) node[below] {$(2x,2y)$};} \; , \quad \tikz[scale=.3,baseline=(current bounding box.center)] {\draw (-1,0) {--++(1,1)--++(1,0)--++(-1,-1)--++(-1,0)
}; \draw (0,0) {--++(1,1)--++(1,0)--++(-1,-1)--++(-1,0)
}; \draw (0,1) {--++(1,0)--++(0,1)--++(-1,0)--++(0,-1)
}; \draw (1,1) {--++(1,0)--++(0,1)--++(-1,0)--++(0,-1)
}; \filldraw (0,0) circle(5pt); \draw (0,0) node[below] {$(2x,2y)$};},\end{aligned}$$ depending on whether $(\xi,\eta) \in \mathcal{F}_{j,\alpha}$, $j=1,...,6$, respectively. Corollary \[coro: frozen\] describes the situation at the boundary of the liquid region, and is consistent with these observations.
Outline of the rest of the paper
--------------------------------
The proofs of Propositions \[prop: frozen\], \[prop:saddle\], \[prop:hightemp\] and \[prop: s on the Riemann surface\] are rather direct and are presented in Section \[section: easy proofs\]. In Section \[section: first steps of the steepest descent of Y\] we follow an idea of [@DK] and perform an eigendecomposition of the matrix valued weight. The eigenvalues and eigenvectors are naturally related to a $2$-sheeted Riemann surface $\mathcal{M}$. The proof of Theorem \[thm: correlation kernel final scalar expression\] is given in Section \[section: reducing the size\], and relies on the fact that $\mathcal{M}$ is of genus $0$. The proof of Theorem \[thm:main\] is done via a saddle point analysis in Section \[section: saddle point analysis\], after considerable preparations have been carried out in Sections \[section: lozenge probabilities\]–\[section: phase functions\]:
- In Section \[section: lozenge probabilities\], we use Theorem \[thm: correlation kernel final scalar expression\] to find double contour formulas for the lozenges in terms of scalar OPs. We also use the symmetry in our model to conclude that it is sufficient to prove Theorem \[thm:main\] for the lower left quadrant of the liquid region.
- In Section \[section: steepest descent for $U$\], we will perform a Deift/Zhou [@DZ] steepest descent analysis on the RH problem for $U$. This analysis goes via a series of transformations $U \mapsto T \mapsto S \mapsto R$. The first transformation $U \mapsto T$ uses a so-called $g$-function which is obtained in Section \[section: g-function\].
- In Section \[section: phase functions\], we study the level set $\mathcal{N}_{\Phi} = \{ \zeta \in \mathbb{C}: {\text{\upshape Re\,}}\Phi(\zeta) = {\text{\upshape Re\,}}\Phi(s) \}$, which is of crucial importance to find the contour deformations that we need to consider for the saddle point analysis.
As mentioned in Remark \[rem:alpha is not 1\], we will always assume that $\alpha \in (0,1)$, even though it will not be written explicitly.
Proofs of Propositions \[prop: frozen\], \[prop:saddle\], \[prop:hightemp\] \[item a in prop mapping s\] and \[prop: s on the Riemann surface\] {#section: easy proofs}
===============================================================================================================================================
Proof of Proposition \[prop: frozen\] (the model is frozen as $\alpha \to 0$) {#section: model is frozen as alpha to 0}
-----------------------------------------------------------------------------
Proposition \[prop: frozen\] can be proved in a straightforward manner by considering successive maximization problems that can be solved simply by an inspection of $\mathcal{G}_{n}$. We assume that $n = 2N$ for a certain positive integer $N$ (the proof for $n$ odd is slightly different and we omit it). Let $\mathcal{V}_{2N}$ be the set of vertices that belong to $\mathcal{G}_{2N}$. It can be explicitly written as $$\begin{aligned}
\mathcal{V}_{2N} = \mathcal{H}_{2N} \cap \big( \mathbb{Z} \times (\tfrac{1}{2}+\mathbb{Z}) \big).\end{aligned}$$ The bottom left vertex has coordinates $(0,\frac{1}{2})$ and the top right vertex is $(4N,4N-\frac{1}{2})$. We first consider the problem of finding non-intersecting paths[^2] $$\begin{aligned}
\mathfrak{p}_{j}^{(0,2)} : \{0,1,2\} \to \mathcal{V}_{2N}, \qquad j = 0,\ldots,2N-1,\end{aligned}$$ which maximize the product $\prod_{j=0}^{2N-1} w_{\smash{\mathfrak{p}_{j}^{(0,2)}}}$. This problem can be directly solved by an inspection of $\mathcal{G}_{2N}$ (see Figure \[fig: 2x2 periodic weightings\] for $\mathcal{G}_{4}$). Its solution is unique, given by $$\begin{aligned}
\mathfrak{p}_{j}^{(0,2)}(0) = \mathfrak{p}_{j}^{(0,2)}(1) = j + \frac{1}{2}, \qquad \mathfrak{p}_{j}^{(0,2)}(2) = j + \frac{3}{2}, \qquad j = 0,\ldots,2N-1,\end{aligned}$$ and satisfies $\prod_{j=0}^{2N-1} w_{\smash{\mathfrak{p}_{j}^{(0,2)}}} = \alpha^{N}$. Figure \[fig: proof frozen\] (left) illustrates the solution for $N = 3$.
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(3,5.5) circle (0.5mm); (3.5,5.5) circle (0.5mm); (4,5.5) circle (0.5mm); (4.5,5.5) circle (0.5mm); (5,5.5) circle (0.5mm); (5.5,5.5) circle (0.5mm); (6,5.5) circle (0.5mm);
(0,0)–(0.5,0)–(1,0.5); (0,0.5)–(0.5,0.5)–(1,1)–(1.5,1)–(2,1.5); (0,1)–(0.5,1)–(1,1.5)–(1.5,1.5)–(2,2)–(2.5,2)–(3,2.5); (0,1.5)–(0.5,1.5)–(1,2)–(1.5,2)–(2,2.5)–(2.5,2.5)–(3,3); (0,2)–(0.5,2)–(1,2.5)–(1.5,2.5)–(2,3); (0,2.5)–(0.5,2.5)–(1,3);
(6,3.5)–(5.5,3)–(5,3); (6,4)–(5.5,3.5)–(5,3.5)–(4.5,3)–(4,3); (6,4.5)–(5.5,4)–(5,4)–(4.5,3.5)–(4,3.5); (6,5)–(5.5,4.5)–(5,4.5);
(1,0.5)–(3.5,0.5)–(6,3); (2,1.5)–(3.5,1.5)–(5,3); (3,2.5)–(3.5,2.5)–(4,3); (3,3)–(3.5,3.5)–(4,3.5); (2,3)–(3.5,4.5)–(5,4.5); (1,3)–(3.5,5.5)–(6,5.5);
Next, we consider the problem of finding non-intersecting paths $$\begin{aligned}
\mathfrak{p}_{j}^{(2,4)} : \{2,3,4\} \to \mathcal{V}_{2N}, \qquad j = 0,\ldots,2N-1,\end{aligned}$$ which maximize the product $\prod_{j=0}^{2N-1} w_{\smash{\mathfrak{p}_{j}^{(2,4)}}}$ (without the constraint that $\mathfrak{p}_{j}^{(2,4)}(2) = \mathfrak{p}_{j}^{(0,2)}(2)$). This problem can again be solved by a direct inspection of $\mathcal{G}_{2N}$. There is not a unique solution; we note that there are $4$ choices for $\mathfrak{p}_{0}^{(2,4)}$ and $4$ choices for $\mathfrak{p}_{2N-1}^{(2,4)}$. All solutions of this problem satisfy $w_{\smash{\mathfrak{p}_{0}^{(2,4)}}} = w_{\smash{\mathfrak{p}_{2N-1}^{(2,4)}}} = 1$ and $$\begin{aligned}
\label{lol34}
& \mathfrak{p}_{j}^{(2,4)}(2) = \mathfrak{p}_{j}^{(2,4)}(3) = j + \frac{3}{2}, \qquad \mathfrak{p}_{j}^{(2,4)}(4) = j + \frac{5}{2}, \qquad j=1,\ldots,2N-2.\end{aligned}$$ Furthermore, $\prod_{j=0}^{2N-1} w_{\smash{\mathfrak{p}_{j}^{(2,4)}}} = \prod_{j=1}^{2N-2} w_{\smash{\mathfrak{p}_{j}^{(2,4)}}} = \alpha^{N-1}$. The paths are represented in Figure \[fig: proof frozen\] (middle). More generally, given $k \in \{1,\ldots,2N\}$, the systems of non-intersection paths $$\begin{aligned}
\label{non-intersecting path level k:frozen}
& \mathfrak{p}_{j}^{(2k-2,2k)} : \{2k-2,2k-1,2k\} \to \mathcal{V}_{2N}, \qquad j = 0,\ldots,2N-1,\end{aligned}$$ which maximize the product $\prod_{j=0}^{2N-1} w_{\smash{\mathfrak{p}_{j}^{(2k-2,2k)}}}$ are all such that $$\begin{aligned}
\label{lol6}
\mathfrak{p}_{j}^{(2k-2,2k)}(2k-2) = \mathfrak{p}_{j}^{(2k-2,2k)}(2k-1) = j + \frac{2k-1}{2}, \qquad \mathfrak{p}_{j}^{(2k-2,2k)}(2k) = j + \frac{2k+1}{2},\end{aligned}$$ where is valid for $$\begin{aligned}
\label{indice lol6}
\begin{cases}
j=k-1,\ldots,2N-k, & \mbox{if } k \in \{1,2,\ldots,N\}, \\
j = 2N+1-k,\ldots,k-2, & \mbox{if } k \in \{N+2,N+3,\ldots,2N\},
\end{cases}\end{aligned}$$ and we have $$\begin{aligned}
\prod_{j=0}^{2N-1} w_{\mathfrak{p}_{j}^{(2k-2,2k)}} = \begin{cases}
\prod_{j=k-1}^{2N-k} w_{\mathfrak{p}_{j}^{(2k-2,2k)}} = \alpha^{N+1-k}, & \mbox{if } k \in \{1,2,\ldots,N\}, \\
\prod_{j=2N+1-k}^{k-2} w_{\mathfrak{p}_{j}^{(2k-2,2k)}} = \alpha^{k-1-N}, & \mbox{if } k \in \{N+2,N+3,\ldots,2N\}.
\end{cases}\end{aligned}$$ The paths – are again represented in Figure \[fig: proof frozen\] (middle). To summarize, we have solved $2N$-consecutive maximization problems, and all solutions $\{\mathfrak{p}_{j}^{(2k-2,2k)}\}_{j=0,...,2N-1}^{k=1,...,2N}$ satisfy and have the maximal product $$\begin{aligned}
\prod_{k=1}^{2N}\prod_{j=0}^{2N-1} w_{\mathfrak{p}_{j}^{(2k-2,2k)}} = \prod_{k=1}^{N} \alpha^{N+1-k} \prod_{k=N+2}^{2N} \alpha^{k-1-N} = \alpha^{N^2}.\end{aligned}$$ Since the set of all $\{\mathfrak{p}_{j}^{(2k-2,2k)}\}_{j=0,...,2N-1}^{k=1,...,2N}$ is larger than the set of all non-intersecting paths $\{\mathfrak{p}_{j}:\{0,1,...,4N\}\to \mathcal{V}_{2N}\}_{j=0,...2N-1}$, it follows that $$\begin{aligned}
\max_{\mathcal{T}}\mathrm{W}(\mathcal{T}) \leq \alpha^{N^2}.\end{aligned}$$ Furthermore, if there exists a tiling $\mathcal{T}_{\max}$ satisfying $\mathrm{W}(\mathcal{T}_{\max}) = \alpha^{N^2}$, then by the associated non-intersecting paths $\{\mathfrak{p}_{j}^{\star}:\{0,1,...,4N\}\to \mathcal{V}_{2N}\}_{j=0,...2N-1}$ necessarily satisfy $$\begin{aligned}
\label{lol7}
\mathfrak{p}_{j}^{\star}(2k-2) = \mathfrak{p}_{j}^{\star}(2k-1) = j + \frac{2k-1}{2}, \qquad \mathfrak{p}_{j}^{\star}(2k) = j + \frac{2k+1}{2},\end{aligned}$$ for $j,k$ as in . The existence and uniqueness of $\{\mathfrak{p}_{j}^{\star}\}_{j=0,...,2N-1}$ (and hence, of $\mathcal{T}_{\max}$) follow after solving successively $2N$ maximization problems (see also the middle and right parts of Figure \[fig: proof frozen\]):
1. \[itemm 1\] By , $\mathfrak{p}_{N}^{\star}(2N) = 2N+\frac{1}{2}$ and $\mathfrak{p}_{N}^{\star}(2N+2) = 2N+\frac{3}{2}$, and therefore we must have $\mathfrak{p}_{N}^{\star}(2N+1) \in \{2N+\frac{1}{2},2N+\frac{3}{2}\}$. Recalling the weightings , the choice that maximizes $w_{\smash{\mathfrak{p}_{N}^{\star}}}$ is $\mathfrak{p}_{N}^{\star}(2N+1)=2N+\frac{3}{2}$. Note that the two edges that have been added are $$\begin{aligned}
\big((2N,\mathfrak{p}_{N}^{\star}(2N)),(2N+1,\mathfrak{p}_{N}^{\star}(2N+1))\big) \quad\mbox{and}\quad \big((2N+1,\mathfrak{p}_{N}^{\star}(2N+1)),(2N+2,\mathfrak{p}_{N}^{\star}(2N+2))\big),\end{aligned}$$ and both have weight $1$.
2. Again by , we have $\mathfrak{p}_{N+1}^{\star}(2N-2) = 2N+\frac{1}{2}$ and $\mathfrak{p}_{N+1}^{\star}(2N+4) = 2N+\frac{7}{2}$. Taking into account that $\mathfrak{p}_{N+1}^{\star}$ does not intersect $\mathfrak{p}_{N}^{\star}$ (found in \[itemm 1\]), it follows from that there is a unique set $\{\mathfrak{p}_{N+1}(2N+j)\}_{j=-1}^{3}$ such that the path $\mathfrak{p}_{N+1}^{\star}$ maximizes $w_{\smash{\mathfrak{p}_{N+1}^{\star}}}$, which is given by $$\begin{aligned}
& \mathfrak{p}_{N+1}^{\star}(2N+j)=2N+j+\frac{1}{2}, \quad j=-1,0,1 \\
& \mathfrak{p}_{N+1}^{\star}(2N+j)=2N+2+\frac{1}{2}, \quad j=2,3.\end{aligned}$$ This determines uniquely $\mathfrak{p}_{N+1}^{\star}$, and we again verify that all the added edges have weight $1$.
3. \[itemm 3\] Similarly, for the successive values of $k=N+2,...,2N-1$, we consider the problem of maximizing $w_{\smash{\mathfrak{p}_{k}^{\star}}}$ among all paths $\mathfrak{p}_{k}^{\star}$ satisfying and not intersecting $\mathfrak{p}_{k-1}^{\star}$. As in the previous steps, we conclude that there is a unique solution, and such that all the added edges have weight $1$.
4. In a similar way as in the steps \[itemm 1\]-\[itemm 3\], for the successive values of $k=N-1,N-2,...,0$, we find that there is a unique way of finding a $\mathfrak{p}_{k}^{\star}$ which maximizes $w_{\smash{\mathfrak{p}_{k}^{\star}}}$, such that it satisfies and does not intersect $\mathfrak{p}_{k+1}^{\star}$. Again, we find that all the added edges have weight $1$.
Proof of Proposition \[prop:saddle\] {#subsection: proof of prop 3.2}
------------------------------------
By , the saddles are the zeros of the polynomial $M$ given by $$\begin{gathered}
M(\zeta) = (\zeta -r_{1})^{2}(\zeta -r_{2})^{2}(\zeta -r_{3})^{2}(\zeta -r_{+})(\zeta -r_{-}) - \\
\Big[ (\zeta-1)(\zeta +1)(\zeta-\alpha c)(\zeta-\tfrac{\alpha}{c})\eta - (\zeta-\alpha)(\zeta+\alpha)(\zeta-c)(\zeta-\tfrac{1}{c})\xi \Big]^{2}.\end{gathered}$$ Since the coefficients of $M$ are real, Proposition \[prop:saddle\] follows if $M$ has at least $6$ zeros on the real line. This can be proved by a direct inspection of the values of $M(\zeta)$ at $\zeta = -\infty,r_{1},0,\alpha c, r_{2}, \frac{\alpha}{c}, c, r_{3}, c^{-1},+\infty$: $$\begin{aligned}
& M(r_{1}) = - \frac{\alpha}{c^{2}}(1-\alpha)^{2}(c+\sqrt{\alpha})^{2} (\alpha c + \sqrt{\alpha})^{2} (\eta + \xi)^{2}, \qquad M(0) = \alpha^{4} (1-(\eta-\xi)^{2}), \\
& M(\alpha c) = (1-\alpha)^{8} c^{8}(1-\xi^{2}), \hspace{1.1cm} M(r_{2}) = - \frac{\alpha (1-\alpha)^{10}c^{8}}{(c+\sqrt{\alpha})^{8}}\big( (1+\alpha^{2})c+\sqrt{\alpha}(1+\alpha) \big)^{2}(\xi-2\eta)^{2}, \\
& M(\alpha c^{-1}) = \alpha^{4}(1-\alpha)^{8}(1-\xi^{2}), \qquad M(c) = (1-\alpha)^{8} c^{8} (1-\eta^{2}), \\
& M(r_{3}) = -\frac{\alpha (1-\alpha)^{10}c^{8}}{(\alpha c + \sqrt{\alpha})^{8}}\big( (1+\alpha^{2})c + \sqrt{\alpha}(1+\alpha) \big)^{2}(\eta - 2 \xi)^{2}, \qquad M(c^{-1}) = \frac{(1-\alpha)^{8}}{\alpha^{4}}(1-\eta^{2}).\end{aligned}$$ Since $(\xi,\eta) \in \mathcal{H}^{\mathrm{o}}$, where $$\begin{aligned}
\mathcal{H}^{\mathrm{o}}= \left\{(\xi,\eta) \mid -1 < \xi < 1, \ -1 < \eta < 1,\ -1 < \eta-\xi < 1 \right\},\end{aligned}$$ the leading coefficients of $M$ is $1-(\xi-\eta)^{2}>0$. We conclude the following:
1. if $\eta \neq -\xi$, $M$ has at least one simple root on $(-\infty,r_{1})$ and at least one simple root on $(r_{1},0)$,
2. if $\eta \neq \frac{\xi}{2}$, $M$ has at least one simple root on $(\alpha c,r_{2})$ and at least one simple root on $(r_{2},\frac{\alpha}{c})$,
3. if $\eta \neq 2\xi$, $M$ has at least one simple root on $(c,r_{3})$ and at least one simple root on $(r_{3},\frac{1}{c})$.
Finally, other computations show that $M'(r_{1}) = 0$ if $\eta = -\xi$, that $M'(r_{2}) = 0$ if $\eta = \frac{\xi}{2}$ and that $M'(r_{3}) = 0$ if $\eta = 2 \xi$. So $M$ has at least $6$ real zeros (counting multiplicities) for each $(\xi,\eta) \in \mathcal{H}^{\mathrm{o}}$.
Proof of Propositions \[prop:hightemp\] and \[prop: s on the Riemann surface\]
------------------------------------------------------------------------------
We start with the proof of Proposition \[prop: s on the Riemann surface\]. By rearranging the terms in , we see that the saddles are the solutions to $$\begin{aligned}
\left[ \frac{1}{2\zeta} - \frac{1}{2}\left( \frac{1}{\zeta - \alpha c} + \frac{1}{\zeta - \alpha c^{-1}} \right) \right]\xi + \left[ -\frac{1}{2\zeta} + \frac{1}{2}\left( \frac{1}{\zeta-c} + \frac{1}{\zeta-c^{-1}} \right) \right]\eta = w,\end{aligned}$$ where $w$ satisfies $w^{2} = \mathcal{Q}(\zeta)$. This can be rewritten as $$\begin{aligned}
\label{lol23}
\frac{-(\zeta - \alpha)(\zeta +\alpha)(\zeta -c)(\zeta - \frac{1}{c})}{(\zeta - \alpha c)(\zeta - \frac{\alpha}{c})(\zeta - 1)(\zeta + 1)}\xi + \eta = \frac{2\zeta (\zeta -c)(\zeta - \frac{1}{c})}{(\zeta - 1)(\zeta + 1)}w.\end{aligned}$$ Taking the real and imaginary parts of , and recalling that $\xi,\eta \in \mathbb{R}$, we get $$\begin{aligned}
\label{lol24}
\begin{pmatrix}
{\text{\upshape Re\,}}\left( \frac{-(\zeta - \alpha)(\zeta +\alpha)(\zeta -c)(\zeta - \frac{1}{c})}{(\zeta - \alpha c)(\zeta - \frac{\alpha}{c})(\zeta - 1)(\zeta + 1)} \right) & 1 \\
{\text{\upshape Im\,}}\left( \frac{-(\zeta - \alpha)(\zeta +\alpha)(\zeta -c)(\zeta - \frac{1}{c})}{(\zeta - \alpha c)(\zeta - \frac{\alpha}{c})(\zeta - 1)(\zeta + 1)} \right) & 0
\end{pmatrix}\begin{pmatrix}
\xi \\ \eta
\end{pmatrix} = \begin{pmatrix}
{\text{\upshape Re\,}}\left( \frac{2\zeta (\zeta -c)(\zeta - \frac{1}{c})}{(\zeta - 1)(\zeta + 1)}w \right) \\
{\text{\upshape Im\,}}\left( \frac{2\zeta (\zeta -c)(\zeta - \frac{1}{c})}{(\zeta - 1)(\zeta + 1)}w \right)
\end{pmatrix}.\end{aligned}$$ Since $$\begin{aligned}
\frac{-(\zeta - \alpha)(\zeta +\alpha)(\zeta -c)(\zeta - \frac{1}{c})}{(\zeta - \alpha c)(\zeta - \frac{\alpha}{c})(\zeta - 1)(\zeta + 1)} = -1 + \frac{a_{1}}{\zeta-1} + \frac{a_{2}}{\zeta + 1} + \frac{a_{3}}{\zeta - \alpha c} + \frac{a_{4}}{\zeta - \frac{\alpha}{c}},\end{aligned}$$ with $a_{1},a_{2},a_{3},a_{4}>0$, we have $$\begin{aligned}
\label{im is neg in diffeo}
{\text{\upshape Im\,}}\frac{-(\zeta - \alpha)(\zeta +\alpha)(\zeta -c)(\zeta - \frac{1}{c})}{(\zeta - \alpha c)(\zeta - \frac{\alpha}{c})(\zeta - 1)(\zeta + 1)} < 0, \quad \mbox{ for } {\text{\upshape Im\,}}\zeta > 0.\end{aligned}$$ Thus, the $2\times 2$ matrix at the left-hand-side of is invertible, and we get . This shows that $(\xi,\eta) \mapsto \big(s(\xi,\eta;\alpha),w(\xi,\eta;\alpha)\big)$ is a bijection from $\mathcal{L}_{\alpha}$ to $\mathcal R_{\alpha}^+$. This mapping is clearly differentiable, and therefore it is a diffeomorphism. Replacing $(s,w) \mapsto (s,-w)$ in the right-hand-side of , we see that the left-hand-side becomes $(\xi,\eta) \mapsto (-\xi,-\eta)$. This implies the symmetry $s(\xi,\eta;\alpha) = s(-\xi,-\eta;\alpha)$. It remains to prove that $(\xi,\eta) \in \mathcal{L}_{\alpha}^{l}$ is mapped to a point $\big(s(\xi,\eta;\alpha),w(\xi,\eta;\alpha)\big)$ lying in the upper half plane of the first sheet. The proof of this claim is splitted in the next two lemmas.
\[lemma: im is 0 for diffeo\] We have ${\text{\upshape Im\,}}\left(\frac{2\zeta(\zeta-c)(\zeta-c^{-1})}{(\zeta-1)(\zeta+1)}\mathcal{Q}(\zeta)^{1/2}\right) = 0$ if and only if $\zeta \in \mathbb{R}\cup \overline{\Sigma_{1}}$.
Consider the function $f$ defined by $$\begin{aligned}
f(\zeta) := \frac{(\zeta-r_{1})^{2}(\zeta-r_{2})^{2}(\zeta-r_{3})^{2}(\zeta-r_{+})(\zeta-r_{-})}{(\zeta-1)^{2}(\zeta+1)^{2}(\zeta-\alpha c)^{2}(\zeta-\alpha c^{-1})^{2}}.\end{aligned}$$ By the fundamental theorem, for each $x \in [0,+\infty)$, there are $8$ solutions $\zeta \in \mathbb{C}$ to $f(\zeta) = x$. The claim follows if we show that all these solutions lie on $\mathbb{R}\cup \Sigma_{1}$. First, note that the function $f$ is positive on the real line, has poles at $-1,\alpha c,\alpha c^{-1},1$, and zeros at $r_{1},r_{2},r_{3}$. Since $-1<r_{1}<\alpha c < r_{2}<\alpha c^{-1} < r_{3}<1$, the equation $f(\zeta) = x$ has at least $6$ real solutions (counting multiplicities) for each $x \in [0,+\infty)$. Furthermore, $f(\zeta) \to 1$ as $\zeta \to \pm \infty$, $f$ has a local minimum at $c^{-1} + R_{1}$, and $f(c^{-1}+R_{1}e^{it})<1$. Therefore, $f(\zeta) = x$ has $8$ solutions on $\mathbb{R}$ for each $x \in [f(c^{-1}+R_{1}),+\infty)$. It remains to show that there are two solutions on $\overline{\Sigma_{1}}$ whenever $x \in [0,f(c^{-1}+R_{1})]$. Writing $\zeta = c^{-1} + R_{1} e^{it} \in \gamma_{1}$, $t \in [-\pi,\pi]$, some computations show that $$\begin{aligned}
f(c^{-1} + R_{1}e^{it}) = \frac{2(\cos t-\cos \theta_{1} )\left( \cos t + \frac{\alpha^{2}+(2-\alpha)\sqrt{1-\alpha + \alpha^{2}}}{2(1-\alpha)} \right)^{2}\cos^{2}(\frac{t}{2})}{\left( \cos t + \frac{\sqrt{1-\alpha + \alpha^{2}}}{1-\alpha} \right)^{2} \left( \cos t + \frac{2-\alpha + \alpha^{2}}{2\sqrt{1-\alpha + \alpha^{2}}} \right)^{2}}.\end{aligned}$$ So $t \mapsto f(c^{-1} + R_{1}e^{it})$ is even, positive and decreases from $f(c^{-1}+R_{1})$ to $0$ as $t$ increases from $0$ to $\theta_{1}$, which finishes the proof.
\[lemma: proof of part (a)\] Let $(s,w) \in \mathcal{R}_{\alpha}^{+}$ such that $w = \mathcal{Q}(s)^{1/2}$ (i.e. $(s,w)$ is in the first sheet). Then, $\xi = \xi(s,w;\alpha) < 0$.
Using together with , we infer that $\xi$ has the same sign as $$\label{eq:signofxi}
- {\text{\upshape Im\,}}\left(\frac{2s(s-c)(s-c^{-1})}{(s-1)(s+1)}w\right).$$ By Lemma \[lemma: im is 0 for diffeo\], is $0$ if and only if $s \in \overline{\Sigma_{1}}$, which implies that the sign of is constant for $s \in \mathbb{C}^{+}\setminus \overline{\Sigma_{1}}$. From the expansion $$\begin{aligned}
-\frac{2s(s-c)(s-c^{-1})}{(s-1)(s+1)}w = 1 + \frac{a}{s} + \bigO(s^{-2}), \qquad \mbox{as } s \to \infty,\end{aligned}$$ where $a>0$, we conclude that is negative for all $s$ sufficiently large and lying in $\mathbb{C}^{+}$, and the claim follows.
This finishes the proof of Proposition \[prop: s on the Riemann surface\] and Proposition \[prop:hightemp\] \[item a in prop mapping s\]. In principle, it is also possible to use to prove parts \[item b in prop mapping s\]–\[item f in prop mapping s\] of Proposition \[prop:hightemp\], but it leads to more involve analyses. However, by rearranging the terms in , we can find other expressions than for the mapping $(s,w) \mapsto (\xi,\eta)$ that lead to simpler proofs of \[item b in prop mapping s\]–\[item f in prop mapping s\]. We only sketch the proof of \[item e in prop mapping s\]. First, we rewrite as $$\begin{aligned}
\xi + \frac{-(\zeta-1)(\zeta+1)(\zeta-\alpha c)(\zeta-\alpha c^{-1})}{(\zeta-\alpha)(\zeta+\alpha)(\zeta-c)(\zeta-c^{-1})}\eta = \frac{-2\zeta(\zeta-\alpha c)(\zeta-\alpha c^{-1})}{(\zeta-\alpha)(\zeta+\alpha)}w,\end{aligned}$$ which implies $$\begin{aligned}
\begin{pmatrix}
1 & {\text{\upshape Re\,}}\left( \frac{-(\zeta-1)(\zeta+1)(\zeta-\alpha c)(\zeta-\alpha c^{-1})}{(\zeta-\alpha)(\zeta+\alpha)(\zeta-c)(\zeta-c^{-1})} \right) \\
0 & {\text{\upshape Im\,}}\left( \frac{-(\zeta-1)(\zeta+1)(\zeta-\alpha c)(\zeta-\alpha c^{-1})}{(\zeta-\alpha)(\zeta+\alpha)(\zeta-c)(\zeta-c^{-1})} \right)
\end{pmatrix} \begin{pmatrix}
\xi \\ \eta
\end{pmatrix} = \begin{pmatrix}
{\text{\upshape Re\,}}\left( \frac{-2\zeta(\zeta-\alpha c)(\zeta-\alpha c^{-1})}{(\zeta-\alpha)(\zeta+\alpha)}w \right) \\
{\text{\upshape Im\,}}\left( \frac{-2\zeta(\zeta-\alpha c)(\zeta-\alpha c^{-1})}{(\zeta-\alpha)(\zeta+\alpha)}w \right)
\end{pmatrix}.\end{aligned}$$ Next, we verify that $$\begin{aligned}
{\text{\upshape Im\,}}\left( \frac{-(\zeta-1)(\zeta+1)(\zeta-\alpha c)(\zeta-\alpha c^{-1})}{(\zeta-\alpha)(\zeta+\alpha)(\zeta-c)(\zeta-c^{-1})} \right)>0, \qquad \mbox{for } {\text{\upshape Im\,}}\zeta > 0,\end{aligned}$$ which implies that $\eta = \eta(\zeta,w;\alpha)$ has the same sign as $$\begin{aligned}
{\text{\upshape Im\,}}\left( \frac{-2\zeta(\zeta-\alpha c)(\zeta-\alpha c^{-1})}{(\zeta-\alpha)(\zeta+\alpha)}w \right).\end{aligned}$$ Finally, in a similar way as in Lemma \[lemma: im is 0 for diffeo\], we show that this quantity is $0$ if and only if $\zeta \in \mathbb{R}\cup \Sigma_{\alpha}$, which proves part \[item e in prop mapping s\]. We omit the proofs of parts \[item b in prop mapping s\], \[item c in prop mapping s\], \[item d in prop mapping s\] and \[item f in prop mapping s\].
Analysis of the RH problem for $Y$ {#section: first steps of the steepest descent of Y}
==================================
In order to describe the behavior of $Y$ as $N \to + \infty$, one needs to control the $2 \times 2$ upper right block of the jumps, which is $A(z)^{2N}z^{-2N}$. To do this, we follow an idea of Duits and Kuijlaars [@DK] and proceed with the eigendecomposition of $A$. Then, we use this factorization to perform a first transformation $Y \mapsto X$ on the RH problem.
Eigendecomposition of $A$
-------------------------
The matrix $A(z)$ defined in has the following eigenvalues $$\label{eq:lambda12}
\lambda_{1,2}(z) = \frac{1+\alpha^{2}}{2}(1+z) \pm \frac{1-\alpha^{2}}{2}\sqrt{(z-z_{+})(z-z_{-})}, \qquad z \in \mathbb{C}\setminus [z_{-},z_{+}],$$ where the $+$ and $-$ signs read for $\lambda_{1}$ and $\lambda_{2}$, respectively, and $z_{+}$ and $z_{-}$ are given by $$\begin{aligned}
z_\pm = \frac{-(1+\alpha^{2}) \pm 2 \sqrt{\alpha(1-\alpha + \alpha^{2})}}{(1-\alpha)^{2}},\end{aligned}$$ and satisfy $z_{-}<-1<z_{+}<0$ and $z_{+}z_{-} = 1$. We define the square root $\sqrt{(z-z_{+})(z-z_{-})}$ such that it is analytic in $\mathbb{C}\setminus [z_{-},z_{+}]$, with an asymptotic behavior at $\infty$ given by $$\begin{aligned}
\sqrt{(z-z_{+})(z-z_{-})} = z + \bigO(1), \qquad \mbox{as } z \to \infty.\end{aligned}$$ The eigenvectors of $A$ are in the columns of the following matrix: $$\begin{aligned}
& E(z) = \frac{1}{1+\alpha} \begin{pmatrix} 1+\alpha & 1+\alpha \\
\lambda_1(z) - (\alpha^{2}+z) & \lambda_2(z) - (\alpha^{2}+z)
\end{pmatrix} \label{def of E} \\
& = \begin{pmatrix}
1 & 1 \\
\frac{1-\alpha}{2}\big( 1-z+\sqrt{(z-z_{+})(z-z_{-})} \big)
& \frac{1-\alpha}{2}\big( 1-z-\sqrt{(z-z_{+})(z-z_{-})} \big) \nonumber
\end{pmatrix},\end{aligned}$$ and we have the factorization $$\label{eigenvalue eigenvector decomposition of A}
A(z) = E(z) \Lambda(z) E(z)^{-1},$$ where $\Lambda(z) = \operatorname{diag}(\lambda_{1}(z),\lambda_{2}(z))$ is the matrix of eigenvalues. The matrix $E(z)$ is analytic for $z \in \mathbb C \setminus [z_{-},z_{+}]$, and satisfies $$\begin{aligned}
& E_{+}(z) = E_{-}(z)\sigma_{1}, & & z \in (z_{-},z_{+}), \label{jumps for E} \\
& E(z) = \begin{pmatrix}
1 & 1 \\
\frac{1-\alpha+\alpha^{2}}{1-\alpha}+\bigO(z^{-1}) & -(1-\alpha) z + \bigO(1)
\end{pmatrix} & & \mbox{ as } z \to \infty, \label{asymp for E}\end{aligned}$$ where $\sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$.
First transformation $Y \mapsto X$
----------------------------------
The first transformation of the RH problem diagonalizes the $2 \times 2$ upper right block of the jumps, and is defined by $$\label{def X}
X(z) = Y(z) \begin{pmatrix} E(z) & 0_2 \\ 0_2 & E(z) \end{pmatrix}.$$
By standard arguments [@Deift], we have $\det Y \equiv 1$. Note however that the $Y \mapsto X$ transformation does not preserve the unit determinant. Indeed, since $\det E(z) = -(1-\alpha)\sqrt{(z-z_{+})(z-z_{-})}$, we have $\det X(z) = (1-\alpha)^{2}(z-z_{+})(z-z_{-})$.
Using the jumps for $E$ given by , we verify that $X$ satisfies the following RH problem.
### RH problem for $X$ {#rh-problem-for-x .unnumbered}
- $X : \mathbb{C}\setminus (\gamma\cup [z_{-},z_{+}]) \to \mathbb{C}^{4\times 4}$ is analytic, where we recall that $\gamma$ is a close contour surrounding $0$ once in the positive direction.
- The jumps for $X$ are given by $$\begin{aligned}
& X_{+}(z) = X_{-}(z) \begin{pmatrix}
I_{2} & \frac{\Lambda^{2N}(z)}{z^{2N}} \\ 0_{2} & I_{2}
\end{pmatrix}, & & \mbox{ for } z \in \gamma \setminus \mathcal{Z}, \\
& X_{+}(z) = X_{-}(z) \begin{pmatrix}
\sigma_{1} & 0_{2} \\ 0_{2} & \sigma_{1}
\end{pmatrix}, & & \mbox{ for } z \in (z_{-},z_{+})\setminus \mathcal{Z},\end{aligned}$$ where $\mathcal{Z} := \gamma \cap [z_{-},z_{+}]$. Depending on $\gamma$, $\mathcal{Z}$ can be the empty set, a finite set, or and infinite set. If $\mathcal{Z}$ contains of one or several intervals, then on these intervals the jumps are $$\begin{aligned}
& X_{+}(z) = X_{-}(z) \begin{pmatrix}
\sigma_{1} & 0_{2} \\ 0_{2} & \sigma_{1}
\end{pmatrix}\begin{pmatrix}
I_{2} & \frac{\Lambda^{2N}(z)}{z^{2N}} \\ 0_{2} & I_{2}
\end{pmatrix}.\end{aligned}$$
- As $z \to \infty$, we have $X(z) = \left(I_{4} + \bigO(z^{-1})\right) \begin{pmatrix}
z^{N}E(z) & 0_{2} \\ 0_{2} & z^{-N}E(z)
\end{pmatrix}$.\
As $z \to z_{-}$ or as $z \to z_{+}$, $X(z) = \bigO(1)\begin{pmatrix} E(z) & 0_2 \\ 0_2 & E(z) \end{pmatrix}$.
Proof of Theorem \[thm: correlation kernel final scalar expression\] {#section: reducing the size}
====================================================================
First, we use the factorization of $A$ obtained in together with the transformation $Y \mapsto X$ given by , to rewrite the formulas – as follows $$\begin{aligned}
& \big[ K(2x+\epsilon_{x},2y+j,2x+\epsilon_{x},2y+i) \big]_{i,j=0}^{1} = \frac{1}{(2\pi i)^{2}}\int_{\gamma}\int_{\gamma} \begin{pmatrix}
\alpha^{2} & \alpha \\ w & 1
\end{pmatrix}^{\epsilon_{x}} \label{kernel X} \\ & \hspace{2cm} \times E(w)\frac{\Lambda(w)^{2N-x-\epsilon_{x}}}{w^{2N-y}}\mathcal{R}^{X}(w,z)\frac{\Lambda(z)^{x}}{z^{y+1}}E(z)^{-1} \begin{pmatrix}
1 & 1 \\ \alpha z & 1
\end{pmatrix}^{\epsilon_{x}} dzdw, \nonumber\end{aligned}$$ where $\mathcal{R}^{X}$ is given by $$\label{def of Rcal X}
\mathcal{R}^{X}(w,z) = E^{-1}(w)\mathcal{R}^{Y}(w,z)E(z) = \frac{1}{z-w} \begin{pmatrix}
0_{2} & I_{2}
\end{pmatrix}X^{-1}(w)X(z) \begin{pmatrix}
I_{2} \\ 0_{2}
\end{pmatrix}.$$ The property of $\mathcal{R}^{Y}$ implies the following reproducing property for $\mathcal{R}^{X}$: $$\label{reproducing property X}
\frac{1}{2\pi i} \int_{\gamma} P(w)E(w) \frac{\Lambda(w)^{2N}}{w^{2N}}\mathcal{R}^{X}(w,z)dw = P(z)E(z),$$ for every $2 \times 2$ matrix valued polynomial $P$ of degree $\leq N-1$.
Now, we introduce the Riemann surface $\mathcal{M}$ associated to the eigenvalues and eigenvectors of $A$. This Riemann surface is of genus $0$ and therefore there is a one-to-one map between it and the Riemann sphere (called the $\zeta$-plane).
The Riemann surface $\mathcal{M}$ and the $\zeta$-plane {#subsection: 5.1}
-------------------------------------------------------
The Riemann surface $\mathcal{M}$ is defined by $$\mathcal{M} = \{(z,y)\in \mathbb{C}\times \mathbb{C}: y^2 = (z-z_{+})(z-z_{-}) \},$$ and has genus zero. We represent it as a two-sheeted covering of the $z$-plane glued along $[z_{-},z_{+}]$. On the first sheet we require $y = z + \bigO(1)$ as $z \to \infty$, and on the second sheet we require $y = -z+ \bigO(1)$ as $z \to \infty$. To shorten the notations, a point $(z,y)$ lying on the Riemann surface will simply be denoted by $z$ when there is no confusion, that is, we will omit the $y$-coordinate. If we want to emphasize that the point $(z,y)$ is on the $j$-th sheet, $j \in \{1,2\}$, then we will use the notation $z^{(j)}$. With this convention, the two points at infinity are denoted by $\infty^{(1)}$ and $\infty^{(2)}$. The function $y$ satisfies $$\begin{aligned}
& y(\tfrac{1}{\alpha^{(2)}}) = - \frac{1+\alpha^{2}}{\alpha(1-\alpha)}, & & y(\alpha^{(2)}) = - \frac{1+\alpha^{2}}{1-\alpha}, \label{y computation} \\
& y(0^{(2)}) = -1, & & y(0^{(1)}) = 1. \nonumber\end{aligned}$$ The functions $\lambda_{1}(z)$ and $\lambda_{2}(z)$ are defined on the $z$-plane, and together they define a function $\lambda$ on $\mathcal{M}$ as follows: $$\lambda\big((z,y)\big) = \left\{ \begin{array}{l l}
\lambda_{1}(z), & \mbox{if } (z,y) \mbox{ is on the first sheet}, \\
\lambda_{2}(z), & \mbox{if } (z,y) \mbox{ is on the second sheet}.
\end{array} \right.$$ This is a meromorphic function on $\mathcal{M}$ with two simple poles at $\infty^{(1)}$ and $\infty^{(2)}$ and no other poles. Using , we verify that $\lambda$ has two simple zeros at $\alpha^{(2)}$ and $\frac{1}{\alpha^{(2)}}$, and since $\mathcal{M}$ has genus $0$, $\lambda$ has no other zeros. From , the matrix $E$ can also be extended to the full Riemann surface as follows $$\begin{aligned}
E\big((z,y)\big) & = \begin{pmatrix}
1 & 1 \\
\frac{1-\alpha}{2}(1-z+y) & \frac{1-\alpha}{2}(1-z-y)
\end{pmatrix} \\
& = \begin{cases}
E(z), & \mbox{if }(z,y) \mbox{ is on the first sheet}, \\
E(z)\sigma_{1}, & \mbox{if } (z,y) \mbox{ is on the second sheet}.
\end{cases}\end{aligned}$$ The function $\zeta = \zeta(z)$ defined by $$\label{def of zeta map}
\zeta = \,\frac{2z -(z_{+}+z_{-}) + 2 y}{z_+-z_{-}},$$ is a conformal and bijective map from $\mathcal{M}$ to the Riemann sphere. The first sheet of $\mathcal{M}$ is mapped by to the subset $\{\zeta \in \mathbb{C}\cup\{\infty\}:|\zeta| > 1\}$ of the $\zeta$-plane, and the second sheet is mapped to $\{\zeta \in \mathbb{C}\cup\{\infty\}: |\zeta| < 1\}$. The inverse function $z=z(\zeta)$ is given by $$\label{def zeta}
z = \frac{z_{+}+z_{-}}{2}+\frac{z_{+}-z_{-}}{4}\left( \zeta + \zeta^{-1} \right),$$ where $z$ is on the first sheet if $|\zeta|>1$ and on the second sheet if $|\zeta|<1$. By definition, the above function $z(\zeta)$ vanishes at $\zeta(0^{(1)})$ and $\zeta(0^{(2)})$. Since it has simple poles at $\zeta = 0$ and $\zeta = \infty$, and since $z(\zeta) = \frac{z_{+}-z_{-}}{4}\zeta + \bigO(1)$ as $\zeta \to \infty$, can be rewritten as $$\label{z factorized in terms of zeta}
z = \frac{z_{+}-z_{-}}{4 \zeta}(\zeta-\zeta(0^{(1)}))(\zeta-\zeta(0^{(2)})).$$ The functions $z(\zeta)$ and $\zeta(z)$ satisfy $$\begin{aligned}
& z(1) = z_{+}, & & z(-1) = z_{-}, & & z(\infty) = \infty^{(1)}, & & z(0) = \infty^{(2)}, \\
& \zeta(z_{+}) = 1, & & \zeta(z_{-}) = -1, & & \zeta(\infty^{(1)}) = \infty, & & \zeta(\infty^{(2)}) = 0.\end{aligned}$$ Also, we note that as $z \in \mathcal{M}$, ${\text{\upshape Im\,}}z = 0$, $z \notin (z_{-},z_{+})$, follows the straight line segments $[\infty^{(1)},z_{-}]$, $[z_{-},\infty^{(2)}]$, $[\infty^{(2)},z_{+}]$, $[z_{+},\infty^{(1)}]$, the function $\zeta(z)$ increases from $-\infty$ to $+\infty$. In particular, we have $$\begin{aligned}
\zeta(z_{-}) < \zeta(\infty^{(2)}) < \zeta(\tfrac{1}{\alpha^{(2)}}) < \zeta(\alpha^{(2)}) < \zeta(0^{(2)}) < \zeta(z_{+}) < \zeta(0^{(1)}).\end{aligned}$$ The following identities will be useful later, and can be verified by direct computations: $$\begin{aligned}
& y = \frac{z_{+}-z_{-}}{4}\left( \zeta-\zeta^{-1} \right), \qquad \frac{dz}{y} = \frac{d\zeta}{\zeta}, \label{dz y in terms of zeta} \\
& \lambda = \frac{z_{+}-z_{-}}{4 \zeta} (\zeta - \zeta(\tfrac{1}{\alpha^{(2)}}))(\zeta-\zeta(\alpha^{(2)})), \label{lambda in terms of zeta} \\
& \frac{d\lambda}{dz} = \frac{\zeta^{2}- \alpha^{2}}{\zeta^2-1}, \label{lambda prime in terms of zeta} \\
& \frac{dz}{d\zeta} = \frac{z_{+}-z_{-}}{4\zeta}\left( \zeta-\zeta^{-1} \right). \label{dz_sur_dzeta in terms of zeta}\end{aligned}$$ We define $c$ by $$\begin{aligned}
c = \frac{z_{+}-z_{-}}{-(z_{+}+z_{-})+2\sqrt{z_{+}z_{-}}} = \sqrt{\frac{\alpha}{1-\alpha +\alpha^{2}}} < 1.\end{aligned}$$ From straightforward calculations using , we have $$\begin{aligned}
& \zeta(\tfrac{1}{\alpha^{(2)}}) = \alpha c, & & \zeta(\alpha^{(2)}) = \alpha c^{-1}, \\
& \zeta(0^{(2)}) = c, & & \zeta(0^{(1)}) = c^{-1},\end{aligned}$$ and $$\begin{aligned}
\label{lambda - alpha2 - z in terms of zeta}
\lambda(z) - \alpha^{2} - z = \frac{1+\alpha^{3}}{1-\alpha}\frac{\zeta -c}{\zeta}.\end{aligned}$$
The reproducing kernel $\mathcal{R}^{M}$ {#subsection: 5.2}
----------------------------------------
For $w^{(j)}$ on the $j$-th sheet of $\mathcal{M}$ and $z^{(k)}$ on the $k$-th sheet, we define $\mathcal{R}^{\mathcal{M}}(w^{(j)},z^{(k)})$ by $$\label{def of reproducing kernel M}
\mathcal{R}^{\mathcal{M}}(w^{(j)},z^{(k)}) = y(w^{(j)}) e_{j}^{T} \mathcal{R}^{X}(w,z) e_{k},$$ where $e_{1} = \begin{pmatrix}
1 \\ 0
\end{pmatrix}$ and $ e_{2} = \begin{pmatrix}
0 \\ 1
\end{pmatrix} $. Note that $\mathcal{R}^{\mathcal{M}} : \mathcal{M}_{*} \times \mathcal{M}_{*} \to \mathbb{C}$ is scalar valued, with $\mathcal{M}_{*} = \mathcal{M}\setminus\{\infty^{(1)},\infty^{(2)}\}$. It is convenient for us to consider formal sums of points on $\mathcal{M}$, which are called *divisors* in the literature. More precisely, a divisor $D$ can be written in the form $$\begin{aligned}
D = \sum_{j=1}^{k}n_{j}z_{j}, \qquad k \geq 1, \quad n_{j} \in \mathbb{Z}, \quad z_{j} \in \mathcal{M},\end{aligned}$$ and we say that $D \geq 0$ if $n_{1},\ldots,n_{k} \geq 0$. Let $f$ be a non-zero meromorphic function on $\mathcal{M}$, let $z_{1},\ldots,z_{k_{1}}$ be its zeros of multiplicities $n_{1},\ldots,n_{k_{1}}$, respectively, and let $z_{k_{1}+1},\ldots,z_{k_{2}}$ be its poles of order $n_{k_{1}+1},\ldots,n_{k_{2}}$, respectively. The divisor $$\begin{aligned}
\mbox{div}(f) := n_{1}z_{1}+...+n_{k_{1}}z_{k_{1}} - n_{k_{1}+1}z_{k_{1}+1} - ... - n_{k_{2}} z_{k_{2}}\end{aligned}$$ is called the divisor of $f$. Given a divisor $D$, we define $L(-D)$ as the vector space of meromorphic functions on $\mathcal{M}$ given by $$\begin{aligned}
L(-D) = \{ f: \mbox{div}(f) \geq -D \mbox{ or } f \equiv 0 \}.\end{aligned}$$ The following divisors will play an important role: $$\begin{aligned}
& D_{N} = (N-1) \cdot \infty^{(1)} + N \cdot \infty^{(2)}, \\
& D_{N}^{*} = N \cdot \infty^{(1)} + (N-1) \cdot \infty^{(2)}.\end{aligned}$$ Thus $L_{N} := L(-D_{N})$ is the vector space of meromorphic functions on $\mathcal{M}$, with poles at $\infty^{(1)}$ and $\infty^{(2)}$ only, such that the pole at $\infty^{(1)}$ is of order at most $N-1$, and the pole at $\infty^{(2)}$ is of order at most $N$. Similarly we define $L_{N}^{*} = L(-D_{N}^{*})$. From the Riemann-Roch theorem, we have $$\begin{aligned}
\mbox{dim } L_{N} = \mbox{dim } L_{N}^{*} = 2N,\end{aligned}$$ since there is no holomorphic differential (other than the zero differential) on a Riemann surface of genus $0$.
\[lemma: reproducing kernel Riemann surface\] We have
- $z \mapsto \mathcal{R}^{\mathcal{M}}(w,z) \in L_{N}$ for every $w \in \mathcal{M}_{*}$,
- $w \mapsto \mathcal{R}^{\mathcal{M}}(w,z) \in L_{N}^{*}$ for every $z \in \mathcal{M}_{*}$,
- $\mathcal{R}^{\mathcal{M}}$ is a reproducing kernel for $L_{N}$ in the sense that $$\label{reproducing property M}
\frac{1}{2\pi i} \int_{\gamma_{\mathcal{M}}} f(w) \frac{\lambda^{2N}(w)}{w^{2N}} \mathcal{R}^{\mathcal{M}}(w,z) \frac{dw}{y(w)} = f(z)$$ for every $f \in L_{N}$, where $\gamma_{\mathcal{M}}$ is a close contour surrounding once $0^{(1)}$ and $0^{(2)}$ on the Riemann surface $\mathcal{M}$ in the positive direction (in particular $\gamma_{\mathcal{M}}$ visits both sheets).
Using the definitions of $\mathcal{R}^{\mathcal{M}}$ and $\mathcal{R}^{X}$ given by and , we can rewrite $\mathcal{R}^{\mathcal{M}}$ as $$\label{kernel Riemann f and g}
\mathcal{R}^{\mathcal{M}}(w,z) = \frac{\sum_{j=1}^{4}g_{j}(w)f_{j}(z)}{z-w}, \qquad w,z \in \mathcal{M}_{*},$$ where $$f_{j}(z) = \left\{ \begin{array}{l l}
X_{j1}(z), & \mbox{if } z = z^{(1)}, \\
X_{j2}(z), & \mbox{if } z = z^{(2)},
\end{array} \right.$$ and $$\begin{aligned}
\label{def of gj in reducing}
g_{j}(w) = y(w) \left\{ \begin{array}{l l}
(X^{-1})_{3j}(w), & \mbox{if } w = w^{(1)}, \\
(X^{-1})_{4j}(w), & \mbox{if } w = w^{(2)}.
\end{array} \right.\end{aligned}$$ From properties (a) and (b) of the RH problem for $X$, the functions $f_{j}$ are analytic in $\mathcal{M}_{*}$. By combining the large $z$ asymptotics of $E(z)$ (given by ) with property (c) of the RH problem for $X$, we obtain $$\begin{aligned}
X(z) \begin{pmatrix}
I_{2} \\ 0_{2}
\end{pmatrix} = \begin{pmatrix}
\bigO(z^{N}) & \bigO(z^{N}) \\
\bigO(z^{N}) & \bigO(z^{N+1}) \\
\bigO(z^{N-1}) & \bigO(z^{N}) \\
\bigO(z^{N-1}) & \bigO(z^{N})
\end{pmatrix}, \quad \mbox{as } z \to \infty,\end{aligned}$$ from which we conclude that the functions $f_{j}$’s have poles of order at most $N$ at $\infty^{(1)}$ and at most $N+1$ at $\infty^{(2)}$. Therefore, we have shown that $$\begin{aligned}
f_{j} \in L(-(D_{N}+\infty^{(1)}+\infty^{(2)})), \qquad j = 1,2,3,4.\end{aligned}$$ The numerator in is, for each fixed $w \in \mathcal{M}_{*}$ a linear combination of the functions $f_{j}$, so belong to $L(-(D_{N}+\infty^{(1)}+\infty^{(2)}))$ as a function of $z$. By definitions of $\mathcal{R}^{\mathcal{M}}$ and $\mathcal{R}^{X}$, the numerator vanishes for $z = w^{(1)}$ and for $z = w^{(2)}$. Thus the division by $z-w$ in does not introduce any poles, but it reduces the order of the poles at $\infty^{(1)}$ and $\infty^{(2)}$ by one, and therefore $z \mapsto \mathcal{R}^{\mathcal{M}}(w,z) \in L_{N}$ as claimed in part (a). Now we turn to the proof of part (b). First, we note that $$\begin{aligned}
E(w)^{-1} = \frac{-\frac{1}{1-\alpha}}{\sqrt{(w-z_{+})(w-z_{-})}} \begin{pmatrix}
\frac{1-\alpha}{2}\big( 1-w-\sqrt{(w-z_{+})(w-z_{-})} \big) & -1 \\
- \frac{1-\alpha}{2}\big( 1-w+\sqrt{(w-z_{+})(w-z_{-})} \big) & 1
\end{pmatrix}.\end{aligned}$$ Therefore, since $\det Y \equiv 1$, by using condition (c) of the RH problem for $X$, we have $$\begin{aligned}
X^{-1}(w) = \begin{pmatrix}
E^{-1}(w) & 0_{2} \\
0_{2} & E^{-1}(w)
\end{pmatrix} \begin{pmatrix}
\bigO(1) & \bigO(1) \\
\bigO(1) & \bigO(1)
\end{pmatrix} \qquad \mbox{as } z \to z_{\star} \in \{z_{+},z_{-}\},\end{aligned}$$ and we conclude from that the functions $g_{j}$ are also analytic in $\mathcal{M}_{*}$. On the other hand, by using the asymptotics $Y(w)=I_{4}+\bigO(w^{-1})$ as $w \to \infty$ together with the fact that $\det Y \equiv 1$, we can obtain asymptotics for $X^{-1}(w)$ as $w \to \infty$ using . After some simple computations, we get $$y(w) \begin{pmatrix}
0_{2} & I_{2}
\end{pmatrix}X^{-1}(w) = \begin{pmatrix}
\bigO(w^{N}) & \bigO(w^{N}) & \bigO(w^{N+1}) & \bigO(w^{N}) \\
\bigO(w^{N-1}) & \bigO(w^{N-1}) & \bigO(w^{N}) & \bigO(w^{N})
\end{pmatrix},$$ from which it follows that $$\begin{aligned}
g_{j} \in L(-(D_{N}^{*}+\infty^{(1)}+\infty^{(2)})), \qquad j = 1,2,3,4.\end{aligned}$$ We conclude the proof of part (b) as in part (a), by noting that $\mathcal{R}^{\mathcal{M}}(w,z)$ in has no pole at $z=w$ (on any sheet). Finally, let us take $P(w) = p(w) e_{1}^{T} = p(w) \begin{pmatrix}
1 & 0
\end{pmatrix}$ in , with $p$ a scalar polynomial satisfying $\mbox{deg }p \leq N-1$. Since $e_{1}^{T} E(w) = \begin{pmatrix}
1 & 1
\end{pmatrix} = e_{1}^{T} + e_{2}^{T}$, it gives $$\begin{aligned}
p(z)\begin{pmatrix}
1 & 1
\end{pmatrix} = \frac{1}{2\pi i} \int_{\gamma}p(w)(e_{1}^{T}+e_{2}^{T}) \frac{\Lambda(w)^{2N}}{w^{2N}}\mathcal{R}^{X}(w,z)dw.\end{aligned}$$ By multiplying the above from the right by $e_{k}$, we obtain $$\begin{aligned}
p(z) = & \; \frac{1}{2\pi i} \int_{\gamma}p(w) \frac{\lambda_{1}(w)^{2N}}{w^{2N}}e_{1}^{T}\mathcal{R}^{X}(w,z)e_{k}dw + \frac{1}{2\pi i} \int_{\gamma}p(w) \frac{\lambda_{2}(w)^{2N}}{w^{2N}}e_{2}^{T}\mathcal{R}^{X}(w,z)e_{k}dw.\end{aligned}$$ We denote $\gamma^{(1)}$ and $\gamma^{(2)}$ for the projections of $\gamma$ on the first and second sheet of $\mathcal{M}$, respectively. Using , the above can be rewritten as $$\begin{aligned}
p(z) = & \; \frac{1}{2\pi i} \int_{\gamma^{(1)}}p(w) \frac{\lambda(w)^{2N}}{w^{2N}}\mathcal{R}^{\mathcal{M}}(w,z^{(k)})\frac{dw}{y(w)} + \frac{1}{2\pi i} \int_{\gamma^{(2)}}p(w) \frac{\lambda(w)^{2N}}{w^{2N}}\mathcal{R}^{\mathcal{M}}(w,z^{(k)})\frac{dw}{y(w)},\end{aligned}$$ for every $z \in \mathbb{C}$ and for any $k \in \{1,2\}$. The two integrals combine to one integral over a contour $\gamma_{\mathcal{M}}$ surrounding both $0^{(1)}$ and $0^{(2)}$ on $\mathcal{M}$ in the positive direction, and thus $$\begin{aligned}
\label{lol8}
p(z) = \frac{1}{2 \pi i} \int_{\gamma_{\mathcal{M}}}p(w) \frac{\lambda(w)^{2N}}{w^{2N}} \mathcal{R}^{\mathcal{M}}(w,z) \frac{dw}{y(w)}, \qquad \mbox{deg } p \leq N-1,\end{aligned}$$ for every $z \in \mathcal{M}_{*}$. Let us now take $P(w) = p(w)e_{2}^{T} = p(w) \begin{pmatrix}
0 & 1
\end{pmatrix}$ in , and note that $$e_{2}^{T}E(w) = \frac{1}{1+\alpha} \begin{pmatrix}
\lambda_{1}(w)-\alpha^{2}-w & \lambda_{2}(w)-\alpha^{2}-w
\end{pmatrix}.$$ The two above entries together define the meromorphic function $w \in \mathcal{M} \mapsto \frac{1}{1+\alpha}(\lambda(w)-\alpha^{2} - w)$ on $\mathcal{M}$. By proceeding in a similar way as for , we obtain this time $$\begin{aligned}
p(z)(\lambda(z)-(\alpha^{2}+z)) = \frac{1}{2 \pi i} \int_{\gamma_{\mathcal{M}}}p(w)(\lambda(w)-(\alpha^{2}+w)) \frac{\lambda(w)^{2N}}{w^{2N}} \mathcal{R}^{\mathcal{M}}(w,z) \frac{dw}{y(w)},\end{aligned}$$ for all scalar polynomials $p$ with $\mbox{deg }p \leq N-1$ and for all $z \in \mathcal{M}_{*}$. Therefore, for any function $f$ in the form $$\begin{aligned}
f(z) = p_{1}(z) + p_{2}(z)(\lambda(z)-\alpha^{2}-z)\end{aligned}$$ with $p_{1}$, $p_{2}$ two polynomials of degree $\leq N-1$, we have proved that $$\begin{aligned}
f(z) = \frac{1}{2 \pi i} \int_{\gamma_{\mathcal{M}}}f(w) \frac{\lambda(w)^{2N}}{w^{2N}} \mathcal{R}^{\mathcal{M}}(w,z) \frac{dw}{y(w)}.\end{aligned}$$ Let $L := \{f:f(z) = p_{1}(z) + p_{2}(z)(\lambda(z)-\alpha^{2}-z)\mbox{ with }p_{1},p_{2} \mbox{ two polynomials of degree } \leq N-1\}$. Since $z \mapsto \lambda-\alpha^{2}-z$ has a simple pole at $\infty^{(2)}$ (and no other poles), we conclude that $L \subseteq L_{N}$. Note also that $\mbox{dim } L = \mbox{dim } L_{N} = 2N$, and thus we have $L=L_{N}$. This finishes the proof.
The reproducing kernel $\mathcal{R}^{U}$ {#subsection: 5.3}
----------------------------------------
To ease the notations, we define $z = z(\zeta)$ and $w = w(\omega)$ by $$\begin{aligned}
& z = \frac{z_{+}+z_{-}}{2}+\frac{z_{+}-z_{-}}{4}\left( \zeta + \zeta^{-1} \right), \qquad \; \; \zeta \in \mathbb{C}\cup \{\infty\}, \; z \in \mathcal{M}, \label{z in terms of zeta lol} \\
& w = \frac{z_{+}+z_{-}}{2}+\frac{z_{+}-z_{-}}{4}\left( \omega + \omega^{-1} \right), \qquad \omega \in \mathbb{C}\cup \{\infty\}, \; w \in \mathcal{M}, \label{w in terms of omega lol}\end{aligned}$$ with the same convention as in , that is, $z$ (resp. $w$) is on the first if $|\zeta|>1$ (resp. $|\omega|>1$), and on the second sheet if $|\zeta|<1$ (resp. $|\omega|<1$). We define $\mathcal{R}^{U}$ in terms of $\mathcal{R}^{\mathcal{M}}$ as follows $$\label{def of Reproducing kernel C}
\mathcal{R}^{U}(\omega,\zeta) = \omega^{N-1}\zeta^{N}\mathcal{R}^{\mathcal{M}}(w(\omega),z(\zeta)).$$
\[prop: reprod Rcal U\] Let $W$ and $c$ be defined as in . $\mathcal{R}^{U}$ is a bivariate polynomial of degree $\leq 2N-1$ in both $\omega$ and $\zeta$. It satisfies $$\label{reproducing property C}
\frac{1}{2\pi i}\int_{\gamma_{\mathbb{C}}} p(\omega) W(\omega) \mathcal{R}^{U}(\omega,\zeta)d\omega = p(\zeta)$$ for every scalar polynomial $p$ of degree $\leq 2N-1$, where $\gamma_{\mathbb{C}}$ is a closed curve in the complex plane going around $c$ and $c^{-1}$ once in the positive direction, but not going around $0$.
From part (a) of Lemma \[lemma: reproducing kernel Riemann surface\], for each $w \in \mathcal{M}_{*}$, the function $z \mapsto \mathcal{R}^{\mathcal{M}}(w,z)$ is meromorphic on $\mathcal{M}$, with a pole of order at most $N-1$ at $\infty^{(1)}$ and a pole of order at most $N$ at $\infty^{(2)}$. Since $z(0) = \infty^{(2)}$ and $z(\infty) = \infty^{(1)}$, we conclude that for each $\omega \in \mathbb{C}$, the function $\zeta \mapsto \mathcal{R}^{\mathcal{M}}(w(\omega),z(\zeta))$ is meromorphic on $\mathbb{C}\cup \{\infty\}$, with a pole of order at most $N-1$ at $\infty$ and another pole of order at most $N$ at $0$. Therefore, for each $\omega \in \mathbb{C}$, the function $\zeta \mapsto \mathcal{R}^{U}(\omega,\zeta)$ is a polynomial of degree at most $2N-1$. From part (b) of Lemma \[lemma: reproducing kernel Riemann surface\], we conclude similarly that for each $\zeta \in \mathbb{C}$, the function $\omega \mapsto \mathcal{R}^{U}(\omega,\zeta)$ is a polynomial of degree at most $2N-1$. So we have proved that $\mathcal{R}^{U}$ is a bivariate polynomial of degree $\leq 2N-1$ in both $\omega$ and $\zeta$.
Now, we turn to the proof of . It can be directly verified from (see also ) that $\omega(0^{(1)}) = c^{-1}$, $\omega(0^{(2)})=c$, $(\partial_{\omega}w)(c^{-1})>0$ and $(\partial_{\omega}w)(c)<0$. In particular, the map $w \mapsto \omega(w)$ is conformal in small neighborhoods of $0^{(1)}$ and $0^{(2)}$. Since conformal maps preserve orientation, the curve $\gamma_{\mathcal{M}}$ which surrounds both $0^{(1)}$ and $0^{(2)}$ once in the positive direction, is mapped by $w \mapsto \omega(w)$ onto a curve $\gamma_{\mathbb{C}}$ on the complex plane, which surrounds $c$ and $c^{-1}$ once in the positive direction. Furthermore, since $\omega(\infty^{(2)}) = 0$, the curve $\gamma_{\mathbb{C}}$ does not surround $0$. By changing variables $(w,z) \mapsto (\omega,\zeta)$ in , and by using , and , we obtain $$\begin{array}{r c l}
f(z(\zeta)) & = & {\displaystyle}\frac{1}{2\pi i} \int_{\gamma_{\mathbb{C}}}f(w(\omega)) \frac{\lambda^{2N}(w(\omega))}{w(\omega)^{2N}}\mathcal{R}^{\mathcal{M}}(w(\omega),z(\zeta)) \frac{dw(\omega)}{y(w(\omega))}, \\
& = & {\displaystyle}\frac{1}{2\pi i} \int_{\gamma_{\mathbb{C}}}f(w(\omega)) \bigg( \frac{(\omega-\alpha c)(\omega-\alpha c^{-1})}{(\omega-c)(\omega-c^{-1})} \bigg)^{2N} \mathcal{R}^{\mathcal{M}}(w(\omega),z(\zeta))\frac{d\omega}{\omega},
\end{array}$$ for every $f \in L_{N}$. Since $f \in L_{N}$, the function $\zeta \mapsto f(z(\zeta))$ is meromorphic on the Riemann sphere, with a pole of degree at most $N$ at $\zeta = 0$ and a pole of degree at most $N-1$ at $\zeta = \infty$. In other words, $\zeta \mapsto \zeta^{N}f(z(\zeta)) =: p(\zeta)$ is a polynomial of degree at most $2N-1$. By multiplying the above equality by $\zeta^{N}$, we thus have $$p(\zeta) = \frac{1}{2\pi i}\int_{\gamma_{\mathbb{C}}} \frac{p(\omega)}{\omega^{N}} \bigg( \frac{(\omega-\alpha c)(\omega-\alpha c^{-1})}{(\omega-c)(\omega-c^{-1})} \bigg)^{2N} \mathcal{R}^{\mathcal{M}}(w(\omega),z(\zeta))\zeta^{N}\frac{d\omega}{\omega}.$$ We obtain the claim after substituting in the above expression.
Now, we prove formula , which expresses $\mathcal{R}^{U}$ in terms of the solution $U$ to the $2 \times 2$ RH problem presented in Section \[subsection: new formula for the kernel\].
The reproducing kernel $\mathcal{R}^{U}$ defined by can be rewritten in terms of $U$ as follows $$\label{reproducing kernel in terms of U}
\mathcal{R}^{U}(\omega,\zeta) = \frac{1}{\zeta-\omega} \begin{pmatrix}
0 & 1
\end{pmatrix} U^{-1}(\omega)U(\zeta) \begin{pmatrix}
1 \\ 0
\end{pmatrix}.$$
By [@DK Lemma 4.6 (c)], there is a unique bivariate polynomial $\mathcal{R}^{U}$ of degree $\leq 2N-1$ in both $\omega$ and $\zeta$ which satisfies . Therefore, it suffices to first replace $\mathcal{R}^{U}$ in the left-hand-side of by $$\begin{aligned}
\frac{1}{\zeta-\omega} \begin{pmatrix}
0 & 1
\end{pmatrix} U^{-1}(\omega)U(\zeta) \begin{pmatrix}
1 \\ 0
\end{pmatrix},\end{aligned}$$ and then to verify that still holds with this replacement. The rest of the proof goes exactly as in [@DK Proposition 4.9], so we omit it.
Proof of formula
-----------------
Now, using the results of Sections \[subsection: 5.1\], \[subsection: 5.2\] and \[subsection: 5.3\], we give a proof for formula . From –, for $x \in \{1,\ldots,2N-1\}$, $y \in \mathbb{Z}$ and $\epsilon_{x} \in \{0,1\}$, we have $$\begin{aligned}
& \big[ K(2x+\epsilon_{x},2y+j,2x+\epsilon_{x},2y+i) \big]_{i,j=0}^{1} \nonumber \\
& = \frac{1}{(2\pi i)^{2}}\int_{\gamma}\int_{\gamma}\begin{pmatrix}
\alpha^{2} & \alpha \\
w & 1
\end{pmatrix}^{\epsilon_{x}}\frac{A(w)^{2N-x-\epsilon_{x}}}{w^{2N-y}}\mathcal{R}^{Y}(w,z)\frac{A(z)^{x}}{z^{y+1}}\begin{pmatrix}
1 & 1 \\
\alpha z & 1
\end{pmatrix}^{\epsilon_{x}} dzdw, \label{K in terms of Rcal Y unified}\end{aligned}$$ where $\gamma$ is a close contour surrounding $0$ once in the positive direction. The proof consists of using the successive transformations $\mathcal{R}^{Y} \mapsto \mathcal{R}^{X} \mapsto \mathcal{R}^{\mathcal{M}} \mapsto \mathcal{R}^{U}$. We first use the eigendecomposition \[eigenvalue eigenvector decomposition of A\] of $A$ and the $\mathcal{R}^{Y} \mapsto \mathcal{R}^{X}$ transformation given in to rewrite as $$\begin{aligned}
& \big[ K(2x+\epsilon_{x},2y+j,2x+\epsilon_{x},2y+i) \big]_{i,j=0}^{1} \nonumber \\
& = \frac{1}{(2\pi i)^{2}}\int_{\gamma}\int_{\gamma} \begin{pmatrix}
\alpha^{2} & \alpha \\
w & 1
\end{pmatrix}^{\epsilon_{x}} E(w)\frac{\Lambda(w)^{2N-x-\epsilon_{x}}}{w^{2N-y}} \mathcal{R}^{X}(w,z) \frac{\Lambda(z)^{x}}{z^{y+1}}E(z)^{-1}\begin{pmatrix}
1 & 1 \\
\alpha z & 1
\end{pmatrix}^{\epsilon_{x}} dzdw. \label{K in terms of Rcal X unified}\end{aligned}$$ Using , we can write $E(w)$ and $E(z)^{-1}$ as $$\begin{aligned}
E(w) & \; = \begin{pmatrix}
1 & 1 \\
\frac{\lambda(w^{(1)})-\alpha^{2}-w}{1+\alpha} & \frac{\lambda(w^{(2)})-\alpha^{2}-w}{1+\alpha}
\end{pmatrix}, \qquad w \in \mathbb{C}, \label{lol3} \\
E(z)^{-1} & \; = \frac{1}{1-\alpha}\begin{pmatrix}
\frac{(1+\alpha^{3})z}{y(z^{(1)})(\lambda(z^{(1)})-\alpha^{2}-z)} & \frac{1}{y(z^{(1)})} \\
\frac{(1+\alpha^{3})z}{y(z^{(2)})(\lambda(z^{(2)})-\alpha^{2}-z)} & \frac{1}{y(z^{(2)})}
\end{pmatrix}, \qquad z \in \mathbb{C}, \label{lol4}\end{aligned}$$ where we have also used the relation $$(\lambda_{1}-\alpha^{2}-z)(\lambda_{2}-\alpha^{2}-z) = -(1+\alpha)(1+\alpha^{3})z$$ to obtain . The identities and allow to rewrite the integrand of by noting that $$\begin{aligned}
&E(w)\frac{\Lambda(w)^{2N-x-\epsilon_{x}}}{w^{2N-y}} \mathcal{R}^{X}(w,z) \frac{\Lambda(z)^{x}}{z^{y+1}}E(z)^{-1} = \sum_{j,k=1}^{2}\begin{pmatrix}
1 \\
\frac{\lambda(w^{(j)})-\alpha^{2}-w}{1+\alpha}
\end{pmatrix}\lambda(w^{(j)})^{2N-x-\epsilon_{x}} \\
& \times e_{j}^{T} \frac{y(w^{(j)})\mathcal{R}^{X}(w,z)}{w^{2N-y}z^{y+1}}e_{k} \lambda(z^{(k)})^{x} \begin{pmatrix}
\frac{(1+\alpha^{3})z}{(1-\alpha)(\lambda(z^{(k)})-\alpha^{2}-z)} & \frac{1}{1-\alpha}
\end{pmatrix}\frac{1}{y(w^{(j)})y(z^{(k)})}.\end{aligned}$$ Therefore, using also the $\mathcal{R}^{X} \mapsto \mathcal{R}^{\mathcal{M}}$ transformation given by , we obtain $$\begin{aligned}
& \big[ K(2x+\epsilon_{x},2y+j,2x+\epsilon_{x},2y+i) \big]_{i,j=0}^{1} = \frac{1}{(2\pi i)^{2}}\int_{\gamma_{\mathcal{M}}}\int_{\gamma_{\mathcal{M}}} \begin{pmatrix}
\alpha^{2} & \alpha \\
w & 1
\end{pmatrix}^{\epsilon_{x}}
\begin{pmatrix}
1 \\ \frac{\lambda(w)-\alpha^{2}-w}{1+\alpha}
\end{pmatrix} \\
& \lambda(w)^{2N-x-\epsilon_{x}} \frac{\mathcal{R}^{\mathcal{M}}(w,z)}{w^{2N-y}z^{y+1}} \lambda(z)^{x} \begin{pmatrix}
\frac{(1+\alpha^{3})z}{(1-\alpha)(\lambda(z)-\alpha^{2}-z)} & \frac{1}{1-\alpha}
\end{pmatrix}\begin{pmatrix}
1 & 1 \\
\alpha z & 1
\end{pmatrix}^{\epsilon_{x}}\frac{dzdw}{y(w)y(z)},\end{aligned}$$ where $\gamma_{\mathcal{M}}$ is a close contour surrounding once $0^{(1)}$ and $0^{(2)}$ on $\mathcal{M}$ in the positive direction. By performing the change of variables $w = w(\omega)$ and $z = z(\zeta)$ as in –, using the factorization and , the identity , and also the $\mathcal{R}^{\mathcal{M}} \mapsto \mathcal{R}^{U}$ transformation given by , we get $$\begin{aligned}
& \big[ K(2x+\epsilon_{x},2y+j,2x+\epsilon_{x},2y+i) \big]_{i,j=0}^{1} = \frac{1}{(2\pi i)^{2}}\int_{\gamma_{\mathbb{C}}}\int_{\gamma_{\mathbb{C}}} \begin{pmatrix}
\alpha^{2} & \alpha \\
w & 1
\end{pmatrix}^{\epsilon_{x}}
\begin{pmatrix}
1 \\ \frac{\lambda(w)-\alpha^{2}-w}{1+\alpha}
\end{pmatrix} \label{lol10} \\
& \bigg( \frac{(\omega - \alpha c)(\omega - \alpha c^{-1})}{\omega(\omega-c)(\omega - c^{-1})} \bigg)^{2N} \mathcal{R}^{U}(\omega,\zeta) \frac{\omega^{N}w^{y}\lambda(z)^{x}}{\zeta^{N+1}z^{y+1}\lambda(w)^{x+\epsilon_{x}}} \begin{pmatrix}
\frac{(1+\alpha^{3})z}{(1-\alpha)(\lambda(z)-\alpha^{2}-z)} & \frac{1}{1-\alpha}
\end{pmatrix}\begin{pmatrix}
1 & 1 \\
\alpha z & 1
\end{pmatrix}^{\epsilon_{x}}d\zeta d\omega, \nonumber\end{aligned}$$ where $\gamma_{\mathbb{C}}$ is a closed curve surrounding $c$ and $c^{-1}$ once in the positive direction, such that it does not surround $0$. On the other hand, using again and , we have $$\begin{aligned}
\frac{w^{y}\lambda(z)^{x}}{z^{y}\lambda(w)^{x}} = \frac{(\omega-c)^{y}(\omega-c^{-1})^{y}}{(\zeta-c)^{y}(\zeta-c^{-1})^{y}} \frac{(\zeta - \alpha c)^{x} (\zeta - \alpha c^{-1})^{x}}{(\omega - \alpha c)^{x} (\omega - \alpha c^{-1})^{x}} \frac{\omega^{x-y}}{\zeta^{x-y}}.\end{aligned}$$ By using the definition of $W$, we can rewrite as $$\begin{aligned}
& \big[ K(2x+\epsilon_{x},2y+j,2x+\epsilon_{x},2y+i) \big]_{i,j=0}^{1} = \frac{1}{(2\pi i)^{2}}\int_{\gamma_{\mathbb{C}}}\int_{\gamma_{\mathbb{C}}} H_{K}(\omega,\zeta;\epsilon_{x}) \label{new formula for the kernel in the thm 2} \\
& W(\omega) \mathcal{R}^{U}(\omega,\zeta) \frac{\omega^{N+x-y}}{\zeta^{N+x-y}} \frac{(\omega-c)^{y}(\omega-c^{-1})^{y}}{(\zeta-c)^{y}(\zeta-c^{-1})^{y}} \frac{(\zeta - \alpha c)^{x} (\zeta - \alpha c^{-1})^{x}}{(\omega - \alpha c)^{x} (\omega - \alpha c^{-1})^{x}}d\zeta d\omega, \nonumber\end{aligned}$$ where $H_{K}(\omega,\zeta;\epsilon_{x})$ is defined for $\omega, \zeta \in \mathbb{C}$ and $\epsilon_{x} \in \{0,1\}$ by $$\begin{aligned}
\label{lol9}
H_{K}(\omega,\zeta;\epsilon_{x}) = & \; \frac{1}{\zeta z \lambda(w)^{\epsilon_{x}}} \begin{pmatrix}
\alpha^{2} & \alpha \\
w & 1
\end{pmatrix}^{\epsilon_{x}} \begin{pmatrix}
1 \\ \frac{1-\alpha + \alpha^{2}}{1-\alpha} \frac{\omega - c}{\omega}
\end{pmatrix} \begin{pmatrix}
\frac{\alpha}{c(1-\alpha)^{2}}(\zeta - c^{-1}) & \frac{1}{1-\alpha}
\end{pmatrix}\begin{pmatrix}
1 & 1 \\
\alpha z & 1
\end{pmatrix}^{\epsilon_{x}}. \end{aligned}$$ Using the identities $$\begin{aligned}
& z = \frac{\alpha (\zeta - c)(\zeta - c^{-1})}{c(1-\alpha)^{2}\zeta}, & & w = \frac{\alpha (\omega - c)(\omega - c^{-1})}{c(1-\alpha)^{2}\omega}, & & \lambda(w) = \frac{\alpha (\omega - \alpha c)(\omega - \alpha c^{-1})}{c(1-\alpha)^{2}\omega},\end{aligned}$$ it is a simple computation to verify that can be rewritten as –. This finishes the proof.
Lozenge probabilities {#section: lozenge probabilities}
=====================
This section is about the lozenge probabilities $P_{j}(x,y)$, $j=1,2,3$, defined in . In Subsection \[subsection: double contour formula for Pj\], we use Theorem \[thm: correlation kernel final scalar expression\] to find double contour formulas for $P_{j}(x,y)$, $j=1,2,3$, in terms of $\mathcal{R}^{U}$. In the rest of this section, we follow [@CDKL Section 7] and use the symmetries in our model to restrict our attention to the lower left part $\eta \leq \frac{\xi}{2} \leq 0$ of the liquid region for the proof of Theorem \[thm:main\].
Double contour formulas {#subsection: double contour formula for Pj}
-----------------------
Formula for the kernel can be rewritten as $$\begin{gathered}
\label{main formula for the kernel written compactly}
\big[ K(2x+\epsilon_{x},2y+j,2x+\epsilon_{x},2y+i) \big]_{i,j=0}^{1} \\
= \frac{1}{(2\pi i)^{2}}\int_{\gamma_{\mathbb{C}}}\int_{\gamma_{\mathbb{C}}} H_{K}(\omega,\zeta;\epsilon_{x}) W(\omega) \mathcal{R}^{U}(\omega,\zeta) \frac{\omega^{N}}{\zeta^{N}}q(\omega,\zeta)^{y} \tilde{q}(\omega,\zeta)^{x} d\zeta d\omega,\end{gathered}$$ where $q$ and $\tilde{q}$ are given by $$\begin{aligned}
\label{def of q and qtilde}
& q(\omega,\zeta) := \frac{\zeta (\omega-c)(\omega-c^{-1})}{\omega(\zeta-c)(\zeta-c^{-1})}, \qquad \tilde{q}(\omega,\zeta) = \frac{\omega(\zeta - \alpha c)(\zeta - \alpha c^{-1})}{\zeta(\omega - \alpha c)(\omega - \alpha c^{-1})}.\end{aligned}$$ The double contour formulas for $P_{j}(x,y)$, $j=1,2,3$, are obtained via a series of lemmas. Let us first recall that the paths $\mathfrak{p}_j:\{0,1,\ldots,4N\} \to \mathbb{Z}+\tfrac{1}{2}$, $j=0,\ldots,2N-1$ are defined in via . We define the height function $h:\{0,1,\ldots,4N\}\times \mathbb Z \to \mathbb{N}_{\geq 0}$ by $$\begin{aligned}
\label{def of h}
h(x,y)= \#\{ j \mid \mathfrak{p}_j(x)< y \}.\end{aligned}$$ Lemma \[lem:height\_to\_lozenge\] below is identical to [@CDKL Lemma 7.2] and allows to recover the lozenges from the height function.
\[lem:height\_to\_lozenge\] For $x \in \{0,1,\ldots, 4N\}$ and $y \in \mathbb Z$, the following identities hold: $$\begin{aligned}
h(x,y+1)-h(x+1,y+1) &=
\begin{cases}
1, & \text{ there is a lozenge \tikz[scale=.3,baseline=(current bounding box.center)] { \draw (0,0) {
--++(1,1)--++(0,1)--++(-1,-1)
--++(0,-1)
}; \filldraw circle(5pt); \node[below] (i) at (0,-.2) {\tiny{$(x,y)$}};} }\\[-10pt]
0, & \text{ otherwise.}
\end{cases} \label{diff loz 1} \\
h(x+1,y+1)-h(x,y) &=
\begin{cases}
1, & \text{ there is a lozenge \tikz[scale=.3,baseline=(current bounding box.center)] { \draw (0,0) {--++(1,0)--++(0,1)--++(-1,0)--++(0,-1)
}; \filldraw circle(5pt); \node[below] (i) at (0,-.2) {\tiny{$(x,y)$}};} }\\[-5pt]
0, & \text{ otherwise.}
\end{cases} \label{diff loz 2} \\
h(x,y+1)-h(x,y) & =
\begin{cases}
0, & \text{ there is a lozenge \tikz[scale=.3,baseline=(current bounding box.center)] { \draw (0,0) {--++(1,1)--++(1,0)--++(-1,-1)--++(-1,0)
}; \filldraw (1,0) circle(5pt); \node[below] (i) at (1,-.2) {\tiny{$(x,y)$}};} }\\[-5pt]
1, & \text{ otherwise.}
\end{cases} \label{diff loz 3}\end{aligned}$$
This is an immediate consequence of and .
The next lemma establishes a double integral formula for the expectation value of the height function.
\[lem:doubleintegralheight\] For $x \in \{1,2,\ldots, 2N-1\}$, $y \in \mathbb Z$ and $\epsilon_{x},\epsilon_{y} \in \{0,1\}$, we have $$\begin{gathered}
\label{formula for expectation h in lemma}
\mathbb{E}[h(2x+\epsilon_{x},2y+\epsilon_{y})] = \frac{1}{(2\pi i)^2} \int_{\tilde{\gamma}_{\mathbb{C}}}d\zeta\int_{\gamma_{\mathbb{C}}} \frac{d\omega}{{q(\omega,\zeta)-1}} \mathcal{R}^{U}(\omega,\zeta) W(\omega) \\
\times \frac{\omega^{N}}{\zeta^{N}}q(\omega,\zeta)^{y}\tilde{q}(\omega,\zeta)^{x} \big(q(\omega,\zeta)^{\epsilon_{y}}H_{K}(\omega,\zeta;\epsilon_{x})_{11}+H_{K}(\omega,\zeta;\epsilon_{x})_{22} \big). \end{gathered}$$ where $\gamma_{\mathbb{C}}$ is a closed curve surrounding both $c$ and $c^{-1}$, but not surrounding $0$, and $\tilde{\gamma}_{\mathbb{C}}$ is a deformation of $\gamma_{\mathbb{C}}$ lying in the bounded region delimited by $\gamma_{\mathbb{C}}$, such that $|q(\zeta,\omega)|>1$ whenever $\zeta \in \tilde{\gamma}_{\mathbb{C}}$ and $\omega \in \gamma_{\mathbb{C}}$.
Let $\mathcal{X}(\tilde{x},\tilde{y})$ be the random variable that counts the number of paths going through the point $(\tilde{x},\tilde{y})$, $\tilde{x},\tilde{y} \in \{0,1,...,4N\}$. Since $\mathcal{X}(\tilde{x},\tilde{y}) \in \{0,1\}$, we have $\mathbb{P}(\mathcal{X}(\tilde{x},\tilde{y}) = 1) = \mathbb{E}(\mathcal{X}(\tilde{x},\tilde{y}))$. Also, note that the identity with $k=1$ is equivalent to $\mathbb{P}(\mathcal{X}(\tilde{x},\tilde{y}) = 1) = K(\tilde{x},\tilde{y},\tilde{x},\tilde{y})$. Thus, by definition of $h$, we get $$\begin{aligned}
\mathbb{E}[h(2x+\epsilon_{x},2y)] & = \sum_{k<y} \big[ K(2x+\epsilon_{x},2k,2x+\epsilon_{x},2k) + K(2x + \epsilon_{x},2k + 1,2x + \epsilon_{x},2k + 1) \big] \nonumber \\
& = \sum_{k<y} \mbox{Tr} \big[ K(2x+\epsilon_{x},2k+j,2x+\epsilon_{x},2k+i) \big]_{i,j=0}^{1}. \label{lol13}\end{aligned}$$ Let us define $\tilde{\gamma}_{\mathbb{C}} := C(c,r) \cup C(c^{-1},r)$, where $C(a,r)$ denotes a circle oriented positively centered at $a$ of radius $r$. We see from that $|q(\omega,\zeta)| \to + \infty$ as $\zeta$ tends to $c$ or $c^{-1}$. Thus, by choosing $r$ sufficiently small, we can make sure that $\tilde{\gamma}_{\mathbb{C}}$ lies in the interior region of $\gamma_{\mathbb{C}}$, and that $$\begin{aligned}
\left| q(\omega,\zeta) \right|>1+\epsilon, \qquad \mbox{for all } \zeta \in \tilde{\gamma}_{\mathbb{C}} \mbox{ and } \omega \in \gamma_{\mathbb{C}},\end{aligned}$$ for a certain $\epsilon > 0$. Therefore, uniformly for $\zeta \in \tilde{\gamma}_{\mathbb{C}}$ and $\omega \in \gamma_{\mathbb{C}}$, we have $$\begin{aligned}
\label{lol14}
& \sum_{k<y} q(\omega,\zeta)^{k} = \frac{q(\omega,\zeta)^{y}}{q(\omega,\zeta)-1}.\end{aligned}$$ The statement with $\epsilon_{y} = 0$ follows after combining , and . Then, with $\epsilon_{y} = 1$ follows from $$\begin{aligned}
\mathbb{E}[h(2x+\epsilon_{x},2y+1)] = \mathbb{E}[h(2x+\epsilon_{x},2y)] + K(2x+\epsilon_{x},2y,2x+\epsilon_{x},2y).\end{aligned}$$
The double contour formulas for $P_{j}$, $j=1,2,3$ are stated in the following proposition.
For $x \in \{1,2,\ldots, 2N-1\}$ and $y \in \mathbb Z$, we have $$\begin{aligned}
& P_{1}(x,y) = \frac{1}{(2\pi i)^{2}}\int_{\gamma_{\mathbb{C}}}\int_{\gamma_{\mathbb{C}}} H_{1}(\omega,\zeta) W(\omega) \mathcal{R}^{U}(\omega,\zeta) \frac{\omega^{N}}{\zeta^{N}}q(\omega,\zeta)^{y} \tilde{q}(\omega,\zeta)^{x} d\zeta d\omega, \label{P1 double contour} \\
& P_{2}(x,y) = \frac{1}{(2\pi i)^{2}}\int_{\gamma_{\mathbb{C}}}\int_{\gamma_{\mathbb{C}}} H_{2}(\omega,\zeta) W(\omega) \mathcal{R}^{U}(\omega,\zeta) \frac{\omega^{N}}{\zeta^{N}}q(\omega,\zeta)^{y} \tilde{q}(\omega,\zeta)^{x} d\zeta d\omega, \label{P2 double contour} \\
& P_{3}(x,y) = \begin{pmatrix}
1 & 1 \\ 1 & 1
\end{pmatrix} - \frac{1}{(2\pi i)^{2}}\int_{\gamma_{\mathbb{C}}}\int_{\gamma_{\mathbb{C}}} H_{3}(\omega,\zeta) W(\omega) \mathcal{R}^{U}(\omega,\zeta) \frac{\omega^{N}}{\zeta^{N}}q(\omega,\zeta)^{y} \tilde{q}(\omega,\zeta)^{x} d\zeta d\omega, \label{P3 double contour}\end{aligned}$$ where $H_{1}$, $H_{2}$ and $H_{3}$ are given by $$\begin{aligned}
& H_{1}(\omega,\zeta) = \begin{pmatrix} \frac{\alpha c (\omega -c) (\omega-c^{-1})}{(\zeta-c)(\zeta-c^{-1})\omega (\omega - \alpha c)} & \frac{(\zeta-\alpha c)(\omega -c)(\omega-c^{-1})}{(\zeta-c)(\zeta-c^{-1})(\omega-\alpha c)(\omega - \alpha c^{-1})} \\
\frac{\omega -c}{(\zeta-c)(\omega - \alpha c)} & \frac{\alpha (\zeta - \alpha c)(\omega-c)}{c \zeta (\zeta -c)(\omega-\alpha c)(\omega-\alpha c^{-1})}
\end{pmatrix} \label{def of H1} \\
& H_{2}(\omega,\zeta) = \begin{pmatrix}
\frac{c(1-\alpha)(\omega-c)}{\alpha(\zeta-c)(\zeta-c^{-1})(\omega-\alpha c)} & \frac{(1-\alpha)(\zeta-\alpha c)(\omega-c)}{c(\zeta-c)(\zeta-c^{-1})(\omega-\alpha c)(\omega-\alpha c^{-1})} \\
\frac{(1-\alpha)c}{(\zeta-c)(\omega - \alpha c)} & \frac{\alpha c (1-\alpha) (\zeta- \alpha c)\omega}{\zeta (\zeta-c)(\omega - \alpha c)(\omega - \alpha c^{-1})}
\end{pmatrix}, \label{def of H2} \\
& H_{3}(\omega,\zeta) = \begin{pmatrix}
\frac{\omega -c}{c \omega (\zeta -c)(\zeta -c^{-1})} & \frac{(\zeta-\alpha c)(\omega-c)}{(\zeta -c)(\zeta -c^{-1})(\omega - \alpha c)} \\
\frac{1}{\zeta -c} & \frac{c(\zeta - \alpha c)}{\zeta (\zeta -c) (\omega - \alpha c)}
\end{pmatrix}. \label{def of H3}\end{aligned}$$
Recall that $\mathcal{P}_{j}$, $j=1,2,3$ are defined by . By , for $\epsilon_{x},\epsilon_{y} \in \{0,1\}$, we have $$\begin{aligned}
& \mathcal{P}_{3}(2x+\epsilon_{x},2y+\epsilon_{y}) = 1 - K(2x+\epsilon_{x},2y+\epsilon_{y},2x+\epsilon_{x},2y+\epsilon_{y}).\end{aligned}$$ Noting that $$\begin{aligned}
H_{3}(\omega,\zeta) = \begin{pmatrix}
H_{K}(\omega,\zeta;0)_{22} & H_{K}(\omega,\zeta;1)_{22} \\
H_{K}(\omega,\zeta;0)_{11} & H_{K}(\omega,\zeta;1)_{11}
\end{pmatrix} = \begin{pmatrix}
\frac{\omega -c}{c \omega (\zeta -c)(\zeta -c^{-1})} & \frac{(\zeta-\alpha c)(\omega-c)}{(\zeta -c)(\zeta -c^{-1})(\omega - \alpha c)} \\
\frac{1}{\zeta -c} & \frac{c(\zeta - \alpha c)}{\zeta (\zeta -c) (\omega - \alpha c)}
\end{pmatrix},\end{aligned}$$ formula follows by combining with . The proof of and requires more work and relies on Lemmas \[lem:height\_to\_lozenge\] and \[lem:doubleintegralheight\]. First, we note the following direct consequences of : $$\begin{aligned}
& \mathbb{E}[h(2x+\epsilon_{x},2y+1+\epsilon_{y})] = \frac{1}{(2\pi i)^2} \int_{\tilde{\gamma}_{\mathbb{C}}}d\zeta\int_{\gamma_{\mathbb{C}}} \frac{d\omega}{{q(\omega,\zeta)-1}} \mathcal{R}^{U}(\omega,\zeta) W(\omega) \nonumber \\
& \hspace{1.5cm} \times \frac{\omega^{N}}{\zeta^{N}}q(\omega,\zeta)^{y}\tilde{q}(\omega,\zeta)^{x} \big(q(\omega,\zeta)H_{K}(\omega,\zeta;\epsilon_{x})_{11}+q(\omega,\zeta)^{\epsilon_{y}}H_{K}(\omega,\zeta;\epsilon_{x})_{22} \big), \label{formula for expectation h number 2} \\
& \mathbb{E}[h(2x+1+\epsilon_{x},2y+1+\epsilon_{y})] = \frac{1}{(2\pi i)^2} \int_{\tilde{\gamma}_{\mathbb{C}}}d\zeta\int_{\gamma_{\mathbb{C}}} \frac{d\omega}{{q(\omega,\zeta)-1}} \mathcal{R}^{U}(\omega,\zeta) W(\omega) \nonumber \\
& \hspace{1cm} \times \frac{\omega^{N}}{\zeta^{N}}q(\omega,\zeta)^{y}\tilde{q}(\omega,\zeta)^{x+\epsilon_{x}} \big(q(\omega,\zeta)H_{K}(\omega,\zeta;1-\epsilon_{x})_{11}+q(\omega,\zeta)^{\epsilon_{y}}H_{K}(\omega,\zeta;1-\epsilon_{x})_{22} \big). \label{formula for expectation h number 3}\end{aligned}$$ Using , and , we get $$\begin{aligned}
& \mathcal{P}_{1}(2x+\epsilon_{x},2y+\epsilon_{y}) = \mathbb{E}[h(2x+\epsilon_{x},2y+1+\epsilon_{y})]-\mathbb{E}[h(2x+1+\epsilon_{x},2y+1+\epsilon_{y})] \nonumber \\
& = \frac{1}{(2\pi i)^2} \int_{\tilde{\gamma}_{\mathbb{C}}}d\zeta\int_{\gamma_{\mathbb{C}}} \frac{d\omega}{{q(\omega,\zeta)-1}} \mathcal{R}^{U}(\omega,\zeta) W(\omega) \frac{\omega^{N}}{\zeta^{N}}q(\omega,\zeta)^{y} \tilde{q}(\omega,\zeta)^{x} \nonumber \\
& \times \Big( q(\omega,\zeta) H_{K}(\omega,\zeta;\epsilon_{x})_{11}- q(\omega,\zeta)\tilde{q}(\omega,\zeta)^{\epsilon_{x}}H_{K}(\omega,\zeta;1-\epsilon_{x})_{11} \nonumber \\
& + q(\omega,\zeta)^{\epsilon_{y}}H_{K}(\omega,\zeta;\epsilon_{x})_{22}- q(\omega,\zeta)^{\epsilon_{y}}\tilde{q}(\omega,\zeta)^{\epsilon_{x}}H_{K}(\omega,\zeta;1-\epsilon_{x})_{22}\Big). \label{lol15}\end{aligned}$$ It is a direct computation to verify that the integrand has no pole at $\zeta = \omega$ for any $\epsilon_{x},\epsilon_{y} \in \{0,1\}$, so that $\tilde{\gamma}_{\mathbb{C}}$ can be deformed back to $\gamma_{\mathbb{C}}$. We obtain after writing in the matrix form . Finally, using , and , we get $$\begin{aligned}
& \mathcal{P}_{2}(2x+\epsilon_{x},2y+\epsilon_{y}) = \mathbb{E}[h(2x+1+\epsilon_{x},2y+1+\epsilon_{y})]-\mathbb{E}[h(2x+\epsilon_{x},2y+\epsilon_{y})] \\
& = \frac{1}{(2\pi i)^2} \int_{\tilde{\gamma}_{\mathbb{C}}}d\zeta\int_{\gamma_{\mathbb{C}}} \frac{d\omega}{{q(\omega,\zeta)-1}} \mathcal{R}^{U}(\omega,\zeta) W(\omega) \frac{\omega^{N}}{\zeta^{N}}q(\omega,\zeta)^{y} \tilde{q}(\omega,\zeta)^{x} \\
& \times \Big( q(\omega,\zeta)\tilde{q}(\omega,\zeta)^{\epsilon_{x}} H_{K}(\omega,\zeta;1-\epsilon_{x})_{11}- q(\omega,\zeta)^{\epsilon_{y}}H_{K}(\omega,\zeta;\epsilon_{x})_{11} \\
& + q(\omega,\zeta)^{\epsilon_{y}}\tilde{q}(\omega,\zeta)^{\epsilon_{x}}H_{K}(\omega,\zeta;1-\epsilon_{x})_{22}- H_{K}(\omega,\zeta;\epsilon_{x})_{22}\Big).\end{aligned}$$ Another direct computation shows that the integrand has no pole at $\zeta = \omega$ for any $\epsilon_{x},\epsilon_{y} \in \{0,1\}$, so that $\tilde{\gamma}_{\mathbb{C}}$ can be deformed back to $\gamma_{\mathbb{C}}$. The formula is then obtained by rewriting the above in the matrix form .
Symmetries {#subsection: symmetries}
----------
Let $H(\omega,\zeta)$ be a $2 \times 2$ meromorphic function in both $\zeta$ and $\omega$, whose only possible poles in each variable are at $0$, $\alpha c$, $\alpha c^{-1}$, $c$ and $c^{-1}$. Furthermore, we assume that all the poles of $H$ are of order $1$ and that $H(\omega,\zeta)$ is bounded as $\zeta$ and/or $\omega$ tend to $\infty$. For $x \in \{1,2,\ldots, 2N-1\}$ and $y \in \mathbb Z$, we define $$\begin{aligned}
\label{mathcalI}
\mathcal{I}(x,y;H) = \frac{1}{(2\pi i)^{2}}\int_{\gamma_{\mathbb{C}}} \int_{\gamma_{\mathbb{C}}} H(\omega, \zeta) W(\omega) \mathcal{R}^{U}(\omega,\zeta) \frac{\omega^{N}}{\zeta^{N}}q(\omega,\zeta)^{y} \tilde{q}(\omega,\zeta)^{x} d\zeta d\omega.\end{aligned}$$ Since the poles of $H$ are of order at most $1$, recalling , the only poles of the integrand are at $0$, $c$ and $c^{-1}$, in both the $\zeta$ and $\omega$ variables. The following star-operation will play an important role for a symmetry property of $\mathcal{I}$: $$\begin{aligned}
\label{star operation}
\zeta^{\star} = c^{-1} + \frac{R_{1}^{2}}{\zeta -c^{-1}}. $$ Let $\gamma_{1}$ be the circle centered at $c^{-1}$ of radius $R_{1} = \frac{1-\alpha}{\sqrt{\alpha}}$. The star-operation maps $\gamma_{1}$ into itself, but reverses the orientation. Furthermore, it satisfies $(\zeta^{\star})^{\star} = \zeta$ for all $\zeta \in \mathbb{C}\cup \{\infty\}$. We start by proving some symmetries for $\mathcal{R}^{U}$.
The reproducing kernel $\mathcal{R}^{U}$ satisfies two symmetries.
1. We have $$\begin{aligned}
\label{first sym for RU}
\mathcal{R}^{U}(\omega,\zeta) = \mathcal{R}^{U}(\zeta,\omega), \qquad \omega,\zeta \in \mathbb{C}.\end{aligned}$$
2. We have $$\label{eq:RNsymmetry}
\mathcal{R}^{U} \left( \omega^{\star},\zeta^{\star} \right) = \frac{R_{1}^{4N-2}\mathcal{R}^{U}(\omega,\zeta)}{(\omega - c^{-1})^{2N-1} (\zeta - c^{-1})^{2N-1}} , \qquad \omega,\zeta \in \mathbb{C} \setminus \{c^{-1}\}.$$
Since $\det U \equiv 1$, it follows from that $$\begin{aligned}
\label{RU first column}
\mathcal{R}^{U}(\omega,\zeta) = \frac{U_{11}(\omega)U_{21}(\zeta)-U_{11}(\zeta)U_{21}(\omega)}{\zeta-\omega},\end{aligned}$$ from which we deduce . Now we prove (b). Note that the first column of $U$ only contains polynomials, which are independent of the choice of the contour $\gamma_{\mathbb{C}}$ that appears in the formulation of the RH problem for $U$. Therefore, $\mathcal{R}^{U}$ is independent of the choice of $\gamma_{\mathbb{C}}$ as well by . Since $\gamma_{1}$ encloses both $c$ and $c^{-1}$, and does not enclose $0$, $\gamma_{1}$ is a valid choice of contour. We use the freedom we have in the choice of $\gamma_{\mathbb{C}}$ by letting $U$ be the solution to the RH problem for $U$ associated to the contour $\gamma_{1}$. We can verify by direct computations that $$\begin{aligned}
\label{W symmetry}
W(\zeta^{\star}) = \frac{(\omega - c^{-1})^{4N}}{R_{1}^{4N}}W(\zeta),\end{aligned}$$ so that $$\widehat{U}(\zeta) := \begin{pmatrix} R_{1}^{2N} & 0 \\ 0 & -R_{1}^{-2N} \end{pmatrix} U(\tfrac{1}{c})^{-1} U\left( \zeta^{\star} \right)
\begin{pmatrix}
\frac{(\zeta - c^{-1})^{2N}}{R_{1}^{2N}} & 0 \\ 0 & - \frac{ R_{1}^{2N}}{(\zeta - c^{-1})^{2N}}
\end{pmatrix}$$ also satisfies the conditions of the RH problem for $U$. By uniqueness of the solution of this RH problem, we infer that $U(\zeta) = \widehat{U}(\zeta)$. After replacing $(\omega,\zeta)$ by $(\omega^{\star},\zeta^{\star})$ in and using the relations $\widehat{U}(\zeta) = U(\zeta)$ and $\frac{\zeta-\omega}{\zeta^{\star}-\omega^{\star}}=-\frac{(\zeta-c^{-1})(\omega-c^{-1})}{R_{1}^{2}}$, we obtain .
\[prop:symmetries\] The double integral $\mathcal{I}(x,y;H)$ satisfies two symmetries.
1. The following $(x,y)\mapsto (2N-x,2N-y)$ symmetry holds $$\begin{aligned}
\label{eq:Isymmetry1}
\mathcal I(2N-x,2N-y; H) & = \mathcal I(x,y; \widehat{H}), \end{aligned}$$ with $$\label{eq:hatH}
\widehat{H}(\omega,\zeta) = H(\zeta,\omega).$$
2. The following $(x,y)\mapsto (x,N+x-y)$ symmetry holds $$\begin{aligned}
\label{eq:Isymmetry2}
\mathcal I(x,N+x-y; H) & = \mathcal I(x,y; \widetilde{H})\end{aligned}$$ with $$\label{eq:tildeH}
\widetilde{H}(\omega,\zeta) = \frac{R_{1}^{2}H\left(\omega^{\star},\zeta^{\star}\right)}{(\omega-c^{-1})(\zeta-c^{-1})}.$$
\(a) From , we verify that $$\begin{aligned}
\label{lol16}
\frac{\omega^{N}}{\zeta^{N}}q(\omega,\zeta)^{2N-y}\tilde{q}(\omega,\zeta)^{2N-x} = \frac{\zeta^{N}}{\omega^{N}} \frac{W(\zeta)}{W(\omega)}q(\zeta,\omega)^{y}\tilde{q}(\zeta,\omega)^{x}.\end{aligned}$$ Replacing $(x,y)$ in by $(2N-x,2N-y)$, and then using , we get $$\begin{aligned}
\label{mathcalI sym1}
\mathcal{I}(2N-x,2N-y;H) = \frac{1}{(2\pi i)^{2}}\int_{\gamma_{\mathbb{C}}} \int_{\gamma_{\mathbb{C}}} W(\zeta) \mathcal{R}^{U}(\omega,\zeta) \frac{\zeta^{N}}{\omega^{N}}q(\zeta,\omega)^{y} \tilde{q}(\zeta,\omega)^{x} H(\omega, \zeta) d\zeta d\omega.\end{aligned}$$ Recalling , the identity follows after interchanging variables in .
\(b) Note that $\gamma_{1}$ encloses both $c$ and $c^{-1}$, and does not enclose $0$, so we can (and do) deform $\gamma_{\mathbb{C}}$ to $\gamma_{1}$ in . We first replace $(x,y)$ by $(x,N+x-y)$ in , and then perform the change of variables $\zeta \mapsto \zeta^{\star}$ and $\omega \mapsto \omega^{\star}$. This gives $$\begin{aligned}
\mathcal{I}(x,N+x-y) = \frac{1}{(2\pi i)^{2}}\int_{\gamma_{1}} \int_{\gamma_{1}} W(\omega^{\star}) \mathcal{R}^{U}(\omega^{\star},\zeta^{\star}) \frac{\omega^{\star \, N}}{\zeta^{\star \, N}} q(\omega^{\star},\zeta^{\star})^{N+x-y}\tilde{q}(\omega^{\star},\zeta)^{x} H(\omega^{\star},\zeta^{\star}) d\zeta^{\star}d\omega^{\star}.\end{aligned}$$ It is a long but direct computation to verify that $$\begin{aligned}
& q(\omega^{\star},\zeta^{\star})^{N+x-y} = q(\omega,\zeta)^{y-x-N}, & & \tilde{q}(\omega^{\star},\zeta^{\star})^{x} = q(\omega,\zeta)^{x} \tilde{q}(\omega,\zeta)^{x}, \\
& \frac{\omega^{\star \, N}}{\zeta^{\star \, N}} = \frac{(\omega -c)^{N}(\zeta-c^{-1})^{N}}{(\omega -c^{-1})^{N}(\zeta-c)^{N}}, & & H(\omega^{\star},\zeta^{\star})d\zeta^{\star}d\omega^{\star} = \frac{H(\omega^{\star},\zeta^{\star})R_{1}^{4} d\zeta d \omega}{(\zeta-c^{-1})^{2}(\omega-c^{-1})^{2}}.\end{aligned}$$ Recalling also and , follows by deforming back $\gamma_{1}$ to the original contour $\gamma_{\mathbb{C}}$ (in each variable).
We recall that $s(\xi,\eta;\alpha)$ is defined for $(\xi,\eta) \in \mathcal L_{\alpha}$ as the unique solution of lying in the upper half-plane, and that $\mathcal{Q}$ is defined by . These quantities will appear naturally in the analysis of the next sections. For now, we simply note the following symmetries for $s(\xi,\eta;\alpha)$.
\[prop:saddlesymmetry\] Let $(\xi,\eta) \in \mathcal L_{\alpha}$. Then also $(-\xi,-\eta) \in \mathcal L_{\alpha}$, $(\xi,\xi-\eta) \in \mathcal L_{\alpha}$ and $$\begin{aligned}
\label{eq:saddlesymmetry1}
s(-\xi,-\eta;\alpha) & = s(\xi,\eta;\alpha) \\
s(\xi,\xi-\eta;\alpha) & = \label{eq:saddlesymmetry2}
\left( \overline{s(\xi,\eta;\alpha)} \right)^{\star},\end{aligned}$$ where $\star$ denotes the star-operation defined in .
The symmetry is part of Proposition \[prop:hightemp\] and has already been proved in Section \[section: easy proofs\]. It remains to prove . We define the function $f$ as follows $$\begin{aligned}
f(\zeta;\xi,\eta) = -\frac{\xi-\eta}{2}\frac{1}{\zeta} + \frac{\xi}{2}\left( \frac{1}{\zeta-\alpha c} + \frac{1}{\zeta - \alpha c^{-1}} \right) - \frac{\eta}{2}\left( \frac{1}{\zeta -c} + \frac{1}{\zeta - c^{-1}} \right),\end{aligned}$$ so that can be rewritten as $$\begin{aligned}
\label{eq determining the saddles}
f(\zeta;\xi,\eta)^{2} = \mathcal{Q}(\zeta).\end{aligned}$$ Note that both $f$ and $\mathcal{Q}$ depend on $\alpha$, even though this is not indicated in the notation. It is a long but direct computation to verify that $$\begin{aligned}
\label{sym for Q and f}
\frac{R_{1}^{4}}{(\zeta-c^{-1})^{4}}\mathcal{Q}( \zeta^{\star}) = \mathcal{Q}(\zeta), \qquad \mbox{ and } \qquad - \frac{R_{1}^{2}}{(\zeta - c^{-1})^2} f ( \zeta^{\star}; \xi,\eta ) = f(\zeta;\xi,\xi-\eta).\end{aligned}$$ By definition of $s(\xi,\eta;\alpha)$, we have $f(s(\xi,\eta;\alpha);\xi,\eta)^{2} = \mathcal{Q}(s(\xi,\eta;\alpha))$, so the symmetry implies that $$\begin{aligned}
\label{lol17}
f(s(\xi,\eta;\alpha)^{\star};\xi,\xi-\eta)^{2} = \mathcal{Q}(s(\xi,\eta;\alpha)^{\star}).\end{aligned}$$ Since the star operation maps the upper half-plane to the lower half-plane, $s(\xi,\eta;\alpha)^{\star}$ lies in the lower half-plane. Therefore, applying the conjugate operation in , and noting that $\overline{f(\zeta)} = f(\overline{\zeta})$ and $\overline{\mathcal{Q}(\zeta)} = \mathcal{Q}(\overline{\zeta})$, we infer that $(\xi,\xi-\eta) \in \mathcal L_{\alpha}$ if and only if $(\xi,\eta) \in \mathcal L_{\alpha}$, and that holds.
Preliminaries to the asymptotic analysis {#subsection: preliminaries to asymp}
----------------------------------------
\[prop:doubleintegrallimit\] Let $\{(x_{N},y_{N}\}_{N \geq 1}$ be a sequence satisfying with $(\xi,\eta) \in \mathcal{L}_{\alpha}$, such that $\eta \leq \frac{\xi}{2}\leq 0$. Let $(\omega,\zeta) \mapsto H(\omega,\zeta)$ be a $2 \times 2$ meromorphic function in both $\zeta$ and $\omega$, whose only possible poles in each variable are at $0$, $\alpha c$, $\alpha c^{-1}$, $c$ and $c^{-1}$. Furthermore we assume that all the poles of $H$ are of order $1$ and that $H(\omega,\zeta)$ is bounded as $\zeta$ and/or $\omega$ tend to $\infty$. Then $\mathcal I(x_{N},y_{N}; H)$ defined in has the limit $$\label{eq:Hintegral}
\lim_{N \to \infty} \mathcal I(x_{N},y_{N};H) = \frac{1}{2\pi i} \int_{\overline{s}}^s H(\zeta,\zeta) d\zeta$$ where $s = s(\xi,\eta;\alpha)$, and the integration path is from $\overline{s}$ to $s$ and lies in $\mathbb C \setminus (-\infty,c^{-1}]$.
The proof of Proposition \[prop:doubleintegrallimit\] will be given in Section \[section: saddle point analysis\], after considerable preparations have been carried out in Sections \[section: g-function\]-\[section: phase functions\].
Proposition \[prop:doubleintegrallimit\] only covers the lower left quadrant $\eta \leq \frac{\xi}{2}\leq 0$ of the liquid region. The next lemma shows that this is sufficient.
\[lemma:from one quadrant to the four\] Assume Proposition \[prop:doubleintegrallimit\] holds true. Then the statement of Proposition \[prop:doubleintegrallimit\] still holds without the assumption that $\eta \leq \frac{\xi}{2}\leq 0$. That is, it holds for all $(\xi,\eta) \in \mathcal{L}_{\alpha}$.
If $\{(x_{N},y_{N}\}_{N \geq 1}$ is a sequence satisfying with $(\xi,\eta) \in \mathcal{L}_{\alpha} \cap \{\eta \geq \frac{\xi}{2}\geq 0\}$, then $\{(2N-x_{N},2N-y_{N}\}_{N \geq 1}$ satisfies with $(-\xi,-\eta)$ lying in the lower left quandrant of $\mathcal{L}_{\alpha}$. Therefore, Proposition \[prop:doubleintegrallimit\] applies to the sequence $\{(2N-x_{N},2N-y_{N}\}_{N \geq 1}$, and we rely on the symmetries and to conclude $$\begin{gathered}
\label{lol20}
\lim_{N \to \infty} \mathcal I(x_{N},y_{N};H) = \lim_{N \to \infty} \mathcal I(2N-x_{N},2N-y_{N};\widehat{H}) \\ = \frac{1}{2\pi i} \int_{\overline{s(-\xi,-\eta;\alpha)}}^{s(-\xi,-\eta;\alpha)} \widehat{H}(\zeta,\zeta) d\zeta = \frac{1}{2\pi i} \int_{\overline{s(\xi,\eta;\alpha)}}^{s(\xi,\eta;\alpha)} H(\zeta,\zeta) d\zeta,\end{gathered}$$ where we have also used for the last equality. Now, if $\{(x_{N},y_{N}\}_{N \geq 1}$ is a sequence satisfying with $(\xi,\eta) \in \mathcal{L}_{\alpha} \cap \{\eta \geq \frac{\xi}{2}\leq 0\}$, then $\{(x_{N},N+x_{N}-y_{N}\}_{N \geq 1}$ satisfies with $(\xi,\xi-\eta)$ lying in the lower left quandrant of $\mathcal{L}_{\alpha}$, so that Proposition \[prop:doubleintegrallimit\] applies. Using the symmetries and , we arrive at $$\begin{gathered}
\label{lol21}
\lim_{N \to \infty} \mathcal I(x_{N},y_{N};H) = \lim_{N \to \infty} \mathcal I(x_{N},N+x_{N}-y_{N};\widetilde{H}) = \frac{1}{2\pi i} \int_{\overline{s(\xi,\xi-\eta;\alpha)}}^{s(\xi,\xi-\eta;\alpha)} \widetilde{H}(\zeta,\zeta) d\zeta \\ = \frac{1}{2\pi i} \int_{s(\xi,\eta;\alpha)^{*}}^{\overline{s(\xi,\eta;\alpha)}^{*}} \frac{R_{1}^{2}H\left(\zeta^{\star},\zeta^{\star}\right)}{(\zeta-c^{-1})^{2}}d\zeta = \frac{1}{2\pi i} \int_{\overline{s(\xi,\eta;\alpha)}}^{s(\xi,\eta;\alpha)} H(\zeta,\zeta) d\zeta,\end{gathered}$$ where, for the last equality, we have applied the change of variables $\zeta \to \zeta^{*}$ stated in . The claim for the last quadrant $(\xi,\eta) \in \mathcal{L}_{\alpha} \cap \{\eta \leq \frac{\xi}{2}\geq 0\}$ follows by combining with .
Proposition \[prop:doubleintegrallimit\] implies Theorem \[thm:main\].
By – and , for $x \in \{1,2,\ldots, 2N-1\}$ and $y \in \mathbb Z$, we can write $$\begin{aligned}
& \mathcal{P}_{j}(x,y) = \mathcal{I}(x,y;H_{j}), \qquad j=1,2, \\
& \mathcal{P}_{3}(x,y) = \begin{pmatrix}
1 & 1 \\ 1 & 1
\end{pmatrix} - \mathcal{I}(x,y;H_{3}),\end{aligned}$$ where the functions $H_{j}$, $j=1,2,3$, are defined in –. Let $\{(x_{N},y_{N}\}_{N \geq 1}$ be a sequence satisfying with $(\xi,\eta) \in \mathcal{L}_{\alpha}$. By Lemma \[lemma:from one quadrant to the four\], we do not need to assume $\eta \leq \frac{\xi}{2}\leq 0$ to invoke Proposition \[prop:doubleintegrallimit\]. Applying Proposition \[prop:doubleintegrallimit\] with $H = H_{3}$, we obtain $$\begin{aligned}
\label{lol19}
\lim_{N \to \infty} P_{3}(x_{N},y_{N}) = \begin{pmatrix}
1 & 1 \\ 1 & 1
\end{pmatrix} - \frac{1}{2\pi i} \int_{\overline{s}}^s H_{3}(\zeta,\zeta) d\zeta.\end{aligned}$$ From , we see that $$\begin{aligned}
\label{lol18}
H_{3}(\zeta,\zeta) = \begin{pmatrix}
\frac{1}{\zeta - c^{-1}} - \frac{1}{\zeta} & \frac{1}{\zeta - c^{-1}} \\
\frac{1}{\zeta -c} & \frac{1}{\zeta - c} - \frac{1}{\zeta}
\end{pmatrix},\end{aligned}$$ and since the path going from $\overline{s}$ to $s$ does not cross $(-\infty,c^{-1}]$, we get after substituting in and carrying out the integration. Similarly, using –, we have $$\begin{aligned}
& H_{1}(\zeta,\zeta) = \begin{pmatrix}
\frac{1}{\zeta - \alpha c} - \frac{1}{\zeta} & \frac{1}{\zeta - \alpha c^{-1}} \\
\frac{1}{\zeta - \alpha c} & \frac{1}{\zeta - \alpha c^{-1}} - \frac{1}{\zeta}
\end{pmatrix} \quad \mbox{ and } \quad H_{2}(\zeta,\zeta) = \begin{pmatrix}
\frac{1}{\zeta - c^{-1}}-\frac{1}{\zeta - \alpha c} & \frac{1}{\zeta - c^{-1}} - \frac{1}{\zeta - \alpha c^{-1}} \\
\frac{1}{\zeta -c} - \frac{1}{\zeta -\alpha c} & \frac{1}{\zeta -c} - \frac{1}{\zeta - \alpha c^{-1}}
\end{pmatrix},\end{aligned}$$ and we obtain – after applying Proposition \[prop:doubleintegrallimit\] with $H = H_{1}$ and $H=H_{2}$, respectively.
$g$-function {#section: g-function}
============
In Section \[section: steepest descent for $U$\], we will perform a Deift/Zhou [@DZ] steepest descent analysis on the RH problem for $U$. The first transformation $U \mapsto T$ consists of normalizing the RH problem and requires considerable preparation. This transformation uses a so-called $g$-function [@Deift], which is of the form $$\begin{aligned}
\label{def of g}
g(\zeta) = \int_{\operatorname{supp}(\mu)}\log(\zeta - \xi) d\mu(\xi),\end{aligned}$$ where $\mu$ is a probability measure, $d\mu$ is its density, and $\operatorname{supp}\mu$ is its (bounded and oriented) support. For any choice of $\mu$, the $g$-function satisfies $$\begin{aligned}
& g(\zeta) = \log(\zeta) + \bigO(\zeta^{-1}), & & \mbox{as } \zeta \to \infty,\end{aligned}$$ so that $U(\zeta)e^{-2Ng(\zeta)\sigma_{3}}$ is normalized at $\infty$, in the sense that $U(\zeta)e^{-2Ng(\zeta)\sigma_{3}} = I_{2} + \bigO(\zeta^{-1})$ as $\zeta \to \infty$. Also, we note that in the definition of $U$, the contour $\gamma_{\mathbb{C}}$ can be chosen arbitrarily, as long as it is a closed curve surrounding $c^{-1}$ and $c$ once in the positive direction, which does not surround $0$. However, in order to successfully perform an asymptotic analysis on the RH problem for $U$, we need to choose $\mu$ and $\gamma_{\mathbb{C}}$ appropriately so that the jumps for $T$ have “good properties".
In this section, we find the key ingredients for the $Y \mapsto T$ transformation of Section \[section: steepest descent for $U$\], that is, we find a $g$-function (built in terms of $\mu$) and a relevant contour $\gamma_{\mathbb{C}}$. Let us rewrite $W$ as follows $$W(\zeta) = \bigg( \frac{(\zeta-\alpha c)(\zeta-\alpha c^{-1})}{\zeta (\zeta-c)(\zeta-c^{-1})} \bigg)^{2N} = e^{-2NV(\zeta)},$$ where the potential $V$ is given by $$\label{def potential}
V(\zeta) = \log \zeta + \log (\zeta - c) + \log (\zeta - c^{-1}) - \log (\zeta - \alpha c) - \log (\zeta - \alpha c^{-1})$$ and we take the principal branch for the logarithms. We require $g$ and $\gamma_{\mathbb{C}}$ to satisfy the following criteria (we define $\operatorname{supp}(\mu)$ as an open set for convenience):
1. \[item a\] $\gamma_{\mathbb{C}}$ is a closed curve surrounding $c^{-1}$ and $c$ once in the positive direction, but not surrounding $0$.
2. \[item b\] $e^{g}$ is analytic in $\mathbb{C}\setminus \overline{\operatorname{supp}(\mu)}$, where $\operatorname{supp}(\mu)$ is an open oriented curve satisfying $\operatorname{supp}(\mu) \subset \gamma_{\mathbb{C}}$.
3. \[item c\] The $g$-function satisfies $$\begin{aligned}
& g_{+}(\zeta) + g_{-}(\zeta) - V(\zeta) + \ell = 0, & & \mbox{for } \zeta \in \operatorname{supp}(\mu), \label{jump relation for g on the support} \\
& {\text{\upshape Re\,}}\big( g_{+}(\zeta) + g_{-}(\zeta) - V(\zeta) + \ell \big) < 0, & & \mbox{for } \zeta \in \gamma_{\mathbb{C}}\setminus \overline{\operatorname{supp}(\mu)}, \label{real part for g negative on gamma setminus S} \\
& {\text{\upshape Im\,}}\big( g_{+}(\zeta) - \tfrac{V(\zeta)}{2} + \tfrac{\ell}{2}\big), & & \mbox{is decreasing along }\operatorname{supp}(\mu), \label{real part for g positive in neighborhood of the support}\end{aligned}$$ for some constant $\ell \in \mathbb{C}$, and where $V$ is given by .
In approximation theory, the equality together with the inequality are usually refered to as the Euler-Lagrange variational conditions [@ST], and $\ell$ is the Euler-Lagrange constant. A measure $\mu$ satisfying – is called the equilibrium measure [@ST] in the external field $V$, because it is the unique minimizer of $$\begin{aligned}
\tilde{\mu} \mapsto \iint \log \frac{1}{|s-t|}d\tilde{\mu}(s)d\tilde{\mu}(t) + {\text{\upshape Re\,}}\int V(s) d\tilde{\mu}(s)\end{aligned}$$ among all probability measures $\tilde{\mu}$ supported on $\operatorname{supp}(\mu)$. Here we require in addition that is satisfied. This extra-condition characterizes $\operatorname{supp}(\mu)$ as a so-called $S$-curve [@Stahl; @GR; @Rak; @MFR; @KS; @MFR2].
Definition of $\mathcal{Q}$ and related computations
----------------------------------------------------
By taking the derivative in , we have $$\begin{aligned}
\label{lol5}
g_{+}'(\zeta) + g_{-}'(\zeta) - V'(\zeta) = 0, \qquad \zeta \in \operatorname{supp}(\mu),\end{aligned}$$ and by condition (b), $g'$ is analytic in $\mathbb{C}\setminus \overline{\operatorname{supp}(\mu)}$. Therefore, the function $$\begin{aligned}
\label{def of Q}
\mathcal{Q}(\zeta) := \left( g^{\prime}(\zeta) - \frac{V^{\prime}(\zeta)}{2} \right)^{2}\end{aligned}$$ is meromorphic on $\mathbb{C}$. By , we get $$\begin{aligned}
\label{def of V'}
V'(\zeta) = \frac{1}{\zeta} + \frac{1}{\zeta-c^{-1}} + \frac{1}{\zeta-c} - \frac{1}{\zeta- \alpha c^{-1}} - \frac{1}{\zeta - \alpha c},\end{aligned}$$ from which we conclude that $\mathcal{Q}$ has a double zero at $\infty$, and double poles at $0$, $\alpha c$, $\alpha c^{-1}$, $c$ and $c^{-1}$. Since a meromorphic function on the Riemann sphere (genus $0$) has as many poles as zeros, $\mathcal{Q}$ has eight other zeros. As $\zeta \to \infty$, we have $g'(\zeta) = \zeta^{-1} + \bigO(\zeta^{-2})$, from which we get $\mathcal{Q}(\zeta) = 2^{-2} \zeta^{-2} + \bigO(\zeta^{-3})$. Therefore, $\mathcal{Q}$ can be written in the form $$\label{def xi prime without the cut yet}
\mathcal{Q}(\zeta) = \frac{\Pi(\zeta)}{4\zeta^{2} (\zeta-\alpha c)^{2} (\zeta-\alpha c^{-1})^{2} (\zeta-c)^{2} (\zeta-c^{-1})^{2}},$$ where $\Pi$ is a monic polynomial of degree $8$ which remains to be determined. If we assume that $g'(\zeta)$ remains bounded for $\zeta \in \mathbb{C}$, then we can deduce from and the leading order term for $\mathcal{Q}(\zeta)$ as $\zeta \to \zeta_{\star} \in \{0,\alpha c, \alpha c^{-1}, c, c^{-1}\}$: $$\begin{aligned}
& \mathcal{Q}(\zeta) = 2^{-2}\zeta^{-2} + \bigO(\zeta^{-1}), & & \mbox{ as } \zeta \to 0, \label{Q 0} \\
& \mathcal{Q}(\zeta) = 2^{-2}(\zeta - \alpha c)^{-2} + \bigO((\zeta - \alpha c)^{-1}), & & \mbox{ as } \zeta \to \alpha c, \label{Q alpha} \\
& \mathcal{Q}(\zeta) = 2^{-2}(\zeta - \alpha c^{-1})^{-2} + \bigO((\zeta - \alpha c^{-1})^{-1}), & & \mbox{ as } \zeta \to \alpha c^{-1}, \label{Q beta alpha} \\
& \mathcal{Q}(\zeta) = 2^{-2}(\zeta - c)^{-2} + \bigO((\zeta - c)^{-1}), & & \mbox{ as } \zeta \to c, \label{Q 0 2} \\
& \mathcal{Q}(\zeta) = 2^{-2}(\zeta - c^{-1})^{-2} + \bigO((\zeta - c^{-1})^{-1}), & & \mbox{ as } \zeta \to c^{-1}. \label{Q 0 1}\end{aligned}$$ By combining these asymptotics with , we get $$\begin{aligned}
& \Pi(0) = \alpha^{4}, & & \Pi(\alpha c) = (1-\alpha)^{8}c^{8}, & & \Pi(\alpha c^{-1}) = {\displaystyle}(1-\alpha)^{8}\alpha^{4}, \nonumber \\
& \Pi(c) = (1-\alpha)^{8}c^{8}, & & \Pi(c^{-1}) = (1-\alpha)^{8}\alpha^{-4}. \label{sqrt pi8}\end{aligned}$$ This gives $5$ linear equations for the $8$ unknown coefficients of $\Pi$, which is not enough to determine $\Pi$ (and hence, $\mathcal{Q}$). Therefore, one needs to make a further assumption: we assume that we can find $\Pi$ in the form $$\label{assumption on Pi}
\Pi(\zeta) = (\zeta-r_{1})^{2}(\zeta-r_{2})^{2}(\zeta-r_{3})^{2}(\zeta-r_{+})(\zeta-r_{-}).$$ As we will see, Assumption implies that $\operatorname{supp}(\mu)$ consists of a single curve (“one-cut regime"). This assumption is justified if we can: 1) find $r_{1}$, $r_{2}$, $r_{3}$, $r_{+}$, $r_{-}$ so that holds and 2) construct a $g$-function via which satisfies the properties \[item a\]–\[item b\]–\[item c\].
Substituting in , we obtain $5$ *non-linear* equations for the $5$ unknowns $r_{1}$, $r_{2}$, $r_{3}$, $r_{+}$, $r_{-}$. This system turns out to have quite a few solutions – we need to select “the correct one". Let us define $r_{1}$, $r_{2}$, $r_{3}$, $r_{+}$, $r_{-}$ by –. It is a simple computation to verify that indeed holds in this case. We will show in Subsection \[subsection: mu and g function\] that this definition of $r_{1}$, $r_{2}$, $r_{3}$, $r_{+}$, $r_{-}$ is “the correct solution" to , in the sense that it allows to construct a $g$-function satisfying the properties \[item a\]–\[item b\]–\[item c\].
Let us briefly comment on how to find –. Unfortunately, we were not able to solve analytically the non-linear system obtained after substituting into . Instead, we have solved numerically (using the Newton–Raphson method) this system for every values of $\alpha \in (0,1)$. As already mentioned, the system possesses several solutions. In order to ensure numerical convergence to “the correct solution", we choose starting values of $r_{1}$, $r_{2}$ and $r_{3}$ so that holds. The expressions – have then been guessed by an inspection of the plots of $r_{1}(\alpha)$, $r_{2}(\alpha)$, $r_{3}(\alpha)$, $r_{+}(\alpha)$, $r_{-}(\alpha)$.
Critical trajectories of $\mathcal{Q}$ {#subsection: critical trajectories of Q}
--------------------------------------
In this subsection, we study the critical trajectories of $\mathcal{Q}$, which are relevant to define the $g$-function and study its properties.
Let $t \mapsto \zeta(t)$, $t \in [a,b]$ be a smooth parametrization of a curve $\sigma$, satisfying $\zeta'(t) \neq 0$ for all $t \in (a,b)$. $\sigma$ is a [*trajectory*]{} of the quadratic differential $\mathcal{Q}(\zeta) d\zeta^2$ if $\mathcal{Q}(\zeta(t)) \zeta'(t)^2 < 0$ for every $t \in (a,b)$, and an [*orthogonal trajectory*]{} if $\mathcal{Q}(\zeta(t)) \zeta'(t)^2 > 0$ for every $t \in (a,b)$. $\sigma$ is [*critical*]{} if it contains a zero or a pole of $\mathcal{Q}$. Note that these definitions are independent of the choice of the parametrization.
Since $r_{+}$ and $r_{-}$ are simple zeros of $\mathcal{Q}$, there are three critical trajectories (and also three orthogonal critical trajectories) emanating from each of the points $r_{\pm}$. Recall the definitions of $\gamma_{0},\gamma_{\alpha},\gamma_{1},\Sigma_{0},\Sigma_{\alpha}$ and $\Sigma_{1}$ given in Subsection \[subsection: saddle points and liquid region\].
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\[lemma:Sigma0 and Sigma alpha\] The arcs $\overline{\Sigma_{0}}$, $\overline{\Sigma_{\alpha}}$ and $\overline{\Sigma_{1}}$ are three critical trajectories of $\mathcal{Q}(\zeta) d\zeta^2$ joining $r_{-}$ with $r_{+}$, and $\gamma_{0} \setminus \Sigma_{0}$, $\gamma_{\alpha} \setminus\Sigma_{\alpha}$ and $\gamma_{1} \setminus\Sigma_{1}$ are each the union of two critical orthogonal trajectories of $\mathcal{Q}(\zeta) d\zeta^2$. An illustration is shown in Figure \[fig: crit traj alpha 04\].
Let $t \mapsto \zeta = \zeta(t) = c^{-1} + R_{1}e^{it}$, $t \in [-\pi,\pi]$, be a parametrization of $\gamma_{1}$. Writing $r_{\pm} = c^{-1} + R_{1}e^{\pm i \theta_{1}}$ with $\theta_{1} \in (\frac{2\pi}{3},\pi)$, and noting that $\zeta' = iRe^{it}$, we have $$\begin{aligned}
(\zeta-r_{+})(\zeta-r_{-}) = 2 R_{1}^{2} e^{i t}(\cos t - \cos \theta_{1}) \quad \mbox{ and } \quad \frac{(\zeta')^{2}}{(\zeta-c^{-1})^{2}} = -1.\end{aligned}$$ Therefore, we get $$\begin{aligned}
\mathcal{Q}(\zeta)(\zeta')^{2} & \; = (\zeta')^{2} \frac{(\zeta-r_{1})^{2}(\zeta-r_{2})^{2}(\zeta-r_{3})^{2}(\zeta-r_{+})(\zeta-r_{-})}{4\zeta^{2} (\zeta-\alpha c)^{2} (\zeta-\alpha c^{-1})^{2} (\zeta-c)^{2} (\zeta-c^{-1})^{2}} \nonumber \\
& \; = - R_{1}^{2}e^{it} \frac{(\zeta-r_{1})^{2}(\zeta-r_{2})^{2}(\zeta-r_{3})^{2}(\cos t - \cos \theta_{1})}{2\zeta^{2} (\zeta-\alpha c)^{2} (\zeta-\alpha c^{-1})^{2} (\zeta-c)^{2}}. \label{Q on circle form 1}\end{aligned}$$ Using , we show that $(\zeta - r_{2}) = 2R_{1}e^{\frac{it}{2}}\cos \frac{t}{2}$, and $$\begin{aligned}
(\zeta - r_{1})(\zeta - r_{3}) & \; = 2R_{1}^{2}e^{it}\left( \cos t + \frac{\alpha^{2} + (2-\alpha)\sqrt{1-\alpha + \alpha^{2}}}{2(1-\alpha)} \right), \\
\zeta (\zeta-c) & \; = 2R_{1}^{2}e^{it} \left( \cos t + \frac{2-3\alpha + 2\alpha^{2}}{2(1-\alpha)\sqrt{1-\alpha + \alpha^{2}}} \right), \\
(\zeta - \alpha c)(\zeta - \alpha c^{-1}) & \; = 2R_{1}^{2} e^{it} \left( \cos t + \frac{2-\alpha + \alpha^{2}}{2\sqrt{1-\alpha + \alpha^{2}}} \right).\end{aligned}$$ Substituting the above expressions in , we find $$\begin{aligned}
\label{Q on circle form 2}
\mathcal{Q}(\zeta)(\zeta')^{2} = \frac{(\cos \theta_{1} - \cos t)\cos^{2} \frac{t}{2} \left( \cos t + \frac{\alpha^{2} + (2-\alpha)\sqrt{1-\alpha + \alpha^{2}}}{2(1-\alpha)} \right)^{2}}{2\left( \cos t + \frac{2-3\alpha + 2\alpha^{2}}{2(1-\alpha)\sqrt{1-\alpha + \alpha^{2}}} \right)^{2}\left( \cos t + \frac{2-\alpha + \alpha^{2}}{2\sqrt{1-\alpha + \alpha^{2}}} \right)^{2}}.\end{aligned}$$ We verify by direct computations that $$\begin{aligned}
\frac{\alpha^{2} + (2-\alpha)\sqrt{1-\alpha + \alpha^{2}}}{2(1-\alpha)} > \frac{2-3\alpha + 2\alpha^{2}}{2(1-\alpha)\sqrt{1-\alpha + \alpha^{2}}} > \frac{2-\alpha + \alpha^{2}}{2\sqrt{1-\alpha + \alpha^{2}}} > 1,\end{aligned}$$ and thus the right-hand-side of is negative for $t \in (-\theta_{1},\theta_{1})$, positive for $t \in (-\pi,-\theta_{1})\cup(\theta_{1},\pi)$ and zero for $t = -\pi,-\theta_{1},\theta_{1},\pi$. We conclude that $\overline{\Sigma_{1}}$ is a critical trajectory and that $\gamma_{1}\setminus \Sigma_{1}$ is the union of two orthogonal critical trajectories. The statement about $\overline{\Sigma_{\alpha}}$, $\gamma_{\alpha}\setminus \Sigma_{\alpha}$, $\overline{\Sigma_{0}}$, $\gamma_{0}\setminus \Sigma_{0}$ can be proved a similar way, and we provide less details. For $\zeta = \zeta(t) = R_{0}e^{it}$, $t \in [-\pi,\pi]$, after long but straightforward computations, we obtain $$\begin{aligned}
\label{Q on circle gamma_0}
\mathcal{Q}(\zeta)(\zeta')^{2} = \frac{(\cos \theta_{0} - \cos t)\cos^{2} \frac{t}{2} \left( \cos t - \frac{(1+\alpha)\sqrt{1-\alpha + \alpha^{2}} - (1-\alpha)^{2}}{2\alpha} \right)^{2}}{2\left( \cos t - \frac{1-\alpha + 2\alpha^{2}}{2\alpha\sqrt{1-\alpha + \alpha^{2}}} \right)^{2}\left( \cos t - \frac{2-\alpha + \alpha^{2}}{2\sqrt{1-\alpha + \alpha^{2}}} \right)^{2}}.\end{aligned}$$ Since $$\begin{aligned}
\frac{1-\alpha + 2\alpha^{2}}{2\alpha\sqrt{1-\alpha + \alpha^{2}}} > \frac{(1+\alpha)\sqrt{1-\alpha + \alpha^{2}} - (1-\alpha)^{2}}{2\alpha} > \frac{2-\alpha + \alpha^{2}}{2\sqrt{1-\alpha + \alpha^{2}}} > 1,\end{aligned}$$ we infer that $\overline{\Sigma_{0}}$ is a critical trajectory and that $\gamma_{0}\setminus \Sigma_{0}$ is the union of two orthogonal critical trajectories. For $\zeta = \zeta(t) = \alpha c^{-1}+ R_{\alpha}e^{it}$, $t \in [-\pi,\pi]$, we obtain $$\begin{aligned}
\label{Q on circle gamma_a}
\mathcal{Q}(\zeta)(\zeta')^{2} = \frac{(\cos t - \cos \theta_{\alpha})\sin^{2} \frac{t}{2} \left( \cos t + \frac{1-(1-2\alpha)\sqrt{1-\alpha + \alpha^{2}}}{2\alpha (1-\alpha)} \right)^{2}}{2\left( \cos t - \frac{1-\alpha + 2\alpha^{2}}{2\alpha\sqrt{1-\alpha + \alpha^{2}}} \right)^{2}\left( \cos t + \frac{2-3\alpha + 2\alpha^{2}}{2(1-\alpha)\sqrt{1-\alpha + \alpha^{2}}} \right)^{2}}\end{aligned}$$ with $$\begin{aligned}
\frac{1-(1-2\alpha)\sqrt{1-\alpha + \alpha^{2}}}{2\alpha (1-\alpha)}> \frac{2-3\alpha + 2\alpha^{2}}{2(1-\alpha)\sqrt{1-\alpha + \alpha^{2}}} > 1, \quad \frac{1-\alpha + 2\alpha^{2}}{2\alpha\sqrt{1-\alpha + \alpha^{2}}}>1.\end{aligned}$$ Therefore, we deduce from an inspection of that $\overline{\Sigma_{\alpha}}$ is a critical trajectory and that $\gamma_{\alpha}\setminus \Sigma_{\alpha}$ is the union of two orthogonal critical trajectories. This finishes the proof.
Branch cut structure and the zero set of $\phi$
-----------------------------------------------
As can be seen in , $g'$ can be expressed as $$\begin{aligned}
\label{lol12}
g'(\zeta) = \frac{V'(\zeta)}{2} + \mathcal{Q}(\zeta)^{1/2},\end{aligned}$$ for a certain branch of $\mathcal{Q}(\zeta)^{1/2}$. To obtain a $g$-function with the desired properties (a)-(b)-(c), it turns out that the branch cut needs to be taken along the critical trajectory $\Sigma_{1}$ (as in Subsection \[subsection: saddle points and liquid region\]).
\[def:Q\^[1/2]{}\] We define $\mathcal{Q}^{1/2}$ as $$\begin{aligned}
\label{def of Q^{1/2}}
\mathcal{Q}(\zeta)^{1/2} = \frac{(\zeta-r_{1})(\zeta-r_{2})(\zeta-r_{3})\sqrt{(\zeta-r_{+})(\zeta-r_{-})}}{2\zeta (\zeta-\alpha c) (\zeta-\alpha c^{-1}) (\zeta-c) (\zeta-c^{-1})},\end{aligned}$$ where the branch cut for $\sqrt{(\zeta-r_{+})(\zeta-r_{-})}$ is taken on $\Sigma_{1}$ such that $$\begin{aligned}
\sqrt{(\zeta-r_{+})(\zeta-r_{-})} = \zeta + \bigO(1), \qquad \mbox{as } \zeta \to \infty.\end{aligned}$$
It will also be convenient to define a primitive of $\mathcal{Q}^{1/2}$.
\[def:phi\] We define $\phi:\mathbb{C}\setminus \big( (-\infty,c^{-1}] \cup \{c^{-1}+R_{1}e^{it}: -\pi \leq t \leq \theta_{1} \} \big) \to \mathbb{C}$ by $$\label{def of phi}
\phi(\zeta) = \int_{r_{+}}^{\zeta} \mathcal{Q}(\xi)^{1/2}d\xi,$$ where the path of integration does not intersect $(-\infty,c^{-1}] \cup \{c^{-1}+R_{1}e^{it}: -\pi \leq t \leq \theta_{1} \}$.
We first state some basic properties of $\phi$. By –, $\mathcal{Q}^{1/2}$ has simple poles at $0$, $\alpha c$, $\alpha c^{-1}$, $c$ and $c^{-1}$, and the residues are real. Also, since $\Sigma_{1}$ is a critical trajectory of $\mathcal{Q}$, we have $\phi_{\pm}(\zeta) \in i \mathbb{R}$ for $\zeta \in \Sigma_{1}$. Therefore, ${\text{\upshape Re\,}}\phi$ is single-valued and continuous in $\mathbb C \setminus \{0,\alpha c, \alpha c^{-1}, c, c^{-1}\}$, and ${\text{\upshape Re\,}}\phi$ is also harmonic in $\mathbb C \setminus (\Sigma_1 \cup \{0,\alpha c, \alpha c^{-1}, c, c^{-1}\})$. Finally, by combining Definition \[def:Q\^[1/2]{}\] with –, we have $$\label{eq:Nphinearpoles}
\begin{aligned}
\phi(\zeta) & = - \frac{1}{2} \log \zeta + \bigO(1) \text{ as } \zeta \to 0, &
\lim_{\zeta \to 0} {\text{\upshape Re\,}}\phi(\zeta) = + \infty, \\
\phi(\zeta) & = \frac{1}{2} \log(\zeta-\alpha c) + \bigO(1) \text{ as } \zeta \to \alpha c, &
\lim_{\zeta \to \alpha c} {\text{\upshape Re\,}}\phi(\zeta) = - \infty, \\
\phi(\zeta) & = \frac{1}{2} \log(\zeta -\alpha c^{-1}) + \bigO(1) \text{ as } \zeta \to \alpha c^{-1}, &
\lim_{\zeta \to \alpha c^{-1}} {\text{\upshape Re\,}}\phi(\zeta) = - \infty, \\
\phi(\zeta) & = -\frac{1}{2} \log(\zeta -c) + \bigO(1) \text{ as } \zeta \to c, &
\lim_{\zeta \to c} {\text{\upshape Re\,}}\phi(\zeta) = + \infty, \\
\phi(\zeta) & = -\frac{1}{2} \log(\zeta -c^{-1}) + \bigO(1) \text{ as } \zeta \to c^{-1}, &
\lim_{\zeta \to c^{-1}} {\text{\upshape Re\,}}\phi(\zeta) = + \infty, \\
\phi(\zeta) & = \frac{1}{2} \log(\zeta) + \bigO(1) \text{ as } \zeta \to \infty, &
\lim_{\zeta \to \infty} {\text{\upshape Re\,}}\phi(\zeta) = + \infty.
\end{aligned}$$
In the rest of this subsection, we determine the zero set $\mathcal N_\phi$ of $\phi$. This will be useful in Subsection \[subsection: mu and g function\] to establish the \[item a\]-\[item b\]-\[item c\] properties of the $g$-function. Let us define $$\label{eq:Nphi}
\mathcal N_\phi = \{ z \in \mathbb{C} : {\text{\upshape Re\,}}\phi(z)=0\}.$$
\[lem:Nphihigh\] We have $$\begin{aligned}
\label{zero set of phi explicitly determined}
\mathcal{N}_{\phi} = \Sigma_{0} \cup \Sigma_{\alpha} \cup \Sigma_{1}.\end{aligned}$$ In particular, $\mathcal{N}_{\phi}$ divides the complex plane in three regions. The sign of ${\text{\upshape Re\,}}\phi$ in these regions is as shown in Figure \[fig: crit traj alpha 04\].
By Lemma \[lemma:Sigma0 and Sigma alpha\], it holds that $$\begin{aligned}
\label{eq subseteq}
\mathcal{N}_{\phi} \supseteq \Sigma_{0} \cup \Sigma_{\alpha} \cup \Sigma_{1}.\end{aligned}$$ We now prove the inclusion $\subseteq$. We first show that $$\begin{aligned}
\label{mathcalN on the real line}
\mathcal{N}_{\phi} \cap \mathbb{R} = (\Sigma_{0} \cup \Sigma_{\alpha} \cup \Sigma_{1})\cap \mathbb{R} = \{\alpha c^{-1} - R_{\alpha}, R_{0}, c^{-1} + R_{1} \}.\end{aligned}$$ By Definitions \[def:Q\^[1/2]{}\] and \[def:phi\], $\phi' = Q_{\alpha}^{1/2}$ changes sign when it crosses each of the nine points $r_{1}$, $0$, $\alpha c$, $r_{2}$, $\alpha c^{-1}$, $c$, $r_{3}$, $c^{-1}$, $c^{-1}+R_{1}$. Since $\phi'(\zeta) = 2^{-1}\zeta^{-1} + \bigO(\zeta^{-2})$ as $\zeta \to \infty$, we have $\phi' > 0$ on the intervals $$\begin{aligned}
(r_{1},0), \quad (\alpha c,r_{2}), \quad (\alpha c^{-1},c), \quad (r_{3},c^{-1}), \quad (c^{-1}+R_{1},+\infty),\end{aligned}$$ and $\phi' < 0$ on the intervals $$\begin{aligned}
(-\infty,r_{1}), \quad (0,\alpha c), \quad (r_{2},\alpha c^{-1}), \quad (c,r_{3}), \quad (c^{-1},c^{-1}+R_{1}).\end{aligned}$$ By , we have $$\begin{aligned}
{\text{\upshape Re\,}}\phi(\alpha c^{-1} - R_{\alpha}) = {\text{\upshape Re\,}}\phi (R_{0}) = {\text{\upshape Re\,}}\phi(c^{-1}+R_{1}) = 0,\end{aligned}$$ so ${\text{\upshape Re\,}}\phi$ admits no other zeros on $(0,\alpha c) \cup (\alpha c^{-1},c) \cup (c^{-1},+\infty)$. On the intervals $(-\infty,0)$ and $(c,c^{-1})$, ${\text{\upshape Re\,}}\phi$ admits a local minimum at $r_{1}$ and $r_{3}$, respectively, and on the interval $(\alpha c, \alpha c^{-1})$, it admits a local maximum at $r_{2}$. Thus holds true if we show that $$\begin{aligned}
\label{pos and neg real part at r1 r2 r3}
{\text{\upshape Re\,}}\phi(r_{1}) > 0, \qquad {\text{\upshape Re\,}}\phi(r_{2}) < 0, \quad \mbox{ and } \quad {\text{\upshape Re\,}}\phi(r_{3}) > 0.\end{aligned}$$ By Lemma \[lemma:Sigma0 and Sigma alpha\], ${\text{\upshape Re\,}}\phi$ is strictly monotone on each of the curves $(\gamma_{0}\setminus \Sigma_{0}) \cap \mathbb{C}^{+}$, $(\gamma_{\alpha}\setminus \Sigma_{\alpha}) \cap \mathbb{C}^{+}$ and $(\gamma_{1}\setminus \Sigma_{1}) \cap \mathbb{C}^{+}$. The expressions , and , together with Definition \[def:Q\^[1/2]{}\], allow to conclude that ${\text{\upshape Re\,}}\phi$ is strictly increasing on $(\gamma_{0}\setminus \Sigma_{0}) \cap \mathbb{C}^{+}$ oriented from $r_{+}$ to $r_{1}$, strictly increasing on $(\gamma_{\alpha}\setminus \Sigma_{\alpha}) \cap \mathbb{C}^{+}$ oriented from $r_{+}$ to $r_{3}$, and strictly decreasing on $(\gamma_{1}\setminus \Sigma_{1}) \cap \mathbb{C}^{+}$ oriented from $r_{+}$ to $r_{2}$. In particular this proves , and thus .
Assume $\mathcal N_{\phi}$ if of the form $\Sigma_{0} \cup \Sigma_{\alpha} \cup \Sigma_{1} \cup \sigma$ for a certain curve $\sigma$ distinct from $\Sigma_{0}$, $\Sigma_{\alpha}$ and $\Sigma_{1}$. Since $\phi_{\pm}'(\zeta) \neq 0$ for $\zeta \in \Sigma_{1}$, we must have $\sigma \cap \Sigma_{1} = \emptyset$. Also, in view of , $\sigma$ cannot intersect the real axis. Then $\sigma$ must be a closed contour in $\mathbb{C}\setminus \big( \mathbb{R}\cup \Sigma_{1} \big)$, and the max/min principle for harmonic functions would then imply that ${\text{\upshape Re\,}}\phi$ in constant on the whole bounded region delimited by $\sigma$. By , ${\text{\upshape Re\,}}\phi$ is clearly not constant on such domain, so we arrive at a contradiction, and we conclude that $\mathcal{N}_{\phi} = \Sigma_{0} \cup \Sigma_{\alpha} \cup \Sigma_{1}$.
Thus $\mathcal{N}_{\phi}$ divides the complex plane in three regions in which ${\text{\upshape Re\,}}\phi$ does not change sign. The signs in each of these regions is then determined immediately by (or equivalently, by ).
Definition and properties of $g$ {#subsection: mu and g function}
--------------------------------
\[def:mu0g\] We define the measure $\mu$ by $$\begin{aligned}
d\mu(\zeta) & = \frac{1}{\pi i} \mathcal{Q}_{-}(\zeta)^{1/2} d\zeta \nonumber \\
& = \frac{1}{\pi i} \frac{(\zeta-r_{1})(\zeta-r_{2})(\zeta-r_{3})\sqrt{(\zeta-r_{+})(\zeta-r_{-})}_{-}}{2\zeta (\zeta-\alpha c) (\zeta-\alpha c^{-1}) (\zeta-c) (\zeta-c^{-1})} d\zeta,
\qquad \zeta \in \Sigma_1, \label{def of mu}\end{aligned}$$ where $\Sigma_1 = \operatorname{supp}(\mu)$ is given by , and is oriented from $r_{-}$ to $r_{+}$; so $\mathcal{Q}_{-}(\zeta)^{1/2}$ denotes the limit of $\mathcal{Q}(\xi)^{1/2}$ as $\xi \to \zeta \in \Sigma_1$ with $\xi$ in the exterior of the circle $\gamma_1$.
The measure $\mu$ defined in is a probability measure.
We compute $\int_{\Sigma_{1}} d\mu$ by residue calculation. Since $\mathcal{Q}_{+} = - \mathcal{Q}_{-}$, we have $$\begin{aligned}
\int_{\Sigma_{1}} d\mu(\zeta) = \frac{1}{2\pi i}\int_{\mathcal{C}} \mathcal{Q}(\zeta)^{1/2}d\zeta,\end{aligned}$$ where $\mathcal{C}$ is a closed curve surrounding $\Sigma_{1}$ once in the positive direction, but not surrounding any of the poles of $\mathcal{Q}$. By deforming $\mathcal{C}$ into another contour $\widetilde{\mathcal{C}}$ surrounding $0$, $\alpha c$, $\alpha c^{-1}$, $c$ and $c^{-1}$, we pick up some residues: $$\begin{aligned}
\label{int of mu with some residue}
\int_{\Sigma_{1}} d\mu(\zeta) = -\sum_{\zeta_{\star}\in \mathcal{P}}\mbox{Res} \left( \mathcal{Q}(\zeta)^{1/2}, \zeta = \zeta_{\star} \right) + \frac{1}{2\pi i}\int_{\widetilde{\mathcal{C}}} \mathcal{Q}(\zeta)^{1/2}d\zeta\end{aligned}$$ where $\mathcal{P}=\{0,\alpha c,\alpha c^{-1},c,c^{-1}\}$. By combining Definition \[def:Q\^[1/2]{}\] with –, we have $$\begin{aligned}
& \mbox{Res} \left( \mathcal{Q}(\zeta)^{1/2}, \zeta = 0 \right) = - \frac{1}{2}, & & \mbox{Res} \left( \mathcal{Q}(\zeta)^{1/2}, \zeta = \alpha c \right) = \frac{1}{2}, \nonumber \\
& \mbox{Res} \left( \mathcal{Q}(\zeta)^{1/2}, \zeta = \alpha c^{-1} \right) = \frac{1}{2}, & & \mbox{Res} \left( \mathcal{Q}(\zeta)^{1/2}, \zeta = c \right) = - \frac{1}{2}, \nonumber \\
& \mbox{Res} \left( \mathcal{Q}(\zeta)^{1/2}, \zeta = c^{-1} \right) = - \frac{1}{2}, \label{residues of Q^1/2}\end{aligned}$$ and since $\mathcal{Q}(\zeta)^{1/2} = \frac{1}{2\zeta} + \bigO(\zeta^{-2})$ as $\zeta \to \infty$, we find $$\begin{aligned}
\label{eq:mu0total}
\int_{\Sigma_{1}}d\mu(\zeta) = 1.\end{aligned}$$ It remains to show that $\mu$ has a positive density on $\Sigma_{1}$. Let $\zeta(t) = c^{-1}+R_{1} e^{i t}$, $-\theta_{1} < t < \theta_{1}$, be a parametrization of $\Sigma_1$. Consider the function $$\label{eq:mu0mass}
t \mapsto \int_{r_-}^{\zeta(t)} d\mu = \frac{1}{\pi i} \int_{r_-}^{\zeta(t)} \mathcal{Q}_{-}(\xi)^{1/2} d\xi,$$ whose derivative is given by $$\begin{aligned}
\label{lol11}
\frac{1}{\pi i} \mathcal{Q}_{-}(\zeta(t))^{1/2} \zeta'(t).\end{aligned}$$ Since $\mathcal{Q}(\zeta(t)) (\zeta'(t))^2 < 0$ for $t \in (-\theta_{1}, \theta_{1})$ by Lemma \[lemma:Sigma0 and Sigma alpha\], is real and non-zero. Note also that the function vanishes for $t = - \theta_{1}$ and equals $1$ for $t= \theta_{1}$ by . Therefore is strictly positive.
\[def: g\] The $g$-function is defined by $$\label{def of g function}
g(\zeta) = \int_{\Sigma_1} \log(\zeta-\xi) d\mu(\xi), \qquad \zeta \in {{\mathbb C}}\setminus
\left((-\infty, r_{2}] \cup \{c^{-1}+R_{1} e^{i t} : -\pi \leq t \leq \theta_{1} \} \right),$$ where for each $\xi \in \Sigma_1$, the function $\zeta \mapsto \log(\zeta-\xi)$ has a branch cut along $(-\infty, r_{2}] \cup
\{c^{-1}+R_{1} e^{i t} : -\pi \leq t \leq \arg \xi\}$ and behaves like $\log(\zeta-\xi) = \log |\zeta| + \bigO(\zeta^{-1}) $, as $\zeta \to +\infty$.
We define the variational constant $\ell \in {{\mathbb C}}$ by $$\begin{aligned}
\label{def of ell}
\ell = - 2g(r_{+}) + V(r_{+}).\end{aligned}$$
The next proposition shows, among other things, that Definition \[def: g\] for $g$ is consistent with , and that $g$ satisfies .
\[prop:phi and g relations\] The functions $g$ and $\phi$ are related by $$\begin{aligned}
\label{phi and g relation}
\phi(\zeta) = g(\zeta) - \frac{V(\zeta)}{2} + \frac{\ell}{2}, \qquad \zeta \in \mathbb{C}\setminus \big( (-\infty,r_{2}] \cup \{c^{-1}+R_{1} e^{i t} : -\pi \leq t \leq \theta_{1} \} \big)\end{aligned}$$ and we have $$\begin{aligned}
& g_{+}(\zeta) + g_{-}(\zeta) - V(\zeta) + \ell = 0, & & \mbox{for } \zeta \in \Sigma_{1}, \label{prop g add} \\
& g_{+}(\zeta)-g_{-}(\zeta) = 2\phi_{+}(\zeta) = - 2 \phi_{-}(\zeta), & & \mbox{for } \zeta \in \Sigma_{1}. \label{prop g sous}\end{aligned}$$ Furthermore, the $g$-function satisfies the properties \[item a\]–\[item b\]–\[item c\] listed at the beginning of Section \[section: g-function\] with $\gamma_{\mathbb{C}} = \gamma_{1}$.
We first prove . For a fixed $\zeta \in \mathbb{C}\setminus \Sigma_{1}$, we have $$\begin{aligned}
& g'(\zeta) = \int_{\Sigma_{1}} \frac{d\mu(\xi)}{\zeta-\xi} = \frac{1}{2\pi i} \int_{\mathcal{C}} \frac{Q(\xi)^{1/2}}{\zeta - \xi}d\xi,\end{aligned}$$ where $\mathcal{C}$ is a closed curve surrounding $\Sigma_{1}$ once in the positive direction, but not surrounding any of the poles of $\mathcal{Q}$, and not surrounding $\zeta$. By deforming $\mathcal{C}$ into another contour $\widetilde{\mathcal{C}}$ surrounding $0$, $\alpha c$, $\alpha c^{-1}$, $c$, $c^{-1}$ and $\zeta$, we obtain $$\begin{aligned}
\label{int of g' with some residue}
\int_{\Sigma_{1}} \frac{d\mu(\xi)}{\zeta - \xi} = -\sum_{\xi_{\star}\in \mathcal{P}}\mbox{Res} \left( \frac{\mathcal{Q}(\xi)^{1/2}}{\zeta- \xi}, \xi = \xi_{\star} \right) + \mathcal{Q}(\zeta)^{1/2} + \frac{1}{2\pi i}\int_{\widetilde{\mathcal{C}}} \frac{\mathcal{Q}(\xi)^{1/2}}{\zeta - \xi}d\xi,\end{aligned}$$ where $\mathcal{P}=\{0,\alpha c,\alpha c^{-1},c,c^{-1}\}$. By deforming $\widetilde{\mathcal{C}}$ to $\infty$, noting that $\mathcal{Q}(\xi)^{1/2} = \bigO(\xi^{-1})$ as $\xi \to \infty$, the integral on the right-hand-side of is $0$. The sum can be evaluated using the residues , and we get $$\begin{aligned}
\int_{\Sigma_{1}} \frac{d\mu(\xi)}{\zeta - \xi} = \frac{1}{2\zeta} - \frac{1}{2(\zeta - \alpha c)} - \frac{1}{2(\zeta - \alpha c^{-1})} + \frac{1}{2(\zeta -c)} + \frac{1}{2(\zeta - c^{-1})} + \mathcal{Q}(\zeta)^{1/2}.\end{aligned}$$ Using and $\phi' = \mathcal{Q}^{1/2}$, the above can be rewritten as $$\begin{aligned}
g'(\zeta) = \frac{V'(\zeta)}{2} + \phi'(\zeta), \qquad \zeta \in \mathbb{C}\setminus \Sigma_{1}.\end{aligned}$$ Integrating this identity from $r_{+}$ to $\zeta$ along a path that does not intersect $(-\infty,c^{-1}]\cup \{c^{-1}+R_{1}e^{i t}:-\pi \leq t < \theta_{1}\}$, we obtain $$\begin{aligned}
g(\zeta) - g(r_{+}) = \frac{V(\zeta)}{2} - \frac{V(r_{+})}{2} + \phi(\zeta),\end{aligned}$$ where we have used $\phi(r_{+}) = 0$. Then follows from the definition of $\ell$ given by . Since $\mathcal{Q}_{+}^{1/2} = -\mathcal{Q}_{-}^{1/2}$ on $\Sigma_{1}$, by we have $$\begin{aligned}
& \phi_{+}(\zeta) + \phi_{-}(\zeta) = 0, \qquad \mbox{for } \zeta \in \Sigma_{1},\end{aligned}$$ from which and follow. The circle $\gamma_{1}$ encloses both $c$ and $c^{-1}$, and $0$ lies in the exterior of $\gamma_{1}$, so criterion \[item a\] is fulfilled. For $\zeta \in (-\infty,r_{2}]\cup \{c^{-1}+R_{1}e^{it}:-\pi \leq -\theta_{1} \}$, we have $$\begin{aligned}
g_{+}(\zeta)-g_{-}(\zeta) = \int_{\Sigma_{1}} \big( \log_{+}(\zeta-\xi)-\log_{-}(\zeta-\xi) \big) d\mu(\xi) = 2 \pi i \int_{\Sigma_{1}} d \mu(\xi) = 2\pi i,\end{aligned}$$ so $e^{g}$ is analytic in $\mathbb{C}\setminus \overline{\Sigma_{1}}$ and criterion \[item b\] is also fulfilled. For $\zeta \in \gamma_{1}\setminus \overline{\Sigma_{1}}$, by and Lemma \[lem:Nphihigh\], we have $$\begin{aligned}
{\text{\upshape Re\,}}\big( g_{+}(\zeta) + g_{-}(\zeta) - V(\zeta) + \ell \big) = {\text{\upshape Re\,}}\big( \phi_{+}(\zeta) + \phi_{-}(\zeta) \big) = 2 \, {\text{\upshape Re\,}}\phi(\zeta) < 0,\end{aligned}$$ as required in . Finally, by Definitions \[def:phi\] and \[def:mu0g\], for $\zeta \in \Sigma_{1}$ we have $$\begin{aligned}
{\text{\upshape Im\,}}\Big( g_{+}(\zeta) - \frac{V(\zeta)}{2} + \frac{\ell}{2}\Big) = {\text{\upshape Im\,}}\phi_{+}(\zeta) = {\text{\upshape Im\,}}\int_{r_{+}}^{\zeta} \mathcal{Q}_{+}^{1/2}(\xi)d\xi = \pi \int_{\zeta}^{r_{+}} d\mu(\xi)\end{aligned}$$ which is strictly decreasing as $\zeta$ goes from $r_{-}$ to $r_{+}$. So holds as well, and hence \[item c\], which finishes the proof.
Steepest descent for $U$ {#section: steepest descent for $U$}
========================
In this section, we will perform an asymptotic analysis of the RH problem for $U$ as $N \to + \infty$, by means of the Deift/Zhou steepest descent method [@DZ]. As mentioned in Section \[section: g-function\], the relevant contour to consider for the RH problem for $U$ is $\gamma_{\mathbb{C}} = \gamma_{1}$. The analysis is split in a series of transformations $U \mapsto T \mapsto S \mapsto R$. The first transformation $U \mapsto T$ of Section \[subsection: g function transformation\] uses the $g$-function obtained in Section \[section: g-function\] to normalize the RH problem at $\infty$. The opening of the lenses $T \mapsto S$ is realised in Section \[subsection: opening of the lenses\]. The last step $S\mapsto R$ requires some preparations that are done in Section \[subsection: parametrices\]: it consists of constructing approximations (called “parametrices") for $S$ in different regions of the complex plane. Finally, the $S \mapsto R$ transformation is carried out in Section \[subsection: small norm\].
First transformation: $U \mapsto T$ {#subsection: g function transformation}
-----------------------------------
We normalize the RH problem with the following transformation $$\label{def of T}
T(\zeta) = e^{N\ell\sigma_{3}}U(\zeta)e^{-2Ng(\zeta)\sigma_{3}}e^{- N\ell \sigma_{3}},$$ where $g$ and $\ell$ are defined in Definition \[def: g\]. Using , we can write the jumps for $T$ in terms of the function $\phi$ of Definition \[def:phi\]. From and , we find that $T$ satisfies the following RH problem.
### RH problem for $T$ {#rh-problem-for-t .unnumbered}
- $T : \mathbb{C}\setminus \gamma_{1} \to \mathbb{C}^{2\times 2}$ is analytic.
- By using , and , the jumps for $T$ are given by $$\begin{aligned}
& T_{+}(\zeta) = T_{-}(\zeta) \begin{pmatrix}
e^{-4N\phi_{+}(\zeta)} & 1 \\
0 & e^{-4N\phi_{-}(\zeta)}
\end{pmatrix}, & & \mbox{ for } \zeta \in \Sigma_{1} \subset \gamma_{1}, \label{jumps for T inside support} \\
& T_{+}(\zeta) = T_{-}(\zeta) \begin{pmatrix}
1 & e^{2N(\phi_{+}(\zeta)+\phi_{-}(\zeta))} \\
0 & 1
\end{pmatrix}, & & \mbox{ for } \zeta \in \gamma_{1} \setminus \overline{\Sigma_{1}}. \label{jumps for T outside support}\end{aligned}$$
- As $\zeta \to \infty$, we have $T(\zeta) = I + \bigO(\zeta^{-1})$.
As $\zeta$ tends to $r_{+}$ or $r_{-}$, $T(\zeta)$ remains bounded.
The following estimates for $T$ will be important for the saddle point analysis of Section \[section: saddle point analysis\].
\[prop:TandTinvsmall\] We have $T(\zeta) = \bigO(N^{1/6})$ and $T^{-1}(\zeta) = \bigO(N^{1/6})$ as $N \to \infty$, uniformly for $\zeta \in \mathbb C \setminus \gamma_1$. In addition, for every $\delta > 0$ fixed, we have $T(\zeta) = \bigO(1)$ and $T^{-1}(\zeta) = \bigO(1)$ as $N \to \infty$ uniformly for $$\label{eq:zawayfrombranchpoints}
\zeta \in \{ \zeta \in \mathbb C \setminus \gamma_{1} : |\zeta-r_+| \geq \delta, |\zeta-r_-| \geq \delta \}.$$
The rest of this section is devoted to the proof of Proposition \[prop:TandTinvsmall\].
Second transformation: $T \mapsto S$ {#subsection: opening of the lenses}
------------------------------------
Note that for $\zeta \in \Sigma_{1}$, the jumps for $T$ can be factorized as follows: $$\begin{aligned}
\label{factorization jumps}
\begin{pmatrix}
e^{-4N\phi_{+}(\zeta)} & 1 \\
0 & e^{-4N\phi_{-}(\zeta)}
\end{pmatrix} = \begin{pmatrix}
1 & 0 \\ e^{-4N\phi_{-}(\zeta)} & 1
\end{pmatrix} \begin{pmatrix}
0 & 1 \\ -1 & 0
\end{pmatrix} \begin{pmatrix}
1 & 0 \\ e^{-4N\phi_{+}(\zeta)} & 1
\end{pmatrix},\end{aligned}$$ where we used $\phi_{+}(\zeta) + \phi_{-}(\zeta) = 0$ for $\zeta \in \Sigma_{1}$. We define the lenses $\gamma_{+}$ and $\gamma_{-}$ by $$\begin{aligned}
& \gamma_{+} := \gamma_{\alpha} \setminus \overline{\Sigma_{\alpha}} \qquad \mbox{ and } \qquad \gamma_{-} := \gamma_{0} \setminus \overline{\Sigma_{0}},\end{aligned}$$ see also Figure \[fig:opening of the lenses\].
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The $T \mapsto S$ transformation is given by $S(\zeta) = T(\zeta) \mathcal{W}(\zeta)$, where $$\label{T to S transformation}
\mathcal{W}(\zeta) = \left\{ \begin{array}{l l}
\begin{pmatrix}
1 & 0 \\ -e^{-4N\phi(\zeta)} & 1
\end{pmatrix}, & \mbox{for } \zeta \mbox{ in the bounded region delimited by } \overline{\Sigma_{1} \cup \gamma_{+}}, \\
\begin{pmatrix}
1 & 0 \\ e^{-4N\phi(\zeta)} & 1
\end{pmatrix}, & \mbox{for } \zeta \mbox{ in the unbounded region delimited by } \overline{\Sigma_{1} \cup \gamma_{-}}, \\
I, & \mbox{otherwise}.
\end{array} \right.$$ $S$ satisfies the following RH problem.
### RH problem for $S$ {#rh-problem-for-s .unnumbered}
- $S : \mathbb{C}\setminus (\gamma_{1}\cup \gamma_{+} \cup \gamma_{-}) \to \mathbb{C}^{2\times 2}$ is analytic.
- The jumps for $S$ are given by $$\begin{aligned}
& S_{+}(\zeta) = S_{-}(\zeta) \begin{pmatrix}
0 & 1 \\
-1 & 0
\end{pmatrix}, & & \mbox{ for } \zeta \in \Sigma_{1}, \\
& S_{+}(\zeta) = S_{-}(\zeta) \begin{pmatrix}
1 & 0 \\
e^{-4N\phi(\zeta)} & 1
\end{pmatrix}, & & \mbox{ for } \zeta \in \gamma_{+} \cup \gamma_{-}, \\
& S_{+}(\zeta) = S_{-}(\zeta) \begin{pmatrix}
1 & e^{2N(\phi_{+}(\zeta)+\phi_{-}(\zeta))} \\
0 & 1
\end{pmatrix}, & & \mbox{ for } \zeta \in \gamma_{1} \setminus \overline{\Sigma_{1}}.\end{aligned}$$
- As $\zeta \to \infty$, we have $S(\zeta) = I + \bigO(\zeta^{-1})$.
As $\zeta$ tends to $r_{+}$ or $r_{-}$, $S(\zeta)$ remains bounded.
Parametrices {#subsection: parametrices}
------------
In this subsection, we find good approximations to $S$ in different regions of the complex plane. By Lemma \[lem:Nphihigh\], ${\text{\upshape Re\,}}\phi(\zeta) > 0$ for $\zeta \in \gamma_{+}\cup \gamma_{-}$, ${\text{\upshape Re\,}}\phi(\zeta) < 0$ for $\zeta \in \gamma_{1} \setminus \overline{\Sigma_{1}}$, and ${\text{\upshape Re\,}}\phi(\zeta) = 0$ for $\zeta \in \Sigma_{1}$. So the jumps for $S$ on $\gamma_{+}\cup\gamma_{-} \cup (\gamma_{1}\setminus \overline{\Sigma_{1}})$ are exponentially close to the identity matrix matrix as $N \to \infty$, uniformly outside fix neighborhoods of $r_{-}$ and $r_{+}$. By ignoring these jumps, we are left with the following RH problem, whose solution is denoted $P^{(\infty)}$. We will show in Subsection \[subsection: small norm\] that $P^{(\infty)}$ is a good approximation to $S$ away from $r_{+}$ and $r_{-}$.
### RH problem for $P^{(\infty)}$ {#rh-problem-for-pinfty .unnumbered}
- $P^{(\infty)} : \mathbb{C}\setminus \overline{\Sigma_{1}} \to \mathbb{C}^{2\times 2}$ is analytic.
- The jumps for $P^{(\infty)}$ are given by $$\begin{aligned}
& P^{(\infty)}_{+}(\zeta) = P^{(\infty)}_{-}(\zeta) \begin{pmatrix}
0 & 1 \\
-1 & 0
\end{pmatrix}, & & \mbox{ for } \zeta \in \Sigma_{1}.\end{aligned}$$
- As $\zeta \to \infty$, we have $P^{(\infty)}(\zeta) = I + \bigO(\zeta^{-1})$.
As $\zeta \to \zeta_{\star} \in \{r_{+},r_{-}\}$, $P^{(\infty)}(\zeta) = \bigO((\zeta-\zeta_{\star})^{-1/4})$.
The condition on the behavior of $P^{(\infty)}(\zeta)$ as $\zeta \to \zeta_{\star}\in\{r_{+},r_{-}\}$ has been added to ensure existence of a solution. This RH problem is independent of $N$, and its unique solution is given by $$P^{(\infty)}(\zeta) = \begin{pmatrix}
\frac{1}{2}(a(\zeta)+a(\zeta)^{-1}) & \frac{1}{2i}(a(\zeta)-a(\zeta)^{-1}), \\[0.2cm]
\frac{1}{-2i}(a(\zeta)-a(\zeta)^{-1}) & \frac{1}{2}(a(\zeta)+a(\zeta)^{-1})
\end{pmatrix},$$ where $a(\zeta) := \left(\frac{\zeta-r_{+}}{\zeta-r_{-}}\right)^{1/4}$ is analytic in $\mathbb{C}\setminus \Sigma_{1}$ and such that $a(\zeta) \sim 1$ as $\zeta \to \infty$.
Note that $P^{(\infty)}$ is not a good approximation to $S$ in small neighborhoods of $r_{+},r_{-}$; this can be seen from the behaviors $$\begin{aligned}
& S(\zeta)=\bigO(1) \quad \mbox{ and } \quad P^{(\infty)}(\zeta) = \bigO((\zeta-\zeta_{\star})^{-1/4}), & & \mbox{as } \zeta \to \zeta_{\star}\in\{r_{+},r_{-}\}.\end{aligned}$$ Let $\delta > 0$ in Proposition \[prop:TandTinvsmall\] be fixed, and let $\mathcal{D}_{r_{+}}$ and $\mathcal{D}_{r_{-}}$ be small open disks of radius $\delta/2$ centered at $r_{+}$ and $r_{-}$, respectively. We now construct local approximations $P^{(r_{+})}$ and $P^{(r_{-})}$ (called “local parametrices") to $S$ in $\mathcal{D}_{r_{+}}$ and $\mathcal{D}_{r_{-}}$, respectively. We require $P^{(r_{\pm})}$ to satisfy the same jumps as $S$ inside $\mathcal{D}_{r_{\pm}}$, to remain bounded as $\zeta \to r_{\pm}$, and to satisfy the matching condition $$\begin{aligned}
\label{matching conditions}
P^{(r_{\pm})}(\zeta) = (I + \bigO(N^{-1}))P^{(\infty)}(\zeta), \qquad \mbox{as } N \to + \infty,\end{aligned}$$ uniformly for $\zeta \in \partial \mathcal{D}_{r_{\pm}}$. The density of $\mu$ vanishes like a square root at the endpoints $r_{+}$ and $r_{-}$, and therefore $P^{(r_{\pm})}$ can be built in terms of Airy functions [@DKMVZ1]. These constructions are well-known and standard, so we do give the details. What is important for us is that $$\label{eq:PandPinvgrow}
P^{(r_{\pm})}(z) = \bigO(N^{\frac{1}{6}}), \quad P^{(r_{\pm})}(z)^{-1} = \bigO(N^{\frac{1}{6}}) \quad \text{ as } N \to \infty,$$ uniformly for $z \in \mathcal D_{r_{\pm}}$.
Small norm RH problem $R$ {#subsection: small norm}
-------------------------
The final transformation $S \mapsto R$ of the steepest descent is defined by $$\label{S to R transformation}
R(\zeta) = \left\{ \begin{array}{l l}
S(\zeta)P^{(\infty)}(\zeta)^{-1}, & \mbox{ for } \zeta \in \mathbb{C}\setminus (\overline{\mathcal{D}_{r_{+}}\cup \mathcal{D}_{r_{-}}}), \\
S(\zeta)P^{(r_{+})}(\zeta)^{-1}, & \mbox{ for } \zeta \in \mathcal{D}_{r_{+}}, \\
S(\zeta)P^{(r_{-})}(\zeta)^{-1}, & \mbox{ for } \zeta \in \mathcal{D}_{r_{-}}.
\end{array} \right.$$ Since $S$ and $P^{(r_{\pm})}$ satisfy the same jumps inside $\mathcal{D}_{r_{\pm}}$, $R$ is analytic inside $(\mathcal{D}_{r_{+}}\setminus \{r_{+}\}) \cup (\mathcal{D}_{r_{-}}\setminus \{r_{-}\})$. Furthermore, $S$ and $P^{(r_{\pm})}$ remain bounded near $r_{\pm}$, so the singularities of $R$ at $r_{\pm}$ are removable. We conclude that $R$ is analytic in $$\begin{aligned}
\label{analyticity of R}
\mathbb C \setminus \Big(\big( (\gamma_1 \cup \gamma_{+} \cup \gamma_{-})\setminus (\mathcal{D}_{r_{+}} \cup \mathcal{D}_{r_{-}})\big) \cup \partial \mathcal{D}_{r_+} \cup \partial \mathcal D_{r_-}\Big).\end{aligned}$$ By , the jumps $R_{-}^{-1}R_{+}$ are $\bigO(N^{-1})$ on $\partial \mathcal{D}_{r_{+}} \cup \partial \mathcal{D}_{r_{+}}$, and by Lemma \[lem:Nphihigh\], $R_{-}^{-1}R_{+} = \bigO(e^{-c N})$ on $(\gamma_1 \cup \gamma_{+} \cup \gamma_{-})\setminus (\mathcal{D}_{r_{+}} \cup \mathcal{D}_{r_{-}})$ for a certain $c > 0$. It follows by standard theory [@DKMVZ1; @DKMVZ] that $$\begin{aligned}
\label{lol35}
R(\zeta) = I + \bigO(N^{-1}), \qquad \mbox{as } N \to + \infty,\end{aligned}$$ uniformly for $\zeta$ in the domain . In particular, $R$ and $R^{-1}$ remain bounded as $N \to \infty$.
Inverting the transformations and , we get $$\begin{aligned}
T(\zeta) = R(\zeta) \times \left\{ \begin{array}{l l}
P^{(\infty)}(\zeta), & \mbox{ for } \zeta \in \mathbb{C}\setminus (\mathcal{D}_{r_{+}}\cup \mathcal{D}_{r_{-}}) \\
P^{(r_{+})}(\zeta), & \mbox{ for } \zeta \in \mathcal{D}_{r_{+}} \\
P^{(r_{-})}(\zeta), & \mbox{ for } \zeta \in \mathcal{D}_{r_{-}}
\end{array} \right\} \times \mathcal{W}(\zeta)^{-1}.\end{aligned}$$ By Lemma \[lem:Nphihigh\], $\mathcal{W}(\zeta)$ and $\mathcal{W}(\zeta)^{-1}$ are bounded as $N \to + \infty$, uniformly for $\zeta \in \mathbb{C}$. Proposition \[prop:TandTinvsmall\] follows then straightforwardly by using the estimates and .
Phase functions $\Phi$ and $\Psi$ {#section: phase functions}
=================================
In Section \[section: saddle point analysis\], we will prove Proposition \[prop:doubleintegrallimit\] via a saddle point analysis of the double contour integral . As it will turn out, the dominant part of the integrand as $N \to + \infty$ will be in the form $e^{2N(\Phi(\zeta;\xi,\eta)-\Phi(\omega;\xi,\eta))}$, for a certain function $\Phi$ which is described below. The analytic continuation of $\Phi$ to the second sheet of $\mathcal{R}_{\alpha}$ is denoted $\Psi$ – it will also play a role in the saddle point analysis and is presented below.
The content of this section is a preparation for the saddle point analysis of Section \[section: saddle point analysis\]. We will study the level set $$\begin{aligned}
\label{mathcalNPhi}
\mathcal{N}_{\Phi} = \{ \zeta \in \mathbb{C}: {\text{\upshape Re\,}}\Phi(\zeta) = {\text{\upshape Re\,}}\Phi(s) \},\end{aligned}$$ and also find the relevant contour deformations to consider.
Preliminaries
-------------
We start with a definition.
\[def: Phi and Psi\] For $(\xi,\eta) \in \mathcal H$ and $\zeta \in \mathbb{C}\setminus \big( (-\infty,c^{-1}] \cup \{c^{-1}+R_{1}e^{it}:-\pi \leq t \leq \theta_{1}\} \big)$, we define $\Phi$ and $\Psi$ by $$\begin{aligned}
\nonumber
\Phi(\zeta) & = \Phi(\zeta;\xi,\eta) \\
& = g(\zeta) - \frac{1+\xi-\eta}{2} \log \zeta + \frac{1+\xi}{2}\log \big( (\zeta-\alpha c)(\zeta-\alpha c^{-1}) \big) - \frac{1+\eta}{2}\log \big( (\zeta-c)(\zeta-c^{-1}) \big) + \frac{\ell}{2} \nonumber \\
\label{Phidef}
& = \phi(\zeta) - \frac{\xi-\eta}{2} \log \zeta + \frac{\xi}{2}\log \big( (\zeta-\alpha c)(\zeta-\alpha c^{-1}) \big) - \frac{\eta}{2}\log \big( (\zeta-c)(\zeta-c^{-1}) \big), \\
\Psi(\zeta) & = \Psi(\zeta;\xi,\eta) = -\Phi(\zeta;-\xi,-\eta)
\\ \label{Psidef}
& = -\phi(\zeta) - \frac{\xi - \eta}{2} \log \zeta + \frac{\xi}{2}\log \big( (\zeta-\alpha c)(\zeta-\alpha c^{-1}) \big) - \frac{\eta}{2}\log \big( (\zeta-c)(\zeta-c^{-1}) \big),\end{aligned}$$ where we have used and to write .
In the formulas that will be used in Section \[section: saddle point analysis\], $\Phi$ and $\Psi$ will always appear in the form $$\begin{aligned}
e^{\pm 2N\Phi(\zeta;\xi_{N},\eta_{N})}, \quad e^{\pm 2N\Psi(\zeta;\xi_{N},\eta_{N})}, \qquad \mbox{ with } \qquad \xi_{N} = \frac{x}{N}-1, \quad \eta_{N} = \frac{y}{N}-1,\end{aligned}$$ for certain integers $x,y \in \{1,\ldots,2N-1\}$. In this case, we verify that $\zeta \mapsto e^{\pm 2N\Phi(\zeta;\xi_{N},\eta_{N})}$ and $\zeta \mapsto e^{\pm 2N\Psi(\zeta;\xi_{N},\eta_{N})}$ have no jumps along $(-\infty,c^{-1}] \cup \{c^{-1} + R_{1}e^{i t}: -\pi\leq t \leq - \theta_{1}\}$. Also, for any $(\xi,\eta) \in \mathcal{H}$, ${\text{\upshape Re\,}}\Phi$ and ${\text{\upshape Re\,}}\Psi$ are harmonic on $\mathbb{C}\setminus (\Sigma_{1} \cup \{0,\alpha c,\alpha c^{-1},c,c^{-1}\})$, and well-defined and continuous on $\mathbb{C}\setminus \{0,\alpha c,\alpha c^{-1},c,c^{-1}\}$. For $(\xi,\eta) \in \mathcal{H}^{\mathrm{o}}$, we note the following basic properties of $\Phi$:
\[eq:NPhinearpoles\] $$\begin{aligned}
\Phi(\zeta) & = - \frac{1+\xi-\eta}{2} \log \zeta + \bigO(1) \text{ as } \zeta \to 0, &
\lim_{\zeta \to 0} {\text{\upshape Re\,}}\Phi(\zeta) = + \infty, \\
\Phi(\zeta) & = \frac{1+\xi}{2} \log(\zeta-\alpha c) + \bigO(1) \text{ as } \zeta \to \alpha c, &
\lim_{\zeta \to \alpha c} {\text{\upshape Re\,}}\Phi(\zeta) = - \infty, \\
\Phi(\zeta) & = \frac{1+\xi}{2} \log(\zeta -\alpha c^{-1}) + \bigO(1) \text{ as } \zeta \to \alpha c^{-1}, &
\lim_{\zeta \to \alpha c^{-1}} {\text{\upshape Re\,}}\Phi(\zeta) = - \infty, \\
\Phi(\zeta) & = -\frac{1+\eta}{2} \log(\zeta -c) + \bigO(1) \text{ as } \zeta \to c, &
\lim_{\zeta \to c} {\text{\upshape Re\,}}\Phi(\zeta) = + \infty, \\
\Phi(\zeta) & = -\frac{1+\eta}{2} \log(\zeta -c^{-1}) + \bigO(1) \text{ as } \zeta \to c^{-1}, &
\lim_{\zeta \to c^{-1}} {\text{\upshape Re\,}}\Phi(\zeta) = + \infty, \\
\Phi(\zeta) & = \frac{1-\xi + \eta}{2} \log(\zeta) + \bigO(1) \text{ as } \zeta \to \infty, &
\lim_{\zeta \to \infty} {\text{\upshape Re\,}}\Phi(\zeta) = + \infty,\end{aligned}$$
and similarly
\[eq:NPsinearpoles\] $$\begin{aligned}
& \lim_{\zeta \to 0} {\text{\upshape Re\,}}\Psi(\zeta) = \lim_{\zeta \to c} {\text{\upshape Re\,}}\Psi(\zeta) = \lim_{\zeta \to c^{-1}} {\text{\upshape Re\,}}\Psi(\zeta) = \lim_{\zeta \to \infty} {\text{\upshape Re\,}}\Psi(\zeta) = - \infty, \\
& \lim_{\zeta \to \alpha c} {\text{\upshape Re\,}}\Psi(\zeta) = \lim_{\zeta \to \alpha c^{-1}} {\text{\upshape Re\,}}\Psi(\zeta) = + \infty\end{aligned}$$
Since the saddle points are the solutions to , it follows from , and that they are also the zeros of $\Phi'$ and $\Psi'$. For the saddle point analysis, it will be important to know: 1) the sign of $|s-c^{-1}|-R_{1}$ and 2) whether $\Phi'(s)=0$ or $\Psi'(s) = 0$. We summarize the different cases in the next lemma.
\[lem:Ldivision\] Let $(\xi,\eta) \in \mathcal L_{\alpha}$ and $s = s(\xi,\eta;\alpha)$. Then we have
1. $\Phi'(s) = 0$ and $|s-c^{-1}| < R_{1}$ if and only if $\xi < 0$ and $\eta < \frac{\xi}{2}$,
2. $\Phi'(s)=0$ and $|s-c^{-1}| > R_{1}$ if and only if $\xi < 0$ and $\eta > \frac{\xi}{2}$,
3. $\Psi'(s)=0$ and $|s-c^{-1}| < R_{1}$ if and only if $\xi > 0$ and $\eta > \frac{\xi}{2}$,
4. $\Psi'(s)=0$ and $|s-c^{-1}| > R_{1}$ if and only if $\xi > 0$ and $\eta < \frac{\xi}{2}$,
5. $|s-c^{-1}| = R_{1}$ if and only if $\xi = 0$ or $\eta = \frac{\xi}{2}$.
This is an immediate consequence of Propositions \[prop:hightemp\] and \[prop: s on the Riemann surface\].
The level set $\mathcal{N}_{\Phi}$ {#subsection: the set NPhi}
----------------------------------
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(-5,-4.6)–(-5,4.6);
We study the set $$\begin{aligned}
\mathcal{N}_{\Phi} = \{ z \in \mathbb{C}: {\text{\upshape Re\,}}\Phi(z) = {\text{\upshape Re\,}}\Phi(s) \},\end{aligned}$$ in case $\eta \leq \frac{\xi}{2}<0$. We have represented $\mathcal{N}_{\Phi}$ for different values of $(\alpha,\xi,\eta)$ in Figures \[fig: cases 1 and 2 for Phi\], \[fig: cases 3 and 4 for Phi\] and \[fig: case 5 for Phi\]. There are in total eight saddles which are the zeros of $\Phi'$ and $\Psi'$. From –, both $\Phi'$ and $\Psi'$ vanish at least once on each of the intervals $(-\infty,0)$, $(\alpha c,\alpha c^{-1})$, and $(c,c^{-1})$. This determines the location of $6$ saddles. The remaining two are $s$ and $\overline{s}$, and we already know from Lemma \[lem:Ldivision\] (a) and (e) that $\Phi'(s) = 0 = \Phi'(\overline{s})$. Therefore, $\Phi' \neq 0$ on $(0,\alpha c) \cup (\alpha c^{-1},c)$. Since $\Phi'(\zeta) \in \mathbb{R}$ for $\zeta \in \mathbb{R}\setminus \{0,\alpha c, \alpha c^{-1},c,c^{-1}\}$, this implies by that $\mathcal{N}_{\Phi}$ intersects exactly once each of these two intervals.
We show with the next two lemmas that the set $\mathcal{N}_{\Phi} \cap (\overline{\Sigma_{\alpha}\cup \Sigma_{1}})\cap \mathbb{C}^{+}$ is either the empty set or a singleton.
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at (0,0) ; at (2.1,0) [ $\bullet$]{}; at (-0.38,0) [ $\bullet$]{}; at (-1.05,0) [ $\bullet$]{}; at (-2.035,0) [ $\bullet$]{}; at (-3.13,0) [ $\bullet$]{};
(\[shift=(-156:3.6cm)\]2.1,0) arc (-156:156:3.6cm);
(\[shift=(95:1.45cm)\]-1.05,0) arc (95:265:1.45cm);
at (-1.25,0.6) [ $\bullet$]{}; at (-1.25,-0.6) [ $\bullet$]{}; (-1.25,0.6) to \[out=40, in=90\] (0.5,0) to \[out=-90, in=-40\] (-1.25,-0.6); (-1.25,0.6) to \[out=-50, in=90\] (-0.9,0) to \[out=-90, in=50\] (-1.25,-0.6); (-1.25,0.6) to \[out=-160, in=90\] (-1.6,0) to \[out=-90, in=160\] (-1.25,-0.6); (-1.25,0.6) to \[out=130, in=180\] (2.4,4.5) to \[out=0, in=90\] (7,0) to \[out=-90, in=0\] (2.4,-4.5) to \[out=-180, in=-130\] (-1.25,-0.6); at (-1.6,0.8) [$+$]{}; at (-0.9,1) [$-$]{}; at (-0.7,0.5) [$+$]{}; at (-1.3,0.3) [$-$]{};
at (1.9,0.5) [$+$]{};
(\[shift=(-180:0.25cm)\]-2.035,0) arc (-180:180:0.25cm); (\[shift=(-180:0.7cm)\]1.9,0) arc (-180:180:0.7cm);
(-4.8,-4.6)–(-4.8,4.6);
For $\zeta \in \mathbb{C}\setminus \{0,\alpha c, \alpha c^{-1},c,c^{-1}\}$, we define the following functions $$\begin{aligned}
f_{1}(\zeta) = \log \frac{(\zeta-c)(\zeta-c^{-1})}{\zeta}, \quad f_{2}(\zeta) = \log \frac{\zeta}{(\zeta - \alpha c)(\zeta - \alpha c^{-1})}, \quad f_{3}(\zeta) = \log \frac{(\zeta-c)(\zeta-c^{-1})}{(\zeta - \alpha c)(\zeta - \alpha c^{-1})}.\end{aligned}$$
\[lem:LogsIncSigmaMinus1\] If $\zeta$ moves along $(\overline{\Sigma_{\alpha}\cup \Sigma_{1}}) \cap \mathbb C^+$ from left to right, then
- ${\text{\upshape Re\,}}f_{1}$ is strictly decreasing on $\Sigma_{\alpha}\cap \mathbb{C}^{+}$ and constant on $\Sigma_{1}\cap \mathbb{C}^{+}$,
- ${\text{\upshape Re\,}}f_{2}$ is constant on $\Sigma_{\alpha}\cap \mathbb{C}^{+}$ and strictly decreasing on $\Sigma_{1}\cap \mathbb{C}^{+}$,
- ${\text{\upshape Re\,}}f_{3}$ is strictly decreasing.
A long and tedious computation shows that $\frac{d}{dt} {\text{\upshape Re\,}}f_{1}(\alpha c^{-1} + R_{\alpha} e^{-i t})$ has the same sign as $\sin t$. In particular, ${\text{\upshape Re\,}}f_{1}(\zeta)$ is strictly decreasing along $\Sigma_{\alpha}\cap \mathbb{C}^{+}$ as $\zeta$ moves from left to right. Another (and simpler) computation gives $$\begin{aligned}
\frac{d}{dt}f_{1}(c^{-1} + R_{1}e^{-it}) = -i \frac{\cos t + \frac{\sqrt{1-\alpha + \alpha^{2}}}{1-\alpha}}{\cos t + \frac{2-3\alpha + 2 \alpha^{2}}{2 (1-\alpha)\sqrt{1-\alpha + \alpha^{2}}}}.\end{aligned}$$ This expression is purely imaginary, so ${\text{\upshape Re\,}}f_{1}$ is constant on $\Sigma_{1}$. The proofs for $f_{2}$ and $f_{3}$ are similar, so we omit them.
\[cor:RePhi\] For $\eta \leq \frac{\xi}{2} < 0$, the function $\zeta \mapsto {\text{\upshape Re\,}}\Phi(\zeta)$ is strictly decreasing as $\zeta$ moves along $(\overline{\Sigma_{\alpha} \cup \Sigma_1}) \cap \mathbb C^+$ from left to right.
We know from Lemma \[lem:Nphihigh\] that ${\text{\upshape Re\,}}\phi = 0$ on $\Sigma_{\alpha} \cup \Sigma_1$. Therefore, from the expression for $\Phi$, for $\zeta \in \Sigma_{\alpha} \cup \Sigma_1$ we have $$\begin{aligned}
{\text{\upshape Re\,}}\Phi (\zeta) & = - \frac{\xi-\eta}{2} \log |\zeta| + \frac{\xi}{2}\log \big| (\zeta-\alpha c)(\zeta-\alpha c^{-1}) \big| - \frac{\eta}{2}\log \big| (\zeta-c)(\zeta-c^{-1}) \big| \nonumber \\
& = \left( \frac{\xi}{4}-\frac{\eta}{2} \right) {\text{\upshape Re\,}}f_{1}(\zeta) - \frac{\xi}{4} ({\text{\upshape Re\,}}f_{2}(\zeta) + {\text{\upshape Re\,}}f_{3}(\zeta)). \label{eq:RePhionSigma}\end{aligned}$$ The claim follows from Lemma \[lem:LogsIncSigmaMinus1\], because $\xi < 0$ and $\frac{\xi}{2} - \eta \geq 0$.
at (0,0) ; at (2.1,0) [ $\bullet$]{}; at (-0.38,0) [ $\bullet$]{}; at (-1.05,0) [ $\bullet$]{}; at (-2.035,0) [ $\bullet$]{}; at (-3.13,0) [ $\bullet$]{};
(\[shift=(-156:3.6cm)\]2.1,0) arc (-156:156:3.6cm);
(\[shift=(95:1.45cm)\]-1.05,0) arc (95:265:1.45cm);
at (-1.25,0.4) [ $\bullet$]{}; at (-1.25,-0.4) [ $\bullet$]{}; (-1.25,0.4) to \[out=40, in=90\] (0.5,0) to \[out=-90, in=-40\] (-1.25,-0.4); (-1.25,0.4) to \[out=-50, in=90\] (-0.9,0) to \[out=-90, in=50\] (-1.25,-0.4); (-1.25,0.4) to \[out=-160, in=90\] (-1.6,0) to \[out=-90, in=160\] (-1.25,-0.4); (-1.25,0.4) to \[out=130, in=0\] (-4.8,2.5) to \[out=180, in=90\] (-7,0) to \[out=-90, in=-180\] (-4.8,-2.5) to \[out=0, in=-130\] (-1.25,-0.4); at (-1.6,0.6) [$+$]{}; at (-0.9,0.8) [$-$]{}; at (-0.7,0.3) [$+$]{}; at (-1.3,0.1) [$-$]{};
at (1.9,0.5) [$+$]{};
(\[shift=(-180:0.25cm)\]-2.035,0) arc (-180:180:0.25cm); (\[shift=(-180:0.7cm)\]1.9,0) arc (-180:180:0.7cm);
plot \[smooth cycle, tension=0.7\] coordinates [(-8.5,0) (-6,3) (2,4.5) (7,0) (2,-4.5) (-6,-3)]{}; at (-5,4) [$+$]{};
#### Notation.
For a given closed curve $\sigma$, we denote $\mbox{int}(\sigma)$ for the open and bounded region delimited by $\sigma$.
Since $\Phi'(s) = 0$, there are four curves $\{\Gamma_{j}\}_{j=1}^{4}$ emanating from $s$ that belongs to $\mathcal{N}_{\Phi}$. By Corollary \[cor:RePhi\], $\mathcal{N}_{\Phi} \cap (\overline{\Sigma_{\alpha}\cup \Sigma_{1}})\cap \mathbb{C}^{+}$ is either the empty set or a singleton, so at least three of the $\Gamma_j$’s, say $\Gamma_1, \Gamma_2, \Gamma_3$, do not intersect $(\overline{\Sigma_{\alpha}\cup \Sigma_{1}}) \cap \mathbb C^+$. The curves $\Gamma_{j}$, $j =1,2,3$ cannot lie entirely in $\mathbb{C}^{+}$; otherwise the max/min principle for harmonic functions would imply that ${\text{\upshape Re\,}}\Phi$ is constant within the region $\mbox{int}(\Gamma_{j})$. Therefore, $\Gamma_{j}$, $j =1,2,3$ have to intersect $\mathbb{R}$. Note that $\overline{\Phi(\zeta)} = \Phi(\overline{\zeta})$ implies that $\mathcal{N}_{\Phi}$ is symmetric with respect to $\mathbb{R}$. In particular, the curves $\Gamma_{j}$, $j=1,2,3$ join $s$ with $\overline{s}$. The next lemma states that $\Gamma_{4}$ is not contained in the region $\mbox{int}(\overline{\Sigma_{\alpha}\cup \Sigma_{1}})$.
\[lemma: Gamma4 intersects at one point\] $\mathcal{N}_{\Phi} \cap (\overline{\Sigma_{\alpha}\cup \Sigma_{1}}) \cap \mathbb{C}^{+}$ is a singleton.
Assume on the contrary that $\Gamma_{4}$ lies entirely in $\mbox{int}(\overline{\Sigma_{\alpha}\cup \Sigma_{1}})$, and denote $p_{j}$ for the intersection point of $\Gamma_{j}$ with $\mathbb{R}$. We assume without loss of generality that $p_{1}<p_{2}<p_{3}<p_{4}$. There is at most one $p_{j}$ inside each of the intervals $$\begin{aligned}
(\alpha c^{-1} - R_{\alpha},\alpha c), \quad (\alpha c, \alpha c^{-1}), \quad (\alpha c^{-1},c), \quad (c,c^{-1}), \quad (c^{-1},c^{-1}+R_{1}),\end{aligned}$$ otherwise we again find a contradiction using the max/min principle for harmonic functions. Thus, there are five $5$ possibilities for the location of the $p_{j}$’s, and each of them leads to a contradiction. Let us treat the case $$\begin{aligned}
\label{the pj location fake case}
p_{1} \in (\alpha c, \alpha c^{-1}), \quad p_{2} \in (\alpha c^{-1},c), \quad p_{3} \in (c,c^{-1}), \quad p_{4} \in (c^{-1},c^{-1}+R_{1}).\end{aligned}$$ Since ${\text{\upshape Re\,}}(\Phi(\zeta)-\Phi(s))$ changes sign as $\zeta$ crosses $\mathcal{N}_{\Phi}\setminus \{s,\overline{s}\}$, by we must have $$\begin{aligned}
\label{two expressions for mathcalN}
\mathcal{N}_{\Phi} = \sigma_{1} \cup \sigma_{2} \cup \bigcup_{j=1}^{4} \Gamma_{j},\end{aligned}$$ where $\sigma_{1}$ is a closed curve surrounding either $\alpha c$ or $\alpha c^{-1}$, such that $\mbox{int}(\sigma_{1}) \cap \mathcal{N}_{\Phi} = \emptyset$, and $\sigma_{2}$ is a closed curve surrounding either $c$ or $c^{-1}$, such that $\mbox{int}(\sigma_{2}) \cap \mathcal{N}_{\Phi} = \emptyset$. Since $\mathcal{N}_{\Phi}$ intersects both $(0,\alpha c)$ and $(\alpha c^{-1},c)$ exactly once, $\sigma_{1}$ surrounds $\alpha c$ and $\sigma_{2}$ surrounds $c^{-1}$. Then, the max/min principle implies that ${\text{\upshape Re\,}}\Phi$ is constant on $\mbox{int}(\overline{\Gamma_{3}\cup \Gamma_{4}})\setminus \mbox{int}(\sigma_{2})$, which is a contradiction. The four other cases than can be treated similarly, so we omit the proofs.
Lemma \[lemma: Gamma4 intersects at one point\] states that $\Gamma_{4}$ crosses $\Sigma_{\alpha}\cup \Sigma_{1}$ exactly once. We know from that ${\text{\upshape Re\,}}\Phi(\zeta) \to + \infty$ as $\zeta \to \infty$, so $\Gamma_{4}$ intersects the real line, and then by symmetry ends at $\overline{s}$. So each of the $\Gamma_{j}$’s intersects $\mathbb{R}$. We denote $p_{j}$ for the intersection point of $\Gamma_{j}$ with $\mathbb{R}$, and choose the ordering such that $p_1 < p_2 < p_3$. We recall that ${\text{\upshape Re\,}}(\Phi(\zeta)-\Phi(s))$ is harmonic for $\zeta \in \mathbb{C}\setminus (\Sigma_{1} \cup \{0,\alpha c,\alpha c^{-1},c,c^{-1}\})$ and changes sign as $\zeta$ crosses $\mathcal{N}_{\Phi}\setminus \{s,\overline{s}\}$. Therefore, by , the region $\mbox{int}(\overline{\Gamma_{1} \cup \Gamma_{2}})$ must contain at least one of the singularities $\alpha c$ and $\alpha c^{-1}$, and $\mbox{int}(\overline{\Gamma_{2} \cup \Gamma_{3}})$ must contain at least one of the singularities $c$ and $c^{-1}$. There are still quite a few cases that can occur. The figures provide a fairly good overview (though not complete) of what can happen:
1. In Figure \[fig: cases 1 and 2 for Phi\] (left), $\alpha c, \alpha c^{-1}, c \in \mbox{int}(\overline{\Gamma_{1}\cup \Gamma_{2}})$, $c^{-1} \in \mbox{int}(\overline{\Gamma_{2}\cup \Gamma_{3}})$.
2. In Figures \[fig: cases 1 and 2 for Phi\] (right) and \[fig: cases 3 and 4 for Phi\] (left), $\alpha c, \alpha c^{-1} \in \mbox{int}(\overline{\Gamma_{1}\cup \Gamma_{2}})$ and $c,c^{-1} \in \mbox{int}(\overline{\Gamma_{2}\cup \Gamma_{3}})$.
3. In Figures \[fig: cases 3 and 4 for Phi\] (right) and \[fig: case 5 for Phi\], $\alpha c^{-1} \in \mbox{int}(\overline{\Gamma_{1}\cup \Gamma_{2}})$ and $c \in \mbox{int}(\overline{\Gamma_{2}\cup \Gamma_{3}})$.
Furthermore, $\Gamma_{4}$ intersects both $\Sigma_{1}$ and $(c^{-1}+R_{1},+\infty)$ in Figure \[fig: cases 1 and 2 for Phi\], intersects both $\Sigma_{\alpha}$ and $(c^{-1}+R_{1},+\infty)$ in Figure \[fig: cases 3 and 4 for Phi\], and intersects both $\Sigma_{\alpha}$ and $(-\infty,\alpha c^{-1} - R_{\alpha})$ in Figure \[fig: case 5 for Phi\]. There are also some obvious intermediate cases which are not illustrated by a figure. In all cases, we can find contours $\gamma_{\zeta}^{\star}$ and $\gamma_{\omega}^{\star}$ as described in the following proposition. These contours are illustrated for two different situations in Figures \[fig: case 1 contour\] and \[fig: case 2 contour\] (left).
at (0,0) ; at (2.1,0) [ $\bullet$]{}; at (-0.38,0) [ $\bullet$]{}; at (-1.05,0) [ $\bullet$]{}; at (-2.035,0) [ $\bullet$]{}; at (-3.13,0) [ $\bullet$]{};
(\[shift=(-156:3.6cm)\]2.1,0) arc (-156:156:3.6cm);
(\[shift=(95:1.45cm)\]-1.05,0) arc (95:265:1.45cm);
at (1.3,2.5) [ $\bullet$]{}; at (1.3,-2.5) [ $\bullet$]{}; (1.3,2.5) to \[out=10, in=90\] (4.5,0) to \[out=-90, in=-10\] (1.3,-2.5); (1.3,2.5) to \[out=-80, in=90\] (1.05,0) to \[out=-90, in=80\] (1.3,-2.5); (1.3,2.5) to \[out=-170, in=45\] (-1,1.3) to \[out=-135, in=80\] (-2.3,0) to \[out=-90, in=135\] (-1,-1.3) to \[out=-45, in=170\] (1.3,-2.5); (1.3,2.5) to \[out=100, in=-60\] (0.95,3.4) to \[out=30, in=90\] (6.1,0) to \[out=-90, in=-30\] (0.95,-3.4) to \[out=60, in=-100\] (1.3,-2.5); (\[shift=(-180:0.45cm)\]-0.28,0) arc (-180:180:0.45cm);
(1.3,2.5) to \[out=135, in=35\] (-1,1.5) to \[out=-135, in=80\] (-2.4,0) to \[out=-90, in=135\] (-1,-1.5) to \[out=-35, in=-135\] (1.3,-2.5); (1.3,2.5) to \[out=-35, in=90\] (2.5,0) to \[out=-90, in=35\] (1.3,-2.5); at (2.8,0) [$\gamma_{\omega}^{\star}$]{};
(1.3,2.5) to \[out=-125, in=90\] (-1.35,0) to \[out=-90, in=125\] (1.3,-2.5); (1.3,2.5) to \[out=55, in=90\] (5.2,0) to \[out=-90, in=-55\] (1.3,-2.5); at (4.9,0) [$\gamma_{\zeta}^{\star}$]{};
\[prop:contoursexist\] Let $(\xi,\eta) \in \mathcal{L}_{\alpha}$ with $\eta < \frac{\xi}{2} < 0$. There exist contours $\gamma_{\zeta}^{\star}$ and $\gamma_{\omega}^{\star}$ such that
- $\gamma_{\omega}^{\star} \subset \mbox{int}(\overline{\Sigma_\alpha \cup \Sigma_{1}})$, it surrounds $\alpha c$ and $\alpha c^{-1}$, and it goes through $s$ and $\overline{s}$ in such a way that $$\begin{aligned}
{\text{\upshape Re\,}}\Phi(\omega) > {\text{\upshape Re\,}}\Phi(s), \qquad \omega \in \gamma_{\omega}^{\star} \setminus \{s, \overline{s}\},\end{aligned}$$
- $\gamma_\zeta^{\star} \subset \mbox{int}(\gamma_1)$, surrounds $c$ and $c^{-1}$, and it goes through $s$ and $\overline{s}$ in such a way that $$\begin{aligned}
{\text{\upshape Re\,}}\Phi(\zeta) < {\text{\upshape Re\,}}\Phi(s), \qquad \zeta \in \gamma_{\zeta}^{\star} \setminus \{s, \overline{s}\}.\end{aligned}$$
If $\eta = \frac{\xi}{2}$, we know from Proposition \[prop:hightemp\] \[item b in prop mapping s\] that $s$ lies on $\gamma_{1}\setminus \overline{\Sigma_{1}}$. For the saddle point analysis, we will need $\gamma_{\zeta}^{\star}$ lying inside $\gamma_{1}$ (not necessarily strictly inside). To prove existence of such a contour $\gamma_{\zeta}^{\star}$, we need to know that ${\text{\upshape Re\,}}\Phi(\zeta)-{\text{\upshape Re\,}}\Phi(s)$ is strictly negative for $\zeta \in \gamma_{1}\setminus \overline{\Sigma_{1}}$ (at least in small neighborhoods of $s$ and $\overline{s}$).
at (0,0) ; at (2.1,0) [ $\bullet$]{}; at (-0.38,0) [ $\bullet$]{}; at (-1.05,0) [ $\bullet$]{}; at (-2.035,0) [ $\bullet$]{}; at (-3.13,0) [ $\bullet$]{};
(\[shift=(-156:3.6cm)\]2.1,0) arc (-156:156:3.6cm);
(\[shift=(95:1.45cm)\]-1.05,0) arc (95:265:1.45cm);
at (-1.25,0.6) [ $\bullet$]{}; at (-1.25,-0.6) [ $\bullet$]{}; (-1.25,0.6) to \[out=40, in=90\] (0.5,0) to \[out=-90, in=-40\] (-1.25,-0.6); (-1.25,0.6) to \[out=-50, in=90\] (-0.9,0) to \[out=-90, in=50\] (-1.25,-0.6); (-1.25,0.6) to \[out=-160, in=90\] (-1.6,0) to \[out=-90, in=160\] (-1.25,-0.6); (-1.25,0.6) to \[out=130, in=180\] (2.4,4.5) to \[out=0, in=90\] (7,0) to \[out=-90, in=0\] (2.4,-4.5) to \[out=-180, in=-130\] (-1.25,-0.6);
(\[shift=(-180:0.25cm)\]-2.035,0) arc (-180:180:0.25cm); (\[shift=(-180:0.7cm)\]1.9,0) arc (-180:180:0.7cm);
(-1.25,0.6) to \[out=85, in=90\] (3,0) to \[out=-90, in=-85\] (-1.25,-0.6); (-1.25,0.6) to \[out=-95, in=90\] (-1.25,0) to \[out=-90, in=95\] (-1.25,-0.6); at (3.3,0) [$\gamma_{\zeta}^{\star}$]{};
(-1.25,0.6) to \[out=-205, in=90\] (-2.4,0) to \[out=-90, in=205\] (-1.25,-0.6); ($(-2,0.6)$) – ++(-170:0.001); (-1.25,0.6) to \[out=-5, in=90\] (-0.6,0) to \[out=-90, in=5\] (-1.25,-0.6); at (-0.5,0.35) [$\gamma_{\omega}^{\star}$]{};
at (0,0) ; at (2.1,0) [ $\bullet$]{}; at (-0.38,0) [ $\bullet$]{}; at (-1.05,0) [ $\bullet$]{}; at (-2.035,0) [ $\bullet$]{}; at (-3.13,0) [ $\bullet$]{};
(\[shift=(-156:3.6cm)\]2.1,0) arc (-156:156:3.6cm);
(\[shift=(95:1.45cm)\]-1.05,0) arc (95:265:1.45cm);
at ($(2.1,0)+(168:3.6)$) [ $\bullet$]{}; at ($(2.1,0)+(-168:3.6)$) [ $\bullet$]{}; ($(2.1,0)+(168:3.6)$) to \[out=40, in=90\] (0.5,0) to \[out=-90, in=-40\] ($(2.1,0)+(-168:3.6)$); ($(2.1,0)+(168:3.6)$) to \[out=-50, in=90\] (-0.9,0) to \[out=-90, in=50\] ($(2.1,0)+(-168:3.6)$); ($(2.1,0)+(168:3.6)$) to \[out=-160, in=90\] (-2.25,0) to \[out=-90, in=160\] ($(2.1,0)+(-168:3.6)$); ($(2.1,0)+(168:3.6)$) to \[out=130, in=180\] (2.4,4.5) to \[out=0, in=90\] (7,0) to \[out=-90, in=0\] (2.4,-4.5) to \[out=-180, in=-130\] ($(2.1,0)+(-168:3.6)$);
(\[shift=(-180:0.7cm)\]1.9,0) arc (-180:180:0.7cm);
(\[shift=(161:3.6cm)\]2.1,0) arc (161:199:3.6cm); ($(2.1,0)+(161:3.6)$) to \[out=0, in=90\] (3,0) to \[out=-90, in=-0\] ($(2.1,0)+(-161:3.6)$); at (3.3,0) [$\gamma_{\zeta}^{\star}$]{};
($(2.1,0)+(168:3.6)$) to \[out=-205, in=90\] (-2.4,0) to \[out=-90, in=205\] ($(2.1,0)+(-168:3.6)$); ($(-2.2,0.55)$) – ++(-145:0.001); ($(2.1,0)+(168:3.6)$) to \[out=-5, in=90\] (-0.6,0) to \[out=-90, in=5\] ($(2.1,0)+(-168:3.6)$); at (-0.5,0.35) [$\gamma_{\omega}^{\star}$]{}; (-4,-5)–(-4,5);
Let $\eta = \frac{\xi}{2}<0$. For $\zeta \in \gamma_{1}\setminus (\overline{\Sigma_{1}} \cup \{s\}) \cap \mathbb{C}^{+}$, we have ${\text{\upshape Re\,}}\Phi(\zeta) < {\text{\upshape Re\,}}\Phi(s)$.
Let $\zeta = c^{-1} + R_{1}e^{it}$. For $t \in (\theta_{1},\pi)$, we have $$\begin{aligned}
\label{lol25}
{\text{\upshape Re\,}}(\Phi'(\zeta)d\zeta) = \frac{-\cos(\frac{t}{2})\left( \sqrt{\cos \theta_{1}-\cos t}(\cos t + a_{1}) + \frac{\xi}{2}(\cos t + a_{2})\sqrt{1-\cos t} \right)}{\sqrt{2}(\cos t + \frac{2-\alpha + \alpha^{2}}{2\sqrt{1-\alpha + \alpha^{2}}})(\cos t + \frac{2-3\alpha + 2\alpha^{2}}{2(1-\alpha)\sqrt{1-\alpha + \alpha^{2}}})},\end{aligned}$$ where $a_{1},a_{2}$ are given by $a_{1} = \frac{\alpha^{2} + (1-\alpha)\sqrt{1-\alpha + \alpha^{2}}}{2(1-\alpha)}$ and $a_{2} = \frac{2-3\alpha + 2\alpha^{2} + \alpha^{3}}{2(1-\alpha)\sqrt{1-\alpha + \alpha^{2}}}$ and satisfy $a_{1}>a_{2}>1$. The expression vanishes if and only if $$\begin{aligned}
\label{lol26}
\frac{\sqrt{\cos \theta_{1} - \cos t}}{\sqrt{1-\cos t}} = - \frac{\xi}{2} \frac{\cos t + a_{2}}{\cos t + a_{1}}.\end{aligned}$$ Since the left-hand-side is strictly decreasing, and the right-hand-side is strictly increasing as $t$ decreases from $\pi$ to $\theta_{1}$, there is a unique $\zeta = c^{-1}+R_{1}e^{it}$, $t \in (\theta_{1},\pi)$, such that ${\text{\upshape Re\,}}(\Phi'(\zeta)d\zeta) = 0$, and this must be $s$. This implies that ${\text{\upshape Re\,}}\Phi(\zeta) - {\text{\upshape Re\,}}\Phi(s)$ is of constant sign on $\gamma_{1}\setminus (\overline{\Sigma_{1}} \cup \{s\}) \cap \mathbb{C}^{+}$. By , ${\text{\upshape Re\,}}(\Phi'(\zeta)d\zeta) > 0$ at $t = \theta_{1}$ (recall that $\xi < 0$), so the claim is proved.
Therefore, we can find contours $\gamma_{\zeta}^{\star}$ and $\gamma_{\omega}^{\star}$ as described in Proposition \[prop:contoursexist eta = xi/2\], see also Figure \[fig: case 2 contour\] (right).
\[prop:contoursexist eta = xi/2\] Let $(\xi,\eta) \in \mathcal{L}_{\alpha}$ with $\eta = \frac{\xi}{2} < 0$. There exist contours $\gamma_{\zeta}^{\star}$ and $\gamma_{\omega}^{\star}$ such that
- $\gamma_{\omega}^{\star} \subset \mbox{int}(\overline{\Sigma_\alpha \cup \Sigma_{1}})$, it surrounds $\alpha c$ and $\alpha c^{-1}$, and it goes through $s$ and $\overline{s}$ in such a way that $$\begin{aligned}
{\text{\upshape Re\,}}\Phi(\omega) > {\text{\upshape Re\,}}\Phi(s), \qquad \omega \in \gamma_{\omega}^{\star} \setminus \{s, \overline{s}\},\end{aligned}$$
- $\gamma_\zeta^{\star} \subset \overline{\mbox{int}(\gamma_1)}$, surrounds $c$ and $c^{-1}$, and it goes through $s$ and $\overline{s}$ in such a way that $$\begin{aligned}
{\text{\upshape Re\,}}\Phi(\zeta) < {\text{\upshape Re\,}}\Phi(s), \qquad \zeta \in \gamma_{\zeta}^{\star} \setminus \{s, \overline{s}\}.\end{aligned}$$
Saddle point analysis {#section: saddle point analysis}
=====================
In this section, we prove Proposition \[prop:doubleintegrallimit\] by means of a saddle point analysis that mainly follows the lines of [@CDKL]. This analysis relies mostly on Sections \[section: steepest descent for $U$\]–\[section: phase functions\] and is only valid for $(\xi,\eta)$ in the lower left part of the liquid region, that is for $(\xi,\eta) \in \mathcal{L}_{\alpha} \cap \{\eta \leq \tfrac{\xi}{2} \leq 0\}$. We divide the proof in three subcases: $\eta \leq \tfrac{\xi}{2} < 0$, $\eta < \tfrac{\xi}{2} = 0$ and $\eta = \xi = 0$.
By adapting the analysis of this section and of Section \[section: phase functions\], it is possible to carry out similar saddle point analyses when $(\xi,\eta)$ lies in the other quadrants of the liquid region. Note however that this is not needed, thanks to the symmetries of Subsection \[subsection: symmetries\] (see also Proposition \[prop:doubleintegrallimit\]).
The case $\eta \leq \tfrac{\xi}{2} < 0$
---------------------------------------
The double integral $\mathcal{I}$ is defined in . The associated two contours of integration can be chosen freely, as long as they are closed curves surrounding $c$ and $c^{-1}$ once in the positive direction, and not surrounding $0$. From now, it will be convenient to take different contours in the $\zeta$ and $\omega$ variables, so we indicate this in the notation by rewriting as $$\begin{aligned}
\label{mathcalI zeta omega}
\mathcal{I}(x,y;H) = \frac{1}{(2\pi i)^{2}}\int_{\gamma_{\zeta}}d\zeta \int_{\gamma_{\omega}}d\omega H(\omega, \zeta) W(\omega) \mathcal{R}^{U}(\omega,\zeta) \frac{\omega^{N}}{\zeta^{N}}q(\omega,\zeta)^{y} \tilde{q}(\omega,\zeta)^{x}.\end{aligned}$$ Only the first column of $U$ appears in , which is independent of the choice of the contour $\gamma_{\mathbb{C}}$ associated to the RH problem for $U$. However, by using the jumps for $U$, we will find (just below) another formula for $\mathcal{I}$ in terms of the second column of $U$. Therefore, the choice of $\gamma_{\mathbb{C}}$ will matter. To be able to use the steepest descent of Section \[section: steepest descent for $U$\], we assume from now that $\gamma_{\mathbb{C}}=\gamma_{1}$. Recall that $T$ is expressed in terms of $U$ via , and define $$\begin{aligned}
\label{lol22}
\widetilde{\mathcal{R}}^{T}(\omega,\zeta) = \begin{pmatrix}
1 & 0
\end{pmatrix} T^{-1}(\omega)T(\zeta) \begin{pmatrix}
1 \\ 0
\end{pmatrix}.\end{aligned}$$ By Proposition \[prop:TandTinvsmall\], $\widetilde{\mathcal{R}}^{T}(\omega,\zeta)$ is uniformly bounded as $\zeta$ and $\omega$ stay bounded away from $r_{+}$ and $r_{-}$. We will need the analytic continuation in $\omega$ of $\widetilde{\mathcal{R}}^{T}(\omega,\zeta)$ from the interior of $\gamma_{1}$ to the bounded region delimited by $\overline{\Sigma_{1} \cup \Sigma_{\alpha}}$ (see Figure \[fig: crit traj alpha 04\]). We denote it $\widetilde{\mathcal{R}}^{T,a}(\omega,\zeta)$, and by it is given by $$\begin{aligned}
\label{def of RTa}
\widetilde{\mathcal{R}}^{T,a}(\omega,\zeta) = \begin{cases}
\begin{pmatrix}
1 & 0
\end{pmatrix} T^{-1}(\omega)T(\zeta) \begin{pmatrix}
1 \\ 0
\end{pmatrix}, & |\omega-c^{-1}| < R_{1}, \, \zeta \in \mathbb{C}\setminus \gamma_{1}, \\[0.4cm]
\begin{pmatrix}
1 & -e^{4N\phi(\omega)}
\end{pmatrix} T^{-1}(\omega)T(\zeta) \begin{pmatrix}
1 \\ 0
\end{pmatrix}, & \omega \in \mbox{int}\big( (\gamma_{1}\setminus \Sigma_{1})\cup \Sigma_{\alpha}\big), \, \zeta \in \mathbb{C}\setminus \gamma_{1}.
\end{cases}\end{aligned}$$ By Lemma \[lem:Nphihigh\], ${\text{\upshape Re\,}}\phi(\omega) < 0$ for $\omega \in \mbox{int}\big( (\gamma_{1}\setminus \Sigma_{1})\cup \Sigma_{\alpha}\big)$, so $\widetilde{\mathcal{R}}^{T,a}(\omega,\zeta)$ remains bounded as $N \to +\infty$, uniformly for $\zeta$ and $\omega$ bounded away from $r_{+}$ and $r_{-}$, as long as $\omega \in \mbox{int}(\overline{\Sigma_{1}}\cup \Sigma_{\alpha})$. Our next goal is to prove the following.
\[prop:deformationhigh\] Let $(x,y)$ be coordinates inside the hexagon, such that $\xi:=\frac{x}{N}-1$ and $\eta:=\frac{y}{N}-1$ satisfy $(\xi,\eta) \in \mathcal L_{\alpha}$ with $\eta \leq \frac{\xi}{2} <0$. Take $\gamma_{\zeta}^{\star}$ and $\gamma_{\omega}^{\star}$ as in Proposition \[prop:contoursexist\] if $\eta < \frac{\xi}{2}$, and as in Proposition \[prop:contoursexist eta = xi/2\] if $\eta = \frac{\xi}{2}$ (see also Figures \[fig: case 1 contour\] and \[fig: case 2 contour\]). Then the double contour integral is equal to $$\label{eq:deformationhigh}
\mathcal I(x,y;H) =
\frac{1}{2\pi i} \int_{\overline{s}}^s H(\zeta,\zeta) d\zeta + \frac{1}{(2\pi i)^2} \int_{\gamma_{\zeta}^{\star}} d\zeta
\int_{\gamma_{\omega}^{\star}} \frac{d\omega}{\omega-\zeta}H(\omega,\zeta)
\widetilde{\mathcal{R}}^{T,a}(\omega,\zeta) e^{2N(\Phi(\zeta;\xi,\eta)-\Phi(\omega;\xi,\eta))}.$$
\[remark: it is analytic in zeta\] By Proposition \[prop:contoursexist eta = xi/2\], $\gamma_{\zeta}^{\star}$ intersects $\gamma_{1}\setminus \overline{\Sigma_{1}}$ whenever $\eta = \frac{\xi}{2}$. We do not indicate whether we take the $+$ or $-$ boundary values in the integrand of . This is without ambiguity, because $$\begin{aligned}
\zeta \mapsto T(\zeta)\begin{pmatrix}
1 \\ 0
\end{pmatrix}e^{2N\Phi(\zeta;\xi,\eta)}\end{aligned}$$ has no jumps on $\gamma_{1}$ (this can be verified using –).
Take $\gamma_{\omega} = \gamma_{1}$ and $\gamma_{\zeta}$ lying strictly inside $\gamma_{1}$ in . From the jumps for $U$ , we have $$\begin{aligned}
W(\omega)\begin{pmatrix}
0 & 1
\end{pmatrix}U(\omega)^{-1} = \begin{pmatrix}
1 & 0
\end{pmatrix}U_{-}(\omega)^{-1} - \begin{pmatrix}
1 & 0
\end{pmatrix}U_{+}(\omega)^{-1}, \qquad \omega \in \gamma_{1}.\end{aligned}$$ Inserting this in , and using the $U \mapsto T$ transformation , we get $$\begin{aligned}
\mathcal{I}(x,y;H) & = \frac{1}{(2\pi i)^{2}}\int_{\gamma_{\zeta}}d\zeta \int_{\gamma_{\omega}=\gamma_{1}}\frac{d\omega}{\omega-\zeta} H(\omega, \zeta) \widetilde{\mathcal{R}}_{+}^{T}(\omega,\zeta) e^{2N(g(\zeta)-g_{+}(\omega))} \frac{\omega^{N}}{\zeta^{N}}q(\omega,\zeta)^{y} \tilde{q}(\omega,\zeta)^{x} \nonumber \\
& - \frac{1}{(2\pi i)^{2}}\int_{\gamma_{\zeta}}d\zeta \int_{\gamma_{\omega}=\gamma_{1}}\frac{d\omega}{\omega-\zeta} H(\omega, \zeta) \widetilde{\mathcal{R}}_{-}^{T}(\omega,\zeta) e^{2N(g(\zeta)-g_{-}(\omega))} \frac{\omega^{N}}{\zeta^{N}}q(\omega,\zeta)^{y} \tilde{q}(\omega,\zeta)^{x}, \label{mathcalI splitting}\end{aligned}$$ where $\widetilde{\mathcal{R}}_{+}^{T}(\omega,\zeta)$ and $\widetilde{\mathcal{R}}_{-}^{T}(\omega,\zeta)$ denote the limits of $\widetilde{\mathcal{R}}^{T}(\omega',\zeta)$ as $\omega' \to \omega$ from the interior and exterior of $\gamma_{1}$, respectively.
For $x,y \in \{1,2,\ldots,2N-1\}$, we define $$\begin{aligned}
\label{integrand in omega and zeta}
m(\omega,\zeta) = \frac{1}{\omega-\zeta} H(\omega, \zeta) \widetilde{\mathcal{R}}^{T}(\omega,\zeta) e^{2N(g(\zeta)-g(\omega))} \frac{\omega^{N}}{\zeta^{N}}q(\omega,\zeta)^{y} \tilde{q}(\omega,\zeta)^{x}.$$ The boundary values of $m$ appear in the integrand of . We recall that $q$ and $\tilde{q}$ are defined in , that $H$ satisfies the conditions stated in Proposition \[prop:doubleintegrallimit\], and that $g(\omega)$ is bounded for $\omega$ in compact subsets and satifies $g(\omega) \sim \log(\omega)$ as $\omega \to \infty$. Therefore, the following properties hold:
1. \[item i\] The function $\zeta \mapsto m(\omega,\zeta)$ is analytic in $\mathbb{C} \setminus \{\omega, 0, c, c^{-1}\}$,
2. \[item ii\] The function $\omega \mapsto m(\omega,\zeta)$ is analytic in $(\mathbb{C}\cup\{\infty\}) \setminus (\{\zeta, \alpha c, \alpha c^{-1}\} \cup \gamma_{1})$.
The statement that $\omega \mapsto m(\omega,\zeta)$ is analytic at $\infty$ deserves a little computation: since $x,y \in \{1,2,\ldots,2N-1\}$, we have $m(\omega,\zeta) = \bigO(\omega^{-1-2N+N-y+x}) = \bigO(\omega^{-2})$ as $\omega \to \infty$.
If $\eta < \frac{\xi}{2}$, Proposition \[prop:contoursexist\] states that $\gamma_{\zeta}$ lies strictly inside $\gamma_{1}$, so in this case we can (and do) take $\gamma_{\zeta} = \gamma_{\zeta}^{\star}$ in . If $\eta = \frac{\xi}{2}$, we know from Proposition \[prop:contoursexist eta = xi/2\] that $\gamma_{\zeta}^{\star}$ intersects $\gamma_{1}\setminus \overline{\Sigma_{1}}$. In this case, we let $\gamma_{\zeta}$ in tend to $\gamma_{\zeta}^{\star}$ from the interior of $\gamma_{1}$. In what follows, we will abuse notation and simply write $\gamma_{\zeta}^{\star}$. We will also omit the boundary values in the $\zeta$-variable, see Remark \[remark: it is analytic in zeta\] (or \[item i\]).
Let us deform $\gamma_{\omega}$ from $\gamma_{1}$ to $\Sigma_{1} \cup \overline{\Sigma_{\alpha}}$ in each of the two integrals of . For each deformation, we pick up a residue at $\omega = \alpha c$. These residues cancel each other and we get $$\begin{aligned}
\mathcal{I}(x,y;H) & = \frac{1}{(2\pi i)^{2}}\int_{\gamma_{\zeta}^{\star}}d\zeta \int_{\gamma_{\omega}=\Sigma_{1} \cup \overline{\Sigma_{\alpha}}}\frac{d\omega}{\omega-\zeta} H(\omega, \zeta) \widetilde{\mathcal{R}}_{+}^{T,a}(\omega,\zeta) e^{2N(g(\zeta)-g_{+}(\omega))} \frac{\omega^{N}}{\zeta^{N}}q(\omega,\zeta)^{y} \tilde{q}(\omega,\zeta)^{x} \nonumber \\
& - \frac{1}{(2\pi i)^{2}}\int_{\gamma_{\zeta}^{\star}}d\zeta \int_{\gamma_{\omega}=\Sigma_{1} \cup \overline{\Sigma_{\alpha}}}\frac{d\omega}{\omega-\zeta} H(\omega, \zeta) \widetilde{\mathcal{R}}_{-}^{T}(\omega,\zeta) e^{2N(g(\zeta)-g_{-}(\omega))} \frac{\omega^{N}}{\zeta^{N}}q(\omega,\zeta)^{y} \tilde{q}(\omega,\zeta)^{x}. \end{aligned}$$ By \[item ii\], the integrand of the second integral has no poles in the exterior region of $\Sigma_{1} \cup \overline{\Sigma_{\alpha}}$, so by deforming $\gamma_{\omega}$ at $\infty$, we find that this integral is $0$. Therefore, we simply get $$\begin{aligned}
\mathcal{I}(x,y;H) & = \frac{1}{(2\pi i)^{2}}\int_{\gamma_{\zeta}^{\star}}d\zeta \int_{\gamma_{\omega}=\Sigma_{1} \cup \overline{\Sigma_{\alpha}}}\frac{d\omega}{\omega-\zeta} H(\omega, \zeta) \widetilde{\mathcal{R}}_{+}^{T,a}(\omega,\zeta) e^{2N(g(\zeta)-g_{+}(\omega))} \frac{\omega^{N}}{\zeta^{N}}q(\omega,\zeta)^{y} \tilde{q}(\omega,\zeta)^{x}.\end{aligned}$$ This formula can be written in terms of $\Phi$ (see Definition \[def: Phi and Psi\]) as follows: $$\begin{aligned}
\label{mathcalI in terms of Phi}
\mathcal{I}(x,y;H) & = \frac{1}{(2\pi i)^{2}}\int_{\gamma_{\zeta}^{\star}}d\zeta \int_{\gamma_{\omega}=\Sigma_{1} \cup \overline{\Sigma_{\alpha}}}\frac{d\omega}{\omega-\zeta} H(\omega, \zeta) \widetilde{\mathcal{R}}_{+}^{T,a}(\omega,\zeta) e^{2N(\Phi(\zeta;\xi,\eta)-\Phi_{+}(\omega;\xi,\eta))},\end{aligned}$$ where $\xi := x/N-1$ and $\eta := y/N-1$. Finally, we deform $\gamma_{\omega}$ into $\gamma_{\omega}^{\star}$. This gives the right-most term of plus a residue at $\omega = \zeta$ (by \[item ii\]). After a small computation, we find that this residue is the first term on the right-hand-side of . This finishes the proof.
Let $\{(x_{N},y_{N}\}_{N \geq 1}$ be a sequence satisfying with $(\xi,\eta) \in \mathcal{L}_{\alpha}\cap \{\eta \leq \frac{\xi}{2}< 0\}$, and define $\xi_{N} := x_{N}/N-1$ and $\eta_{N}:= y_{N}/N-1$. By , we have $\xi_{N} \to \xi$ and $\eta_{N} \to \eta$ as $N \to + \infty$. If $\eta = \frac{\xi}{2}$, we assume for simplicity that $(\xi_{N},\eta_{N}) \in \mathcal{L}_{\alpha}\cap \{\eta \leq \frac{\xi}{2}< 0\}$ for all large enough $N$. Replacing $(x,y)$ in by $(x_{N},y_{N})$, we get $$\label{lol27}
\mathcal I(x_{N},y_{N};H) -\frac{1}{2\pi i} \int_{\overline{s_{N}}}^{s_{N}} H(\zeta,\zeta) d\zeta =
\frac{1}{(2\pi i)^2} \int_{\gamma_{\zeta}^{\star}} d\zeta
\int_{\gamma_{\omega}^{\star}} \frac{d\omega}{\omega-\zeta}H(\omega,\zeta)
\widetilde{\mathcal{R}}^{T,a}(\omega,\zeta) e^{2N(\Phi_{N}(\zeta)-\Phi_{N}(\omega))},$$ where $s_{N} = s(\xi_{N},\eta_{N};\alpha)$, $\Phi_{N}(\zeta) := \Phi(\zeta;\xi_{N},\eta_{N})$ and the contours $\gamma_{\zeta}^{\star}$ and $\gamma_{\omega}^{\star}$ also depend on $N$, even though this is not indicated in the notation. Since $\gamma_{\zeta}^{\star}$ and $\gamma_{\omega}^{\star}$ do not pass through $r_{+}$ and $r_{-}$, Proposition \[prop:TandTinvsmall\] implies that $$\begin{aligned}
\widetilde{\mathcal{R}}^{T,a}(\omega,\zeta) = \bigO(1), \qquad \mbox{as } N \to + \infty \; \mbox{ uniformly for all } \zeta \in \gamma_{\zeta}^{\star} \mbox{ and } \omega \in \gamma_{\omega}^{\star}.\end{aligned}$$ We also know from Propositions \[prop:contoursexist\] and \[prop:contoursexist eta = xi/2\] that $$\begin{aligned}
{\text{\upshape Re\,}}\Phi_{N}(\zeta) < {\text{\upshape Re\,}}\Phi_{N}(s_{N}) < {\text{\upshape Re\,}}\Phi_{N}(\omega), \qquad \mbox{for all } \zeta \in \gamma_{\zeta}^{\star}\setminus \{s_{N},\overline{s_{N}}\},\omega \in \gamma_{\omega}^{\star}\setminus \{s_{N},\overline{s_{N}}\},\end{aligned}$$ which implies that the right-hand-side of is $$\begin{aligned}
\label{lol28}
\frac{1}{(2\pi i)^2} \hspace{-0.05cm} \int_{\gamma_{\zeta}^{\star} \cap D_{\epsilon}} \hspace{-0.2cm} d\zeta
\hspace{-0.05cm} \int_{\gamma_{\omega}^{\star}\cap D_{\epsilon}} \frac{d\omega}{\omega-\zeta}H(\omega,\zeta)
\widetilde{\mathcal{R}}^{T,a}(\omega,\zeta) e^{2N(\Phi_{N}(\zeta)-\Phi_{N}(\omega))} + \bigO(e^{-C_{1}N}), \quad \mbox{as } N \to \infty,\end{aligned}$$ for a certain $C_{1}>0$, and where $D_{\epsilon}$ is the union of two small disks of radii $\epsilon > 0$ surrounding $s$ and $\overline{s}$. Since $s_{N}$ and $\overline{s_{N}}$ are simple zeros of $\Phi_{N}'$, we have the estimates $$\begin{aligned}
& {\text{\upshape Re\,}}(\Phi_{N}(\zeta)-\Phi_{N}(s_{N})) < -C_{2}|\zeta - s_{N}|^{2}, & & \mbox{for } \zeta \in \gamma_{\zeta}^{\star}\setminus \{s_{N},\overline{s_{N}}\}, \\
& {\text{\upshape Re\,}}(\Phi_{N}(\omega)-\Phi_{N}(s_{N})) \geq C_{2}|\omega - s_{N}|^{2}, & & \mbox{for } \omega \in \gamma_{\omega}^{\star}\setminus \{s_{N},\overline{s_{N}}\},\end{aligned}$$ for a certain $C_{2}>0$. Therefore, the left-most term in is, in absolute value, $$\begin{aligned}
\leq C_{3}\iint_{|x|^{2}+|y|^{2} \leq \epsilon^{2}}\frac{e^{-4C_{2}N(x^{2}+y^{2})}}{\sqrt{x^{2}+y^{2}}}dxdy = 2\pi C_{3} \int_{0}^{\epsilon}e^{-4C_{2}Nr^{2}}dr \leq C_{4} N^{-\frac{1}{2}} \label{lol32}\end{aligned}$$ for certain $C_{3},C_{4}>0$ and for all large enough $N$. Therefore, $$\begin{aligned}
\mathcal I(x_{N},y_{N};H) -\frac{1}{2\pi i} \int_{\overline{s_{N}}}^{s_{N}} H(\zeta,\zeta) d\zeta = \bigO(N^{-1/2}), \qquad \mbox{as } N \to + \infty,\end{aligned}$$ which give .
The case $\xi = 0$ and $\eta < 0$
---------------------------------
Let us briefly recall first the situation for $(\xi',\eta') \in \mathcal{L}$, such that $\eta' < \frac{\xi'}{2}<0$. In this case, the set $\mathcal{N}_{\Phi}$ contains four curves emanating from $s$: three of these curves, namely $\Gamma_{1}$, $\Gamma_{2}$ and $\Gamma_{3}$, lie in $\mbox{int}(\overline{\Sigma_{1}\cup \Sigma_{\alpha}})$, the other curve $\Gamma_{4}$ intersects once $\overline{\Sigma_{1}\cup \Sigma_{\alpha}}$. Denote $p_{j}$ for the intersection of $\Gamma_{j}$ with $\mathbb{R}$, and recall that the ordering for $\Gamma_{1}$, $\Gamma_{2}$ and $\Gamma_{3}$ is such that $p_{1}<p_{2}<p_{3}$.
As $(\xi',\eta') \to (0,\eta)$ with $\eta < 0$ (see Figures \[fig: cases 1 and 2 for Phi\] and \[fig: cases xi=0 for Phi and Psi\] (left)), we know from Proposition \[prop:hightemp\] that $s(\xi',\eta';\alpha)$ tends to a point $s=s(0,\eta;\alpha)$ lying on $\Sigma_{1}$. In this limit, both $\Gamma_{3}$ and $\Gamma_{4}$ tend to the arc $$\begin{aligned}
\Sigma_{s}:= \{c^{-1} + R_{1}e^{it} : -\arg s \leq t \leq \arg s\} \subset \Sigma_{1},\end{aligned}$$ and a part of $\Gamma_{1}$ tends to $\Sigma_{1}\setminus \Sigma_{s}$. Thus, the case $\xi = 0$ gives less freedom for the contour deformations and the saddle point analysis is more involved. To handle this case, we need information about both $\mathcal{N}_{\Phi}$ and $\mathcal{N}_{\Psi}$, where $$\begin{aligned}
\mathcal{N}_{\Psi} := \{\zeta \in \mathbb{C} : {\text{\upshape Re\,}}\Psi(\zeta) = \Psi(s)\}.\end{aligned}$$ The sets $\mathcal{N}_{\Phi}$ and $\mathcal{N}_{\Psi}$ are represented in Figure \[fig: cases xi=0 for Phi and Psi\] for a particular choice of the parameters. We have the following.
For $\xi = 0$, we have $\Sigma_{1} \subset \mathcal{N}_{\Phi}$ and $\Sigma_{1} \subset \mathcal{N}_{\Psi}$
Since ${\text{\upshape Re\,}}\phi(\zeta) = 0$ for $\zeta \in \Sigma_{1}$, by Definition \[def: Phi and Psi\] and we have $$\begin{aligned}
{\text{\upshape Re\,}}\Phi(\zeta) = {\text{\upshape Re\,}}\Psi(\zeta) = -\frac{\eta}{2} {\text{\upshape Re\,}}f_{1}(\zeta),\end{aligned}$$ and by Lemma \[lem:LogsIncSigmaMinus1\] this expression is constant for $\zeta \in \Sigma_{1}$.
We choose $\gamma_{\zeta}^{\star}$ and $\gamma_{\omega}^{\star}$ as follows (see also Figure \[fig: cases xi=0 for Phi and Psi contour\]):
- $\gamma_{\omega}^{\star} \subset \overline{\mbox{int}(\overline{\Sigma_\alpha \cup \Sigma_{1}})}$, is such that $(\Sigma_{1}\setminus \Sigma_{s}) \subset \gamma_{\omega}^{\star}$, surrounds $\alpha c$ and $\alpha c^{-1}$, and it satisfies $$\begin{aligned}
{\text{\upshape Re\,}}\Phi(\omega) > {\text{\upshape Re\,}}\Phi(s), \qquad \omega \in \gamma_{\omega}^{\star} \setminus (\overline{\Sigma_{1}\setminus\Sigma_{s}}),\end{aligned}$$
- $\gamma_\zeta^{\star} \subset \overline{\mbox{int}(\gamma_1)}$, is such that $\Sigma_{s} \subset \gamma_\zeta^{\star}$, surrounds $c$ and $c^{-1}$, and it satisfies $$\begin{aligned}
{\text{\upshape Re\,}}\Phi(\zeta) < {\text{\upshape Re\,}}\Phi(s), \qquad \zeta \in \gamma_{\zeta}^{\star} \setminus \Sigma_{s}.\end{aligned}$$
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(-8.5,-5)–(-8.5,5);
Let $\{(x_{N},y_{N}\}_{N \geq 1}$ be a sequence satisfying with $(\xi,\eta) \in \mathcal{L}_{\alpha}\cap \{\eta < \frac{\xi}{2}= 0\}$, and define $\xi_{N} := x_{N}/N-1$ and $\eta_{N}:= y_{N}/N-1$. For convenience, we assume that $(\xi_{N},\eta_{N}) \in \mathcal{L}_{\alpha}$ satisfies $\eta_{N}<0$ and $\xi_{N} = 0$ for all $N$. The proof of Proposition \[prop:deformationhigh\] still goes through with the above choice of $\gamma_{\zeta}^{\star}$ and $\gamma_{\omega}^{\star}$, and as in we obtain $$\label{lol29}
\mathcal I(x_{N},y_{N};H) -\frac{1}{2\pi i} \int_{\overline{s}}^{s} H(\zeta,\zeta) d\zeta =
\frac{1}{(2\pi i)^2} \int_{\gamma_{\zeta}^{\star}} d\zeta
\int_{\gamma_{\omega}^{\star}} \frac{d\omega}{\omega-\zeta}H(\omega,\zeta)
\widetilde{\mathcal{R}}^{T,a}(\omega,\zeta) e^{2N(\Phi(\zeta)-\Phi(\omega))},$$ where $s = s(\xi_{N},\eta_{N};\alpha)$, $\Phi(\zeta) = \Phi(\zeta;\xi_{N},\eta_{N})$, and the contours $\gamma_{\zeta}^{\star}$ and $\gamma_{\omega}^{\star}$ depend on $N$. We also take the $+$ boundary value in whenever $\omega \in \gamma_{1}$. Since ${\text{\upshape Re\,}}\Phi(\zeta) = {\text{\upshape Re\,}}\Phi(s)$ for all $\zeta \in \Sigma_{s}$ and ${\text{\upshape Re\,}}\Phi(\omega) = {\text{\upshape Re\,}}\Phi(s)$ for all $\omega \in \overline{\Sigma_{1}\setminus \Sigma_{s}}$, we need additional deformation of contours.
We first treat the contour deformations in the $\zeta$-variable. Recall the definition of $\widetilde{\mathcal{R}}^{T,a}$. For $\zeta \in \Sigma_{s}$, we use $\Phi_{+}(\zeta) = \Psi_{-}(\zeta)$ and the jumps for $T$ to obtain $$\begin{aligned}
\label{lol30}
e^{2N\Phi(\zeta)}T(\zeta) \begin{pmatrix}
1 \\ 0
\end{pmatrix} = e^{2N(\Phi_{+}(\zeta)-2\phi_{+}(\zeta))}T_{+}(\zeta) \begin{pmatrix}
0 \\ 1
\end{pmatrix} - e^{2N\Psi_{-}(\zeta)}T_{-}(\zeta)\begin{pmatrix}
0 \\ 1
\end{pmatrix}.\end{aligned}$$ We substitute in , and then split the integral over $\Sigma_{s}\subset \gamma_{\zeta}$ in into two parts. For the second term in , the contour $\Sigma_{s}$ is deformed outwards to $\Sigma_{s,\mathrm{out}}$, see Figure \[fig: cases xi=0 for Phi and Psi further deformations\]. Because $\Psi_{\pm}(\zeta)=\Phi_{\mp}(\zeta)$ for $\zeta \in \Sigma_{1}$, and since $\Sigma_{1}\subset \mathcal{N}_{\Phi}\cap \mathcal{N}_{\psi}$, the signs of $$\begin{aligned}
{\text{\upshape Re\,}}\big(\Phi(\zeta + \epsilon (\zeta-c^{-1}))-\Phi(s)\big) \qquad \mbox{ and } \qquad {\text{\upshape Re\,}}\big(\Psi(\zeta-\epsilon (\zeta-c^{-1})))-\Psi(s)\big)\end{aligned}$$ are different for all $\zeta \in \Sigma_{1}$, provided $\epsilon = \epsilon(\zeta) \in \mathbb{R}$ is small enough ($\epsilon$ non necessarily positive), see also the signs around $\Sigma_{1}$ in Figure \[fig: cases xi=0 for Phi and Psi\]. In particular, we have ${\text{\upshape Re\,}}\Psi(\zeta) < {\text{\upshape Re\,}}\Psi(s)$ for $\zeta \in \Sigma_{s,\mathrm{out}}$. For the first term in , the dominant part is $e^{2N(\Phi_{+}(\zeta)-2 \phi_{+}(\zeta))}$, and by Definition \[def: Phi and Psi\], we have $\Psi = \Phi-2\phi$. Therefore, we deform $\Sigma_{s}$ inwards to $\Sigma_{s,\mathrm{in}}$, and this contour is chosen such that ${\text{\upshape Re\,}}\Psi(\zeta) < {\text{\upshape Re\,}}\Psi(s)$ for $\zeta \in \Sigma_{s,\mathrm{in}}$, see Figure \[fig: cases xi=0 for Phi and Psi further deformations\].
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at ($(2.1,0)+(118:3.6)$) [ $\bullet$]{}; at ($(2.1,0)+(-118:3.6)$) [ $\bullet$]{}; at ($(2.1,0)+(156:3.6)$) [ $\bullet$]{}; at ($(2.1,0)+(-156:3.6)$) [ $\bullet$]{}; at ($(2.1,0)+(118:3.9)$) [$s$]{}; at ($(2.1,0)+(-118:3.9)$) [$\overline{s}$]{}; at ($(2.1,0)+(156:3.9)$) [$r_{+}$]{}; at ($(2.1,0)+(-156:3.9)$) [$r_{-}$]{};
($(2.1,0)+(118:3.6)$) to \[out=-62, in=90\] (1.05,0) to \[out=-90, in=62\] ($(2.1,0)+(-118:3.6)$); ($(2.1,0)+(156:3.6)$) to \[out=-135, in=90\] (-2.15,0) to \[out=-90, in=135\] ($(2.1,0)+(-156:3.6)$); (\[shift=(-180:0.45cm)\]-0.28,0) arc (-180:180:0.45cm);
In the $\omega$-variable, we simply analytically continue the integrand and deform $\Sigma_{1}\setminus \Sigma_{s}$ outwards to $(\gamma_{1}\setminus \Sigma_{s})_{\mathrm{ext}}$, see Figure \[fig: cases xi=0 for Phi and Psi further deformations\]. This contour is chosen such that ${\text{\upshape Re\,}}\Psi(\omega) > {\text{\upshape Re\,}}\Psi(s)$ for $\omega \in (\gamma_{1}\setminus \Sigma_{s})_{\mathrm{ext}}$. Since $\Phi_{+}(\omega) = \Psi_{-}(\omega)$ on $\Sigma_1$, the exponential factor of the integrand is $e^{-2N\Psi(\omega)}$ there. Also, for $\omega \in \Sigma_{1}\setminus \Sigma_{s}$, by we have $$\begin{aligned}
\label{lol31}
\begin{pmatrix} 1 & 0 \end{pmatrix} T_{+}^{-1}(\omega) = \begin{pmatrix} e^{-4N \phi_-(\omega)} & -1 \end{pmatrix} T_{-1}^{-1}(\omega),\end{aligned}$$ and we know from Lemma \[lem:Nphihigh\] that $e^{-4N \phi(\omega)}$ remains bounded for $\omega \in (\gamma_{1}\setminus \Sigma_{s})_{\mathrm{ext}}$.
The result of the above deformations is that the integrand is uniformally exponentially small on the contours, as long as $\zeta$ stays away from $s,\overline{s}$, and that $\omega$ stays away from $s,\overline{s},r_{+},r_{-}$. By a similar analysis as the one done in , we show that the contribution to when $\zeta$ and $\omega$ are close to $s$ or $\overline{s}$ is $\bigO(N^{-\frac{1}{2}})$ as $N \to + \infty$. When $\omega$ is close to $r_{\pm}$, we know by Proposition \[prop:TandTinvsmall\] that $T^{-1}(\omega) = \bigO(N^{1/6})$. Since $\Phi'(r_{\pm}) \neq 0 \neq \Psi'(r_{\pm})$, the contribution to when $\zeta$ is close to $s$ or $\overline{s}$ and simultaneously $\omega$ close to $r_{+}$ or $r_{-}$ is $$\begin{aligned}
\leq C_{1}N^{\frac{1}{6}}\iint_{|x|^{2}+|y|^{2} \leq \epsilon^{2}} e^{-C_{2}N(|x|+y^{2})}dxdy \leq C_{3} N^{-\frac{17}{6}}\end{aligned}$$ for certain constant $C_{1},C_{2},C_{3}>0$ and all large enough $N$. In particular, this proves .
The case $\xi = 0$ and $\eta = 0$
---------------------------------
At the center of the hexagon, we have $s = s(0,0;\alpha) = r_{+}$, $\overline{s} = r_{-}$, and $\Phi = -\Psi = \phi$ (see also Definition \[def: Phi and Psi\]). The sets $\mathcal{N}_{\Phi}$ and $\mathcal{N}_{\Psi}$ are then given by Lemma \[lem:Nphihigh\]: $$\begin{aligned}
\mathcal{N}_{\Phi} = \mathcal{N}_{\Psi} = \mathcal{N}_{\phi} = \Sigma_{0} \cup \Sigma_{\alpha} \cup \Sigma_{1}.\end{aligned}$$ Note that for $(\xi',\eta')=(0,\eta') \in \mathcal{L}_{\alpha}$ with $\eta'<0$, part of contour $\gamma_{\omega}^{\star}$ lies in the region $\mbox{int}(\overline{\Sigma_{\alpha} \cup \Gamma_{1}})$, see Figures \[fig: cases xi=0 for Phi and Psi\] and \[fig: cases xi=0 for Phi and Psi further deformations\]. As $\eta'\to \eta = 0$, $\Gamma_{1}$ tends to $\Sigma_{\alpha}$, so we need additional contour deformations to handle this case. Consider the contours $\gamma_{\zeta}^{\star}:= \gamma_{1}$ and $\gamma_{\omega}^{\star} = \gamma_{\alpha}$. By Lemma \[lem:Nphihigh\], we have $$\begin{aligned}
& {\text{\upshape Re\,}}\Phi(\omega) > 0, \quad \mbox{for } \omega \in \gamma_\alpha \setminus \overline{\Sigma_{\alpha}} & & {\text{\upshape Re\,}}\Phi(\omega) = 0, \quad \mbox{for } \omega \in \Sigma_{\alpha}, \\
& {\text{\upshape Re\,}}\Phi(\zeta) < 0, \quad \mbox{for } \zeta \in \gamma_{1} \setminus \overline{\Sigma_{1}} & & {\text{\upshape Re\,}}\Phi(\zeta) = 0, \quad \mbox{for } \zeta \in \Sigma_1.\end{aligned}$$
at (0,0) ; at (2.1,0) [ $\bullet$]{}; at (-0.38,0) [ $\bullet$]{}; at (-1.05,0) [ $\bullet$]{}; at (-2.035,0) [ $\bullet$]{}; at (-3.13,0) [ $\bullet$]{};
(\[shift=(-118:3.6cm)\]2.1,0) arc (-118:118:3.6cm); ($(2.1,0)+(118:3.6)$) to \[out=-84.5, in=90\] (-0.9,0) to \[out=-90, in=84.5\] ($(2.1,0)+(-118:3.6)$); ($(2.1,0)+(118:3.6)$) to \[out=118-45, in=90\] (6.5,0) to \[out=-90, in=-118+45\] ($(2.1,0)+(-118:3.6)$); at (7,1.6) [$\Sigma_{s,\mathrm{out}}$]{}; ($(2.1,0)+(118:3.6)$) to \[out=118-45-60, in=90\] (5,0) to \[out=-90, in=-118+45+60\] ($(2.1,0)+(-118:3.6)$); at (4,1.6) [$\Sigma_{s,\mathrm{in}}$]{}; (\[shift=(-156:3.6cm)\]2.1,0) arc (-156:-118:3.6cm); (\[shift=(118:3.6cm)\]2.1,0) arc (118:156:3.6cm); ($(2.1,0)+(118:3.6)$) to \[out=118+45, in=-156-75\] ($(2.1,0)+(156:3.6)$); ($(2.1,0)+(-118:3.6)$) to \[out=-118-45, in=156+75\] ($(2.1,0)+(-156:3.6)$); ($(2.1,0)+(118:3.6)$) to \[out=-39.5, in=90\] (2.5,0) to \[out=-90, in=39.5\] ($(2.1,0)+(-118:3.6)$); ($(2.1,0)+(156:3.6)$) to \[out=-150, in=90\] (-2.3,0.1) to \[out=-90, in=90\] (-2.3,0) to \[out=-90, in=90\] (-2.3,-0.1) to \[out=-90, in=150\] ($(2.1,0)+(-156:3.6)$); at (-2.2,2.6) [$(\gamma_{1}\setminus \Sigma_{s})_{\mathrm{out}}$]{}; at (-2.2,-2.6) [$(\gamma_{1}\setminus \Sigma_{s})_{\mathrm{out}}$]{}; (\[shift=(95:1.45cm)\]-1.05,0) arc (95:265:1.45cm);
at ($(2.1,0)+(118:3.6)$) [ $\bullet$]{}; at ($(2.1,0)+(-118:3.6)$) [ $\bullet$]{}; at ($(2.1,0)+(156:3.6)$) [ $\bullet$]{}; at ($(2.1,0)+(-156:3.6)$) [ $\bullet$]{}; at ($(2.1,0)+(118:3.9)$) [$s$]{}; at ($(2.1,0)+(-118:3.9)$) [$\overline{s}$]{}; at ($(2.1,0)+(156:3.9)$) [$r_{+}$]{}; at ($(2.1,0)+(-156:3.9)$) [$r_{-}$]{};
($(2.1,0)+(118:3.6)$) to \[out=-62, in=90\] (1.05,0) to \[out=-90, in=62\] ($(2.1,0)+(-118:3.6)$); ($(2.1,0)+(156:3.6)$) to \[out=-135, in=90\] (-2.15,0) to \[out=-90, in=135\] ($(2.1,0)+(-156:3.6)$); (\[shift=(-180:0.45cm)\]-0.28,0) arc (-180:180:0.45cm);
For simplicity, we consider the sequence $\{(x_{N},y_{N})=(N,N)\}_{N \geq 1}$, so that $\xi_{N} := x_{N}/N-1 =0$ and $\eta_{N}:= y_{N}/N-1 =0$ for all $N$. In the same way as done in Proposition \[prop:deformationhigh\], we find $$\label{lol33}
\mathcal I(x_{N},y_{N};H) -\frac{1}{2\pi i} \int_{r_{-}}^{r_{+}} H(\zeta,\zeta) d\zeta =
\frac{1}{(2\pi i)^2} \int_{\gamma_{\zeta}^{\star}} d\zeta
\int_{\gamma_{\omega}^{\star}} \frac{d\omega}{\omega-\zeta}H(\omega,\zeta)
\widetilde{\mathcal{R}}^{T,a}(\omega,\zeta) e^{2N(\Phi(\zeta)-\Phi(\omega))},$$ and we take the $+$ boundary value in whenever $\omega \in \Sigma_{\alpha}$.
For $\zeta \in \Sigma_{1}$, we use to split the integrand into two parts, and again we deform the integral associated to the first term slightly inwards, and the other one slightly outwards. As a result, both deformed integrals have exponentially decaying integrands.
For $\omega \in \Sigma_{\alpha}$, $\widetilde{\mathcal{R}}^{T,a}(\omega,\zeta)$ is given by the second line of , and thus the dominant $\omega$-part in the integrand is $$\begin{aligned}
e^{-2N \Phi(\omega)} \begin{pmatrix} 1 & - e^{4N \phi(\omega)} \end{pmatrix} T^{-1}(\omega) = e^{-2N \phi(\omega)} \begin{pmatrix} 1 & 0 \end{pmatrix} T^{-1}(\omega) - e^{2N \phi(\omega)} \begin{pmatrix} 0 & 1 \end{pmatrix} T^{-1}(\omega).\end{aligned}$$ For the first term, we deform $\Sigma_{\alpha}$ outwards so that ${\text{\upshape Re\,}}\phi(\omega) > 0$, and for the first term, we deform $\Sigma_{\alpha}$ inwards so that ${\text{\upshape Re\,}}\phi(\omega) < 0$.
On the deformed contours, the integrand in uniformly exponentially small, as long as $\zeta$ and $\omega$ are bounded away from $r_{+}$ and $r_{-}$. For $\zeta$ and $\omega$ close to $r_{\pm}$, by Proposition \[prop:TandTinvsmall\] we have $T(\zeta) = \bigO(N^{1/6})$ and $T^{-1}(\omega) = \bigO(N^{1/6})$. The contribution to when $\zeta$ and $\omega$ are close to $r_{+}$ and $r_{-}$ is thus bounded by $$\begin{aligned}
\leq C_{1}N^{\frac{1}{3}}\iint_{|x|^{2}+|y|^{2} \leq \epsilon^{2}}\frac{e^{-C_{2}N(x^{2}+y^{2})}}{\sqrt{x^{2}+y^{2}}}dxdy \leq C_{3} N^{-\frac{1}{6}}\end{aligned}$$ for certain $C_{1},C_{2},C_{3}>0$ and for all large enough $N$. This finishes the proof of Proposition \[prop:doubleintegrallimit\].
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[^1]: The quantities $N,M$ and $L$ in the notation of [@DK] are equal to $N$, $N$, and $4N$ in our notation.
[^2]: We say that $\mathfrak{p}:\mathcal{I}\to \frac{1}{2}+\mathbb{Z}$ is a *path* if $\mathcal{I} \subset \mathbb{N}$ and for each $x,x+1 \in \mathcal{I}$, we have $\mathfrak{p}(x+1)-\mathfrak{p}(x) \in \{0,1\}$.
|
---
abstract: 'In this work, we introduce [[VQA 360$^\circ$]{}]{}, a novel task of visual question answering on 360$^\circ$ images. Unlike a normal field-of-view image, a 360$^\circ$ image captures the entire visual content around the optical center of a camera, demanding more sophisticated spatial understanding and reasoning. To address this problem, we collect the first [[VQA 360$^\circ$]{}]{}dataset, containing around 17,000 real-world image-question-answer triplets for a variety of question types. We then study two different VQA models on [[VQA 360$^\circ$]{}]{}, including one conventional model that takes an equirectangular image (with intrinsic distortion) as input and one dedicated model that first projects a 360$^\circ$ image onto cubemaps and subsequently aggregates the information from multiple spatial resolutions. We demonstrate that the cubemap-based model with multi-level fusion and attention diffusion performs favorably against other variants and the equirectangular-based models. Nevertheless, the gap between the humans’ and machines’ performance reveals the need for more advanced [[VQA 360$^\circ$]{}]{}algorithms. We, therefore, expect our dataset and studies to serve as the benchmark for future development in this challenging task. Dataset, code, and pre-trained models are available online.[^1]'
author:
- |
Shih-Han Chou$^{1,2}$, Wei-Lun Chao$^{3}$, Wei-Sheng Lai$^{5}$, Min Sun$^{2}$, Ming-Hsuan Yang$^{4,5}$\
$^{1}$University of British Columbia $^{2}$National Tsing Hua University\
$^{3}$The Ohio State University $^{4}$University of California at Merced $^{5}$Google
bibliography:
- 'ref.bib'
title: 'Visual Question Answering on 360$\mathbf{^\circ}$ Images'
---
Introduction {#sec:intro}
============
Visual question answering (VQA) has attracted significant attention recently across multiple research communities. In this task, a machine needs to visually perceive the environment, understand human languages, and perform multimodal reasoning—all of them are essential components to develop modern AI systems. Merely in the past three years, more than two dozen datasets have been published, covering a wide variety of scenes, language styles, as well as reasoning difficulties [@Antol_2015_ICCV; @gao2015you; @goyal2017making; @johnson2017clevr; @malinowski2014nips; @ren2015exploring; @zhu2016visual7w]. Together with those datasets are over a hundred algorithms being developed, consistently shrinking the gap between humans’ and machines’ performance [@ben2017mutan; @fukui2016multimodal; @kafle2017analysis; @kazemi2017show; @kim2016hadamard].
Despite such an explosive effort, existing work is constrained in the way a machine visually perceives the world. Specifically, nearly all the datasets use normal field-of-view (NFOV) images taken by consumer cameras. Convolutional neural networks (CNNs) that are carefully designed for such images [@he2016deep; @VGG] have been necessary to extract powerful visual features. Nevertheless, NFOV images are not the only way, and very likely not the most efficient way, for a machine to interact with the world. For example, considering a 360$^\circ$ horizontally surrounding scene, the NFOV of a consumer camera can only capture an $18\%$ portion [@su2016activity]. Such a fact, together with the reduced price of 360$^\circ$ cameras (e.g., Ricoh Theta S, Samsung Gear 360, and GoPro Omni), has motivated researchers to dig into 360$^\circ$ vision [@cheng2018cube; @chou2017self; @hu2017deep; @su2017learning]. We could imagine every robot to be equipped with a 360$^\circ$ camera in the near future. It is thus desirable to extend VQA to such an informative visual domain.
In this work, we make the first attempt toward VQA on 360$^\circ$ images ([[VQA 360$^\circ$]{}]{}). Two major challenges immediately emerge. First, modern deep learning algorithms are heavily data consuming, yet so far, there is no publicly available dataset for [[VQA 360$^\circ$]{}]{}. Second, 360$^\circ$ (i.e., equirectangular) images have intrinsic distortion and larger spatial coverage, requiring a novel way to process visual inputs and perform sophisticated spatial reasoning. Specifically, a machine needs to understand the spatial information in questions, search answers across the entire 360$^\circ$ scene, and finally aggregate the information to answer. To resolve the first challenge, we collect the first real [[VQA 360$^\circ$]{}]{}dataset, using 360$^\circ$ images from real-world scenes. Our dataset contains about 17,000 image-question-answer triplets with human-annotated answers (see an example in [Figure \[fig:problem\]]{}). We have carefully taken the bias issue [@goyal2017making; @kafle2017analysis], which many existing VQA datasets suffer, into account in designing our dataset. We thus expect our dataset to benefit the development of this novel task. In addition, we study two models to address [[VQA 360$^\circ$]{}]{}. On the one hand, we use equirectangular images as input, similar to conventional VQA models on NFOV images. On the other hand, to alleviate spatial distortion, we represent an input 360$^\circ$ image by six cubemaps [@greene1986environment]. Each map has its own spatial location and suffers less distortion (cf. Figure \[fig:tocube\]). We develop a multi-level attention mechanism with spatial indexing to aggregate information from each cubemap while performing reasoning. In this way, a machine can infer answers at multiple spatial resolutions and locations, effectively addressing the algorithmic challenge of [[VQA 360$^\circ$]{}]{}. Moreover, cubemap-based architecture is flexible to take existing (pre-trained) VQA models as backbone feature extractors on cubemaps, effectively fusing multimodal information and overcoming the limited data issue.
We conduct extensive empirical studies to evaluate multiple variants of these models. The superior performance by the cubemap-based model demonstrates the need to explicitly consider intrinsic properties of [[VQA 360$^\circ$]{}]{}, both visually and semantically. By analyzing the gap between the machine’s and the human’s performance, we further suggest future directions to improve algorithms for [[VQA 360$^\circ$]{}]{}.
Our contributions in this work are two-fold:
We define a novel task named [[VQA 360$^\circ$]{}]{}. We point out the intrinsic difficulties compared to VQA on NFOV images. We further collect the first real [[VQA 360$^\circ$]{}]{}dataset, which is designed to include complicated questions specifically for 360$^\circ$ images.
We comprehensively evaluate two kinds of VQA models for [[VQA 360$^\circ$]{}]{}, including one that can effectively handle spatial distortion while performing multi-level spatial reasoning. We then point out future directions for algorithm design for [[VQA 360$^\circ$]{}]{}.
![**360$\mathbf{^\circ}$ image and cubemaps.** A equirectangular 360$^\circ$ image can be represented by six cubemaps, each corresponding to a spatial location, to reduce spatial distortion.[]{data-label="fig:tocube"}](fig_tocube.pdf){width="0.95\linewidth"}
-12.5pt
Related Work {#sec:related}
============
Visual Question Answering requires comprehending and reasoning with visual (image) and textual (question) information [@zeng2017leveraging]. The mainstream of model architectures is to first learn the joint image-question representation and then predict the answer through multi-way classification. In the first stage, two mechanisms, *visual attention* [@anderson2018bottom; @xu2015show; @lu2016hierarchical] and *multimodal fusion* [@fukui2016multimodal; @ben2017mutan], have been widely explored. For example, the stacked attention networks (SANs) [@yang2016stacked] was developed to perform multi-round attention for higher-level visual understanding. On the other hand, Fukui et al. [@fukui2016multimodal] proposed the Multimodal Compact Bilinear pooling (MCB) to learn a joint representation, and Ben et al. [@ben2017mutan] developed a tensor-based Tucker decomposition to efficiently parameterize the bilinear interaction. Recently, several work [@chen2019uniter; @li2019visualbert; @lu2019vilbert; @su2019vl] extended BERT [@devlin2018bert] by developing new pre-training tasks to learn (bidirectional) transformers [@vaswani2017attention] for joint image and text representations.
Despite the variety of architectures, most of existing methods directly apply CNNs to the whole NFOV image to extract (local) features, which may not be suitable to 360$^\circ$ images. In this paper, we explore a different architecture to extract CNN features from the cubemap representations of a 360$^\circ$ image and then fuse features across cubemaps. The cubemap-based model shares some similarity to [@anderson2018bottom; @yang2016stacked], yet we apply multiple-rounds of attentions to different spatial resolutions, one within and one across cubemaps, so as to achieve better spatial understanding.
There have been over two dozen of VQA datasets on NFOV images published in recent years. Most of them aim for open-ended answering [@Antol_2015_ICCV; @goyal2017making; @krishna2017visual], providing for a pair of image and question with one or multiple correct answers [@chao2018being; @zhu2016visual7w]. An alternative setting is multiple-choice answering: a set of candidate answers are provided for each question, in which one of them is correct. Our [[VQA 360$^\circ$]{}]{}dataset belongs to the first category but focuses on a very different input domain, 360$^\circ$ images.
We note that there are two emerging VQA tasks, embodied QA [@das2018embodied] and interactive QA [@gordon2018iqa], that require a machine to interact with the 3D environment (e.g., turn right or move closer). Our dataset and task are different, from two aspects. First, we work on real-world scenes, while both of them are on synthetic ones. Second, we take 360$^\circ$ images as input while they take NFOV images. A machine there has to take actions to explore the environment, being less efficient.
With the growing popularity of virtual reality (VR) and augmented reality (AR), 360$^\circ$ images and videos have attracted increasing attention lately. One of the interesting problems is to automatically navigate a 360$^\circ$ video [@hu2017deep; @su2017learning; @su2016activity] or create a fast-forward summary [@lai2017semantic]. Other research topics include 360$^\circ$ video stabilization [@kopf2016360], compression [@su2018learning], saliency prediction [@cheng2018cube], depth estimation [@de2018eliminating], and object detection [@chou2019360; @su2017learning]. Recently, Chou [@chou2017self] study visual grounding to localize objects in a 360$^\circ$ video for a given narrative, while Chen [@chen2019touchdown] explore natural language navigation in 360$^\circ$ street environments. In contrast to these tasks, VQA on 360$^\circ$ images requires further inferring the answers according to questions, demanding more sophisticated reasoning of the scene.
2.6pt
[c|l|l|l]{} & & &\
Scene &
-------------------------------------
What room is depicted in the image?
-------------------------------------
&
-------------------------------------
What room is depicted in the image?
-------------------------------------
&
-------------
bedroom/...
-------------
\
Exist &
----------------------------------------------------
Is/Are there (a) $<$obj1$>$ ?
+ in the $<$scene$>$
+ $<$direc$>$
+ $<$direc$>$ of the $<$obj2$>$
+ $<$direc$>$ of the $<$obj2$>$ in the $<$scene$>$
----------------------------------------------------
&
------------------------------------------------------------------
Is there a chair in the kitchen?
Is there a chair at my right side?
Is there a chair at the right side of the window?
Is there a chair at the right side of the window in the kitchen?
------------------------------------------------------------------
&
--------
yes/no
--------
\
Counting &
----------------------------------------------------
How many $<$obj1$>$ are ?
+ in the $<$scene$>$
+ $<$direc$>$
+ $<$direc$>$ of the $<$obj2$>$
+ $<$direc$>$ of the $<$obj2$>$ in the $<$scene$>$
----------------------------------------------------
&
---------------------------------------------------------------------
How many chairs are in the kitchen?
How many chairs are at my right side?
How many chairs are at the right side of the window?
How many chairs are at the right side of the window in the kitchen?
---------------------------------------------------------------------
&
-----------
0/1/2/...
-----------
\
Property &
----------------------------------------------------
What is the ($<$color$>$) $<$obj1$>$ made of?
What is the color of the $<$obj1$>$ ?
+ in the $<$scene$>$
+ $<$direc$>$
+ $<$direc$>$ of the $<$obj2$>$
+ $<$direc$>$ of the $<$obj2$>$ in the $<$scene$>$
----------------------------------------------------
&
[@l@]{}\
\
What is the red sofa in the bedroom made of?\
What is the red sofa at my right side made of?\
What is the color of the sofa at the right of the window?\
What is the color of the sofa at the right of the window in the bedroom?
&
[@l@]{}\
\
plastic/wood/...\
red/brown/...
\
Spatial &
-------------------------------------------------
Where can I find the $<$obj1$>$?
Which side of the $<$obj1$>$ is the $<$obj2$>$?
+ $<$color$>$
+ $<$material$>$
-------------------------------------------------
&
[@l@]{}\
\
Where can I find the white flowers?\
Which side if the white chair is the wooden door?
&
[@l@]{}\
\
in front of you/...\
right side/...
\
-10pt
-5pt
[[VQA 360$^\circ$]{}]{}Dataset {#sec:dataset}
==============================
We first present the proposed [[VQA 360$^\circ$]{}]{}dataset to give a clear look at the task and its intrinsic challenges. We begin with the dataset construction, including image collection, question generation, and answer annotation. We then provide detailed statistics for our [[VQA 360$^\circ$]{}]{}dataset.
Images Collection {#subsec:imagecollect}
-----------------
We focus on indoor scenes as they are usually more dense with contents such as objects, which are suitable for developing algorithms for sophisticated reasoning. In contrast, outdoor scenes, like those in [@hu2017deep; @lai2017semantic; @su2018learning; @su2016activity], capture certain (ego-centric) activities and are of sparse contents, which are more suitable for summarization or navigation.
We collect 360$^\circ$ images of indoor scenes from two publicly accessible datasets, Stanford 2D-3D [@2017arXiv170201105A] and Matterport3D [@Matterport3D]. Both datasets provide useful side information such as scene types and semantic segmentation, which benefit question generation. There are about $23$ different scenes, including common areas in houses (e.g., bathroom, kitchen, bedroom, etc.) and workplaces (e.g., office, conference room, auditorium, etc.). To maximize the image diversity, we discard images captured in the same room but with different viewpoints. In total, we collect $744$ images from the Stanford 2D-3D dataset and $746$ images from the Matterport3D dataset. All the 360$^\circ$ images are stored in the equirectangular format and resized to $1024 \times 512$. The equirectangular projection maps latitude and longitude of a sphere to the horizontal and vertical lines (e.g., a point at the top of the sphere is mapped to a straight line in an equirectangular image), which inevitably introduces heavy spatial distortion.
Question Generation
-------------------
We design several question templates (c.f. Table \[tab:Qtemplete\]), together with the semantic segmentation and scene types associated with each 360$^\circ$ image[^2], to automatically generate questions. Our templates contain five different types: “scene”, “exist”, “counting”, “property” and “spatial”. While imposing templates limit the diversity of questions, the main purpose of our dataset is to promote VQA on a new visual domain that has larger spatial coverage and complexity. As illustrated in Figure \[fig:problem\], a 360$^\circ$ image can easily contain multiple objects distributed at multiple locations. *We thus specifically design the question templates—either include spatial specifications or ask for spatial reasoning—to disambiguate the questions and encourage machines to acquire better spatial understanding.* For instance, to answer “What is the color of the vase at the right of pictures?” in Figure \[fig:problem\], a machine needs to first find the pictures (rightmost), look to the right to find the vase, and return the color[^3]. To answer “Which side of the TV is the pictures?”, a machine needs to detect the TV and picture, and then return their relative spatial information in the scene. Both examples require visual and spatial understanding at multiple resolutions and locations, which are scarce in existing VQA datasets on NFOV images (see the supplementary material for details). On average, we create $11$ questions per image.
Answer Annotations & Question Refinements {#subsec:annotation}
-----------------------------------------
We resort to human annotators to provide precise answers. We ask $20$ in-house annotators to answer the questions in our dataset. To avoid synonyms words and to ease the process, we offer candidate answers according to the question types for annotators to select directly. Annotators can also type free-form answers if none of the candidates is applicable. We note that the automatically generated questions might be irrelevant to the image or lead to ambiguous answers[^4]. In such cases, we instruct the annotators to slightly modify the questions—e.g., by adding spatial specifications—to make them image-related or identifiable. We also instruct annotators to draw bounding boxes (for a subset of image-question pairs), which indicate specific objects or locations associated with the answer. Such information facilitates the analysis of model performances.
Training Validation Test
--------------------- ---------- ------------ ------
$\#$images 743 148 599
QA pairs 8227 1756 6962
$\#$unique answers 51 51 53
$\#$Scene type Q 765 150 614
$\#$Counting type Q 1986 495 1934
$\#$Existed type Q 2015 417 1655
$\#$Property type Q 1355 322 1246
$\#$Spatial type Q 2106 372 1513
: **Summary of 360$\mathbf{^\circ}$ VQA dataset.** We summarize the number of images, QA pairs, and unique answers in each split of our dataset. We also provide a detailed statistic for each type of question.[]{data-label="tab:dataset_statistic"}
-12.5pt
-5pt
![ **Distribution of answers.** We balance our dataset such that the answers of the same question type appear uniformly (e.g., “yes/no”, “0/1”, and “right side/left side”).[]{data-label="fig:statistic"}](statistics_V2.pdf){width="0.9\linewidth"}
-12.5pt
-5pt
Dataset Statistics {#subsec:datasetstatistics}
------------------
Our [[VQA 360$^\circ$]{}]{}dataset consists of $1,490$ images and $16,945$ question-answer pairs, which are split into the training, validation, and test sets with $50\%$, $10\%$, and $40\%$ of images, respectively. We summarize the statistics in Table \[tab:dataset\_statistic\] and show the distribution of the top $20$ answers in Figure \[fig:statistic\]. We note that each question type has at least $2$ corresponding answers in the top $20$ ones. Moreover, those from the same type have the similar number of presence (e.g., “yes/no”, “0/1”, “right/left side”), preventing a machine from cheating by predicting the dominant answer. For question types with a few unique answers, we make sure that the unique answers appear almost uniformly to minimize dataset bias.
[[VQA 360$^\circ$]{}]{}Models {#sec:algorithm}
=============================
{width=".85\linewidth"}
-12.5pt
In this section, we study two VQA models, including one dedicated to resolving inherent challenges in [[VQA 360$^\circ$]{}]{}. Given a question $q$ and an image $i$, a machine needs to generate the answer $a$. One common VQA model is to first extract visual features $f_i = \mathcal{F}_I(i)$ and question features $f_q = \mathcal{F}_Q(q)$, followed by multimodal representations $g_{iq} = \mathcal{G}(f_i, f_q)$. The multimodal representations are then inputted into a classifier $\mathcal{C}(\cdot)$ of $K$ classes, corresponding to the top $K$ frequent answers, to generate the answer $a$. Representative choices for $\mathcal{F}_I(\cdot)$ and $\mathcal{F}_Q(\cdot)$ are CNN and RNN models [@yang2016stacked], respectively.
Equirectangular-based Models {#subsec:equi}
----------------------------
As the most common format to store and display a 360$^\circ$ image is the equirectangular projection into a 2D array, we can indeed directly apply existing (pre-trained) VQA models for [[VQA 360$^\circ$]{}]{}. We take the Multimodal Low-rank Bilinear Attention Network ([[<span style="font-variant:small-caps;">MLB</span>]{}]{}) model [@kim2016hadamard] as an example, which adopts an efficient bilinear interaction for $\mathcal{G}(f_i, f_q)$. We first extract the visual features $f_i$ by the pre-trained ResNet-152 [@he2016deep] and adopt the Gated Recurrent Units (GRU) [@chung2014empirical; @kiros2015skip] to extract the question features $f_q$. We then input the resulting $g_{iq}=\mathcal{G}(f_i,f_q)$ into a fully-connected layer with $K$ output units to build a $K$-way classifier $\mathcal{C}(\cdot)$. We optimize the whole network using the training set of our [[VQA 360$^\circ$]{}]{}dataset and set $K$ to be the number of unique training answers (i.e., $51$).
The [[<span style="font-variant:small-caps;">MLB</span>]{}]{}model $\mathcal{G}(f_i,f_q)$ pre-trained on the [VQA-1]{} [@Antol_2015_ICCV] dataset requires $f_i$ to retain a $14\times 14$ spatial resolution, equivalent to inputting a $448\times 448$ image to the ResNet. We thus adopt a few strategies, including cropping or resizing the original 360$^\circ$ image, or inputting the original image while resizing the output ResNet features into a $14\times 14$ spatial resolution by an average pooling layer. We analyze these strategies in Section \[sec:exp\].
While the above strategies allow us to exploit VQA models pre-trained on much larger NFOV datasets (e.g., [VQA-1]{} [@Antol_2015_ICCV]), applying CNNs directly on 360$^\circ$ images suffers the inherent spatial distortion [@su2017learning]. On the other hand, adopting specifically designed spherical convolutions [@su2017learning] prevents us from leveraging existing models and pre-trained weights. An intermediate solution that takes both concerns into account is thus desirable.
Moreover, existing VQA models like [[<span style="font-variant:small-caps;">MLB</span>]{}]{} [@kim2016hadamard] and SAN [@yang2016stacked] only consider a single visual resolution when performing feature aggregation in $\mathcal{G}(f_i, f_q)$. For 360$^\circ$ images that cover a large spatial range, a more sophisticated mechanism that involves multiple resolutions of feature aggregation is required. To this end, we propose a cubemap-based model to simultaneously tackle the above challenges.
Cubemap-based Models
--------------------
To reduce spatial distortion, we first represent a 360$^\circ$ image by six non-overlapping cubemaps, $\{i^{(j)}\}_{j=1}^J$, via the perspective projection (c.f. Figure \[fig:tocube\]; see the supplementary material for details). Each cubemap corresponds to a specific portion of the 360$^\circ$ image with less distortion. Collectively, the cubemaps together can recover the original image. This representation naturally leads to a bottom-up architecture that begins with the local region understanding and then global reasoning (cf. Figure \[fig:model\]).
In the first stage, we can apply any existing VQA models, e.g., [[<span style="font-variant:small-caps;">MLB</span>]{}]{} [@kim2016hadamard], to each cubemap individually, resulting in $J$ local multimodal representations: $$\begin{aligned}
g_{iq}^{(j)}=\mathcal{G}(f_{i^{(j)}}, f_q)\,,\end{aligned}$$ where $f_{i^{(j)}}$ denotes the visual features of the $j$-th cubemap.
In the second stage, the main challenge is to effectively aggregate information from cubemaps. While average and max pooling have been widely used, they simply ignore the location associated with each cubemap. We thus resort to the attention mechanism: $$\begin{aligned}
& g_{i} = \sum_{j=1}^J \alpha^{(j)}g_{iq}^{(j)},\hspace{10pt}\text{s.t.}~\alpha^{(j)}\geq 0 ,~ \sum_j \alpha^{(j)} = 1. \label{eq:att}\end{aligned}$$ The attention weight $\alpha^{(j)}$ can be computed according to information of each cubemap, *including its location*, making aggregation more flexible. As many existing VQA models already apply the attention mechanism *within* the input images [@kim2016hadamard; @yang2016stacked] (e.g., a cubemap in our cases), the attention to aggregate *across* cubemaps is actually the second-level of attention but on a coarse resolution.
We apply Tucker fusion $\mathcal{T}(\cdot,\cdot)$ [@ben2017mutan] to compute the attention weights according to the cubemap feature $g_{iq}^{(j)}$, location indicator $l^{(j)}$, and question feature $f_q$: Tucker fusion has been shown effective and efficient in fusing information from multiple modalities. The resulting $\alpha^{(j)}$ is as follows, $$\alpha^{(j)} =\text{softmax}\{\mathcal{T}([l^{(j)}, g_{iq}^{(j)}], f_q)\}\,, \label{eq:Tucker}$$ where $[\cdot,\cdot]$ means concatenation. The softmax is performed over $j\in\{1,\cdots,J\}$. We use a one-hot vector $l^{(j)}$ to encode the cubemap location. In this way, the attention weights can zoom into the cubemap location mentioned in the question.
The attention weighs by ; however, do not explicitly consider spatial relationship across cubemaps. For a question like “Is there a chair at the right side of the window?”, we would expect the model to first attend to the cubemap that contain the window, and then *shift* its attention to the cubemap at the right. To incorporate such a capability, we learn a diffusion matrix $M(f_q)$ conditioned on the question $f_q$: the entry $M(f_q)_{u,v}$ indicates how much attention to be shifted from the cubemap $v$ to $u$. The resulting formula for $g_i$ in becomes: $$\begin{aligned}
g_{i}=\sum_{u=1}^J \left(\sum_{v=1}^J M(f_q)_{u,v}\alpha^{(v)}\right) g_{iq}^{(u)},
\text{s.t.} \sum_{u=1}^J M(f_q)_{u,v}=1.
\label{eq:TuckerM}\end{aligned}$$ The resulting feature $g_i$ in or then undergoes another Tucker fusion to extract higher-level image-question interactions before inputted into the classifier $\mathcal{C}(\cdot)$. We can also replace $g_{iq}^{(j)}$ in or by the concatenation of $g_{iq}^{(j)}$ and $l^{(j)}$ to incorporate location cues into $g_i$. This strategy is, however, meaningless to average or max pooling—it simply results in an all-one vector. We illustrate the overall model architecture in [Figure \[fig:model\]]{}. More details are included in the supplementary material.
Experimental Results {#sec:exp}
====================
Setup
-----
The cubemap-based model can take any existing VQA model as the backbone. We choose the [[<span style="font-variant:small-caps;">MLB</span>]{}]{}model [@kim2016hadamard], a bilinear multimodal fusion and attention model. We experiment with other VQA backbones [@ben2017mutan; @singh2018pythia] in the supplementary material to demonstrate the applicability of the cubemap-based models.
We remove the fully-connected layer of the original [[<span style="font-variant:small-caps;">MLB</span>]{}]{}model to extract multimodal features. We apply the pre-trained [[<span style="font-variant:small-caps;">MLB</span>]{}]{}model to each cubemap of size $448\times 448$, and consider the following three different *aggregation schemes* before performing the final answer prediction.
[[<span style="font-variant:small-caps;">Cubemap-Avgpool</span>]{}]{}: apply average pooling on $g_{iq}^{(j)}$.
[[<span style="font-variant:small-caps;">Tucker</span>]{}]{}: attention weights by Tucker fusion in .
[[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{}: attention weights by Tucker fusion followed by the diffusion in .
We consider four ways to apply [[<span style="font-variant:small-caps;">MLB</span>]{}]{}on the equirectangular images.
[[<span style="font-variant:small-caps;">Central-crop</span>]{}]{}: resize the shorter size of the image to $448$ to preserve the aspect ratio and then crop the image to $448 \times 448$ to extract ResNet features.
[[<span style="font-variant:small-caps;">Resize</span>]{}]{}: resize the image into $448 \times 448$ without any cropping and extract ResNet features.
[[<span style="font-variant:small-caps;">ResNet-Avgpool</span>]{}]{}: resize the shorter size of the image to $448$ and apply an average pooling layer on the ResNet output to obtain $14 \times 14$ resolution features.
[[<span style="font-variant:small-caps;">Direct-split</span>]{}]{}: split an equirectangular image into $2 \times 3$ patches, resize each to $448 \times 448$ and apply [[<span style="font-variant:small-caps;">MLB</span>]{}]{}, and then apply [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{}to aggregate information for predicting the answer.
Note that the [[<span style="font-variant:small-caps;">Direct-split</span>]{}]{}and [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{}models have the same architecture but different inputs.
We provide [[<span style="font-variant:small-caps;">Q-type prior</span>]{}]{}, a model that outputs the most frequent answer of each question type.
Model Variants Overall avg Avg by type Scene Exist Counting Property Spatial
-------------------------------------------------------------------- ------------------------------------------------------------------------ ------------- ------------- ----------- ----------- ----------- ----------- -----------
[[<span style="font-variant:small-caps;">Q-type prior</span>]{}]{} - 33.50 31.71 25.41 55.47 33.56 21.99 22.14
Equirectangular-based [[<span style="font-variant:small-caps;">Central-crop</span>]{}]{} 53.39 54.07 60.66 75.00 47.10 50.16 37.45
Equirectangular-based [[<span style="font-variant:small-caps;">Resize</span>]{}]{} 54.21 55.77 68.46 75.66 **47.31** 51.48 35.96
Equirectangular-based [[<span style="font-variant:small-caps;">ResNet-Avgpool</span>]{}]{} 54.47 56.14 69.34 76.81 46.32 50.96 37.25
Equirectangular-based$^\star$ [[<span style="font-variant:small-caps;">ResNet-Avgpool</span>]{}]{} 54.15 55.55 67.48 77.17 46.17 49.04 37.90
Equirectangular-based [[<span style="font-variant:small-caps;">Direct-split</span>]{}]{} 54.77 56.59 71.36 75.75 46.68 49.56 39.62
Cubemap-based [[<span style="font-variant:small-caps;">Cubemap-Avgpool</span>]{}]{} 54.60 56.23 69.17 76.22 46.79 **51.72** 37.26
Cubemap-based [[<span style="font-variant:small-caps;">Tucker</span>]{}]{} 57.71 59.07 69.89 **77.23** 46.53 48.24 53.47
Cubemap-based [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{} **58.66** **60.26** **72.01** 76.34 46.84 50.12 **55.98**
Cubemap-based$^\star$ [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{} 54.09 55.54 67.65 76.16 45.91 48.60 39.39
-10pt
We first pre-train the backbone [[<span style="font-variant:small-caps;">MLB</span>]{}]{}model on the [VQA-1]{} [@Antol_2015_ICCV] dataset, which contains over $100,000$ NFOV images and $300,000$ question-answer pairs for training. Then, we plug the pre-trained model in all the compared models and fine-tune the models on our [[VQA 360$^\circ$]{}]{}training set for $150$ epochs. We optimize our models with the ADAM [@kingma2014adam] optimizer and select the model with the best performance on the validation set. We use the top-1 accuracy for evaluation. We report two types of accuracy: the average accuracy i) over all the questions, and ii) over question types.
Analysis and Discussions
------------------------
Table \[tab:exp\] summarizes the results on [[VQA 360$^\circ$]{}]{}test set. The cubemap-based model with [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{}for attention weights performs favorably against other models, demonstrating the effectiveness of multi-level and *diffused* attention on top of cubemaps representation for [[VQA 360$^\circ$]{}]{}. In the following, we discuss several key observations. The top row (Q-type prior) in Table \[tab:exp\] examines the dataset bias, which predicts the most frequent answer of each question type. The inferior results suggest a low language bias in our dataset. Specifically, for “exist” type questions that only have two valid answers each (i.e, “yes” or “no”), using language prior is close to random guess. Machines need to rely on images to answer.
As shown in Table \[tab:exp\], the [[<span style="font-variant:small-caps;">ResNet-Avgpool</span>]{}]{}model outperforms the [[<span style="font-variant:small-caps;">Central-crop</span>]{}]{}and [[<span style="font-variant:small-caps;">Resize</span>]{}]{}, indicating the poor applicability of cropping and resizing to 360$^\circ$ images. Since 360$^\circ$ images have large spatial coverage, in which objects might be of small sizes, resizing will miss those small objects while central cropping will lose $50\%$ of the image content.
One major issue of applying existing VQA models directly to the 360$^\circ$ images is the spatial distortion. This is justified by the fact that all the equirectangular-based models are outperformed by all the cubemap-based models (except the [[<span style="font-variant:small-caps;">Cubemap-Avgpool</span>]{}]{}one) on the overall performance. Specifically, by comparing the [[<span style="font-variant:small-caps;">Direct-split</span>]{}]{}and [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{}, whose main difference is the input, the $3\sim 4\%$ performance gap clearly reflects the influence of distortion. By looking into different question types, we also observe consistent improvements by applying cubemaps.
Comparing the models with $\star$ (trained from scratch) and without $\star$ (with pre-training), the pre-trained weights (from the VQA-1 dataset) benefits the overall performance, especially for the cubemap-based models.
Applying cubemaps resolves one challenge of [[VQA 360$^\circ$]{}]{}: spatial distortion. We argue that a sophisticated way to aggregate cubemaps features to support spatial reasoning is essential to further boost the performance. This is shown from the improvement by [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{}or [[<span style="font-variant:small-caps;">Tucker</span>]{}]{}, compared to [[<span style="font-variant:small-caps;">Cubemap-Avgpool</span>]{}]{}: the former two apply attention mechanisms guided by questions and cubemap locations for multi-level attention. Specifically, [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{}outperforms [[<span style="font-variant:small-caps;">Cubemap-Avgpool</span>]{}]{}by a notable $3.4\%$ at Avg. by Q type, mostly from the “spatial” question type. [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{}with spatial *diffusion* also outperforms [[<span style="font-variant:small-caps;">Tucker</span>]{}]{}in all the question types.
{width="0.83\linewidth"}
Model Avg. Avg. by Q type Spatial
----------------------------------------------------------------------------- ------- ---------------- ---------
[[<span style="font-variant:small-caps;">Tucker</span>]{}]{}(w/o) 53.81 53.81 36.09
[[<span style="font-variant:small-caps;">Tucker</span>]{}]{}(w/) 57.71 59.07 53.47
[[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{}(w/o) 54.91 56.51 39.13
[[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{}(w/) 58.66 60.26 55.98
: **Comparison of w/ and w/o location feature.**[]{data-label="tab:exp_indicator"}
-12.5pt
Model Overall Scene Exist Counting Property Spatial
--------- --------- ------- ------- ---------- ---------- ---------
Human 84.05 88.95 91.79 71.58 89.97 85.25
Machine 59.80 68.89 77.12 49.65 45.81 61.97
: **Results of human evaluation.** We also include the machine’s performance on the same 1,000 questions to analyze the humans’ and machines’ gap.[]{data-label="tab:exp_human"}
-10pt
Concatenating $l^{(j)}$ with $g_{iq}^{(j)}$ in (\[eq:att\]) and (\[eq:TuckerM\]) enables our model to differentiate cubemaps. Table \[tab:exp\_indicator\] compares the [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{}and [[<span style="font-variant:small-caps;">Tucker</span>]{}]{}with/without $l^{(j)}$. The location indicator leads to consistent improvement, especially on the “spatial” type questions.
We conduct a user study on our [[VQA 360$^\circ$]{}]{}dataset. We sample $1,000$ image-question-answer triplets from the test set and ask at least two different users to answer each question. To ease the process, we give users five candidate answers, including the correct answer and four other answers that are semantically related to the question. There are a total of $50$ unique users participating in the user study. We note that the annotators labeling our dataset are not involved in the human evaluation to avoid any bias.
We summarize the results of human evaluation and the machine’s prediction[^5] in Table \[tab:exp\_human\]. Humans achieve a $84.05\%$ overall accuracy, which is at the same level as many existing VQA datasets [@Antol_2015_ICCV; @chao2018being; @yu2015visual] and is much higher than another dataset on indoor images [@malinowski2014nips], justifying the quality of our [[VQA 360$^\circ$]{}]{}dataset. Among the five question types, humans perform relatively poorly on “counting”, which makes sense due to the complicated contents of $360^\circ$ images and the possible small objects. Overall, there is about $\sim 25\%$ performance gap between human and machines. The gap is larger especially on “counting”, “property”, and “spatial” types, suggesting the directions to improve algorithms so as to match humans’ inference abilities.
We present qualitative results in Figure \[fig:result2\]. Besides showing the predicted answers, we visualize the attention weights across cubemaps (by the digits) and within cubemaps (by the heat maps). The cubemap-based model with [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{}can zoom in to the cubemaps related to the questions, capture the answer regions, and aggregate them to predict the final answers. Take the question “Which side of the window is the painting?” for example (the top-left one of [Figure \[fig:result2\]]{}). The model puts high attention on the cubemaps with windows and pictures and is able to infer the relative location. For the question “What room is depicted in the image?” (the top-right of [Figure \[fig:result2\]]{}), the model distributes attention to all cubemaps except the top and bottom ones to learn information through them. We also show a failure case in the bottom-right of [Figure \[fig:result2\]]{}. The question asks “Which side of the door is the whiteboard?”. However, the model mistakenly recognizes the window as the white board and incorrectly answers “right side”.
Discussion and Conclusion {#sec:conclusion}
=========================
We introduce [[VQA 360$^\circ$]{}]{}, a novel VQA task on a challenging visual domain, 360$^\circ$ images. We collect the first [[VQA 360$^\circ$]{}]{}dataset and experiment with multiple VQA models. We then present a multi-level attention model to effectively handle spatial distortion (via cubemaps) and perform sophisticated reasoning. Experimental results demonstrate the need to explicitly model intrinsic properties of 360$^\circ$ images, while the noticeable gap between humans’ and machines’ performance reveals the difficulty of reasoning on 360$^\circ$ images compared to NFOV images.
We surmise that the gap may partially be attributed to the hand-crafted cubemap cropping. On one end, objects appear around the cubemap boundaries may be splitted. On the other end, it requires specifically designed mechanisms (e.g., attention diffusion (\[eq:TuckerM\])) to reason the spatial relationship among cubemaps. These issues likely explain the human-machine gap at the “counting” and “spatial” questions. Thus, to advance [[VQA 360$^\circ$]{}]{}, we suggest developing image-dependent cropping that detects objectness regions from the equirectangular images. We also suggest developing a back-projection-and-inference mechanism that back-projects the detected objects into the 360$^\circ$ environment and performs reasoning accordingly. Besides, the current questions are generated (or initialized) by templates. A future work is to include more human efforts to increase the question diversity. We expect our dataset and studies to serve as the benchmark for the future developments.
This work is supported in part by NSF CAREER (\# 1149783) and MOST 108-2634-F-007-006 Joint Research Center for AI Technology and All Vista Healthcare, Taiwan.
Supplementary Material
======================
In this section, we present additional results to complement the main paper.
Section \[sec:data\_suppl\]: Details on data collection (cf. Section 3 in the main paper).
Section \[sec:approach\_suppl\]: Implementation details of the proposed model (cf. Section 4.2 and 5.1 in the main paper).
Section \[sec:addition\_exp\]: Additional experimental results on the backbone VQA model and the answer prediction strategy for the cubemap-based models (cf. Section 5 in the main paper).
Section \[sec:result\_suppl\]: Additional qualitative results (cf. Section 5.3 in the main paper).
Data Collection {#sec:data_suppl}
---------------
#### Question generation.
We design templates with place holders (cf. $<\cdot>$ in Table 2 of the main paper) to automatically generate questions. We fill in $<$obj$>$ and $<$scene$>$ according to the semantic segmentation and scene types given by the Stanford 2D-3D [@2017arXiv170201105A] and Matterport3D [@Matterport3D] datasets. We fill in $<$color$>$ of $<$obj$>$ according to the corresponding pixel values. For $<$direc$>$ of $<$obj$>$, we derive it from the corresponding cubemap location. To generate questions with either “no” or “0” as the answer, we fill in the combinations of $<$obj$>$, $<$color$>$, and $<$direc$>$ that are not shown in the images. We provide specific guidance (cf. Figure 2 of the main paper) for the annotators to identify the direction and location in a $360^\circ$ image. We note that human annotators are allowed to modify the questions to make them less ambiguous or more related to the image contents. In details, we instruct human annotators to modify the questions by following the templates in Table 1 in the main paper. This flexibility also increases the diversity of the questions in our [[VQA 360$^\circ$]{}]{}dataset.
#### Question types.
Our templates can be categorized into five different types: “scene”, “exist”, “counting”, “property” and “spatial”.
“Scene” type: related to scene or room types, e.g., kitchen, office, etc.
“Exist” type: related to object presences and positions.
“Counting’ type: for object counting and may involve object attributes and positions.
“Property” type: for object attributes, e.g., color and material.
“Spatial” type: related to objects’ relative positions and the photographer’s relative position.
The “exist”, “counting”, “property”, and “spatial” type questions generally require a model to infer answers from multiple locations (potentially across the entire scene) in a $360^\circ$ image.
#### NFOV vs. 360$^\circ$ images.
The demanding of spatial reasoning plays the key difference for Visual QA on 360$^\circ$ images (as mentioned in Section 3.2 in the main paper). Therefore, the proposed dataset includes questions for spatial reasoning or with spatial cues, either by the templates or by the annotators. Figure \[fig:rebuttal\] shows examples used in the VQA2 paper [@goyal2017making]. This is also evidenced by less than $3\%$ of the questions belonging to the “where” type in VQA2. In contrast, objects in 360$^\circ$ images are highly distributed (even behind the observer), and we have more than $15\%$ “where” type questions.
{width=".7\linewidth"}
Implementation Details {#sec:approach_suppl}
----------------------
We provide the implementation details of the proposed model, following the notations introduced in Section 4.1 in the main paper. We focus on [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{}, together with [[<span style="font-variant:small-caps;">MLB</span>]{}]{} [@kim2016hadamard] as the backbone VQA model and fusion aggregation to predict the answer (cf. Section 4.2 and Figure 4 in the main paper).
[Step 1]{}: Extract the question feature $f_q = \mathcal{F}_Q(q)$.
[Step 2]{}: Extract the visual feature $f_{i^{(j)}} = \mathcal{F}_I(i^{(j)})$ for each cubemap $j$.
[Step 3]{}: Extract the local multimodal feature $g_{iq}^{(j)}=\mathcal{G}(f_{i^{(j)}}, f_q)$ for each cubemap $j$, where $\mathcal{G}(\cdot,\cdot)$ is the [[<span style="font-variant:small-caps;">MLB</span>]{}]{}model without the last fully-connected layer.
[Step 4]{}: Compute the attention weight $\alpha^{(j)}$ for each cubemap $j$, $\alpha^{(j)} \propto \exp\{\mathcal{T}([l^{(j)}, g_{iq}^{(j)}], f_q)\}$, where $\mathcal{T}$ is the Tucker fusion module [@ben2017mutan] with an output dimension $1$ and $l^{(j)}$ is a one-hot location feature. We note that the Tucker fusion’s output dimensionality is adjustable by adding a fully-connected layer.
[Step 5]{}: Generate a diffusion matrix $M(f_q)$ conditioned on the question $f_q$.
[Step 6]{}: Compute the aggregated feature over cubemaps, $g_{i} = \sum_{j=1}^J\left(\sum_{v=1}^J M(f_q)_{j,v}\alpha^{(v)}\right)[l^{(j)}, g_{iq}^{(j)}]$, where we concatenate the location feature $l^{(j)}$ with the multimodal feature $g_{iq}^{(j)}$.
[Step 7]{}: Extract a higher-level multimodal feature $g_{iq}^{(\text{higher})} =\mathcal{T}^{(\text{higher})}(g_{i}, f_q)$, where $\mathcal{T}^{(\text{higher})}$ is the Tucker fusion module with a multi-dimensional output.
[Step 8]{}: Feed $g_{iq}^{(\text{higher})}$ in the classifier $\mathcal{C}(\cdot)$, implemented by a fully-connected layer, to predict the answer.
#### Cubemap projection.
The cube mapping projection is a commonly used method to project an equirectangular image onto NFOV planes [@facebookcube; @cheng2018cube; @el2016streaming; @WangACCV18]. Specifically, there are six cube faces (top, front, left, behind, right and bottom) to fill the whole sphere as shown in Figure \[fig:tocube2\] in the main paper.
![**360$\mathbf{^\circ}$ image and cubemaps.** A 360$^\circ$ image can be represented by six cubemaps, each corresponding to a specific spatial location, to reduce the spatial distortion.[]{data-label="fig:tocube2"}](fig_tocube.pdf){width=".8\linewidth"}
-12.5pt
We use the implementation in [@cheng2018cube] to project the equirectangular images onto cubemaps.
Additional Experimental Results {#sec:addition_exp}
-------------------------------
We provide additional comparisons on the backbone VQA models and the cubemap-based models.
-5pt \[tab:exp\_mlb\]
Model Variants Backbone Overall avg Avg by type Scene Exist Counting Property Spatial
----------------------- ------------------------------------------------------------------------ ---------- ------------- ------------- ------- ------- ---------- ---------- ---------
Equirectangular-based [[<span style="font-variant:small-caps;">ResNet-Avgpool</span>]{}]{} MLB 54.47 56.14 69.34 76.81 46.32 50.96 37.25
Cubemap-based [[<span style="font-variant:small-caps;">Cubemap-Avgpool</span>]{}]{} MLB 55.03 56.89 71.41 76.14 47.72 52.77 36.42
Cubemap-based [[<span style="font-variant:small-caps;">Tucker</span>]{}]{} MLB 57.71 59.07 69.89 77.23 46.53 48.24 53.47
Cubemap-based [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{} MLB 58.66 60.26 72.01 76.34 46.84 50.12 55.98
Equirectangular-based [[<span style="font-variant:small-caps;">ResNet-Avgpool</span>]{}]{} MUTAN 52.05 53.35 65.08 73.18 46.12 47.00 35.38
Cubemap-based [[<span style="font-variant:small-caps;">Cubemap-Avgpool</span>]{}]{} MUTAN 53.56 53.56 69.07 74.86 46.37 50.72 35.45
Cubemap-based [[<span style="font-variant:small-caps;">Tucker</span>]{}]{} MUTAN 54.06 55.29 65.13 74.59 45.06 48.20 43.49
Cubemap-based [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{} MUTAN 54.08 55.80 69.82 75.57 46.32 49.32 37.96
Equirectangular-based - Pythia 50.37 49.59 45.02 43.51 72.69 47.63 39.09
Cubemap-based [[<span style="font-variant:small-caps;">Cubemap-Avgpool</span>]{}]{} Pythia 50.90 51.47 56.37 43.99 70.92 48.68 37.37
Cubemap-based [[<span style="font-variant:small-caps;">Tucker</span>]{}]{} Pythia 51.88 51.34 49.84 46.32 75.48 47.87 37.21
Cubemap-based [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{} Pythia 53.06 52.43 50.41 48.34 75.26 49.00 39.13
-5pt \[tab:exp\_fusion\]
Model Variants Ans. Prediction Overall avg Avg by type Scene Exist Counting Property Spatial
--------------- ------------------------------------------------------------------------ -------------------- ------------- ------------- ------- ------- ---------- ---------- ---------
Cubemap-based [[<span style="font-variant:small-caps;">Cubemap-Avgpool</span>]{}]{} Aggregation 55.03 56.89 71.41 76.15 47.72 52.77 36.42
Cubemap-based [[<span style="font-variant:small-caps;">Cubemap-Avgpool</span>]{}]{} Fusion Aggregation 54.60 56.23 69.17 76.22 46.79 51.72 37.26
Cubemap-based [[<span style="font-variant:small-caps;">Tucker</span>]{}]{} Aggregation 54.12 54.94 62.97 75.24 46.22 46.63 43.62
Cubemap-based [[<span style="font-variant:small-caps;">Tucker</span>]{}]{} Fusion Aggregation 57.71 59.07 69.89 77.23 46.53 48.24 53.47
Cubemap-based [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{} Aggregation 55.21 56.52 66.67 75.75 47.39 52.81 39.99
Cubemap-based [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{} Fusion Aggregation 58.66 60.26 72.01 76.34 46.84 50.12 55.98
Comparisons on Backbone VQA Models {#sec:model_suppl}
----------------------------------
We compare two VQA pre-trained models, MLB [@kim2016hadamard], MUTAN [@ben2017mutan] and Pythia [@singh2018pythia; @singh2019TowardsVM], as the backbone of the proposed method. Table \[tab:exp\_mlb\] summarizes the results of an equirectangular model, [[<span style="font-variant:small-caps;">ResNet-Avgpool</span>]{}]{}, as well as three cubemap-based models, [[<span style="font-variant:small-caps;">Cubemap-Avgpool</span>]{}]{}, [[<span style="font-variant:small-caps;">Tucker</span>]{}]{}, and [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{}. We observe a similar trend as discussed in Section 5.2 of the main paper: the cubemap-based methods generally outperforms the equirectangular-based models, while the cubemap-based [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{}model with multi-level attention performs favorably against other variants. Note that the state-of-the-art model on NFOV images, Pythia [@singh2018pythia; @singh2019TowardsVM], does not perform better than the MLB and MUTAN models. The possible reasons are: 1) the object detector is not generalized well to the [[VQA 360$^\circ$]{}]{}dataset, and 2) the cubemap project sometimes split an object into multiple parts. These observations indicate the potential future directions on exploring the adaptive cubemap projections or object detection on $360^{\circ}$ images.
Comparisons on Answer Prediction Strategies {#sec:exp_suppl}
-------------------------------------------
As mentioned in Section 4 of the main paper, we fuse the multimodal feature $g_i$ with the question feature $f_q$ (which is named as Fusion Aggregation) before inputting to the classifier for predicting the answer. Here we study another simpler strategy – the multimodal feature $g_i$ is directly inputted into the classifier for prediction – which is named as Aggregation. In Table \[tab:exp\_fusion\], we compare these two answer prediction strategies on the cubemap-based [[<span style="font-variant:small-caps;">Cubemap-Avgpool</span>]{}]{}, [[<span style="font-variant:small-caps;">Tucker</span>]{}]{}, and [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{}models.
For the [[<span style="font-variant:small-caps;">Cubemap-Avgpool</span>]{}]{}model, having a higher-level fusion degrades the performance. However, for the [[<span style="font-variant:small-caps;">Tucker</span>]{}]{}and [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{}models, the fusion aggregation clearly improves the overall performance. Since both the [[<span style="font-variant:small-caps;">Tucker</span>]{}]{}and [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{}use the location indicators as one of the features (see Step 6 in Section \[sec:approach\_suppl\]), the fusion aggregation is necessary to associate the question to certain cubemaps so as to answer questions such as “Which side of the tv is the pictures?” in Figure 1 in the main paper. We note that, as shown in Table 4 of the main paper, adding the location feature leads to notable improvements on “spatial” type questions.
Additional Qualitative Results {#sec:result_suppl}
------------------------------
We provide more qualitative results using our cubemap-based model with [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{}in Figure \[fig:resultsupp\] and Figure \[fig:resultsupp2\]. For each $360^\circ$ image, we show both the correct predictions and failure cases. We observe two notable failure cases — colors and properties — both require accurately locating the objects according to the questions, especially for small objects. We suggest that further improvements can be achieved by advanced object detection in 360$\mathbf{^\circ}$ images.
{width="\linewidth"}
{width="\linewidth"}
[^1]: http://aliensunmin.github.io/project/360-VQA/
[^2]: We can obtain room types and objects appearing in the scenes.
[^3]: There are three vases in Figure \[fig:problem\]. Adding spatial specifications is thus necessary, and different specifications will lead to different answers.
[^4]: For instance, if there are two chairs with different colors, a question “What is the color of the chair?” will lead to ambiguous answers.
[^5]: We use our best cubemap-based model [[<span style="font-variant:small-caps;">Tucker&Diffusion</span>]{}]{}.
|
---
abstract: |
------------------------------------------------------------------------
In many statistical problems, several estimators are usually available for interval estimation of a parameter of interest, and hence, the selection of an appropriate estimator is important. The criterion for a good estimator is to have a high coverage probability close to the nominal level and a shorter interval length. However, these two concepts are in opposition to each other: high and low coverages are associated with longer and shorter interval lengths respectively. Some methods, such as bootstrap calibration, modify the nominal level to improve the coverage and thereby allow the selection of intervals based on interval lengths only. Nonetheless, these methods are computationally expensive. In this paper, we propose an index which offers an easy to compute approach of comparing confidence interval estimators based on a compromise between the coverage probability and the confidence interval length. We illustrate that the confidence interval index has range of values within the neighbourhood of the range of the coverage probability, \[0,1\]. In addition, a good confidence interval estimator has an index value approaching 1; and a bad confidence interval has an index value approaching 0. A simulation study was conducted to assess the finite sample performance of the index. The proposed index is illustrated with a practical example from the literature.\
------------------------------------------------------------------------
MSC: 62F99 , 62G99\
*Keywords*: Confidence interval; Empirical coverage probability ; Confidence interval length; Bootstrap calibration
author:
- |
Richard Minkah[^1]\
Department of Statistics and Actuarial Science, College of Basic and Applied Science University of Ghana, Accra, Ghana\
*[email protected]*\
and\
Tertius de Wet\
Department of Statistics and Actuarial Science, Stellenbosch University, Stellenbosch, South Africa\
title: 'Comparison of Confidence Interval Estimators: An Index Approach'
---
\#1
Introduction
============
Most statistical problems involve the estimation of some unknown parameter, $\theta,$ of a population from an observed sample using an estimator, $\hat{\theta}$ [@Banik2010; @Gulhar2012; @Pires2008; @Zaane2012]. In order to provide a complete description of the information in the sample about $\theta,$ a confidence interval is usually constructed. The key concepts associated with confidence intervals are the coverage probability and interval length. The former is the proportion of times the confidence interval encloses $\theta$ under many replications; and the latter refers to the difference between the upper and the lower confidence limits. These two key concepts are related: longer confidence intervals have higher coverage probabilities approaching the nominal level and shorter confidence intervals have lower coverage probabilities. In statistics, one is often faced with a number of confidence intervals for a parameter arising from different estimators or methods of estimation, and a decision has to be made on the “best" method of estimation. Since these two key concepts are in opposition to each other, that is, better coverage probability goes with weaker length and vice versa, it is useful to have some practical way of combining these measures. In this paper, such an easy to compute measure is proposed and applied to two well-known problems as well as a practical example from the literature.
Suppose we have a sample, $\bm{x}=\{x_1,...,x_n\},$ drawn from an unknown distribution function, $F.$ Let $\ell(\alpha;\bm{x})=\left(\ell_L\left(\alpha;\bm{x}\right),\ell_U\left(\alpha;\bm{x}\right)\right)$ be the $\alpha$-level confidence interval for the unknown parameter $\theta$. Also, denote by $L(\alpha),$ the average of the confidence interval length between the upper confidence limit $\ell_U(\alpha;\bm{x})$ and the lower confidence limit, $\ell_L(\alpha;\bm{x}).$ Furthermore, let the coverage probability be given by $\eta(\alpha)=P\left(\theta \in \ell(\alpha;\bm{x})\right).$
The estimation of $\eta(\alpha)$ is an important issue for statisticians and the goal is to obtain a confidence interval estimator with estimated coverage probability (usually referred to as empirical coverage probability), $\hat{\eta}(\alpha),$ equal to the nominal coverage, $1-\alpha$ [@Agresti2000; @Loh1987; @Loh1991]. However, it is often the case that $\hat{\eta}(\alpha)$ is not exactly equal to $1-\alpha.$ A requirement for a good confidence interval estimator is to have a short interval length and a coverage probability equal to or approximately equal to the nominal coverage. As a result, confidence interval estimator selection can be done by a comparison of the intervals’ coverage probabilities and lengths. However, this can be subjective especially if several interval estimators are involved and a compromise is sought between coverage probability and interval length.
A handful of methods to overcome the difficulties in comparing confidence interval estimators rely on an adjustment of the interval lengths such that each interval gives a coverage probability close or equal to the nominal level, $1-\alpha.$ In that case, the comparison of the confidence interval estimators can be done using the confidence interval lengths only. Examples of these methods in the literature include bootstrap calibration [@Loh1987; @Martin1990] and prepivoting [@Beran1987; @Lee2003]. The basic idea underlying the bootstrap calibration is to obtain $\beta~ (\beta<\alpha)$ such that the resultant interval’s coverage probability equals $1-\alpha$ i.e. $\eta(\beta)=1-\alpha$. Also, prepivoting involves the transformation of the lower (and/or upper) confidence level(s) by using its estimated bootstrap distribution function. This has an important application in reducing the coverage error of bootstrap confidence intervals. In addition, prepivoting can be iterated and this automatically moves the empirical coverage closer to the desired level, $1-\alpha.$ However, in practice, these bootstrap-based procedures generally require computationally costly nested bootstraps (i.e. bootstrapping from the bootstrapped data) from the data. For example, in the case of a double bootstrap, the first bootstrap sample needs $B_1$ resamples from the data and then resampling $B_2$ times from each of single bootstrap samples. Thus, the computational cost involves $B_1\times B_2$ samples in addition to the confidence interval calculations. Also, in [@Kilian2000], the authors were constrained in terms of the number of estimators for impulse responses in large Vector Autoregressive Models due to the prohibitive computational cost. Even for the limited confidence interval estimators considered, in cases like the bias-corrected and accelerated bootstrap method, the computing time required for the evaluation of the estimators was over one year.
Furthermore, [@Beran1987] shows that the coverage precision increases with increasing levels of resamples. Thus, the level of resamples can be done until a point where the coverage is approximately equal to $1-\alpha.$ At this stage, the comparison of interval estimators can be done on the interval lengths only. However, in applications, this is limited by the huge computing power and time needed for such levels of bootstrap. As a result, some work has been done to reduce the computational burden involved in the use of these bootstrap based procedures. Among these, [@Loh1987] and [@Loh1988] proposed a linear and a nonlinear interpolation respectively to reduce the level of bootstrap replications in calibration. Also, [@Nankervis2005] provides an algorithm for the double bootstrap, illustrated above, to reduce the $B_1\times B_2$ total resamples to an appreciable level. These algorithms have varying degrees of success in implementation. Nevertheless, in practical application of these methods, a determination is needed of the benefits of higher levels of resamples against the computational cost. In this paper, we propose an index which offers a straightforward approach of comparing confidence interval estimators without the use of Monte Carlo simulation or analytical derivations. The index is based on a compromise between the coverage probability and the confidence interval length. The need for such an index arose from a recent very large simulation study of comparing different estimators of tail index in extreme value theory. Running the simulation to obtain a variety of confidence intervals based on the different estimators, was already computationally very intensive. Applying a further computationally intensive calibration or double bootstrap, would have been too costly in terms of computing time and resources. Hence, the proposed index was developed as a computationally inexpensive compromise between coverage probability and confidence interval length.
The rest of the paper is organised as follows. In section \[sec1\], the bootstrap calibration method is presented. The proposed confidence interval index is presented in Section \[sec2\]. In Section 4, we conduct a simulation study to assess the finite sample performance of the proposed index on four popular confidence interval estimators of the mean from a symmetric and skewed distribution. In addition, several confidence interval estimators of the binomial proportion are examined using the index. Section \[sec4\] deals with an application of the index on a study of the performance of several confidence interval estimators of the coefficient of variation from [@Gulhar2012]. Finally, we present some concluding remarks in Section \[sec5\].
Bootstrap Calibration {#sec1}
=====================
[@Loh1991] catalogs some procedures for generating confidence intervals with improved coverage probabilities. These include Edgeworth expansion (analytical) and bootstrapping (simulation). The author states, with references, that given that the Edgeworth expansion and bootstrap procedures are valid, both produce results that have the same asymptotic error rates. In particular, the bootstrap procedure implements the Edgeworth correction through simulation in an automatic fashion. In view of this, we consider the bootstrap calibration method of [@Loh1987] only.
Let $x_1,\ldots,x_n$ be a random sample of size $n$ from the distribution function $F.$ We consider the estimation of the $100(1-\alpha)\%$ 2-sided normal-theory confidence interval of the mean, $\theta=\theta(F),$ given by
$$[\theta_L,~\theta_U]=\left[\hat{\theta}+\frac{z_{\alpha/2}\hat{\sigma}}{\sqrt{n}},~\hat{\theta}-\frac{z_{\alpha/2}\hat{\sigma}}{\sqrt{n}} \right].$$
Here, $\hat{\sigma}$ and $z_{\alpha/2}=\Phi^{-1}(\alpha/2)$ are the unbiased estimate of the variance and the quantile of the standard normal distribution, $\Phi,$ respectively.
If $F$ is normally distributed and $n$ large, the estimated coverage probability, $\hat{\eta}(\alpha),$ will be close to $1-\alpha.$ However, for smaller $n$ and non-normal distributions, $\hat{\eta}(\alpha)$ may differ substantially from $1-\alpha.$ The idea of calibration introduced by [@Loh1987] is to replace $\alpha$ with $\beta~ (\beta<\alpha)$ such that $$\label{Cal}
\hat{\eta}(\beta)\approxeq1-\alpha.$$ This invariably implies several or possibly infinite search for $\beta,$ satisfying (\[Cal\]), and each of this searches is accompanied by bootstrapping samples to obtain $\hat{\eta}(\beta).$ Thus, the method seems impractical. However, [@Loh1987] and [@Loh1988] respectively proposed a linear and a smooth nonlinear interpolation, to reduce the level of bootstrap resampling needed for the seemingly infinite search for $\beta$ to be replaced with just one level. Thus, the calibration is obtained by generating $B$ bootstrap replications.
Let ${\bm{x}}^{*}=\{x_{1}^{*},\ldots,x_{n}^{*}\}$ be a bootstrap sample from $\bm{x}=\{x_1,x_2,\ldots,x_n\}.$ Also, let $\hat{\theta}^*=\hat{\theta}(\bm{x}^{*}),~\hat{\sigma}^*=\hat{\sigma}(\bm{x}^{*})$ and $t^*_j=\sqrt{n}\left(\hat{\theta}^{*}_{j}-\hat{\theta}\right)/\hat{\sigma}^*$ the $t$ statistic computed from the j*th* bootstrap sample. [@Loh1991] defines $\hat{\lambda}_j=1-\Phi\left(|t^*_j|\right)$ and $\beta$ is taken as the $\alpha$-quantile of $\left(\hat{\lambda}_1,\ldots,\hat{\lambda}_B\right).$ In addition, the author argues that the calibration method above is equivalent to the bootstrap root method of [@Beran1987].
The implementation of the calibration method leads to intervals with error rates comparable to bootstrap $t,$ and the accelerated bias-corrected percentile method. However, it is known that these confidence interval estimation methods have limitations with respect to coverage probability and interval lengths [@Efron1993].
The Index {#sec2}
=========
We introduce an index which offers a straightforward approach and avoids the computational burden in the bootstrap-based methods in comparing confidence interval estimators. In addition, the index abstracts the information provided by the confidence interval length and coverage probability, thereby making it a standalone value for comparative purposes. The idea behind the proposed index was to obtain a value that is simple, easy to interpret, and takes into account confidence interval length and coverage probability. In addition, the index is expected to have a range within the neighborhood of the desired coverage probability and hence, can easily be reported together (e.g. graphically) for comparative purposes.
Consider $R$ confidence interval estimators and let $\bm{\eta}=\{\eta_1,\ldots,\eta_R\}'$ and $\bm{L}=\{L_1,\ldots,L_R\}'$ denote the vectors of realised coverage probabilities and average interval lengths respectively.
The confidence interval index, $I$, is defined as
$$\label{Index1}
I(L_j,\eta_j;\alpha)=k_\alpha\left(1-\frac{1}{2}\left(\frac{1+H(\eta_j;\alpha)}{1+\left(\frac{\eta_j}{1+L_j}\right)}\right)\right),~~ L\geq0,~ 0\leq\eta_j\leq1, ~j=1,\ldots,R,$$
where $k_\alpha$ is a constant depending on the significance level, $\alpha.$ Here, $H$ is a loss function which describes the penalty incurred by the deviation of the empirical coverage probability from $1-\alpha$. In this study, we choose $H$ as a simple absolute loss function defined by
$$\label{Loss}
H(\eta_j;\alpha)=|1-\alpha-\eta_j|, ~0\le \eta \le 1,~j=1\ldots,R.$$
Consequently, using (\[Loss\]), the scaling parameter is taken as $$\label{k_alpha}
k_\alpha=\frac{4-2\alpha}{3-2\alpha},$$ to obtain the range of values of $I(L_j,\eta_j;\alpha)$ within the neighbourhood of the desired coverage probability. To derive the range of values of the index, $I(L_j,\eta_j;\alpha),$ we examine the limit at four extreme cases:
1. $L_j\to 0, \eta_j \to 0 ~ \Longrightarrow ~I(L_j,\eta_j;\alpha) \to \frac{k_\alpha\alpha}{2}.$
2. $L_j\to\infty, \eta_j \to 0 ~ \Longrightarrow ~I(L_j,\eta_j;\alpha) \to \frac{k_\alpha\alpha}{2}.$
3. $L_j\to \infty, \eta_j \to 1-\alpha ~ \Longrightarrow ~I(L_j,\eta_j;\alpha) \to \frac{k_\alpha}{2}.$
4. $L_j\to 0, \eta_j \to 1-\alpha ~ \Longrightarrow ~I(L_j,\eta_j;\alpha) \to 1.$
Thus, $I(L_j,\eta_j;\alpha)$ has a range $\left[k_\alpha\alpha/2,~ 1 \right].$ A bad confidence interval estimator (i.e. an interval with low coverage probability and large interval length) corresponds to cases $\textbf{I}$ and $\textbf{II},$ with $I(L_j,\eta_j;\alpha) \to k_\alpha\alpha/2 .$ On the other hand, a good confidence interval estimator (i.e. case $\textbf{IV}$) has $I(L_j,\eta_j;\alpha) \to 1.$ We note that the range of $I(L_j,\eta_j;\alpha)$ can be transformed to the desirable range of the coverage probability, \[0,1\], via an affine function $f(x)=2x/(2-k_\alpha\alpha)-k_\alpha\alpha/(2-k_\alpha\alpha),$ for increased interpretability.
From the aforementioned limits, we conclude that generally a higher value of the index means a better confidence estimator of the parameter $\theta.$ That is, such an estimator has coverage probability close to the nominal value and shorter interval lengths. In addition, as the index penalises for deviation from the nominal level and larger interval lengths, estimators with small coverage probabilities and/or large interval lengths generally have smaller confidence interval index. Therefore, in using this index for comparative purposes, the estimator with largest index value will be chosen ahead of the smaller values. In the subsequent sections, we take $\alpha=0.05$ and, thus, $I(L_j,\eta_j)\in [0.034, 1.000],~~j=1,\ldots,R.$
We note that, other loss functions can be chosen for this purpose, for example, quadratic, a Huber function [@Huber1992], among others and appropriate values of $k_\alpha$ determined analytically. For example, if we consider the case of the square loss function, $H(\eta_j;\alpha)=\left(1-\alpha-\eta_j\right)^2, ~0\le \eta \le 1,~j=1\ldots,R.$ The value of $k_\alpha$ can be taken as in (\[k\_alpha\]). However, the limits of $I(L_j,\eta_j;\alpha)$ corresponding to cases I, II, III and IV are respectively $k_\alpha\alpha(2-\alpha)/2$, $k_\alpha\alpha(2-\alpha)/2$, $k_\alpha/2$ and 1. Thus, the range of the resulting $I(L_j,\eta_j;\alpha),$ is $[\alpha(2-\alpha)^2/(3-2\alpha),1].$ In the case of the two loss functions considered, the rationale behind the choice of $k_\alpha,$ is to obtain a range of values of $I(L_j,\eta_j;\alpha)$ in the neighbourhood of the range of the coverage probability for ease of interpretation. Lastly, the effect of the choice of loss function is reflected in the range of values of $I(L_j,\eta_j;\alpha).$
Simulation Study {#sec3}
================
In this section, we study the performance of the confidence interval index, $I,$ through a simulation study. In this regard, we assess the performance of several confidence interval estimators of the mean from a symmetric and skewed (or asymmetric) distributions. In addition, several estimators of the binomial proportion are assessed using $I.$
Confidence Interval Index for the Mean {#CI_Mean}
--------------------------------------
We present a simulation study on the estimation of the mean from a symmetric and a skewed distribution in the two subsections that follow. In the former, we considered samples generated from a normal distribution and the latter from a lognormal distribution.
To study the behaviour of the estimators, samples of size, $n~ (n=10, 50, 100,\\ 200, 500,~1000)$ were generated from a normal or a lognormal distribution with mean, $\mu,$ and variance, $\sigma^2.$ The parameter of interest is the population mean, $\mu$, which is estimated by the sample mean, $\bar{x}$. The 95% two-sided confidence interval of $\mu$ was constructed using four different methods, namely, the normal theory interval, the Johnson $t$ interval ( unlike the normal theory interval, adjust for positive and negative skewness in a data set by shifting the endpoints right and left respectively. The Johnson $t$ interval is given by $\left(\bar{x}+ \hat{\kappa}_3/6\sqrt{n}\left(1+2t_{\alpha}^2\right)\right)\pm t_{\alpha}s/\sqrt{n},$ where $\hat{\kappa}_3$ is the estimate of the population skewness $E\left(X-\mu\right)^3/\sigma^3,~~t_\alpha$ is the $\alpha$ quantile of the $t$ distribution with $n-1$ degrees of freedom and $s$ is the sample standard deviation.) [@Johnson1978] and the bootstrap-based intervals-the bootstrap percentile and the Bias-Corrected and accelerated (BCa) [@Efron1993 Chapters 12 and 14].
The following procedure was used to compute the index and its summary statistics:
- Generate $N (N=1000)$ samples each of size $n$ from $N(\mu,\sigma^2).$\[AL1\]
- Draw $B (B=1000)$ bootstrap samples from each sample in A1 and use these to compute $B$ bootstrap confidence intervals (i.e. bootstrap percentile and the BCa) of the mean. Compute the average of the $B$ interval lengths, $L,$ and the empirical coverage probability $\hat{\eta}$ for both bootstrap interval types separately. \[AL2\]
- Compute the confidence interval for the mean using the normal theory interval and Johnson $t$ interval using each of the $N$ samples in A1. Calculate the average of the $N$ interval lengths, $L,$ and the empirical coverage probability $\hat{\eta}$ for the two interval types. \[AL3\]
- Repeat A1-A3 a large number of times $R (R=5000),$ to obtain the pairs $\{(\hat{\eta}_1,L_1),\ldots,(\hat{\eta}_R,L_R)\}$ and, hence, the confidence interval index, $I^{(i,j)},~i=1,\ldots,4,~j=1,\ldots,R.$\[AL4\]
- Compute summary statistics for the indexes, $I^{(i,.)}~i=1,\ldots,4.$ \[AL5\]
### Mean of a Symmetric Distribution {#SecSym}
Table \[tab1\] shows the summary statistics of the index for the four interval types computed for observations from $N(2,1).$ It can be seen that, as the sample size increases, $I$ tends to 1: the confidence interval estimators improve with increasing sample size. This is expected in line with the weak law of large numbers: $\bar{x}$ approaches $\mu$ as $n\to \infty$.
-------- ------------------ --------------- ------------- ---------------------- ---------
$n$ Basic Statistics Normal Theory Johnson $t$ Bootstrap Percentile BCa
Mean 0.8555 0.8600 0.8443 0.8401
$10$ Skewness -0.4151 -2.0374 -0.2722 -0.3029
Kurtosis 0.5392 8.1841 0.1896 0.3133
St. dev 0.0075 0.0033 0.0084 0.0089
Mean 0.8895 0.8913 0.8816 0.8793
$50$ Skewness -0.4417 -1.4934 -0.0191 -0.2255
Kurtosis -0.3837 2.5216 -0.0874 -0.0693
St. dev 0.0061 0.0031 0.0069 0.0074
Mean 0.9229 0.9231 0.9179 0.9158
$100$ Skewness -1.1576 -1.5628 -0.4305 -0.2758
Kurtosis 0.9175 2.7156 -0.1391 -0.3270
St. dev 0.0042 0.0029 0.0062 0.0064
Mean 0.9708 0.9709 0.9693 0.9691
$500$ Skewness -1.6225 -1.7231 -0.7722 -0.8726
Kurtosis 2.7522 3.2814 -0.2012 0.1894
St. dev 0.0021 0.0020 0.0033 0.0033
Mean 0.9777 0.9778 0.9745 0.9732
$1000$ Skewness -2.1632 -2.2197 -1.0131 -0.7991
Kurtosis 6.7939 7.2921 0.8867 0.4052
St. dev 0.0029 0.0029 0.0048 0.0053
-------- ------------------ --------------- ------------- ---------------------- ---------
: Summary statistics for the Confidence Interval Index of the Mean
\[tab1\]
In addition, for smaller sample sizes (i.e. $n\le50$), the Johnson $t$ interval has the largest $I$ values in most cases followed by the normal interval. Generally, these two estimators provide better confidence intervals as they have larger index values for measures of location, smaller variability, large negative skewness and large peakedness. In the case of large sample sizes (i.e. $n>50$), there is not much difference between the performance of the normal theory and Johnson $t$ interval estimators of the mean. The bootstrap percentile is the next best confidence interval estimator of the mean followed by the BCa interval estimator based on the summary statistics. Since the sample mean is an unbiased estimator of the population mean, the percentile interval is expected, as shown in the simulation study, to give better intervals in terms of coverage and interval lengths. We remark that the simulation was carried out for larger sample variances and the results show wider interval length leading to smaller values of the index. Due to space consideration, the results are not included but can be obtained from the authors upon request.
Furthermore, we consider the performance of $I$ in relation to bootstrap calibration of [@Loh1987] and [@Loh1988]. Again, the estimation of the mean of a normal distribution is considered. Here, we considered smaller sample sizes where the empirical coverage probability tends to be smaller than the nominal level, $1-\alpha.$ In that case, calibration can be used to increase the empirical coverage probability to approximately equal to $1-\alpha.$ Our aim in this case is to assess the conclusions reached for calibrated intervals in relation to the index. The results of the simulation study for observations from $N(2,1)$ are presented in Table \[Calibration\].
-------------- ---------------------- ------- ------- ------- -- ------- ------- -------
Sample size Estimator CP L $I$ CP L $I$
\*[$n=10$]{} Normal theory 0.907 1.196 0.849 0.922 1.290 0.852
Johnson $t$ 0.936 1.380 0.855 0.960 1.518 0.853
Bootstrap Percentile 0.884 1.128 0.838 0.906 1.219 0.846
BCa 0.889 1.146 0.840 0.906 1.235 0.845
\*[$n=15$]{} Normal theory 0.931 0.988 0.878 0.945 1.045 0.883
Johnson $t$ 0.950 1.081 0.883 0.950 1.081 0.883
Bootstrap Percentile 0.918 0.953 0.873 0.929 1.004 0.876
BCa 0.917 0.959 0.872 0.931 1.013 0.876
\*[$n=20$]{} Normal theory 0.930 0.866 0.887 0.942 0.890 0.893
Johnson $t$ 0.946 0.924 0.892 0.952 0.953 0.892
Bootstrap Percentile 0.922 0.842 0.884 0.931 0.865 0.888
BCa 0.919 0.847 0.882 0.929 0.870 0.886
\*[$n=30$]{} Normal theory 0.951 0.708 0.912 0.951 0.708 0.912
Johnson $t$ 0.962 0.739 0.907 0.962 0.739 0.907
Bootstrap Percentile 0.950 0.695 0.914 0.950 0.695 0.914
BCa 0.946 0.699 0.911 0.946 0.699 0.911
\*[$n=50$]{} Normal theory 0.950 0.550 0.928 0.950 0.550 0.928
Johnson $t$ 0.953 0.564 0.926 0.953 0.564 0.926
Bootstrap Percentile 0.941 0.545 0.923 0.958 0.575 0.923
BCa 0.945 0.545 0.926 0.954 0.572 0.925
-------------- ---------------------- ------- ------- ------- -- ------- ------- -------
: Calibrated and non-calibrated interval estimators for the mean
\[Calibration\]
For smaller sample sizes $(n\le20),$ the Johnson $t$ interval has empirical coverage probabilities close to the nominal level of 0.95. Calibration of such interval leads to overestimation of the coverage probability. Therefore, we failed to calibrate interval estimators with empirical coverage probability close to $0.95.$ The performance of the Johnson $t$ confidence interval estimator is expected as it adjusts for the skewness in the data (in particular for small sample sizes where skewness is prevalent). However, this interval consistently has the largest interval length compared with the normal theory, bootstrap percentile and BCa. The index values for the Johnson $t$ interval are the largest, and thus, can be considered as the most appropriate confidence interval estimator of the mean.
As the sample size increases, the normal theory interval and the Johnson $t$ interval estimators outperform the other intervals in terms of coverage probability. Also, the normal theory interval estimator has interval lengths fairly competitive to the bootstrap-based intervals and outperforms the Johnson $t$ interval. This can be seen from the index of the normal theory interval having the largest values.
In general, we note that calibration as demonstrated in Table \[Calibration\] does not necessarily bring the empirical coverage up to the desired level of $1-\alpha.$ Thus, calibration, although expensive, is not always attractive. We find that the conclusions of the confidence interval index for the non-calibrated intervals agree mostly with that of the calibrated intervals. Again, if the coverage probability is close to the nominal value, calibration leads to overestimation of the coverage probability. However, the index penalises such intervals for the deviation from the nominal level, and thus, discriminates good intervals from the bad ones.
### Mean of a Skewed Distribution {#SecSkew}
In this section, we consider the performance of the confidence interval index for the estimation of the mean of a skewed distribution. Observations were generated from a lognormal distribution with mean $0.$ Since the skewness of the lognormal distribution depends only on the variance, we took variances of 3, 2 and 0.2 corresponding to skewness of 23.732, 6.185 and 1.516 respectively. The results for these three values are shown in Tables \[skew1\], \[skew2\] and \[skew3\] respectively.
Firstly, for a largely skewed distribution, it is evident that the normal theory interval has the smallest average confidence interval indexes but relatively larger standard deviations. The normal intervals are symmetric, and hence, has a challenge when it is used to provide a confidence interval for the mean of a heavily-skewed distribution. On the other hand, the BCa interval records the best performance as it has the largest average confidence interval indexes. This results from the fact that for a heavily-skewed distribution, large bias is expected but the BCa interval corrects for bias and skewness and hence, provides better intervals that enclose the actual parameter being estimated.
-------------- ------------------ --------------- ------------- ---------------------- ---------
$n$ Basic Statistics Normal Theory Johnson $t$ Bootstrap Percentile BCa
\*[$10$]{} Mean 0.4067 0.4252 0.4091 0.4522
Skewness 0.1708 0.1861 0.2562 0.1138
Kurtosis 0.0329 0.1244 0.0155 0.0543
St. dev 0.0154 0.0152 0.0152 0.0150
\*[$50$]{} Mean 0.5101 0.5142 0.5190 0.5646
Skewness -0.0834 -0.0800 0.0123 0.0611
Kurtosis -0.0276 0.0814 -0.0294 -0.0544
St. dev 0.0143 0.0144 0.0146 0.0128
\*[$100$]{} Mean 0.5485 0.5508 0.5568 0.5952
Skewness -0.0954 -0.1354 -0.1217 -0.0438
Kurtosis 0.1251 0.1313 0.3955 -0.1864
St. dev 0.0134 0.0133 0.0133 0.0122
\*[$500$]{} Mean 0.6209 0.6215 0.6255 0.6454
Skewness 0.0489 0.0370 0.0436 0.1057
Kurtosis -0.3218 -0.3116 -0.2307 0.0324
St. dev 0.0112 0.0112 0.0109 0.0105
\*[$1000$]{} Mean 0.6487 0.6491 0.6511 0.6638
Skewness -0.0177 -0.0281 0.1117 -0.0156
Kurtosis -0.1513 -0.1528 0.0376 -0.2113
St. dev 0.0109 0.0108 0.0113 0.0104
-------------- ------------------ --------------- ------------- ---------------------- ---------
: Summary statistics for the Confidence Interval Index of the Mean from $Lognormal(0,3)$
\[skew1\]
Secondly, in the case of a moderately skewed distribution, the Johnsons-$t$ interval estimator is by far the best estimator of the mean of the lognormal distribution with larger index values. This is followed by the BCa interval estimator especially for smaller sample sizes where skewness is high. However, as the sample size increases, the normal interval estimator improves and surpasses the BCa with larger confidence interval indexes.
Thirdly, for the case of low skewness, i.e. $\sigma^2=0.2,$ the performance relatively follows a similar pattern but with some notable differences. The Johnson $t$ remains the best estimator but the performance of the normal theory interval improves significantly for large sample sizes giving large values of the index. At $n=1000,$ there is not much difference between the two estimators. However, the performance of the BCa interval reduces as its index values are smaller compared with the other estimators. This may be attributed to the low skewness of the distribution, and hence, being close to a symmetric distribution similar to the normal distribution presented in Section \[SecSym\].
Lastly, the index values increase with decreasing variance of the lognormal distribution (i.e. decreasing skewness) signifying better performances than for the case of larger variances (i.e. increasing skewness). This is to be expected as the confidence interval estimators give intervals that have better coverage and smaller interval lengths when skewness is small. This is in conformity with earlier studies that compared estimators of the mean of a skewed distribution based on interval length and coverage probability [@Banik2010]. Therefore, in general, the confidence interval index works well for selecting confidence interval estimators of the mean of a skewed distribution.
The Confidence Interval Index for the Binomial Proportion
---------------------------------------------------------
In this section, we consider the estimation of the binomial proportion by sampling from a binomial distribution. This issue usually arises in applied statistics e.g. incidence rates (in medical science), proportion of defective items (in manufacturing), among others. In addition, unlike the confidence interval for the mean, that of the proportion enables us to measure the performance of the index on non-symmetric intervals.
Assume $X$ is binomially distributed with parameters $n$ and $p,$ written, $X\sim Bin(n,p).$ Here, the estimator, $\hat{p},$ is the maximum likelihood estimator given by $\hat{p}=X/n.$ This estimator is consistent and, since, the expected value of $X$ is equal to $np,$ it is also unbiased.
The most basic form of an interval estimate for the proportion, $p,$ is the Wald interval, $$\label{Wald}
\hat{p}\pm Z_{\alpha/2}\sqrt{\hat{p}(1-\hat{p})/n}$$ [@Leemis1996; @Brown2001; @Brown2002]. The properties of this interval estimator have been studied extensively in the literature. Its performance is known to be erratic with respect to coverage probability. In addition, recommendations concerning the values of $n$ and $p$ where this interval is appropriate are conflicting [@Leemis1996]. Several attempts have been made to obtain better confidence interval estimators of the binomial proportion. For example, [@Agresti2000] proposed an amendment of the interval (\[Wald\]) by defining $\hat{p}$ as $(X+2)/(n+2).$ Some further modifications and other intervals that are not based on the normality assumption are presented in Table \[Est\_Tab\].
[width=]{}
Estimator Confidence Interval Reference
----------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------
Exact $\left(\mbox{Beta}\left(\alpha/2;X,n-X+1\right),~ \mbox{Beta}\left(1-\alpha/2;X,n-X+1\right)\right) $ [@Agresti2000]
Wald $\hat{p}\pm Z_{\alpha/2}\sqrt{\hat{p}(1-\hat{p})/n}$ [@Brown2001]
Arcsin $\sin{^2\left(\arcsin\sqrt{\frac{X-1/2}{n}}\pm\frac{Z_{\alpha/2}}{2\sqrt{n}}\right)}$ [@Pires2008]
Arcsin.CC $\sin{^2\left(\arcsin\sqrt{\frac{X-1/8}{n+3/4}}\pm\frac{Z_{\alpha/2}}{2\sqrt{n+1/2}}\right)}$ [@Pires2008]
Pois $\left(\frac{1}{2n}\chi^2_{2X,1-\alpha/2},~ \frac{1}{2n}\chi^2_{2(X+1),1-\alpha/2}\right)$ [@Leemis1996]
BCG $\frac{X+Z_{\alpha/2}^2/2}{n+Z_{\alpha/2}^2}\pm Z_{\alpha/2}\sqrt{\frac{X}{n^2}\left(1-\frac{X}{n}\right)}$ [@Brown2002]
Wils $\frac{X+Z_{\alpha/2}^2/2}{n+Z_{\alpha/2}^2}\pm \sqrt{\frac{nZ_{\alpha/2}^2}{\left(n+Z_{\alpha/2}^2\right)^2}\left(\frac{X}{n}\left(1-\frac{X}{n}\right)+\frac{Z_{\alpha/2}^2}{4n}\right)}$ [@Wilson1927]
Wils.CC $\left(\frac{2X+Z_{\alpha/2}^2-1-Z_{\alpha/2}\sqrt{Z_{\alpha/2}^2-2-1/n+4X\left(1-X/n+1/n\right)}}{2\left(n+Z_{\alpha/2}^2\right)},~\frac{2X+Z_{\alpha/2}^2+1+Z_{\alpha/2}\sqrt{Z_{\alpha/2}^2+2-1/n+4X\left(1-X/n-1/n\right)}}{2\left(n+Z_{\alpha/2}^2\right)}\right)$ [@Wilson1927]
AgreC $\frac{X+Z_{\alpha/2}^2/2}{n+Z_{\alpha/2}^2}\pm Z_{\alpha/2}\sqrt{\frac{X+Z_{\alpha/2}^2/2}{\left(n+Z_{\alpha/2}^2\right)^2}\left(1-\frac{X+Z_{\alpha/2}^2/2}{n+Z_{\alpha/2}^2}\right)}$ [@Agresti1998]
Ag.add4 $\frac{X+2}{n+4}\pm Z_{\alpha/2}\sqrt{\frac{X+2}{(n+4)^2}\left(1-\frac{X+2}{n+4}\right)}$ [@Agresti2000]
mid-P $\left(\mbox{Beta}\left(\alpha/2;X+1/2,n-X+1/2\right), \mbox{Beta}\left(1-\alpha/2;X+1/2,n-X+1/2\right)\right) $ [@Agresti2000]
: Confidence intervals for the binomial proportion
\[Est\_Tab\]
In the present study, we generated samples of size, $n,$ and proportion, $p,$ from a binomial distribution. Each estimator in Table \[Est\_Tab\] is used to obtain a confidence interval for $p.$ We repeat the process $R~ (R=1000)$ times and obtain the average confidence interval length and coverage probability. We then compute diagnostic checks on these intervals using the index $I$ in (\[Index1\]).
Tables \[Prop\_Tab1\]-\[Prop\_Tab3\] show the results of the simulation for combinations of $n$ and $p.$
[width=]{}
\[Prop\_Tab1\]
[width=]{}
\[Prop\_Tab2\]
[width=]{}
\[Prop\_Tab3\]
We find that the confidence interval length improves with increasing sample size. In addition, most of the empirical coverage probabilities of the estimators become much closer to the nominal level of 0.95 as the sample size increases. In particular, the Pois estimator for estimating $p=0.1,$ improves drastically from $\hat{\eta}=0.556$ to $\hat{\eta}=0.942$ respectively for sample sizes 10 and 100. Together with the corresponding confidence interval lengths, the values of index for the Pois estimator increases from 0.6940 (compared with the best estimator’s index of 0.9395) to 0.9715 (joint best with Arc.CC) for sample sizes 10 and 100 respectively. However, for other values of $p$ (more generally $p>0.1$), the Pois estimator overestimates the coverage probability and has larger confidence intervals relative to the other estimators. Therefore, the Pois estimator has smaller index values compared with the other estimators, and hence, is not appropriate for the estimation of $p.$
Furthermore, the Exact estimator overestimates the coverage probability in all cases. In addition, it has large confidence interval lengths and these are shown in its index values. This is consistent with results reported in [@Pires2008]. In most cases, estimators such as Wilson, AgreC, Ag.add4 and midP have relatively good coverage properties and interval lengths and are shown in their $I$ values usually approaching 1.
In general, for the estimation of a proportion, the index is able to distinguish between estimators that are appropriate or not based on their interval lengths and coverage probabilities.
Application {#sec4}
===========
To illustrate the application of our index, we consider the paper by [@Gulhar2012]. The authors compared several confidence interval estimators for the coefficient of variation (CV). The coefficient of variation is defined as the variability of a random variable relative to its mean. It is usually expressed as a percentage. The confidence interval estimators of the CV were compared based on their interval lengths and the empirical coverage probabilities. The authors used separate plots for the coverage probabilities and the interval lengths across different sample sizes, CV values and distributions. We take a different approach in this paper by constructing plots showing simultaneously the coverage probabilities and the confidence interval index, $I.$
[width=]{}
Abbreviation Confidence Interval type
-------------- ----------------------------------------------------------------------------------
NP.BS Bootstrap Percentile
PBS Bootstrap-$t$
Mill [@Miller1991] interval from asymptotic normal approximation
BSMill Modified median Miller estimator based on critical values from bootstrap samples
BS C.P [@Curto2009] modified median estimator based on BS sample
S.K [@Sharma1994] interval from inverted CV
C.P [@Curto2009] *iid* assumption interval
McK [@McKay1932] Interval from chi-square approximation
MMcK Modified McKay’s interval [@Vangel2012]
Panich [@Panich2009] modified McKay’s interval
Prop [@Gulhar2012] interval from chi-square approximation
MedMill Median Modified Miller Estimator
MedMcK Median Modified of McKay interval
MedMMcK Median Modification of Modified McKay’s interval
Med C.P Median Modified [@Curto2009] interval
: List of confidence interval estimators in [@Gulhar2012]
\[Abbrev\]
The various confidence intervals considered and their abbreviations are presented in Table \[Abbrev\]. We compute the confidence interval indexes for the estimators in Table 8 using the values in Table 4 of [@Gulhar2012 page 63]. In addition, we assess the conclusions reached in that paper with that of the computed confidence interval indexes. The confidence interval indexes for each combination of $n$ and CV are shown in Tables \[A1\]-\[A4\] in Appendix B. In addition, the plots of the coverage probabilities and the corresponding confidence interval index values are presented in Figure \[fig2\]. We can now make inferences from the graphs and compare these with the conclusions reached in [@Gulhar2012].
\
\
\
\
Firstly, it can be seen that the most visible estimator that performs badly is the S.K estimator. It has mostly low coverage probability and this is reflected in it having smaller values on the index. It must be noted that some of the corresponding interval lengths of the S.K estimator were 2 to 8 times shorter than the other interval lengths. However, having a shorter interval length with low coverage probability is not practically desirable. As the sample size increases, there is a remarkable increase in the performance of the S.K estimator especially for $CV=0.5.$ Therefore, we can conclude that, in this case, the index discriminates the bad estimator from the good ones even though shorter interval lengths were recorded.
Secondly, [@Gulhar2012 page 57] concludes that “By $n = 100,$ almost all intervals are performing at a similar level (Figure 1). All C.P intervals (C.P, Med C.P, and BS C.P) over exceeded the expected coverage probability of 95% and reached 100% and are clear outliers". From Figure \[fig2\], it can easily be seen from the bottom panel (i.e. for $n=100$) that the index values for these estimators are smaller compared to the other estimators: this indicates that the C.P-based estimators are inappropriate for the estimation of the CV relative to the other estimators.
Thirdly, we can plot the index against sample size and CV values as an alternative to the four cases: CP against sample size; CP against CV; Interval length against $n;$ and interval length against CV. These graphs, not shown here, lead to the same conclusions obtained in [@Gulhar2012].
In general, the index values are consistent with the conclusions from the CP and the interval lengths. Therefore, the index provides a useful, but computationally inexpensive method for measuring the relative performance of the estimators of confidence intervals for CV.
Conclusion {#sec5}
==========
In this paper, an index for measuring the performance of confidence interval estimators was proposed. The index is based on the traditional trade-off between confidence interval length and empirical coverage probability. Unlike the confidence interval length which has range, ${\mathbb{R}}^+$, the index has range of values within that of the coverage probability. We showed that index values close to 1 indicate a good confidence interval estimator whereas values far removed from 1 indicate a bad confidence interval estimator. Thus, it can easily be superimposed on a plot of coverage probabilities to aid in the selection of estimators with good coverage probabilities and interval lengths. The index can be used alone or to complement the coverage probability for measuring the performance of confidence interval estimators.
In all the simulations and practical application, we assessed the performance of estimators through the sizes of the values of the index. However, an issue of practical importance is the statistical difference between indexes. In practice, we propose that a hypothesis of equality or otherwise can be performed on any observed differences between indexes. Since the sampling distribution of the index remains an open problem, a non-parametric or the estimation of standard errors based on resampling methods can be used in such a test.
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Appendix A
==========
-------------- ------------------ --------------- ------------- ---------------------- ---------
$n$ Basic Statistics Normal Theory Johnson $t$ Bootstrap Percentile BCa
\*[$10$]{} Mean 0.6574 0.6719 0.6547 0.6652
Skewness -0.0870 -0.0719 -0.0700 -0.1204
Kurtosis -0.0205 0.2760 0.1860 -0.0512
St. dev 0.0111 0.0104 0.0115 0.0108
\*[$50$]{} Mean 0.7645 0.7669 0.7640 0.7645
Skewness -0.2761 -0.3045 -0.2162 -0.1518
Kurtosis -0.0811 -0.0616 -0.2474 -0.1753
St. dev 0.0092 0.0090 0.0093 0.0089
\*[$100$]{} Mean 0.8033 0.8048 0.8018 0.8005
Skewness -0.1532 -0.1717 -0.1574 -0.1004
Kurtosis -0.1021 -0.1106 -0.0155 -0.1015
St. dev 0.0083 0.0081 0.0080 0.0083
\*[$500$]{} Mean 0.8806 0.8810 0.8780 0.8757
Skewness -0.6314 -0.7137 -0.2949 -0.4518
Kurtosis -0.3605 -0.1775 -0.6620 -0.0593
St. dev 0.0055 0.0053 0.0065 0.0065
\*[$1000$]{} Mean 0.9064 0.9066 0.9038 0.9024
Skewness -1.0806 -1.1117 -0.6161 -0.5282
Kurtosis 0.6637 0.7065 -0.3380 -0.4309
St. dev 0.0045 0.0043 0.0061 0.0058
-------------- ------------------ --------------- ------------- ---------------------- ---------
: Summary statistics for the Confidence Interval Index of the Mean from $Lognormal(0,1)$
\[skew2\]
-------------- ------------------ --------------- ------------- ---------------------- ---------
$n$ Basic Statistics Normal Theory Johnson $t$ Bootstrap Percentile BCa
\*[$10$]{} Mean 0.9426 0.9549 0.9346 0.9332
Skewness -0.0624 -1.1730 0.0143 -0.0753
Kurtosis -0.0272 1.1878 -0.0062 -0.0660
St. dev 0.0050 0.0030 0.0054 0.0055
\*[$50$]{} Mean 0.9777 0.9793 0.9761 0.9759
Skewness -0.8278 -1.7621 -0.5530 -0.4509
Kurtosis 0.2735 3.6100 -0.0649 -0.2337
St. dev 0.0034 0.0023 0.0039 0.0040
\*[$100$]{} Mean 0.9842 0.9849 0.9833 0.9831
Skewness -1.3544 -1.7882 -0.8309 -0.8662
Kurtosis 2.1611 4.0482 0.3911 0.6067
St. dev 0.0026 0.0019 0.0030 0.0031
\*[$500$]{} Mean 0.9919 0.9920 0.9916 0.9915
Skewness -1.6490 -1.7769 -1.4613 -1.4610
Kurtosis 2.8838 3.5023 1.9419 2.5344
St. dev 0.0021 0.0021 0.0024 0.0025
\*[$1000$]{} Mean 0.9938 0.9938 0.9934 0.9933
Skewness -1.7466 -1.7604 -1.5275 -1.4532
Kurtosis 3.3484 3.3595 2.3416 1.8059
St. dev 0.0020 0.0020 0.0024 0.0025
-------------- ------------------ --------------- ------------- ---------------------- ---------
: Summary statistics for the Confidence Interval Index of the Mean from $Lognormal(0,0.2)$
\[skew3\]
Appendix B
==========
[width=]{}
\[A1\]
[width=]{}
\[A2\]
[width=]{}
\[A3\]
[width=]{}
\[A4\]
[^1]: R. Minkah thanks the UG-Carnegie NGAA and UG-BANGA Africa projects for providing financial support.
|
---
abstract: 'The two-photon dressing of a “three-level ladder” system, here the ground state, the exciton and the biexciton of a semiconductor quantum dot, leads to new eigenstates and allows one to manipulate the time ordering of the paired photons without unitary post processing. We show that, after spectral post-selection of the single dressed states, the time ordering of the cascaded photons can be removed or conserved. Our joint experimental and theoretical study demonstrates the high potential of a “ladder” system to be a versatile source of orthogonally polarized, bunched or antibunched pairs of photons.'
author:
- 'Samir Bounouar\*$^1$, Max Strauß$^1$, Alexander Carmele$^2$, Peter Schnauber$^1$, Alexander Thoma$^1$, Manuel Gschrey$^1$, Jan-Hindrik Schulze$^1$, André Strittmatter$^1$, Sven Rodt$^1$, Andreas Knorr$^2$ and Stephan Reitzenstein$^1$'
title: 'Path-controlled time reordering of paired photons in a dressed three-level cascade'
---
Resonance fluorescence , i.e. scattering of radiation by atomic systems irradiated by a resonant or quasiresonant laser field, has become a very active field of research in quantum optics in recent years. It has been first predicted [@mollow] and demonstrated [@schuda; @wu] that for a two-level system driven by a high intensity laser, the fluorescence spectrum is characterized by three components, namely the Mollow triplet. After the demonstration that the overall statistics of the generated photons is subpoissonian [@kimble], the correlation of the different components showed the heralding of the photons coming from the different sidebands [@cohen; @shrama]. These features were later suc- cessfully reproduced thanks to the resonant excitation of the excitonic transition in quantum dots (QD) acting as “artifcial atoms” [@muller; @atature; @shih; @ulhaq; @xu2; @He]. Further observations, in the context of coherent control, such as the Rabi oscillations [@kamada; @Ramsay], Ramsey interferences [@jayakumar], Autler-Townes splittings [@Xu; @kabuss] confirmed this “atom-like” behaviour.
The involvement of the biexciton state $|XX\rangle$ of a QD forms with the exciton state $|X\rangle$ and the ground state $|G\rangle$ a Ladder-type system. The latter is highly attractive for the field of quantum communication since it proved to be able to provide on demand polarization-entangled [@ondem] and time-bin entangled photons [@timebin] and gives access to multiple applications [@Pan; @laussy]. Polarization entanglement is usually considered when the excitonic fine structure splitting is smaller than the radiative linewidth of the involved transitions [@Young], making the two recombination paths of $|XX\rangle$ indistinguishable. This is experimentally very difficult to obtain with standard InGaAs QD, for which even since even uni-axially applied strain tuning is not sufficient to fulfill this condition [@jons]. Another proposed alternative is to tune the dot spectrum to have coincidence of colors across generations, rather than within generations [@gershoni]. It relies on the the manipulation of the photons quantum chronology so that the “which-path” information contained by the time ordering of the photons is erased. The underlying unitary transformation is challenging and was originally proposed to be performed with bulky linear optics [@knill].
In this letter, we present a robust and universally implementable scheme to perform this task through the resonant two-photon driving of a Ladder-type system. Interestingly, this system which was recently uncovered experimentally [@hargart; @ardelt], suffered so far from severe experimental restrictions which prevented one from detecting and analyzing the emitted photons in the time domain, for atoms as well as for QDs. We solve this issue by using determinsitcally fabricated QD-microlenses [@gschrey] in combination with high efficiency laser suppression. This allows for the practical use of the generated paired or single photons generated by this excitation scheme. The photons coming from the different dressed states and the perpendicularly polarized remaining bare exciton are correlated. Due the coupling of the field, photons within an emited pair have no ordering in their arrival times, which is in contrast to the natural time ordering of the bare excitonic and biexcitonic photons from the radiative cascade. The non-linear nature of the coherent driving is reflected through two-photon Rabi oscillations observed in the time domain. Detuning of the laser allows for a highly efficient preparation of a particular dressed state and forces the system along a certain path in the dressed radiative cascade. Under these conditions the time ordering of the photons is reestablished. These features are in very good agreements with theoretically calculated correlations in the dressed states basis.
Our experiments are carried out on self-assembled InGaAs/GaAs QDs grown by metal-organic chemical vapor deposition (MOCVD) and embedded in deterministically fabricated microlenses [@gschrey]. These nanophotonic structures allow for an efficient and broadband collection of the excitonic and biexcitonic photons. They also play a positive role on the focusing of the excitation laser on the target QD, improving the signal to noise ratio of the observed transitions. In comparison to planar structures with usually very poor extraction efficiency this leads to major advantages especially with respect to photon-correlation measurements [@ardelt]. The microlens sample is placed in a Helium flow cryostat and cooled at a temperature of 5 K. By using a cross-polarization configuration, the laser is efficiently suppressed in order to observe the emission lines with very reduced laser background.
![(a) Typical emission spectra of the two-photon resonantly excited QD: for an excitation power $P_o=100\, \mu\text{W}$ (upper spectrum), for an excitation power of $P=38\, P_o$ (lower spectrum). (b) Left: Scheme of the relevant QD bare excitonic states. Right: Scheme of the dressed states resulting from the two-photon laser coupling to the excitonic states. (c) Upper graph: Spectra of the lines $L_+$, $L_0$, $R_+$ and $R_0$ as a function of the power. Lower graph: spectra of the same lines for different laser detunings to the two-photon resonance. (d) Autocorrelation of the $R_0$ line photons for a laser detuning of $52 \mu eV$. []{data-label="fig:gull"}](Graph1_brandneu.eps){width="\linewidth"}
Fig. 1 (a) shows typical spectra of the QD $|G\rangle$-$|X\rangle$-$|XX\rangle$ system being strongly driven by a narrow external cavity tunable laser in resonance to the virtual state of the two-photon resonance for two different excitation powers (upper spectrum for $P_o=100\, \mu$W and lower spectrum for $P=38\, P_o=3.8\, m$W respectively) (i.e. $E_l=E_X-\Delta E_b/2=E_{XX}+\Delta E_b/2$, with $E_l$ the laser energy, $E_X$ the exciton energy and $\Delta E_b$ the biexciton binding energy ). The exciting laser is vertically polarized in order to only involve the vertical excitonic state $|V\rangle$ in the coherent driving and leave the horizontally polarized exciton state $|H\rangle$ unexcited, as can be seen in the underlying level scheme presented in Fig. 1(b). The strong coupling of the laser to the $|G\rangle-|V\rangle$ and $|V\rangle-|B\rangle$ transitions lead to new eigenstates labelled $|+\rangle$, $|-\rangle$ and $|0\rangle$. In particular, the states $|+\rangle$ and $|0\rangle$ can be written:
$$\begin{aligned}
\label{eq:solve}
|0\rangle=\frac{1}{\sqrt{2+(\frac{E_+}{\hbar\Omega})^2}}\Big(|G\rangle+ E_+/(\hbar\Omega)|V\rangle+|XX\rangle\Big)\\
|+\rangle=\frac{1}{\sqrt{2}}(|XX\rangle-|G\rangle)\end{aligned}$$
$\Omega$ is the (single photon) Rabi frequency and $E_+$ is the eigenenergy of the state $|+\rangle$ (calculated in the supplementary material). The expression of these states show that the state $|+\rangle$ is a simple superposition of $|G\rangle$ and $|XX\rangle$, independant from the Rabi frequency. Its eigenergy is therefore constant with the field strength. On the other hand, the state $|0\rangle$ is a function of the Rabi frequency: its composition and energy change with the excitation power [@delvalle]. The resulting emission lines are schematized on Fig. 1 (b). The two lower energy lines labeled $L_0$ and $L_+$ correspond to two transitions from the dressed states of the manifold associated with n+2 interacting photons ($|+\rangle_{n+2}$ and $|0\rangle_{n+2}$) to the bare excitonic state $|H\rangle$. The two lines labeled $R_0$ and $R_+$ on the high energy side correspond to transitions from $|H\rangle$ to the dressed states of the manifold associated with n interacting photons ($|+\rangle_{n}$ and $|0\rangle_{n}$). Fig. 1 (c) (upper panel) shows the splitting of the dressed states as a function of the excitation power. The increase of the splitting with the excitation power is characteristic of the driving of a ladder system with a resonant laser. As expected from the theory, the energy of the dressed state $|+\rangle$ ($E_+$) stays unchanged, whereas the state $|0\rangle$ ($E_0)$ is shifted to higher energies [@delvalle]. Interestingly, no broadening of the dressed state emission lines above the experimental resolution can be observed, which is here a major difference with the phonon-mediated decoherence of the Mollow sidebands [@weiler]. This interesting and unexpected feature is most probably related to the fact that in contrast to the standard 2-level system, here the detected transition is not directly driven. Further studies are required to understand the underlying physics in more detail. Fig. 1 (c) (lower panel) shows the spectra of the lines around the original $|X\rangle$ and $|XX\rangle$ energies as the laser is swept through the two-photon resonance, at fixed excitation power. A clear anticrossing-like behavior is observed for the two doublets, evidencing the coherent interaction between the laser and the excitonic system. By laser detunings ($\lvert \Delta_{laser} \rvert> 60$ $\mu$eV), one can prepare a particular dressed state with a great fidelity, and force the cascaded emission of the photon pairs to take a priviledged path through the dressed states ladder. When the laser is detuned on the higher energy side of the two-photon excitation, the lines $L_+$ and $R_0$ are very intense whereas $L_0$ and $R_+$ are suppressed. The same way, when the laser is detuned towards the low energy side $L_0$ and $R_+$ are the privileged transitions whereas $L_+$ and $R_0$ are suppressed.\
![\[fig1\] (a) Intensity of the dressed states ($L_++L_0$) after the rise of the 20 ns long laser pulse, (b) Intensity of the excitonic emission ($R_++R_0$) after the rise of the same 20 ns long laser pulse](Rabi_Kombi_G1.eps){width="\linewidth"}
The coherent nature of the two-photon driving is confirmed by the observation of Rabi oscillations in the time domain. A two-photon resonant laser pulse at a repetition rate of 17 MHz and 20 ns long (much longer than the lifetime of the transitions $\tau_{(L_0,L_+)}=314$ ps) excites the quantum dot under quasi-CW pumping. The photons emitted by the lines $L_0$ and $L_+$ are detected together by a single photon couting module (SPCM) with 40 ps temporal resolution. The delays between the arrival times of the photons and the trigger (provided by the pulse generator) are measured and stored in a correlation histogram, allowing one to follow the time evolution of the dressed state population under the resonant CW excitation. Fig. 2 (a) shows the resulting measurement. After a non-linear rise of the detection probability, some damped oscillations can be observed, probing the beating of the dressed states between $|XX\rangle$ and $|G\rangle$. These oscillations were until now only probed through picosecond pulsed excitation, as a function of the pulse area [@stuffler]. This measurement is here fully accounting for the two-photon resonant driving of the biexciton, where the intermediate exciton state is detuned and therefore adiabatically eliminated leading to an effective two-photon Rabi frequency of $\Omega^2/\Delta E_b$. This measurement provides a dynamical picture of the dressed states population during the coherent driving, whereas in the pulsed configuration one actually measures the occupation of the bare biexciton a long time after the end of the pulse, i.e. after the phonon-mediated relaxation from the dressed states to the bare excitonic states, as the system is not coherently driven any more. Fig. 2 (c) shows the extracted Rabi frequencies as a function of the excitation power. Interestingly, because of the non-linear nature of the two-photon driving, the evolution of the Rabi frequency with respect to the excitation power is linear while, in the case of a two-level system coherently driven by a strongly coupled laser, it scales with square root of the power [@wei]. The same measurement was made with the $R_0$ and $R_+$ photons, as shown in Fig. 2 (b). In contrast to the dressed states, no oscillation is here observed: the $|H\rangle$ exciton is not coherently driven and the oscillations probed on the occupation probability of the dressed states are washed out through the radiative relaxation towards $|H\rangle$. These measurements are in good qualitative agreement with the theory presented in the supplementary information [@sup] (section V.I). Michelson-type interference measurements gives acces to the Fourier transform (defined as $g^{(1)}(t)$) of the spectrum. The first order correlation functions $g^{(1)}(t)$, measured for the ($ L_0+L_+$) lines and for the ($R_0+R_+$) lines are shown in the inset of Fig. 2 (c). They both exhibit a gaussian profil and their fit gives a coherence time of 390 ps $\pm20$ ps for the ($ L_0+L_+$) photons and of 420 ps $\pm$ 30 ps for ($ R_0+R_+$) photons. The former value is particularly interesting if one considers the very short lifetime of the biexcitonic transitions. These measurements are performed at low power, and the splitting of the dressed states is evaluated to be smaller than $10 \mu eV$. This explains the corresponding oscillations in $g^{(1)}(t)$ with a low bound of the oscillation period of 400 ps are not visible to significant damping at this time-scale.
In order to understand the dynamics of the dressed states and the time ordering of the emitted photons under resonant driving, we correlate the different lines of the emission spectrum with a cross-correlation setup of temporal resolution of 140 ps (full width half-maximum).
We first checked the autocorrelation of the individual lines $L_0$ and $R_0$, (see inset Fig.1 (d)). As expected, they both exhibit antibunchings. Interestingly, with the laser detuned (by 52 $\mu eV$) from the two-photon resonance ($L_+$ suppressed), the correlation of the $L_0$ line is showing a deep antibunching with $g^{(2)}_{L_0}(0)$ value as low as 0.07 limited by non-ideal laser suppression. This indicates that this scheme could be operated as an efficient coherently tunable single photon source, with no need of usually applied strain or temperature-tuning. Here the two-photon driving gives the appealing advantage to offer spectral fine-tuning of coherent single photons emitted far from the laser energy. This could be a substantial improvement for quantum memory schemes requesting the coupling of a quantum dot single photons to an atomic transition [@ulrich], or to the exploration of cavity strong coupling regimes with Mollow triplet sidebands [@kim]. The measured characteristic antibunching time ($\tau_p=350\, ps$) corresponds to the time needed by the system to repopulate the next higher two-photon manifold allowing for another two-photon radiative cascade via the non-driven exciton state $|H\rangle$ (see Fig 1 (b)).
In the following, we address the correlations between the single emission lines. Fig. 3 shows the cross-correlation of the single dressed state lines in two-photon resonance. For the sake of consistency, negative delays always describe the emission of a “biexcitonic state” ($L_+$ or $L_0$) after triggering from a “$R_+$ or $R_0$” photon while positive delays correspond to the detection of an “excitonic” photon after a “biexcitonic” photon. Fig. 3 (a) shows the correlation between $L_+$ and $R_+$. Bunchings are observed for positive and negative delays, meaning that the two successive emitted photons are emitted from the same direct radiative cascade without time-ordering. This means that an “excitonic”’ photon can also trigger a “biexcitonic”’ photon, which is not possible when the radiative cascade is bare. This is due to the fact that this correlated emission of paired photons involves the same dressed state as an initial state and as a final state. The relatively large observed bunching ($g^{(2)}_{L_+,R_+}(0)=4$) indicates that the heralded photons are strongly correlated, presenting obvious advantages in the future realization of Franson-type experiments without the need of tuning the system in a compromise between high count rate and strong correlation, as it is the case for a two-level system [@franss]. Fig 3 (b) presents the cross-correlation between the $L_0$ and $R_+$ photons. In this case, photon antibunching is observed for the negative delays. This shows that the original time ordering of the photons is conserved. This result is explained by the orthogonality of the involved dressed states as described in Eq. 1 and illustrates the versatility of the resonantly-driven radiative cascade: by tuning the spectral detection windows, one has the opportunity to select either time ordered or time unordered paired photons rendering our system an ideal testbed for any time-bin critical experiments.
The laser field is now detuned ($\Delta_{laser}= -63$ $\mu$eV ) from the two-photon resonance. As shown on Fig. 1 (c), under detuned driving, the eigenstates of the system are modified and thus their emission energies. A path though the dressed radiative cascade is priviledged, which has some further consequences on the time ordering of the emitted photons. An example is shown in the inset of Fig. 4. With a detuning $\Delta_{laser}= -63$ $\mu$eV, the lines $L_0$ and $R_+$ become very week whereas the lines $R_0$ and $L_+$ are very bright. The system is actually “forced”’ to emit preferentially through the path ($|+\rangle$-$|H\rangle$-$|0\rangle$) and the state $|+\rangle$ is deterministically prepared by the laser field. Fig 4 (a) shows the correlation between the two strong lines of the spectrum $R_0$ and $L_+$. As for the case of strictly resonant exictation, an antibunching can be observed for small negative delays. Despite the fact that the composition of the $|+\rangle$ and $|0\rangle$ states are changed, they remain, as eigenstates, orthogonal to each other. Fig. 4 (b) shows the correlation between the bright line $R_0$ and the weak line $L_0$. Whereas a bunching can be observed for the positive delays, neither a bunching nor an antibunching can be seen for the negative delays. The detuning of the laser changes the composition of the dressed states, making the emission from the state $|0\rangle$ unprobable. As a consequence, the detection of a $L_0$ photon triggers with a large probability a $R_0$ photon, whereas the detection probability of a $L_0$ photon after the triggering of a $R_0$ photon becomes so weak that its statistics is poissonian. This effect can be seen as an analogy to the time ordering between Mollow triplet sidebands under detuned excitation [@shrama; @muller]. Therefore a time ordering is reintroduced between photons which used to be unordered under resonant excitation.
In conclusion, we demonstrated the coherent control over a three-level ladder system and evidenced the two-photon oscillation of its dressed states in the time domain. We show that the strong coupling to the laser field induces a removal of the photon ordering or its conservation, depending on which dressed states are chosen as initial and final states. Providing that the quantum dot photons can be detected in both polarization components, the time reordering of the photons can be used to complete the “which-path” information erasure *in situ*, without unitary post processing. The large degree of correlation shown by the heralded photons can be used for efficient Franson-type interferometry [@franss]. It can also be applied to the generation of entangled photons through a resonant cavity mode under resonant excitation [@delvalle]. In the field of quantum communication, it could be directly used in order to enlarge the success probability of quantum teleportation of photonic qubits generated with QDs [@shields].
The research leading to these results has received funding from from the European Research Council (ERC) under the European Union’s Seventh Framework ERC Grant Agreement No. 615613 and from the German Research Foundation via Project No. RE2974/4-1 and RE2974/12-1. We also acknowledge support from the Deutsche Forschungsgemeinschaft (DFG) through SFB 787. A. C. gratefully acknowledges support from the SFB 910: “Control of self-organizing nonlinear systems”.
\*S.B. and M.S. contributed equally to this work.
###
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abstract: 'We study black hole radiation inside black holes within the framework of quantum gravity. First, we review on our previous work of a canonical quantization for a spherically symmetric geometry where one of the spatial coordinates is treated as the time variable, since we think of the interior region of a black hole. Based on this formalism, under physically plausible assumptions, we solve the Wheeler-De Witt equation inside the black hole, and show that the mass-loss rate of an evaporating black hole due to thermal radiation is equivalent to the result obtained by Hawking in his semi-classical approach. A remarkable point is that our assumptions make the momentum constraint coincide with the Hamiltonian constraint up to an irrelevant overall factor. Furthermore, for comparison, we solve the Wheeler-De Witt equation outside the black hole as well, and see that the mass-loss rate of an evaporating black hole has the same expression. The present analysis suggests that the black hole radiation comes from the black hole singularity. We also comment on the Birkhoff theorem in quantum gravity.'
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=cmr5 \#1[\#1\^[\^]{}]{}
\#1
DPUR/TH/43\
March, 2015\
[**Hawking Radiation inside Black Holes in Quantum Gravity** ]{}
Ichiro Oda [^1]
Department of Physics, Faculty of Science, University of the Ryukyus,\
Nishihara, Okinawa 903-0213, Japan.\
Introduction
============
There has been a recent revival of an interest in the interior of a black hole. For instance, there has been an active debate on whether the AdS/CFT correspondence could describe the physics of the interior of a black hole or not [@Pol1; @Pol2; @Papa1]. This problem is closely related to the information loss paradox in physics of black holes. Moreover, it has been more recently pointed out that black holes have a very large interior: For a stellar black hole, the volume inside the black hole is larger than that of our universe [@Chris; @Beng]. This study is also relevant to the information loss paradox since there might be a lot of real estate available inside a black hole for stocking micro-states associated with a black hole entropy. [^2]
In the 1990’s, there were also active studies of the interior region of a black hole despite that the interior is physically of no relevance for external observers outside the horizon. In those days, the interest in the interior of a black hole has been triggered by development of the understanding of the internal geometry near the Cauchy horizon inside the Reissner-Nordstrom black hole, what is called, the mass inflation [@Poisson; @Ori; @Brady], and the phenomenon of the smearing of a black hole singularity in quantum gravity [@Hosoya1]. As one of motivations behind these studies, there was an expectation that since both the Cauchy horizon and a spacetime singularity exhibit highly pathological behavior in the classical theory of general relativity, and quantum effects would play a dominant role, studies of the physics inside the horizon of a black hole might give us some important clues for constructing a theory of quantum gravity.
Stimulated with the interest in the interior of a black hole in the 1990’s, we have already formulated a canonical formalism of a system with a spherically symmetric black hole holding in the interior region bounded by the apparent horizon and the singularity [@Hosoya2], which is a natural generalization of the canonical formalism holding in the exterior region covering the spacetime between the apparent horizon and the spatial infinity [@Hajicek]. In this region, the Killing vector $\frac{\partial}{\partial t}$ becomes spacelike whereas it is timelike in the exterior region. Consequently, one has to foliate the interior of a black hole with a family of spacelike hypersurfaces, for instance, $r = const$.
As one of applications of this canonical formalism, following Tomimatsu’s idea [@Tomimatsu], we have considered black hole radiation in quantum gravity where it was shown that the mass-loss rate due to the black hole radiation is equal to that evaluated by Hawking in the semiclassical approximation [@Hawking]. [^3] In this work, we have focused on the vicinity of the apparent horizon where one component $\gamma$ in the metric tensor becomes zero so we had to adopt a regularization such that $\gamma$ takes a small but finite value.
This regularization is clearly so unwelcoming that one should dispense with it. In this article, without considering the vicinity of the apparent horizon at the beginning, making more physically plausible assumptions, we will derive the Hawking radiation in the interior region within the framework of quantum gravity. This modification of the model setting makes it possible to calculate the expectation value of the mass-loss rate without any pathology. Moreover, we will also consider the exterior region of a black hole and do the same job in order to make a comparison of black hole radiation between the interior and the exterior of the horizon.
The analysis of both the interior and exterior regions of the apparent horizon of a dynamical black hole suggests that the thermal radiation of the back hole comes from the spacetime singularity. This fact might imply that the resolution of the information loss paradox needs the understanding of the physics of the spacetime singularity of a black hole. As a bonus, we will comment on the Birkhoff theorem [@Birkhoff] in quantum gravity.
This article is organized as follows: In the next section, after mentioning notation and conventions, we review on the canonical formalism of a system with a spherically symmetric black hole in the interior region bounded by the apparent horizon and the singularity [@Hosoya2]. In Section 3, we apply the canonical formalism reviewed in Section 2 for the calculation of the mass-loss rate due to black hole radiation. In Section 4, we also calculate the mass-loss rate in the exterior region bounded by the apparent horizon and the spatial infinity. The final section contains a conclusion.
Review of canonical formalism inside black hole
===============================================
Before delving into details, let us explain our notation and conventions. We mainly follow notation and conventions by Misner et al.’s textbook [@MTW], for instance, the flat Minkowski metric $\eta_{\mu\nu} = diag(-, +, +, +)$, the Riemann curvature tensor $R^\mu \, _{\nu\alpha\beta} =
\partial_\alpha \Gamma^\mu_{\nu\beta} - \partial_\beta \Gamma^\mu_{\nu\alpha} + \Gamma^\mu_{\sigma\alpha}
\Gamma^\sigma_{\nu\beta} - \Gamma^\mu_{\sigma\beta} \Gamma^\sigma_{\nu\alpha}$, and the Ricci tensor $R_{\mu\nu} = R^\alpha \, _{\mu\alpha\nu}$. Throughout this article, we adopt the natural units $c = \hbar = G = 1$. In this units, all quantities become dimensionless.
Let us start with the review of a canonical formalism of a spherically symmetric system with a black hole, which has been presented in our previous work [@Hosoya2], but compared to this work we will change the notation slightly for the purpose of the comparison with the canonical formalism in the exterior geomery (For instance, we have exchanged the role between the lapse function $\alpha$ and the shift function $\beta$).
As a first step, one needs to select arbitrary spherically symmetric spacelike hypersurfaces to foliate the spacetime. The key point here is that the radial coordinate plays the role of time in the interior of the horizon in the spherically symmetric coordinate system. As a simple choice, it is convenient to take the $x^1 = const$ hypersurfaces to slice the interior region. In the next section, we will choose the simplest case, $x^1 = r$.
The four-dimensional action which we consider in this paper takes the following form: $$\begin{aligned}
S = \int d^4 x \sqrt{- ^{(4)}g} \left[ \frac{1}{16 \pi} \, ^{(4)}R - \frac{1}{4 \pi} \, ^{(4)}g^{\mu\nu}
(D_\mu \Phi)^\dagger D_\nu \Phi - \frac{1}{16 \pi} F_{\mu\nu} F^{\mu\nu} \right],
\label{Action 1}\end{aligned}$$ where $\Phi$ is a complex scalar field and its covariant derivative is given by $$\begin{aligned}
D_\mu \Phi = \partial_\mu \Phi + i e A_\mu \Phi,
\label{Cov-derivative}\end{aligned}$$ with $e$ and $A_\mu$ being the electric charge of $\Phi$ and the $U(1)$ gauge field, respectively. Moreover, as usual, $F_{\mu\nu}$ is the field strength defined as $$\begin{aligned}
F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.
\label{Field strength}\end{aligned}$$ To clarify the four-dimensional meaning we put the suffix $(4)$ in front of the metric tensor and the scalar curvature. As a final note, the Greek indices $\mu, \nu, \cdots$ take the four-dimensional values 0, 1, 2 and 3 whereas the Latin ones $a, b, \cdots$ do the two-dimensional values 0 and 1. Of course, it is straightforward to include the other matter fields as well as the cosmological constant in the action (\[Action 1\]) even if we limit ourselves to the action for simplicity.
The most general spherically symmetric ansatz for the four-dimensional line element is of form $$\begin{aligned}
^{(4)}ds^2 &=& ^{(4)}g_{\mu\nu} dx^\mu dx^\nu, \nonumber\\
&=& g_{ab} dx^a dx^b + \phi^2 ( d\theta^2 + \sin^2 \theta d\varphi^2 ),
\label{Line element}\end{aligned}$$ where the two-dimensional metric $g_{ab}$ and the radial function $\phi$ are the functions of only the two-dimensional coordinates $x^a$. The substitution of the ansatz (\[Line element\]) into the action (\[Action 1\]) and then integration over the angular variables $(\theta, \varphi)$ produces the two-dimensional effective action $$\begin{aligned}
S &=& \frac{1}{2} \int d^2 x \sqrt{-g} \left[ 1 + g^{ab} \partial_a \phi \partial_b \phi
+ \frac{1}{2} R \phi^2 \right] \nonumber\\
&-& \int d^2 x \sqrt{-g} \left[ \phi^2 g^{ab} (D_a \Phi)^\dagger D_b \Phi
+ \frac{1}{4} \phi^2 F_{ab} F^{ab} \right],
\label{2D action}\end{aligned}$$ where we have assumed that $A_a$ and $\Phi$ are also the functions of the two-dimensional coordinates $x^a$ and set $A_\theta = A_\varphi = 0$.
Next let us rewrite the action (\[2D action\]) in the ADM form. As remarked before, we will regard the $x^1$ spatial coordinate as time to cover the interior of a black hole by spacelike hypersurfaces. The appropriate ADM splitting of (1+1)-dimensional spacetime is given by $$\begin{aligned}
g_{ab} = \left(
\begin{array}{cc}
\gamma & \beta \\
\beta & \frac{\beta^2}{\gamma} - \alpha^2
\end{array}
\right).
\label{ADM}\end{aligned}$$ The normal unit vector $n^a$ which is orthogonal to the hypersurfaces $x^1 = const$ reads $$\begin{aligned}
n^a = \left( \frac{\beta}{\alpha \gamma}, \, - \frac{1}{\alpha} \right).
\label{Normal unit}\end{aligned}$$ The induced metric on the hypersurfaces, that is, the projection operator over $x^1 = const$ hypersurfaces, is given by $$\begin{aligned}
h^{ab} = g^{ab} + n^a n^b.
\label{Projection}\end{aligned}$$ It is easy to check that $h^{ab}$ is indeed the projection operator by inserting (\[ADM\]) and (\[Normal unit\]) to (\[Projection\]).
The extrinsic curvature $K_{ab}$, its trace $K$ and the scalar curvature $R$ are given by [@Wald] $$\begin{aligned}
K_{ab} &=& K_{ba} = h_a \, ^c \nabla_c n_b, \nonumber\\
K &=& g^{ab} K_{ab} = \nabla_a n^a = \frac{1}{\sqrt{-g}} \partial_a ( \sqrt{-g} \, n^a ),
\nonumber\\
R &=& 2 n^a \partial_a K + 2 K^2 - 2 \nabla_c ( n^a \nabla_a n^c ).
\label{Extrinsic 1}\end{aligned}$$ Using Eqs. (\[ADM\])-(\[Extrinsic 1\]), a straightforward calculation reveals us $$\begin{aligned}
K &=& - \frac{\gamma'}{2 \alpha \gamma} + \frac{\dot{\beta}}{\alpha \gamma}
- \frac{\beta}{2 \alpha \gamma^2} \dot{\gamma}, \nonumber\\
R &=& 2 n^a \partial_a K + 2 K^2 - \frac{2}{\alpha \sqrt{\gamma}} \partial_0
\left( \frac{\dot{\alpha}}{\sqrt{\gamma}} \right),
\label{Extrinsic 2}\end{aligned}$$ where $\frac{\partial}{\partial x^0} = \partial_0$ and $\frac{\partial}{\partial x^1} = \partial_1$ are also denoted as an overdot and a prime, respectively. With the help of these equations, one can cast the action (\[2D action\]) to the form $$\begin{aligned}
S &\equiv& \int d^2 x L \nonumber\\
&=& \int d^2 x \Biggl[ \frac{1}{2} \alpha \sqrt{\gamma} \Biggl\{ 1 - (n^a \partial_a \phi)^2
+ \frac{1}{\gamma} \dot{\phi}^2 - K n^a \partial_a (\phi^2)
+ \frac{\dot{\alpha}}{\alpha \gamma} \partial_0 (\phi^2) \Biggr\} \nonumber\\
&+& \alpha \sqrt{\gamma} \phi^2 \left\{ | n^a D_a \Phi |^2 - \frac{1}{\gamma} | D_0 \Phi |^2 \right\}
+ \frac{1}{2} \alpha \sqrt{\gamma} \phi^2 E^2 \Biggr] \nonumber\\
&+& \int d^2 x \left[ \frac{1}{2} \partial_a ( \alpha \sqrt{\gamma} K n^a \phi^2 )
- \frac{1}{2} \partial_0 \left( \frac{\dot{\alpha}}{\sqrt{\gamma}} \phi^2 \right) \right],
\label{2D action 2}\end{aligned}$$ where we have defined $E$ as $$\begin{aligned}
E = \frac{1}{\sqrt{-g}} F_{01} = \frac{1}{\alpha \sqrt{\gamma}} (\dot{A}_1 - A'_0).
\label{E}\end{aligned}$$ Now the differentiation of the action (\[2D action 2\]) with respect to the spatial derivative of the canonical variables $\Phi ( \Phi^\dagger ), \phi, \gamma$ and $A_0$ leads to the corresponding canonical conjugate momenta $p_\Phi ( p_{\Phi^\dagger} ), p_\phi, p_\gamma$ and $p_A$ $$\begin{aligned}
p_\Phi &=& - \sqrt{\gamma} \phi^2 n^a ( D_a \Phi )^\dagger, \quad
p_\phi = \sqrt{\gamma} n^a \partial_a \phi + \sqrt{\gamma} K \phi, \nonumber\\
p_\gamma &=& \frac{1}{4 \sqrt{\gamma}} n^a \partial_a (\phi^2), \quad
p_A = - \phi^2 E.
\label{Momenta}\end{aligned}$$ Then, the Hamiltonian, which is defined as $$\begin{aligned}
H = \int d x^0 \left( p_\Phi \Phi' + p_{\Phi^\dagger} \Phi'^\dagger + p_\phi \phi'
+ p_\gamma \gamma' + p_A A_0' - L \right),
\label{Hamiltonian 1}\end{aligned}$$ is expressed in terms of a linear combination of constraints as expected from diffeomorphism invariance $$\begin{aligned}
H = \int d x^0 \left( \alpha H_0 + \beta H_1 + A_1 H_2 \right),
\label{Hamiltonian 2}\end{aligned}$$ where $\alpha, \beta$ and $A_1$ are non-dynamical Lagrange multiplier fields, and the Hamiltonian constraint, the momentum one and the constraint associated with the $U(1)$ gauge transformation are respectively given by $$\begin{aligned}
H_0 &=& \frac{1}{\sqrt{\gamma} \phi^2} p_\Phi p_{\Phi^\dagger} - \frac{\sqrt{\gamma}}{2}
- \frac{\dot{\phi}^2}{2 \sqrt{\gamma}} + \partial_0 \left( \frac{\partial_0 (\phi^2)}{2 \sqrt{\gamma}} \right)
+ \frac{\phi^2}{\sqrt{\gamma}} | D_0 \Phi |^2
\nonumber\\
&-& \frac{2 \sqrt{\gamma}}{\phi} p_\phi p_\gamma
+ \frac{2 \gamma \sqrt{\gamma}}{\phi^2} p_\gamma^2 + \frac{\sqrt{\gamma}}{2 \phi^2} p_A^2,
\nonumber\\
H_1 &=& \frac{1}{\gamma} \left[ p_\Phi D_0 \Phi + p_{\Phi^\dagger} (D_0 \Phi)^\dagger \right]
+ \frac{1}{\gamma} p_\phi \dot{\phi} - 2 \dot{p}_\gamma - \frac{1}{\gamma} p_\gamma \dot{\gamma},
\nonumber\\
H_2 &=& - i e \left( p_\Phi \Phi - p_{\Phi^\dagger} \Phi^\dagger \right) - \dot{p}_A.
\label{Constraints 1}\end{aligned}$$ As is well known, the action can be written as the first-order ADM canonical form by the dual Legendre transformation $$\begin{aligned}
S = \int d x^1 \left[ \int d x^0 \left( p_\Phi \Phi' + p_{\Phi^\dagger} \Phi'^\dagger
+ p_\phi \phi' + p_\gamma \gamma' + p_A A_0' \right) - H \right].
\label{Legendre}\end{aligned}$$ In order to obtain the correct Hamiltonian which yields the Einstein equations through the Hamilton equations, it is necessary to supplement surface terms to the Hamiltonian (\[Hamiltonian 2\]) [@Regge]. In the formalism at hand, since we take the variation of all the fields to be zero at boundaries, we do not have to add any surface terms to the Hamiltonian.
Black hole radiation in the interior region
===========================================
We are now ready to apply the canonical formalism constructed in the previous section for understanding the Hawking radiation [@Hawking] from the viewpoint of the internal region of a black hole in quantum gravity. A similar analysis was performed in our previous work [@Hosoya2] where only the region near the apparent horizon was considered from the outset. In the present article, we first work with the whole region bounded between the apparent horizon and the spacetime singularity, make important assumptions on some variables, derive the Wheeler-De Witt equation, and solve it analytically. After performing this procedure, we impose the condition of the vicinity of the apparent horizon for compatibility with field equations. We will see that this method nicely overcomes the problem of the vanishing $\gamma$ variable near the apparent horizon. In the next section, we will apply the same method to the exterior region bounded between the apparent horizon and the spatial infinity.
To consider the simplest model of the Hawking radiation, let us switch off the $U(1)$ gauge field and treat with the neutral scalar field by which the gauge constraint $H_2$ is identically vanishing and the Hamitonian and momentum constraints reduce to the simpler form $$\begin{aligned}
H_0 &=& - \frac{\sqrt{\gamma}}{2} - \frac{\dot{\phi}^2}{2 \sqrt{\gamma}}
+ \partial_0 \left( \frac{\partial_0 (\phi^2)}{2 \sqrt{\gamma}} \right)
+ \frac{\phi^2}{\sqrt{\gamma}} ( \partial_0 \Phi )^2
- \frac{2 \sqrt{\gamma}}{\phi} p_\phi p_\gamma
+ \frac{2 \gamma \sqrt{\gamma}}{\phi^2} p_\gamma^2,
\nonumber\\
H_1 &=& \frac{1}{\gamma} p_\Phi \partial_0 \Phi
+ \frac{1}{\gamma} p_\phi \dot{\phi} - 2 \dot{p}_\gamma - \frac{1}{\gamma} p_\gamma \dot{\gamma}.
\label{Constraints 2}\end{aligned}$$ Moreover, we will use the ingoing Vaidya metric [@Vaidya] to describe the black hole radiation. The treatment of the case of the outgoing Vaidya metric can be made in a perfectly analogous manner. We therefore define the two-dimensional coordinates $x^a$ as $$\begin{aligned}
x^a = ( x^0, x^1 ) = ( v - r, r ),
\label{Advanced time}\end{aligned}$$ where $v$ is the advanced time coordinate. Now let us fix the two-dimensional diffeomorphisms by the gauge conditions $$\begin{aligned}
g_{ab} = \left(
\begin{array}{cc}
\gamma & \beta \\
\beta & \frac{\beta^2}{\gamma} - \alpha^2
\end{array}
\right)
= \left(
\begin{array}{cc}
- \left( 1 - \frac{2M}{r} \right) & \frac{2M}{r} \\
\frac{2M}{r} & 1 + \frac{2M}{r}
\end{array}
\right).
\label{Gauge}\end{aligned}$$ From the gauge conditions (\[Gauge\]), the two-dimensional line element takes the form of the Vaidya metric [@Vaidya] $$\begin{aligned}
ds^2 &=& g_{ab} dx^a dx^b, \nonumber\\
&=& - \left( 1 - \frac{2M}{r} \right) dv^2 + 2 dv dr.
\label{2D line element}\end{aligned}$$ For a dynamical black hole, we make use of the local definition of the horizon, namely, the apparent horizon, rather than the global one, the event horizon. Note that the apparent horizon is now defined as $$\begin{aligned}
x^1 = r = 2 M (x^0, x^1),
\label{Apparent horizon}\end{aligned}$$ where $M = M (x^0, x^1)$ plays the role of the mass function of a black hole.
Since we treat with a massless scalar field which moves along the null geodesics, it is natural to assume that the scalar field $\Phi$ depends on only the null coordinate $v$, by which the mass function $M$ also becomes the function of the advanced time coordinate $v$. Furthermore, the radial function $\phi$ is assumed to be a radial coordinate $r$. Thus, under the situation at hand, we assume $$\begin{aligned}
\Phi \approx \Phi(v), \quad M \approx M(v), \quad \phi \approx r.
\label{Assumption}\end{aligned}$$ Henceforth, we shall use the simbol $\approx$ to indicate the equalities holding under the assumptions (\[Assumption\]). It will turn out that these assumptions play a critical role in simplifying the diffeomorphism constraints and lead to a solvable model of a quantum black hole.
Here it is valuable to check whether the above assumptions (\[Assumption\]) to be compatible with the field equations as follows: The field equations obtained from the action (\[2D action\]) are given by $$\begin{aligned}
&{}& - \frac{2}{\phi} \nabla_a \nabla_b \phi + \frac{2}{\phi} g_{ab} \nabla_c \nabla^c \phi
+ \frac{1}{\phi^2} g_{ab} \partial_c \phi \partial^c \phi - \frac{1}{\phi^2} g_{ab} =
2 \left( \partial_a \Phi \partial_b \Phi - \frac{1}{2} g_{ab} \partial_c \Phi \partial^c \Phi \right),
\nonumber\\
&{}& \frac{1}{\sqrt{-g}} \partial_a \left( \sqrt{-g} g^{ab} \partial_b \phi \right)
- \frac{1}{2} R \phi = - \phi \partial_a \Phi \partial^a \Phi,
\nonumber\\
&{}& \partial_a \left( \sqrt{-g} \phi^2 g^{ab} \partial_b \Phi \right) = 0.
\label{Field equations}\end{aligned}$$ The Vaidya metric (\[2D line element\]) gives us the metric tensor $$\begin{aligned}
g_{ab} = \left(
\begin{array}{cc}
g_{vv} & g_{vr} \\
g_{rv} & g_{rr}
\end{array}
\right)
= \left(
\begin{array}{cc}
- \left( 1 - \frac{2M}{r} \right) & 1 \\
1 & 0
\end{array}
\right).
\label{Vaidya}\end{aligned}$$ To check the compatibility of the assumptions (\[Assumption\]) with the field equations (\[Field equations\]), let us make an ansatz that the variables have the form $$\begin{aligned}
M \approx M(v), \quad \phi \approx r,
\label{Ansatz}\end{aligned}$$ but the scalar field is still the function of both $v$ and $r$, i.e., $\Phi \approx \Phi(v, r)$. The first field equation in (\[Field equations\]) is satisfied if the following relations hold: $$\begin{aligned}
\partial_r \Phi \approx 0, \quad \partial_v \Phi \approx \frac{\sqrt{\partial_v M}}{r}.
\label{Solution 1}\end{aligned}$$ Then, we find the second equation in (\[Field equations\]) to be satisfied automatically. Finally, using Eq. (\[Solution 1\]), the third field equation in (\[Field equations\]) reduces to the nontrivial equation $$\begin{aligned}
2 r \partial_v \partial_r \Phi + 2 \partial_v \Phi + r \left( 1 - \frac{2 M}{r} \right)
\partial_r^2 \Phi \approx 0.
\label{Solution 2}\end{aligned}$$ This equation (\[Solution 2\]) as well as the latter relation in (\[Solution 1\]) require us to limit ourselves to working with the vicinity of the apparent horizon (\[Apparent horizon\]) [@Hosoya2]. In other words, for the consistency of the field equations, in addition to the assumptions (\[Assumption\]), one has to supplement one more assumption $$\begin{aligned}
r \approx 2 M(v).
\label{Assumption 2}\end{aligned}$$ To put differently, with the assumption (\[Assumption 2\]) we study physics of a black hole near the apparent horizon inside a black hole.
After all, given the assumptions (\[Assumption\]) and (\[Assumption 2\]), the field equations become $$\begin{aligned}
\partial_r \Phi &\approx& 0, \nonumber\\
\partial_v \Phi &\approx& \frac{\sqrt{\partial_v M}}{r} \approx \frac{\sqrt{\partial_v M}}{2 M}, \nonumber\\
\partial_r \partial_v \Phi &\approx& \partial_v \partial_r \Phi \approx - \frac{\sqrt{\partial_v M}}{r^2}
\approx - \frac{\sqrt{\partial_v M}}{4 M^2}.
\label{Solution 3}\end{aligned}$$ Then, the solution is found to be $$\begin{aligned}
\Phi (v, r) = \left( 1 - \frac{2M}{r} \right)^2 \frac{1}{4 \sqrt{\partial_v M}}
+ \int^v dv \frac{\sqrt{\partial_v M}}{2 M}.
\label{Solution 4}\end{aligned}$$ As a result, we have $$\begin{aligned}
\Phi(v, r) \approx \int^v dv \frac{\sqrt{\partial_v M}}{2 M},
\label{Solution 5}\end{aligned}$$ which means that one can set $\Phi(v, r) \approx \Phi(v)$ in the vicinity of the apparent horizon. In this sense, the assumptions (\[Assumption\]), if the assumption (\[Assumption 2\]) is added, are at least classically consistent with the field equations (\[Field equations\]).
Next we will turn our attention to quantum theory. In quantum gravity, following Dirac [@Dirac], one must impose the constraints (\[Constraints 2\]) on the wave functional $\Psi$ as operator equations to find the physical state. In general, it is very difficult to solve such constraint equations at the same time. In some specific situations, however, the constraints become tractable. For instance, in quantum cosmology, the fundamental equation is the Wheeler-De Witt equation, $H_0 \Psi = 0$, which is the operator equation associated with the Hamiltonian constraint, since the momentum constraint becomes trivial in cosmology owing to the translation invariance of the universe. Of course, solving the Wheeler-De Witt equation is still a tough work, so some people try to simplify the equation by considering minisuperspace.
In this article, our strategy for solving the operator equations $H_0 \Psi = 0, H_1 \Psi = 0$ is similar to minisuperspace approach in the sense that we set up the assumptions (\[Assumption\]) to simplify the constraints. However, we do not impose the assumption (\[Assumption 2\]) a priori since this assumption is not only unnecessary in order to simplify the constraint equations but also leads to a problem of the dynamical variable $\gamma$ being zero. Indeed, it is remarkable that the assumptions (\[Assumption\]) not only make the momentum constraint agree with the Hamiltonian one up to an overall factor but also reduce the unified operator equation, which is nothing but the Wheeler-De Witt equation, to be a solvable equation in an analytical manner.
With the two-dimensional coordinates (\[Advanced time\]), the derivative operators take the form $$\begin{aligned}
\left( \frac{\partial}{\partial x^0}, \frac{\partial}{\partial x^1} \right)
\equiv \left( \partial_0, \partial_1 \right) = \left( \partial_v, \partial_v + \partial_r \right).
\label{Derivative}\end{aligned}$$ With the help of Eqs. (\[Normal unit\]) and (\[Gauge\]), in case of a real scalar field and vanishing gauge field the assumptions (\[Assumption\]) reduce the canonical conjugate momenta (\[Momenta\]) to the form $$\begin{aligned}
p_\Phi &\approx& - \phi^2 \partial_v \Phi, \quad
p_\phi \approx - \frac{1}{\frac{2M}{\phi} - 1} \partial_v M + 1 - \frac{M}{\phi}, \nonumber\\
p_\gamma &\approx& - \frac{\phi}{2} \approx - \frac{r}{2}.
\label{Momenta 2}\end{aligned}$$ Then, the remarkable point is that the momentum constraint becomes identical with the Hamiltonian constraint up to an irrelevant overall factor $$\begin{aligned}
\sqrt{\gamma} H_0 &\approx& - \gamma H_1 \nonumber\\
&\approx& \frac{1}{\phi^2} p_\Phi^2 + \left( \frac{2M}{\phi} - 1 \right)
\left( p_\phi - 1 + \frac{M}{\phi} \right).
\label{Constraints 3}\end{aligned}$$ This compatibility between the momentum and the Hamiltonian constraints justifies the assumptions (\[Assumption\]) in quantum gravity.
The constraint (\[Constraints 3\]) as an operator equation on the wave functional $\Psi$ gives rise to the Wheeler-De Witt equation $$\begin{aligned}
\left[ - \frac{1}{\phi^2} \frac{\partial^2}{\partial \Phi^2} + \left( \frac{2M}{\phi} - 1 \right)
\left( - i \frac{\partial}{\partial \phi} - 1 + \frac{M}{\phi} \right) \right] \Psi = 0.
\label{WDW 1}\end{aligned}$$ It is worthwhile to rewrite this Wheeler-De Witt equation as follows: $$\begin{aligned}
i \frac{\partial \Psi}{\partial T} = \left[ p_\Phi^2 - \frac{2M^2}{e^{2MT} + 1} \tanh(MT) \right] \Psi,
\label{WDW 2}\end{aligned}$$ where we have defined $T = - \frac{1}{2M} \log (\frac{2M}{\phi} - 1)$. This Wheeler-De Witt equation can be interpreted as the Schrodinger equation with the Hamiltonian $H = p_\Phi^2 - \frac{2M^2}{e^{2MT} + 1} \tanh(MT)$ and the time $T$ in the superspace at hand. It is of interest to note that the superspace time $T$ “stops” on the apparent horizon owing to gravitational time dilation. On the other hand, the Hamiltonian has a problematic behavior in that it is not positive semi-definite.
Now it is easy to find a special solution of the Wheeler-De Witt equation (\[WDW 1\]) by the method of separation of variables. The result is given by $$\begin{aligned}
\Psi = \left( B e^{\sqrt{A} \Phi} + C e^{- \sqrt{A} \Phi} \right)
e^{ i \left[ \phi - M \log \phi - \frac{A}{2M} \log ( \frac{2M}{\phi} - 1 ) \right] },
\label{WDW-solution 1}\end{aligned}$$ where $A, B$ and $C$ are integration constants. Provided that the expectation value $< \cal{O} >$ of an operator $\cal{O}$ is defined as $$\begin{aligned}
< {\cal{O}} > = \frac{1}{\int d \Phi |\Psi|^2} \int d \Phi \Psi^* {\cal{O}} \Psi,
\label{Exp 1}\end{aligned}$$ one can calculate the expectation value of mass-loss rate $< \partial_v M >$ by using (\[Momenta 2\]) or (\[Constraints 3\]) $$\begin{aligned}
< \partial_v M > = - \frac{A}{\phi^2}.
\label{Mass-loss 1}\end{aligned}$$ At this stage, let us substitute the condition (\[Assumption 2\]) for the consistency of field equations. Then, we obtain $$\begin{aligned}
< \partial_v M > = - \frac{k^2}{4 M^2},
\label{Mass-loss 2}\end{aligned}$$ where we have set $A = k^2$. This result precisely coincides with that by Hawking in his semiclassical approach. Accordingly, our result shows that a black hole completely evaporates within a finite time. However, it is worth stressing the difference between the Hawking approach and the present one: In the Hawking semiclassical approach, the gravitational field is fixed as a classical background, and the matter field is treated only quantum-mechanically. By contrast, our formulation is purely quantum-mechanical even if we have imposed physically plausible assumptions on the scalar field, mass function and the radial field.
Two remarks are in order. First of all, recall that in our previous article [@Hosoya2] we have derived the same result for the expectation value of the mass-loss rate (\[Mass-loss 2\]) in the interior near the apparent horizon, but we have encountered a difficulty of the dynamical variable $\gamma$ becoming zero, by which various equalities become singular. Thus we further had to take a regularization such that $\gamma$ is not strictly zero but takes a small but finite value. It is worth mentioning that in the present formulation this artificial regularization is avoided by imposing the condition (\[Assumption 2\]) only at the final stage.
Second, one should comment on the boundary condition on the wave functional $\Psi$. Our physical state, which satisfies the Wheeler-De Witt equation, takes the form $$\begin{aligned}
\Psi = \left( B e^{|k| \Phi} + C e^{- |k| \Phi} \right)
e^{ i \left[ \phi - M \log \phi - \frac{k^2}{2M} \log ( \frac{2M}{\phi} - 1 ) \right] }.
\label{WDW-solution 2}\end{aligned}$$ This physical state does not satisfy the Dirichlet boundary condition $\Psi \rightarrow 0$ for $|\Phi|
\rightarrow \infty$, nor is its norm $\int d \Phi |\Psi|^2$ finite. Of course, these requirements might be too strict since we do not have any physical principle to pick up the appropriate boundary conditions, and the present formulation does not provide any information on the correct definition of the inner product. Near the spacetime singularity, because of the huge quantum effects, the matter field $\Phi$ would fluctuate so strongly that the Dirichlet boundary condition seems to be appropriate to suppress such a unwieldy behavior of the physical state.
Black hole radiation in the exterior region
===========================================
Now we will move on to an application of our idea for the Hawking radiation in the exterior region of a black hole in quantum gravity. The same problem has been already attacked by Tomimatsu [@Tomimatsu] where the assumption (\[Assumption 2\]) is fully utilized from scratch. In this section, we will use only the assumptions (\[Assumption\]) to find the physical state, and impose the condition (\[Assumption 2\]) at the final stage in interpreting the mass-loss rate of a dynamical black hole.
The argument proceeds in a perfectly analogous way to the case of the interior region of a black hole. In the exterior, the ADM splitting of (1+1)-dimensional spacetime is of form [@Hajicek] $$\begin{aligned}
g_{ab} = \left(
\begin{array}{cc}
\frac{\beta^2}{\gamma} - \alpha^2 & \beta \\
\beta & \gamma
\end{array}
\right).
\label{ADM 2}\end{aligned}$$ The normal unit vector $n^a$ to the Cauchy hypersurfaces $x^0 = const$ reads $$\begin{aligned}
n^a = \left( \frac{1}{\alpha}, \, - \frac{\beta}{\alpha \gamma} \right).
\label{Normal unit 2}\end{aligned}$$ The trace of the extrinsic curvature is calculated to be $$\begin{aligned}
K = \frac{\dot{\gamma}}{2 \alpha \gamma} - \frac{\beta'}{\alpha \gamma}
+ \frac{\beta}{2 \alpha \gamma^2} \gamma'.
\label{Extrinsic 3}\end{aligned}$$ In case of a real scalar field and vanishing gauge field, the canonical conjugate momenta $\pi_\Phi, \pi_\phi$, and $\pi_\gamma$ are now given by $$\begin{aligned}
\pi_\Phi &\approx& \phi^2 \partial_v \Phi, \quad
\pi_\phi \approx \frac{1}{1 + \frac{2M}{\phi}} \partial_v M + \frac{\frac{2M^2}{\phi^2}}{1 + \frac{2M}{\phi}},
\nonumber\\
\pi_\gamma &\approx& \frac{M}{1 + \frac{2M}{\phi}}.
\label{Momenta 3}\end{aligned}$$ The momentum constraint turns out to become proportional to the Hamiltonian constraint again $$\begin{aligned}
\sqrt{\gamma} H_0 &\approx& \gamma H_1 \nonumber\\
&\approx& \frac{1}{\phi^2} \pi_\Phi^2 - \left( 1 + \frac{2M}{\phi} \right) \pi_\phi
+ \frac{2M^2}{\phi^2}.
\label{Constraints 4}\end{aligned}$$ This compatibility between the momentum and the Hamiltonian constraints justifies the assumptions (\[Assumption\]) in quantum gravity as well.
An imposition of the constraint (\[Constraints 4\]) as an operator equation on the wave functional $\Psi$ produces the Wheeler-De Witt equation $$\begin{aligned}
\left[ - \frac{1}{\phi^2} \frac{\partial^2}{\partial \Phi^2} + i \left( 1 + \frac{2M}{\phi} \right)
\frac{\partial}{\partial \phi} + \frac{2M^2}{\phi^2} \right] \Psi = 0.
\label{WDW 3}\end{aligned}$$ Then, a special solution of the Wheeler-De Witt equation (\[WDW 3\]) is given by $$\begin{aligned}
\Psi = \left( B e^{\sqrt{A} \Phi} + C e^{- \sqrt{A} \Phi} \right)
e^{ i \frac{A - 2M^2}{2M} \log ( 1 + \frac{2M}{\phi} ) },
\label{WDW-solution 3}\end{aligned}$$ where $A, B$ and $C$ are integration constants. As before, the expectation value of mass-loss rate $< \partial_v M >$ is calculated to be $$\begin{aligned}
< \partial_v M > = - \frac{A}{\phi^2}.
\label{Mass-loss 3}\end{aligned}$$ After the assumption (\[Assumption 2\]) is inserted to this result, we obtain $$\begin{aligned}
< \partial_v M > = - \frac{k^2}{4 M^2},
\label{Mass-loss 4}\end{aligned}$$ where we have defined $A = k^2$ again. This result precisely coincides with that obtained in the interior region of a black hole.
Thus we have shown that the result of the mass-loss rate of a black hole owing to the Hawking radiation is equivalent between the interior and the exterior of a black hole. This fact suggests that the Hawking radiation comes from the spacetime singularity as expected. Of course, the physical state satisfying the Wheeler-De Witt equation is different between the interior and the exterior regions of a black hole, but the reason is connected with the fact that the Vaidya metric is not the maxmally extended, complete geometry and has a coordinate singularity at the horizon.
As a final comment, let us consider the Birkhoff theorem within the present framework. In classical general relativity, the Birkhoff theorem holds in the spherically symmetric geometry, thereby prohibiting the existence of the graviton. In this article, we have also considered the spherically symmetric geometry, so it is natural to ask ourselves if we could get some information on the Birkhoff theorem in quantum gravity. The reparametrization invariance in two dimensions allows the dynamical variable $\gamma$ and the radial function $\phi$ to remain as physical degrees of freedom except for the matter field $\Phi$. As seen in the arguments done thus far, the variable $\phi$ is removed via the assumptions (\[Assumption\]) so it does not play the role of a dynamical variable. This reduction of the dynamical degree of freedom could be understood as a result of the momentum constraint. In this context, it is important to notice that the condition (\[Assumption 2\]) is not needed to reduce the momentum constraint. On the other hand, via the gauge conditions (\[Gauge\]), the variable $\gamma$ is related to the mass function $M(v)$ whose change rate with respect to the advanced time $v$ is evaluated in this article. In this sense, the gravitational mode is left in the form of the mass function for a dynamical black hole in quantum gravity even if there is no explicit gravitational wave.
Conclusion
==========
In this article, we have made assumptions (\[Assumption\]), thereby the momentum constraint becoming identical with the Hamiltonian one up to an irrelevant overall factor. Moreover, these assumptions made it possible to solve the Wheeler-De Witt equation in an analytical way. As mentioned before, imposing these assumptions on the matter field, mass function and the radial field can be understood naturally from the physical viewpoint: The massless scalar field propagates along the null geodesics, and the mass function of a black hole receives influences from such a scalar field, so they are the functions of only the variable $v$, $\Phi = \Phi(v), M = M(v)$. The radial function plays the role of the radius of a black hole at the primitive level, so that it is natural to take $\phi = r$. It seems that these assumptions have more profound mathematical meaning rather than mere technical devices. It is known that the advanced time coordinate $v$, i.e., the tortoise coordinate, makes the $r-t$ plane look “conformal” so that conformal field theory can be applied [@Motl]. Then, our assumptions $\Phi = \Phi(v), M = M(v)$ can be interpreted as the holomorphic (or analytic, or chiral) condition, thereby making the complicated constraints associated with two-dimensional diffeomorphisms be tractable and solvable analytically.
One of motivations in this paper was to avoid an artificial regularization, which was adopted in our previous paper [@Hosoya2], such that $\gamma$ takes a small but finite value, and then derive the mass-loss rate of a black hole owing to the Hawking radiation in a more reasonable way. Indeed, without this regularization we have succeeded in deriving the same result as that in [@Hosoya2]. In retrospect, the analysis in this article seems to give the regularization a sound foundation.
There are a lot of works to be done in future. Firstly, it is interesting to generalize the present formulation to the Reissner-Nordstrom black hole where we have to pay attention to the constraint associated with the $U(1)$ gauge transformation. Secondly, it might be possible to relax the assumption $\phi = r$ since this assumption is somewhat ad hoc in that $\phi$ has a possibility of having a more general function of the $r$ coordinate. Thirdly, in a recent progress on large interior of a black hole [@Chris], the most dominant contribution comes from $r = \frac{3}{2} M$ hypersurface, which is very close to the horizon $r = 2M$. Thus, it would be interesting to construct a model of quantum black holes holding near $r = \frac{3}{2} M$ rather than the horizon for understanding this issue in a quantum-mechanical way. Finally, it would be valuable to formulate the present formulation in the Kruskal-Szekeres coordinate system since we can consider both the interior and the exterior regions of a black hole at the same time, and might have a smoothly interpolating physical state in the both regions. [^4] We wish to return these problems in future.
[**Acknowledgements**]{}
This work is supported in part by the Grant-in-Aid for Scientific Research (C) No. 25400262 from the Japan Ministry of Education, Culture, Sports, Science and Technology.
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[^1]: E-mail address: [email protected]
[^2]: See Ref. [@Kawai] for a phenomenological description of a spherically symmetric black hole.
[^3]: See also related works [@Oda1; @Hosoya3].
[^4]: The Kruskal-Szekeres coordinate system holds only for eternal black holes, so some extension of which might be necessary in taking account of a dynamical black hole.
|
---
abstract: 'Bolometers are cryogenic calorimeters which feature excellent energy resolution, low energy threshold, high detection efficiency, flexibility in choice of materials, particle identification capability if operated as hybrid devices. After thirty years of rapid progresses, they represent nowadays a leading technology in several fields: particle and nuclear physics, X-ray astrophysics, cosmology. However, further and substantial developments are required to increase the sensitivity to the levels envisioned by future researches. A review of the challenges to be addressed and potentialities of bolometers in the search for rare nuclear decays is given, with particular emphasis to the neutrinoless double beta decay physics case.'
address:
- 'Sapienza Università di Roma, P.le Aldo Moro 2 Roma, 00185,Italy'
- |
INFN, Sezione di Roma, P.le Aldo Moro 2, 00185, Rome, Italy\
[email protected]
author:
- Fabio Bellini
bibliography:
- 'bellini.bib'
title: Potentialities of the future technical improvements in the search of rare nuclear decays by bolometers
---
Introduction {#sec:intro}
============
The need for improved energy resolution and sensitivity to smaller energy depositions led to the proposal to use cryogenic techniques as an instrument for detecting low energy particles.\
In 1984 three different groups suggested low temperature detectors as an instrument to investigate fundamental problems in nuclear and astroparticle physics. Fiorini and Niinikowsky [@Fiorini:1983yj] proposed cryogenic calorimeters for the study of the double beta decay and of the neutrino mass measurement. Drukier and Stodoloky [@drukier1] studied the use of superconducting detectors for the search for the coherent neutrino scattering off nuclei. Finally McCammon and coworkers [@McCammon] indicated X-ray astrophysics as a possible field where cryogenic devices might have an important impact.\
In the last 30 years, Low Temperature Detectors (LTD) experienced a very rapid growth and reached a level of maturity and versatility such to represent one of the leading technology in different research fields nowadays. Examples are: dark matter [@pirroreview], neutrinoless double beta decays [@Poda:2017jnl], neutrino mass measurement [@Nucciotti:2015rsl], rare nuclear decays and processes [@demarcillac], X-ray astrophysics [@Ullom], cosmological microwave background precision measurements [@pirroreview].\
Nevertheless to face the challenges imposed by the aforementioned researches and increase the sensitivity of the current experiments, further and substantial technological developments are necessary.\
This review describes the potentialities of the future technical improvements in the search for rare nuclear decays by bolometers. Although the term bolometer was originally used for a detector measuring the intensity of electromagnetic radiation by heating up the detector itself, through the text it will be used as synonymous of cryogenic calorimeter.
This paper is organized as follows: in section \[sec:herc\] the basic principles of cryogenic detectors are summarized together with the main advantages and drawbacks compared to traditional devices. In section \[DBD\] the double beta decay physics case is analyzed: actual limits and future developments are discussed. In particular section \[BAPI\] describes the effort for the implementation of active background identification techniques while needs for reduction of the environmental radioactivity is discussed in section \[err\]. Finally section \[rare\] highlights the use of bolometers for other rare [$\alpha$]{} and [$\beta$]{} nuclear decays as well as electron capture processes.
High Energy Resolution Calorimeters {#sec:herc}
===================================
Conventional Calorimeters {#sec:conv}
-------------------------
A calorimeter is a device which is sensitive to the energy deposited by a single particle.\
All conventional calorimeters share the same principle: a ionizing particle interacting with a solid medium, deposits part of its energy into the medium itself. The released energy $E$ produces out-of-equilibrium excitation quanta: electron-hole pairs or photons. The quanta are collected as completely as possible before they decay into an undetectable channels. To get good energy resolution, the detector response must be uniform throughout the detection volume so that the fraction of energy released into the desired channel and the collection efficiency are the same for all events. The number of excited quanta is proportional to $E$ and inversely proportional to the mean energy necessary to produce each of them $w$; poissonian statistical fluctuations in the number of created quanta represent the ultimate limit of the technology.\
When energy resolution matters, the choice in conventional detectors is limited to germanium or silicon devices. For example, in silicon, $w$=3.6 eV and the best energy resolution obtained in semiconductor X-rays detectors is about 125 eV Full With Half Maximum (FWHM) at 6 keV [@quaglia]. The excitation energy $w$ is about three times the band gap $E_g$. The presence of several modes with an excitation energy less than $E_g$ and momentum conservation, that requires lattice vibration excitations (phonons), implies that 70% of the energy goes into undetectable channel. The statistical contribution to the energy resolution $\Delta E_{rms}$ is: $$\Delta E_{rms} = \sqrt{wFE},
\label{risoSi}$$ where F is the Fano factor [@Fano; @Klein], which reduces the overall random spread when multiples excitation mechanisms play a role. On the other hand, the maximum phonon energy in Si is only 60 meV; much more phonons than electron-hole pairs are produced. The possibility to detect such phonons overcomes the limit imposed on the energy resolution by poissonian fluctuations and allows much smaller energy depositions to be detected.\
This is the rationale behind the cryogenic calorimeter technique.\
Bolometers {#sec:basic}
----------
A bolometer is solid state device composed by an absorber, connected through a thermal link to a heat sink, and equipped with a temperature sensor (thermometer) for the conversion of phonons into an electrical signal. Different thermometers are commonly used depending on the specific application: high resistivity doped semiconductors (Neutron Transmutation Doped Thermistors [@Hallerf; @Haller]), paramagnetic sensors (Metallic Magnetic Calorimeters [@porst]), superconducting materials sensors (Kinetic Inductance Detector [@day], Transition Edge Sensors [@Ullom], Superconducting Tunnel Junctions [@Enns], Superheated Superconducting Granules [@Enns]). Several reviews on bolometers have been published [@Enns; @bolo-review1; @bolo-review2; @Twerenbold; @McCammon200411; @Ullom] which include a very general description of different LTDs making using of different sensor approaches. Here we recall only the basic principles.
Bolometers can be operated as equilibrium or non-equilibrium devices. In the former all the released energy $E$ degrades into heat while in the latter out-of-equilibrium (ballistic) phonons are collected.
In the most simple model, $E$ is fully thermalized and the temperature variation is $\Delta T=E/C(T)$ where $C(T)$ is the detector heat capacity at a working temperature $T$. In order to have the highest temperature variation, $C$ must be as low as possibile. This leads to the necessity to operate the detector at temperatures well below one Kelvin and to select materials in order to avoid contributions that increase the heat capacity. Several material-related characteristics contribute to the specific heat: the lattice contribution that is proportional to $({T/T_D})^3$ where $T_D$ is the Debye temperature of the material; the electron contribution that depends on ${T/T}_F$ where $T_F$ indicates the Fermi temperature; the paramagnetic component which is proportional to $1/T$. It is clear that the paramagnetic contribution is very dangerous, but also the use of conductors could be limited by the specific heat of electrons. For a superconductor the electron heat decreases as $\exp(-2T_c/T)$, where $T_c$ is the superconducting critical temperature. For specific applications, superconductors could represent absorbers of interest in which the excited quanta are the quasi-particles induced by the breaking of Cooper pairs.
Typical phonons excitations are limited by the Debye cut frequency and lie in the range of tens of meV for most materials. The ultimate energy resolution could be therefore very high and limited only by thermodynamic fluctuations due to the random exchange of phonons with the thermal bath. It has been shown in Ref. that $\Delta E_{rms}$ is: $$\Delta E_{rms} = \sqrt{\xi C_0K_BT_0^2},
\label{risoBo}$$ where $K_B$ is the Boltzmann constant, $T_0$ is the heat sink temperature and $\xi$ is a parameter that depends on the thermometer characteristic which is one in the ideal case but can reach values up to ten. Using Eq. \[risoBo\] a resolution of the order few eV can be reached. In reality several contributions can deteriorate the resolution: Johnson noise of the sensor and polarization network, phonon noise due to the temperature gradient, electronic noise of the amplifier, microphonic noise, metastable electron-holes state or long lived non-thermal phonons. Anyhow, using a suitable thermometer and with an appropriate electronic readout, energy resolutions of few eV are reachable.\
It’s remarkable that $\Delta E_{rms}$ does not depend on $E$, on the thermal conductance G of the thermal link and of the detector time constant $\tau$=G/C. The feature remains valid even including signal and noise power spectra and more refined analyses if the assumption of fully thermalization is achieved [@bolo-review2]. This opens the window to operate very massive detectors with different absorber materials provided they are kept at sufficiently low temperature.\
On the other hand, cryogenic particle detectors sensitive to ballistic phonons, are faster relative to equilibrium devices, since thermal equilibrium often takes a very long time to establish at low temperatures, and with no restrictions on the equilibration time, they offer even more flexibility in choice of materials. They are subject to branching statistics as well as ionization detectors but the number of excited quanta is much larger. They may suffer from position dependence and/or the lifetime and detection efficiency of excitations but for applications that require large volumes of dielectric material and do not need exceptionally good energy resolution, the speed advantage may outweigh other considerations.
### Strenghts and Weaknesses {#sec:limitation}
To summarize, the main advantages of bolometers compared to the well established semiconductor ionization technology are:
better energy resolutions;
enhanced sensitivity to low energy release;
wide flexibility of materials usable for the absorber. This characteristic is of primary importance when a particular isotope is needed as the absorber or the source of particles.
Despite these advantages, bolometers show limitations and practical challenges.
They need very a complicated apparatus necessary to maintain very low temperatures. Despite the improvements in cryogenic techniques, the size of the experimental volume is limited to a cubic meter. Moreover the cryostat must be very stable since the signal is represented by a very tiny temperature rise (hundreds of $\mathrm{\mu}$K for one MeV of released energy $E$) compared to the thermic bath and temperature fluctuations may limit the detector response. Furthermore unwanted noise, introduced by the cryogenic apparatus itself, could jeopardize the excellent energy resolution and affect the energy threshold.
The slowness of the bolometric response, that can extends up to several seconds for equilibrium devices, represents a concern in the search for rare processes. Pile-up events, induced by cosmic rays and by natural radioactivity in the bolometer itself and surrounding material, requires operation in deep underground sites, protection against external radioactivity and a careful selection of radio-pure materials for the detector itself.
In rare nuclear processes a small signal must be resolved over a large background. Full thermalized bolometers are almost equally sensitive to any kind of particle, despite the way energy is released. Electrons, [$\alpha$]{} particles, and nuclear recoils, depositing the same amount of energy in the detector, produce a pulse with the same amplitude and shape.\
In addiction the response is unaffected by the impact point of the event. If this feature allows for an excellent energy resolution, it makes however impossible to distinguish bulk from near surface particle interactions. In other words bolometers don’t have a dead layer at the surface, they are fully sensitive in their volume.\
The lack of particle identification and the impossibility to tag surface events makes the external background reduction a paramount concern. This represents the most serious limitation that searches, carried out with bolometric techniques, are facing.
### Hybryd bolometers {#sec:hybryd}
To overcome the aforementioned limits, hybrid bolometers were developed, in which a double readout is exploited. The temperature increase is measured in parallel with ionization, scintillation or Cherenkov light detection. This possibility permits the discrimination between events that release energy with different efficiencies in different detectable channels. Typical examples are neutrons which interact mainly by nuclear recoils, with only a very small fraction of energy that goes into the ionization channels or [$\alpha$]{} particles that are quenched in the scintillation channel. This idea was initially developed for dark matter searches and lead to realization of very sensitive detectors for both heat-ionization and heat-scintillation devices [@cdms; @Agnese:2014aze; @Agnese:2013ixa; @Armengaud:2016cvl; @Angloher:2015ewa; @Angloher:2014myn; @Angloher:2016ooq; @Angloher:2016hbv].
Recently hybrid bolometers come to play an important role in the search for Majorana neutrinos through the neutrinoless double beta decay. The discovery potential of future experiment is closely related to successful implementation of this technology.
Double Beta Decay {#DBD}
=================
The [${0\nu\beta \beta}$]{} physics case
----------------------------------------
The neutrinoless double beta decay [${0\nu\beta \beta}$]{} [@Furry] is a transition, in which a nucleus (A,Z) decays into its isobar (A,Z+2) with the simultaneous emission of two electrons. Both the parent and the daughter nucleus must be more bound than the intermediate one (A,Z+1) in order to avoid the occurrence of the sequence of two single beta decays. Such a condition, due to the pairing term, is fulfilled in nature for 35 even-even nuclei [@giuntibook].\
This process violates the lepton number by two units; it’s not allowed by the Standard Model of interactions but it’s envisaged in many of its extensions in which neutrinos are their own antiparticles [@giuntibook]. Its discovery would ascertain unambiguously the nature of neutrinos as Majorana fermions [@PhysRevD.25.2951], would constrain the absolute neutrino mass scale and provide support to leptogenesis theories [@Luty:1992un]. In the standard paradigma [@giuntibook; @pdg] the decay is mediated only by the exchange of three virtual light neutrinos between two charged weak interaction vertices. The chirality mismatch imposed by the V-A structure of the ElectroWeak theory leads to an amplitude proportional to a linear combination of the three neutrino masses. The absolute value of the neutrino masses is unknown yet but their sum is constrained to be less than 0.66 eV at 95% C.L. from cosmological observations [@pdg; @Cremonesi:2013vla]. On the other hand, the squared mass differences are well measured from neutrino oscillation experiments.\
Three possible orderings are therefore conceivable: normal hierarchy (NH), in which m$_{\nu1}$ <m$_{\nu2}$ <m$_{\nu3}$, inverted hierarchy (IH) where m$_{\nu3}$ <m$_{\nu1}$ <m$_{\nu2}$, and the quasi-degenerate hierarchy (QD), for which mass differences are tiny compared to their absolute values.\
Being a second-order weak interaction process and due the smallness of neutrinos masses, extraordinary long lifetimes ($\tau$>10$^{25}$yr) are expected for the [${0\nu\beta \beta}$]{} decay.\
Despite decades of experimental search it has not been observed so far. Actual limits on the mean lifetimes $\tau$ are in the range of $10^{24-26}$ yr [@Cremonesi:2013vla; @DellOro:2016tmg]; running experiments are deeply probing the QD parameter space and have the possibility to start to scan the IH region [@Cremonesi:2013vla].\
The goal of the next generation experiments is to completely cover the IH mass scheme and to have a high chance to assess the neutrino nature in their first operational stages [@Agostini:2017jim].\
The main signature of the [${0\nu\beta \beta}$]{} decay is a peak in the sum energy spectrum of the electrons at the transition energy of the reaction (commonly referred as [${Q_{\beta\beta}}$]{}). A typical [${Q_{\beta\beta}}$]{} for nuclei of experimental interest lies in the two-three MeV energy range. The signal peak must be resolved on top of a continuum background induced by natural and anthropogenic radioactive decay chains and cosmogenic induced activity. Consequently, the main task in [${0\nu\beta \beta}$]{} searches is to decrease the background in the Region Of Interest (ROI).\
The requirements to achieve the IH coverage ($\tau \sim 10^{27-28}$ yr) descends consequently: about a few thousand moles of the isotope under study (several hundreds of kgs) must be measured in combination with a background close to zero at the ton $\times$ year exposure scale and a FWHM energy resolution better than $0.5$% [@Cremonesi:2013vla; @Artusa:2014wnl].\
Bolometers are natural candidate detector for this purpose. They show excellent energy resolution, high detection efficiency thanks to the source-equal-detector approach, scalability to the ton scale and can be made of different materials allowing the search in several candidate nuclei.\
The state of the art in the bolometric search for the [${0\nu\beta \beta}$]{} is represented by the CUORE (Cryogenic Underground Observatory for Rare Events) experiment [@Artusa:2014lgv]. CUORE recently demonstrated [@Alduino:2017ehq] that a thousand TeO$_2$ bolometers can be successfully operated at a temperature of about ten mK studying the decay of the $^{130}$Te ($Q_{\beta\beta}$ $\sim$ 2527 keV [@Redshaw:2009zz; @scielzo09; @Rahaman:2011zz]). Despite the improvement compared to its predecessor CUORICINO [@Andreotti:2010vj] and CUORE-0 [@Aguirre:2014lua; @Alfonso:2015wka; @Alduino:2016vjd], the CUORE background is currently of the order of $10^{-2}$ [counts keV$^{-1}$kg$^{-1}$ y$^{-1}$]{}[@Alduino:2017ehq] as expected from simulations [@Alduino:2017qet].\
Its sensitivity is mainly limited by energy-degraded $\alpha$’s, emitted by surface contaminations on the crystals and on the copper supporting structure [@Bucci:2009fk; @Alduino:2017qet]. High energy (four - six MeV) [$\alpha$]{} particles, in fact, lose only part of their energy in the crystal or surrounding materials and give rise to a continuum of events in the ROI. The amount of surface contamination is less than a few 10$^{-8}$ counts h$^{-1}$ keV$^{-1}$ cm$^{-2}$ and can not be measured or screened with any standard technique.\
As stated in section \[sec:limitation\] the impossibility to disentangle particles on the crystal surface or external to it and the lack of particle identification represents the actual limits of bolometers in the search of [${0\nu\beta \beta}$]{} decay.\
Given the enormous effort already devoted to surface treatment, it is unlikely that the required reduction in the background level can be achieved by improving the radio-purity of the detector materials alone. A major role to overcome this limit consists in the development of new technologies for active background suppression.\
This is the goal of the CUPID project (CUORE Upgrade with Particle ID) [@Wang:2015raa; @Wang:2015taa] that aims at enhancing the sensitivity for a bolometric experiment by two orders of magnitude increasing the source mass and reducing the backgrounds using isotopically enriched bolometers with particle identification. The pursued approaches are based on different physical principles and different techniques and will be detailed in section \[BAPI\]. The reduction of the background induced by sources other than surface ones are reported in section \[err\].
Other second order weak processes
---------------------------------
In the [${0\nu\beta \beta}$]{} case discussed so far, the decay was assumed to proceed to the ground state of the final nucleus. Given the short range of MeV electrons in a solid medium (few mm) the majority of the candidate events consists in a monochromatic energy release contained in a single crystal. The decay can also proceed to the excited states of the final nucleus. Although the longer predicted half life for these cases, they could be of experimental interest because the de-excitation [$\gamma$]{} of the final nucleus give rise to a multi-crystals signature and therefore to a strongly reduced background.\
However the exchange of Majorana neutrinos between the two weak vertices can occur in transitions with a nuclear charge change of $\Delta$Z=-2 through $0\nu \beta^+ \beta^+$, $0\nu$EC$\beta^+$, $0\nu$ECEC decays modes, where EC is the electron capture acronym. In the last mode no leptons are available to carry away the released energy and the decay can happen through radiative [@Sujkowski] or resonant decay [@Bernabeu]. The ECEC mode is therefore typically suppressed but enhancements (up to a factor 10$^6$) may happen when there is degeneracy of the initial and final excited atomic states (resonance condition). The $\Delta$Z=-2 processes are interesting because they can provide an insight into the [${0\nu\beta \beta}$]{} mechanism since they are dominated by right-handed weak currents.\
The most sensitive probe is represented by the [${0\nu\beta \beta}$]{} process to the ground state and in the rest of the review only this decay is considered. Anyhow the potentialities of the future experiments are related to the background reduction capability and this aspect leads to significative improvements for all modes, irrespective of their signature.
The [${0\nu\beta \beta}$]{} process is always accompanied by the [${2\nu\beta \beta}$]{} decay which is allowed by the Standard Model since two neutrinos are emitted in the finale state. This second order weak process has been observed for 11 nuclei [@Barabash:2015eza] and represent the rarest nuclear decays ever measured. The experimental signature is weaker than the [${0\nu\beta \beta}$]{} mode and consists in a broad spectrum from zero up the the [${Q_{\beta\beta}}$]{} value (neutrinos emitted at rest) and, as for the [${0\nu\beta \beta}$]{} case, most of events are contained in a single crystals. The extraction of the signal is therefore challenging because it must be disentangled from several background sources and particle identification is not going to play a fundamental role since backgrounds are mainly [$\beta$]{}/[$\gamma$]{} i.e. same particle type of the signal. On the other hand the limited size of actual cryogenic apparatuses imposes the use of bolometers enriched in the isotope under investigation, therefore the [${2\nu\beta \beta}$]{} signal to background ratio will dramatically increase thus allowing better measurements of the [${2\nu\beta \beta}$]{} shape and intensity. Finally it must be stressed the [${2\nu\beta \beta}$]{} represent the irreducible background for the [${0\nu\beta \beta}$]{} search as discussed in section \[err\].
Bolometers with Active Background Suppression {#BAPI}
=============================================
Scintillating bolometers {#sb}
------------------------
The use of heat-scintillation hybrid bolometers for the [${0\nu\beta \beta}$]{} search was proposed in 2005 [@Pirro:2005ar].\
In a scintillating bolometer [@nsvecchioarticoloCaF2; @Pirro:2005ar; @Giuliani2012] a small fraction (between one per cent and one per mill) of the released energy $E$ is converted into scintillation light. This eventually escapes the crystal and is absorbed by a thin bolometer working as light detector.\
The ratio between the two signals (scintillation/heat) depends on the particle type; [$\beta$]{}/[$\gamma$]{} particles have a light yield (LY) which is typically different from the LY of [$\alpha$]{} interactions or neutrons that are quenched. Consequently, the dual readout allows particle identification.
In addition, the freedom in the choice of the absorber provides the unique opportunity of selecting the [${0\nu\beta \beta}$]{} isotopes with high [${Q_{\beta\beta}}$]{}[@Arnaboldi:2010tt; @Gironi:2009ay; @Arnaboldi:2010jx].\
This is a very important aspect. The most prominent natural high-energy [$\gamma$]{}s, induced by the [$^{208}\mathrm{Tl}$]{} decay, are distributed up to 2615 keV while only rare [$\gamma$]{} decays coming from the $^{214}$Bi decay populate the above region and up to 3270 keV. Selecting a [${0\nu\beta \beta}$]{} candidate with a [${Q_{\beta\beta}}$]{} larger then 2615 keV will leads to a [$\gamma$]{} background reduction in the ROI by about one order of magnitude as can be inferred from the [$\gamma$]{} spectrum measured at the Laboratori Nazionali del Gran Sasso [@Bucci:2009fk].\
With a proper absorber choice, scintillating bolometers can simultaneously get rid of both [$\alpha$]{} induced background and the most intense natural [$\gamma$]{} radioactivity.\
In the last ten years several scintillating bolometers were operated with remarkable results using as [${0\nu\beta \beta}$]{} emitters: $^{82}$Se([${Q_{\beta\beta}}$]{}=2998 keV[@wang2016]), $^{100}$Mo([${Q_{\beta\beta}}$]{}=3034 keV[@wang2016]), $^{116}$Cd ([${Q_{\beta\beta}}$]{}=2813 keV[@wang2016]). The effort was devoted not only to the development of the light detector technology but also to the production of very pure enriched crystals (see section \[err\]).
Three small scale pilot experiments using scintillating and isotopically enriched crystals are (or close to) being operated as final demonstrator in view of a next-generation experiment. CUPID-0 [@Artusa:2016maw], formerly LUCIFER [@Beeman:2013sba], is currently running using an array of 24 enriched Zn$^{82}$Se crystals; LUMINEU/CUPID-0-Mo [@Armengaud:2016dqg; @Armengaud:2017hit] will start data taking in January 2018 with an array of 20 Li$_2$$^{100}$MoO$_4$ crystals; AMORE [@Kim:2015pua] is operating 5 $^{48-dep}$Ca$^{100}$MoO$_4$ bolometers but it foresees the use of other molybdates such as Zn$^{100}$MoO$_4$ or Li$_2$$^{100}$MoO$_4$ for the final experiment. The first two mentioned experiments are part of the CUPID R&D. No demonstrators for $^{116}$CdWO$_4$ bolometers are on going or planned despite the good results obtained [@Danevich:2016xcm]; hence only results for $^{100}$Mo and $^{82}$Se are discussed.
One of the greatest advantages of scintillating bolometers is that the amount of collected light is high enough to be recorded using a germanium slab operated as bolometer and equipped with a standard Neutron Transmutation Doped (NTD) germanium thermistor. NTDs are heavily doped semiconductors, obtained by thermal neutron irradiation [@Hallerf; @Haller] and with impurity concentrations slightly below the metal-insulator transition. In this variable-range-hopping regime, their resistivity depends exponentially on the temperature. They are high impedance devices (1-100 $M\Omega$), read out in constant current biasing mode and matched to room temperature JFET amplifiers. They are commonly used for the heat channel readout. The possibility to use the same sensor with minimal modification also for the light channel represents a clear advantage in terms of reliability, robustness and impact on the cryogenic infrastructure in view of a reuse of the existing CUORE infrastructure for a future experiment. Those light detectors have been extensively characterized [@Beeman:2013zva] and the technology could be considered already mature. Light detectors show a baseline RMS noise of hundreds of eV and allow particle identification even in the case of the worst scintillators (LY$\sim$1 keV/MeV) [@Artusa:2016maw; @Armengaud:2017hit; @Beeman:2013zva].
The ratio of the light signal associated to an [$\alpha$]{} interaction (QF) to [$\beta$]{}/[$\gamma$]{} on for events with the same heat energy release is defined Quenching Factor (QF). It is of the order of 0.2 for most of the studied compounds with the only exception of the ZnSe which exhibits a QF of 3-6 [@Arnaboldi:2010jx; @Beeman:2013vda; @Beeman:2013sba; @Artusa:2016maw].
The discrimination capability of a scintillating bolometer is usually parametrized by a quantity, the Discrimination Power (DP), defined as $$\mathrm{DP} = \left|\mu_{\beta /\gamma}-\mu_{\alpha}\right|/\sqrt{\sigma_{\beta/\gamma}^2+\sigma_\alpha^2},
\label{dp}$$
where $\mu$ and $\sigma$ denote the average value and the standard deviation of the $\alpha$ or [$\beta$]{}/[$\gamma$]{} distributions respectively and quantities are computed at [${Q_{\beta\beta}}$]{} since they might depend on the energy. The DP can be the light/heat ratio but can also be applied to any pulse shape variable.
ZnSe and some molybdates, in fact, show a peculiar feature: the thermal pulse induced by an [$\alpha$]{} particle has a slightly faster decay time than that induced by [$\beta$]{}/[$\gamma$]{} interactions [@Gironi:2009ay; @Beeman:2012jd; @Armengaud:2017hit; @Artusa:2016maw; @Gironi:2010hs; @Beeman:2013vda; @Arnaboldi:2010jx; @Arnaboldi:2010gj; @Beeman:2012gg; @ZnMo4poda; @Casali:2017zvs; @Kim:2017xrs; @Kim:2015pua].
![Shape parameter of a light detector as a function of the energy released in a Zn$^{82}$Se bolometer. The energy scale is calibrated on [$\beta$]{}/[$\gamma$]{} and thus referred as keVee. The lines indicate the 2$\sigma$ (continuous) and 3$\sigma$ (dashed) [$\beta$]{}/[$\gamma$]{} and [$\alpha$]{} bands. [$\alpha$]{} events produced by a smeared Sm source (below 3MeVee) and by contaminations of the crystal bulk (peaks above 5MeVee) can be easily rejected, in particular in the region of interest for the $^{82}$Se [${0\nu\beta \beta}$]{}(dashed vertical green line). Inset: time development of light pulses produced by [$\beta$]{}/[$\gamma$]{} and a [$\alpha$]{} interactions with energy of about 2.6 MeV. A DP of 12 is obtained at the $^{82}$Se [${Q_{\beta\beta}}$]{}. Figure adapted from Ref. with kind permission of the European Physical Journal (EPJ).[]{data-label="znse"}](bw_ZnSe){width="8.0cm"}
![Light Yield-vs-heat scatter-plot obtained with AmBe neutron source with a 151 g Li$_2$MoO$_4$ scintillating bolometer. A clear separation between [$\beta$]{}/[$\gamma$]{} and [$\alpha$]{} interactions is visible. The [$\beta$]{}/[$\gamma$]{} band exceeds the natural $^{208}$Tl end-point because of the prompt de-excitation [$\gamma$]{}s following $^9$Be([$\alpha$]{},n)$^{12}$C$^{\star}$ reaction. The cluster of events in the [$\alpha$]{} region is caused by the reaction $^6$Li(n,t)[$\alpha$]{}. Figure adapted from Ref. with kind permission of the European Physical Journal (EPJ).[]{data-label="limo"}](bw_Limo){width="8.0cm"}
This is a tiny (few percent) effect that allows to discern the nature of the interacting particle without light detection thus greatly simplifying the detector assembly and readout. This effect could be ascribed to the long scintillation decay time (of the order of hundreds microseconds) and the high percentage of non-radiative de-excitations of the scintillation channels, that produce delayed phonons [@GironiPSD]. However a very good signal-to-noise ratio for all the channels is required because of the smallness of the effect and the discrimination power was not always reproducible in different experimental measurements. Further developments are necessary before considering a reliable technology.\
In the case of ZnSe bolometers, a pulse shape difference, more discriminating than the one on the heat bolometer, is seen on the light channel [@Beeman:2013vda]. This is currently used by the CUPID-0 collaboration [@Artusa:2016maw] to avoid the leakage of the [$\alpha$]{} band of the LY into the [$\beta$]{}/[$\gamma$]{} band as observed in the light-vs-heat scatter plot [@Beeman:2013vda]. Examples of distributions of discriminating variables are reported in Fig. \[znse\] and Fig. \[limo\].\
The requirements for a bolometric experiment to assess the IH hierarchy mass region imply a background level in the ROI of $10^{-4}$ [counts kg$^{-1}$ y$^{-1}$]{}[@Beeman:2011bg]. This requires a rejection factor on [$\alpha$]{}s better than 99.9% while preserving a signal efficiency greater then 90% [@Beeman:2011bg]. A DP of 3.1 or greater is necessary to satisfy this criterion. Table \[tab:PID\] reports the the [$\beta$]{}/[$\gamma$]{} LY, QF and DP for all the compounds for which a pilot experiment is running or in construction phase; all of them exhibit a DP exceeding 9, much above the requested threshold.
Cherenkov light in TeO$_2$ bolometers {#nonsb}
-------------------------------------
The TeO$_2$ bolometers used by the CUORE collaboration do not show any significant scintillation. A tiny light signal was observed in 2004 [@coron] and seems to find confirmation recently [@Berge:2017nys]. In any case the light yield is low ($\sim$20 eV) and negligible compared to another and more important process that takes place: the emission of light through the Cherenkov effect.
![Detected light versus calibrated heat in a CUORE-like TeO$_2$ bolometer read out with a CUPID-0 NTD Ge thermometer. The mean light is clearly energy dependent for the [$\beta$]{}/[$\gamma$]{} peaks (circles below 3 MeV) and compatible with zero for the [$\alpha$]{} decay of the $^{210}$Po (triangle). TeO$_2$ does not scintillate, however Cherenkov light is produced by [$\beta$]{}/[$\gamma$]{} interactions (circles) and not by [$\alpha$]{} ones (triangle). Figure adapted from Ref. with kind permission of the European Physical Journal (EPJ). []{data-label="Cherenkov"}](Fig4d){width="9cm"}
In 2010 the use of the Cherenkov light in TeO$_2$ bolometers was suggested as a tool for particle identification [@TabarellideFatis:2009zz]. The threshold for Cherenkov emission in TeO$_2$ is around 50 keV for electrons and around 400 MeV for [$\alpha$]{}s. At the energy scale of interest for [${0\nu\beta \beta}$]{}, the signal electrons emit light while [$\alpha$]{} particles do not. Several tests were done on small [@Bellini:2012rc; @Willers:2014eoa; @Gironi:2016nae] and large crystals [@Beeman:2011yc; @Schaffner:2014caa; @Casali:2014vvt; @Casali:2015gya; @Artusa:2016mat; @Berge:2017nys] to characterize the discrimination power.\
The challenge of this method is the detection of the extremely small amount of light emitted by electrons at the $^{130}$Te [${0\nu\beta \beta}$]{} energy ([${Q_{\beta\beta}}$]{}$\sim$2.5 MeV) that is of the order of 100 eV [@Casali:2013bva; @Casali:2014vvt; @Casali:2016luq], i.e. comparable to the noise resolution of NTD-based standard light detectors used in scintillating bolometers (see Fig. \[Cherenkov\]). A signal-to-noise ratio greater than 5 is needed to reach [$\alpha$]{}/([$\beta$]{}/[$\gamma$]{}) separation allowing for a 99.9% rejection of the [$\alpha$]{} background [@Casali:2014vvt]. Attempts to increase the light collection [@Casali:2014vvt] do not lead to significative results, this implies that a light detector technology with a noise level below 20 eV RMS is mandatory. Furthermore the light detectors must be robust and reproducible in view of a ton-scale experiment with about 1000 bolometers, made of radio-pure materials and possibly have a multiplexed readout to avoid a large heat-load on the cryogenic apparatus. Finally the light detector must have an active area comparable to the top bolometer face (about 20 cm$^2$) in order to maximize the light collection. A carefully optimized Ge bolometer with a NTD-Ge sensor [@Coron2004] achieved the required performance in terms of resolution but reproducibility and robustness are far to be demonstrated.
The next three sections are dedicated to the status and perspectives of the light detector technologies under development.
### Transition edge sensors and metallic magnetic calorimeters {#tes}
A technology, able to reach the desired energy resolution on a large area light detector, does exist and is currently implemented by the CRESST dark matter experiment [@Angloher:2011uu; @Angloher:2014myn; @Angloher:2015ewa]. A half mm thick sapphire disc, with a one micron layer of silicon on it, is equipped with thin tungsten Transition Edge Sensor (TES) coupled to an aluminum absorber. A TES sensor is a resistive device that operates at the critical temperature $T_c$ of the superconductor so that the resistivity changes sharply from zero to a finite value in a very narrow temperature interval. TESs are biased at a constant voltage and their low impedance (in the few m$\Omega$ - $\Omega$ range) imposes the use of Superconducting Quantum Interference Device (SQUID) amplifiers. They are intrinsically fast devices with a bandwidth of MHz or more. This offers, in addition to the excellent energy resolution, two advantages: the pulse shape sensitivity is significantly improved and time resolution better than one ms can be achieved. A CRESST light detector, coupled to a CUORE-style bolometer, demonstrated an event-by-event basis [$\alpha$]{}/([$\beta$]{}/[$\gamma$]{}) separation [@Schaffner:2014caa] (see Fig. \[Che\_tes\]) but scaling of the technology to a thousand detectors requires a dedicated development on the reproducibility of the technology (e.g. uniformity of transition temperature across many channels) at temperatures of about ten mK and on the readout multiplexing capability to reduce the wiring complexity and the heat load. Solutions exist in the astrophysics community [@Nucciotti:2015rsl] but the portability to the [${0\nu\beta \beta}$]{} research field is not trivial. They are based on the use of RF-SQUIDs coupled to superconducting coplanar waveguide (CPW) GHz resonators and homodyne detection. Tuning the resonators at different frequencies, it is possibile to multiplex several RF carriers (see section \[mkid\]). This approach, called Microwave Multiplexing ($\mu$MUX), has been demonstrated for two channels [@Noroozian] and is quickly developing [@Dicker] but it has been shown to work up to now only for compact arrays of micro-calorimeters with mass much less than one milligram.
![Background data of the TeO$_2$ bolometer is shown in the light yield-energy plane. Light yield obtained with a massive (285 g) TeO$_2$ bolometer and TES-equipped Silicon on Sapphire light detector. Two distributions can be noted: a band due to [$\beta$]{}/[$\gamma$]{} interactions as well as the less populated band at zero light yield due to [$\alpha$]{} particles from a degraded [$\alpha$]{} source. The bands which indicate the region expected for [$\beta$]{}/[$\gamma$]{} events are shown in form of central probability bands. The dotted lines are $\pm 1.28 \sigma$ contours whereas the solid lines are $\pm 3 \sigma$ contours, thus 99.8% of all [$\beta$]{}/[$\gamma$]{} events are expected to be contained within the two solid contour lines. A DP of 3.7 is achieved. The [$\alpha$]{} particle distribution appears at a light yield of zero, separated from the populated [$\beta$]{}/[$\gamma$]{} band. The dashed vertical line indicates the Q-value of $^{130}$Te of 2530 keV. Figure adapted from Ref. with permission from Elsevier. []{data-label="Che_tes"}](bw_Ch_tes){width="9cm"}
Other TES implementations are under study in the [${0\nu\beta \beta}$]{} community [@Wang:2015taa]. The first aims at reaching a lower $T_c$ making use of the proximity effect in bilayer films of superconductor and normal conductor (e.g. Ir-Au, Ir-Pt, Mo-Au, etc). The second makes use of NbSi, which is a superconductor for an appropriate stoichiometric ratio with an intrinsic high resistivity in the normal state (1-5 M$\Omega$). This would allow the use of the same conventional electronics, based on JFETs, as for NTDs when they are operated within the transition. This solution does not provide all the advantages related to the low-impedance TESs, but it is possible to get a temperature sensitivity up to ten times higher of that achieved by NTDs keeping the same front-end electronics, and so with a minimal impact on the readout structure.\
Another class of sensors, which share similar characteristics with TESs, is represented by Metallic Magnetic Calorimeters (MMC). They base their principle on the strong temperature dependence of the magnetization in paramagnetic sensors and are typically made of Au:Er. A variation of the magnetic moment can be read out with high sensitivity using meander-shaped thin-film pickup coils and SQUID magnetometers. This effect, already exploited with outstanding results in X-ray spectroscopy [@porst], can be used to develop exceptionally sensitive thermometers. They are very fast sensors (rise-time below 50 $\mu$s) and can reach an energy resolution better than ten eV. Because of these two features, their multiplexed readout is even more demanding than that of TESs and the only feasible approach is $\mu$MUX [@kempfs]. MMCs are adopted by the AMORE collaboration [@Kim:2017xrs] although the amount of scintillation light produced by the $^{48-dep}$Ca$^{100}$MoO$_4$ crystal does not require a very sensitive light detector. No multiplexing readout is applied in this case.
### Kinetic inductance detectors {#mkid}
The working principle of a Kinetic Inductance Detectors (KID) is based on the change of its kinetic inductance when the density of Cooper pairs is modified [@day]. In superconducting materials the Cooper pairs, characterized by a binding energy smaller than 1$\,$meV, move through the lattice without scattering. If a RF electromagnetic field is applied, the pairs oscillate and acquire a kinetic inductance. The inductor is inserted into a high quality factor ($Q>10^3$) RLC circuit giving rise to a resonator with a resonant frequency $f_0=1/2\pi\sqrt{LC}$. An energy release $E$, able to break Coopers pairs into quasi-particles, changes the kinetic inductance and thus the transfer function and could be inferred by the variations in phase and amplitude of the transmitted signal. KID detectors are a leading technology in astroparticle physics [@Monfardini:2011yh; @Mazin:2013wvi] and their use in the [${0\nu\beta \beta}$]{} field was proposed by the CALDER project [@Battistelli:2015vha].\
Their strengths are: several KIDs can be coupled to the same feedline and can be multiplexed by making them resonate at slightly different frequencies since $f_0$ can be easily changed by slightly modifying the layout of the capacitor and/or inductor of the circuit; the readout electronics is quite simple and operated at room temperature, with the exception for a low noise cryogenic amplifier; performances do not depend critically on the working temperature, provided it is well below the critical temperature of the superconductor.\
The main drawback is that dimensions must be smaller than the wavelength of the excitation signal, so that the current in the inductor is uniform and the signal does not depend on the position of the energy release. Their size is limited to a few mm$^2$, by the optimal range (1-4 GHz) of already available electronics and the number of resonators that can be coupled to the same line. To reach a large surface light detector, KIDs are deposited on silicon substrate as in CRESST light detectors [@Angloher:2011uu]. Photons impinging on the back side of the chip produce ballistic phonons which scatter through the substrate and reach the KID on the opposite surface [@Moore:2012au; @Swenson:2010yf]. To compensate the efficiency loss with respect to direct absorption, a few KIDs per light detector are needed. In the last 3 year the CALDER [@Battistelli:2015vha] project developed and tested several KID detectors using Aluminum as superconducting material [@Cardani:2015tqa; @Casali:2015bhk; @Colantoni:2016alu; @Martinez:2016rks; @Vignati:2016adb; @Colantoni:2016tpk; @Bellini:2016lgg; @Casali:2017yro]. With a 4 mm$^2$ single KID resonator on a 2x2 cm$^2$ substrate, an energy resolution of about 80 eV has been achieved [@Bellini:2016lgg].
The energy resolution of KIDs scales as T$_c/\sqrt{QL}$[@Zmui; @McCammon]. Too boost it to the desired level, different superconductors with optimized T$_c$ and L were investigated. The first large area (25 cm$^2$) detector made with Al+Ti+Al KID is being measured and results will be published soon.
### Neganov-Trofimov-Luke effect {#nl}
If a light detector comprises a semiconductor substrate, its baseline noise resolution could be enhanced exploiting the Neganov-Trofimov-Luke (NFL) effect [@luke; @neganov]. An electric field applied to the device, in fact, accelerates electron and holes generated by an energy release $E$ inside the detector itself. The work done by the field on the charges produces an enhancement of the thermal signal recorded by the thermometer attached to the semiconductor wafer. The total energy $E_t$ dissipated is $$E_{t} = E(1+\frac{eV}{w}),
\label{NL}$$ where e is the electron charge, V is the applied drift voltage across the electrodes, and $w$ is the mean energy needed to create an electron-hole pair. The amplification is independent of any other source of noise and allows to lower the baseline noise resolution and decrease the energy threshold. This mechanism is well known and used in dark matter searches [@stark; @Isaila:2011kp; @cdms; @edel] and in the last two years has been successfully applied to detect the Cherenkov light in TeO$_2$ bolometers with different devices with both germanium and silicon absorbers.[@Willers:2014eoa; @Casali:2015gya; @Artusa:2016mat; @Gironi:2016nae; @Biassoni:2015eij].\
Recently a complete event-by-event [$\alpha$]{}/([$\beta$]{}/[$\gamma$]{}) separation in a full-size TeO$_2$ CUORE bolometer coupled to a NTD-based germanium light detector with NTL amplification has been achieved [@Berge:2017nys]. In this case the electrodes, a set of concentric Al rings on a side, generate an electric field parallel to the surface that allows to decrease the charge trapping probability thanks to the short path length of the charges to the electrodes. This represent a fundamental result in view of an application in CUPID [@Wang:2015raa] since it could be adopted with minimal modification of the entire readout with the respect to the one actually in use in the CUORE experiment.
Devices with silicon absorbers and TES [@Willers:2014eoa] and NTD [@Gironi:2016nae; @Biassoni:2015eij] sensors were also developed. In the case of NTD sensors, the advantages compared to germanium absorber hinge on the wider range of processing technologies for silicon, that potentially allows the integration of thermal sensors and mechanical suspension structures. A further advantage of silicon over germanium is the fact that specific heat of silicon is a factor four smaller than germanium, opening the possibility of building substantially larger detectors without compromising the signal amplitude which is inversely proportional to the heat capacity of the device. Promising results have been obtained with 2x2 cm$^2$ area detectors[@Biassoni:2015eij] and first 5x5 cm$^2$ sample are under measurement. Plans to use the NFL effect with KID sensors for single photon counting are also under investigation.
Surface sensitive bolometers {#shape}
----------------------------
Tagging surface events is difficult, as bolometers are fully sensitive device in the volume and often present a single response to any type of fast energy deposition, irrespective of its nature and location. Even when [$\alpha$]{} background could be reduced at the desired level, a non negligible contribution could be represented by single [$\beta$]{} particles emitted in decays of $^{214}$Bi as well as by $^{210,208}$Tl decays that emit electron and [$\gamma$]{}s in coincidence producing a single event which escapes delayed coincidences tagging (see section \[err\]).
An alternative approach, to those aiming of hybrid bolometers, consists into achieving impact-point sensitivity making use of superconducting Al films (about ten $\mu$m) deposited on the detector surface which can modify the signal shape of surface events [@Schnagl:2000]. The physical principle is the following: athermal phonons generated by a particle that releases its energy within a few mm from the surface ([$\alpha$]{} or [$\beta$]{}s) break Cooper pairs in the superconducting film and produce quasi-particles, which have a long lifetime (on the order of milliseconds) in high purity aluminum. The quasi-particles recombination produces ballistic phonons, that will add a delayed component to the leading edge of the signal read out by the sensor on the main bolometric absorber. For bulk events, instead, the athermal phonon population reaching the Al film is more degraded in energy and less efficient in producing quasi-particles. Surface events will have therefore longer rise-time compared to bulk interactions. This mechanism has been evidenced in Ref. and the proof of principle of this technique for [${0\nu\beta \beta}$]{} decay detector has been demonstrated in a TeO$_2$ bolometer with deposited Al film and using fast phonon sensors based on NbSi films, with rise times on the order of one ms [@Nones:2012].
Unfortunately, the current NbSi sensor technology is unsuitable for [${0\nu\beta \beta}$]{} search because the important component of athermal phonons in the signal induces position dependent amplitude and thus deteriorate the energy resolution. The recently ERC-approved project CROSS [@Giuliani:2017] aims at achieving surface-to-bulk signal separation with the use of NTD sensors, i.e. with a heat pulse rise-time on the order of tens of milliseconds. This might be possible as the excellent signal-to-noise ratio characterizing the typical CUORE readout has the potential to highlight even tiny pulse-shape differences. This technique could be applied also to scintillating bolometers since once the surface $\alpha$ background is rejected, the dominant contribution arises from surface $\beta$s contaminations [@Artusa:2014wnl].
Environmental Radioactivity Reduction {#err}
=====================================
The particle identification techniques, on which detector developments are mainly focused, aim at reducing to negligible levels the effect of surface contaminations of detector materials that represents the dominant background in CUORE. However, the reduction of surface contamination effects can’t by itself ensure the achievement of a two orders of magnitude background reduction as foreseen in CUPID. Other different sources, as bulk contaminations of crystals, copper supporting structure, lead shields, and the -small parts- as glue, bonding wires or readout cables and pads, can contribute to the ROI counting rate at levels of $\sim 10^{-3}$ [counts kg$^{-1}$ y$^{-1}$]{}[@Artusa:2014wnl].\
All of this results in serious restrictions in the use of materials. Stringent purification protocols for crystal production must be developed and all the materials close to the detectors have to be fabricated from radio-pure materials and assembled in radon free environment with dedicated radio-pure tools. A special attention has also to be paid to avoid cosmogenic activation. An exhaustive list of low background techniques exploited in this research field can be found in Ref. .\
The bulk activity of the crystal absorber must be controlled to a level such to not spoil the background index the ROI. Internal [$\alpha$]{} decays from U/Th chains can not contribute to background since they give rise to sharp peaks with energies Q$_{\alpha} >$ 4 MeV, i.e. far above the ROI. Internal $\beta$ decays with Q$_{\beta} >$ 3 MeV could represent instead a worrisome background due to their continuum spectrum. They are generated by $^{214}$Bi ($^{238}$U chain) and its daughter $^{210}$Tl and by $^{208}$Tl ($^{232}$Th chain) as reported in the scheme of Fig. \[BiPo\].
![Left: decay scheme of $^{214}$Bi. The short life of its daughter, $^{214}$Po, causes the pile-p of the [$\beta$]{} emitted by $^{214}$Bi and the [$\alpha$]{} particle produced by $^{214}$Po. Right: decay scheme of $^{208}$Tl. The delayed coincidence with the [$\alpha$]{} particle produced by its mother, $^{212}$Bi, allows to suppress this background source. Illustration is a courtesy of Laura Cardani. []{data-label="BiPo"}](214BiDecay.pdf "fig:"){width="6.5"}![Left: decay scheme of $^{214}$Bi. The short life of its daughter, $^{214}$Po, causes the pile-p of the [$\beta$]{} emitted by $^{214}$Bi and the [$\alpha$]{} particle produced by $^{214}$Po. Right: decay scheme of $^{208}$Tl. The delayed coincidence with the [$\alpha$]{} particle produced by its mother, $^{212}$Bi, allows to suppress this background source. Illustration is a courtesy of Laura Cardani. []{data-label="BiPo"}](208Tl_decay.pdf "fig:"){width="6.5cm"}
Given the slowness of the bolometric response the $^{214}$Bi $\beta$ decay is followed by the fast $^{214}$Po [$\alpha$]{} decay and their energy is summed up far from the ROI (Bi-Po events). The $^{208}$Tl and $^{210}$Tl decays instead to stable $^{208}$Pb and $^{210}$Pb respectively; anyway they could be tagged by delayed coincidence with the primary [$\alpha$]{} decay. In order to limit the dead time introduced with this technique ([$\beta$]{} half life of the order of minutes) the total activity of U/Th contamination in the bulk must be kept at the level of the $\mu$Bq/kg.\
Another important aspect which must be taken into account is that the increase of the sensitive mass demands the production of high-quality, radio-pure enriched crystals. The difficulty in operating cryogenic systems with an experimental value larger than the existing CUORE one makes the isotope enrichment the only viable way for enhancing the mass of the isotope of interest. The use of enriched material has consequences on the purification/crystallization chain. The enriched material could have residual chemical impurities which may require additional purification stages to get high-quality crystals. Enrichment is generally done by gas centrifugation in facilities used to separate different isotopes without special radio-purity concern and therefore an additional purification is required. Furthermore the enriched material is expensive, and the growth procedure must be adapted in order to reduce as much as possible the irrecoverable losses of the initial charge. Crystal bulk contaminations [@Poda:2017jnl] are actually at the level of few to ten $\mu$Bq/kg and are approaching the target value to reduce the internal background to a harmless level.\
Even if it would be possibile to get rid of U and Th contaminations, the [${2\nu\beta \beta}$]{} decay induces an irriducibile background for the [${0\nu\beta \beta}$]{} search. The end-point of the [${2\nu\beta \beta}$]{} spectrum (when neutrinos are emitted at rest) contribute to the background in the ROI since all the available energy is carried out by the two electrons expect for the two negligible neutrino masses. The ratio of [${2\nu\beta \beta}$]{} to [${0\nu\beta \beta}$]{} event rate depends on the [${2\nu\beta \beta}$]{} half life and on the energy resolution and could be assumed negligible in the case of IH region for an energy resolution better than one per cent [@Artusa:2014wnl]. On the other hand, accidental pile-up of two [${2\nu\beta \beta}$]{} events in the same detector within a time window smaller than the typical time response of the detector can produce a signal that mimics a [${0\nu\beta \beta}$]{} decay. This contribution can be suppressed using the leading edge of the thermal response that range from microseconds (athermal sensors) to milliseconds (thermal sensors). This turns out to be problematic only in the case of $^{100}$Mo which has the fastest observed [${2\nu\beta \beta}$]{} decay and can contribute to ROI with a background of the level of 10$^{-2}$ [counts keV$^{-1}$kg$^{-1}$ y$^{-1}$]{} [@Beeman:2011bg; @Chernyak:2012zz; @Chernyak:2014ska] in case of slow NTD thermal sensors to 10$^{-4}$ [counts keV$^{-1}$kg$^{-1}$ y$^{-1}$]{}in case of fast sensors sensitive to athermal phons as MMC [@Luqman:2016okt]. The discrimination capability depends on the slope-to-noise ratio [@spieler] and it has been demonstrated that the use of light detector with NLF signal amplification, could lower this background down to 6$\times$10$^{-5}$ [counts keV$^{-1}$kg$^{-1}$ y$^{-1}$]{}[@Chernyak:2016aps].
In the CUORE background budget [@Alduino:2017qet], no positive indication of bulk contaminations in the other detector elements have been obtained, However, current upper limits could translate to potentially dangerous counting rates for the CUPID background target. One order of magnitude improvement in the sensitivities of presently screening technology is therefore mandatory.\
The screening techniques commonly used are: HPGe (High Purity Germanium detector), ICPMS (Inductively Coupled Plasma Mass Spectrometry, NAA (Neutron Activation Analysis). NAA and HPGe measurements can reach a sensitivity of the order of $\mu$Bq/kg on $^{232}$Th in copper, the material used as supporting structure for the crystal absorber. The sensitivity is limited by the mass of the copper sample that can’t be increased [*ad libitum*]{} due to the self-absorption of the [$\gamma$]{} lines inside the sample. It could be increased making use of pre-concentration of contaminants through chemical treatment of materials which is equivalent to a mass increase of the sample. The technique is used in ICPMS measurements but can also be applied to NAA or HPGe spectroscopy. However, it requires a dedicated study for each material as well as a very careful control of systematics.\
Some detectors parts used in the form of foils, as super-insulation or flat cables, are not suitable for HPGe due to their small mass neither for NAA or ICPMS which have restrictive conditions on the material that can be analyzed. In these cases, surface alpha spectroscopy through Si surface barrier diodes proved to reach competitive sensitivities. An alternative technique consists of using a bolometric detector for the measurement of surface/bulk contamination; TeO$_2$ slabs can be used to realize a sandwich-like detector where samples are inserted in-between thin bolometers. Given the better energy resolution, the lower energy threshold and the superior radio-purity [@Cardani:2012xq], this approach could reach a sensitivity up to a factor 100 higher than the actual ones and could provide information on the X-ray emission of the samples providing a complementary information for contamination identification.
Other Rare Nuclear Decays and Processes {#rare}
=======================================
[$\alpha$]{} decays
-------------------
The discovery of the [$\alpha$]{} decay of $^{209}$Bi with a half-life of 1.9$\times$10$^{19}$ yr in 2003 [@demarcillac] renewed the interest in the field of rare [$\alpha$]{} decays as a fundamental tool for the study of the structure of nuclei and for a better understanding of the theoretical framework of nuclear models [@XuRen].
The possibility to produce massive bolometers with a wide choice of materials has very clear advantages. A significant amount of the nucleus of interest could be embedded in the detector itself, i.e. the detector and the decay source coincide. As a consequence the decay is full contained in detector thus resulting in excellent detector efficiency. This aspect is of primary importance for rare [$\alpha$]{} decays search also because of the short (few microns) range of a MeV [$\alpha$]{} particle in a solid medium. In addition the high energy resolution and the capability to identify the interacting nature of the particle in the detector, as discussed in section \[sb\], leads to tremendous background suppression, especially for rare decays with transition energy lower than 2.6 MeV (as for some lead isotopes) which would otherwise be overwhelmed by the near background from [$\gamma$]{} emission of [$^{208}\mathrm{Tl}$]{}. Bolometers allow to measure half lives much longer than the age of the Universe. They led to the conclusive test on the identification the [$\alpha$]{} decay of $^{209}$Bi [@demarcillac] and the discovery of its decays to the first excited level [@Beeman:2011kv; @cardanirari], to the discovery of $^{180}W$ [@Cozzini] and $^{151}$Eu [@Casali:2013zzr] [$\alpha$]{} decays and to set the most stringest lower limits on the half life of the [$\alpha$]{} decays of lead isotopes [@Beeman:2013Pb].\
An alternative approach consists in the use of a well known scintillating bolometer doped with the isotope under investigation. This allows to select a vey radio-pure crystal with high light yield and excellent [$\alpha$]{}/([$\beta$]{}/[$\gamma$]{}) separation and to study more elements as Sm, Nd, Os, Hf, Pt. The drawback is that the mass of the candidate isotope is limited to few grams. This technique was recently proposed and used for a precise measurement of the half life and transition energy of the $^{148}$Sm [$\alpha$]{} decay in a ZnWO$_4$ crystal [@Casali:2016vbw].
[$\beta$]{} decays
------------------
The improvement of the sensitivity in the search for rare [$\alpha$]{} decays had an impact also on the study of extreme [$\beta$]{} decay as the ones generated by small decay energies or initial and final nuclear states with large angular momentum difference.\
The knowledge of rare [$\beta$]{} decay existence and their spectral shape is fundamental since they can represent a background for the [${0\nu\beta \beta}$]{} search. The $^{214}$Bi, for example, has a $Q_\beta$= 3270 keV thus exceeding the [${Q_{\beta\beta}}$]{} of the more studied [${0\nu\beta \beta}$]{} isotopes (see section \[err\]). In about 19% of the cases it decays to the ground state of $^{214}$Po with change in angular momentum J and parity $\pi$, $\Delta J^{\Delta \pi}=1^-$ (first forbidden non-unique transition). Its shape is not well measured experimentally neither theoretically predicted, moreover it must be taken into account that forbidden [$\beta$]{} shape can significantly deviate from known allowed spectra.
But there is a more important feature related to forbidden [$\beta$]{} decays: their shape could be used to infer the ratio of the weak axial to vector coupling constant g$_A$/g$_V$ in nuclear decays.\
The [${0\nu\beta \beta}$]{} half life is proportional to the fourth power of g$_A$. The recent analyses of nuclear models in [$\beta$]{} and [${2\nu\beta \beta}$]{} decays indicate that the value of g$_A$ could be quenched, up to a ratio of g$_{free}$/g$_A$ $\sim$4, where g$_{free}$ = 1.27 is the free value of g$_A$ inferred from the neutron decay. This could potentially translates into a two order of magnitude difference in the sensitivity of [${0\nu\beta \beta}$]{} experiment. This [*naive*]{} expectation has been very recently scaled back to a factor between two and six if a consistent approach is used for the calculation of the [${2\nu\beta \beta}$]{} and [${0\nu\beta \beta}$]{} decays [@Suhonen:2017rjf]. Nevertheless the measurement of g$_A$ is of pivotal importance. This could be inferred by the shape of $\Delta J^{\Delta \pi}=4^+$ non-unique forbidden [$\beta$]{} decays as for $^{113}$Cd and $^{115}$In [@Haaranen:2016rzs; @Haaranen:2017ovc; @Kostensalo:2017xxq; @Kostensalo:2017jgw; @Suhonen:2017krv]. For such decays the shape of energy spectrum relies on the sum of different nuclear matrix elements with different phase space factors which include $g_A$ and $g_V$ and their values could be extracted by the comparison between theoretical and experimental spectra. While in the case $^{113}$Cd the spectral shape has been characterized [@Belli], only and old measurement [@Pfe] exists for the shape of $^{115}$In. To perform a clean and reliable measurement a 10 g LiInSe$_2$ scintillating bolometer is currently in data taking at the Modane underground laboratory [@Tretyak:2017zqd].
Electron capture processes
--------------------------
Bolometers can play in important role also in the search of other rare nuclear processes as the rare electron capture. When the source of the decay is embedded in the bolometer, in fact, a signal corresponding to the total binding energy of the captured electron can be measured with very high efficiency because X-rays/Auger electrons following the atomic de-excitation are fully contained. Moreover the excellent energy resolution is a powerful tool to discern externally generated [$\gamma$]{} from X-ray and electron cascades.\
As example the electron capture of $^{123}$Te is predicted but not yet observed. The best limit, obtained with a TeO$_2$ bolometer is $T_{1/2}>5.0\times10^{19}$ y [@Alessandrello:2002ag]. A previous observation [@Alessandrello:1996zz] was confuted and explained as the electron capture in $^{121}$Te, an isotope created by neutron capture on the 0.09% natural abundant $^{120}$Te isotope at see level. The importance of this measurement relies on the fact that it could be used to constrain and test nuclear models used to estimate intensities for rare electroweak decays [@civitarese], models that in some case foresee a suppression of the rate up to six orders of magnitude [@broglia97]. The CUORE experiment, with is huge mass compared to its predecessors, will be able to improve the results by orders of magnitude and possibly to discover the electron capture of $^{123}$Te.
Conclusions
===========
Bolometers are cryogenic calorimeters which base their principle on the phonon detection. They exhibit: excellent energy resolution, low energy threshold, high detection efficiency, wide choice of material for the calorimeter absorber. These characteristics make them one of the best performing instruments in several fields: double beta decay search, neutrino mass measurement, dark matter search, CMB precision measurement, high resolution X-ray detection, rare nuclear process detection.\
This review is focused on the bolometric applications in the field of rare nuclear processes: in particular on the neutrinoless double beta decay search. The demand for the increase of the experimental sensitivity imposes a series of technical challenges and improvements of the actual technology. In particular, experiments aiming at covering the inverted hierarchy region of the neutrino mass scheme and possibly discovering the Majorana neutrino nature, need to lower the background in the region of interest to the level of 10$^{-4}$ [counts kg$^{-1}$ y$^{-1}$]{}and increase the source mass. This implies a manifold effort: the development of passive methods for background reduction and new screening techniques, the growth of very radio-pure enriched crystals and the implementation of reliable active background rejection techniques.\
The first pilot demonstrators, using enriched scintillator bolometers for particle identification, are already in data taking while the development of detectors for the Cherenkov light detection in TeO$_2$ bolometers is rapidly growing and entering into its final phase. On the other hand, the successful operation of the CUORE experiment (988 massive bolometers) ensures that hundred of kgs of isotopes can be studied in a stable and reliable cryogenic system.\
Despite three decades have passed since they have been conceived, bolometers are still a very active field and performances are continuously improving. The viability of a next generation experiment in an almost background-free environment, is within the reach if actual R&D will be successful.\
Moreover, the superior bolometric features renewed in the last five years the interest in rare nuclear processes as a tool for the comprehension of nuclear models. The improvements in terms of performances and radio-purity of material, requested by the neutrinoless double beta decay search, are beneficial for all the rare nuclear process searches and will boost the sensitivity to unprecedented levels.\
|
---
abstract: 'We present new constraints on the dark matter-induced annual modulation signal using [1.7years]{} of COSINE-100 data with a total exposure of [97.7kg$\cdot$years]{}. The COSINE-100 experiment, consisting of 106kg of NaI(Tl) target material, is designed to carry out a model-independent test of DAMA/LIBRA’s claim of WIMP discovery by searching for the same annual modulation signal using the same NaI(Tl) target. The crystal data show a 2.7 cpd/kg/keV background rate on average in the 2–6 keV energy region of interest. Using a $\chi$-squared minimization method we observe best fit values for modulation amplitude and phase of [0.0092$\pm$0.0067]{} cpd/kg/keV and [127.2$\pm$45.9]{}d, respectively.'
bibliography:
- 'cosine100modulation.bib'
title: 'Search for a Dark Matter-Induced Annual Modulation Signal in NaI(Tl) with the COSINE-100 Experiment'
---
Cosmological observations give strong evidence that 27% of the energy content of the Universe exists in the form of nonluminous dark matter [@Ade:2013zuv], unaccounted for by the standard model of particle physics [@pdg2018]. One theoretically favored model of dark matter posits the existence of weakly interacting massive particles (WIMPs) [@Lee; @Goodman] that interact only through the gravitational and weak scale forces and have a mass on the GeV to TeV scale [@WIMP; @Bertone]. Within the context of the standard halo model, there will be an annual modulation in the dark matter–nucleon interaction rate with a period of one year [@Colloquium; @Freese:1987wu; @Lewin:1996]. One experiment, DAMA/LIBRA, observes annual modulations in the detected event rate with a significance exceeding 12$\sigma$, which they attribute to the presence of dark matter [@Bernabei:2005hj; @Bernabei:2013xsa; @DAMAPhase2]. DAMA/LIBRA’s observation is inconsistent with other experiments under most well-motivated WIMP dark matter models [@LUX2017; @XENON2018; @PandaXII2017; @CDMS; @CRESST; @PICO; @KIMS; @SIMPLE; @XMASS2018]; however, none of these other experiments have used the same target material as DAMA, thallium-doped sodium iodide \[NaI(Tl)\] scintillating crystals. Thus, these comparisons are necessarily dependent on the particular model of WIMP-nucleus scattering and the assumed WIMP halo structure.
The COSINE-100 experiment aims to resolve this tension in the field by performing a model-independent test of DAMA’s observation using the same detector material, NaI(Tl), as DAMA. Previously, we have performed a model-dependent test of DAMA and found that DAMA’s observed annual modulation cannot be explained by spin-independent WIMP-nucleus scattering in the context of the standard halo model [@COSINE_WIMP]. Additionally, there are several other experiments aimed at performing model-independent tests of DAMA, including DM-Ice17 [@BarbosaDeSouza:2017], KIMS [@KIMS_NaI], SABRE [@SABRE], and ANAIS-112 [@ANAIS2019EPJC; @ANAIS2019], which has recently reported its first result.
COSINE-100 is located at the Yangyang Underground Laboratory (Y2L) in South Korea, with $>$700m of rock overburden. It consists of eight NaI(Tl) crystals with a total mass of 106kg immersed in 2200l of liquid scintillator (LS) that reduces internal and external backgrounds [@KIMS_LS]. Each NaI(Tl) crystal is optically coupled to two photomultiplier tubes (PMTs), each of which detects scintillation photons with the signals recorded as 8$\mu$s waveforms [@COSINE_daq]. These eight crystals are referred to as Crystal 1 (C1) to Crystal 8 (C8). C1, C5, and C8 are excluded from this analysis due to their high background (about twice that of the other crystals), high noise rate (C1), and low light yield (C5 and C8), for a total effective mass of [61.3kg]{}. The detector is surrounded by passive and active shielding that includes, from the inside out, copper plates of 3cm in total thickness, 20cm of lead, and 3cm of 37 plastic scintillator panels for cosmic ray muon tagging [@COSINE_muon]. More details of the experimental apparatus are presented in Ref. [@COSINE_detector].
Data taking for COSINE-100 began in September 2016, and the analysis presented here covers an exposure of [1.7years]{} years, spanning from [October 21, 2016]{} to [July 18, 2018]{}. Several datasets from C2 and C7 are excluded due to excessive noise levels. The total exposure used in this analysis corresponds to [97.7kg$\cdot$years]{}.
The overall stability of the detector is closely monitored to ensure that neither environmental nor detector effects can create an artificial dark matter signal [@COSINE_detector]. Humidity and temperature of the detector room are maintained at 40.0$\pm$3%RH (relative humidity) and 23.5$\pm$0.3$^{\circ}$C, respectively. Gas boiloff from liquid nitrogen is introduced into the space above the liquid scintillator inside the inner copper chamber at a rate of 3 l/min to purge radon and prevent contact between the LS and oxygen or water vapor, which maintains a high scintillator light yield. The humidity inside the shielding structure is kept at $<$5%RH and the high heat capacity helps to keep the temperature within the liquid stable at 24.2$\pm$0.1$^{\circ}$C. The radon level in the detector room is measured at 36$\pm$10 Bq/m$^{3}$. The time dependence of temperature, humidity, radon, and cosmic ray muons [@COSINE_muon] is shown in Fig. \[fig:env\_monitor\]. The spikes in Fig. \[fig:env\_monitor\](a) are due to power outages or air conditioning failures; these periods are excluded from the data. The effects of temperature and radon level on the pulse shape, light yield, and overall performance of the NaI(Tl) detectors and of the full detector were reported in Ref. [@Schneid1977]. A monitoring of fast neutrons inside the detector room has recently begun in Summer 2018 [@COSINE_neutron].
The gain of the PMTs is monitored by measuring the position of the 46.5keV peak from $^{210}$Pb decays that occur in the NaI(Tl) crystal bulk. The gain is tracked and modeled as a piecewise linear function in time. Observed gain shifts over time are corrected for in each PMT. After correction, the 46.5keV peak is stable to within 0.1% on average. We assess the efficacy of this gain correction method within the 2–6keV region of interest by measuring the position of the 3.2keV decay peak from $^{40}$K over time; the position of the decay peak is stable to within $<$2% on average in the dataset used in the analysis.
![COSINE-100’s environmental parameters as a function of time. (a) Detector room and near-crystal temperature. (b) Relative humidity for the detector room and the top volume of acrylic box, at the top of the LS. Note that the measurement taken at the top of the LS began on day 450. (c) The radon level in the detector room air. (d) Rate of muons passing through the detector over time. Here, the rate is binned in 30-day intervals. []{data-label="fig:env_monitor"}](figures/env_monitor.pdf){width="\columnwidth"}
![Efficiency-corrected and time-integrated energy spectra for the five crystals used in this analysis between 2–20keV (top panels) and signal selection efficiency evaluated using $^{60}$Co calibration data (bottom panel). The efficiencies at 2keV are $>$60% for all crystals. The primary sources of background in the crystals are $^{210}$Pb and $^{40}$K, which are lower for Crystal 6 and Crystal 7. These spectra are obtained using the full dataset considered in this analysis.[]{data-label="fig:eff_spectrum"}](figures/eff_spectrum.png){width="0.9\columnwidth"}
Events that trigger more than one crystal, pulses with pulse shapes that are inconsistent with a NaI(Tl) scintillation signal, e.g., PMT related noise, are rejected [@KIMS_NaI; @COSINE_detector; @ANAIS2014]. We use two boosted decision trees, which are multivariate analysis algorithms (BDTs) [@BDT], to remove PMT-related and other noise events, which we call BDT1 and BDT2. BDT1 is used to remove PMT-induced noise and is based on the amplitude-weighted average time of a pulse, the ratios of the leading- and trailing-edge charge sums relative to total charge, and the difference of deposited charges between the two PMTs [@COSINE_EFT]. It is trained with a sample of signal-rich, energy-weighted events from a $^{60}$Co calibration run for signal, and single-hit events from the WIMP-search physics-run data for noise, with the latter mostly triggered by PMT noise events. The second BDT, BDT2, includes weighted higher-order time moments and eliminates intermittent PMT discharge-triggered events that have slower pulse decay times. The event selection technique and criteria are described more in detail in Refs. [@COSINE_detector; @COSINE_WIMP].
The same BDT selections were applied to the Compton-scattered low energy events from a $^{60}$Co calibration run to estimate the event selection efficiency. The efficiency is the ratio of events that survive the selection to the total number of signal events. Uncertainties on the efficiency follow binomial statistics. Figure \[fig:eff\_spectrum\] shows the event selection efficiency as a function of energy, along with the efficiency-corrected, 2–20keV spectra of the five crystals used in this analysis. The spectra are well modeled with a <span style="font-variant:small-caps;">GEANT</span>4-based simulation [@geant; @geant_2; @COSINE_background]; the 3.2keV $^{40}$K peak is clearly visible in C2 and C4, whereas the overall background levels in C6 and C7 are lower than in other crystals because of their lower $^{210}$Pb and $^{40}$K contamination levels.
![Rate vs. time for Crystals 2, 3, 4, 6, and 7 from [October 21, 2016]{} to [July 18, 2018]{} for the 2–6keV energy region binned in 15-day intervals. The histograms show the result of the fit described in the text. Solid blue arrows indicate the peak date in the modulation as reported by DAMA/LIBRA [@DAMAPhase2]. Data taking was suspended for calibrations at the end of 2016 as indicated by the shaded region.[]{data-label="fig:rate_versus_time"}](figures/rate_versus_time.png){width="\columnwidth"}
In order to confirm our background understanding and account for possible systematic effects that could appear over time, we investigated a control sample of multiple-hit events in the 2–20keV energy region with statistics comparable to that in the region of interest (ROI) of 2–6keV. These are events in which multiple NaI(Tl) crystals are triggered or a single crystal is triggered along with the LS and, thus, cannot be attributable to typical WIMP dark matter interactions. They comprise 20% of the total signal event sample. We also consider the possibility that certain event types that are removed during event selection could cause a modulation signal. The noise events observed in the COSINE-100 detector are systematically categorized and studied to understand how their removal affects the signal region counting rate over time. This study confirmed none of the cut individually show a modulation in the removed events and have negligible impact on the modulation of signal events.
The event rates as functions of time are modeled as: $$\label{eq:fit}
\textrm{Rate} = C+ p_{0}\exp{(-\frac{ln2t}{p_{1}})} + A\cos{\frac{2\pi(t-t_{0})}{T}},$$ where $C$ is a constant offset constrained by background modeling as described in Ref. [@COSINE_background], and $p_{0}$ and $p_{1}$ are the amplitude and half-life for an exponentially decaying background, which models cosmogenically activated backgrounds. The modulation is described by $A$, $T=365.25$ d, and $t_{0}$, its amplitude, period, and phase, respectively.
The data from all crystals were fit simultaneously with the same amplitude and phase amongst all crystals but allowing for different exponential decaying and constant background components to account for the varying background levels across different crystals. Figure \[fig:rate\_versus\_time\] shows the COSINE-100 event rates over time for the 2–6keV ROT in the crystals used in this analysis, where recorded 670 events/day on average, i.e. 2.7 cpd/kg/keV. We performed $\chi$-squared minimization fits for the modulation amplitude with the period fixed at 365.25 d with the phase as a free parameter and, also, with it fixed at the halo-model expectation value of 152.5 d and the DAMA/LIBRA-observed value of 145 d. Initially, we performed a blinded analysis by only analyzing $\sim$9% of the data, evenly distributed in time. However, during unblinding, we observed a large number of anomalous noise events within the signal region. This led us to develop BDT2 in order to remove these anomolous events and to reanalyze the data unblinded. The best fit to the 2–6keV range has a modulation amplitude of [0.0092$\pm$0.0067]{}cpd/kg/keV with a phase of [127.2$\pm$45.9]{}d. A log-likelihood parameter estimation of the annual modulation with amplitude and phase as free parameters shows that the current data from COSINE-100 is consistent with both the DAMA/LIBRA annual modulation result and the null hypothesis of no modulation at the 68.3% C.L. as shown in Fig \[fig:chisq2D\]. A Feldman-Cousins method [@Feldman] was also used to crosscheck the result, and returned a consistent C.L.
![The COSINE-100 best fit and 68.3%, 95.5%, and 99.7% C.L. contours as functions of modulation amplitude and phase relative to January 1, for a fixed period of 365.25 d. A Feldman-Cousins technique is used as a crosscheck and resultant 68.3% C.L. is shown. The amplitude and phase reported by DAMA/LIBRA in the 2–6keV energy interval with statistical uncertainties (blue cross) and the phase expected from the standard halo model (June 2) are overlaid for comparison. Top and side panels show the dependence of $\Delta\chi^{2}$ on phase and amplitude, respectively, along with two-sided significance levels.[]{data-label="fig:chisq2D"}](figures/chisq2D.png){width="\columnwidth"}
Table \[tab:resultNumbers\] summarizes the result of the various fitting scenarios used for the 2–6keV energy interval. The period is fixed at 365.25 d (one year) for all scenarios, whereas the phase is either floated freely or fixed at 152.5 d as expected from the standard halo model. COSINE-100 is the only NaI(Tl) experiment with a LS veto surrounding the crystals providing additional capabilities for rejection of external background. As a crosscheck, we show the annual modulation fit results both with and without the LS veto. The LS veto removes backgrounds and improves the uncertainties on the annual modulation amplitudes by 4%.
Configuration $\chi^{2}$ DOF *p*-value Amplitude (cpd/kg/keV) Phase (d)
---------------------------- ------------------- ------------------- ------------------- ------------------------ ----------------
COSINE-100 175.3 174 0.457 0.0092$\pm$0.0067 127.2$\pm$45.9
DAMA/LIBRA (Phase1+Phase2) $\cdot\cdot\cdot$ $\cdot\cdot\cdot$ $\cdot\cdot\cdot$ 0.0096$\pm$0.0008 145$\pm$5
COSINE-100 175.6 175 0.473 0.0083$\pm$0.0068 152.5 (fixed)
COSINE-100 (Without LS) 194.7 175 0.147 0.0024$\pm$0.0071 152.5 (fixed)
ANAIS-112 48.0 53 0.67 -0.0044$\pm$0.0058 152.5 (fixed)
DAMA/LIBRA (Phase1+Phase2) 71.8 101 0.988 0.0095$\pm$0.0008 152.5 (fixed)
\[tab:resultNumbers\]
![Modulation amplitude as a function of energy in 1keV bins for the 1.7 year COSINE-100 single-hit (red closed circle) and multiple-hit (orange open circle) events. DAMA/LIBRA phase 1 (blue) and phase 2 (green) from Ref. [@DAMAPhase2] are also shown for reference. The period and phase are fixed at 365.25 d and 152.5 d. Horizontal error bars represent the width of the energy bins used for the analysis. Vertical error bars are $\pm 1\sigma$ errors on the binned modulation fit amplitudes.[]{data-label="fig:modulationVsEnergy"}](figures/amp_versus_energy_DAMA12.png){width="\columnwidth"}
The best fit modulation amplitudes as a function of energy with 1keV energy bins are shown in Fig. \[fig:modulationVsEnergy\]. These fits were performed with a fixed period of one year and the phase fixed at 152.5 d. In summary, we report the results from the search for a dark matter–induced annual modulation signal in NaI(Tl) based on 1.7 years of COSINE-100 data. A fit to the 2–6 keV energy range returns a modulation amplitude of [0.0092$\pm$0.0067]{}cpd/kg/keV with a phase of [127.2$\pm$45.9]{}d. At 68.3% C.L., this result is consistent with both the null hypothesis and DAMA/LIBRA’s 2–6keV best fit value. We expect COSINE-100 will attain 3$\sigma$ coverage of DAMA region within five years of data exposure. Future searches with will utilize a larger dataset and lower energy threshold of at least 1keV with improved event selection efficiency and are expected to reduce the required exposure for 3$\sigma$ coverage.
We thank the Korea Hydro and Nuclear Power (KHNP) Company for providing underground laboratory space at Yangyang. This work is supported by: the Institute for Basic Science (IBS) under project code IBS- R016-A1 and NRF-2016R1A2B3008343, Republic of Korea; UIUC campus research board, the Alfred P. Sloan Foundation Fellowship, NSF Grants No. PHY-1151795, No. PHY-1457995, and No. DGE-1122492, WIPAC, the Wisconsin Alumni Research Foundation, United States; STFC Grants No. ST/N000277/1 and No. ST/K001337/1, United Kingdom; and Grant No. 2017/02952-0 FAPESP, CAPES Finance Code 001, Brazil. We thank P.T. Surukuchi for helpful discussions.
|
---
abstract: 'Quantitatively assessing relationships between latent variables and observed variables is important for understanding and developing generative models and representation learning. In this paper, we propose latent-observed dissimilarity (LOD) to evaluate the dissimilarity between the probabilistic characteristics of latent and observed variables. We also define four essential types of generative models with different independence/conditional independence configurations. Experiments using tractable real-world data show that LOD can effectively capture the differences between models and reflect the capability for higher layer learning. They also show that the conditional independence of latent variables given observed variables contributes to improving the transmission of information and characteristics from lower layers to higher layers.'
author:
- 'Yasushi Terazono[^1]'
title: 'A latent-observed dissimilarity measure'
---
Introduction {#sec_intro}
============
Models with latent variables have been proposed and investigated for explaining, understanding, or classifying observed data. If a model is a generative model, observed data are modeled to be as if they were generated by latent variables through parameterized probability distributions. Popular criteria for learning generative models include likelihood or posterior probability, which both evaluate the probability of the given observed data or parameters. Another kind of criteria is mutual information. Mutual information has been used to learn non-linear generative models [@Pinchaud2011NIPS_mutual_information] in which relationships between observed and latent variables are directly evaluated. It has also been used to learn linear encoding (recognition) models [@Baldi1995IEEE-NN_LNN; @Obradovic1998NeuralComputation_infomax_ica].
The relationships between observed and latent variables have greater importance in more complex generative models, e.g., deep learning models [@Hinton2006science_autoencoder; @Hinton2006NeuralComputation_DBN]. In the pre-training of deep belief networks (DBNs), one of the models or techniques of deep learning, posterior samples of latent variables in the lower layer are used as samples of observed variables in the next, higher layer. For successive layer learning to be possible, latent variables should possess properties that enable such learning. It is crucial and fundamental for multiple layer learning theory to assess which observed variable properties are preserved, discarded, or modified in latent variables. For this purpose, it is necessary to have good measures that capture the capability of higher layer learning and to know the configurations of models suitable for higher layer learning. Unfortunately, mutual information is not an adequate measure for this purpose. The maximization of mutual information is known to yield independent latent variables under certain conditions [@Obradovic1998NeuralComputation_infomax_ica], however, if latent variables are independent of each other, successive learning exploiting their correlations becomes impossible.
In this paper, we propose a novel measure to capture the dissimilarity between latent and observed variables in two-layer models. We refer to the proposed measure as latent-observed dissimilarity (LOD). The key idea is to define a “virtual-latent” probability mass function (pmf) over observed variables, using the conditionally expected information of latent variables. This definition provides us with a new pmf for which we can measure the dissimilarity from the original pmf. The dissimilarity between these two pmfs can be regarded as the dissimilarity between the latent and observed variables, since the defined pmf reflects the conditionally expected information of latent samples, while the original pmf reflects the self-information of observed samples. We applied LOD to four essential types of two-layer models: 1) a single-latent-variable model (SL), 2) a multi-latent-variable model whose latent variables are independent of each other (IL), 3) a multi-latent-variable model whose latent variables are conditionally independent given observed variables (CI), and 4) a multi-latent-variable model whose latent variables are independent of each other and conditionally independent given observed variables (ICI). These four types cover the major possible combinations of independence or conditional independence in two-layer models. In our experiments, LOD clearly reflected the difference between these four model types. LOD was also shown to reflect the latent layer’s capability for higher layer learning. Our experiments also revealed that the conditional independence of latent variables given observed variables, particularly for CI models, contributes to the improvement of higher layer learning, improving LOD and the mutual information between lower and higher layers.
Latent-observed dissimilarity {#sec_lod}
=============================
Definition of LOD {#ss_def_lod}
-----------------
Let $p_{\rm G}\lra{X,Y}$ denote the probability mass function (pmf) of a generative model where $X$ denotes observed variables and $Y$ denotes latent variables. When an observation $X$ is received, its self information under a model $p_{\rm G}$ is given as $-\log p_{\rm G}\lra{X}$. We first define the corresponding expected information for latent variables. Let $f\lra{X}$ denote the expected information of $Y$ given $X$, $$\begin{aligned}
f\lra{X}
& = E_{Y|X}\lrc{- \log {p_{\rm G}\lra{Y}}} \label{eq_fx_0} \\
& = - \textstyle\sum_Y p_{\rm G}\lra{Y|X}\log p_{\rm G}\lra{Y} \label{eq_fx}\end{aligned}$$ where $f\lra{X}$ may be said to be the expected surprise of the latent layer given $X$, while $-\log p\lra{X}$ is the surprise of the observed layer given $X$.
We then define a pmf $q\lra{X}$ based on $f\lra{X}$. To measure the distance between some pmf and $f\lra{X}$, preprocessing is necessary because the function $f\lra{X}$ is not guaranteed to be a pmf. Based on the fact that $f\lra{X}$ represents the expected information, we define the following pmf, $$\begin{gathered}
\label{eq_qx}
q\lra{X}
= \frac{\exp\lra{- f\lra{X}}}{C},\end{gathered}$$ where $C = \textstyle\sum_X \exp\lra{- f\lra{X}}$. Let $\tilde{p}\lra{X}$ denote a data distribution. That is, we assume $\frac{1}{T}\sum_{t=1}^T g\lra{X\lra{t}}=\sum_X \tilde{p}\lra{X}g\lra{X}$ for any function $g$. Using $q\lra{X}$, we define the dissimilarity between the observed and latent variables for a dataset using KL-divergence, $$\begin{aligned}
{\rm LOD}\lra{X,Y}
&= D\lra{\tilde{p}\lra{X}||q\lra{X}}. \label{eq_SH_0}
$$
Characteristics of LOD {#ss_char}
----------------------
#### Single variable example.
We now study the differences between LOD and mutual information using single variable examples.
The proposed measure, LOD, behaves differently from the mutual information of $X$ and $Y$. When the joint probability of $X$ and $Y$ is defined by $p_{\rm G}\lra{X,Y}$, the mutual information $I\lra{X;Y}$ between $X$ and $Y$ is $$\begin{aligned}
I \lra{X;Y}
& = \sum_X p_{\rm G}\lra{X} D_Y \lra{p_{\rm G}\lra{Y|X} || p_{\rm G}\lra{Y}}, \label{eq_MI_1}\end{aligned}$$ where $D$ denotes the Kullback-Leibler divergence. A more data-based evaluation is possible if the data distribution $\tilde{p}\lra{X}$ is employed $$\begin{gathered}
\label{eq_MI_2}
{\rm MI}\lra{X,Y}
= \textstyle\sum_X \tilde{p}(X) D_Y \lra{p_{\rm G}\lra{Y|X} \| p_{\rm G}\lra{Y}}.\end{gathered}$$ We also refer to MI as (data-based) mutual information. Consider the difference between LOD and MI in the simplest case. Consider a model consisting of a single observed variable and a single latent variable. Let $X\in\lrb{x_1,x_2,\ldots,x_6}$ and $Y\in\lrb{y_1,y_2,\ldots,y_3}$. Define the probabilities $\tilde{p} \lra{X=x_1}, \tilde{p} \lra{X=x_2}, \ldots, \tilde{p} \lra{X=x_6} $ as $1/21, 2/21, \ldots, 6/21$, respectively. For simplicity, we assume the mapping from $X$ to $Y$ to be deterministic, so each $p_{\rm G}\lra{Y|X}$ is either $0$ or $1$. From among all possible $p_{\rm G}\lra{Y|X}$ under this assumption, let $p_1\lra{Y,X}$ denote the one that realizes the best LOD, and let $p_2\lra{Y,X}$ denote the one that realizes the best MI. The joint and marginal probabilities of $p_1$ and $p_2$ as well as the transformed probabilities $q\lra{X}$ are shown in Table \[tb\_p1\_p2\_6\]. Note that since $p_1\lra{X}=p_2\lra{X}=\tilde{p}\lra{X}$ by assumption, the log likelihood is maximized for both $p_1$ and $p_2$, as $\sum_X \tilde{p}\lra{X}\log\tilde{p}\lra{X}
=\sum_X\tilde{p}\lra{X}\log p_1\lra{X}
=\sum_X\tilde{p}\lra{X}\log p_2\lra{X}$. The scores of LOD and MI are shown in Table \[tb\_p1\_p2\_score\_6\].
$p_1(X,Y)$ $x_1$ $x_2$ $x_3$ $x_4$ $x_5$ $x_6$ $p_1 (Y)$
------------ ------- ------- ------- ------- ------- ------- -----------
$y_1$ $a$ $2a$ $0$ $0$ $0$ $0$ $3a$
$y_2$ $0$ $0$ $3a$ $4a$ $0$ $0$ $7a$
$y_3$ $0$ $0$ $0$ $0$ $5a$ $6a$ $11a$
$p_1(X)$ $a$ $2a$ $3a$ $4a$ $5a$ $6a$
$q_1(X)$ $3b$ $3b$ $7b$ $7b$ $11b$ $11b$
: Joint and marginal probabilities of $p_1$ and $p_2$, and transformed probabilities $q\lra{X}$. Top: the best similarity assignment. Bottom: the best mutual information assignment. Note that $a=1/21$, $b=1/42$.[]{data-label="tb_p1_p2_6"}
$p_2(X,Y)$ $x_1$ $x_2$ $x_3$ $x_4$ $x_5$ $x_6$ $p_2 (Y)$
------------ ------- ------- ------- ------- ------- ------- -----------
$y_1$ $a$ $0$ $0$ $0$ $0$ $6a$ $7a$
$y_2$ $0$ $2a$ $0$ $0$ $5a$ $0$ $7a$
$y_3$ $0$ $0$ $3a$ $4a$ $0$ $0$ $7a$
$p_2(X)$ $a$ $2a$ $3a$ $4a$ $5a$ $6a$
$q_2(X)$ $7b$ $7b$ $7b$ $7b$ $7b$ $7b$
: Joint and marginal probabilities of $p_1$ and $p_2$, and transformed probabilities $q\lra{X}$. Top: the best similarity assignment. Bottom: the best mutual information assignment. Note that $a=1/21$, $b=1/42$.[]{data-label="tb_p1_p2_6"}
\[tb\_p1\_p2\_score\_6\]
$p_1$ $p_2$
----- ---------- ---------
LOD $0.0137$ $0.129$
MI $0.983$ $1.10$
: Scores for $p_1$ and $p_2$. LOD: smaller is better. MI: larger is better.
From these results, we can confirm the differences between the minimum LOD model and the minimum MI model. The model $p_1 \lra{X,Y}$ that minimizes ${\rm LOD}$ provides a $q_1\lra{X}$ that has a distribution similar to $\tilde{p}\lra{X}$. The model $q_2\lra{X}$ from $p_2$ that minimizes MI is far from similar, though the fact that MI is minimized in $p_2$ means that knowing $Y$ in the $p_2$ model reduces the uncertainty of $X$ more than in the $p_1$ model.
#### Sizes of latent/observed space.
The proposed dissimilarity measure LOD achieves zero when $-\log\tilde{p}\lra{X}=f\lra{X}$. However, there are other cases where LOD also achieves zero. An illustrative case is the *expanding* case where the size of the latent space in the model is an integer multiplication of the size of the observed space. Let $K_A$ denote the total number of states of observed variables, $K_A = \prod_i K_i$, and let $L_A$ denote the total number of states of latent variables, $L_A = \prod_j L_j$. Suppose $L_A=\alpha K_A$ by an integer $\alpha\ge 1$ and $p_{\rm G}\lra{X,Y}$ is defined as $$\begin{gathered}
p_{\rm G}\lra{Y=l|X=k}=
\begin{cases}
1/\alpha, & \text{if } \alpha\lra{k-1}+1\le l \le \alpha k .\\
0, & \text{otherwise}.
\end{cases}\nonumber\end{gathered}$$ This leads to $p_{\rm G}\lra{Y={\alpha\lra{k-1}+l}}=p\lra{X=k}/\alpha$ for $l=1,\ldots,\alpha$. In this case, $f\lra{X=k}=\log p_{\rm G}\lra{X=k} - \log \alpha$, and hence $q\lra{X=k}=p_{\rm G}\lra{X=k}$, yielding ${\rm LOD}=0$. The *shrinking* case, where $K_A=\beta L_A$ by an integer $\beta\ge 1$, is also possible, which we shall omit the explanation. The *expanding*/*shrinking* cases show an invarance aspect of LOD, which imply the potential advantage of LOD as an optimization criterion for the expansion and reduction of latent representation spaces.
Models {#sec_models}
======
In this section, the model types used in our experiments (Section \[sec\_exp\]) are defined. These model types differ in the independence or conditional independence of their latent variables. By comparing these models in our experiments, we hope to determine which configurations affect the relationships between observed, latent, and higher latent variables.
We consider the unsupervised learning of two-layer generative models with four different configurations of latent and observed variables. One of the layers is of observed, or manifest variables, $X$, and the other is of latent, or hidden variables, $Y$. The stochastic variables $X$ and $Y$ are assumed to be finite and discrete, and $X$ and $Y$ may consist of multiple variables. Let $N_x$ be the number of observation variables and $N_y$ be that of latent variables. In addition, let $K_i$, $i=1,\ldots,N_x$ be the number of states $X_i$ can take and $L_j$, $j=1,\ldots,N_y$ be the number that $Y_j$ can take. We denote a model probability by $p_{\rm G}\lra{X,Y}$.
The models and satisfied constraints are summarized in Table \[tb\_model\_constraint\].
Constraint SL IL CI ICI
----------------------------------------- ---- ---- ---- -----
$p\lra{X|Y}=\textstyle\prod_i p(X_i|Y)$
$p\lra{Y|X}=\textstyle\prod_j p(Y_j|X)$ () -
$p\lra{Y}=\textstyle\prod_j p(Y_j)$ () -
: Models and satisfied constraints.[]{data-label="tb_model_constraint"}
-0.1in
#### Single-label models (SL).
The most simple of these models has a single latent variable where each observed variable is conditioned only by the latent variable. A Bayesian network representation of this model is shown in . This model is a type of mixture model and is called a latent class or naive Bayes model in different contexts. The model assumes the conditional independence of $X$ given $Y$, $$\begin{gathered}
\label{eq_x_ci}
p_{\rm G}\lra{X|Y} = \textstyle\prod_{i=1}^{N_x} p(X_i|Y) .\end{gathered}$$ The joint probability of the model is $$\begin{gathered}
\label{eq_lcm}
p\lra{X,Y} = \lrb{ \textstyle\prod_i p_{\rm G}\lra{X_i|Y} } p_{\rm G}\lra{Y}.
$$ We define each conditional probability by a conditional probability table, $$\begin{gathered}
\label{eq_cpt_xoz}
p_{\rm G}(X_i=x|Y=y) = \Theta_{i,x}^y.\end{gathered}$$ We call this model the single label model (SL). If $L$, the number of values $Y$ can take, is sufficiently large, say $L \ge \textstyle\prod_i K_i$, then the model can realize any $p\lra{X}$.
#### Independent label models (IL).
There are several ways to add more latent variables to single-label models. One is to add latent variables as indicated in . Though the extension seems simple and straightforward in the graphical representation, the graph indicates the additional assumption that $Y$ is independent, that is, $p_{\rm G}\lra{Y}=\prod_j p_{\rm G}\lra{Y_j}$. The joint probability is thus $$\begin{gathered}
\label{eq_mllcm}
p_{\rm G}(X,Y) = \lrba{ \textstyle\prod_{i=1}^{N_x} p_{\rm G}(X_i|Y) } \lrba{ \textstyle\prod_{j=1}^{N_y} p_{\rm G}(Y_j)}.
$$ Models in this form have been proposed in different contexts, including the probabilistic formulation of the quick medical reference network (QMR-DT) [@Shwe1990UAI_QMR; @Jaakkola1999JAIR_variational_QMR], and the partially observed bipartite network (POBN) used for the analysis of transcriptional regulatory networks [@Alvarez2011BMC_POBN]. These models usually further restrict the form of probability. In this paper, however, we do not restrict $p_{\rm G}(X_i|Y)$ and $p_{\rm G}(Y_j)$ to some specific form. We define each conditional probability by a conditional probability table, $p_{\rm G}(X_i=x|Y_j=y) = \Theta_{i,x}^{j,y}$, and $p(Y_j)$ is defined as $p(Y_j=y)=\Phi_{j,y}$. We call this model the independent label model (IL).
![Bayesian network representations of single label and independent label models.[]{data-label="fig_lcms"}](fig_lcm "fig:"){width="35mm"} \[fig\_lcm\]
![Bayesian network representations of single label and independent label models.[]{data-label="fig_lcms"}](fig_mllcm "fig:"){width="35mm"} \[fig\_mllcm\]
#### Conditionally independent label models (CI).
If independence is not assumed on multiple latent variables, a model takes the form shown in . However, since $Z$ is latent and unsupervised learning is assumed, models of this form are just equivalent to “large” single-label models. A possible constraint other than independence is conditional independence of $Z$ given $X$. $$\begin{aligned}
p_{\rm G}(X,Y)
& = \lrba{ \textstyle\prod_i p_{\rm G}(X_i|Y) } p(Y) \label{eq_holo_01} \\
& = \lrba{ \textstyle\prod_j p_{\rm G}(Y_j|X) } p(X). \label{eq_holo_02}
$$ That is, latent variables are conditionally independent given observed variables, while observed variables are conditionally independent given latent variables. These two kinds of conditional independence are impossible to capture in a single Bayesian network representation; two Bayesian networks are necessary to illustrate two-way conditional independence. illustrates conditional independence in a generative model and illustrates it in a recognition model. We call this model a conditionally independent label model (CI).
![Two Bayesian network representations of a single probability model.[]{data-label="fig_ci"}](fig_ci_dec "fig:"){width="35mm"} \[fig\_ci\_dec\]
![Two Bayesian network representations of a single probability model.[]{data-label="fig_ci"}](fig_ci_enc "fig:"){width="35mm"} \[fig\_ci\_enc\]
Joint probabilities satisfying these two-way constraints do exist. An example class is that of the restricted Boltzmann machines (RBMs) [@Smolensky1986Book_harmony_theory; @Hinton2002NeuralComputation_PoE_CD]. In an RBM, a joint probability of $X$ and $Y$ is defined as $p_{\rm G}\lra{X,Y} = \frac{1}{D}
\exp
\lraa{
-\myvect{a}X - \myvect{b}Y- Y^{\rm T}\myvec{W}X
}$, where $D$ is the normalizing constant. This is often called a partition function. Constraints and are consistently satisfied by RBMs. If the generative part of a model is defined in the most general form, that is, if it is parameterized as $p_{\rm G}\lra{X_i=x_i|Y=y}=\Theta_{i,x_i}^{y}$ and $p_{\rm G}\lra{Y=y}=\Phi_y$, the parameters $\Theta_{i,x_i}^{y}$ and $\Phi_y$ should be constrained to satisfy the recognition conditional independence . It is almost impossible to solve such constraints analytically; however, a numerical, and perhaps approximate, satisfaction of the constraints is possible through the framework of (stochastic) Helmholtz machines (HMs) and the wake-sleep algorithm [@Hinton1995Science_wake-sleep; @Dayan1995NeuralComp_Helmholtz-Machine; @Dayan1996NN_helmholtz-machine].
#### Independent and conditionally independent label models
If the independence and conditional independence constraints are assumed simultaneously, the model satisfies $$\begin{aligned}
p_{\rm G}(X,Y)
& = \lrba{ \textstyle\prod_i p(X_i|Y) } \lrba{ \textstyle\prod_j p(Y_j) } \label{eq_ici_01} \\
& = \lrba{ \textstyle\prod_j p(Y_j|X) } p(X). \label{eq_ici_02}\end{aligned}$$ We call this model the independent and conditionally independent label model (ICI). Learning and (approximate) realization of this class of models are also possible using the wake-sleep algorithm.
If $X$ and $Y$ are continuous and linearly mapped each other, i.e., $X=\myvec{A}Y$ and $Y=\myvec{W}X$ where $\myvec{A}$ and $\myvec{W}$ are matrices, the model represents independent component analysis (ICA) [@Obradovic1998NeuralComputation_infomax_ica]. In ICA, only $\myvec{W}$ is learned using some independence criterion. The relationship between ICA and Helmholtz machines has been investigated in, for example, [@Xu1998Neurocomputing_Ying-Yang] and [@Ohata2003ICA_ICA_Helmholtz].
Experiments {#sec_exp}
===========
Two-layer models {#ss_exp_twolayer}
----------------
We first considered the two-layer models described in Section \[sec\_models\]. The models were trained on patches from images in the MNIST handwritten digits database[@LeCun_MNIST].
#### Experimental settings.
We preprocessed the images by quantizing them to three levels per pixel. From each 28 $\times$ 28 pixel image, a 2 $\times$ 2 pixel image patch was taken from a fixed location. Thus, $N_x=4$, $K_1=\cdots=K_4=3$ for observed variable $X$. We used all the training samples in the database, so the number of samples $T$ was 60000. Thus, a patch set consisted of 60000 samples of four observed variables, where each variable is a “trit” (i.e., takes one of three values). Eight non-overlapping locations were employed to yield eight such patch sets. To avoid the local minimum problem, twenty trials were made for each patch set, changing the initial parameters for the EM and the wake-sleep algorithm, and the trial with the best log likelihood $(1/T)\sum_{t=1}^T \log p\lra{X\lra{t}}$ was chosen for each patch set.
The four kinds of models described in Section \[sec\_models\] were tested. For the IL, CI, and ICI models, $L_j$ $(j=1,\ldots,N_y)$ were fixed to two, and $N_y$ was varied from one to six. For the SL model, $L$, the number of values $Y$ could take were $2^1, 2^2, \ldots, 2^6$. The SL and IL models were trained using the EM algorithm, while the CI and ICI models were trained using the wake-sleep algorithm.
After learning, we evaluated the learned models using the following quantities: a) log likelihood $(1/T)\sum_{t=1}^T \log p_{\rm G}\lra{X\lra{t}}$, b) data-based mutual information MI , c) the proposed dissimilarity measure LOD . To remove any large deviation caused by different patch sets, an offset removal procedure was performed as follows. Let $V\lra{m,s,n}$ denote the raw evaluation values, where $m$ denotes the model, $s$ denotes model size, and $n$ denotes patch set number. 1) The average of values of the smallest model in the series was measured over the patch sets, $V_a\lra{m} = \frac{1}{N} \sum_n V\lra{m,1,n}$. 2) From the evaluated values of a patch set model, the value of the smallest model was subtracted, $V_r\lra{m,s,n} = V\lra{m,s,n} - V\lra{m,1,n}$. 3) The average $V_a$ was then added back to $V_r$, $V_q\lra{m,s,n} = V_a\lra{m} + V_r\lra{m,s,n}$. Means and standard deviations were calculated for $V_q$ using $\frac{1}{N}\sum_n V_q\lra{m,s,n}$ and plotted.
{width="0.9\columnwidth"} \[fig\_aa\_sh\]
{width="0.9\columnwidth"} \[fig\_aa\_mi\]
{width="0.9\columnwidth"} \[fig\_aa\_ll\]
#### Results: LOD.
shows LOD scores for the tested models. CI has a lower LOD than the other models for $N_y\ge 3$. The graphs are, as a whole, decreasing for $N_y$, but monotonic decrease holds only for CI. For $N_y\ge4$, four types kept the order of ${\rm CI}<{\rm SL}<{\rm ICI}<{\rm IL}$. This suggests that the conditional independence of latent variables given observed variables improves LOD because the essential difference between CI and SL as well as between ICI and IL is the conditional independence. Compared to MI and log likelihood, LOD clearly captured the difference between model types. The difference between LOD and log likelihood () indicates that the minimization of LOD may lead to a model different from the maximum log likelihood model. The incorporation of LOD into log likelihood as a regularization may also be a future topic of discussion.
#### Results: Mutual Information.
shows the mutual information between the latent and observed variables for the tested models. All models show a monotonic increase of mutual information for $N_y$. For $N_y\ge4$, models appear to form two groups: CI and ICI, and SL and IL. The CI-ICI group took larger values than the SL-IL group, and in the CI-ICI group, CI was larger. These phenomena can be explained as follows. First, the conditional independence of the latent variables contributed to a larger MI. Secondly, the independence assumption on latent variables did not affect MI as much as it affected LOD. In fact, recalling the equivalence of ICA and mutual information maximization [@Obradovic1998NeuralComputation_infomax_ica], the independence assumption probably does not disturb the increase of mutual information.
#### Results: log likelihood.
shows the log likelihood of the tested models. All models showed almost equally high likelihood for the same model size; of these, ICI had a slightly lower value. This is because ICI is the most restricted model among these four types and the log likelihood was the objective of the optimization. For LOD, MI, and log likelihood, CI almost always yielded the best results. This supports the incorporation of conditional independence into models to improve the information transmission from the observed to latent variables without penalizing the log likelihood too much.
Learning of the higher (third) Layer {#ss_higher}
------------------------------------
Next, we performed learning of SL models on top of the two-layer models learned in \[ss\_exp\_twolayer\], and evaluated how the characteristics of the lower layers are preserved or reflected in the higher layers.
#### Learning and evaluation procedures.
Let us refer to the two-layer models learned in \[ss\_exp\_twolayer\] as the “lower” models, and denote their probability as $p_{\rm L}\lra{X,Y}$. After learning these lower models, a learning process similar to greedy layer-wise learning in deep belief networks [@Hinton2006NeuralComputation_DBN] was carried out. We applied each model’s posterior distribution $p_{\rm L}\lra{Y|X}$ to the dataset used in \[ss\_exp\_twolayer\] to derive $\tilde{p}\lra{Y}
:=\sum_X \tilde{p}\lra{X}p_{\rm L}\lra{Y|X}
=(1/T)\sum_t p_{\rm L}\lra{Y|X=\myvec{x}\lra{t}}$. For the derived $\tilde{p}\lra{Y}$ of each model, we learned a “higher” SL model, $p_{\rm H}\lra{Y,Z}=\lraa{\prod_j p_{\rm H}\lra{Y_j|Z}}p_{\rm H}\lra{Z}$, to maximize $\sum_Y \tilde{p}\lra{Y}\log p_{\rm H}\lra{Y}$, where $Z$ denotes a set of the third layer latent variables. The learning of $p_{\rm H}\lra{Y,Z}$ based on $\tilde{p}\lra{Y}$ is essentially equivalent to the learning based on the samples $Y$ from $p_{\rm L}\lra{Y|X}$ for the dataset; however, as model sizes are assumed to be small and tractability is ensured, we can directly store and calculate $\tilde{p}\lra{Y}$ and do not need the actual samples from $p_{\rm L}\lra{Y|X}$.
The learning procedure yields the higher two-layer SL models $p_{\rm H}\lra{Y,Z}$ on top of the lower two-layer model $p_{\rm L}\lra{X,Y}$. We evaluated the correlations between the lower model score $S\lra{X,Y}$ for $p_{\rm L}\lra{X,Y}$ and the connected model score $S\lra{X,Z}$ for $p_{\rm C}\lra{X,Y,Z}$, where the score was either LOD or MI. The probability of a connected model $p_{\rm C}$ is defined by $$\begin{gathered}
\label{eq_p_connected}
p_{\rm C}\lra{X,Y,Z}
= p_{\rm H}\lra{Z|Y}p_{\rm L}\lra{Y|X}\tilde{p}\lra{X}.\end{gathered}$$ In , the lower and higher models are used as encoders, because here we are focusing on how the higher layers preserve the characteristics of the lower layers and not on the generative properties of the models. In the four lower model types, higher model learning is impossible for the lower SL models as they are, since SL models only have a single latent variable. To make the learning of higher models possible, the lower SL models were converted into multiple latent variable models as follows. For the models whose number of states of $Y$ was $2^{m}$, a corresponding model with $m$ binary latent variables as in Figure \[fig\_ci\_dec\] was defined. Let $Y'=\lraa{Y'_1,\ldots,Y'_{m}}$ denote its latent variables. The states of $Y$ can be mapped to the states of $Y'$ in a bijective (one-to-one and onto) manner. Once such a bijection is determined, the $m$-latent variables model and the SL model are equivalent as generative models for $X$. To determine a bijection for each lower model, we first prepared twenty random bijections as the candidates. For each bijection, learning a higher SL model with a single binary latent variable was performed, and the bijection yielding the largest mean log likelihood $\sum_Y \tilde{p}\lra{Y} \log p_{\rm H}\lra{Y}$ was selected from the twenty candidates.
#### Experimental settings.
The experiment was configured as follows. The number of datasets was eight, as in \[ss\_exp\_twolayer\], and the lower models with $N_y=3,4,5,6$ were used. For each lower model, SL models with $K_z=2,3,\ldots,2^{N_y-2}$ were learned. The number of the models used was thus $8\times \lra{1+3+7+15} = 208$ for each lower model type (SL, IL, CI, and ICI). Higher SL models were learned using the EM algorithm, which we ran twenty times with different initial values, picking the run that gave the best log likelihood. For the lower and higher models, LOD and mutual information were evaluated using for between $X$-$Y$ and $X$-$Z$.
#### Results.
Figure \[fig\_lod1\_lod3\] shows the relations between ${\rm LOD}\lra{X,Y}$ and ${\rm LOD}\lra{X,Z}$. Figure \[fig\_mi1\_mi3\] shows the relations between ${\rm MI}\lra{X,Y}$ and ${\rm MI}\lra{X,Z}$. Their correlations are shown in Table \[tb\_corr\_s1\_s3\].
In Figures \[fig\_lod1\_lod3\] and \[fig\_mi1\_mi3\], CI models achieved the lowest $X$-$Z$ dissimilarity and the highest $X$-$Z$ mutual information among the four model types. This indicates that latent variables encoded by CI models keep more aspects of the information of the observed variables than the other model types do. From Table \[tb\_corr\_s1\_s3\], the CI models had larger correlation coefficients than those of SL models for both LOD and MI. This relationship was also true for the ICI models and IL models. The capability of $Y$ to provide information to $Z$ was improved by the incorporation of the conditional independence of the latent variables given observed variables.
LOD for $\lra{X,Y}$ and $\lra{X,Z}$ showed significant $\lra{p<0.05}$ correlations for all of the four model types, whereas MI showed significant correlations only for CI and ICI models. These results indicates that, along with dissimilarity itself, LOD also represents how well similarity can be transmitted to the higher layer, whereas MI does not necessarily represent such a capability of transmission.
{width="0.9\columnwidth"} \[fig\_lod1\_lod3\]
{width="0.9\columnwidth"} \[fig\_mi1\_mi3\]
0.15in
------- --------- ---------- ----------- ----------
MODEL $r$ $p$ $r$ $p$
(SL) $0.157$ $0.037$ $-0.0664$ $0.381$
(IL) $0.373$ $<0.001$ $-0.0586$ $0.440$
(CI) $0.410$ $<0.001$ $0.510$ $<0.001$
(ICI) $0.602$ $<0.001$ $0.156$ $0.039$
------- --------- ---------- ----------- ----------
: Correlations between ${\rm Score}\lra{X,Y}$ and ${\rm Score}\lra{X,Z}$. Score is either LOD or MI. $r$ means the correlation coefficient, and $p$ means the p-value. See Figures \[fig\_lod1\_lod3\] and \[fig\_mi1\_mi3\] for the source data.[]{data-label="tb_corr_s1_s3"}
Conclusions {#sec_conclu}
===========
We proposed latent-observed dissimilarity (LOD), a dissimilarity measure between latent and observed variables in generative models, to evaluate the relationships between latent and observed variables. LOD compares the self-information of an observation with the expected information of a latent layer given that observation. We numerically evaluated four types of two-layer models (SL, IL, CI, and ICI) using log likelihood, mutual information, and LOD. The results suggested an advantage of using LOD as a measure for multi-layer learning; the LOD between observed and latent variables had significant correlation with the LOD between observed and higher layer latent variables for all four types of models, while mutual information had significant correlation only for CI models. The results also suggested the conditional independence of latent variables given observed variables facilitates the transmission of a layer’s characteristics to the higher layers. This fact sheds new light on the advantages of conditional independence, of which usually only its computational advantage is emphasized.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work was supported by MEXT KAKENHI Grant Number 23240019.
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[^1]: Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo
|
---
abstract: 'We study the eigenvalue distributions of the Conjugate Kernel and Neural Tangent Kernel associated to multi-layer feedforward neural networks. In an asymptotic regime where network width is increasing linearly in sample size, under random initialization of the weights, and for input samples satisfying a notion of approximate pairwise orthogonality, we show that the eigenvalue distributions of the CK and NTK converge to deterministic limits. The limit for the CK is described by iterating the Marcenko-Pastur map across the hidden layers. The limit for the NTK is equivalent to that of a linear combination of the CK matrices across layers, and may be described by recursive fixed-point equations that extend this Marcenko-Pastur map. We demonstrate the agreement of these asymptotic predictions with the observed spectra for both synthetic and CIFAR-10 training data, and we perform a small simulation to investigate the evolutions of these spectra over training.'
author:
- |
Zhou Fan\
Department of Statistics and Data Science\
Yale University\
`[email protected]`
- |
Zhichao Wang\
Department of Mathematics\
University of California, San Diego\
`[email protected]`
bibliography:
- 'NTKspectrum.bib'
title: 'Spectra of the Conjugate Kernel and Neural Tangent Kernel for Linear-Width Neural Networks'
---
Introduction {#sec:introduction}
============
Recent progress in our theoretical understanding of neural networks has connected their training and generalization to two associated kernel matrices. The first is the *Conjugate Kernel (CK)* or the equivalent Gaussian process kernel [@neal1995bayesian; @williams1997computing; @cho2009kernel; @daniely2016toward; @poole2016exponential; @schoenholz2017deep; @lee2018deep]. This is the gram matrix of the derived features produced by the final hidden layer of the network. The network predictions are linear in these derived features, and the CK governs training and generalization in this linear model.
The second is the *Neural Tangent Kernel (NTK)* [@jacot2018neural; @du2019gradienta; @allen2019convergence]. This is the gram matrix of the Jacobian of in-sample predictions with respect to the network weights, and was introduced to study full network training. Under gradient-flow training dynamics, the in-sample predictions follow a differential equation governed by the NTK. We provide a brief review of these matrices in Section \[sec:model\].
The spectral decompositions of these kernel matrices are related to training and generalization properties of the underlying network. Training occurs most rapidly along the eigenvectors of the largest eigenvalues [@advani2017high], and the eigenvalue distribution may determine the trainability of the model and the extent of implicit bias towards simpler functions [@xiao2019disentangling; @yang2019fine]. It is thus of interest to understand the spectral properties of these matrices, both at random initialization and over the course of training.
Summary of contributions
------------------------
In this work, we apply techniques of random matrix theory to derive an exact asymptotic characterization of the eigenvalue distributions of the CK and NTK at random initialization, in a multi-layer feedforward network architecture. We study a “linear-width” asymptotic regime, where each hidden layer has width proportional to the training sample size. We impose an assumption of approximate pairwise orthogonality for the training samples, which encompasses general settings of independent samples that need not have independent entries.
We show that the eigenvalue distributions for both the CK and the NTK converge to deterministic limits, depending on the eigenvalue distribution of the training data. The limit distribution for the CK at each intermediate hidden layer is a Marcenko-Pastur map of a linear transformation of that of the previous layer. The limit for the NTK may be described by a recursively defined sequence of fixed-point equations that extend this Marcenko-Pastur map.
We demonstrate the agreement of these asymptotic limits with the observed spectra on both synthetic and CIFAR-10 training data of moderate size. We conclude by examining empirically the evolutions of these spectra during training, on simple examples of learning a single neuron and learning a binary classifier for two classes in CIFAR-10. In these examples, the bulk eigenvalue distributions of the CK and NTK undergo small elongations, and isolated principal components emerge that are highly predictive of the training labels.
Related literature
------------------
Under linear-width asymptotics, the limit CK spectrum for one hidden layer was characterized in [@pennington2017nonlinear] for training data with i.i.d. Gaussian entries. For activations satisfying ${\mathbb{E}}_{\xi \sim {\mathcal{N}}(0,1)}[\sigma'(\xi)]=0$, [@pennington2017nonlinear] conjectured that this limit is a Marcenko-Pastur law also in multi-layer networks, and this was proven under a more general subgaussian assumption in [@benigni2019eigenvalue]. [@louart2018random] studied the one-hidden-layer CK with general training data, and [@liao2018spectrum] specialized this to Gaussian mixture models. These works [@louart2018random; @liao2018spectrum] showed that the limit spectrum is a Marcenko-Pastur map of the inter-neuron covariance. We build on this insight by analyzing this covariance across multiple layers, under approximate orthogonality of the training samples. This orthogonality condition is similar to that of [@adlam2019random], which recently studied the one-hidden-layer CK with a bias term. This condition is also more general than the assumption of i.i.d. entries, and we describe in Appendix \[appendix:penningtonreduction\] the reduction to the one-hidden-layer result of [@pennington2017nonlinear], as this reduction is not immediately clear.
We believe that our characterization of the limit NTK spectrum is new in the linear-width regime even for one hidden layer. The equivalent spectrum of the covariance matrix $JJ^\top$, which is one of two components of the Hessian of the training loss, was studied for one hidden layer in [@pennington2017geometry; @pennington2018spectrum] in a slightly different setting. [@pennington2017geometry; @pennington2018spectrum] considered an output dimension that is also proportional to $n$, and [@pennington2018spectrum] further studied the expectation of $JJ^\top$ over the input samples $X$, rather than $JJ^\top$ itself.
The spectrum of a gram matrix $X^\top X$ is equivalent (up to the addition/removal of 0’s) to $XX^\top$, which is the sample covariance matrix for linear regression using the features $X$. As recognized in [@dicker2016ridge; @pennington2017nonlinear; @louart2018random], its Stieltjes transform is directly related to the in-sample training error of ridge regression using $X$. Thus our results have direct bearing on the training error for random features regression using the derived features of the final layer or of the Jacobian $J=\nabla_\theta f_\theta(X)$. Analysis of generalization error uses related techniques but is more involved, as this requires understanding the joint spectral limit of $XX^\top$ with its expectation [@dobriban2018high]. This was carried out for the one-hidden-layer CK in [@hastie2019surprises; @mei2019generalization], for inputs with i.i.d. Gaussian entries or with uniform distribution on the sphere.
Many properties of the CK and NTK have been established in the limit of infinite width and fixed sample size $n$. In this limit, both the CK [@neal1995bayesian; @williams1997computing; @daniely2016toward; @lee2018deep] and the NTK [@jacot2018neural; @lee2019wide; @yang2019scaling] at random initialization converge to fixed $n \times n$ kernel matrices. The associated random features regression models converge to kernel linear regression in the RKHS of these limit kernels. Furthermore, network training occurs in a “lazy” regime [@chizat2019lazy], where the NTK remains constant throughout training [@jacot2018neural; @du2019gradienta; @du2019gradientb; @allen2019convergence; @lee2019wide; @arora2019exact]. Spectral properties of the CK, NTK, and Hessian of the training loss have been previously studied in this infinite-width limit in [@poole2016exponential; @sagun2018empirical; @xiao2019disentangling; @karakida2019universal; @geiger2019jamming; @jacot2019asymptotic]. Limitations of lazy training and these equivalent kernel regression models have been studied theoretically and empirically in [@chizat2019lazy; @arora2019exact; @yehudai2019power; @ghorbani2019limitations; @ghorbani2019linearized; @liang2019risk], suggesting that trained neural networks of practical width are not fully described by this type of infinite-width kernel equivalence. The asymptotic behavior is different in the linear-width regime that we study in our work: For example, for the simple linear activation $\sigma(x)=x$, the infinite-width limit of the CK at random initialization is the input Gram matrix $X^\top X$, whereas its limit spectrum under linear-width asymptotics has an additional noise component from iterating the Marcenko-Pastur map.
In the linear-width regime, the CK and NTK are expected to evolve over training, as feature learning is expected to occur. Our results characterize these spectra only at random initialization of the weights. Recent work has studied the evolution of the NTK in an entrywise sense [@huang2019dynamics; @dyer2019asymptotics], and we believe it is an interesting open question to translate this understanding to a more spectral perspective.
Background {#sec:background}
==========
Neural network model and kernel matrices {#sec:model}
----------------------------------------
We consider a fully-connected, feedforward neural network with input dimension $d_0$, hidden layers of dimensions $d_1,\ldots,d_L$, and a scalar output. For an input ${\mathbf{x}}\in {\mathbb{R}}^{d_0}$, we parametrize the network as $$\label{eq:NNfunc}
f_\theta({\mathbf{x}})={\mathbf{w}}^\top \frac{1}{\sqrt{d_L}} \sigma\bigg(
W_L \frac{1}{\sqrt{d_{L-1}}}\sigma \Big(\ldots
\frac{1}{\sqrt{d_2}}\sigma\Big(W_2\frac{1}{\sqrt{d_1}} \sigma(W_1 {\mathbf{x}})
\Big)\Big)\bigg) \in {\mathbb{R}}.$$ Here, $\sigma:{\mathbb{R}}\to {\mathbb{R}}$ is the activation function (applied entrywise) and $$W_\ell \in {\mathbb{R}}^{d_\ell \times d_{\ell-1}} \quad
\text{ for } 1 \leq \ell \leq L, \qquad
{\mathbf{w}}\in {\mathbb{R}}^{d_L}$$ are the network weights. We denote by $\theta=(W_1,\ldots,W_L,{\mathbf{w}})$ the weights across all layers. The scalings by $1/\sqrt{d_\ell}$ reflect the “NTK-parametrization” of the network [@jacot2018neural]. We discuss alternative scalings and an extension to multi-dimensional outputs in Section \[sec:extensions\].
Given $n$ training samples ${\mathbf{x}}_1,\ldots,{\mathbf{x}}_n \in {\mathbb{R}}^{d_0}$, we denote the matrices of inputs and post-activations by $$X \equiv X_0
=\begin{pmatrix} {\mathbf{x}}_1 & \dots & {\mathbf{x}}_n \end{pmatrix} \in {\mathbb{R}}^{d_0 \times n},
\quad X_\ell=\frac{1}{\sqrt{d_\ell}}
\sigma\left(W_\ell X_{\ell-1}\right) \in
{\mathbb{R}}^{d_\ell \times n}\quad \text{ for } 1\leq\ell\leq L.$$ Then the in-sample predictions of the network are given by $f_\theta(X)=(f_\theta({\mathbf{x}}_1),\ldots,f_\theta({\mathbf{x}}_n))
={\mathbf{w}}^\top X_L \in {\mathbb{R}}^{1 \times n}$. The [**Conjugate Kernel (CK)**]{} is the matrix $$K^{\text{CK}}=X_L^\top X_L \in {\mathbb{R}}^{n \times n}.$$ More generally, we will call $X_\ell^\top X_\ell$ the conjugate kernel at the intermediate layer $\ell$. Fixing the matrix $X_L$, the CK governs training and generalization in the linear regression model ${\mathbf{y}}={\mathbf{w}}^\top X_L$. For very wide networks, $K^{\text{CK}}$ may be viewed as an approximation of its infinite-width limit,[^1] and regression using $X_L$ is an approximation of regression in the RKHS defined by this limit kernel [@rahimi2008random].
We denote the Jacobian matrix of the network predictions with respect to the weights $\theta$ as $$J=\nabla_\theta f_\theta(X)=
\begin{pmatrix} \nabla_\theta f({\mathbf{x}}_1) & \cdots & \nabla_\theta f({\mathbf{x}}_n)
\end{pmatrix} \in {\mathbb{R}}^{\dim(\theta) \times n}.$$ The [**Neural Tangent Kernel (NTK)**]{} is the matrix $$\label{eq:NTK}
K^{{\text{NTK}}}=J^\top J=\big(\nabla_\theta f_\theta(X)\big)^\top
\big(\nabla_\theta f_\theta(X)\big) \in {\mathbb{R}}^{n \times n}.$$ Under gradient-flow training of the network weights $\theta$ with training loss $\|{\mathbf{y}}-f_\theta(X)\|^2/2$, the time evolutions of residual errors and in-sample predictions are given by $$\label{eq:training}
\frac{d}{dt}\Big({\mathbf{y}}-f_{\theta(t)}(X)\Big)=-K^{\text{NTK}}(t) \cdot
\Big({\mathbf{y}}-f_{\theta(t)}(X)\Big), \quad
\frac{d}{dt}f_{\theta(t)}(X)
=K^{\text{NTK}}(t) \cdot \Big({\mathbf{y}}-f_{\theta(t)}(X)\Big)$$ where $\theta(t)$ and $K^{\text{NTK}}(t)$ are the parameters and NTK at training time $t$ [@jacot2018neural; @du2019gradienta]. Denoting the eigenvalues and eigenvectors of $K^{\text{NTK}}(t)$ by $(\lambda_{\alpha}(t),{\mathbf{v}}_{\alpha}(t))_{{\alpha}=1}^n$, and the spectral components of the residual error by $r_{\alpha}(t)={\mathbf{v}}_{\alpha}(t)^\top ({\mathbf{y}}-f_{\theta(t)}(X))$, these training dynamics are expressed spectrally as $${\mathbf{v}}_{\alpha}(t)^\top \frac{d}{dt}
\Big({\mathbf{y}}-f_{\theta(t)}(X)\Big)=-\lambda_{\alpha}(t)r_{\alpha}(t), \qquad
\frac{d}{dt}f_{\theta(t)}(X)=
\sum_{{\alpha}=1}^n \lambda_{\alpha}(t)r_{\alpha}(t) \cdot {\mathbf{v}}_{\alpha}(t).$$ Hence, $\lambda_{\alpha}(t)$ controls the instantaneous rate of decay of the residual error in the direction of ${\mathbf{v}}_{\alpha}(t)$. For very wide networks, $K^{\text{NTK}}$, $\lambda_{\alpha}$, and ${\mathbf{v}}_{\alpha}$ are all approximately constant over the entirety of training [@jacot2018neural; @du2019gradienta; @du2019gradientb; @allen2019convergence; @chizat2019lazy]. This yields the closed-form solution $r_{\alpha}(t) \approx r_{\alpha}(0)e^{-t \lambda_{\alpha}}$, so that the in-sample predictions $f_{\theta(t)}(X)$ converge exponentially fast to the observed training labels ${\mathbf{y}}$, with a different exponential rate $\lambda_{\alpha}$ along each eigenvector ${\mathbf{v}}_{\alpha}$ of $K^{\text{NTK}}$.
Eigenvalue distributions, Stieltjes transforms, and the Marcenko-Pastur map {#sec:stieltjes}
---------------------------------------------------------------------------
We will derive almost-sure weak limits for the empirical eigenvalue distributions of random symmetric kernel matrices $K \in {\mathbb{R}}^{n \times n}$ as $n \to \infty$. Throughout this paper, we will denote this as $${\operatorname{lim\;spec}}K=\mu$$ where $\mu$ is the limit probability distribution on ${\mathbb{R}}$. Letting $\{\lambda_{\alpha}\}_{{\alpha}=1}^n$ be the eigenvalues of $K$, this means $$\label{eq:weakconverge}
\frac{1}{n}\sum_{{\alpha}=1}^n f(\lambda_{\alpha}) \to {\mathbb{E}}_{\lambda \sim \mu} [f(\lambda)]$$ a.s. as $n \to \infty$, for any continuous bounded function $f:{\mathbb{R}}\to {\mathbb{R}}$. Intuitively, this may be understood as the convergence of the “bulk” of the eigenvalue distribution of $K$.[^2] We will also show that $\|K\| \leq C$ a.s., for a constant $C>0$ and all large $n$. Then (\[eq:weakconverge\]) in fact holds for any continuous function $f:{\mathbb{R}}\to {\mathbb{R}}$, as such a function must be bounded on $[-C,C]$.
We will characterize the probability distribution $\mu$ and the empirical eigenvalue distribution of $K$ by their Stieltjes transforms. These are defined, respectively, for a spectral argument $z \in {\mathbb{C}}^+$ as[^3] $$m_\mu(z)=\int \frac{1}{x-z}d\mu(x), \qquad
m_K(z)=\frac{1}{n}\sum_{{\alpha}=1}^n \frac{1}{\lambda_{\alpha}-z}
=\frac{1}{n}{\operatorname{Tr}}(K-z{\operatorname{Id}})^{-1}.$$ The pointwise convergence $m_K(z) \to m_\mu(z)$ a.s. over $z \in {\mathbb{C}}^+$ implies ${\operatorname{lim\;spec}}K=\mu$. For $z=x+i\eta \in {\mathbb{C}}^+$, the value $\pi^{-1}{\operatorname{Im}}m_\mu(z)$ is the density function of the convolution of $\mu$ with the distribution $\operatorname{Cauchy}(0,\eta)$ at $x \in {\mathbb{R}}$. Hence, the function $m_\mu(z)$ uniquely defines $\mu$, and evaluating $\pi^{-1}{\operatorname{Im}}m_\mu(x+i\eta)$ for small $\eta>0$ yields an approximation for the density of $\mu$.
An example of this type of characterization is given by the *Marcenko-Pastur map*, which describes the spectra of sample covariance matrices [@marchenko1967distribution]: Let $X \in {\mathbb{R}}^{d \times n}$ have i.i.d. ${\mathcal{N}}(0,1/d)$ entries, let $\Phi \in {\mathbb{R}}^{n \times n}$ be positive semi-definite, and let $n \to \infty$ such that ${\operatorname{lim\;spec}}\Phi=\mu$ and $n/d \to \gamma \in (0,\infty)$. Then the sample covariance matrix $\Phi^{1/2}X^\top X\Phi^{1/2}$ has an almost sure spectral limit, $$\label{eq:MPmap}
{\operatorname{lim\;spec}}\,\Phi^{1/2}X^\top X\Phi^{1/2}=\rho^{\text{MP}}_\gamma \boxtimes \mu.$$ We will call this limit $\rho^{\text{MP}}_\gamma \boxtimes \mu$ the Marcenko-Pastur map of $\mu$ with aspect ratio $\gamma$. This distribution $\rho^{\text{MP}}_\gamma \boxtimes \mu$ may be defined by its Stieltjes transform $m(z)$, which solves the Marcenko-Pastur fixed point equation [@marchenko1967distribution] $$\label{eq:MPeq}
m(z)=\int \frac{1}{x(1-\gamma-\gamma zm(z))-z}\,d\mu(x).$$
Main results {#sec:results}
============
Assumptions
-----------
We use Greek indices ${\alpha}$, ${\beta}$, etc. for samples in $\{1,\ldots,n\}$, and Roman indices $i$, $j$, etc. for neurons in $\{1,\ldots,d\}$. For a matrix $X \in {\mathbb{R}}^{d \times n}$, we denote by ${\mathbf{x}}_{\alpha}$ its ${\alpha}^\text{th}$ column and by ${\mathbf{x}}_i^\top$ its $i^\text{th}$ row. $\|\cdot\|$ is the $\ell_2$-norm for vectors and $\ell_2 \to \ell_2$ operator norm for matrices. ${\operatorname{Id}}$ is the identity matrix.
\[def:orthogonal\] Let ${\varepsilon},B>0$. A matrix $X \in {\mathbb{R}}^{d \times n}$ is [**$({\varepsilon},B)$-orthonormal**]{} if its columns satisfy, for every ${\alpha}\neq {\beta}\in \{1,\ldots,n\}$, $$\big|\|{\mathbf{x}}_{\alpha}\|^2-1\big| \leq {\varepsilon},
\qquad \big|{\mathbf{x}}_{\alpha}^\top {\mathbf{x}}_{\beta}\big| \leq {\varepsilon},
\qquad \|X\| \leq B, \qquad
\sum_{{\alpha}=1}^n (\|{\mathbf{x}}_{\alpha}\|^2-1)^2 \leq B^2.$$
\[assump:asymptotics\] The number of layers $L \geq 1$ is fixed, and $n,d_0,d_1,\ldots,d_L \to \infty$, such that
(a) The weights $\theta=(W_1,\ldots,W_L,{\mathbf{w}})$ are i.i.d. and distributed as ${\mathcal{N}}(0,1)$.
(b) The activation $\sigma(x)$ is twice differentiable, with $\sup_{x \in {\mathbb{R}}}
|\sigma'(x)|,|\sigma''(x)| \leq \lambda_\sigma$ for some $\lambda_\sigma<\infty$. For $\xi \sim {\mathcal{N}}(0,1)$, we have ${\mathbb{E}}[\sigma(\xi)]=0$ and ${\mathbb{E}}[\sigma^2(\xi)]=1$.
(c) The input $X \in {\mathbb{R}}^{d_0 \times n}$ is $({\varepsilon}_n,B)$-orthonormal in the sense of Definition \[def:orthogonal\], where $B$ is a constant, and ${\varepsilon}_n n^{1/4} \to 0$ as $n \to \infty$.
(d) As $n \to \infty$, ${\operatorname{lim\;spec}}X^\top X=\mu_0$ for a probability distribution $\mu_0$ on $[0,\infty)$, and $\lim n/d_\ell=\gamma_\ell$ for constants $\gamma_\ell \in (0,\infty)$ and each $\ell=1,2,\ldots,L$.
The condition in part (c) will hold under fairly general settings of random input training samples, for example satisfying the following convex concentration property, which is discussed further in [@vu2015random; @adamczak2015note]. This encompasses settings where $\sqrt{d_0} \cdot {\mathbf{x}}_{\alpha}=f({\mathbf{z}}_{\alpha})$, ${\mathbf{z}}_{\alpha}\in {\mathbb{R}}^m$ has independent entries satisfying a log-Sobolev inequality, and $f:{\mathbb{R}}^m \to {\mathbb{R}}^{d_0}$ is any Lipschitz function. Note that the entries of ${\mathbf{x}}_{\alpha}$ may be correlated, and the input spectrum $\mu_0$ is not necessarily the Marcenko-Pastur law.
\[prop:inputisgood\] Let $X=({\mathbf{x}}_1,\ldots,{\mathbf{x}}_n) \in {\mathbb{R}}^{d_0 \times n}$, where ${\mathbf{x}}_1,\ldots,{\mathbf{x}}_n$ are independent training samples satisfying ${\mathbb{E}}[{\mathbf{x}}_{\alpha}]=0$ and ${\mathbb{E}}[\|{\mathbf{x}}_{\alpha}\|^2]=1$. Suppose, for some constant $c_0>0$, that $d_0 \geq c_0n$, and each vector $\sqrt{d_0}\cdot {\mathbf{x}}_{\alpha}$ satifies the convex concentration property $${\mathbb{P}}\Big[\big|\varphi(\sqrt{d_0} \cdot {\mathbf{x}}_{\alpha})-{\mathbb{E}}\varphi(\sqrt{d_0} \cdot {\mathbf{x}}_{\alpha})
\big| \geq t\Big] \leq 2e^{-c_0t^2}$$ for every $t>0$ and every 1-Lipschitz convex function $\varphi:{\mathbb{R}}^{d_0} \to {\mathbb{R}}$. Then for any $k>0$, with probability $1-n^{-k}$, $X$ is $(\sqrt{\frac{C\log n}{d_0}},B)$-orthonormal for some $C,B>0$ depending only on $c_0,k$.
The scaling of $\theta$, together with the scalings in (\[eq:NNfunc\]) and the conditions ${\mathbb{E}}[\sigma(\xi)]=0$ and ${\mathbb{E}}[\sigma^2(\xi)]=1$, ensure that all pre-activations have approximate mean 0 and variance 1. This may be achieved in practice by batch normalization [@ioffe2015batch]. For $\xi \sim {\mathcal{N}}(0,1)$, we define the following constants associated to $\sigma(x)$. We verify in Proposition \[prop:sigmaproperties\] that under Assumption \[assump:asymptotics\](b), we have $b_\sigma^2 \leq 1 \leq a_\sigma$. $$\label{eq:qr}
b_\sigma={\mathbb{E}}[\sigma'(\xi)], \quad a_\sigma={\mathbb{E}}[\sigma'(\xi)^2],
\quad q_\ell=(b_\sigma^2)^{L-\ell}, \quad r_\ell=a_\sigma^{L-\ell},
\quad r_+=\sum_{\ell=0}^{L-1} r_\ell-q_\ell.$$
Spectrum of the Conjugate Kernel {#sec:CK}
--------------------------------
Recall the Marcenko-Pastur map (\[eq:MPmap\]). Let $\mu_1,\mu_2,\mu_3,\ldots$ be the sequence of probability distributions on $[0,\infty)$ defined recursively by $$\label{eq:muell}
\mu_\ell=\rho^{\text{MP}}_{\gamma_\ell} \boxtimes \Big((1-b_\sigma^2)
+b_\sigma^2 \cdot \mu_{\ell-1}\Big).$$ Here, $\mu_0$ is the input limit spectrum in Assumption \[assump:asymptotics\](d), $b_\sigma$ is defined in (\[eq:qr\]), and $(1-b_\sigma^2)+b_\sigma^2 \cdot \mu$ denotes the translation and rescaling of $\mu$ that is the distribution of $(1-b_\sigma^2)+b_\sigma^2 \lambda$ when $\lambda \sim \mu$.
\[thm:CK\] Suppose Assumption \[assump:asymptotics\] holds, and define $\mu_1,\ldots,\mu_L$ by (\[eq:muell\]). Then $${\operatorname{lim\;spec}}X_\ell^\top X_\ell=\mu_\ell \quad
\text{ for each } \ell=1,\ldots,L, \qquad {\operatorname{lim\;spec}}K^{\text{CK}}=\mu_L.$$ Furthermore, $\|K^{\text{CK}}\| \leq C$ a.s. for a constant $C>0$ and all large $n$.
If $\sigma(x)$ is such that $b_\sigma=0$, then each distribution $\mu_\ell$ is simply the Marcenko-Pastur law $\rho_{\gamma_\ell}^{\text{MP}}$. This special case was previously conjectured in [@pennington2017nonlinear] and proven in [@benigni2019eigenvalue], for input data $X$ with i.i.d. entries.
To connect Theorem \[thm:CK\] to our next result on the NTK, let us describe the iteration (\[eq:muell\]) more explicitly using a recursive sequence of fixed-point equations derived from the Marcenko-Pastur equation (\[eq:MPeq\]): Let $m_\ell(z)$ be the Stieltjes transform of $\mu_\ell$, and define $$\tilde{t}_\ell(z_{-1},z_\ell)=\lim_{n \to \infty} \frac{1}{n}
{\operatorname{Tr}}(z_{-1}{\operatorname{Id}}+z_\ell X_\ell^\top X_\ell)^{-1}
=\frac{1}{z_\ell}m_\ell\left(-\frac{z_{-1}}{z_\ell}\right).$$ Applying the Marcenko-Pastur equation (\[eq:MPeq\]) to $m_\ell(-z_{-1}/z_\ell)$, and introducing $\tilde{s}_\ell(z_{-1},z_\ell)=[z_\ell(1-\gamma_\ell+\gamma_\ell z_{-1}
\tilde{t}_\ell(z_{-1},z_\ell))]^{-1}$, one may check that (\[eq:muell\]) may be written as the pair of equations $$\begin{aligned}
\tilde{t}_\ell(z_{-1},z_\ell)&=\tilde{t}_{\ell-1}\bigg(
z_{-1}+\frac{1-b_\sigma^2}{\tilde{s}_\ell(z_{-1},z_\ell)},\;
\frac{b_\sigma^2}{\tilde{s}_\ell(z_{-1},z_\ell)}\bigg),\label{eq:tildet}\\
\tilde{s}_\ell(z_{-1},z_\ell)
&=(1/z_\ell)+\gamma_\ell\Big(\tilde{s}_\ell(z_{-1},z_\ell)-z_{-1}
\tilde{s}_\ell(z_{-1},z_\ell)\tilde{t}_\ell(z_{-1},z_\ell)\Big),\label{eq:tildes}\end{aligned}$$ where (\[eq:tildes\]) is a rearrangement of the definition of $\tilde{s}_\ell$. Applying (\[eq:tildet\]) to substitute $\tilde{t}_\ell(z_{-1},z_\ell)$ in (\[eq:tildes\]), the equation (\[eq:tildes\]) is a fixed-point equation that defines $\tilde{s}_\ell$ in terms of $\tilde{t}_{\ell-1}$. Then (\[eq:tildet\]) defines $\tilde{t}_\ell$ in terms of $\tilde{s}_\ell$ and $\tilde{t}_{\ell-1}$. The limit Stieltjes transform for $K^{\text{CK}}$ is the specialization $m_{\text{CK}}(z)=\tilde{t}_L(-z,1)$.
Spectrum of the Neural Tangent Kernel {#sec:NTK}
-------------------------------------
In the neural network model (\[eq:NNfunc\]), an application of the chain rule yields an explicit form $$K^{\text{NTK}}=X_L^\top X_L+\sum_{\ell=1}^L (S_\ell^\top S_\ell) \odot
(X_{\ell-1}^\top X_{\ell-1})$$ for certain matrices $S_\ell \in {\mathbb{R}}^{d_\ell \times n}$, where $\odot$ is the Hadamard (entrywise) product. We refer to Appendix \[appendix:NTKapprox\] for the exact expression; see also [@huang2019dynamics Eq. (1.7)]. Our spectral analysis of $K^{\text{NTK}}$ relies on the following approximation, which shows that the limit spectrum of $K^{\text{NTK}}$ is equivalent to a linear combination of the CK matrices $X_0^\top X_0,\ldots,X_L^\top X_L$ and ${\operatorname{Id}}$. We prove this in Appendix \[appendix:NTKapprox\].
\[lemma:NTKapprox\] Under Assumption \[assump:asymptotics\], letting $r_+$ and $q_\ell$ be as defined in (\[eq:qr\]), $${\operatorname{lim\;spec}}K^{\text{NTK}}={\operatorname{lim\;spec}}\Big(r_+{\operatorname{Id}}+X_L^\top X_L+\sum_{\ell=0}^{L-1}
q_\ell X_\ell^\top X_\ell\Big).$$
To provide an analytic description of this spectrum, we extend (\[eq:tildet\],\[eq:tildes\]) to characterize the trace of rational functions of $X_0^\top X_0,\ldots,X_L^\top X_L$ and ${\operatorname{Id}}$. Denote the closed lower-half complex plane with 0 removed as ${\mathbb{C}}^*=\overline{{\mathbb{C}}^-} \setminus \{0\}$. For $\ell=0,1,2,\ldots$, we define recursively two sequences of functions $$\begin{aligned}
t_\ell&:({\mathbb{C}}^- \times {\mathbb{R}}^\ell \times {\mathbb{C}}^*) \times {\mathbb{C}}^{\ell+2} \to {\mathbb{C}},
& ({\mathbf{z}},{\mathbf{w}}) \mapsto t_\ell({\mathbf{z}},{\mathbf{w}})\\
s_\ell&:{\mathbb{C}}^- \times {\mathbb{R}}^\ell \times {\mathbb{C}}^* \to {\mathbb{C}}^+,
& {\mathbf{z}}\mapsto s_\ell({\mathbf{z}}).\end{aligned}$$ where ${\mathbf{z}}=(z_{-1},z_0,\ldots,z_\ell) \in {\mathbb{C}}^- \times {\mathbb{R}}^\ell \times {\mathbb{C}}^*$ and ${\mathbf{w}}=(w_{-1},w_0,\ldots,w_\ell) \in {\mathbb{C}}^{\ell+2}$. We will define these functions such that $t_\ell({\mathbf{z}},{\mathbf{w}})$ will be the value of $$\lim_{n \to \infty} n^{-1}{\operatorname{Tr}}(z_{-1}{\operatorname{Id}}+z_0 X_0^\top X_0+\ldots+z_\ell
X_\ell^\top X_\ell)^{-1}(w_{-1}{\operatorname{Id}}+w_0 X_0^\top X_0+\ldots+w_\ell
X_\ell^\top X_\ell).$$ For $\ell=0$, we define the first function $t_0$ by $$\label{eq:t0}
t_0\Big((z_{-1},z_0),(w_{-1},w_0)\Big)
=\int \frac{w_{-1}+w_0x}{z_{-1}+z_0x} d\mu_0(x)$$ For $\ell \geq 1$, we then define the functions $s_\ell$ and $t_\ell$ recursively by $$\begin{aligned}
s_\ell({\mathbf{z}})&=(1/z_\ell)
+\gamma_\ell t_{\ell-1}\big({\mathbf{z}}_{{\text{prev}}}(s_\ell({\mathbf{z}})),\,(1-b_\sigma^2,0,
\ldots,0,b_\sigma^2)\big),\label{eq:sl}\\
t_\ell({\mathbf{z}},{\mathbf{w}})&=(w_\ell/z_\ell)
+t_{\ell-1}\big({\mathbf{z}}_{{\text{prev}}}(s_\ell({\mathbf{z}})),\,{\mathbf{w}}_{{\text{prev}}}\big)\label{eq:tl}\end{aligned}$$ where we write as shorthand $$\begin{aligned}
{\mathbf{z}}_{{\text{prev}}}(s_\ell({\mathbf{z}}))
& \equiv \left(z_{-1}+\frac{1-b_\sigma^2}{s_\ell({\mathbf{z}})},
z_0,\ldots,z_{\ell-2},z_{\ell-1}+\frac{b_\sigma^2}{s_\ell({\mathbf{z}})}\right)
\in {\mathbb{C}}^- \times {\mathbb{R}}^{\ell-1} \times {\mathbb{C}}^*,\label{eq:zprev}\\
{\mathbf{w}}_{{\text{prev}}}& \equiv (w_{-1},\ldots,w_{\ell-1})-(w_\ell/z_\ell) \cdot
(z_{-1},\ldots,z_{\ell-1}) \in {\mathbb{C}}^{\ell+1}.\label{eq:wprev}\end{aligned}$$
\[prop:swelldefined\] For each $\ell \geq 1$ and any ${\mathbf{z}}\in {\mathbb{C}}^- \times
{\mathbb{R}}^\ell \times {\mathbb{C}}^*$, there is a unique solution $s_\ell({\mathbf{z}}) \in
{\mathbb{C}}^+$ to the fixed-point equation (\[eq:sl\]).
Hence, (\[eq:sl\]) defines $s_\ell$ in terms of $t_{\ell-1}$, and this is then used in (\[eq:tl\]) to define $t_\ell$. This is illustrated diagrammatically as $$\begin{matrix}
t_0 & \rightarrow & t_1 & \rightarrow & t_2 &\rightarrow &\cdots\\
\downarrow& \large\nearrow & \downarrow& \large\nearrow & \downarrow& \large\nearrow & \\
s_1 & & s_2 & & s_3 & & \\
\end{matrix}$$
\[thm:NTK\] Under Assumption \[assump:asymptotics\], for any fixed values $z_{-1},z_0,\ldots,z_L \in {\mathbb{R}}$ where $z_L \neq 0$, ${\operatorname{lim\;spec}}(z_{-1}{\operatorname{Id}}+z_0X_0^\top X_0+\ldots+z_L X_L^\top X_L)=\nu$ where $\nu$ is the probability distribution with Stieltjes transform $m_\nu(z)=t_L((-z+z_{-1},z_0,\ldots,z_L),(1,0,\ldots,0))$.
In particular, ${\operatorname{lim\;spec}}K^{\text{NTK}}$ is the probability distribution with Stieltjes transform $$m_{\text{NTK}}(z)=t_L\Big((-z+r_+,q_0,\ldots,q_{L-1},1),(1,0,\ldots,0)\Big).$$ Furthermore, $\|K^{\text{NTK}}\| \leq C$ a.s. for a constant $C>0$ and all large $n$.
This also describes the limit for $K^{\text{CK}}=X_L^\top X_L$, by specializing to $(z_{-1},\ldots,z_L)=(0,\ldots,0,1)$. One may check that $s_\ell(z_{-1},0,\ldots,0,z_\ell)=\tilde{s}_\ell(z_{-1},z_\ell)$ and $t_\ell((z_{-1},0,\ldots,0,z_\ell),(1,0,\ldots,0))=\tilde{t}_\ell(z_{-1},z_\ell)$, where $\tilde{s}_\ell,\tilde{t}_\ell$ are defined by (\[eq:tildet\],\[eq:tildes\]), and (\[eq:sl\],\[eq:tl\]) reduce to (\[eq:tildet\],\[eq:tildes\]) under this specialization.
Extension to multi-dimensional outputs and rescaled parametrizations {#sec:extensions}
--------------------------------------------------------------------
Theorem \[thm:NTK\] pertains to $K^{\text{NTK}}$ for a network with scalar outputs, under the “NTK-parametrization” of network weights in (\[eq:NNfunc\]). We consider here a network with $k$-dimensional output, defined as $$\label{eq:NNfuncmulti}
f_\theta({\mathbf{x}})=W_{L+1}^\top \frac{1}{\sqrt{d_L}} \sigma\bigg(
W_L \frac{1}{\sqrt{d_{L-1}}}\sigma \Big(\ldots
\frac{1}{\sqrt{d_2}}\sigma\Big(W_2\frac{1}{\sqrt{d_1}} \sigma(W_1 {\mathbf{x}})
\Big)\Big)\bigg) \in {\mathbb{R}}^k$$ where $W_{L+1}^\top \in {\mathbb{R}}^{k \times d_L}$. We write the coordinates of $f_\theta$ as $(f_\theta^1,\ldots,f_\theta^k)$, and the vectorized output for all training samples $X \in {\mathbb{R}}^{d_0 \times n}$ as $f_\theta(X)=(f_\theta^1(X),\ldots,f_\theta^k(X)) \in {\mathbb{R}}^{nk}$. We consider the NTK $$\label{eq:NTKmulti}
K^{\text{NTK}}=\sum_{\ell=1}^{L+1}
\tau_\ell \Big(\nabla_{W_\ell} f_\theta(X)\Big)^\top
\Big(\nabla_{W_\ell} f_\theta(X)\Big) \in {\mathbb{R}}^{nk \times nk}.$$ For $\tau_1=\ldots=\tau_{L+1}=1$, this is a flattening of the NTK defined in [@jacot2018neural], and we recall briefly its derivation from gradient-flow training in Appendix \[appendix:NTKmultiderivation\]. We consider general constants $\tau_1,\ldots,\tau_{L+1}>0$ to allow for a different learning rate for each weight matrix $W_\ell$, which may arise from backpropagation in the model (\[eq:NNfuncmulti\]) using a parametrization with different scalings of the weights.
\[thm:NTKmulti\] Fix any $k \geq 1$. Suppose Assumption \[assump:asymptotics\] holds. Then $\|K^{\text{NTK}}\| \leq C$ a.s. for a constant $C>0$ and all large $n$, and ${\operatorname{lim\;spec}}K^{\text{NTK}}$ is the probability distribution with Stieltjes transform $$m_{\text{NTK}}(z)=t_L\Big((-z+\tau \cdot r_+,\;\tau_1q_0,\ldots,\tau_Lq_{L-1},\tau_{L+1}),(1,0,\ldots,0)\Big),
\quad \tau \cdot r_+ \equiv \sum_{\ell=0}^{L-1} \tau_{\ell+1}(r_\ell-q_\ell).$$
Experiments {#sec:experiments}
===========
We describe in Appendix \[appendix:computation\] an algorithm to numerically compute the limit spectral densities of Theorem \[thm:NTK\]. The computational cost is independent of the dimensions $(n,d_0,\ldots,d_L)$, and each limit density below was computed within a few seconds on our laptop computer. Using this procedure, we investigate the accuracy of the theoretical predictions of Theorems \[thm:CK\] and \[thm:NTK\]. Finally, we conclude by examining the spectra of $K^{\text{CK}}$ and $K^{\text{NTK}}$ after network training.
Simulated Gaussian training data
--------------------------------
We consider $n=3000$ training samples with i.i.d. ${\mathcal{N}}(0,1/d_0)$ entries, input dimension $d_0=1000$, and $L=5$ hidden layers of dimensions $d_1=\ldots=d_5=6000$. We take $\sigma(x) \propto \tan^{-1}(x)$, normalized so that ${\mathbb{E}}[\sigma(\xi)^2]=1$. A close agreement between the observed and limit spectra is displayed in Figure \[fig:gaussian\], for both $K^{\text{CK}}$ and $K^{\text{NTK}}$. Intermediate layers are depicted in Appendix \[appendix:alllayers\].
We highlight two qualitative phenomena: The spectral distribution of the NTK (at initialization) is separated from 0, as explained by the ${\operatorname{Id}}$ component in Lemma \[lemma:NTKapprox\]. Across layers $\ell=1,\ldots,L$, there is a merging of the spectral bulk components of the CK, and an extension of its spectral support.
CIFAR-10 training data {#sec:CIFAR}
----------------------
We consider $n=5000$ samples randomly selected from the CIFAR-10 training set [@krizhevsky2009learning], with input dimension $d_0=3072$, and $L=5$ hidden layers of dimensions $d_1=\ldots=d_5=10000$. Strong principal component structure may cause the training samples to have large pairwise inner-products. Thus, we pre-process the training samples by removing the leading 10 PCs. A close agreement between the observed and limit spectra is displayed in Figure \[fig:CIFAR\], for both $K^{\text{CK}}$ and $K^{\text{NTK}}$. Results without removing these leading 10 PCs are presented in Appendix \[appendix:CIFARraw\], where there is close agreement for $K^{\text{CK}}$ but a deviation from the theoretical prediction for $K^{\text{NTK}}$. This suggests that the approximation in Lemma \[lemma:NTKapprox\] is sensitive to large but low-rank perturbations of $X$.
CK and NTK spectra after training {#sec:training}
---------------------------------
We consider $n=1000$ training samples $({\mathbf{x}}_{\alpha},y_{\alpha})$, with ${\mathbf{x}}_{\alpha}$ uniformly distributed on the unit sphere of dimension $d_0=800$, and $y_{\alpha}=\sigma({\mathbf{x}}_{\alpha}^\top {\mathbf{v}})$ for ${\mathbf{v}}\in {\mathbb{R}}^{d_0}$ on the sphere of radius $\sqrt{d_0}$. We train a 3-layer network with widths $d_1=d_2=d_3=800$, without biases, using the Adam optimizer in Keras with learning rate $0.01$, batch size 32, and 300 training epochs. The final mean-squared training error is $10^{-4}$, and the test-sample prediction-$R^2$ is 0.81.
Figure \[fig:training\] depicts the spectra of $K^{\text{CK}}$ and $K^{\text{NTK}}$ for the trained weights $\theta$. Intermediate layers are shown in Appendix \[appendix:alllayers\]. We observe that the bulk spectra of $K^{\text{CK}}$ and $K^{\text{NTK}}$ are elongated from their random initializations. Furthermore, large outlier eigenvalues emerge in both $K^{\text{CK}}$ and $K^{\text{NTK}}$ over training. The corresponding eigenvectors are highly predictive of the training labels ${\mathbf{y}}$, suggesting the emergence of these eigenvectors as the primary mechanism of training in this example. We describe in Appendix \[appendix:CIFARtraining\] a second training example for a binary classification task on CIFAR-10, where similar qualitative phenomena are observed.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research is supported in part by NSF Grant DMS-1916198. We would like to thank John Lafferty and Ganlin Song for helpful discussions regarding the Neural Tangent Kernel.
Numerical solution of the fixed-point equations {#appendix:computation}
===============================================
Theorem \[thm:NTK\] characterizes the limit Stieltjes transform $m(z)$ of matrices such as $K^{\text{CK}}$ and $K^{\text{NTK}}$. By the discussion in Section \[sec:stieltjes\], a numerical approximation to the density functions of the corresponding spectral distributions may be obtained by computing $m(z)$ for $z=x+i\eta$, across a fine grid of values $x \in {\mathbb{R}}$ and for a fixed small imaginary part $\eta>0$. We describe here one possible approach for this computation.
To compute the limit spectrum for $z_{-1}{\operatorname{Id}}+z_0X_0^\top X_0+\ldots+z_LX_L^\top X_L$ and general values $z_{-1},\ldots,z_L \in {\mathbb{R}}$, fix the spectral argument $z=x+i\eta$ and denote $${\mathbf{z}}_L=(-z+z_{-1},z_0,\ldots,z_L),\; {\mathbf{z}}_{L-1}={\mathbf{z}}_{{\text{prev}}}(s_L({\mathbf{z}}_L),{\mathbf{z}}_L),
\;{\mathbf{z}}_{L-2}={\mathbf{z}}_{{\text{prev}}}(s_{L-1}({\mathbf{z}}_{L-1}),{\mathbf{z}}_{L-1}), \text{ etc.}$$ Here, for $s \in {\mathbb{C}}^+$ and ${\mathbf{z}}\in {\mathbb{C}}^- \times {\mathbb{R}}^\ell \times {\mathbb{C}}^*$, the quantity $${\mathbf{z}}_{{\text{prev}}}(s,{\mathbf{z}})=\left(z_{-1}+\frac{1-b_\sigma^2}{s},z_0,\ldots,
z_{\ell-2},z_{\ell-1}+\frac{b_\sigma^2}{s}\right)
\in {\mathbb{C}}^- \times {\mathbb{R}}^{\ell-1} \times {\mathbb{C}}^*$$ is as defined in (\[eq:zprev\]), and we are making its dependence on ${\mathbf{z}}$ explicit. Denote $s_\ell \equiv s_\ell({\mathbf{z}}_\ell)$ for each $\ell=1,\ldots,L$. Observe that, if we are given $s_1,\ldots,s_L$, then the value $t_\ell({\mathbf{z}}_\ell,{\mathbf{w}})$ may be directly computed from (\[eq:tl\]), for any $\ell \in \{0,\ldots,L\}$ and any vector ${\mathbf{w}}\in {\mathbb{C}}^{\ell+2}$. This is because the fixed points needed to compute the arguments ${\mathbf{z}}_{{\text{prev}}}(s_\ell({\mathbf{z}}_\ell),{\mathbf{z}}_\ell)$, ${\mathbf{z}}_{{\text{prev}}}(s_{\ell-1}({\mathbf{z}}_{\ell-1}),{\mathbf{z}}_{\ell-1})$, etc. for the successive evaluations of $t_\ell$, $t_{\ell-1}$, etc. are provided by this given sequence $s_1,\ldots,s_L$.
Thus, we apply an iterative procedure of initializing $s_1^{(0)},\ldots,s_L^{(0)} \in {\mathbb{C}}^+$, and computing the *simultaneous* updates $s_1^{(t+1)},\ldots,s_L^{(t+1)}$ using the previous values $s_1^{(t)},\ldots,s_L^{(t)}$. That is, we iterate the following two steps:
1. Set ${\mathbf{z}}_L^{(t)}={\mathbf{z}}_L$, and compute ${\mathbf{z}}_{L-1}^{(t)}={\mathbf{z}}_{{\text{prev}}}(s_L^{(t)},{\mathbf{z}}_L^{(t)})$, ${\mathbf{z}}_{L-2}^{(t)}={\mathbf{z}}_{{\text{prev}}}(s_{L-1}^{(t)},{\mathbf{z}}_{L-1}^{(t)})$, etc.
2. Compute an update $s_\ell^{(t+1)}$ for the value of $s_\ell({\mathbf{z}}_\ell)$ and each $\ell=1,\ldots,L$, using the right side of (\[eq:sl\]) with ${\mathbf{z}}_\ell^{(t)}$ and ${\mathbf{z}}_\ell^{(t-1)} \equiv {\mathbf{z}}_{{\text{prev}}}(s_\ell^{(t)},{\mathbf{z}}_\ell^{(t)})$ in place of ${\mathbf{z}}_\ell$ and ${\mathbf{z}}_{{\text{prev}}}(s_\ell({\mathbf{z}}_\ell),{\mathbf{z}}_\ell)$.
After this iteration converges to fixed points $s_1^*,\ldots,s_L^*$, we then compute $m(z)=t_L({\mathbf{z}}_L,(1,0,\ldots,0))$ using (\[eq:tl\]) and these fixed points. For each successive value $z=x+i\eta$ along the grid of values $x \in {\mathbb{R}}$, we initialize $s_1^{(0)},\ldots,s_L^{(0)}$ by linear interpolation from the computed fixed points at the preceding two values of $x$ along this grid, for faster computation.
Note that for each value $z=x+i\eta$, if the above iteration converges to fixed points $s_1^*,\ldots,s_L^* \in {\mathbb{C}}^+$, then this procedure computes the correct value for $m(z)$: This is because, denoting $${\mathbf{z}}_{L-1}^*={\mathbf{z}}_{{\text{prev}}}(s_L^*,{\mathbf{z}}_L), \quad
{\mathbf{z}}_{L-2}^*={\mathbf{z}}_{{\text{prev}}}(s_{L-1}^*,{\mathbf{z}}_{L-1}^*), \quad \ldots,
\qquad {\mathbf{z}}_1^*={\mathbf{z}}_{{\text{prev}}}(s_2^*,{\mathbf{z}}_2^*),$$ it may be checked iteratively from (\[eq:sl\],\[eq:tl\]) and the uniqueness guarantee of Proposition \[prop:swelldefined\] that $s_1^*=s_1({\mathbf{z}}_1^*)$, then $s_2^*=s_2({\mathbf{z}}_2^*)$, etc., and finally that $s_L^*=s_L({\mathbf{z}}_L)$. This then means that ${\mathbf{z}}_{L-1}^*={\mathbf{z}}_{{\text{prev}}}(s_L({\mathbf{z}}_L),{\mathbf{z}}_L)={\mathbf{z}}_{L-1}$, then ${\mathbf{z}}_{L-2}^*={\mathbf{z}}_{{\text{prev}}}(s_{L-1}({\mathbf{z}}_{L-1}),{\mathbf{z}}_{L-1})={\mathbf{z}}_{L-2}$, etc., and so $s_\ell^*=s_\ell({\mathbf{z}}_\ell)$ for each $\ell$. Then this method computes the correct value for $m(z)=t_L({\mathbf{z}}_L,(1,0,\ldots,0))$.
We have found in practice that the above iteration occasionally converges to fixed points $s_1,\ldots,s_L$ not belonging to ${\mathbb{C}}^+$ (i.e. this is not a mapping from $({\mathbb{C}}^+)^L$ to $({\mathbb{C}}^+)^L$). If this occurs, we randomly re-initialize $s_1^{(0)},\ldots,s_L^{(0)} \in {\mathbb{C}}^+$, and we have found that the method reaches the correct fixed point within a small number of random initializations.
Proof of $({\varepsilon},B)$-orthonormality for independent input training samples
==================================================================================
We prove Proposition \[prop:inputisgood\]. For convenience, in this section, we denote the input dimension $d_0$ simply as $d$, and we denote the rescaled input by ${\widetilde{X}}=\sqrt{d}\,X$, with columns ${\tilde{\mathbf{x}}}_{\alpha}=\sqrt{d} \cdot {\mathbf{x}}_{\alpha}$.\
[**Bound for $\|{\tilde{\mathbf{x}}}_{\alpha}\|^2$:**]{} Note that ${\mathbb{E}}[\|{\tilde{\mathbf{x}}}_{\alpha}\|^2]=d$. Applying the convex concentration property and [@adamczak2015note Theorem 2.5] with $A={\operatorname{Id}}$, we have for any $t>0$ that $$\label{eq:xnormconcentration}
{\mathbb{P}}\Big[\big|\|{\tilde{\mathbf{x}}}_{\alpha}\|^2-d\big|>t\Big] \leq
2\exp\left(-c\min\left(\frac{t^2}{d},t\right)\right)$$ for a constant $c$ depending only on $c_0$. Applying this for $t=\sqrt{Kd\log n}$ and a union bound, with probability $1-2ne^{-cK\log n}$, $$\label{eq:xgood}
\Big|\|{\tilde{\mathbf{x}}}_{\alpha}\|^2-d\Big| \leq \sqrt{Kd\log n} \quad
\text{ for all } {\alpha}\in [n].$$ Rescaling, this shows $|\|{\mathbf{x}}_{\alpha}\|^2-1| \leq \sqrt{(K\log n)/d}$.\
[**Bound for ${\tilde{\mathbf{x}}}_{\alpha}^\top {\tilde{\mathbf{x}}}_{\beta}$:**]{} Since ${\tilde{\mathbf{x}}}_{\alpha}$ and ${\tilde{\mathbf{x}}}_{\beta}$ are independent, conditional on ${\tilde{\mathbf{x}}}_{\beta}$, we have ${\mathbb{E}}[{\tilde{\mathbf{x}}}_{\alpha}^\top {\tilde{\mathbf{x}}}_{\beta}\mid {\tilde{\mathbf{x}}}_{\beta}]=0$, and the map ${\tilde{\mathbf{x}}}_{\alpha}\mapsto {\tilde{\mathbf{x}}}_{\alpha}^\top {\tilde{\mathbf{x}}}_{\beta}$ is convex and $\|{\tilde{\mathbf{x}}}_{\beta}\|$-Lipschitz. Then the convex concentration property implies, for any $t>0$, $${\mathbb{P}}\Big[|{\tilde{\mathbf{x}}}_{\alpha}^\top {\tilde{\mathbf{x}}}_{\beta}|>t \Big|{\tilde{\mathbf{x}}}_{\beta}\Big] \leq
2e^{-c_0t^2/\|{\tilde{\mathbf{x}}}_{\beta}\|^2}.$$ On the event (\[eq:xgood\]), applying this for $t=\sqrt{Kd\log n}$, this probability is at most $2e^{-cK\log n}$. Taking a union bound, with probability $1-2n^2e^{-cK\log n}$, $$\Big|{\tilde{\mathbf{x}}}_{\alpha}^\top {\tilde{\mathbf{x}}}_{\beta}\Big| \leq \sqrt{Kd\log n}
\quad \text{ for all } {\alpha}\neq {\beta}\in [n].$$ Rescaling, this shows $|{\mathbf{x}}_{\alpha}^\top {\mathbf{x}}_{\beta}| \leq \sqrt{(K\log n)/d}$.\
[**Bound for $\|{\widetilde{X}}\|$:**]{} Fix any unit vector ${\mathbf{v}}=(v_1,\ldots,v_n) \in {\mathbb{R}}^n$. By [@kasiviswanathan2019restricted Lemma C.11], the random vector ${\widetilde{X}}{\mathbf{v}}$ also satisfies the convex concentration property, with a modified constant $c_0'$. Note that ${\mathbb{E}}[\|{\widetilde{X}}{\mathbf{v}}\|^2]=d\|{\mathbf{v}}\|^2=d$. Then, as in (\[eq:xnormconcentration\]), we have $${\mathbb{P}}\Big[|\|{\widetilde{X}}{\mathbf{v}}\|^2-d|>t\Big] \leq
2\exp\left(-c\min\left(\frac{t^2}{d},t\right)\right).$$ Applying this with $t=(B^2/4-1)d$, and taking a union bound over a $1/2$-net ${\mathcal{N}}$ of the unit ball $\{{\mathbf{v}}\in {\mathbb{R}}^n:\|{\mathbf{v}}\|=1\}$ with cardinality $5^n$, we have with probability at least $1-5^n \cdot 2e^{-cB^2d}$ that $$\|{\widetilde{X}}{\mathbf{v}}\| \leq (B/2)\sqrt{d} \quad \text{ for all } {\mathbf{v}}\in {\mathcal{N}}.$$ Since $$\|{\widetilde{X}}\|=\sup_{{\mathbf{v}}:\|{\mathbf{v}}\|=1} \|{\widetilde{X}}{\mathbf{v}}\|
\leq \sup_{{\mathbf{v}}\in {\mathcal{N}}} \|{\widetilde{X}}{\mathbf{v}}\|+\|{\widetilde{X}}\|/2,$$ we have $\|{\widetilde{X}}\| \leq B\sqrt{d}$ on this event. Rescaling, this shows $\|X\|
\leq B$.\
[**Bound for $\sum_{{\alpha}=1}^n (\|{\tilde{\mathbf{x}}}_{\alpha}\|^2-d)^2$:**]{} Define ${\mathbf{z}}=(z_1,\ldots,z_n)$ where ${\mathbf{z}}_{\alpha}=\|{\tilde{\mathbf{x}}}_{\alpha}\|^2-d$. Fixing any unit vector ${\mathbf{v}}=(v_1,\ldots,v_n) \in {\mathbb{R}}^n$, let us first bound ${\mathbf{v}}^\top {\mathbf{z}}$: We have $${\mathbf{v}}^\top {\mathbf{z}}=\sum_{{\alpha}=1}^n v_{\alpha}(\|{\tilde{\mathbf{x}}}_{\alpha}\|^2-d),$$ which has mean 0. Note that integrating the tail bound (\[eq:xnormconcentration\]) yields the sub-exponential condition $${\mathbb{E}}\left[\exp\left(\lambda(\|{\tilde{\mathbf{x}}}_{\alpha}\|^2-d)\right)\right]
\leq \exp(Cd\lambda^2)
\quad \text{ for all } |\lambda| \leq c'$$ and some constants $C,c'>0$. (See e.g. [@boucheron2013concentration Theorem 2.3], applied with $(v,c)=(Cd',C')$ and a large enough constant $C'>0$.) Then, as ${\tilde{\mathbf{x}}}_1,\ldots,{\tilde{\mathbf{x}}}_n$ are independent and $\|{\mathbf{v}}\|^2=1$, also $${\mathbb{E}}[e^{\lambda {\mathbf{v}}^\top {\mathbf{z}}}]
={\mathbb{E}}\left[\exp\left(\lambda \sum_{{\alpha}=1}^n v_{\alpha}(\|{\tilde{\mathbf{x}}}_{\alpha}\|^2-d)\right)\right]
\leq \exp(Cd\lambda^2)
\quad \text{ for all } |\lambda| \leq c'.$$ For any $t>0$, applying this with $\lambda=\min(t/(2Cd),c')$ yields the sub-exponential tail bound $${\mathbb{P}}[{\mathbf{v}}^\top {\mathbf{z}}\geq t] \leq e^{-\lambda t}{\mathbb{E}}[e^{\lambda {\mathbf{v}}^\top {\mathbf{z}}}]
\leq \exp\left(-c\min\left(\frac{t^2}{d},t\right)\right).$$ Now applying this for $t=(B/2)d$, and again taking a union bound over a $1/2$-net ${\mathcal{N}}$ of the unit ball, we have with probability $1-5^n \cdot e^{-cBd}$ that $${\mathbf{v}}^\top {\mathbf{z}}\leq (B/2)d \quad \text{ for all } {\mathbf{v}}\in {\mathcal{N}}.$$ On this event, we have as above that $\|{\mathbf{z}}\| \leq Bd$, so $\|{\mathbf{z}}\|^2 \leq
B^2d^2$. Rescaling, this shows $\sum_{{\alpha}=1}^n (\|{\tilde{\mathbf{x}}}_{\alpha}\|^2-1)^2 \leq B^2$.
Applying all of the above bounds for sufficiently large constants $K,B>0$, we obtain that these hold with probability at least $n^{-k}$, which yields Proposition \[prop:inputisgood\].
Overview of proofs for the main results {#appendix:overview}
=======================================
The proofs of Theorems \[thm:CK\], \[thm:NTK\], and \[thm:NTKmulti\] are contained in the subsequent Appendices \[appendix:orthogonal\]–\[appendix:NTKmulti\]. We provide here an outline of the argument.
We will apply induction across the layers $\ell=1,\ldots,L$, analyzing the post-activation matrix $X_\ell$ of each layer conditional on the previous post-activations $X_0,\ldots,X_{\ell-1}$ (i.e. with respect to only the randomness of $W_\ell$). For the Conjugate Kernel, this will entail analyzing the Stieltjes transform $$\frac{1}{n} {\operatorname{Tr}}(X_L^\top X_L-z{\operatorname{Id}})^{-1}$$ conditional on the previous layers. For the Neural Tangent Kernel, given the approximation in Lemma \[lemma:NTKapprox\], this will entail analyzing the Stieltjes transform $$\frac{1}{n} {\operatorname{Tr}}(A+X_L^\top X_L-z{\operatorname{Id}})^{-1}$$ conditional on the previous layers, where $A$ is a linear combination of $X_0^\top X_0,\ldots,X_{L-1}^\top X_{L-1}$, and ${\operatorname{Id}}$. Note that this matrix $A$ is deterministic conditional on the previous layers.
In Appendix \[appendix:orthogonal\], we carry out a non-asymptotic analysis of $({\varepsilon},B)$-orthonormality. In particular, we show that if the deterministic input $X \equiv X_0$ is $({\varepsilon},B)$-orthonormal, then $X_1$ is $(C{\varepsilon},CB)$-orthonormal with high probability, for a constant $C>0$ depending only on $\lambda_\sigma$. Note that we require the fourth technical condition $$\sum_{{\alpha}=1}^n (\|{\mathbf{x}}_{\alpha}\|^2-1)^2 \leq B^2$$ in Definition \[def:orthogonal\] to ensure that the operator norm $\|X_1\|$ remains of constant order, as otherwise $X_1$ may have a rank-one component whose norm grows slowly with $n$. Applying this result conditionally for every layer, Assumption \[assump:asymptotics\] then implies that $X_0,\ldots,X_L$ are all $(\tilde{{\varepsilon}}_n,\tilde{B})$-orthonormal for modified parameters $(\tilde{{\varepsilon}}_n,\tilde{B})$.
In Appendix \[appendix:singlelayer\], we carry out the analysis of the trace $$\frac{1}{n}{\operatorname{Tr}}(A+\alpha X_1^\top X_1-z{\operatorname{Id}})^{-1}$$ in a single layer, for a deterministic $({\varepsilon}_n,B)$-orthonormal input $X_0$, symmetric matrix $A \in
{\mathbb{R}}^{n \times n}$, and spectral parameters $\alpha \in {\mathbb{C}}^* \equiv
\overline{{\mathbb{C}}^-} \setminus 0$ and $z \in {\mathbb{C}}^+$. We allow $\alpha \in {\mathbb{C}}^*$ (rather than fixing $\alpha=1$), as the subsequent induction argument for the NTK will require this extension. When $A=0$ and $\alpha=1$, this reduces to the analysis in [@louart2018random], and also mirrors the proof of the Marcenko-Pastur equation (\[eq:MPeq\]). For $A \neq 0$, this trace will depend jointly on $A$ and the second-moment matrix $\Phi_1 \in {\mathbb{R}}^{n \times n}$ for the rows of $X_1$. We derive a fixed-point equation in terms of $A$ and $\Phi_1$, which approximates this trace in the $n \to \infty$ limit.
In Appendix \[appendix:CK\], we prove Theorem \[thm:CK\] on the CK, by specializing this analysis to the setting $A=0$ and $\alpha=1$. The inductive loop is closed via an entrywise approximation of the second-moment matrix $\Phi_\ell$ in each layer by a linear combination of $X_{\ell-1}^\top X_{\ell-1}$ and ${\operatorname{Id}}$ in the previous layer. The main argument for this approximation has been carried out in Appendix \[appendix:orthogonal\].
In Appendix \[appendix:NTK\], we prove Theorem \[thm:NTK\] on the NTK. Our analysis reduces the trace of any linear combination of $X_0^\top X_0,\ldots,X_L^\top X_L,{\operatorname{Id}}$ to the trace of a more general rational function of $X_0^\top X_0,\ldots,X_{L-1}^\top X_{L-1},{\operatorname{Id}}$ in the previous layer. In order to close the inductive loop, we analyze the trace of such a rational function across layers, and show that it may be characterized by the recursive fixed-point equations (\[eq:sl\]) and (\[eq:tl\]). In Appendix \[appendix:NTK\], we also establish the approximation in Lemma \[lemma:NTKapprox\] and the existence and uniqueness of the fixed point to (\[eq:sl\]).
Finally, in Appendix \[appendix:NTKmulti\], we prove Theorem \[thm:NTKmulti\], which is a minor extension of Theorem \[thm:NTK\].\
[**Notation.**]{} In the proof, ${\mathbf{v}}^*$ and $M^*$ denote the conjugate transpose. For a complex matrix $M \in {\mathbb{C}}^{n \times n}$, we denote by $${\operatorname{tr}}M=n^{-1}{\operatorname{Tr}}M$$ the normalized matrix trace, by $\|M\|=\sup_{{\mathbf{v}}\in {\mathbb{C}}^n:\|{\mathbf{v}}\|=1} \|M{\mathbf{v}}\|$ the operator norm, and by $\|M\|_F=({\operatorname{Tr}}M^*M)^{1/2}=(\sum_{{\alpha},{\beta}} |M_{{\alpha}{\beta}}|^2)^{1/2}$ the Frobenius norm. Note that we have $$|{\operatorname{tr}}M| \leq \|M\| \leq \|M\|_F, \qquad \|M\|_F \leq
\sqrt{n}\|M\|, \qquad |{\operatorname{tr}}AB| \leq n^{-1}\|A\|_F\|B\|_F.$$
Let us collect here a few basic results, which we will use in the subsequent sections.
\[prop:sigmaproperties\] Under Assumption \[assump:asymptotics\](b), the constants $a_\sigma$ and $b_\sigma$ in (\[eq:qr\]) satisfy $$|b_\sigma| \leq 1 \leq \sqrt{a_\sigma} \leq \lambda_\sigma.$$ For a universal constant $C>0$, the activation function $\sigma$ satisfies $$\label{eq:sigmabound}
|\sigma(x)| \leq \lambda_\sigma(|x|+C) \qquad \text{ for all } x \in {\mathbb{R}}.$$
It is clear from definition that $a_\sigma \leq \lambda_\sigma^2$. By the Gaussian Poincaré inequality, $$1={\mathbb{E}}[\sigma(\xi)^2]={\operatorname{Var}}[\sigma(\xi)]
\leq {\mathbb{E}}[\sigma'(\xi)^2]=a_\sigma.$$ By Gaussian integration-by-parts and Cauchy-Schwarz, $$|b_\sigma|=|{\mathbb{E}}[\sigma'(\xi)]|=|{\mathbb{E}}[\xi \cdot \sigma(\xi)]|
\leq {\mathbb{E}}[\xi^2]^{1/2}{\mathbb{E}}[\sigma(\xi)^2]^{1/2}=1.$$ We have $$\label{eq:sigma0bound}
|\sigma(0)| \leq {\mathbb{E}}[|\sigma(0)-\sigma(\xi)|]+{\mathbb{E}}[|\sigma(\xi)|]
\leq \lambda_\sigma {\mathbb{E}}[|\xi|]+{\mathbb{E}}[\sigma(\xi)^2]^{1/2}
\leq C\lambda_\sigma$$ (the last inequality applying $\lambda_\sigma \geq 1$). Then $|\sigma(x)| \leq |\sigma(0)|+\lambda_\sigma |x| \leq \lambda_\sigma(|x|+C)$.
\[prop:invertible\] Suppose $M=U+iV \in {\mathbb{C}}^{n \times n}$, where the real and imaginary parts $U,V \in {\mathbb{R}}^{n \times n}$ are symmetric, and $V$ is invertible with either $V \succeq c_0{\operatorname{Id}}$ or $V \preceq -c_0{\operatorname{Id}}$ for a value $c_0>0$. Then $M$ is invertible, and $\|M^{-1}\| \leq 1/c_0$.
For any unit vector ${\mathbf{v}}\in {\mathbb{C}}^n$, $$\|M{\mathbf{v}}\|=\|M{\mathbf{v}}\| \cdot \|{\mathbf{v}}\| \geq |{\mathbf{v}}^* M{\mathbf{v}}|
=|{\mathbf{v}}^*U{\mathbf{v}}+i \cdot {\mathbf{v}}^*V{\mathbf{v}}| \geq |{\mathbf{v}}^*V{\mathbf{v}}|,$$ the last step holding because $U,V$ are real-symmetric so that ${\mathbf{v}}^*U{\mathbf{v}}$ and ${\mathbf{v}}^*V{\mathbf{v}}$ are both real. By the given assumption on $V$, we have $|{\mathbf{v}}^*V{\mathbf{v}}| \geq c_0$, so $\|M{\mathbf{v}}\| \geq c_0$ for every unit vector ${\mathbf{v}}\in {\mathbb{C}}^n$. Then $M$ is invertible, and $\|M^{-1}\| \leq 1/c_0$.
\[prop:specapprox\] Let $M,{\widetilde{M}}\in {\mathbb{R}}^{n \times n}$ be any two symmetric matrices satisfying $$\frac{1}{n}\|M-{\widetilde{M}}\|_F^2 \to 0$$ a.s. as $n \to \infty$. If ${\operatorname{lim\;spec}}M=\nu$ for a probability distribution $\nu$ on ${\mathbb{R}}$, then also ${\operatorname{lim\;spec}}{\widetilde{M}}=\nu$.
For fixed $z \in {\mathbb{C}}^+$, let $m(z)={\operatorname{tr}}(M-z{\operatorname{Id}})^{-1}$ and $\tilde{m}(z)={\operatorname{tr}}({\widetilde{M}}-z{\operatorname{Id}})^{-1}$ be the Stieltjes transforms. Then applying $A^{-1}-B^{-1}=A^{-1}(B-A)B^{-1}$, we may bound their difference by $$\begin{aligned}
|m(z)-\tilde{m}(z)|^2&=\frac{1}{n^2}\Big|{\operatorname{Tr}}[(M-z{\operatorname{Id}})^{-1}-({\widetilde{M}}-z{\operatorname{Id}})^{-1}]\Big|^2\\
&=\frac{1}{n^2}\Big|{\operatorname{Tr}}(M-z{\operatorname{Id}})^{-1}({\widetilde{M}}-M)({\widetilde{M}}-z{\operatorname{Id}})^{-1}\Big|^2\\
&\leq \frac{1}{n^2}\|{\widetilde{M}}-M\|_F^2\|(M-z{\operatorname{Id}})^{-1}({\widetilde{M}}-z{\operatorname{Id}})^{-1}\|_F^2\\
&\leq \frac{1}{n}\|{\widetilde{M}}-M\|_F^2\|(M-z{\operatorname{Id}})^{-1}\|^2\|({\widetilde{M}}-z{\operatorname{Id}})^{-1}\|^2\end{aligned}$$ Applying $\|(M-z{\operatorname{Id}})^{-1}\| \leq 1/{\operatorname{Im}}z$ by Proposition \[prop:invertible\], and similarly for ${\widetilde{M}}$, the given condition shows that $m(z)-\tilde{m}(z) \to 0$ a.s., pointwise over $z \in
{\mathbb{C}}^+$. If ${\operatorname{lim\;spec}}M=\nu$, then $m(z) \to m_\nu(z) \equiv \int (x-z)^{-1}d\nu(x)$ a.s., and hence also $\tilde{m}(z) \to m_\nu(z)$ a.s. and ${\operatorname{lim\;spec}}{\widetilde{M}}=\nu$.
Propagation of approximate pairwise orthogonality {#appendix:orthogonal}
=================================================
In this section, we work in the following (non-asymptotic) setting of a single layer: Consider any deterministic matrix $X \in {\mathbb{R}}^{d \times n}$, let $W \in {\mathbb{R}}^{{\check{d}}\times d}$ have i.i.d. ${\mathcal{N}}(0,1)$ entries, and set $$\label{def:vX}
{\widecheck{X}}=\frac{1}{\sqrt{{\check{d}}}}\sigma(WX) \in {\mathbb{R}}^{{\check{d}}\times n}.$$ Note that ${\widecheck{X}}$ has i.i.d. rows with distribution $\sigma({\mathbf{w}}^\top
X)/\sqrt{{\check{d}}}$, where ${\mathbf{w}}\sim {\mathcal{N}}(0,{\operatorname{Id}})$. Define the second-moment matrix of ${\widecheck{X}}$ by $$\label{def:Phi}
\Phi={\mathbb{E}}[{\widecheck{X}}^\top {\widecheck{X}}]
={\mathbb{E}}[\sigma({\mathbf{w}}^\top X)^\top \sigma({\mathbf{w}}^\top X)] \in {\mathbb{R}}^{n \times n}$$ where the expectations are over the standard Gaussian matrix $W$ and standard Gaussian vector ${\mathbf{w}}$. We show in this section the following result.
\[lemma:orthonormalpropagation\] Suppose $X$ is $({\varepsilon},B)$-orthonormal where ${\varepsilon}<1/\lambda_\sigma$. Then for universal constants $C,c>0$, with probability at least $1-2n^2e^{-c{\check{d}}{\varepsilon}^2}-3e^{-cn}$, the matrix ${\widecheck{X}}$ remains $({\widecheck{\varepsilon}},{\widecheck{B}})$-orthonormal with $${\widecheck{\varepsilon}}=C\lambda_\sigma^2{\varepsilon},
\qquad {\widecheck{B}}=C\Big(1+n/{\check{d}}\Big)\lambda_\sigma^2 B.$$
\[cor:orthonormalinduction\] Under Assumption \[assump:asymptotics\], there exist parameters $(\tilde{{\varepsilon}}_n,\tilde{B})$ still satisfying $\tilde{{\varepsilon}}_n n^{1/4} \to 0$, such that a.s. for all large $n$, every matrix $X_0,\ldots,X_L$ is $(\tilde{{\varepsilon}}_n,\tilde{B})$-orthonormal.
Note that increasing ${\varepsilon}_n$ represents a weaker assumption, so we may assume without loss of generality that ${\varepsilon}_n \geq n^{-0.49}$. Then by Lemma \[lemma:orthonormalpropagation\], there is a constant $C_0 \geq 1$ depending on $\lambda_\sigma,\gamma_1,\ldots,\gamma_L$, such that if $X_{\ell-1}$ is $(C_0^{\ell-1}{\varepsilon}_n,C_0^{\ell-1}B)$-orthonormal, then conditional on this event, $X_\ell$ is $(C_0^\ell {\varepsilon}_n,C_0^\ell B)$-orthonormal with probability at least $1-e^{-n^{0.01}}$ for all large $n$. Thus, setting $\tilde{{\varepsilon}}_n=C_0^L{\varepsilon}_n$ and $\tilde{B}=C_0^LB$, with probability at least $1-Le^{-n^{0.01}}$, every matrix $X_0,\ldots,X_L$ is $(\tilde{{\varepsilon}}_n,\tilde{B})$-orthonormal. The almost sure statement then follows from Borel-Cantelli.
In the remainder of this section, we prove Lemma \[lemma:orthonormalpropagation\]. We divide the proof into Lemmas \[lemma:orthog\], \[lemma:normbound\], and \[lemma:colsqbound\] below, which check the individual requirements for $({\widecheck{\varepsilon}},{\widecheck{B}})$-orthonormality of ${\widecheck{X}}$. We denote by $C,C',c,c'>0$ universal constants that may change from instance to instance.
\[lemma:orthog\] If $X$ is $({\varepsilon},B)$-orthonormal where ${\varepsilon}<1/\lambda_\sigma$, then for universal constants $C,c>0$:
(a) For all ${\alpha}\neq {\beta}\in [n]$, $$\begin{aligned}
|\Phi_{{\alpha}{\beta}}-b_\sigma^2 {\mathbf{x}}_{\alpha}^\top {\mathbf{x}}_{\beta}| &\leq C\lambda_\sigma^2 {\varepsilon}^2\label{eq:concentration_offdiagonal}\\
\Big|{\mathbb{E}}_{{\mathbf{w}}\sim {\mathcal{N}}(0,{\operatorname{Id}})}[\sigma({\mathbf{w}}^\top {\mathbf{x}}_{\alpha})]\Big| &\leq C\lambda_\sigma
\Big|\|{\mathbf{x}}_{\alpha}\|^2-1\Big| \leq C\lambda_\sigma {\varepsilon}\label{eq:wxbound}\\
|\Phi_{{\alpha}{\alpha}}-1| & \leq C\lambda_\sigma
\Big|\|{\mathbf{x}}_{\alpha}\|^2-1\Big|\leq C\lambda_\sigma {\varepsilon}\label{eq:concentration_diagonal}\end{aligned}$$
(b) With probability at least $1-2n^2e^{-c{\check{d}}{\varepsilon}^2}$, simultaneously for all ${\alpha}\neq {\beta}\in [n]$, the columns of ${\widecheck{X}}$ satisfy $$\big|\|{\widecheck{\mathbf{x}}}_{\alpha}\|^2-1\big| \leq C\lambda_\sigma^2 {\varepsilon},
\qquad \big|{\widecheck{\mathbf{x}}}_{\alpha}^\top {\widecheck{\mathbf{x}}}_{\beta}\big| \leq C\lambda_\sigma^2 {\varepsilon}.$$
Note that (\[eq:concentration\_offdiagonal\]) establishes an approximation which is second-order in ${\varepsilon}$—this will be important in our later arguments which approximate $\Phi$ in Frobenius norm.
For part (a), observe that $(\zeta_{\alpha},\zeta_{\beta}) \equiv
({\mathbf{w}}^\top {\mathbf{x}}_{\alpha},{\mathbf{w}}^\top {\mathbf{x}}_{\beta})$ is bivariate Gaussian, with mean 0 and covariance $$\Sigma=\begin{pmatrix} \|{\mathbf{x}}_{\alpha}\|^2 & {\mathbf{x}}_{\alpha}^\top {\mathbf{x}}_{\beta}\\
{\mathbf{x}}_{\alpha}^\top {\mathbf{x}}_{\beta}& \|{\mathbf{x}}_{\beta}\|^2 \end{pmatrix}={\operatorname{Id}}+\Delta$$ where $\Delta$ is entrywise bounded by ${\varepsilon}$. Then performing a Gram-Schmidt orthogonalization procedure, for some standard Gaussian variables $\xi_{\alpha},\xi_{\beta}\sim {\mathcal{N}}(0,1)$, we have $$\label{eq:gramschmidt}
\zeta_{\alpha}=u_{\alpha}\xi_{\alpha}, \qquad \zeta_{\beta}=u_{\beta}\xi_{\beta}+v_{\beta}\xi_{\alpha}$$ where $u_{\alpha},u_{\beta}>0$ and $v_{\beta}\in {\mathbb{R}}$ satisfy $|u_{\alpha}-1|,|u_{\beta}-1|,|v_{\beta}| \leq C{\varepsilon}$ for a universal constant $C>0$.
By a Taylor expansion of $\sigma(\zeta)$ around $\zeta=\xi$, there exists a random variable $\eta$ between $\zeta$ and $\xi$ such that $$\label{eq:taylorexpansion}
\sigma(\zeta)=\sigma(\xi)+\sigma'(\xi)(\zeta-\xi)+\frac{1}{2}\sigma''(\eta)(\zeta-\xi)^2.$$ For ${\alpha}\neq {\beta}$, applying this for both $\zeta_{\alpha}$ and $\zeta_{\beta}$, noting that the product of leading terms satisfies ${\mathbb{E}}[\sigma(\xi_{\alpha})\sigma(\xi_{\beta})]=0$, and applying also the bounds $|\sigma'(x)|,|\sigma''(x)| \leq \lambda_\sigma$ where $\lambda_\sigma \geq 1$, it is easy to check that $$\Phi_{{\alpha}{\beta}}={\mathbb{E}}[\sigma(\zeta_{\alpha})\sigma(\zeta_{\beta})]
={\mathbb{E}}\Big[\sigma(\xi_{\alpha}) \cdot
\sigma'(\xi_{\beta})(\zeta_{\beta}-\xi_{\beta})+\sigma(\xi_{\beta}) \cdot
\sigma'(\xi_{\alpha})(\zeta_{\alpha}-\xi_{\alpha}) \Big]+\text{remainder}$$ where this remainder has magnitude at most $C\lambda_\sigma^2{\varepsilon}^2$. For the first term, substituting (\[eq:gramschmidt\]) and applying independence of $\xi_{\alpha}$ and $\xi_{\beta}$, we have $$\begin{aligned}
&{\mathbb{E}}\Big[\sigma(\xi_{\alpha}) \cdot
\sigma'(\xi_{\beta})(\zeta_{\beta}-\xi_{\beta})+\sigma(\xi_{\beta}) \cdot
\sigma'(\xi_{\alpha})(\zeta_{\alpha}-\xi_{\alpha}) \Big]\\
&=(u_{\beta}-1) {\mathbb{E}}[\sigma(\xi_{\alpha})] \cdot {\mathbb{E}}[\sigma'(\xi_{\beta})\xi_{\beta}]
+v_{\beta}{\mathbb{E}}[\sigma(\xi_{\alpha})\xi_{\alpha}] \cdot {\mathbb{E}}[\sigma'(\xi_{\beta})]
+(u_{\alpha}-1) {\mathbb{E}}[\sigma(\xi_{\beta})] \cdot {\mathbb{E}}[\sigma'(\xi_{\alpha})\xi_{\alpha}].\end{aligned}$$ Applying ${\mathbb{E}}[\sigma(\xi)]=0$ and the integration-by-parts identity ${\mathbb{E}}[\sigma(\xi)\xi]={\mathbb{E}}[\sigma'(\xi)]=b_\sigma$, this term equals $v_{\beta}b_\sigma^2$. From (\[eq:gramschmidt\]), we have $u_{\alpha}v_{\beta}={\mathbb{E}}[\zeta_{\alpha}\zeta_{\beta}]={\mathbf{x}}_{\alpha}^\top {\mathbf{x}}_{\beta}$. Since $|u_{\alpha}-1| \leq C{\varepsilon}$ and $|{\mathbf{x}}_{\alpha}^\top {\mathbf{x}}_{\beta}|
\leq {\varepsilon}$, this implies $|v_{\beta}b_\sigma^2-b_\sigma^2 {\mathbf{x}}_{\alpha}^\top {\mathbf{x}}_{\beta}| \leq
Cb_\sigma^2 {\varepsilon}^2 \leq C\lambda_\sigma^2 {\varepsilon}^2$. Combining these yields (\[eq:concentration\_offdiagonal\]). Similarly, from a first-order Taylor expansion analogous to (\[eq:taylorexpansion\]), $$\begin{aligned}
\Big|{\mathbb{E}}[\sigma({\mathbf{w}}^\top {\mathbf{x}}_{\alpha})]\Big|
&=\Big|{\mathbb{E}}[\sigma(\zeta_{\alpha})]-{\mathbb{E}}[\sigma(\xi_{\alpha})]\Big|
\leq C\lambda_\sigma \cdot |u_{\alpha}-1|,\\
|\Phi_{{\alpha}{\alpha}}-1|&=\Big|{\mathbb{E}}[\sigma(\zeta_{\alpha})^2]-{\mathbb{E}}[\sigma(\xi_{\alpha})^2]\Big|
\leq C\max\Big(\lambda_\sigma \cdot |u_{\alpha}-1|,\;\lambda_\sigma^2 \cdot
|u_{\alpha}-1|^2\Big).\end{aligned}$$ The bounds (\[eq:wxbound\]) and (\[eq:concentration\_diagonal\]) follow from the observations $u_{\alpha}^2={\mathbb{E}}[\zeta_{\alpha}^2]=\|{\mathbf{x}}_{\alpha}\|^2$ and $|u_{\alpha}-1| \leq |u_{\alpha}-1| \cdot |u_{\alpha}+1|=|u_{\alpha}^2-1| \leq {\varepsilon}$.
For part (b), let ${\mathbf{w}}_k^\top$ be the $k^\text{th}$ row of $W$. Then by definition of ${\widecheck{X}}$, for any ${\alpha},{\beta}\in [n]$ (including ${\alpha}={\beta}$), $${\widecheck{\mathbf{x}}}_{\alpha}^\top {\widecheck{\mathbf{x}}}_{\beta}=\frac{1}{{\check{d}}} \sum_{k=1}^{{\check{d}}}
\sigma\Big({\mathbf{w}}_k^\top {\mathbf{x}}_{\alpha}\Big)
\sigma\Big({\mathbf{w}}_k^\top {\mathbf{x}}_{\beta}\Big).$$ We apply Bernstein’s inequality: Denote by $\|\cdot\|_{\psi_2}$ and $\|\cdot\|_{\psi_1}$ the sub-Gaussian and sub-exponential norms of a random variable. For any deterministic vector ${\mathbf{x}}\in {\mathbb{R}}^{d}$, the function ${\mathbf{w}}\mapsto \sigma({\mathbf{w}}^\top {\mathbf{x}})$ is $\lambda_\sigma \|{\mathbf{x}}\|$-Lipschitz. Then for ${\mathbf{w}}\sim {\mathcal{N}}(0,{\operatorname{Id}})$ and a universal constant $C>0$, we have by Gaussian concentration-of-measure $$\|\sigma({\mathbf{w}}^\top {\mathbf{x}}_{\alpha})-{\mathbb{E}}[\sigma({\mathbf{w}}^\top {\mathbf{x}}_{\alpha})]\|_{\psi_2}
\leq C\lambda_\sigma \|{\mathbf{x}}_{\alpha}\|.$$ From (\[eq:wxbound\]), $|{\mathbb{E}}[\sigma({\mathbf{w}}^\top {\mathbf{x}}_{\alpha})]| \leq C\lambda_\sigma {\varepsilon}$. Thus (recalling that $|\|{\mathbf{x}}_{\alpha}\|-1| \leq {\varepsilon}$), we have $\|\sigma({\mathbf{w}}^\top {\mathbf{x}}_{\alpha})\|_{\psi_2} \leq C\lambda_\sigma$ for a constant $C>0$, and similarly for ${\mathbf{x}}_{\beta}$. So $$\label{eq:subexpbound}
\|\sigma({\mathbf{w}}^\top {\mathbf{x}}_{\alpha}) \sigma({\mathbf{w}}^\top {\mathbf{x}}_{\beta})\|_{\psi_1}
\leq \|\sigma({\mathbf{w}}^\top {\mathbf{x}}_{\alpha})\|_{\psi_2}
\|\sigma({\mathbf{w}}^\top {\mathbf{x}}_{\beta})\|_{\psi_2} \leq C\lambda_\sigma^2.$$ Applying Bernstein’s inequality (see [@vershynin2018high Theorem 2.8.1]), for a universal constant $c>0$ and any $t>0$, $${\mathbb{P}}\Big[\big|{\widecheck{\mathbf{x}}}_{\alpha}^\top {\widecheck{\mathbf{x}}}_{\beta}-{\mathbb{E}}\big[{\widecheck{\mathbf{x}}}_{\alpha}^\top {\widecheck{\mathbf{x}}}_{\beta}\big] \big|>t \Big]\leq 2\exp\left(-c{\check{d}}\min\left(\frac{t^2}{\lambda_\sigma^4},\frac{t}{\lambda_\sigma^2}\right)\right).$$ Applying this for $t=\lambda_\sigma^2{\varepsilon}$ and taking a union bound over all ${\alpha},{\beta}\in [n]$, we get $$\begin{aligned}
{\mathbb{P}}\Big[\big|{\widecheck{\mathbf{x}}}_{\alpha}^\top {\widecheck{\mathbf{x}}}_{\beta}-{\mathbb{E}}\big[{\widecheck{\mathbf{x}}}_{\alpha}^\top {\widecheck{\mathbf{x}}}_{\beta}\big] \big|\leq
\lambda_\sigma^2 {\varepsilon}\text{ for all } {\alpha},{\beta}\in [n] \Big] \geq 1-2n^2\exp\left(-c{\check{d}}\cdot {\varepsilon}^2\right).\end{aligned}$$ Since ${\mathbb{E}}[{\widecheck{\mathbf{x}}}_{\alpha}^\top {\widecheck{\mathbf{x}}}_{\beta}]=\Phi_{{\alpha}{\beta}}$, part (b) now follows from part (a).
\[lemma:normbound\] If $X$ is $({\varepsilon},B)$-orthonormal where ${\varepsilon}<1/\lambda_\sigma$, then for universal constants $C,c>0$:
(a) $\|\Phi\| \leq C\lambda_\sigma^2 B^2$.
(b) With probability at least $1-2e^{-cn}$, $\|{\widecheck{X}}\| \leq C\Big(1+\sqrt{n/{\check{d}}}\Big)\lambda_\sigma B$.
For part (a), define $$\label{eq:Sigma}
\Sigma={\mathbb{E}}\Big[\sigma({\mathbf{w}}^\top X)^\top \sigma({\mathbf{w}}^\top X)\Big]
-{\mathbb{E}}[\sigma({\mathbf{w}}^\top X)]^\top{\mathbb{E}}[\sigma({\mathbf{w}}^\top X)]$$ where the first term on the right is $\Phi$. Then $$\|\Sigma\|=\sup_{{\mathbf{v}}:\|{\mathbf{v}}\|=1} {\mathbf{v}}^\top \Sigma {\mathbf{v}}=\sup_{{\mathbf{v}}:\|{\mathbf{v}}\|=1} \left|{\mathbb{E}}\Big[\big(\sigma({\mathbf{w}}^\top X)
{\mathbf{v}}\big)^2\Big]-{\mathbb{E}}\Big[\sigma({\mathbf{w}}^\top X){\mathbf{v}}\Big]^2\right|
=\sup_{{\mathbf{v}}:\|{\mathbf{v}}\|=1} {\operatorname{Var}}\big[\sigma({\mathbf{w}}^\top X){\mathbf{v}}\big].$$ We bound this variance using the Gaussian Poincaré inequality: Let us fix ${\mathbf{v}}\in{\mathbb{R}}^n$ with $\|{\mathbf{v}}\|=1$ and define $$F({\mathbf{w}})=\sigma({\mathbf{w}}^\top X){\mathbf{v}}=\sum_{{\alpha}=1}^n v_{\alpha}\sigma({\mathbf{w}}^\top {\mathbf{x}}_{\alpha}).$$ Then, letting ${\mathbf{u}}\in {\mathbb{R}}^n$ be the vector with entries $u_{\alpha}=v_{\alpha}\sigma'({\mathbf{w}}^\top {\mathbf{x}}_{\alpha})$, $$\label{eq:Flipschitz}
\nabla F({\mathbf{w}})=\sum_{{\alpha}=1}^n v_{\alpha}\sigma'({\mathbf{w}}^\top {\mathbf{x}}_{\alpha}) \cdot {\mathbf{x}}_{\alpha}=X{\mathbf{u}}, \qquad \|\nabla F({\mathbf{w}})\| \leq \|X\| \cdot \|{\mathbf{u}}\|
\leq \lambda_\sigma B.$$ Then by the Gaussian Poincaré inequality, ${\operatorname{Var}}[F({\mathbf{w}})] \leq {\mathbb{E}}[\|\nabla F({\mathbf{w}})\|^2] \leq \lambda_\sigma^2B^2$, so $\|\Sigma\| \leq \lambda_\sigma^2 B^2$. In addition, by (\[eq:wxbound\]), the difference between $\Phi$ and $\Sigma$ is a rank-one perturbation controlled by $$\label{eq:rankone_perturbation_bound}
\|\Phi-\Sigma\|=\|{\mathbb{E}}[\sigma({\mathbf{w}}^\top X)]\|^2=\sum_{{\alpha}=1}^n
{\mathbb{E}}[\sigma({\mathbf{w}}^\top {\mathbf{x}}_{\alpha})]^2 \leq C\lambda_\sigma^2
\sum_{{\alpha}=1}^n (\|{\mathbf{x}}_{\alpha}\|^2-1)^2 \leq C \lambda_\sigma^2B^2,$$ the last inequality using the final condition of $({\varepsilon},B)$-orthonormality in Definition \[def:orthogonal\]. This establishes part (a).
For part (b), we apply the concentration result of [@vershynin2010introduction Eq. (5.26)] for matrices with independent sub-Gaussian rows. For any fixed unit vector ${\mathbf{v}}\in {\mathbb{R}}^n$, recall from (\[eq:Flipschitz\]) that $F({\mathbf{w}})=\sigma({\mathbf{w}}^\top X){\mathbf{v}}$ is $\lambda_\sigma
B$-Lipschitz. Then by Gaussian concentration-of-measure, $$\|F({\mathbf{w}})-{\mathbb{E}}[F({\mathbf{w}})]\|_{\psi_2} \leq C\lambda_\sigma B.$$ We have $|{\mathbb{E}}[F({\mathbf{w}})]| \leq \|{\mathbb{E}}[\sigma({\mathbf{w}}^\top X)]\| \leq C\lambda_\sigma B$ by (\[eq:rankone\_perturbation\_bound\]), so also $\|F({\mathbf{w}})\|_{\psi_2} \leq C\lambda_\sigma B$. This holds for any unit vector ${\mathbf{v}}\in {\mathbb{R}}^n$, hence $\|\sigma({\mathbf{w}}^\top X)\|_{\psi_2} \leq C\lambda_\sigma B$ for the vector sub-Gaussian norm. Thus, $\sqrt{d}{\widecheck{X}}/(\lambda_\sigma B)$ has i.i.d.rows whose sub-Gaussian norm is at most a universal constant. Recalling $\Phi={\widecheck{X}}^\top {\widecheck{X}}$ and applying [@vershynin2010introduction Eq. (5.26)] with $A=\sqrt{d}{\widecheck{X}}/(\lambda_\sigma B)$, we obtain for some universal constants $C,c>0$ that $${\mathbb{P}}\left[\|{\widecheck{X}}^\top {\widecheck{X}}-\Phi\|>
\max(\delta,\delta^2)\|\Phi\|\right] \leq 2e^{-ct^2},
\qquad \delta=C\sqrt{n/{\check{d}}}+t/\sqrt{{\check{d}}}.$$ Note that the complementary event $\|{\widecheck{X}}^\top {\widecheck{X}}-\Phi\|
\leq \max(\delta,\delta^2)\|\Phi\|$ implies $$\|{\widecheck{X}}\| \leq \sqrt{(1+\max(\delta,\delta^2))\|\Phi\|}
\leq (1+C'\delta)\sqrt{\|\Phi\|}$$ for a constant $C'>0$. Then choosing $t=\sqrt{n}$ and applying part (a) yields part (b).
\[lemma:colsqbound\] If $X$ is $({\varepsilon},B)$-orthonormal where ${\varepsilon}<1/\lambda_\sigma$, then for universal constants $C,c>0$, with probability at least $1-e^{-cn}$, the columns of ${\widecheck{X}}$ satisfy $$\sum_{{\alpha}=1}^n (\|{\widecheck{\mathbf{x}}}_{\alpha}\|^2-1)^2 \leq C\Big(1+n^2/{\check{d}}^2\Big)\lambda_\sigma^4
B^2.$$
Let us remark that in settings where ${\varepsilon}\gg 1/\sqrt{n}$, applying Lemma \[lemma:orthog\](b) to bound each term $(\|{\widecheck{\mathbf{x}}}_{\alpha}\|^2-1)^2$ separately would not yield a constant-order bound for this sum. The proof below performs a more careful analysis of the combined fluctuations of $(\|{\widecheck{\mathbf{x}}}_{\alpha}\|^2-1)^2$.
Let ${\mathbf{z}}=(z_1,\ldots,z_n) \in {\mathbb{R}}^n$ and ${\mathbf{r}}=(r_1,\ldots,r_n) \in {\mathbb{R}}^n$ be defined as $$z_{\alpha}=\|{\widecheck{\mathbf{x}}}_{\alpha}\|^2-{\mathbb{E}}[\|{\widecheck{\mathbf{x}}}_{\alpha}\|^2],
\qquad r_{\alpha}={\mathbb{E}}[\|{\widecheck{\mathbf{x}}}_{\alpha}\|^2]-1.$$ The quantity to be bounded is $\|{\mathbf{z}}+{\mathbf{r}}\|^2$. Note that $\|{\mathbf{z}}+{\mathbf{r}}\|^2 \leq 2\|{\mathbf{z}}\|^2+2\|{\mathbf{r}}\|^2$. We have $${\mathbb{E}}[\|{\widecheck{\mathbf{x}}}_{\alpha}\|^2]={\mathbb{E}}\left[\frac{1}{{\check{d}}} \sum_{i=1}^{{\check{d}}}
\sigma({\mathbf{w}}_i^\top {\mathbf{x}}_{\alpha})^2 \right]=\Phi_{{\alpha}{\alpha}},$$ so applying (\[eq:concentration\_diagonal\]) from Lemma \[lemma:orthog\], $$\label{eq:rsqbound}
\|{\mathbf{r}}\|^2=\sum_{{\alpha}=1}^n (\Phi_{{\alpha}{\alpha}}-1)^2
\leq C\lambda_\sigma^2 \sum_{{\alpha}=1}^n (\|{\mathbf{x}}_{\alpha}\|^2-1)^2 \leq C\lambda_\sigma^2
B^2.$$ Thus it remains to bound $\|{\mathbf{z}}\|^2$.
Let ${\mathcal{N}}$ be a $1/2$-net of the unit ball $\{{\mathbf{w}}\in {\mathbb{R}}^n:\|{\mathbf{w}}\|=1\}$, of cardinality $|{\mathcal{N}}| \leq 5^n$. Then $$\|{\mathbf{z}}\|=\sup_{{\mathbf{w}}:\|{\mathbf{w}}\| \leq 1}
{\mathbf{w}}^\top {\mathbf{z}}\leq \sup_{{\mathbf{v}}\in {\mathcal{N}}} {\mathbf{v}}^\top {\mathbf{z}}+\|{\mathbf{z}}\|/2,$$ so $\|{\mathbf{z}}\| \leq 2\sup_{{\mathbf{v}}\in {\mathcal{N}}} {\mathbf{v}}^\top {\mathbf{z}}$. For each fixed vector ${\mathbf{v}}=(v_1,\ldots,v_n) \in {\mathcal{N}}$, we have $$\begin{aligned}
{\mathbf{v}}^\top {\mathbf{z}}&=\sum_{{\alpha}=1}^n v_{\alpha}\cdot \frac{1}{{\check{d}}}
\sum_{i=1}^{{\check{d}}} \Big(\sigma({\mathbf{w}}_i^\top {\mathbf{x}}_{\alpha})^2-{\mathbb{E}}[\sigma({\mathbf{w}}_i^\top {\mathbf{x}}_{\alpha})^2]
\Big)\nonumber\\
&=\frac{1}{{\check{d}}}\sum_{i=1}^{{\check{d}}}
\bigg(\sum_{{\alpha}=1}^n
\Big(\sigma({\mathbf{w}}_i^\top {\mathbf{x}}_{\alpha})^2-{\mathbb{E}}[\sigma({\mathbf{w}}_i^\top {\mathbf{x}}_{\alpha})^2]\Big) v_{\alpha}\bigg).
\label{eq:vzprod}\end{aligned}$$ We will bound the sub-exponential norm of each summand $i=1,\ldots,{\check{d}}$ and apply Bernstein’s inequality.
For ${\mathbf{w}}\sim {\mathcal{N}}(0,{\operatorname{Id}})$, denote $${\mathbf{q}}\equiv {\mathbf{q}}({\mathbf{w}})=(q_1,\ldots,q_n)=({\mathbf{w}}^\top {\mathbf{x}}_1,\ldots,{\mathbf{w}}^\top {\mathbf{x}}_n),
\qquad F({\mathbf{q}})=\sum_{{\alpha}=1}^n \Big(\sigma(q_{\alpha})^2
-{\mathbb{E}}[\sigma(q_{\alpha})^2] \Big)v_{\alpha}.$$ Observe that ${\mathbf{q}}({\mathbf{w}})=X^\top {\mathbf{w}}$. Thus we wish to bound the sub-exponential norm of $F({\mathbf{q}}({\mathbf{w}}))$ when ${\mathbf{w}}\sim
{\mathcal{N}}(0,{\operatorname{Id}})$. By the Gaussian Sobolev inequality (see [@adamczak2015concentration Eq. (3)]), for any $p \geq 2$, $$\label{eq:sobolev}
\|F({\mathbf{q}}({\mathbf{w}}))\|_{L^p} \leq \sqrt{p} \cdot \Big\|\|\nabla_{{\mathbf{w}}}
F({\mathbf{q}}({\mathbf{w}}))\|\Big\|_{L^p}$$ where $\|Y\|_{L^p}={\mathbb{E}}[|Y|^p]^{1/p}$ denotes the $L^p$-norm of a random variable (and $\|\nabla_{\mathbf{w}}F({\mathbf{q}}({\mathbf{w}}))\|$ is the usual $\ell_2$ vector norm of the gradient of $F({\mathbf{q}}({\mathbf{w}}))$ in ${\mathbf{w}}$). By the chain rule, $$\nabla_{{\mathbf{w}}} F({\mathbf{q}}({\mathbf{w}}))=X \cdot \nabla_{\mathbf{q}}F({\mathbf{q}}),$$ so $$\|\nabla_{{\mathbf{w}}} F({\mathbf{q}}({\mathbf{w}}))\|^2 \leq \|X\|^2 \|\nabla_{\mathbf{q}}F({\mathbf{q}})\|^2
\leq B^2\|\nabla_{\mathbf{q}}F({\mathbf{q}})\|^2.$$ We have $(\partial/\partial q_{\alpha}) F({\mathbf{q}})=2\sigma(q_{\alpha})\sigma'(q_{\alpha})v_{\alpha}$, so $$\|\nabla_{\mathbf{q}}F({\mathbf{q}})\|^2
=\sum_{{\alpha}=1}^n 4\sigma(q_{\alpha})^2\sigma'(q_{\alpha})^2v_{\alpha}^2
\leq 4\lambda_\sigma^2 \sum_{{\alpha}=1}^n \sigma(q_{\alpha})^2v_{\alpha}^2.$$ Recalling (\[eq:subexpbound\]), we have $\|\sigma(q_{\alpha})^2\|_{\psi_1}
=\|\sigma({\mathbf{w}}^\top {\mathbf{x}}_{\alpha})^2\|_{\psi_1} \leq C\lambda_\sigma^2$. Then $$\left\|\sum_{{\alpha}=1}^n \sigma(q_{\alpha})^2v_{\alpha}^2\right\|_{\psi_1}
\leq C\lambda_\sigma^2 \sum_{{\alpha}=1}^n v_{\alpha}^2=C\lambda_\sigma^2,$$ so $$\Big\|\|\nabla_{{\mathbf{w}}} F({\mathbf{q}}({\mathbf{w}}))\|^2\Big\|_{\psi_1}
\leq C \lambda_\sigma^4B^2.$$ This implies the bound (see [@vershynin2018high Proposition 2.7.1]), for any $p \geq 1$, $$\Big\|\|\nabla_{{\mathbf{w}}} F({\mathbf{q}}({\mathbf{w}}))\|\Big\|_{L^{2p}}^{2p}
={\mathbb{E}}\Big[\|\nabla_{{\mathbf{w}}} F({\mathbf{q}}({\mathbf{w}}))\|^{2p}\Big]
=\Big\|\|\nabla_{{\mathbf{w}}} F({\mathbf{q}}({\mathbf{w}}))\|^2\Big\|_{L^p}^p
\leq (C'\lambda_\sigma^4B^2 \cdot p)^p$$ for a universal constant $C'>0$. Thus, applying this to (\[eq:sobolev\]), we obtain for any $p \geq 2$ $$\|F({\mathbf{q}}({\mathbf{w}}))\|_{L^p} \leq \sqrt{p} \cdot C\lambda_\sigma^2B \sqrt{p}
=C\lambda_\sigma^2B \cdot p.$$ Finally, this implies (see again [@vershynin2018high Proposition 2.7.1]) $\|F({\mathbf{q}}({\mathbf{w}}))\|_{\psi_1} \leq C'\lambda_\sigma^2B$ for a universal constant $C'>0$, which is our desired bound on the sub-exponential norm of $F({\mathbf{q}}({\mathbf{w}}))$.
Applying this and Bernstein’s inequality to (\[eq:vzprod\]), for any $t>0$, $${\mathbb{P}}[{\mathbf{v}}^\top {\mathbf{z}}>t]
\leq \exp\left(-c{\check{d}}\min\left(\frac{t^2}{\lambda_\sigma^4B^2},
\frac{t}{\lambda_\sigma^2B}\right)\right).$$ Setting $$t=C_0\lambda_\sigma^2B \cdot \max(\delta,\delta^2),
\qquad \delta=\sqrt{n/{\check{d}}}$$ for a large enough constant $C_0>0$, and taking the union bound over all $5^n$ vectors ${\mathbf{v}}\in {\mathcal{N}}$, we get $${\mathbb{P}}[\|{\mathbf{z}}\|>2t]
\leq {\mathbb{P}}\left[\sup_{{\mathbf{v}}\in {\mathcal{N}}} {\mathbf{v}}^\top {\mathbf{z}}>t\right] \leq e^{-cn}$$ for a constant $c>0$. Combining with the bound on $\|{\mathbf{r}}\|^2$ in (\[eq:rsqbound\]), we obtain the lemma.
Resolvent analysis for a single layer {#appendix:singlelayer}
=====================================
We consider the same setting of a single layer as in the preceding section. Let ${\widecheck{X}}$ and $\Phi$ be defined by the deterministic input $X \in {\mathbb{R}}^{d \times n}$ and Gaussian matrix $W \in {\mathbb{R}}^{{\check{d}}\times d}$ as in (\[def:vX\]) and (\[def:Phi\]), and define the ($n$-dependent) aspect ratio $$\gamma=n/{\check{d}}.$$ Consider a deterministic real-symmetric matrix $A \in {\mathbb{R}}^{n \times n}$, and two (possibly $n$-dependent) spectral arguments $\alpha \in {\mathbb{C}}^*$ and $z \in {\mathbb{C}}^+$, where ${\mathbb{C}}^*=\overline{{\mathbb{C}}^-} \setminus \{0\}$. We study the matrix $$A+\alpha {\widecheck{X}}^\top {\widecheck{X}}-z{\operatorname{Id}}.$$
We collect here the set of assumptions that we will use in this section.
\[assump:singlelayer\] There are constants $B,C_0,c_0>0$ such that
(a) $\alpha \in {\mathbb{C}}^*$ and $z \in {\mathbb{C}}^+$, and $\gamma,|\alpha|,|z|,{\operatorname{Im}}z \in [c_0,C_0]$.
(b) $X$ is $({\varepsilon}_n,B)$-orthonormal, where ${\varepsilon}_n<n^{-0.01}$.
(c) $A \in {\mathbb{R}}^{n \times n}$ is deterministic and symmetric, satisfying $\|A\| \leq C_0$.
(d) $W$ has i.i.d. ${\mathcal{N}}(0,1)$ entries, and $\sigma(x)$ satisfies Assumption \[assump:asymptotics\](b).
Throughout this section, $C,C',c,c',n_0>0$ denote constants changing from instance to instance that may depend on $\lambda_\sigma$ and the above values $B,C_0,c_0$.
Proposition \[prop:invertible\] ensures that $A+\alpha{\widecheck{X}}^\top{\widecheck{X}}-z{\operatorname{Id}}$ is invertible. Define the resolvent $$\label{eq:resolvent}
R=(A+\alpha {\widecheck{X}}^\top {\widecheck{X}}-z{\operatorname{Id}})^{-1} \in {\mathbb{C}}^{n \times n}$$ and the deterministic ($n$-dependent) parameter $$\label{eq:sbar}
\bar{s}=\alpha^{-1}+\gamma \cdot {\mathbb{E}}[{\operatorname{tr}}R\Phi].$$ The goal of this section is to prove the following result, which approximates this resolvent $R$ by replacing the random matrix $\alpha {\widecheck{X}}^\top {\widecheck{X}}$ with a deterministic matrix $\bar{s}^{-1}\Phi$, and provides an approximate fixed-point equation that defines this parameter $\bar{s}$.
For $A=0$ and $\alpha=1$, we will verify in Appendix \[appendix:CK\] that this result reduces to the Marcenko-Pastur equation (\[eq:MPeq\]).
\[lemma:fixedpoint\] Under Assumption \[assump:singlelayer\], there are constants $C,c,c',n_0>0$ such that for all $n \geq n_0$, any deterministic matrix $M \in {\mathbb{C}}^{n \times n}$, and any $t \in (n^{-1},c')$,
(a) $\displaystyle {\mathbb{P}}\left[\left|{\operatorname{tr}}RM-{\operatorname{tr}}\left(A+\bar{s}^{-1}\Phi-z{\operatorname{Id}}\right)^{-1}M\right|
>\|M\|t\right] \leq Cne^{-cnt^2}$
(b) $\displaystyle {\mathbb{P}}\left[\left|\bar{s}-\big(\alpha^{-1}+\gamma
{\operatorname{tr}}\left(A+\bar{s}^{-1} \Phi-z{\operatorname{Id}}\right)^{-1}\Phi\big)\right|
>t\right] \leq Cne^{-cnt^2}$
Basic bounds
------------
\[prop:basic\] Under Assumption \[assump:singlelayer\], deterministically for some constants $C,c,n_0>0$ and all $n \geq n_0$, $$\|R\| \leq C, \qquad \|\Phi\| \leq C, \qquad
|\bar{s}| \leq C, \qquad {\operatorname{Im}}\bar{s} \geq c.$$ Furthermore, with probability at least $1-2e^{-c'n}$ for a constant $c'>0$, $${\operatorname{Im}}{\operatorname{tr}}R\Phi \geq c.$$
We may write $A+\alpha {\widecheck{X}}^\top {\widecheck{X}}-z{\operatorname{Id}}=U+iV$ where $U=A+({\operatorname{Re}}\alpha){\widecheck{X}}^\top
{\widecheck{X}}-({\operatorname{Re}}z){\operatorname{Id}}$ and $V=({\operatorname{Im}}\alpha) {\widecheck{X}}^\top {\widecheck{X}}^\top-({\operatorname{Im}}z){\operatorname{Id}}$. Both $U$ and $V$ are symmetric, and $V \preceq (-{\operatorname{Im}}z){\operatorname{Id}}$ because ${\operatorname{Im}}\alpha \leq 0$ and ${\operatorname{Im}}z>0$. Then $\|R\| \leq 1/{\operatorname{Im}}z \leq C$ by Proposition \[prop:invertible\].
The bound $\|\Phi\| \leq C$ comes from Lemma \[lemma:normbound\](a) and the $({\varepsilon}_n,B)$-orthonormality assumption for $X$. Then from the definition of $\bar{s}$ in (\[eq:sbar\]) and the bounds $\|R\|,\|\Phi\| \leq C$, we have also $|\bar{s}| \leq C$. For the lower bound for ${\operatorname{Im}}\bar{s}$ and ${\operatorname{Im}}{\operatorname{tr}}R\Phi$, let us write $${\operatorname{tr}}R\Phi={\operatorname{tr}}\left(\frac{R+R^*}{2}\right)\Phi+{\operatorname{tr}}\left(\frac{R-R^*}{2}\right)\Phi.$$ The first trace is real because $R+R^*$ is Hermitian, so $${\operatorname{Im}}{\operatorname{tr}}R\Phi={\operatorname{Im}}{\operatorname{tr}}\left(\frac{R-R^*}{2}\right)\Phi.$$ Denoting $Y=A+\alpha {\widecheck{X}}^\top {\widecheck{X}}-z{\operatorname{Id}}$ and applying the identity $A^{-1}-B^{-1}=A^{-1}(B-A)B^{-1}$, we have $$R-R^*=Y^{-1}-(Y^*)^{-1}
=Y^{-1}(Y^*-Y)(Y^*)^{-1}=R(Y^*-Y)R^*.$$ Then, writing $Y=U+iV$ as above and applying $Y^*-Y=-2iV$, we get $$\begin{aligned}
{\operatorname{Im}}{\operatorname{tr}}R\Phi&={\operatorname{Im}}(-i \cdot {\operatorname{tr}}RVR^*\Phi)\\
&={\operatorname{Re}}\left(-({\operatorname{Im}}\alpha) \cdot {\operatorname{tr}}R {\widecheck{X}}^\top {\widecheck{X}}R^*\Phi
+({\operatorname{Im}}z) \cdot {\operatorname{tr}}RR^*\Phi\right).\end{aligned}$$ Since ${\operatorname{tr}}R{\widecheck{X}}^\top {\widecheck{X}}R^*\Phi={\operatorname{tr}}\Phi^{1/2}R{\widecheck{X}}^\top {\widecheck{X}}R^*\Phi^{1/2}$, where this matrix is positive semi-definite, this trace is real and non-negative. Similarly, ${\operatorname{tr}}RR^*\Phi$ is real and non-negative. Then the above yields the lower bound $${\operatorname{Im}}{\operatorname{tr}}R\Phi \geq {\operatorname{Im}}z \cdot {\operatorname{tr}}RR^*\Phi
\geq {\operatorname{Im}}z \cdot \lambda_{\min}(RR^*) \cdot {\operatorname{tr}}\Phi,$$ where $\lambda_{\min}(RR^*)$ is the smallest eigenvalue of $RR^*$. By (\[eq:concentration\_diagonal\]) and the condition ${\varepsilon}_n<n^{-0.01}$, we have ${\operatorname{tr}}\Phi \geq c$ for a constant $c>0$ and large enough $n_0$. Observe that $\lambda_{\min}(RR^*)=1/\|Y\|^2$, and $\|Y\| \leq \|A\|+|\alpha| \cdot \|{\widecheck{X}}\|^2+|z|$. By Lemma \[lemma:normbound\](b), with probability $1-2e^{-c'n}$, we have $\|{\widecheck{X}}\| \leq C$, so putting this together yields ${\operatorname{Im}}{\operatorname{tr}}R\Phi \geq c$ with this probability. Finally, for the deterministic bound ${\operatorname{Im}}\bar{s} \geq c$, we may apply ${\operatorname{Im}}{\operatorname{tr}}R\Phi \geq c$ on the event where $\|{\widecheck{X}}\| \leq C$ holds, and ${\operatorname{Im}}{\operatorname{tr}}R\Phi \geq 0$ on the complementary event. Taking an expectation and applying the definition (\[eq:sbar\]) yields ${\operatorname{Im}}\bar{s} \geq c$.
Resolvent approximation
-----------------------
We recall the result of [@louart2018random Lemma 1], which establishes concentration of quadratic forms in the rows of ${\widecheck{X}}$. The following is its specialization to standard Gaussian matrices $W$, and stated in our notation.
\[lemma:quadform\] Suppose $\sigma(x)$ is $\lambda_\sigma$-Lipschitz, and let ${\widecheck{\mathbf{x}}}_i^\top$ be a row of ${\widecheck{X}}$. Then for any deterministic matrix $Y \in {\mathbb{R}}^{n\times n}$ with $\|Y\|\le 1$, for some constants $C,c>0$ (depending on $\lambda_\sigma$), and for any $t>0$, $$\label{eq:concentration_quadratic}
{\mathbb{P}}\left(\left|\frac{1}{\gamma}{\widecheck{\mathbf{x}}}_i^\top Y {\widecheck{\mathbf{x}}}_i-{\operatorname{tr}}Y\Phi \right|>t\right)
\leq C\exp\left(-\frac{cn}{\|X\|^2}\min\left(\frac{t^2}{t_0^2},
t\right)\right)$$ where $t_0=|\sigma(0)|+\lambda_\sigma\|X\|\sqrt{1/\gamma}$.
Using this result, we establish the following approximation for the resolvent $R$ in (\[eq:resolvent\]).
\[lemma:resolvent\_remainder\] Consider any deterministic matrix $M \in {\mathbb{C}}^{n \times n}$, and set $$\delta_n={\operatorname{tr}}M-{\operatorname{tr}}R\left(A+\frac{1}{\alpha^{-1}+\gamma {\operatorname{tr}}R\Phi}\Phi-z{\operatorname{Id}}\right)M.$$ Under Assumption \[assump:singlelayer\], there exist constants $C,c,c',n_0>0$ such that for all $n \geq n_0$ and $t \in (n^{-1},c')$, $${\mathbb{P}}[|\delta_n|>\|M\|t] \leq Cne^{-cnt^2}.$$
By rescaling $M$, we may assume that $\|M\| \leq 1$. We have ${\operatorname{Id}}=R(A+\alpha {\widecheck{X}}^\top {\widecheck{X}}-z{\operatorname{Id}})=RA+\alpha R{\widecheck{X}}^\top {\widecheck{X}}-zR$. Writing ${\widecheck{X}}^\top {\widecheck{X}}=\sum_i {\widecheck{\mathbf{x}}}_i{\widecheck{\mathbf{x}}}_i^\top$ (where ${\widecheck{\mathbf{x}}}_i^\top$ is the $i^\text{th}$ row of ${\widecheck{X}}$), multiplying by $M$, and taking the normalized trace ${\operatorname{tr}}=n^{-1}{\operatorname{Tr}}$, $$\begin{aligned}
{\operatorname{tr}}M&={\operatorname{tr}}RAM+\alpha {\operatorname{tr}}R{\widecheck{X}}^\top {\widecheck{X}}M-z{\operatorname{tr}}RM\\
&={\operatorname{tr}}RAM+\frac{\alpha}{n}\sum_{i=1}^{{\check{d}}} {\widecheck{\mathbf{x}}}_i^\top MR{\widecheck{\mathbf{x}}}_i-z{\operatorname{tr}}RM.\end{aligned}$$ Hence $$\delta_n=\frac{\alpha}{n}\sum_{i=1}^{{\check{d}}} {\widecheck{\mathbf{x}}}_i^\top MR{\widecheck{\mathbf{x}}}_i
-\frac{{\operatorname{tr}}R\Phi M}{\alpha^{-1}+\gamma {\operatorname{tr}}R\Phi}.$$
Let us define the leave-one-out resolvent $$R^{(i)}=\left(A+\alpha \sum_{j:j \neq i} {\widecheck{\mathbf{x}}}_j{\widecheck{\mathbf{x}}}_j^\top-z{\operatorname{Id}}\right)^{-1}.$$ We may then decompose $\delta_n$ as $\delta_n=J_1+\gamma J_2$ where (recalling $\gamma=n/{\check{d}}$) $$\begin{aligned}
J_1&=\frac{1}{n}\sum_{i=1}^{{\check{d}}}\left(\alpha {\widecheck{\mathbf{x}}}_i^\top MR{\widecheck{\mathbf{x}}}_i-\frac{\gamma {\operatorname{tr}}R^{(i)}\Phi M}{\alpha^{-1}+\gamma{\operatorname{tr}}R^{(i)}\Phi}\right),\\
J_2&=\frac{1}{n}\sum_{i=1}^{{\check{d}}}\left(\frac{ {\operatorname{tr}}R^{(i)}\Phi
M}{\alpha^{-1}+\gamma{\operatorname{tr}}R^{(i)}\Phi}-\frac{ {\operatorname{tr}}R\Phi M}{\alpha^{-1}
+\gamma{\operatorname{tr}}R\Phi}\right).\end{aligned}$$ Let us denote these summands as $$J_1^{(i)}=\alpha {\widecheck{\mathbf{x}}}_i^\top MR{\widecheck{\mathbf{x}}}_i-\frac{\gamma{\operatorname{tr}}R^{(i)}\Phi M}{
\alpha^{-1}+\gamma{\operatorname{tr}}R^{(i)}\Phi}\quad\text{and}\quad
J_2^{(i)}=\frac{{\operatorname{tr}}R^{(i)}\Phi M}{\alpha^{-1}+\gamma{\operatorname{tr}}R^{(i)}\Phi}-\frac{{\operatorname{tr}}R\Phi M}{\alpha^{-1}+\gamma{\operatorname{tr}}R\Phi}.$$
[**Bound for $J_1$.**]{} Momentarily fix the index $i \in
\{1,\ldots,{\check{d}}\}$. Applying the Sherman-Morrison identity, we have $$\label{eq:Sherman-Morrison}
R=R^{(i)}-\frac{{\alpha}R^{(i)}{\widecheck{\mathbf{x}}}_i{\widecheck{\mathbf{x}}}_i^\top R^{(i)}}{1+{\alpha}{\widecheck{\mathbf{x}}}_i^\top
R^{(i)}{\widecheck{\mathbf{x}}}_i}.$$ Then, introducing $A_1={\widecheck{\mathbf{x}}}_i^\top MR^{(i)}{\widecheck{\mathbf{x}}}_i$ and $A_2={\widecheck{\mathbf{x}}}_i^\top R^{(i)}{\widecheck{\mathbf{x}}}_i$, $$\alpha {\widecheck{\mathbf{x}}}_i^\top MR {\widecheck{\mathbf{x}}}_i=\alpha A_1-\frac{\alpha^2 A_1A_2}{1+\alpha A_2}
=\frac{A_1}{\alpha^{-1}+A_2}.$$ Recall that the rows of ${\widecheck{X}}$ are i.i.d. Let ${\widecheck{X}}^{(i)}$ be the matrix ${\widecheck{X}}$ with the $i^\text{th}$ row ${\widecheck{\mathbf{x}}}_i$ removed, and let ${\mathbb{E}}_{{\widecheck{\mathbf{x}}}_i}[\cdot]$ be the expectation over only ${\widecheck{\mathbf{x}}}_i$ (i.e.conditional on ${\widecheck{X}}^{(i)}$). Observe that $R^{(i)}$ is a function of ${\widecheck{X}}^{(i)}$. Applying Proposition \[prop:basic\] with ${\widecheck{X}}^{(i)}$ in place of ${\widecheck{X}}$, we see that $\|R^{(i)}\|$ and $\|MR^{(i)}\|$ are both bounded by a constant. Then applying Lemma \[lemma:quadform\] conditional on ${\widecheck{X}}^{(i)}$, and recalling the bound (\[eq:sigmabound\]) for $\sigma(0)$, there are constants $C,c>0$ for which $${\mathbb{P}}[|A_k-{\mathbb{E}}_{{\widecheck{\mathbf{x}}}_i}[A_k]|>t]
\leq Ce^{-cn\min(t^2,t)} \qquad \text{ for } k=1,2.$$
Note that $${\mathbb{E}}_{{\widecheck{\mathbf{x}}}_i}[A_1]={\operatorname{Tr}}MR^{(i)}{\mathbb{E}}[{\widecheck{\mathbf{x}}}_i{\widecheck{\mathbf{x}}}_i^\top]=\frac{1}{{\check{d}}}
{\operatorname{Tr}}MR^{(i)}\Phi=\gamma {\operatorname{tr}}R^{(i)}\Phi M.$$ Similarly, ${\mathbb{E}}_{{\widecheck{\mathbf{x}}}_i}[A_2]=\gamma {\operatorname{tr}}R^{(i)}\Phi$, so $$J_1^{(i)}=\frac{A_1}{\alpha^{-1}+A_2}-\frac{{\mathbb{E}}_{{\widecheck{\mathbf{x}}}_i}[A_1]}{\alpha^{-1}+{\mathbb{E}}_{{\widecheck{\mathbf{x}}}_i}[A_2]}.$$ Applying Proposition \[prop:basic\], we have for some constants $C,c,c'>0$, on an event ${\mathcal{E}}({\widecheck{X}}^{(i)})$ of probability $1-2e^{-c'n}$, that $$|{\mathbb{E}}_{{\widecheck{\mathbf{x}}}_i}[A_1]| \leq C, \qquad
|\alpha^{-1}+{\mathbb{E}}_{{\widecheck{\mathbf{x}}}_i}[A_2]| \geq {\operatorname{Im}}(\alpha^{-1}+{\mathbb{E}}_{{\widecheck{\mathbf{x}}}_i}[A_2])
\geq c.$$ Then, for any $t$ such that $t<c/2$, on the event where $|A_1-{\mathbb{E}}_{{\widecheck{\mathbf{x}}}_i}[A_1]| \leq t$, $|A_2-{\mathbb{E}}_{{\widecheck{\mathbf{x}}}_i}[A_2]| \leq t$, and ${\mathcal{E}}({\widecheck{X}}^{(i)})$ all hold, $$\label{eq:J1bound}
\left|J_1^{(i)}\right|
\leq \frac{|A_1-{\mathbb{E}}_{{\widecheck{\mathbf{x}}}_i}[A_1]|}{|\alpha^{-1}+A_2|}+
|{\mathbb{E}}_{{\widecheck{\mathbf{x}}}_i}[A_1]| \cdot \frac{|A_2-{\mathbb{E}}_{{\widecheck{\mathbf{x}}}_i}[A_2]|}
{|\alpha^{-1}+A_2| \cdot |\alpha^{-1}+{\mathbb{E}}_{{\widecheck{\mathbf{x}}}_i}[A_2]|} \leq Ct.$$ Thus, for $t<c'$ and a sufficiently small constant $c'>0$, we have ${\mathbb{P}}[|J_1^{(i)}| \geq t] \leq Ce^{-cnt^2}$. Applying a union bound over $i \in \{1,\ldots,{\check{d}}\}$, this yields ${\mathbb{P}}[|J_1| \geq t] \leq Cne^{-cnt^2}$.\
[**Bound for $J_2$.**]{} Applying the identity $A^{-1}-B^{-1}=A^{-1}(B-A)B^{-1}$, $$R^{(i)}-R=R^{(i)}(R^{-1}-(R^{(i)})^{-1})R=\alpha R^{(i)}{\widecheck{\mathbf{x}}}_i {\widecheck{\mathbf{x}}}_i^\top R.$$ Then, applying also the bounds $\|R\|,\|R^{(i)}\| \leq C$ from Proposition \[prop:basic\], $$|{\operatorname{tr}}(R^{(i)}-R)\Phi M|=\frac{1}{n}|\alpha {\widecheck{\mathbf{x}}}_i^\top R\Phi MR^{(i)}{\widecheck{\mathbf{x}}}_i|
\leq \frac{C\|{\widecheck{X}}\|^2}{n}.$$ Applying Lemma \[lemma:normbound\](b), with probability $1-2e^{-cn}$, this is at most $C/n$ for every $i \in \{1,\ldots,{\check{d}}\}$. Similarly, $|{\operatorname{tr}}(R^{(i)}-R)\Phi| \leq C/n$ with this probability. Applying again $|{\operatorname{tr}}R\Phi M| \leq C$, $|\alpha^{-1}+\gamma {\operatorname{tr}}R\Phi| \geq c$, and an argument similar to (\[eq:J1bound\]), we obtain $|J_2^{(i)}| \leq
C'/n$ for a constant $C'>0$. Taking a union bound over $i \in \{1,\ldots,{\check{d}}\}$, this yields ${\mathbb{P}}[|J_2|>C/n] \leq C'ne^{-cn}$. Combining these bounds for $J_1$ and $J_2$, choosing $t>cn^{-1}$, and re-adjusting the constants yields the lemma.
Proof of Lemma \[lemma:fixedpoint\]
-----------------------------------
We now prove Lemma \[lemma:fixedpoint\] using Lemma \[lemma:resolvent\_remainder\]. Define the random $n$-dependent parameter $$s=\alpha^{-1}+\gamma {\operatorname{tr}}R\Phi,$$ so that $\bar{s}={\mathbb{E}}[s]$. The following establishes concentration of $s$ around $\bar{s}$.
\[lemma:s-bars\_converge\] Under Assumption \[assump:singlelayer\], for some constants $c,n_0>0$, all $n \geq n_0$, and any $t>0$, $${\mathbb{P}}\left[\left|s-\bar{s}\right|>t\right]\leq 2e^{-cnt^2}.$$
Define $F(W)=\gamma {\operatorname{tr}}R\Phi$, where $R$ and ${\widecheck{X}}$ are considered as a function of $W$. Fix any matrices $W,\Delta \in {\mathbb{R}}^{{\check{d}}\times n}$ where $\|\Delta\|_F=1$, and define $W_t=W+t \Delta$. Then, applying $\partial R=-R(\partial (R^{-1}))R$ and $R=R^\top$, $$\begin{aligned}
{\operatorname{vec}}(\Delta)^\top (\nabla F(W))=\frac{d}{dt}\Big|_{t=0} F(W_t)
&=-\gamma {\operatorname{tr}}R\left(\frac{d}{dt}\Big|_{t=0}R^{-1}\right)R\Phi\\
&=-2\gamma \alpha {\operatorname{tr}}R\left({\widecheck{X}}^\top \cdot \frac{d}{dt}\Big|_{t=0}{\widecheck{X}}\right)R\Phi\\
&=-\frac{2\gamma \alpha}{\sqrt{{\check{d}}}}
{\operatorname{tr}}R\left({\widecheck{X}}^\top \cdot \left(\sigma'(WX) \odot
(\Delta X)\right)\right)R\Phi,\end{aligned}$$ where $\odot$ is the Hadamard product, and $\sigma'$ is applied entrywise. Applying Proposition \[prop:basic\], $$\Big|{\operatorname{vec}}(\Delta)^\top (\nabla F(W))\Big|
\leq \frac{C}{\sqrt{{\check{d}}}} \cdot \Big\|R{\widecheck{X}}^\top \cdot
(\sigma'(WX) \odot (\Delta X)) \cdot R\Big\|
\leq \frac{C'}{\sqrt{{\check{d}}}} \cdot \|R{\widecheck{X}}^\top\| \cdot
\|\sigma'(WX) \odot (\Delta X)\|.$$ For the first term, $$\begin{aligned}
\|R{\widecheck{X}}^\top\|^2=\frac{1}{|\alpha|}\|R(\alpha {\widecheck{X}}^\top {\widecheck{X}}) R^*\|
&\leq \frac{1}{|\alpha|}\left(\|R(A+\alpha {\widecheck{X}}^\top {\widecheck{X}}-z{\operatorname{Id}})R^*\|
+\|R(A-z{\operatorname{Id}})R^*\|\right)\\
& \leq \frac{1}{|\alpha|}(\|R\|+\|R\|^2(\|A\|+|z|)) \leq C.\end{aligned}$$ For the second term, $$\|\sigma'(WX) \odot (\Delta X)\| \leq \|\sigma'(WX) \odot (\Delta
X)\|_F \leq \lambda_\sigma \|\Delta X\|_F \leq \lambda_\sigma \|\Delta\|_F \cdot
\|X\| \leq C.$$ Thus $|{\operatorname{vec}}(\Delta)^\top (\nabla F(W))| \leq C/\sqrt{n}$. This holds for every $\Delta$ such that $\|\Delta\|_F=1$, so $F(W)$ is $C/\sqrt{n}$-Lipschitz in $W$ with respect to the Frobenius norm. Then the result follows from Gaussian concentration of measure.
To conclude the proof of Lemma \[lemma:fixedpoint\], we may again assume $\|M\| \leq 1$ by rescaling $M$. Set $${\widetilde{M}}=\left(A+\bar{s}^{-1}\Phi-z{\operatorname{Id}}\right)^{-1}M.$$ Note that $\bar{s}^{-1} \in {\mathbb{C}}^-$, so $\|{\widetilde{M}}\| \leq
\|(A+\bar{s}^{-1}\Phi-z{\operatorname{Id}})^{-1}\| \leq C$ by Proposition \[prop:invertible\]. Applying Lemma \[lemma:resolvent\_remainder\] with ${\widetilde{M}}$, $$\label{eq:applicationM}
{\mathbb{P}}\left[\Big|{\operatorname{tr}}{\widetilde{M}}-{\operatorname{tr}}R\left(A+s^{-1}
\Phi-z{\operatorname{Id}}\right){\widetilde{M}}\Big|>t\right] \leq Cne^{-cnt^2}$$ for all $t \in (n^{-1},c')$. Furthermore, applying the definition of ${\widetilde{M}}$, $$\begin{aligned}
|{\operatorname{tr}}R\left(A+s^{-1} \Phi-z{\operatorname{Id}}\right){\widetilde{M}}-{\operatorname{tr}}RM|
&=\left|{\operatorname{tr}}R\left(\left(A+s^{-1} \Phi-z{\operatorname{Id}}\right)
-\left(A+\bar{s}^{-1} \Phi-z{\operatorname{Id}}\right)\right){\widetilde{M}}\right|\\
&=|s^{-1}-\bar{s}^{-1}| \cdot |{\operatorname{tr}}R\Phi{\widetilde{M}}|
\leq C|s^{-1}-\bar{s}^{-1}|.\end{aligned}$$ Recall that $|\bar{s}| \geq {\operatorname{Im}}\bar{s} \geq c$. Then, on the event where $|s-\bar{s}| \leq t$ and $t<c/2$, we have $|s^{-1}-\bar{s}^{-1}| \leq Ct$. Then applying Lemma \[lemma:s-bars\_converge\], for some constants $c,c'>0$ and all $t \in (0,c')$, $${\mathbb{P}}\left[|{\operatorname{tr}}R\left(A+s^{-1} \Phi-z{\operatorname{Id}}\right){\widetilde{M}}-{\operatorname{tr}}RM|>t \right] \leq 2e^{-cnt^2}.$$ Combining this with (\[eq:applicationM\]) yields Lemma \[lemma:fixedpoint\](a). Specializing Lemma \[lemma:fixedpoint\](a) to $M=\Phi$, we obtain $${\mathbb{P}}\left[\left|s-\left(\alpha^{-1}+\gamma
{\operatorname{tr}}(A+\bar{s}^{-1}\Phi-z{\operatorname{Id}})^{-1}\Phi\right)\right|>t\right]
\leq Cne^{-cnt^2}.$$ Applying again Lemma \[lemma:s-bars\_converge\] to bound $|s-\bar{s}|$, we obtain Lemma \[lemma:fixedpoint\](b).
Analysis for the Conjugate Kernel {#appendix:CK}
=================================
Theorem \[thm:CK\] is a special case of Theorem \[thm:NTK\], but let us provide here a simpler argument. Define, for each layer, the $n \times n$ matrices $$\begin{aligned}
\Phi_\ell&={\mathbb{E}}_{{\mathbf{w}}}\Big[\sigma({\mathbf{w}}^\top X_{\ell-1})^\top
\sigma({\mathbf{w}}^\top X_{\ell-1})\Big]\label{eq:Phiell}\\
\tilde{\Phi}_\ell&=b_\sigma^2 X_{\ell-1}^\top X_{\ell-1}+(1-b_\sigma^2){\operatorname{Id}}\label{eq:tildePhiell}\end{aligned}$$ where ${\mathbb{E}}_{\mathbf{w}}$ denotes the expectation over only the random vector ${\mathbf{w}}\sim
{\mathcal{N}}(0,{\operatorname{Id}})$. Here, $\Phi_\ell,\tilde{\Phi}_\ell$ are deterministic conditional on $X_{\ell-1}$, but are random unconditionally for $\ell \geq 2$. For each fixed $\ell=1,\ldots,L$, we will show $$\label{eq:Phiapprox}
{\operatorname{lim\;spec}}\Phi_\ell={\operatorname{lim\;spec}}\tilde{\Phi}_\ell.$$ Conditional on $X_{\ell-1}$, the spectral limit of $X_\ell^\top X_\ell$ was shown in [@louart2018random] to be a Marcenko-Pastur map of the spectral limit of $\Phi_\ell$—we reproduce a short proof below under our assumptions, by specializing Lemma \[lemma:fixedpoint\] to $\alpha=1$ and $A=0$. Combining with (\[eq:Phiapprox\]) and iterating from $\ell=1,\ldots,L$ yields Theorem \[thm:CK\].
\[lemma:difference\_Phi\] Under Assumption \[assump:asymptotics\], for each $\ell=1,\ldots,L$, almost surely as $n \to \infty$, $$\frac{1}{n}\|\Phi_\ell-\tilde{\Phi}_\ell\|_F^2 \to 0.$$
By Corollary \[cor:orthonormalinduction\], increasing $({\varepsilon}_n,B)$ as needed, we may assume that each matrix $X_0,\ldots,X_L$ is $({\varepsilon}_n,B)$-orthonormal. Denote by $\Phi_\ell[{\alpha},{\beta}]$ and $\tilde{\Phi}_\ell[{\alpha},{\beta}]$ the $({\alpha},{\beta})$ entries of these matrices. Then Lemma \[lemma:orthog\](a) shows for ${\alpha}\neq {\beta}$ that $$|\Phi_\ell[{\alpha},{\beta}]-\tilde{\Phi}_\ell[{\alpha},{\beta}]|
\leq C{\varepsilon}_n^2.$$ For ${\alpha}={\beta}$, applying $\tilde{\Phi}_\ell[\alpha,\alpha]=1-b_\sigma^2+
b_\sigma^2 \|{\mathbf{x}}_{\alpha}^{\ell-1}\|^2$, we have $$|\Phi_\ell[{\alpha},{\alpha}]-\tilde{\Phi}_\ell[{\alpha},{\alpha}]|
\leq |\Phi_\ell[{\alpha},{\alpha}]-1|+b_\sigma^2|\|{\mathbf{x}}^{\ell-1}_{\alpha}\|^2-1|
\leq C{\varepsilon}_n.$$ Then $$\|\Phi_\ell-\tilde{\Phi}_\ell\|_F^2
\leq Cn(n-1){\varepsilon}_n^4+Cn{\varepsilon}_n^2,$$ and the result follows from the condition ${\varepsilon}_n n^{1/4} \to 0$.
By Corollary \[cor:orthonormalinduction\], we may assume that each matrix $X_0,\ldots,X_L$ is $({\varepsilon}_n,B)$-orthonormal. This implies the bounds $\|X_\ell\| \leq C$ and $\|K^{\text{CK}}\| \leq C$ for all large $n$.
For the spectral convergence, suppose by induction that ${\operatorname{lim\;spec}}X_{\ell-1}^\top X_{\ell-1}=\mu_{\ell-1}$, where the base case ${\operatorname{lim\;spec}}X_0^\top X_0=\mu_0$ holds by assumption. Defining $$\nu_\ell=(1-b_\sigma^2)+b_\sigma^2 \cdot \mu_{\ell-1},$$ Proposition \[prop:specapprox\] and Lemma \[lemma:difference\_Phi\] together show that $${\operatorname{lim\;spec}}\Phi_\ell={\operatorname{lim\;spec}}\tilde{\Phi}_\ell=\nu_\ell.$$ Specializing Lemma \[lemma:fixedpoint\](b) to the setting $A=0$, $\alpha=1$, $X=X_{\ell-1}$, and ${\widecheck{X}}=X_\ell$, and choosing $t \equiv t_n$ such that $t_n \to 0$ and $nt_n^2 \gg \log n$, we obtain $$\label{eq:sbarMP}
\Big|\bar{s}-1-(n/d_\ell) {\operatorname{tr}}(\bar{s}^{-1}\Phi_\ell-z{\operatorname{Id}})^{-1}\Phi_\ell\Big| \to 0$$ a.s. as $n \to \infty$, where $$\bar{s}=1+\frac{n}{d_\ell}
{\mathbb{E}}_{W_\ell}[{\operatorname{tr}}(X_\ell^\top X_\ell-z{\operatorname{Id}})^{-1}\Phi_\ell].$$ Here, this expectation is taken over only $W_\ell$ (i.e. conditional on $X_0,\ldots,X_{\ell-1}$).
Proposition \[prop:basic\] verifies that $\bar{s}$ is bounded as $n \to \infty$, so for any subsequence in $n$, there is a further sub-subsequence along which $\bar{s} \to s_0$ for a limit $s_0 \equiv s_0(z)
\in {\mathbb{C}}^+$. Applying $A^{-1}-B^{-1}=A^{-1}(B-A)B^{-1}$ and Propositions \[prop:invertible\] and \[prop:basic\], $$\begin{aligned}
&\Big|{\operatorname{tr}}(\bar{s}^{-1}\Phi_\ell-z{\operatorname{Id}})^{-1}\Phi_\ell-
{\operatorname{tr}}(s_0^{-1}\Phi_\ell-z{\operatorname{Id}})^{-1}\Phi_\ell\Big|\\
&=|s_0^{-1}-s^{-1}| \cdot {\operatorname{tr}}\Big|(s_0^{-1}\Phi_\ell-z{\operatorname{Id}})^{-1}
\Phi_\ell (\bar{s}^{-1}\Phi_\ell-z{\operatorname{Id}})^{-1}\Phi_\ell\Big|\\
&\leq |s_0^{-1}-s^{-1}| \cdot \|(s_0^{-1}\Phi_\ell-z{\operatorname{Id}})^{-1}\| \cdot
\|(\bar{s}^{-1}\Phi_\ell-z{\operatorname{Id}})^{-1}\|\cdot \|\Phi_\ell\|^2\\
&\leq C|s_0^{-1}-s^{-1}|.\end{aligned}$$ Thus, along the sub-subsequence where $\bar{s} \to s_0$, we get $$\label{eq:sbys0}
{\operatorname{tr}}(\bar{s}^{-1}\Phi_\ell-z{\operatorname{Id}})^{-1}\Phi_\ell-
{\operatorname{tr}}(s_0^{-1}\Phi_\ell-z{\operatorname{Id}})^{-1}\Phi_\ell \to 0.$$ We have also $$\label{eq:Phibynu}
{\operatorname{tr}}(s_0^{-1}\Phi_\ell-z{\operatorname{Id}})^{-1}\Phi_\ell \to
\int \frac{x}{s_0^{-1}x-z}d\nu_\ell(x),$$ since the function $x \mapsto x/(s_0^{-1}x-z)$ is continuous and bounded over ${\mathbb{R}}$, and ${\operatorname{lim\;spec}}\Phi_\ell=\nu_\ell$. Thus, taking the limit of (\[eq:sbarMP\]) along this sub-subsequence, the value $s_0$ must satisfy $$\label{eq:s0}
s_0-1-\gamma_\ell \int \frac{x}{s_0^{-1}x-z}\,d\nu_\ell(x)=0.$$
Now applying Lemma \[lemma:fixedpoint\](a) with $M={\operatorname{Id}}$, and taking the limit along this sub-subsequence, by a similar argument we obtain that $$\label{eq:mlz}
{\operatorname{tr}}(X_\ell^\top X_\ell-z{\operatorname{Id}})^{-1} \to
\int \frac{1}{s_0^{-1}x-z}d\nu_\ell(x).$$ Denoting this limit by $m_\ell(z)$, and rewriting (\[eq:s0\]) by applying $$\int \frac{x}{s_0^{-1}x-z}d\nu_\ell(x)
=s_0\int \left(1+\frac{z}{s_0^{-1}x-z}\right)d\nu_\ell(x)=s_0(1+zm_\ell(z)),$$ we get $s_0^{-1}=1-\gamma_\ell-\gamma_\ell zm_\ell(z)$. Applying this back to the definition of $m_\ell(z)$ in (\[eq:mlz\]), this shows that $m_\ell(z)$ satisfies the Marcenko-Pastur equation $$m(z)=\int \frac{1}{x(1-\gamma_\ell-\gamma_\ell z m(z))-z}d\nu_\ell(x),$$ so $m_\ell(z)$ is the Stieltjes transform of $\mu_\ell=\rho_{\gamma_\ell}^{\text{MP}}\boxtimes \nu_\ell
=\rho_{\gamma_\ell}^{\text{MP}}\boxtimes ((1-b_\sigma^2)+b_\sigma^2
\cdot \mu_{\ell-1})$.
We have shown that ${\operatorname{tr}}(X_\ell^\top X_\ell-z{\operatorname{Id}})^{-1} \to m_\ell(z)$ almost surely along this sub-subsequence in $n$. Since, for every subsequence in $n$, there exists such a sub-subsequence, this implies $\lim_{n \to \infty} {\operatorname{tr}}(X_\ell^\top X_\ell-z{\operatorname{Id}})^{-1}=m_\ell(z)$ almost surely. Thus ${\operatorname{lim\;spec}}X_\ell^\top X_\ell=\mu_\ell$, which completes the induction.
Analysis for the Neural Tangent Kernel {#appendix:NTK}
======================================
Spectral approximation and operator norm bound {#appendix:NTKapprox}
----------------------------------------------
We first prove the spectral approximation stated in Lemma \[lemma:NTKapprox\], as well as the operator norm bound $\|K^{\text{NTK}}\| \leq C$. The following form of $K^{\text{NTK}}$ is derived also in [@huang2019dynamics Eq. (1.7)]: Denote by ${\mathbf{x}}^\ell_{\alpha}$ the ${\alpha}^\text{th}$ column of $X_\ell$. For each $\ell=1,\ldots,L$, define the matrix $S_\ell \in {\mathbb{R}}^{d_\ell \times n}$ whose ${\alpha}^\text{th}$ column is given by $$\label{eq:Sell}
{\mathbf{s}}^\ell_{\alpha}=D^\ell_{\alpha}\frac{W_{\ell+1}^\top}{\sqrt{d_\ell}}
D^{\ell+1}_{\alpha}\frac{W_{\ell+2}^\top}{\sqrt{d_{\ell+1}}} D^{\ell+2}_{\alpha}\ldots
\frac{W_L^\top}{\sqrt{d_{L-1}}}
D^L_{\alpha}\frac{{\mathbf{w}}}{\sqrt{d_L}},$$ where we define diagonal matrices indexed by ${\alpha}\in [n]$ and $k \in [L]$ as $$D^k_{\alpha}\equiv {\operatorname{diag}}\Big(\sigma'(W_k {\mathbf{x}}^{k-1}_{\alpha})\Big) \in {\mathbb{R}}^{d_k \times d_k}.$$ Applying the chain rule, we may verify for each input sample ${\mathbf{x}}_{\alpha}$ that $$\nabla_{\mathbf{w}}f_\theta({\mathbf{x}}_{\alpha})={\mathbf{x}}_{\alpha}^L \in {\mathbb{R}}^{d_L},
\quad \nabla_{W_\ell} f_\theta({\mathbf{x}}_{\alpha})
={\mathbf{s}}_{\alpha}^\ell \otimes {\mathbf{x}}_{\alpha}^{\ell-1} \in {\mathbb{R}}^{d_\ell d_{\ell-1}}.$$ Then $$\begin{aligned}
\big(\nabla_{{\mathbf{w}}} f_\theta(X)\big)^\top
\big(\nabla_{{\mathbf{w}}} f_\theta(X)\big)&=X_L^\top X_L,\\
\big(\nabla_{W_\ell} f_\theta(X)\big)^\top
\big(\nabla_{W_\ell} f_\theta(X)\big)&=(S_\ell^\top S_\ell) \odot
(X_{\ell-1}^\top X_{\ell-1}),\end{aligned}$$ where $\odot$ is the Hadamard product. Thus, the NTK is given by $$\label{eq:NTKform}
K^{{\text{NTK}}}=\Big(\nabla_\theta f_\theta(X)\Big)^\top
\Big(\nabla_\theta f_\theta(X)\Big)
=X_L^\top X_L+\sum_{\ell=1}^L (S_\ell^\top S_\ell) \odot
(X_{\ell-1}^\top X_{\ell-1}).$$
\[lemma:onelayerapprox\] Let $X \in {\mathbb{R}}^{d \times n}$ be $({\varepsilon},B)$-orthonormal, let $W \in {\mathbb{R}}^{{\check{d}}\times
d}$ have i.i.d. ${\mathcal{N}}(0,1)$ entries, and let ${\mathbf{x}}_{\alpha},{\mathbf{x}}_{\beta}$ be two columns of $X$ where ${\alpha}\neq {\beta}$. Then for universal constants $C,c>0$ and any $t>0$:
(a) With probability at least $1-2e^{-c{\check{d}}t^2}$, $$\left|\frac{1}{{\check{d}}}{\operatorname{Tr}}\Big({\operatorname{diag}}\big(\sigma'(W {\mathbf{x}}_{\alpha})\big)
{\operatorname{diag}}\big(\sigma'(W {\mathbf{x}}_{\beta})\big)\Big)-b_\sigma^2\right| \leq
C\lambda_\sigma^2({\varepsilon}+t).$$
(b) Let $M \in {\mathbb{R}}^{d \times d}$ be any deterministic symmetric matrix, and denote $$T({\mathbf{x}}_{\alpha},{\mathbf{x}}_{\beta})=\frac{1}{{\check{d}}}{\operatorname{Tr}}\Big({\operatorname{diag}}\big(\sigma'(W{\mathbf{x}}_{\alpha})\big)WMW^\top
{\operatorname{diag}}\big(\sigma'(W{\mathbf{x}}_{\beta})\big)\Big).$$ With probability at least $1-(2{\check{d}}+2)e^{-c\min(t^2{\check{d}},t\sqrt{{\check{d}}})}$, $$\left|T({\mathbf{x}}_{\alpha},{\mathbf{x}}_{\beta})-b_\sigma^2 {\operatorname{Tr}}M\right| \leq C\lambda_\sigma^2
\left({\varepsilon}\sqrt{d}+t\sqrt{d}+t\sqrt{{\check{d}}}\right)\|M\|_F.$$
Furthermore, both (a) and (b) hold with $({\mathbf{x}}_{\alpha},{\mathbf{x}}_{\alpha})$ in place of $({\mathbf{x}}_{\alpha},{\mathbf{x}}_{\beta})$, upon replacing $b_\sigma^2$ by $a_\sigma$.
Write ${\mathbf{w}}_k^\top \in {\mathbb{R}}^d$ for the $k^\text{th}$ row of $W$. Then $$\frac{1}{{\check{d}}}{\operatorname{Tr}}\Big({\operatorname{diag}}\big(\sigma'(W {\mathbf{x}}_{\alpha})\big)
{\operatorname{diag}}\big(\sigma'(W {\mathbf{x}}_{\beta})\big)\Big)
=\frac{1}{{\check{d}}}\sum_{k=1}^{{\check{d}}} \sigma'({\mathbf{w}}_k^\top {\mathbf{x}}_{\alpha})\sigma'({\mathbf{w}}_k^\top {\mathbf{x}}_{\beta}).$$ Applying $\sigma'({\mathbf{w}}_k^\top {\mathbf{x}}_{\alpha})
\sigma'({\mathbf{w}}_k^\top {\mathbf{x}}_{\beta}) \in [-\lambda_\sigma^2,\lambda_\sigma^2]$ and Hoeffding’s inequality, $${\mathbb{P}}\left[\left|\frac{1}{{\check{d}}}\sum_{k=1}^{{\check{d}}} \Big(\sigma'({\mathbf{w}}_k^\top {\mathbf{x}}_{\alpha})
\sigma'({\mathbf{w}}_k^\top {\mathbf{x}}_{\beta})-{\mathbb{E}}[\sigma'({\mathbf{w}}_k^\top {\mathbf{x}}_{\alpha})
\sigma'({\mathbf{w}}_k^\top {\mathbf{x}}_{\beta})]\Big)\right|>\lambda_\sigma^2 t\right] \leq 2e^{-c{\check{d}}t^2}.$$ To bound the mean, recall that $(\zeta_{\alpha},\zeta_{\beta}) \equiv
({\mathbf{w}}_k^\top {\mathbf{x}}_{\alpha},{\mathbf{w}}_k^\top{\mathbf{x}}_{\beta})$ is bivariate Gaussian, which we may write as $$\zeta_{\alpha}=u_{\alpha}\xi_{\alpha}, \qquad \zeta_{\beta}=u_{\beta}\xi_{\beta}+v_{\beta}\xi_{\alpha}$$ as in (\[eq:gramschmidt\]). Here, $\xi_{\alpha},\xi_{\beta}\sim {\mathcal{N}}(0,1)$ are independent, $u_{\alpha},u_{\beta}>0$ and $v_{\beta}\in {\mathbb{R}}$, and these satisfy $|u_{\alpha}-1|,|u_{\beta}-1|,|v_{\beta}| \leq C{\varepsilon}$. Applying the Taylor expansion $$\sigma'(\zeta)=\sigma'(\xi)+\sigma''(\eta)(\zeta-\xi)$$ for some $\eta$ between $\zeta$ and $\xi$, and the conditions ${\mathbb{E}}[\sigma'(\xi)]=b_\sigma$ and $|\sigma''(x)| \leq
\lambda_\sigma$, it is easy to check that $|{\mathbb{E}}[\sigma'(\zeta_{\alpha})\sigma'(\zeta_{\beta})]-b_\sigma^2| \leq
C\lambda_\sigma^2{\varepsilon}$. Then part (a) follows. The statement with $({\mathbf{x}}_{\alpha},{\mathbf{x}}_{\alpha})$ and $a_\sigma$ follows similarly from this Taylor expansion and the bound $|{\mathbb{E}}[\sigma'(\zeta_{\alpha})^2]-a_\sigma| \leq C\lambda_\sigma^2{\varepsilon}$.
For part (b), we write $$T({\mathbf{x}}_{\alpha},{\mathbf{x}}_{\beta})=\frac{1}{{\check{d}}}\sum_{k=1}^{{\check{d}}} \sigma'({\mathbf{w}}_k^\top {\mathbf{x}}_{\alpha})
\sigma'({\mathbf{w}}_k^\top {\mathbf{x}}_{\beta}) \cdot {\mathbf{w}}_k^\top M {\mathbf{w}}_k.$$ By the Hanson-Wright inequality (see [@rudelson2013hanson Theorem 1.1]), $${\mathbb{P}}\Big[|{\mathbf{w}}_k^\top M {\mathbf{w}}_k-{\operatorname{Tr}}M|>\|M\|_F \cdot t\sqrt{{\check{d}}}\Big]
\leq 2e^{-c\min(t^2{\check{d}},t\sqrt{{\check{d}}})}$$ for a constant $c>0$. Then, applying $|\sigma'(x)| \leq \lambda_\sigma$ and a union bound over $k=1,\ldots,{\check{d}}$, with probability at least $1-2{\check{d}}e^{-c\min(t^2{\check{d}},t\sqrt{{\check{d}}})}$, $$\left|T({\mathbf{x}}_{\alpha},{\mathbf{x}}_{\beta})-{\operatorname{Tr}}M \cdot
\frac{1}{{\check{d}}}\sum_{k=1}^{{\check{d}}} \sigma'({\mathbf{w}}_k^\top {\mathbf{x}}_{\alpha})\sigma'({\mathbf{w}}_k^\top {\mathbf{x}}_{\beta})\right|
\leq \|M\|_F \cdot \lambda_\sigma^2 t\sqrt{{\check{d}}}.$$ Then part (b) follows from combining with part (a), and applying ${\operatorname{Tr}}M \leq
\sqrt{d}\|M\|_F$.
\[cor:Sapprox\] Let ${\mathbf{s}}_{\alpha}^\ell$ be as defined in (\[eq:Sell\]), and let $q_\ell,r_\ell$ be the constants in (\[eq:qr\]). Under Assumption \[assump:asymptotics\], for a constant $C>0$, almost surely for all large $n$ and for all $\ell \in [L]$ and ${\alpha}\neq
{\beta}\in [n]$, $$\label{eq:Sabbound}
\Big|{{\mathbf{s}}_{\alpha}^\ell}^\top {\mathbf{s}}_{\beta}^\ell-q_{\ell-1}\Big| \leq C\max({\varepsilon}_n,n^{-0.48}),
\qquad \Big|\|{\mathbf{s}}_{\alpha}^\ell\|^2-r_{\ell-1}\Big| \leq C\max({\varepsilon}_n,n^{-0.48}).$$
By Corollary \[cor:orthonormalinduction\], we may assume that each matrix $X_0,\ldots,X_L$ is $({\varepsilon}_n,B)$-orthonormal. Since a larger value of ${\varepsilon}_n$ corresponds to a weaker assumption, we may assume without loss of generality that ${\varepsilon}_n \geq n^{-0.48}$.
Fix $\ell \in [L]$ and ${\alpha},{\beta}\in [n]$, and define $$\begin{aligned}
M_\ell&=D_{\alpha}^\ell D_{\beta}^\ell\nonumber\\
M_k&=D_{\alpha}^k\frac{W_k}{\sqrt{d_{k-1}}}
\ldots D_{\alpha}^{\ell+1}\frac{W_{\ell+1}}{\sqrt{d_\ell}}D_{\alpha}^\ell D_{\beta}^\ell
\frac{W_{\ell+1}^\top}{\sqrt{d_\ell}}D_{\beta}^{\ell+1}\ldots
\frac{W_k^\top}{\sqrt{d_{k-1}}}D_{\beta}^k \quad \text{ for }
\quad \ell+1 \leq k \leq L.\label{eq:Mell}\end{aligned}$$ Recalling the definition (\[eq:Sell\]) and applying the Hanson-Wright inequality conditional on $W_1,\ldots,W_L$, $$\label{eq:HansonWright}
\left|{{\mathbf{s}}_{\alpha}^\ell}^\top {\mathbf{s}}_{\beta}^\ell-\frac{1}{d_L}{\operatorname{Tr}}M_L\right|
\leq C{\varepsilon}_n\sqrt{n} \cdot \frac{1}{d_L}\|M_L\|_F$$ with probability $1-e^{-c\min({\varepsilon}_n^2n,{\varepsilon}_n\sqrt{n})}
\geq 1-e^{-n^{0.01}}$. Next, for each $k=L,L-1,\ldots,\ell+1$, we apply Lemma \[lemma:onelayerapprox\](b) conditional on $W_1,\ldots,W_{k-1}$, with $t={\varepsilon}_n$, $M=M_{k-1}/d_{k-1}$, $d=d_{k-1}$, and ${\check{d}}=d_k$. Note that $k-1 \geq \ell \geq 1$, so that both $d_{k-1}$ and $d_k$ are proportional to $n$. Then $$\left|\frac{1}{d_k}{\operatorname{Tr}}M_k-b_\sigma^2 \cdot
\frac{1}{d_{k-1}}{\operatorname{Tr}}M_{k-1}\right|
\leq C{\varepsilon}_n \sqrt{n} \cdot \frac{1}{d_{k-1}}\|M_{k-1}\|_F$$ with probability $1-e^{-n^{0.01}}$. Finally, for $k=\ell$, applying Lemma \[lemma:onelayerapprox\](a) conditional on $W_1,\ldots,W_{\ell-1}$ and with $t={\varepsilon}_n$, $$\left|\frac{1}{d_\ell}{\operatorname{Tr}}M_\ell-b_\sigma^2\right| \leq C{\varepsilon}_n$$ with probability $1-e^{-n^{0.01}}$. Combining these bounds, with probability $1-C'e^{-n^{0.01}}$, $$\left|{{\mathbf{s}}_{\alpha}^\ell}^\top {\mathbf{s}}_{\beta}^\ell-(b_\sigma^2)^{L-\ell+1}\right|
\leq \frac{C{\varepsilon}_n}{\sqrt{n}}\left(
\|M_L\|_F+\ldots+\|M_\ell\|_F+\sqrt{n}\right).$$ We also have $\|W_k/\sqrt{d_k}\| \leq C$ for each $k=2,\ldots,L$ with probability $1-C'e^{-cn}$, see e.g. [@vershynin2018high Theorem 4.4.5]. Then, applying $\|D_k\| \leq \lambda_\sigma$, we have $\|M_k\|_F \leq C\sqrt{n}\|M_k\| \leq C'\sqrt{n}$ for every $k=1,\ldots,L$. Then the first bound of (\[eq:Sabbound\]) follows. The second bound of (\[eq:Sabbound\]) is the same, applying Lemma \[lemma:onelayerapprox\] for $({\mathbf{x}}_{\alpha},{\mathbf{x}}_{\alpha})$ instead of $({\mathbf{x}}_{\alpha},{\mathbf{x}}_{\beta})$. The almost sure statement follows from Borel-Cantelli.
\[lemma:NTKFapprox\] Under Assumption \[assump:asymptotics\], almost surely as $n \to \infty$, $$\frac{1}{n}
\left\|K^{\text{NTK}}-\left(r_+{\operatorname{Id}}+X_L^\top X_L+\sum_{\ell=0}^{L-1} q_\ell X_\ell^\top
X_\ell\right)\right\|_F^2 \to 0.$$ Furthermore, for a constant $C>0$, almost surely for all large $n$, $\|K^{\text{NTK}}\| \leq C$.
By Corollary \[cor:orthonormalinduction\], we may assume that each matrix $X_0,\ldots,X_L$ is $({\varepsilon}_n,B)$-orthonormal. Then $$\Big|{{\mathbf{x}}_{\alpha}^{\ell-1}}^\top{\mathbf{x}}_{\beta}^{\ell-1}\Big| \leq {\varepsilon}_n,
\qquad \Big|\|{\mathbf{x}}_{\alpha}^{\ell-1}\|^2-1\Big| \leq {\varepsilon}_n.$$ Increasing ${\varepsilon}_n$ if necessary, we may assume ${\varepsilon}_n \geq n^{-0.48}$. Combining with (\[eq:Sabbound\]), we have for the off-diagonal entries of the Hadamard product that $$\Big|\big((S_\ell^\top S_\ell) \odot (X_{\ell-1}^\top X_{\ell-1})\big)
[{\alpha},{\beta}]-q_{\ell-1} X_{\ell-1}^\top X_{\ell-1}[{\alpha},{\beta}]\Big| \leq C{\varepsilon}_n^2,$$ and for the diagonal entries that $$\begin{aligned}
&\Big|\big((S_\ell^\top S_\ell) \odot (X_{\ell-1}^\top
X_{\ell-1})[{\alpha},{\alpha}]-q_{\ell-1}(X_{\ell-1}^\top X_{\ell-1})[{\alpha},{\alpha}]
-(r_{\ell-1}-q_{\ell-1})\Big|\\
&\leq \Big|\big((S_\ell^\top S_\ell) \odot (X_{\ell-1}^\top
X_{\ell-1})[{\alpha},{\alpha}]-r_{\ell-1}\Big|
+q_{\ell-1}\Big|X_{\ell-1}^\top X_{\ell-1}[{\alpha},{\alpha}]-1\Big|
\leq C{\varepsilon}_n.\end{aligned}$$ Then applying this to (\[eq:NTKform\]), $$\left\|K^{\text{NTK}}-\left(r_+{\operatorname{Id}}+X_L^\top X_L+\sum_{\ell=0}^{L-1} q_\ell X_\ell^\top
X_\ell\right)\right\|_F^2 \leq Cn(n-1){\varepsilon}_n^4+Cn{\varepsilon}_n^2.$$ The first statement of the lemma then follows from the assumption ${\varepsilon}_n n^{1/4} \to 0$.
For the second statement on the operator norm, we have $$\|(S_\ell^\top S_\ell) \odot (X_{\ell-1}^\top X_{\ell-1})\|
\leq \max_{{\alpha}=1}^n \Big|{{\mathbf{s}}^\ell_{\alpha}}^\top {\mathbf{s}}^\ell_{\alpha}\Big|
\cdot \|X_{\ell-1}^\top X_{\ell-1}\|.$$ See [@johnson1990matrix Eq. (3.7.9)], applied with $X=Y=S_\ell$. Then $\|K^{\text{NTK}}\| \leq C$ follows from (\[eq:NTKform\]), the $({\varepsilon}_n,B)$-orthonormality of each matrix $X_{\ell-1}$, and the bound for $\|{\mathbf{s}}_{\alpha}^\ell\|^2$ in (\[eq:Sabbound\]).
Combining Lemma \[lemma:NTKFapprox\] and Proposition \[prop:specapprox\], this proves Lemma \[lemma:NTKapprox\].
Unique solution of the fixed-point equation
-------------------------------------------
Let $A,\Phi \in {\mathbb{R}}^{n \times n}$ be symmetric matrices, where $\Phi$ is positive semi-definite. Let $z \in {\mathbb{C}}^+$, $\alpha \in {\mathbb{C}}^*$, and $\gamma>0$. For $s \in {\mathbb{C}}^+$, define $$S(s)=(A+s^{-1}\Phi-z{\operatorname{Id}})^{-1}, \quad
f_n(s)=\alpha^{-1}+\gamma {\operatorname{tr}}S(s)\Phi.$$
\[lemma:fixedpointfiniten\]
(a) For any $s \in {\mathbb{C}}^+$, setting $S \equiv S(s)$, $${\operatorname{Im}}f_n(s) \geq {\operatorname{Im}}z \cdot \gamma {\operatorname{tr}}S\Phi S^* \geq 0.$$
(b) For any $s_1,s_2 \in {\mathbb{C}}^+$, setting $S_1 \equiv S(s_1)$ and $S_2 \equiv S(s_2)$, $$\begin{aligned}
&|f_n(s_1)-f_n(s_2)|\\
& \leq
|s_1-s_2| \cdot \left(\frac{{\operatorname{Im}}f_n(s_1)-{\operatorname{Im}}z \cdot \gamma {\operatorname{tr}}S_1\Phi S_1^*}
{{\operatorname{Im}}s_1}\right)^{1/2}
\left(\frac{{\operatorname{Im}}f_n(s_2)-{\operatorname{Im}}z \cdot \gamma {\operatorname{tr}}S_2\Phi S_2^*}{{\operatorname{Im}}s_2}\right)^{1/2}\end{aligned}$$
For part (a), let us write $$S\Phi=S\Phi S^*(A+s^{-1}\Phi-z{\operatorname{Id}})^*
=S\Phi S^*A+(1/s^*)S\Phi S^*\Phi-z^* S\Phi S^*.$$ Since $S\Phi S^*$ is Hermitian and positive semi-definite, the quantities ${\operatorname{tr}}S\Phi S^*A$, ${\operatorname{tr}}S\Phi S^* \Phi$, and ${\operatorname{tr}}S\Phi S^*$ are all real, and the latter two are nonnegative. Then $$\label{eq:ImPhiS}
{\operatorname{Im}}f_n(s)={\operatorname{Im}}\alpha^{-1}+\gamma {\operatorname{Im}}{\operatorname{tr}}S\Phi=
{\operatorname{Im}}\alpha^{-1}+\frac{{\operatorname{Im}}s}{|s|^2} \cdot \gamma {\operatorname{tr}}S\Phi S^*\Phi+{\operatorname{Im}}z \cdot
\gamma{\operatorname{tr}}S\Phi S^*.$$ Each term on the right side of (\[eq:ImPhiS\]) is nonnegative, and dropping the first two of these terms yields (a).
For part (b), applying the identity $A^{-1}-B^{-1}=A^{-1}(B-A)B^{-1}$, we have $$S_1-S_2=S_1(s_2^{-1}\Phi-s_1^{-1}\Phi)S_2=\frac{s_1-s_2}{s_1s_2}S_1\Phi S_2,$$ so $$f_n(s_1)-f_n(s_2)=\gamma {\operatorname{tr}}S_1\Phi-\gamma {\operatorname{tr}}S_2\Phi
=\frac{\gamma(s_1-s_2)}{s_1s_2}{\operatorname{tr}}S_1\Phi S_2\Phi.$$ Applying Cauchy-Schwarz to the inner-product $\langle S_1,S_2 \rangle_\Phi={\operatorname{tr}}S_1\Phi S_2^*\Phi$, $$|{\operatorname{tr}}S_1\Phi S_2\Phi|^2
=|\langle S_1,S_2^* \rangle_\Phi|^2
\leq \langle S_1,S_1 \rangle_\Phi \cdot \langle S_2^*,S_2^* \rangle_\Phi
= {\operatorname{tr}}S_1\Phi S_1^*\Phi \cdot {\operatorname{tr}}S_2\Phi S_2^* \Phi.$$ Then $$|f_n(s_1)-f_n(s_2)| \leq |s_1-s_2| \cdot
\left(\frac{\gamma {\operatorname{tr}}S_1\Phi S_1^*\Phi}{|s_1|^2}\right)^{1/2}
\left(\frac{\gamma {\operatorname{tr}}S_2\Phi S_2^* \Phi}{|s_2|^2}\right)^{1/2}.$$ Dropping ${\operatorname{Im}}\alpha^{-1}$ in (\[eq:ImPhiS\]) and applying this to upper-bound $\gamma {\operatorname{tr}}S\Phi S^*\Phi/|s|^2$, part (b) follows.
\[cor:fixedpointunique\] As $n \to \infty$, suppose that $f_n(s) \to f(s)$ pointwise for each $s \in {\mathbb{C}}^+$, the empirical spectral distributions of $\Phi$ and $A$ converge weakly to deterministic limits, and the limit for $\Phi$ is not the point distribution at 0. Then the fixed-point equation $s=f(s)$ has at most one solution $s \in {\mathbb{C}}^+$.
Let us first show that for each $s \in {\mathbb{C}}^+$ and a value $c_0(s)>0$ independent of $n$, $$\label{eq:trSPhiSlowerbound}
\liminf_{n \to \infty} {\operatorname{tr}}S(s)\Phi S(s)^* \geq c_0(s)>0.$$ Denoting $S \equiv S(s)$ and applying the von Neumann trace inequality, $${\operatorname{tr}}S\Phi S^*=\frac{1}{n}{\operatorname{Tr}}\Phi S^*S
\geq \frac{1}{n}\sum_{{\alpha}=1}^n \lambda_{\alpha}(\Phi)\lambda_{n+1-{\alpha}}(S^*S),$$ where $\lambda_1(\cdot) \geq \ldots \geq \lambda_n(\cdot)$ denote the sorted eigenvalues. Since $\Phi$ has a non-degenerate limit spectrum, there is a constant ${\varepsilon}>0$ for which $\lambda_{{\varepsilon}n}(\Phi)>{\varepsilon}$ for all large $n$. (Throughout the proof, ${\varepsilon}n$, ${\varepsilon}n/2$, etc. should be understood as their roundings to the nearest integer.) Then $${\operatorname{tr}}S\Phi S^* \geq {\varepsilon}\cdot \frac{1}{n}\sum_{{\alpha}=1}^{{\varepsilon}n}
\lambda_{n+1-{\alpha}}(S^*S).$$ Denoting by $\sigma_{\alpha}(\cdot)$ the ${\alpha}^\text{th}$ largest singular value, observe that $$\lambda_{n+1-{\alpha}}(S^*S)=\sigma_{n+1-{\alpha}}(S)^2
=\sigma_{\alpha}(A+s^{-1}\Phi-z{\operatorname{Id}})^{-2}.$$ Applying $\sigma_{{\alpha}+{\beta}-1}(A+B) \leq \sigma_{\alpha}(A)+\sigma_{\beta}(B)$, we have $$\sigma_{\alpha}(A+s^{-1}\Phi-z{\operatorname{Id}}) \leq
\sigma_{{\alpha}/2}(A)+|s|^{-1}\sigma_{{\alpha}/2+1}(\Phi)+|z|.$$ Since the spectra of $A$ and $\Phi$ converge to deterministic limits, this implies that there is a constant $C(s)>0$ (also depending on $z$ and ${\varepsilon}$) such that $\sigma_{\alpha}(A+s^{-1}\Phi-z{\operatorname{Id}}) \leq C(s)$ for every ${\alpha}\in [{\varepsilon}n/2,{\varepsilon}n]$ and all large $n$. Thus $${\operatorname{tr}}S\Phi S^* \geq {\varepsilon}\cdot \frac{{\varepsilon}n-{\varepsilon}n/2}{n} \cdot C(s)^{-2}$$ for all large $n$, and this shows the claim (\[eq:trSPhiSlowerbound\]).
Then, taking the limit $n \to \infty$ in Lemma \[lemma:fixedpointfiniten\](b), we get $$|f(s_1)-f(s_2)| \leq |s_1-s_2|
\cdot \left(\frac{{\operatorname{Im}}f(s_1)-{\operatorname{Im}}z \cdot \gamma c_0(s_1)}{{\operatorname{Im}}s_1}\right)^{1/2}
\left(\frac{{\operatorname{Im}}f(s_2)-{\operatorname{Im}}z \cdot \gamma c_0(s_2)}{{\operatorname{Im}}s_2}\right)^{1/2}.$$ If $s_1=f(s_1)$ and $s_2=f(s_2)$, then this yields $|s_1-s_2| \leq |s_1-s_2| \cdot h(s_1,s_2)$ for some quantity $h(s_1,s_2) \in [0,1)$, where $h(s_1,s_2)<1$ strictly because $c_0(s_1),c_0(s_2)>0$. This implies $s_1=s_2$, so the equation $s=f(s)$ has at most one solution $s \in {\mathbb{C}}^+$.
Proof of Proposition \[prop:swelldefined\] and Theorem \[thm:NTK\]
------------------------------------------------------------------
The operator norm bound in Theorem \[thm:NTK\] was shown in Lemma \[lemma:NTKFapprox\]. For the spectral convergence, note that by Lemma \[lemma:NTKapprox\], the limit Stieltjes transform of $K^{\text{NTK}}$ at any $z \in {\mathbb{C}}^+$ is given by $$m_{\text{NTK}}(z)=\lim_{n \to \infty} {\operatorname{tr}}\left((-z+r_+){\operatorname{Id}}+X_L^\top X_L
+\sum_{\ell=0}^{L-1} q_\ell X_\ell^\top X_\ell\right)^{-1},$$ provided that this limit exists and defines the Stieltjes transform of a probability measure. For $${\mathbf{z}}=(z_{-1},\ldots,z_\ell) \in {\mathbb{C}}^- \times {\mathbb{R}}^\ell \times {\mathbb{C}}^*,
\qquad {\mathbf{w}}=(w_{-1},\ldots,w_\ell) \in {\mathbb{C}}^{\ell+2},$$ recall the functions $${\mathbf{z}}\mapsto s_\ell({\mathbf{z}}), \quad ({\mathbf{z}},{\mathbf{w}}) \mapsto t_\ell({\mathbf{z}},{\mathbf{w}})$$ defined recursively by (\[eq:sl\]) and (\[eq:tl\]). Proposition \[prop:swelldefined\] and Theorem \[thm:NTK\] are immediate consequences of the following extended result.
\[lemma:NTKextended\] Under Assumption \[assump:asymptotics\], for each $\ell=1,\ldots,L$:
(a) For every ${\mathbf{z}}\in {\mathbb{C}}^- \times {\mathbb{R}}^\ell \times {\mathbb{C}}^*$, the equation (\[eq:sl\]) has a unique fixed point $s_\ell({\mathbf{z}}) \in {\mathbb{C}}^+$.
(b) For every $({\mathbf{z}},{\mathbf{w}}) \in ({\mathbb{C}}^- \times {\mathbb{R}}^\ell \times {\mathbb{C}}^*) \times
{\mathbb{C}}^{\ell+2}$, almost surely $$\begin{aligned}
&t_\ell({\mathbf{z}},{\mathbf{w}})\nonumber\\
&=\lim_{n \to \infty}
{\operatorname{tr}}\Big(z_{-1}{\operatorname{Id}}+z_0 X_0^\top X_0+\ldots+z_\ell
X_\ell^\top X_\ell\Big)^{-1}\Big(w_{-1}{\operatorname{Id}}+w_0 X_0^\top X_0+\ldots+w_\ell
X_\ell^\top X_\ell\Big).\label{eq:tliscorrect}\end{aligned}$$ In particular, for any $z_{-1},\ldots,z_\ell \in {\mathbb{R}}$ where $z_\ell \neq 0$, $${\operatorname{lim\;spec}}z_{-1}{\operatorname{Id}}+z_0 X_0^\top X_0+\ldots+z_\ell
X_\ell^\top X_\ell=\nu$$ where $\nu$ is a probability measure on ${\mathbb{R}}$ with Stieltjes transform $$m(z)=t_\ell\Big((-z+z_{-1},z_0,\ldots,z_\ell),(1,0,\ldots,0)\Big).$$
By Corollary \[cor:orthonormalinduction\], we may assume that each matrix $X_0,\ldots,X_L$ is $({\varepsilon}_n,B)$-orthonormal.
Define $\Phi_\ell,\tilde{\Phi}_\ell$ by (\[eq:Phiell\]) and (\[eq:tildePhiell\]). For ${\mathbf{z}}=(z_{-1},\ldots,z_\ell)$, let us write as shorthand $${\mathbf{z}}\cdot {\mathbf{X}}^\top {\mathbf{X}}(\ell)
=z_{-1}{\operatorname{Id}}+z_0X_0^\top X_0+\ldots+z_\ell X_\ell^\top X_\ell,$$ where the parenthetical $(\ell)$ signifies the index of the last term in this sum. Let us define similarly ${\mathbf{w}}\cdot {\mathbf{X}}^\top {\mathbf{X}}(\ell)$.
Note that part (b) holds for $\ell=0$, by the assumption ${\operatorname{lim\;spec}}X_0^\top X_0=\mu_0$, the definition of $t_0((z_{-1},z_0),(w_{-1},w_0))$ in (\[eq:t0\]), and the fact that the function $x
\mapsto (w_{-1}+w_0x)/(z_{-1}+z_0x)$ is continuous and bounded over the non-negative real line when $z_{-1} \in {\mathbb{C}}^-$ and $z_0 \in {\mathbb{C}}^*$.
We induct on $\ell$. Suppose that part (b) holds for $\ell-1$. To show part (a) for $\ell$, fix any ${\mathbf{z}}=(z_{-1},\ldots,z_\ell) \in {\mathbb{C}}^- \times {\mathbb{R}}^\ell \times {\mathbb{C}}^*$ (not depending on $n$) and consider the matrix $$\label{eq:RNTK}
R=\Big({\mathbf{z}}\cdot {\mathbf{X}}^\top {\mathbf{X}}(\ell)\Big)^{-1}.$$ We apply the analysis of Appendix \[appendix:singlelayer\], conditional on $X_0,\ldots,X_{\ell-1}$, and with the identifications $${\widecheck{X}}=X_\ell, \qquad X=X_{\ell-1}, \qquad {\check{d}}=d_\ell, \qquad d=d_{\ell-1},$$ $$A=z_0X_0^\top X_0+\ldots+z_{\ell-1}X_{\ell-1}^\top X_{\ell-1}, \qquad
\alpha=z_\ell, \qquad z=-z_{-1}.$$ Observe that $\alpha \in {\mathbb{C}}^*$ and $z \in {\mathbb{C}}^-$. The matrix $R$ in (\[eq:RNTK\]) is exactly $$R=(A+\alpha {\widecheck{X}}^\top {\widecheck{X}}-z{\operatorname{Id}})^{-1}.$$ Since each $X_0,\ldots,X_{\ell-1}$ is $({\varepsilon}_n,B)$-orthonormal, we have $\|A\| \leq C$ for some constant $C>0$ (depending on $z_{-1},\ldots,z_\ell,\lambda_\sigma$). Thus Assumption \[assump:singlelayer\] holds, conditional on $X_0,\ldots,X_{\ell-1}$. Let us define the $n$-dependent parameter $$\bar{s}=\frac{1}{\alpha}+\frac{n}{d_\ell}{\operatorname{tr}}{\mathbb{E}}_{W_\ell}[R \Phi_\ell]$$ where this expectation is over only the weights $W_\ell$. Then, applying Lemma \[lemma:fixedpoint\](b) with a value $t \equiv t_n$ such that $t \to 0$ and $nt^2 \gg \log n$, we obtain $$\label{eq:fixedpointNTKbars}
\Big|\bar{s}-\frac{1}{\alpha}-\frac{n}{d_\ell}
{\operatorname{tr}}(A+\bar{s}^{-1}\Phi_\ell-z{\operatorname{Id}})^{-1}\Phi_\ell\Big| \to 0$$ almost surely as $n \to \infty$.
Proposition \[prop:basic\] shows that $|\bar{s}|$ is bounded, so for any subsequence in $n$, there is a further sub-subsequence where $\bar{s} \to
s_0$ for a limit $s_0 \equiv s_0({\mathbf{z}}) \in {\mathbb{C}}^+$. Let us now replace $\bar{s}$ and $\Phi_\ell$ above by $s_0$ and $\tilde{\Phi}_\ell$: First we have $${\operatorname{tr}}\left(A+\bar{s}^{-1} \Phi_\ell-z{\operatorname{Id}}\right)^{-1}\Phi_\ell-{\operatorname{tr}}\left(A+s_0^{-1} \tilde{\Phi}_\ell-z{\operatorname{Id}}\right)^{-1}\Phi_\ell
\to 0$$ by the same argument as (\[eq:sbys0\]). Then, we have $$\begin{aligned}
&\left|{\operatorname{tr}}\left(A+s_0^{-1} \Phi_\ell-z{\operatorname{Id}}\right)^{-1}\Phi_\ell-{\operatorname{tr}}\left(A+s_0^{-1} \tilde{\Phi}_\ell-z{\operatorname{Id}}\right)^{-1}\Phi_\ell\right|\\
&=\left|s_0^{-1}{\operatorname{tr}}\left(A+s_0^{-1} \Phi_\ell-z{\operatorname{Id}}\right)^{-1}
(\tilde{\Phi}_\ell-\Phi_\ell)\left(A+s_0^{-1} \tilde{\Phi}_\ell-z{\operatorname{Id}}\right)^{-1}
\Phi_\ell\right|\\
&\leq \frac{C}{n}\|\tilde{\Phi}_\ell-\Phi_\ell\|_F
\cdot \left\|(A+s_0^{-1} \tilde{\Phi}-z{\operatorname{Id}})^{-1}
\Phi(A+s_0^{-1} \Phi-z{\operatorname{Id}})^{-1}\right\|_F\\
&\leq \frac{C}{\sqrt{n}}
\|\tilde{\Phi}_\ell-\Phi_\ell\|_F
\cdot \|(A+s_0^{-1} \tilde{\Phi}-z{\operatorname{Id}})^{-1}\|
\cdot \|\Phi\| \cdot \|(A+s_0^{-1} \Phi-z{\operatorname{Id}})^{-1}\| \to 0,\end{aligned}$$ where the convergence to 0 follows from Lemma \[lemma:NTKFapprox\]. Finally, we have $$\begin{aligned}
&\left|{\operatorname{tr}}\left(A+s_0^{-1} \Phi_\ell-z{\operatorname{Id}}\right)^{-1}\Phi_\ell-{\operatorname{tr}}\left(A+s_0^{-1} \Phi_\ell-z{\operatorname{Id}}\right)^{-1}\tilde{\Phi}_\ell\right|\\
&\leq \frac{1}{n}\|(A+s_0^{-1} \Phi_\ell-z{\operatorname{Id}})^{-1}\|_F
\cdot \|\Phi_\ell-\tilde{\Phi}_\ell\|_F
\leq \frac{1}{\sqrt{n}}\|(A+s_0^{-1} \Phi_\ell-z{\operatorname{Id}})^{-1}\|
\cdot \|\Phi_\ell-\tilde{\Phi}_\ell\|_F \to 0.\end{aligned}$$ Applying these approximations to (\[eq:fixedpointNTKbars\]), we have almost surely along this sub-subsequence that $$\label{eq:s0fixedfinal}
\Big|s_0-\frac{1}{\alpha}-\gamma_\ell
{\operatorname{tr}}(A+s_0^{-1}\tilde{\Phi}_\ell-z{\operatorname{Id}})^{-1}\tilde{\Phi}_\ell\Big| \to 0.$$
Now observe from the definitions of $A$, $\tilde{\Phi}_\ell$, and $z$ that $$\begin{aligned}
A+s_0^{-1}\tilde{\Phi}_\ell-z{\operatorname{Id}}&=\Big(z_{-1}+\frac{1-b_\sigma^2}{s_0}\Big){\operatorname{Id}}+\sum_{k=0}^{\ell-2} z_kX_k^\top X_k
+\Big(z_{\ell-1}+\frac{b_\sigma^2}{s_0}\Big)X_{\ell-1}^\top X_{\ell-1},\\
\tilde{\Phi}_\ell&=(1-b_\sigma^2){\operatorname{Id}}+b_\sigma^2 X_{\ell-1}^\top X_{\ell-1}.\end{aligned}$$ Then, applying (\[eq:s0fixedfinal\]) and the induction hypothesis that part (b) holds for $\ell-1$, we obtain that the value $s_0$ must satisfy $$s_0=\frac{1}{\alpha}+\gamma_\ell t_{\ell-1}
\Big({\mathbf{z}}_{{\text{prev}}}(s_0),(1-b_\sigma^2,0,\ldots,0,b_\sigma^2)\Big),$$ where ${\mathbf{z}}_{{\text{prev}}}$ is defined in (\[eq:zprev\]). This shows the existence of a solution (in ${\mathbb{C}}^+$) to the fixed-point equation (\[eq:sl\]).
To show uniqueness, we apply Corollary \[cor:fixedpointunique\]: For any fixed $s \in {\mathbb{C}}^+$, defining $$f_n(s)=\frac{1}{\alpha}+(n/d_\ell){\operatorname{tr}}(A+s^{-1}\Phi_\ell-z{\operatorname{Id}})^{-1}\Phi_\ell,$$ the same arguments as above establish that $$\lim_{n \to \infty} f_n(s)=f(s) \equiv
\frac{1}{\alpha}+\gamma_\ell t_{\ell-1}
\Big({\mathbf{z}}_{{\text{prev}}}(s),(1-b_\sigma^2,0,\ldots,0,b_\sigma^2)\Big).$$ Part (b) holding for $\ell-1$ implies that both $A$ and $\Phi_\ell$ have deterministic spectral limits, where $${\operatorname{lim\;spec}}\Phi_\ell={\operatorname{lim\;spec}}\tilde{\Phi}_\ell$$ by (\[eq:Phiapprox\]). This cannot be the point distribution at 0, because (\[eq:concentration\_diagonal\]) implies that ${\operatorname{tr}}\Phi_\ell \geq 1/2$ for all large $n$, and $\|\Phi_\ell\| \leq C$ so at least $n/(2C)$ eigenvalues of $\Phi_\ell$ exceed $1/2$ for every $n$. Thus, Corollary \[cor:fixedpointunique\] implies that the fixed point $s=f(s)$ is unique. So the fixed point $s_\ell({\mathbf{z}}) \in {\mathbb{C}}^+$ is uniquely defined by (\[eq:sl\]), and this shows part (a) for $\ell$.
By the uniqueness of this fixed point, we have also shown that $s_0=s_\ell({\mathbf{z}})$, where $s_0$ is the limit of $\bar{s}$ along the above sub-subsequence. Since for any subsequence in $n$, there exists a sub-subsequence for this which holds, this shows that $\lim_{n \to \infty} \bar{s}=s_\ell({\mathbf{z}})$ almost surely.
Now, to show that part (b) holds for $\ell$, let us also fix any ${\mathbf{w}}=(w_{-1},\ldots,w_\ell) \in {\mathbb{C}}^{\ell+2}$. Using that $z_\ell \neq 0$, we may write $${\mathbf{w}}\cdot {\mathbf{X}}^\top {\mathbf{X}}(\ell)
=\frac{w_\ell}{z_\ell}\cdot {\mathbf{z}}\cdot {\mathbf{X}}^\top {\mathbf{X}}(\ell)
+{\mathbf{w}}_{{\text{prev}}}\cdot {\mathbf{X}}^\top {\mathbf{X}}(\ell-1),$$ where ${\mathbf{w}}_{{\text{prev}}}$ is as defined in (\[eq:wprev\]). Then $$\label{eq:wreduce}
\Big({\mathbf{z}}\cdot {\mathbf{X}}^\top {\mathbf{X}}(\ell)\Big)^{-1}
\Big({\mathbf{w}}\cdot {\mathbf{X}}^\top {\mathbf{X}}(\ell)\Big)
=\frac{w_\ell}{z_\ell}{\operatorname{Id}}+
\Big({\mathbf{z}}\cdot {\mathbf{X}}^\top {\mathbf{X}}(\ell)\Big)^{-1}
\Big({\mathbf{w}}_{{\text{prev}}}\cdot {\mathbf{X}}^\top {\mathbf{X}}(\ell-1)\Big).$$ We now apply Lemma \[lemma:fixedpoint\](a) conditional on $X_0,\ldots,X_{\ell-1}$, with the same identifications as above and with $$M={\mathbf{w}}_{{\text{prev}}}\cdot {\mathbf{X}}^\top {\mathbf{X}}(\ell-1).$$ Note that $M$ is indeed deterministic conditional on $X_0,\ldots,X_{\ell-1}$, and $\|M\| \leq C$ for a constant $C>0$ (depending on ${\mathbf{z}}$ and ${\mathbf{w}}$) since $X_0,\ldots,X_{\ell-1}$ are $({\varepsilon}_n,B)$-orthonormal. Then, applying Lemma \[lemma:fixedpoint\](a), $${\operatorname{tr}}\Big[\Big({\mathbf{z}}\cdot {\mathbf{X}}^\top {\mathbf{X}}(\ell)\Big)^{-1}
\Big({\mathbf{w}}_{{\text{prev}}}\cdot {\mathbf{X}}^\top {\mathbf{X}}(\ell-1)\Big)\Big]-
{\operatorname{tr}}\Big[(A+\bar{s}^{-1}\Phi_\ell-z{\operatorname{Id}})^{-1}
\Big({\mathbf{w}}_{{\text{prev}}}\cdot {\mathbf{X}}^\top {\mathbf{X}}(\ell-1)\Big)\Big] \to 0.$$ By the same arguments as above, we may replace $\bar{s}$ by $s_0=s_\ell({\mathbf{z}})$ and $\Phi_\ell$ by $\tilde{\Phi}_\ell$. Then, applying this to (\[eq:wreduce\]), $${\operatorname{tr}}\Big[\Big({\mathbf{z}}\cdot {\mathbf{X}}^\top {\mathbf{X}}(\ell)\Big)^{-1}
\Big({\mathbf{w}}\cdot {\mathbf{X}}^\top {\mathbf{X}}(\ell)\Big)\Big]
-\frac{w_\ell}{z_\ell}-
{\operatorname{tr}}\Big[(A+s_\ell({\mathbf{z}})^{-1}\tilde{\Phi}_\ell-z{\operatorname{Id}})^{-1}
\Big({\mathbf{w}}_{{\text{prev}}}\cdot {\mathbf{X}}^\top {\mathbf{X}}(\ell-1)\Big)\Big] \to 0.$$ Finally, applying that part (b) holds for $\ell-1$, this yields $$\lim_{n \to \infty} {\operatorname{tr}}\Big[\Big({\mathbf{z}}\cdot {\mathbf{X}}^\top {\mathbf{X}}(\ell)\Big)^{-1}
\Big({\mathbf{w}}\cdot {\mathbf{X}}^\top {\mathbf{X}}(\ell)\Big)\Big]
=\frac{w_\ell}{z_\ell}+t_{\ell-1}({\mathbf{z}}_{{\text{prev}}}(s_\ell({\mathbf{z}})),{\mathbf{w}}_{{\text{prev}}}),$$ which is the definition of $t_\ell({\mathbf{z}},{\mathbf{w}})$. This establishes (\[eq:tliscorrect\]).
For any fixed $z_{-1},\ldots,z_\ell \in {\mathbb{R}}$ where $z_\ell \neq 0$, and any fixed $z \in {\mathbb{C}}^+$, this implies that the Stieltjes transform of ${\mathbf{z}}\cdot
{\mathbf{X}}^\top {\mathbf{X}}(\ell)$ has the almost sure limit $$m(z)=t_\ell\Big((-z+z_{-1},z_0,\ldots,z_\ell),(1,0,\ldots,0)\Big).$$ So $m(z)$ defines the Stieltjes transform of a sub-probability distribution $\nu$, and the empirical eigenvalue distribution of ${\mathbf{z}}\cdot {\mathbf{X}}^\top {\mathbf{X}}(\ell)$ converges vaguely a.s. to $\nu$. Since $\|{\mathbf{z}}\cdot {\mathbf{X}}^\top {\mathbf{X}}(\ell)\|$ is bounded because $X_0,\ldots,X_L$ are $({\varepsilon}_n,B)$-orthonormal, this limit $\nu$ must in fact be a probability distribution, and the eigenvalue distribution converges weakly to $\nu$. This concludes the induction and the proof.
Multi-dimensional outputs and rescaled parametrizations {#appendix:NTKmulti}
=======================================================
In this section, we provide some motivation for the form of the NTK in (\[eq:NTKmulti\]) for networks with a $k$-dimensional output, and we prove Theorem \[thm:NTKmulti\] regarding its spectrum.
Derivation of (\[eq:NTKmulti\]) from gradient flow training {#appendix:NTKmultiderivation}
-----------------------------------------------------------
Consider gradient flow training of the network (\[eq:NNfuncmulti\]), with training samples $({\mathbf{x}}_{\alpha},{\mathbf{y}}_{\alpha})_{{\alpha}=1}^n$ where ${\mathbf{x}}_{\alpha}\in {\mathbb{R}}^{d_0}$ and ${\mathbf{y}}_{\alpha}\in {\mathbb{R}}^k$, under the general training loss $$F(\theta)=\sum_{{\alpha}=1}^n \mathcal{L}(f_\theta({\mathbf{x}}_{\alpha}),{\mathbf{y}}_{\alpha}).$$ Here, $\mathcal{L}:{\mathbb{R}}^k \times {\mathbb{R}}^k \to {\mathbb{R}}$ is the loss function. We denote by $\nabla \mathcal{L}(f_\theta({\mathbf{x}}_{\alpha}),{\mathbf{y}}_{\alpha}) \in {\mathbb{R}}^k$ the gradient of $\mathcal{L}$ with respect to its first argument, and by $\nabla_{W_\ell} f_\theta({\mathbf{x}}_{\alpha})\in {\mathbb{R}}^{\dim(W_\ell) \times k}$ the Jacobian of $f_\theta({\mathbf{x}}_{\alpha})$ with respect to the weights $W_\ell$.
Consider a possibly reweighted gradient-flow training of $\theta$, where the evolution of weights $W_\ell$ is given by $$\begin{aligned}
\frac{d}{dt}W_\ell(t)&=-\tau_\ell \cdot
\nabla_{W_\ell} F(\theta(t))=-\tau_\ell
\sum_{{\alpha}=1}^n \nabla_{W_\ell} f_{\theta(t)}({\mathbf{x}}_{\alpha}) \cdot \nabla
\mathcal{L}(f_{\theta(t)}({\mathbf{x}}_{\alpha}),{\mathbf{y}}_{\alpha}).\end{aligned}$$ The learning rate for each weight matrix $W_\ell$ is scaled by a constant $\tau_\ell$—this may arise, for example, from reparametrizing the network (\[eq:NNfuncmulti\]) using $\widetilde{W}_\ell=\tau_\ell^{-1} \cdot W_\ell$ and considering gradient flow training for $\widetilde{W}_\ell$. Denoting the vectorization of all training predictions and its Jacobian by $$f_\theta(X)=(f_\theta^1(X),\ldots,f_\theta^k(X)) \in {\mathbb{R}}^{nk},
\qquad \nabla_{W_\ell} f_\theta(X) \in {\mathbb{R}}^{\dim(W_\ell) \times nk},$$ and the corresponding vectorization of $(\nabla \mathcal{L}(f_\theta({\mathbf{x}}_{\alpha}),{\mathbf{y}}_{\alpha}))_{{\alpha}=1}^n$ by $\nabla \mathcal{L}(f_\theta(X),{\mathbf{y}}) \in {\mathbb{R}}^{nk}$, this may be written succinctly as $$\frac{d}{dt}W_\ell(t)=-\tau_\ell \cdot \nabla_{W_\ell} f_{\theta(t)}(X)
\cdot \nabla \mathcal{L}(f_{\theta(t)}(X),{\mathbf{y}}).$$ Then the time evolution of in-sample predictions is given by $$\begin{aligned}
\frac{d}{dt} f_{\theta(t)}(X)
&=\Big(\nabla_\theta f_{\theta(t)}(X)\Big)^\top
\cdot \frac{d}{dt}\theta(t)\\
&=-\sum_{\ell=1}^{L+1} \tau_\ell
\Big(\nabla_{W_\ell} f_{\theta(t)}(X)\Big)^\top
\Big(\nabla_{W_\ell} f_{\theta(t)}(X)\Big) \cdot \nabla
\mathcal{L}(f_{\theta(t)}(X),{\mathbf{y}})\\
&=-K^{\text{NTK}}(t) \cdot \nabla \mathcal{L}(f_{\theta(t)}(X),{\mathbf{y}}),\end{aligned}$$ where $K^{\text{NTK}}$ is the matrix defined in (\[eq:NTKmulti\]). For $\tau_1=\ldots=\tau_{L+1}=1$, this matrix is simply $$K^{\text{NTK}}=\Big(\nabla_\theta f_\theta(X)\Big)^\top
\Big(\nabla_\theta f_\theta(X)\Big) \in {\mathbb{R}}^{nk \times nk},$$ which is a flattening of the neural tangent kernel $K \in {\mathbb{R}}^{n \times n \times k \times k}$ (identified as a map $K:{\mathbb{R}}^{n \times n} \to {\mathbb{R}}^{k \times k}$) that is defined in [@jacot2018neural].
Proof of Theorem \[thm:NTKmulti\]
---------------------------------
The matrix $K^{\text{NTK}}$ in (\[eq:NTKmulti\]) admits a $k \times k$ block decomposition $$K^{\text{NTK}}=\begin{pmatrix} K_{11}^{\text{NTK}}& \cdots & K_{1k}^{\text{NTK}}\\
\vdots & \ddots & \vdots \\
K_{k1}^{\text{NTK}}& \cdots & K_{kk}^{\text{NTK}}\end{pmatrix},
\qquad K_{ij}^{\text{NTK}}=\sum_{\ell=1}^{L+1}
\tau_\ell \Big(\nabla_{W_\ell} f_\theta^i(X)\Big)^\top
\Big(\nabla_{W_\ell} f_\theta^j(X)\Big) \in {\mathbb{R}}^{n \times n}.$$ Writing $$W_{L+1}=\begin{pmatrix} {\mathbf{w}}_1^\top \\ \vdots \\ {\mathbf{w}}_k^\top \end{pmatrix},$$ a computation using the chain rule similar to (\[eq:NTKform\]) verifies that $$K_{ij}^{\text{NTK}}={\mathbf{1}}\{i=j\}\tau_{L+1}X_L^\top X_L+\sum_{\ell=1}^L \tau_\ell
({S_\ell^i}^\top S_\ell^j) \odot (X_{\ell-1}^\top X_{\ell-1})$$ where $S_\ell^i \in {\mathbb{R}}^{d_\ell \times n}$ is the matrix with the same column-wise definition as in (\[eq:Sell\]), replacing ${\mathbf{w}}$ by ${\mathbf{w}}_i$.
\[lemma:NTKFapproxmulti\] Under the assumptions of Theorem \[thm:NTKmulti\], for each $\ell \in [L]$ and any indices $i \neq j \in [k]$, almost surely as $n
\to \infty$, $$\frac{1}{n}\|K_{ij}^{\text{NTK}}\|_F^2 \to 0.$$ Furthermore, for a constant $C>0$, almost surely for all large $n$, $\|K_{ij}^{\text{NTK}}\| \leq C$.
By Corollary \[cor:orthonormalinduction\], we may assume that each $X_0,\ldots,X_L$ is $({\varepsilon}_n,B)$-orthonormal.
Let us fix $i,j,\ell$ and denote the columns of $S_\ell^i$ and $S_\ell^j$ by ${\mathbf{s}}_{\alpha}^{\ell,i}$ and ${\mathbf{s}}_{\beta}^{\ell,j}$ for ${\alpha},{\beta}\in [n]$. We apply the Hanson-Wright inequality conditional on $W_1,\ldots,W_L$, which is similar to (\[eq:HansonWright\]). However, since ${\mathbf{w}}_i$ and ${\mathbf{w}}_j$ are independent, there is no trace term, and we obtain instead $$\Big|{{\mathbf{s}}_{\alpha}^{\ell,i}}^\top {\mathbf{s}}_{\beta}^{\ell,j}\Big| \leq C{\varepsilon}_n\sqrt{n}
\frac{1}{d_L}\|M_L\|_F$$ for both ${\alpha}={\beta}$ and ${\alpha}\neq {\beta}$ with probability $1-e^{-n^{0.01}}$, where $M_L$ is the same matrix as defined in (\[eq:Mell\]). Applying the bound $\|M_L\|_F \leq C\sqrt{n}$ as in the proof of Corollary \[cor:Sapprox\], this yields $$\Big|{{\mathbf{s}}_{\alpha}^{\ell,i}}^\top {\mathbf{s}}_{\beta}^{\ell,j}\Big| \leq C{\varepsilon}_n$$ almost surely for all ${\alpha},{\beta}\in [n]$ and all large $n$. Combining with the $({\varepsilon}_n,B)$-orthonormality of $X_{\ell-1}$, we get for ${\alpha}\neq {\beta}$ that $$\Big|({S_\ell^i}^\top S_\ell^j) \odot (X_{\ell-1}^\top X_{\ell-1})[{\alpha},{\beta}]\Big|
\leq C{\varepsilon}_n^2, \qquad
\Big|({S_\ell^i}^\top S_\ell^j) \odot (X_{\ell-1}^\top X_{\ell-1})[{\alpha},{\alpha}]\Big|
\leq C{\varepsilon}_n.$$ Then $$\|({S_\ell^i}^\top S_\ell^j) \odot (X_{\ell-1}^\top X_{\ell-1})\|_F^2
\leq Cn(n-1){\varepsilon}_n^4+Cn{\varepsilon}_n^2,$$ and the first statement follows from the assumption ${\varepsilon}_n n^{1/4} \to 0$. The second statement on the operator norm follows from the bound $$\|({S_\ell^i}^\top {S_\ell^j}) \odot (X_{\ell-1}^\top X_{\ell-1})\|
\leq \left(\max_{{\alpha}=1}^n \Big|{{\mathbf{s}}^{\ell,i}_{{\alpha}}}^\top
{{\mathbf{s}}^{\ell,i}_{{\alpha}}}\Big|\right)^{1/2}
\left(\max_{{\alpha}=1}^n \Big|{{\mathbf{s}}^{\ell,j}_{\alpha}}^\top
{\mathbf{s}}^{\ell,j}_{\alpha}\Big|\right)^{1/2} \cdot \|X_{\ell-1}^\top X_{\ell-1}\|.$$ See [@johnson1990matrix Eq. (3.7.9)] applied with $X=S_\ell^i$ and $Y=S_\ell^j$. The bound $\|K_{ij}^{\text{NTK}}\| \leq C$ then follows from the $({\varepsilon}_n,B)$-orthonormality of $X_{\ell-1}$ and Corollary \[cor:Sapprox\], applied to $S_\ell^i$ and $S_\ell^j$.
Applying this lemma together with Proposition \[prop:specapprox\], we obtain $${\operatorname{lim\;spec}}K^{\text{NTK}}={\operatorname{lim\;spec}}\begin{pmatrix} K_{11}^{\text{NTK}}& & \\
& \ddots & \\ & & K_{kk}^{\text{NTK}}\end{pmatrix}$$ where the off-diagonal blocks $K_{ij}^{\text{NTK}}$ may be replaced by 0. Then the limit spectral distribution of $K^{\text{NTK}}$ is an equally weighted mixture of those of $K_{11}^{\text{NTK}},\ldots,K_{kk}^{\text{NTK}}$. For each diagonal block $K_{ii}^{{\text{NTK}}}$, the argument of Lemma \[lemma:NTKFapprox\] shows that $${\operatorname{lim\;spec}}K_{ii}^{{\text{NTK}}}={\operatorname{lim\;spec}}\left(\tau \cdot r_+{\operatorname{Id}}+\tau_{L+1}X_L^\top X_L+\sum_{\ell=0}^{L-1}
\tau_{\ell+1}q_\ell X_\ell^\top X_\ell\right).$$ Then by Theorem \[thm:NTK\], each diagonal block $K_{ii}^{\text{NTK}}$ has the same limit spectral distribution, whose Stieltjes transform is given by the function $m_{\text{NTK}}(z)$ in Theorem \[thm:NTKmulti\]. Furthermore, since $\|K_{ii}^{\text{NTK}}\| \leq C$ by Lemma \[lemma:NTKFapprox\] and $\|K_{ij}^{\text{NTK}}\| \leq C$ for $i \neq j$ by Lemma \[lemma:NTKFapproxmulti\], this shows $\|K^{\text{NTK}}\| \leq C$. This establishes Theorem \[thm:NTKmulti\].
Reduction to result of Pennington and Worah [@pennington2017nonlinear] for one hidden layer {#appendix:penningtonreduction}
===========================================================================================
Consider the one-hidden-layer conjugate kernel $$K^{\text{CK}}=X_1^\top X_1=\frac{1}{d_1}\sigma(W_1X)^\top \sigma(W_1X) \in
{\mathbb{R}}^{n\times n}.$$ Define an associated covariance matrix $$\label{eq:CKcompanion}
M=\frac{1}{n}\sigma(W_1X)\sigma(W_1X)^\top \in {\mathbb{R}}^{d_1 \times d_1},$$ and observe that the eigenvalues of $K^{\text{CK}}$ are those of $M$ multiplied by $n/d_1$ and padded by $n-d_1$ additional zeros (or with $d_1-n$ zeros removed, if $n-d_1<0$). [@pennington2017nonlinear Theorem 1] characterizes the limit spectral distribution of $M$ in terms of a quartic equation in its Stieltjes transform, under the additional assumptions that $X$ has i.i.d. ${\mathcal{N}}(0,1/d_0)$ entries and $n/d_0 \to \gamma_0 \in (0,\infty)$.[^4] By Theorem \[thm:CK\], this should be equivalent to the description $$\label{eq:onelayerCK}
{\operatorname{lim\;spec}}K^{\text{CK}}=\rho_{\gamma_1}^{\text{MP}}\boxtimes \Big((1-b_\sigma^2)+b_\sigma^2 \mu_0\Big)$$ for the limit spectrum of $K^{\text{CK}}$, if we specialize to $\mu_0=\rho_{\gamma_0}^{\text{MP}}$ being the Marcenko-Pastur limit of the input gram matrix $X^\top X$. We derive this equivalence in this section.
Let $m_K(z)$ and $m_M(z)$ be the *limit* Stieltjes transforms for $K^{\text{CK}}$ and $M$. For any $z \in {\mathbb{C}}^+$, by the relation between the eigenvalues of $K^{\text{CK}}$ and $M$, $$\begin{aligned}
\frac{1}{n}{\operatorname{Tr}}\left(K^{\text{CK}}-\frac{n}{d_1}z{\operatorname{Id}}\right)^{-1}&=\frac{n-d_1}{n}\left(-\frac{n}{d_1}z\right)^{-1}
+\frac{1}{n}{\operatorname{Tr}}\left(\frac{n}{d_1}M-\frac{n}{d_1}z{\operatorname{Id}}\right)^{-1}\\
&=-\left(1-\frac{d_1}{n}\right)\frac{d_1}{n}\cdot
\frac{1}{z}+\left(\frac{d_1}{n}\right)^2
\cdot \frac{1}{d_1}{\operatorname{Tr}}(M-z{\operatorname{Id}})^{-1}.\end{aligned}$$ Taking the limit on both sides, we obtain the relation between $m_K(z)$ and $m_M(z)$, which is $$\label{eq:CKMrelation}
m_K(\gamma_1 z)=-\left(1-\frac{1}{\gamma_1}\right)\frac{1}{\gamma_1
z}+\frac{1}{\gamma_1^2}m_M(z)
=\frac{1}{\gamma_1^2}\left(m_M(z)+\frac{1-\gamma_1}{z}\right).$$
Following the notation of [@pennington2017nonlinear], let us set $$\label{eq:PWCKnotation}
\phi=1/\gamma_0,\quad \psi=\gamma_1/\gamma_0,
\quad \eta=1={\mathbb{E}}[\sigma(\xi)^2], \quad \zeta=b_\sigma^2.$$ [@pennington2017nonlinear Theorem 1] characterizes $G(z) \equiv -m_M(z)$ as the root of a quartic equation. Defining three $z$-dependent quantities $P,P_\phi,P_\psi$ by $$\label{eq:Pequations}
G(z)=\frac{\psi}{z}P+\frac{1-\psi}{z},
\quad P_\phi=1+(P-1)\phi, \quad P_\psi=1+(P-1)\psi,$$ this quartic equation is expressed as $$\label{eq:quarticP}
P=1+(1-\zeta)tP_\phi P_\psi+\frac{\zeta tP_\phi P_\psi}
{1-\zeta tP_\phi P_\psi} \qquad \text{ where } \qquad t=\frac{1}{z\psi},$$ see [@pennington2017nonlinear Equations (10–12)].
To verify that (\[eq:onelayerCK\]) is equivalent to this equation (\[eq:quarticP\]), note that (\[eq:onelayerCK\]) means the Stieltjes transform $m_K(z)$ is defined by the Marcenko-Pastur equation (\[eq:MPeq\]) as $$\label{eq:MP}
m_K(z)=\int \frac{1}{[(1-b_\sigma^2)+b_\sigma^2 x][1-\gamma_1-\gamma_1 z
m_K(z)]-z}d\mu_0(x).$$ Applying the identity $1-\gamma_1-\gamma_1^2 zm_K(\gamma_1 z)
=-zm_M(z)$ from rearranging (\[eq:CKMrelation\]), and applying also $\zeta=b_\sigma^2$ in (\[eq:PWCKnotation\]), $$\label{eq:CKfixedpoint1}
m_K(\gamma_1 z)=\int \frac{1}{[(1-\zeta)+\zeta x][-zm_M(z)]-\gamma_1 z}
d\mu_0(x).$$ When $X$ has i.i.d. ${\mathcal{N}}(0,1/d_0)$ entries, the limit spectral distribution of $X^\top X$ is the Marcenko-Pastur law $\mu_0=\rho_{\gamma_0}^{\text{MP}}$. The Stieltjes transform $m(z)$ of this law $\mu_0=\rho_{\gamma_0}^{\text{MP}}$ is characterized by the quadratic equation $$1=m(z)[1-\gamma_0-\gamma_0zm(z)-z]$$ (which is the specialization of (\[eq:MPeq\]) when $\mu$ is the point distribution at 1). Defining $$g(a,b)=\int \frac{1}{ax-b}d\mu_0(x)=\frac{1}{a}m\left(\frac{b}{a}\right),$$ we obtain then that $g(a,b)$ satisfies the quadratic equation $$\begin{aligned}
1&=g(a,b)[a-\gamma_0a-\gamma_0bm(b/a)-b]\\
&=g(a,b)[(a-b)-\gamma_0a-\gamma_0ab \cdot g(a,b)].\end{aligned}$$ Applying this with $a=-\zeta zm_M(z)$ and $b=(1-\zeta)zm_M(z)+\gamma_1 z$, the quantity (\[eq:CKfixedpoint1\]) is exactly $g(a,b)$. Thus this equation holds for $g(a,b)=m_K(\gamma_1z)$ and these settings of $(a,b)$, i.e.$$\label{eq:CKfixedpoint2}
1=m_K(\gamma_1 z)\Big(-zm_M(z)-\gamma_1 z+\gamma_0\zeta zm_M(z)
+\gamma_0\zeta zm_M(z)[(1-\zeta)zm_M(z)+\gamma_1 z]m_K(\gamma_1 z)\Big).$$ From the relation (\[eq:CKMrelation\]), we see that this is a quartic equation in $m_M(z)$. Note that the definitions of $P_\psi$ and $P_\phi$ in (\[eq:Pequations\]) may be equivalently written as $$\begin{aligned}
P_\psi&=\psi P+1-\psi=zG(z)=-zm_M(z),\\
P_\phi&=1+\frac{\phi}{\psi}(zG(z)-1)
=\frac{1}{\gamma_1}(-zm_M(z)-1+\gamma_1)
=-\gamma_1 zm_K(\gamma_1 z)\end{aligned}$$ where we have used $G(z)=-m_M(z)$, $\psi/\phi=\gamma_1$ from (\[eq:PWCKnotation\]), and the relation (\[eq:CKMrelation\]). Applying now $\gamma_1 z=(\psi/\phi)z=1/(\phi t)$ and $\gamma_0=1/\phi$, the equation (\[eq:CKfixedpoint2\]) becomes $$\begin{aligned}
1&=-\phi t P_\phi \left(P_\psi-\frac{1}{\phi t}
-\frac{\zeta}{\phi} P_\psi+\frac{\zeta}{\phi} P_\psi\left[
-(1-\zeta)P_\psi+\frac{1}{\phi t}\right]\phi t P_\phi\right)\\
&=-\phi tP_\phi P_\psi+P_\phi+(1-P_\phi)\zeta tP_\phi P_\psi
+\zeta(1-\zeta)\phi(tP_\phi P_\psi)^2.\end{aligned}$$ This may be rearranged as $$(1-P_\phi-\phi)(1-\zeta tP_\phi P_\psi)
=-\phi(1-\zeta tP_\phi P_\psi)-\phi tP_\phi
P_\psi+\zeta(1-\zeta)\phi(tP_\phi P_\psi)^2,$$ and dividing both sides by $-\phi(1-\zeta tP_\phi P_\psi)$ yields $$\frac{1}{\phi}(P_\phi-1)+1
=1+\frac{tP_\phi P_\psi-\zeta(1-\zeta)(tP_\phi P_\psi)^2}{1-\zeta tP_\phi
P_\psi}
=1+(1-\zeta)tP_\phi P_\psi+\frac{\zeta tP_\phi P_\psi}{1-\zeta tP_\phi P_\psi}.$$ Identifying the left side as $P$ by (\[eq:Pequations\]), we obtain (\[eq:quarticP\]) as desired.
Additional simulation results
=============================
Pairwise orthogonality of training samples
------------------------------------------
All pairwise inner-products $\{{\mathbf{x}}_{\alpha}^\top {\mathbf{x}}_{\beta}:1 \leq {\alpha}<{\beta}\leq n\}$, for (a) 5000 CIFAR-10 training samples, (b) 5000 CIFAR-10 training samples with the first 10 PCs removed, and (c) i.i.d. Gaussian training data of the same dimensions. Results for (b) were reported in Section \[sec:CIFAR\], and results for (a) are reported below in Appendix \[appendix:CIFARraw\]. CIFAR-10 training samples were mean-centered and normalized to satisfy ${\mathbf{x}}_{\alpha}^\top 1=0$ and $\|{\mathbf{x}}_{\alpha}\|^2=1$ in (a) and (b).
The pairwise inner-products in (a) span a typical range of $[-0.5,0.5]$. Those in (b) span a range of about $[-0.2,0.2]$, and those in (c) about $[-0.02,0.02]$. Thus, with 10 PCs removed, these inner-products for CIFAR-10 are larger than for i.i.d. Gaussian inputs by a factor of 10. We found in Section \[sec:CIFAR\] that the inner-products of (b) are sufficiently small for the observed spectra to match the theoretical limits of Theorems \[thm:CK\] and \[thm:NTK\].
CK and NTK spectra for CIFAR-10 without removal of leading PCs {#appendix:CIFARraw}
--------------------------------------------------------------
Same plots as Figure \[fig:CIFAR\] for CIFAR-10 training samples, without the removal of the 10 leading PCs. We observe a close agreement of the observed CK spectrum with the limit spectrum of Theorem \[thm:CK\]. However, there is a greater discrepancy of the NTK spectrum with the limit spectrum of Theorem \[thm:NTK\] in this setting.
Observed and limit CK spectra for all layers {#appendix:alllayers}
--------------------------------------------
{width="33.00000%"}{width="33.00000%"}{width="33.00000%"}
{width="33.00000%"}{width="33.00000%"}{width="33.00000%"}
Simulated spectra of the CK matrices $X_\ell^\top
X_\ell$ at all intermediate layers $\ell=1,\ldots,5$, corresponding to the i.i.d. Gaussian training data example of Figure \[fig:gaussian\]. Numerical computations of the limit spectra from Theorem \[thm:CK\] are overlaid in red. We observe a merging of the two bulk spectral components and an extension of the spectral support with increase in layer number.
{width="33.00000%"}{width="33.00000%"}{width="33.00000%"}
{width="33.00000%"}{width="33.00000%"}{width="33.00000%"} The same as above, corresponding to the CIFAR-10 training samples in Appendix \[appendix:CIFARraw\]. (Results with 10 PCs removed look the same.) A close agreement with the limit spectrum described by Theorem \[thm:CK\] is observed at each layer.
{width="33.00000%"}{width="33.00000%"}{width="33.00000%"} Spectra of the CK matrices at all three layers, corresponding to the trained 3-layer network of Section \[sec:training\]. The limit spectra at random initialization of weights are depicted in red, and the two largest eigenvalues of each matrix are depicted by blue arrows.
CK spectrum after training on a CIFAR-10 example {#appendix:CIFARtraining}
------------------------------------------------
We train a binary classifier on $n=10000$ training samples from CIFAR-10, corresponding to classes 0 (airplane) and 1 (automobile). The classifier is a fully-connected network with $L=4$ hidden layers of dimensions $d_1=\ldots=d_4=1000$, with bias terms and a sigmoid activation at each hidden layer and also at the output layer. This network is given by $$f_\theta({\mathbf{x}})=\sigma({\mathbf{w}}^\top {\mathbf{x}}^L+b),
\qquad {\mathbf{x}}^\ell=\frac{1}{\sqrt{d_\ell}}\sigma(W_\ell {\mathbf{x}}^{\ell-1}+\mathbf{b}_\ell)
\quad \text{ for } \quad \ell=1,\ldots,L$$ where $b \in {\mathbb{R}}$ and $\mathbf{b}_\ell \in {\mathbb{R}}^{d_\ell}$ for each $\ell=1,\ldots,L$ are the bias parameters. We use the sigmoid activation function $\sigma(x) \propto
(1-e^{-x})/(1+e^{-x})$, scaled such that ${\mathbb{E}}[\sigma(\xi)^2]=1$. Weights $\theta=(W_1,\ldots,W_4,{\mathbf{w}})$ are initialized to ${\mathcal{N}}(0,1)$, and biases $(\mathbf{b}_1,\ldots,\mathbf{b}_4,b)$ are initialized to 0. Hence, $K^{\text{CK}}$ at random initialization has the same definition as in the main text.
We train the weights and biases using the Adam optimizer in Keras, with learning rate 0.01, batch size 128, and 60 training epochs. To ensure that the leading PCs of the *untrained* kernel matrix $K^{\text{CK}}$ are not too predictive of the training labels, and to better separate the original PCs from those that emerge after training, we remove the leading 5 PCs of the input data before training. The resulting 0–1 classification accuracy on the CIFAR-10 test set is $85.3\%$. (Training without removing these 5 PCs yields a slightly higher test accuracy of $90.7\%$, using the same network architecture.)
\
Panel (a) above shows all eigenvalues of $K^{\text{CK}}$ at random initialization, with the largest being approximately 500. We observe a close agreement with the limit spectrum of Theorem \[thm:CK\]. Panel (b) shows the eigenvalues of $K^{\text{CK}}$ after training. We observe an elongation of the bulk spectral support and the emergence of large outlier eigenvalues, analogous to the synthetic example of Section \[sec:training\].
\
The above figure depicts the information about the training labels that is contained in the top 2 PCs of $K^{\text{CK}}$, (a) before training and (b) after training. Denoting by $\hat{X}_L$ the rank-2 approximation of $X_L$, with columns $\hat{{\mathbf{x}}}_1^L,\ldots,\hat{{\mathbf{x}}}_n^L$ (both before and after training), we re-fit a linear binary classifier $y_{\alpha}=\sigma({\mathbf{w}}^\top \hat{{\mathbf{x}}}_{\alpha}^L+b)$ of the training labels to these columns. The in-sample 0–1 training accuracy of this classifier is 51.4% pre-training and 96.8% post-training, and the figure shows the linear predictions ${\mathbf{w}}^\top \hat{{\mathbf{x}}}_{\alpha}^L+b$ against the training labels $y_{\alpha}$. We observe that the leading principal components of $K^{\text{CK}}$ are not predictive of the training labels before training, but become highly predictive after training.
[^1]: In this paper, we use “conjugate kernel” and “neural tangent kernel” to refer to these matrices for a finite-width network, rather than their infinite-width limits.
[^2]: We caution that this does not imply convergence of the largest and smallest eigenvalues of $K$ to the support of $\mu$, which is a stronger notion of convergence than what we study in this work.
[^3]: Note that some authors use a negative sign convention and define $m_\mu(z)$ as $\int 1/(z-x)d\mu(x)$.
[^4]: In [@pennington2017nonlinear], the $1/\sqrt{d_0}$ scaling is in $W_1$ rather than $X$, but these are clearly the same. We consider $\sigma_w=\sigma_x=1$ and $\eta=1$ in the results of [@pennington2017nonlinear].
|
---
abstract: 'We consider integrals of the form $\int_0^1 \ln{\ln{(\frac{1}{x})}}R{(x)}{\mathrm{d}}x$ again, where $R{(x)}$ is a rational function, and we will explain a way to obtain their evaluation.'
author:
- Alexander Aycock
title: Notes and Remarks on certain logarithmic integrals
---
§1 In mathematics treatises can roughly be divided up into two classes, the first containing those, that expand the boundaries of mathematics and the second containing those, that represent already known things either with new theories in a simpler manner or achieve the same aim with familiar methods in a simpler way.\
§2 This little memoir falls in the second class, because it considers integrals of the form $\int_0^1 \ln{\ln{(\frac{1}{x}})}R{(x)}{\mathrm{d}}x$ again, which have been already been studied by several other mathematicians. Here Vardi [@7], Medina and Moll [@17] and Adamchik [@18] are to be mentioned, but also a lot of the “oder” mathematicians wrote about such integrals. They mainly treated integrals of the form $\int_0^1\frac{Q{(x)}{\mathrm{d}}x}{\ln{x}}$ (where $Q{(x)}$ is a rational function in most cases), which are easily shown to be equivalent to those (at least in certain cases) we want to consider here. We want to mention Euler [@11], Legendre [@1] and Malmstèn [@3]. The latter contributed the most to this kind of integrals - as we will see below - and actually provided everything to get as far as the mathematicians mentioned earlier.
So we also want to study this integrals and evaluate them in a simpler manner and want to explain carefully, how to get there a priori.\
§3 Therefore we will have to look for methods - and will have to explain them -, to obtain these integrals, without using any knowledge from complex theory of functions. It will therefore be convenient to say some things in advance, that will be useful later. These things are, of course, well-known, but the way, to obtain them, that we will present here, seems to be mostly forgotten. And so it will be worth the effort, to describe it.\
§4 We want to begin with the partial fraction decomposition of the circle or hyperbola functions.
We will start with $\frac{\sin{(ax)}}{\sin{(bx)}}$, the factors of the denominator are $k\pi - bx$ and $k\pi + bx$, where $k$ is a natural number, and it is easily seen, that all factors are simple. Now we put $$\frac{\sin{(ax)}}{\sin{(bx)}} = \frac{A}{k\pi - bx} + G(x)$$ where $A$ is a number, we have to determine, and $G{(x)}$ is a function, not involving the factor $k\pi - bx$. We multiply by $\sin{(bx)}$ and find $$\sin{(ax)}=\frac{A\sin{(bx)}}{k\pi-bx}+\sin{(bx)}G{(x)}$$ Letting $x$ tend to $\frac{k\pi}{b}$ the second term on the right-hand side vanishes; the first can be calculated with L’ Hospital’s rule, and we find $$\sin{(\frac{ak\pi}{b})} = -A\cos{(k\pi)}$$ Because $k$ is a natural number, we have $$A = (-1)^{k+1}\sin{(\frac{ak\pi}{b})}$$ And in the same way we will find for the other factor $k\pi + bx$, if we set $$\frac{\sin({ax})}{\sin({bx})} = \frac{A}{k\pi + bx} + Q(x)$$ that $$B = (-1)^{k+1}\sin{(\frac{ak\pi}{b})}$$ And by summing over $k$ from $1$ to infinity $$\frac{\sin{(ax)}}{\sin{(bx)}} = \sum_{k=1}^{\infty}(-1)^{k+1}\sin{(\frac{ak\pi}{b})}(\frac{1}{k\pi-bx}+\frac{1}{k\pi+bx})$$ and by contracting the two denominators $$\frac{\sin{(ax)}}{\sin{(bx)}} = 2\pi\sum_{k=1}^{\infty}(-1)^{k+1}\sin{(\frac{ak\pi}{b})}(\frac{k}{(k\pi)^2-(bx)^2})$$ and if we put $x = iy$ we will obtain $$\frac{\sinh{(ay)}}{\sinh{(by)}} = \frac{e^{ay}-e^{-ay}}{e^{by}-e^{-by}} = 2\pi\sum_{k=1}^{\infty}(-1)^{k+1}\sin{(\frac{ak\pi}{b})}(\frac{k}{(k\pi)^2+(bx)^2})$$
§5 We want to consider another example of this kind, $\frac{\cos{(ax)}}{\sin{(bx)}}$, that slightly differs from the preceding one. Because we see, that the first term of the partial fraction decomposition, because of the zero at $x = 0$ of the denominator and $\cos{(0)} = 1$, will be $\frac{1}{bx}$; for the rest we have exactly the same factors as in the first example. This time we put $$\frac{\cos{(ax)}}{\sin{(bx)}} - \frac{1}{bx} = \frac{A}{k\pi - bx} + R(x)$$ because the first term of the expansion is already known, then we have $$\cos{(ax)}-\frac{\sin{(bx)}}{bx}= \frac{A\sin{(bx)}}{k\pi-bx}+\sin{(bx)}R{(x)}$$ Letting $x$ tend to $\frac{k\pi}{b}$ again we obtain $$A = (-1)^{k+1}\cos{(\frac{ak\pi}{b})}$$ And putting $$\frac{\cos{(ax)}}{\sin{(bx)}} - \frac{1}{bx} = \frac{B}{k\pi + bx} + Q(x)$$ we find $$B = -(-1)^{k+1}\cos{(\frac{ak\pi}{b})}$$ And summing over $k$ from $1$ to infinity again, yields $$\frac{\cos{(ax)}}{\sin{(bx)}} = \frac{1}{bx} + \sum_{k=1}^{\infty}(-1)^{k+1}\cos{(\frac{ak\pi}{b})}(\frac{1}{k\pi - bx} - \frac{1}{k\pi + bx})$$ or after a little simplification $$\frac{\cos{(ax)}}{\sin{(bx)}} = \frac{1}{bx} + \sum_{k=1}^{\infty}(-1)^{k+1}\cos{(\frac{ak\pi}{b})}(\frac{2bx}{(k\pi)^2 - (bx)^2})$$ and the special case $a = b = 1$ gives $$\cot{(x)} = \frac{1}{x} + \sum_{k=1}^{\infty}\frac{2x}{x^2-(k\pi)^2}$$ and in a similar way one can find several more formulas, also for other functions than the circle and hyperbola functions, which subject we will not investigate here. Instead we remark, that, by integrating the last identity and simplifying, we arrive at the famous sine product formula $$\sin{(x)} = x\prod_{k=1}^{\infty}(1-\frac{x^2}{(k\pi)^2})$$ which was given by Euler [@9]. For $x = \frac{\pi}{2}$ we find the Wallis product formula for $\frac{\pi}{2}$ $$\frac{\pi}{2} = \prod_{k=1}^{\infty}(\frac{4k^2}{4k^2-1})$$ §6 Euler [@14] was the first to use the presented method for finding the partial fraction decomposition. At first only for rational functions, later also for the circle and the hyperbola functions. Legendre [@1] also arrived at these result with this method. Despite the simplicity of this method it seems to be mostly forgotten, see Sandifer’s text [@5] for a note on this.\
§7 We want to add one remark. Considering the formula $$\frac{\sin{(ax)}}{\sin{(bx)}} = 2\pi\sum_{k=1}^{\infty}(-1)^{k+1}\sin{(\frac{ak\pi}{b})}(\frac{k}{(k\pi)^2-(bx)^2})$$ it can also be interpretated as a function of the variable $a$. So we want to differentiate with respect to $a$, then we will have $$x\frac{\cos{(ax)}}{\sin{(bx)}} = \frac{2}{b}\sum_{k=1}^{\infty}(-1)^{k+1}\cos{(\frac{ak\pi}{b})}(\frac{(k\pi)^2}{(k\pi)^2-(bx)^2})$$ We already know the value on the left-hand side and writing $(k\pi)^2 = (k\pi)^2-(bx)^2+ (bx)^2$ on the right-hand side, we arrive at the following equation, taking into account the already derived partial fraction decomposition for $\frac{\cos{(ax)}}{\sin{(bx)}}$ $$x[\frac{1}{bx}+\sum_{k=1}^{\infty}(-1)^{k+1}\cos{(\frac{ak\pi}{b})}(\frac{2bx}{(k\pi)^2 - (bx)^2})]= \frac{2}{b}\sum_{k=1}^{\infty}(-1)^{k+1}\cos{(\frac{ak\pi}{b})}+$$ $$\frac{2}{b}\sum_{k=1}^{\infty}(-1)^{k+1}\cos{(\frac{ak\pi}{b})}(\frac{(bx)^2}{(k\pi)^2-(bx)^2})$$ and this gives after a little simplification $$\frac{1}{2} = \sum_{k=1}^{\infty}(-1)^{k+1}\cos{(\frac{ak\pi}{b})}$$ and for $a = 0$ $$\frac{1}{2} = \sum_{k=1}^{\infty}(-1)^{k+1}$$ which series is diverging. But it is well-known, that such series occur very often and in this case we derived it from an identity, which is also known from elsewhere. Therefore we want to see the value of this particular series as $\frac{1}{2}$.\
§8 It will be convenient to note, that Legendre [@1] arrived at the partial fraction decomposition for $\frac{\cos{(ax)}}{\sin{(bx)}}$ using the series $\frac{1}{2} = \sum_{k=1}^{\infty}(-1)^{k+1}$. Euler also explained on several occasions [@8], [@16], how such results should be interpretated and that it makes sense to use such a particular value. These series can be used, if they are interpretated correctly. See Hardy [@6] or Ford [@19], to name some more recent mathematicians, who worked on this subject. We will avoid these series as far as possible, because they require a theory for their explanaition, that cannot be regarded as elementary anymore. But we will nevertheless use the value of the one series, that we found here, later.\
§9 Before we are able to get to our main results, we have to show some identities, that will be useful later.\
We start with the formulae $$\frac{\sin{(a)}}{1+2y\cos{(a)}+y^2} = \sum_{n=1}^{\infty}(-1)^{n-1}y^{n-1}\sin{(na)}$$ and $$\frac{a}{2}= \sum_{n=1}^{\infty}(-1)^{n-1}\frac{\sin{(na)}}{n}$$ which are well-known in the theory of Fourier series, but they can be proven by elementary means and the proof, we will give here, traces back to Euler [@16]. Consider the geometric series $$\frac{1}{1+x} = \sum_{n=0}^{\infty}(-1)^{n}x^{n}$$ And for $x = ye^{i\phi}$ by using the Euler identity $e^{ix}=\cos{(x)}+i\sin{(x)}$ this gives $$\frac{1}{1+y\cos{(\phi)}+iy\sin{(\phi)}} = \sum_{n=0}^{\infty}(-1)^{n}y^{n}e^{in\phi}$$ By using de Moivre’s formula and after a simplification $$\frac{1+y\cos{(\phi)}}{1+2y\cos{(\phi)}+y^2}-\frac{i\sin{(\phi)}}{1+2y\cos{(\phi)}+y^2} = \sum_{n=0}^{\infty}(-1)^{n}y^{n}(\cos{(n\phi)}+i\sin{(n\phi)})$$ Comparing the real and imaginary parts yields $$\frac{1+y\cos{(\phi)}}{1+2y\cos{(\phi)}+y^2}= \sum_{n=0}^{\infty}(-1)^{n}y^{n}\cos{(n\phi)}$$ and $$\frac{\sin{(\phi)}}{1+2y\cos{(\phi)}+y^2}=\sum_{n=1}^{\infty}(-1)^{n-1}y^{n-1}\sin{(n\phi)}$$ The first formula gives the divergent series from above, if we put $y = 1$, and the second is the one, we wanted to demonstrate. To get the formula for $\frac{a}{2}$ we can either integrate the forst formula with respect to $\phi$ or the second with respect to $y$ and put $y = 1$ after the integration. We omit the exact calculation here, because it the one given should be sufficient. We could also derive the identity in exactly the same way as the first from the power series for $\ln{(1+x)}$. But this would lead to the question, how we find the logarithm of a complex number, already requiring a little complex analysis. Therefore we will not do it that way, but mention, that Euler [@16] actually did it this way.\
§10 Another useful identity for evaluating logarithmic integrals is this one $$\int_0^{\infty}\frac{e^{au}-e^{-au}}{e^{\pi u}-e^{-\pi u}}\cos{(uz)}{\mathrm{d}}u= \frac{\sin{(a)}e^{-z}}{1+2e^{-z}\cos{(a)}+e^{-2z}}$$ To show this one, we will need the following integral for positive $n$ $$\int_0^{\infty}\frac{\cos{(nx)}{\mathrm{d}}x}{1+x^2}= \frac{\pi}{2}e^{-n}$$ We only have to cnsider it for positve $n$, because the function is symmetric in $n$ and we will just need it for positive $n$ later anyway. We suppose, that we do not know the answer yet, but we know, the value will be a function of $n$, and so we set $$\int_0^{\infty}\frac{\cos{(nx)}{\mathrm{d}}x}{1+x^2}= f(n)$$ then we will also have by differentiating $$\int_0^{\infty}\frac{x\sin{(nx)}{\mathrm{d}}x}{1+x^2}= -f^{\prime}(n)$$ By integrating by parts one easily confirms the following fomulas $$\int_0^{\infty}e^{-at}\cos{(bt)}{\mathrm{d}}t = \frac{a}{a^2+b^2}$$ and $$\int_0^{\infty}e^{-at}\sin{(bt)}{\mathrm{d}}t = \frac{b}{a^2+b^2}$$ And we find as a special case $$\int_0^{\infty}e^{-xt}\sin{(t)}{\mathrm{d}}t = \frac{1}{1+x^2}$$ Multiplying this equation by $\cos{(nx)}$ and integrating from $0$ to infinity with respect to $x$ yields $$\int_0^{\infty}\int_0^{\infty}\cos{(nx)}e^{-xt}\sin{(t)}{\mathrm{d}}t{\mathrm{d}}x=\int_0^{\infty}\frac{\cos{(nx)}{\mathrm{d}}x}{1+x^2}$$ The right-hand side is $f{(n)}$ amd therefore we have $$f(n) = \int_0^{\infty}(\int_0^{\infty}\cos{(nx)}e^{-xt}{\mathrm{d}}x)\sin{(t)}{\mathrm{d}}t$$ The inner integral can be expressed with the formula above. Using this, this leads to $$f(n)=\int_0^{\infty}\frac{t\sin{(t)}{\mathrm{d}}t}{n^2+t^2}$$ and for $t =ny$ $$f(n)= \int_0^{\infty}\frac{y\sin{(ny)}{\mathrm{d}}y}{1+y^2}=f^{\prime}(n)$$ So we arrived at a differential equation for $f{(n)}$, whose solution is seen to be $Ce^{-n}$, where $C$ is a constant, but we have $$Ce^{-0}= C = \int_0^{\infty}\frac{{\mathrm{d}}y}{1+y^2} = \frac{\pi}{2}$$ which we wanted to prove, because now we have $$\int_0^{\infty}\frac{\cos{(nx)}{\mathrm{d}}x}{1+x^2}= \frac{\pi}{2}e^{-n}$$ §11 Even if this proof was not completely rigorous and some conditions are to be added, it leads to the right result, which was already known to mathematician like Legendre [@1], who also evaluated it without the use of complex function theory. But this result is - of course - easyly confirmed by the calculus of residues nowadays.\
§12 But now we are able, to prove our important identity, from which essentially all other things will flow; this will also seen below. We want to demonstrate now $$\int_0^{\infty}\frac{e^{ax}-e^{-ax}}{e^{\pi x}-e^{-\pi x}}\cos{(nx)}{\mathrm{d}}x= \frac{\sin{(a)}e^{-n}}{1+2e^{-n}\cos{(a)}+e^{-2n}}$$ We start from the integral and attempt to evaluate it, we found the partial fraction for $\frac{e^{ax}-e^{-ax}}{e^{\pi x}-e^{-\pi x}}$ above, if we put $b= \pi$ in that formula. Replacing this expression with its partial fraction decomposition gives $$\int_0^{\infty}\frac{e^{ax}-e^{-ax}}{e^{\pi x}-e^{-\pi x}}\cos{(nx)}{\mathrm{d}}x= \frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k+1}\sin{(ak)}k\int_0^{\infty}\frac{\cos{(nx)}{\mathrm{d}}x}{k^2+x^2}$$ Setting $yk =x$ in the intgral on the right-hand side $$\int_0^{\infty}\frac{e^{ax}-e^{-ax}}{e^{\pi x}-e^{-\pi x}}\cos{(nx)}{\mathrm{d}}x= \frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k+1}\sin{(ak)}\int_0^{\infty}\frac{\cos{(nky)}{\mathrm{d}}y}{1+y^2}$$ This integral can be evaluated and we will have $$\int_0^{\infty}\frac{e^{ax}-e^{-ax}}{e^{\pi x}-e^{-\pi x}}\cos{(nx)}{\mathrm{d}}x= \frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k+1}\sin{(ak)}\frac{\pi}{2}e^{-nk}$$ The arising series can be expressed in finite terms using the results from above and yields $$\int_0^{\infty}\frac{e^{ax}-e^{-ax}}{e^{\pi x}-e^{-\pi x}}\cos{(nx)}{\mathrm{d}}x= \frac{\sin{(a)}e^{-n}}{1+2e^{-n}\cos{(a)}+e^{-2n}}$$ This is the one, we wanted to show and it will be observed later, that it will lead to the disired evaluation of logarithmic integrals, how Malmstèn [@3] realized at first, even if this partiular formula traces back to Legendre [@1].\
§13 In the papers of Vardi [@7], Medina and Moll [@17] and Adamchik [@18] one will see quite fast, that the logarithmic integrals follow from functiona equations of certain Dirichlet series, series of the form $\sum_{k=1}^{\infty}\frac{a{(n)}}{n^{s}}$. So, if we show these, we can claim, to have everything in our, what the mentioned authors proved, concerning the evaluation at least. We just would have to follow their way. Now, so I claim, we can already derive these functional equations with few sketches from the preceding. It will therefore be worth the effort to show these functiona equations, before explaining another related method.\
§14 For this purpose we note, that we have $$\int_0^{\infty}z^{s-1}\cos{(uz)}{\mathrm{d}}z= \frac{\Gamma{(s)}}{u^{s}}\cos{(\frac{s\pi}{2})}$$ This can be seen as follows. We have, where - as usual - $\Gamma{(s)}$ is the well-known function, defined by the integral $\int_0^{\infty}e^{-t}t^{s-1}{\mathrm{d}}t $, if we write $kt$ for $t$ in this integral, $$\frac{\Gamma{(s)}}{k^{s}}=\int_0^{\infty}e^{-kt}t^{s-1}{\mathrm{d}}t$$ and therefore for $k = a+bi$ and using the Euler identity again $$\frac{\Gamma{(s)}}{(a+bi)^{s}}=\int_0^{\infty}e^{-at}\cos{(bt)}t^{s-1}{\mathrm{d}}t -i\int_0^{\infty}e^{-at}\sin{(bt)}t^{s-1}{\mathrm{d}}t$$ And we have, of course, $$\frac{\Gamma{(s)}}{(a+bi)^{s}}\cdot\frac{(a-bi)^{s}}{(a-bi)^{s}}=\frac{\Gamma{(s)}}{(a^2+b^2)^{s}}(a-bi)^{s}$$ writing the complex number in polar coordinates this reduces to $$\frac{\Gamma{(s)}}{(a+bi)^{s}}= \frac{\Gamma{(s)}}{(a^2+b^2)^{\frac{s}{2}}}(\cos{(s\arctan{\frac{b}{a}})}-i\sin{(s\arctan{\frac{b}{a}})})$$ which is the value of our integral; if we compare the real parts, we will obtain the desired identity, after having changed the letters $$\int_0^{\infty}e^{-xz}z^{s-1}\cos{(uz)}{\mathrm{d}}z= \frac{\Gamma{(s)}}{(x^2+u^2)^{\frac{s}{2}}}\cos{(s\arctan{\frac{u}{x}})}$$ Letting $x$ tend to $0$, the argument of the arctan tends to infinity and therefore the arctan tends to $\frac{\pi}{2}$. This shows our formula $$\int_0^{\infty}z^{s-1}\cos{(uz)}{\mathrm{d}}z= \frac{\Gamma{(s)}}{u^{s}}\cos{(\frac{s\pi}{2})}$$ which goes back to Euler [@12] again and gives the Fresnel integrals for $s=\frac{1}{2}$. We now want to multiply the last equation by $\frac{e^{au}-e^{-au}}{e^{\pi u}-e^{-\pi u}}$ and integrate from $0$ to infintity with respect to $u$, which yields $$\int_0^{\infty}\int_0^{\infty}\frac{e^{au}-e^{-au}}{e^{\pi u}-e^{-\pi u}}z^{s-1}\cos{(uz)}{\mathrm{d}}z{\mathrm{d}}u= \Gamma{(s)}\cos{(\frac{s\pi}{2})}\int_0^{\infty}\frac{e^{au}-e^{-au}}{e^{\pi u}-e^{-\pi u}}\frac{{\mathrm{d}}u}{u^{s}}$$ The double integral can be simplified by using the idendity from above, namely, $$\int_0^{\infty}\frac{e^{au}-e^{-au}}{e^{\pi u}-e^{-\pi u}}\cos{(uz)}{\mathrm{d}}u= \frac{\sin{(a)}e^{-z}}{1+2e^{-z}\cos{(a)}+e^{-2z}}$$ and gives $$\int_0^{\infty}\frac{e^{au}-e^{-au}}{e^{\pi u}-e^{-\pi u}}\frac{{\mathrm{d}}u}{u^{s}}=\frac{1}{\Gamma{(s)}\cos{(\frac{s\pi}{2})}}\int_0^{\infty}\frac{z^{s-1}\sin{(a)}e^{-z}{\mathrm{d}}z}{1+2e^{-z}\cos{(a)}+e^{-2z}}$$ For $e^{-z}= \ln{(y)}$ this reduces to $$\int_0^{\infty}\frac{e^{au}-e^{-au}}{e^{\pi u}-e^{-\pi u}}\frac{{\mathrm{d}}u}{u^{s}}=
\frac{1}{\Gamma{(s)}\cos{(\frac{s\pi}{2})}}\int_0^{1}\frac{\ln^{s-1}{(\frac{1}{y})}\sin{(a)}{\mathrm{d}}y}{1+2y\cos{(a)}+y^{2}}$$\
§15 And this is already the fundmaental formula for the functional equations of certain Dirichlet series, which are important for the consideration about logarithmic integrals; this can be seen as follows. We have $$\int_0^{\infty}\frac{e^{au}-e^{-au}}{e^{\pi u}-e^{-\pi u}}\frac{{\mathrm{d}}u}{u^{s}}=
\frac{\sin{(a)}}{\Gamma{(s)}\cos{(\frac{s\pi}{2})}}\int_0^{1}\frac{\ln^{s-1}{(\frac{1}{y})}{\mathrm{d}}y}{1+2y\cos{(a)}+y^{2}}$$ and $$\int_0^{\infty}\frac{e^{-\pi u}(e^{au}-e^{-au})}{1-e^{-2\pi u}}\frac{{\mathrm{d}}u}{u^{s}}=
\frac{\sin{(a)}}{\Gamma{(s)}\cos{(\frac{s\pi}{2})}}\int_0^{1}\frac{\ln^{s-1}{(\frac{1}{y})}{\mathrm{d}}y}{1+2y\cos{(a)}+y^{2}}$$ If we use the series expansions for $\frac{1}{1-e^{-2\pi u}}$, and $\frac{\sin{(a)}}{1+2y\cos{(a)}+y^{2}}$ - we found the series for this second expression above -, we find $$\sum_{n=0}^{\infty}\int_0^{\infty}(e^{-2n\pi u-\pi u -a}-e^{-2n\pi u -\pi u +a})u^{-s}{\mathrm{d}}u= \frac{1}{\Gamma{(s)}\cos{(\frac{s\pi}{2})}}\sum_{n=1}^{\infty}(-1)^{n-1}\sin{(na)}\int_0^{1}\ln^{s-1}{(\frac{1}{y})} y^{n-1} {\mathrm{d}}y$$ All occuring integrals can be expressed in terms of $\Gamma{(s)}$ and this leads to $$\Gamma{(1-s)}\sum_{n=0}^{\infty}(\frac{1}{((2n+1)\pi-a)^{1-s}}-\frac{1}{((2n+1)\pi+a)^{1-s}})=\frac{\Gamma{(s)}}{\Gamma{(s)}\cos{(\frac{s\pi}{2})}}\sum_{n=1}^{\infty}(-1)^{n-1}\frac{\sin{(na)}}{n^{s}}$$ On the right-hand side the $\Gamma{(s)}$ functions cancel, and writing $\pi -a$ instead of $a$, this gives a similar formula $$\Gamma{(1-s)}\sum_{n=0}^{\infty}(\frac{1}{(2n\pi+a)^{1-s}}-\frac{1}{((2n+2)\pi-a)^{1-s}})=\frac{1}{\cos{(\frac{s\pi}{2})}}\sum_{n=1}^{\infty}\frac{\sin{(na)}}{n^{s}}$$ And finally putting $a =\frac{k\pi}{l}$ and writing $1-s$ instead $s$ we arrive a these two formulas $$\sum_{n=0}^{\infty}(\frac{1}{((2n+1)l-k)^{s}}-\frac{1}{((2n+1)l+k)^{s}})= \frac{(\frac{\pi}{l})^{s}}{\sin{(\frac{s\pi}{2})}\Gamma{(s)}}\sum_{n=1}^{\infty}(-1)^{n+1}\frac{\sin{(\frac{nk\pi}{l})}}{n^{1-s}}$$ and $$\sum_{n=0}^{\infty}(\frac{1}{(2nl+k)^{s}}-\frac{1}{((2n+2)l+k)^{s}})= \frac{(\frac{\pi}{l})^{s}}{\sin{(\frac{s\pi}{2})}\Gamma{(s)}}\sum_{n=1}^{\infty}\frac{\sin{(\frac{nk\pi}{l})}}{n^{1-s}}$$ §16 If we use the first series for an example and put $k=1$, $l=2$ and $k=1$, $l=3$ respectively, we obtain these to equations $$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{s}}= \frac{(\frac{\pi}{2})^{s}}{\sin{(\frac{s\pi}{2})}\Gamma{(s)}}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{1-s}}$$ and $$\sum_{n=0}^{\infty}(\frac{1}{(3n+1)^{s}}-\frac{1}{(3n+2)^{s}})= \frac{(\frac{2\pi}{3})^{s}}{\sin{(\frac{s\pi}{2})}\Gamma{(s)}}\sum_{n=0}^{\infty}(\frac{1}{(3n+1)^{1-s}}-\frac{1}{(3n+2)^{1-s}})$$ These and some more were at first given by Malmstèn [@2] and proved rigorously. So Malmstèn was probably the first person to proove functional equation of Dirichlet series, which cannot be praised enough.\
\
§17 It will be observed, that you can add many more, even infinitely many, to those two, we gave here, and certainly all, that were considered by Vardi [@7], Medina and Moll [@17] and Adamchik [@18]. Only the functional equation of the famous Riemann zeta function, $\zeta{(s)}=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$, or equivalently the Dirichlet eta function, $\eta{(s)}=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{s}}$, are missing, both being such examples, that cannot be obtained from our formulas by putting in values.\
§18 But you can perform a little trick, as I demonstrated on another occasion, considering Malmstèn’s paper [@2]. I will nevertheless, because it fits right in, be convenient, to show this little trick again and show $$\eta{(1-s)}=\frac{2^{s}-1}{1-2^{s-1}}\pi^{-s}\cos{(\frac{s\pi}{2})}\Gamma{(s)}\eta{(s)}$$ For the sake of brevity we want to set $$\lambda{(s)}=\sum_{n=0}^{\infty}\frac{1}{(2n+1)^{s}}=\frac{2^{s}-1}{2^{s}-2}\eta{(s)}$$ We will start from our fundamental identity again $$\int_0^{\infty}\frac{e^{au}-e^{-au}}{e^{\pi u}-e^{-\pi u}}\frac{{\mathrm{d}}u}{u^{s}}=
\frac{1}{\Gamma{(s)}\cos{(\frac{s\pi}{2})}}\int_0^{1}\frac{\ln^{s-1}{(\frac{1}{y})}\sin{(a)}{\mathrm{d}}y}{1+2y\cos{(a)}+y^{2}}$$ We want to divide both sides by $\sin{(a)}$ and let $a$ tend to $0$, where we use the limit $$\lim_{a \rightarrow 0}\frac{e^{au}-e^{-au}}{\sin{a}} = 2u$$ which can be derived from L’ Hospitals’s rule. This yields $$2\cos{\left(\frac{\pi s}{2}\right)}\int_0^{\infty}\frac{u^{1-s}{\mathrm{d}}u}{e^{\pi u}-e^{-\pi u}} = \frac{1}{{\Gamma ( s )}}\int_0^1\frac{\ln^{s-1}{(\frac{1}{y})}{\mathrm{d}}y}{(1+y)^2}$$ Multiplying the numerator and the denominator by $e^{-\pi u}$ and using the series expansions for $\frac{1}{1-e^{-2\pi u}}$ and $\frac{1}{(1+y)^{2}}$ we find $$2\cos{\left(\frac{\pi s}{2}\right)}\int_0^{\infty}\sum_{n=0}^{\infty}e^{-(2n+1)\pi u}u^{1-s}{\mathrm{d}}u = \frac{1}{{\Gamma ( s )}}\int_0^1\ln^{s-1}{\left(\frac{1}{y}\right)\sum_{n=1}^{\infty}(-1)^{n+1}n y^{n-1}{\mathrm{d}}y}$$ Interchanging the order of summation and integration, both integrals can be reduced to constant multiples of $\Gamma{(s)}$ and yield $$\frac{2\cos{\left(\frac{\pi s}{2}\right)}}{\pi^{2-s}}\sum_{n=0}^{\infty}\frac{1}{(2n+1)^{2-s}}{{\Gamma ( 2-s )}}
= \frac{1}{{\Gamma ( s )}}\sum_{n=1}^{\infty}(-1)^{n+1}n\frac{{\Gamma ( s )}}{n^s}\;.$$ And with our abbreviations $$\frac{2\cos{\left(\frac{\pi s}{2}\right)}}{\pi^{2-s}}\lambda{(2-s)}{\Gamma ( 2-s )} = \eta{(s-1)}$$ If we finally write $2-s$ instead of $s$ and replace $\lambda{(s)}$ with the identity involving $\eta{(s)}$, we will get $$\eta{(1-s)} = \frac{2^s-1}{1-2^{s-1}}\pi^{-s}\cos{\left(\frac{\pi s}{2}\right)}{\Gamma ( s )}\eta{(s)}$$ which equation we wanted to show. We want to note after all this, that this functional equation and the one for $\beta{(s)}= \sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{s}}$ were already known by Euler [@15] and he found it with the help of divergent series. Further you can show by replacing $\eta{(s)}$ with the relation to $\zeta{(s)}$ - we have $\zeta{(s)}= \frac{\eta{(s)}}{1-2^{1-s}}$ , of course - the functional equation for $\zeta{(s)}$, namely $$\zeta{(1-s)}=\frac{2}{(2\pi)^{s}}\Gamma{(s)}\cos{(\frac{s\pi}{2})}\zeta{(s)}$$ which was at first proved rigorously by Riemann in his famous memoir [@2].\
§19 Now we see, that we have everything in our hands, that Vardi [@7], Medina and Moll [@17] and Adamchik [@18] needed, to obtain their results, we only representes everything with integrals, so that we could end the paper right here, having provided a a priori method to evaluate the logarithmic integrals.\
§20 But these integrals can also be calculated in a different way, as special values of the Fourier series expansion for $\ln{\Gamma{(x)}}$. The Fourier series was given by Kummer [@4] in 1847, but also by Malmstèn [@3] one year earlier in an entirely different way, which we will essentially present here, on the one hand, because it involves logarithmic integrals - and therefore reveals the connection of these intgrals to the Fourier series - and on the other hand, because it deserves to be mentionend, containing some interesting manipulations. We go on to the proof.\
§21 We want to derive the Fourier series for $\ln{\frac{\Gamma{(\frac{1}{2}+x)}}{\Gamma{(\frac{1}{2}-x)}}}$. For this we note the following identities, which are straight-forward to prove, see also Legendre [@1] and Malmstèn [@3]. $$\int_0^{1}\frac{y^{a}-y^{b}}{1-y^2}{\mathrm{d}}y = \frac{1}{2}[\psi{(\frac{1}{2}(b+1))}-\psi{(\frac{1}{2}(a+1))}]$$ where $\psi{(x)}$ is the Digamma function and can be defined by $$\psi{(x)}= -\gamma +\int_0^{1}\frac{1-t^{x-1}}{1-t}{\mathrm{d}}t= \frac{{\mathrm{d}}}{{\mathrm{d}}x}\ln{\Gamma{(x)}}$$ From this we find $$\psi{(\frac{1}{2})}=-\gamma -2\ln{2}= -\gamma -\ln{4}$$ and we also have $$\int_0^{1}\ln{\ln{(\frac{1}{y})}}y^{n-1}{\mathrm{d}}y= -\frac{\gamma+\ln{n}}{n}$$ which can be shown by differentiating the following formula with respect to $s$ and setting $s=1$ afterwards $$\int_0^{1}\ln^{s-1}{(\frac{1}{y})}y^{n-1}{\mathrm{d}}y= \frac{\Gamma{(s)}}{n^s}$$ where $\gamma$ is, as above, $-\Gamma^{\prime}{(1)}$. Actually, of course, $\gamma$ is the famous Euler-Mascheroni constant and is defined by the following limit $$\gamma = \lim_{n \rightarrow \infty}(\sum_{k=1}^{n}\frac{1}{k}-\ln{n}) = 0,577215664901...$$ and it can be shown, that we also have $\gamma= -\Gamma^{\prime}{(1)}$. But, because we will neither use the exact value nor the limit definition, and only the value $-\Gamma^{\prime}{(1)}$, we can omit this particular proof.\
§22 Having said all this in advance, we can derive the Fourier series expansion. From the preceeding we already have $$\cos{(\frac{s\pi}{2})}\int_0^{\infty}\frac{e^{au}-e^{-au}-2au}{e^{\pi u}-e^{-\pi u}}\frac{{\mathrm{d}}u}{u^{s}}=\frac{1}{\Gamma{(s)}}\int_0^{1}(\frac{\ln^{s-1}{(\frac{1}{y})}\sin{(a)}}{1+2y\cos{(a)}+y^{2}}-\frac{a\ln^{s-1}{(\frac{1}{y})}}{(1+y)^2}){\mathrm{d}}y$$ We differentiate with respect to $s$ and obtain $$-\frac{\pi}{2}\sin{(\frac{s\pi}{2})}\int_0^{\infty}\frac{e^{au}-e^{-au}-2au}{e^{\pi u}-e^{-\pi u}}\frac{{\mathrm{d}}u}{u^{s}}-\cos{(\frac{s\pi}{2})}\int_0^{\infty}\frac{e^{au}-e^{-au}-2au}{e^{\pi u}-e^{-\pi u}}\frac{\ln{u}{\mathrm{d}}u}{u^{s}}$$ $$=-\frac{\Gamma^{\prime}{(s)}}{\Gamma{(s)}}\int_0^{1}(\frac{\ln^{s-1}{(\frac{1}{y})}\sin{(a)}}{1+2y\cos{(a)}+y^{2}}-\frac{a\ln^{s-1}{(\frac{1}{y})}}{(1+y)^2}){\mathrm{d}}y+$$ $$\frac{1}{\Gamma{(s)}}\int_0^{1}\ln{\ln{(\frac{1}{y})}}\ln^{s-1}{(\frac{1}{y})}(\frac{\sin{(a)}}{1+2y\cos{(a)}+y^{2}}-\frac{a}{(1+y)^2}){\mathrm{d}}y$$ Because the second integral on the richt-hand side of the equation keeps bounded as $s$ tends to $1$, we get, letting $s$ tend to $1$ $$-\frac{\pi}{2}\int_0^{\infty}\frac{e^{au}-e^{-au}-2au}{e^{\pi u}-e^{-\pi u}}\frac{{\mathrm{d}}u}{u}=\gamma\int_0^{1}(\frac{\sin{(a)}}{1+2y\cos{(a)}+y^{2}}-\frac{a}{(1+y)^2}){\mathrm{d}}y$$ $$+\int_0^{1}\frac{\ln{\ln{(\frac{1}{y})}}\sin{(a)}{\mathrm{d}}y}{1+2y\cos{(a)}+y^{2}}-a\int_0^{1}\frac{\ln{\ln{(\frac{1}{y})}}{\mathrm{d}}y}{(1+y)^{2}}$$ The first integral on the right-hand side of the equation is seen to vanish by direct calculation, for the second we want for the sake of brevity write $F{(a)}$. Then our equation becomes $$-\frac{\pi}{2}\int_0^{a}\frac{{\mathrm{d}}}{{\mathrm{d}}a}(\int_0^{\infty}\frac{e^{au}-e^{-au}-2au}{e^{\pi u}-e^{-\pi u}}\frac{{\mathrm{d}}u}{u}){\mathrm{d}}a=-a\int_0^{1}\frac{\ln{\ln{(\frac{1}{y})}}{\mathrm{d}}y}{(1+y)^{2}}+F(a)$$ And if we expand $\frac{1}{(1+y)^2}$ into a series $$-\frac{\pi}{2}\int_0^{a}\int_0^{\infty}\frac{e^{au}-e^{-au}-2}{e^{\pi u}-e^{-\pi u}}{\mathrm{d}}u{\mathrm{d}}a= a\sum_{n=1}^{\infty}(-1)^{n}n\int_{0}^{1}\ln{\ln{(\frac{1}{y})}}y^{n-1}{\mathrm{d}}y+ F(a)$$ If we set $e^{-\pi u} =y$ and write $y^0+y^0$ instead of $2$ and evaluate the integral on the right-hand side, with the identity mentioned earlier, we will have $$-\frac{1}{2}\int_0^{a}\int_0^{1}\frac{y^{\frac{a}{\pi}}+y^{-\frac{a}{\pi}}-y^0-y^0}{1-y^2}{\mathrm{d}}y{\mathrm{d}}a=a\sum_{n=1}^{\infty}(-1)^{n+1}n\frac{\gamma+\ln{n}}{n}$$ The inner integral, depending on $y$, can be expressed in terms of $\psi{(x)}$ using the identity from above, the occuring sum $\sum_{n=1}^{\infty}(-1)^{n+1}$ is equal to $\frac{1}{2}$, as we saw earlier. Therefore $$-\frac{1}{2}\int_0^{a}\frac{1}{2}(2\psi{(\frac{1}{2})}-\psi{(\frac{1}{2}+\frac{a}{2\pi})}-\psi{(\frac{1}{2}-\frac{a}{2\pi})}){\mathrm{d}}a=\frac{\gamma a}{2}+\frac{a}{2}\sum_{n=1}^{\infty}(-1)^{n+1}\ln{n}+F(a)$$ The occuring sum, involving logarithms, leads to this infinite product $$\prod_{k=1}^{\infty}(\frac{2k-1}{2k})$$ But it can be reduces to the Wallis product formula as follows $$\prod_{k=1}^{\infty}(\frac{2k-1}{2k})=\sqrt{\prod_{k=1}^{\infty}(\frac{4k^2}{4k^2-1})}=\sqrt{\frac{2}{\pi}}$$ Simplyfying and using these values gives $$-\frac{1}{4}\cdot 2\psi{(\frac{1}{2})}a+\frac{2\pi}{4}\ln{\Gamma{(\frac{1}{2}+\frac{a}{2\pi})}}-\frac{2\pi}{4}\ln{\Gamma{(\frac{1}{2}-\frac{a}{2\pi})}}=\frac{\gamma a}{2}+\frac{a}{2}\ln{\sqrt{\frac{2}{\pi}}}+F(a)$$ Solving for $F{(a)}$ and subsituting the value for $\psi{(\frac{1}{2})}$, yields after some simplification $$F(a)=\int_0^{1}\frac{\ln{\ln{(\frac{1}{y})}}\sin{(a)}{\mathrm{d}}y}{1+2y\cos{(a)}+y^{2}}=\frac{\pi}{2}\ln{(\frac{(2\pi)^{\frac{a}{\pi}}\ln{\Gamma{(\frac{1}{2}+\frac{a}{2\pi})}}}{\ln{\Gamma{(\frac{1}{2}-\frac{a}{2\pi})}}})}$$ If we apply the series for $\frac{\sin{(a)}{\mathrm{d}}y}{1+2y\cos{(a)}+y^2}$, we obtain $$\sum_{n=1}^{\infty}(-1)^{n-1}\sin{(na)}\int_0^{1}\ln{\ln{(\frac{1}{y})}}y^{n-1}{\mathrm{d}}y=\frac{\pi}{2}\ln{(\frac{(2\pi)^{\frac{a}{\pi}}\ln{\Gamma{(\frac{1}{2}+\frac{a}{2\pi})}}}{\ln{\Gamma{(\frac{1}{2}-\frac{a}{2\pi})}}})}=F(a)$$ The integral is known again and leads to $$F(a)=-\gamma\sum_{n=1}^{\infty}(-1)^{n-1}\frac{\sin{(na)}}{n}-\sum_{n=1}^{\infty}(-1)^{n-1}\frac{\sin{(na)}\ln{n}}{n}$$ We saw the first series to be $\frac{a}{2}$. If we use this and let $a =2\pi x$ and solve for $\ln{\frac{\Gamma{(\frac{1}{2}+x)}}{\Gamma{(\frac{1}{2}-x)}}}$, we arrive at the following Fourier series expansion $$\ln{\frac{\Gamma{(\frac{1}{2}+x)}}{\Gamma{(\frac{1}{2}-x)}}}=-2x(\gamma+\ln{(2\pi)})+\frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\frac{\ln{k}}{k}\sin{(2k\pi x)}$$ And this series enables us, to evaluate a lot of logarithmic integrals, because we know see the connection between the integrals and the series from all the things we derived. So it will be convenient to offer some examples, that it can be seen better.\
§23 The first example should be Vardi’s integral, as the following integral is now called, $$\int_0^{1}\frac{\ln{\ln{(\frac{1}{y})}}{\mathrm{d}}y}{1+y^{2}}=\frac{\pi}{2}\ln{(\frac{(2\pi)^{\frac{1}{2}}\ln{\Gamma{(\frac{3}{4})}}}{\ln{\Gamma{(\frac{1}{4})}}})}$$ We can use the integral for the evaluation $$F(a)=\int_0^{1}\frac{\ln{\ln{(\frac{1}{y})}}\sin{(a)}{\mathrm{d}}y}{1+2y\cos{(a)}+y^{2}}=\frac{\pi}{2}\ln{(\frac{(2\pi)^{\frac{a}{\pi}}\ln{\Gamma{(\frac{1}{2}+\frac{a}{2\pi})}}}{\ln{\Gamma{(\frac{1}{2}-\frac{a}{2\pi})}}})}$$ we get formula 1 in Vardi’s paper, because $a=\frac{\pi}{2}$ already leads directly to the desired integral. To get more formulas, we just have to put in more values for $a$, we want to take $a=\frac{\pi}{3}$, this will give $$\int_0^{1}\frac{\ln{\ln{(\frac{1}{y})}}{\mathrm{d}}y}{1+y+y^{2}}=\frac{\pi}{\sqrt{3}}\ln{(\frac{(2\pi)^{\frac{1}{3}}\ln{\Gamma{(\frac{2}{3})}}}{\ln{\Gamma{(\frac{1}{3})}}})}$$ and for $a=\frac{2\pi}{3}$ we will get this formula $$\int_0^{1}\frac{\ln{\ln{(\frac{1}{y})}}{\mathrm{d}}y}{1-y+y^{2}}=\frac{\pi}{\sqrt{3}}\ln{(\frac{(2\pi)^{\frac{2}{3}}\ln{\Gamma{(\frac{7}{6})}}}{\ln{\Gamma{(-\frac{1}{6})}}})}$$ And this, with the known properties of $\ln{\Gamma{(x)}}$, reduces to the following value, also given by Vardi in his paper [@7]. $$\int_0^{1}\frac{\ln{\ln{(\frac{1}{y})}}{\mathrm{d}}y}{1+y+y^{2}}=\frac{2\pi}{\sqrt{3}}[\frac{5}{6}\ln{(2\pi)}-\ln{\Gamma(\frac{1}{6})}]$$ We would for example use the reflection formula for $\Gamma{(x)}$, namely $$\Gamma{(x)}\Gamma{(1-x)}=\frac{\pi}{\sin{\pi x}}$$ which was given by Euler [@13], which we did not show and therefore will not use here.\
§24 It will be convenient, to add one more example, that involves a rational function, whose denominator is raised to the second power. For this purpose we consider our integral again $$F(a)=\int_0^{1}\frac{\ln{\ln{(\frac{1}{y})}}\sin{(a)}{\mathrm{d}}y}{1+2y\cos{(a)}+y^{2}}=\frac{\pi}{2}\ln{(\frac{(2\pi)^{\frac{a}{\pi}}\ln{\Gamma{(\frac{1}{2}+\frac{a}{2\pi})}}}{\ln{\Gamma{(\frac{1}{2}-\frac{a}{2\pi})}}})}$$ If we differentiate it with respect to $a$ again, we obtain $$\int_0^{1}\frac{\ln{\ln{(\frac{1}{y})}}((1+y^2)\cos{(a)}+2y){\mathrm{d}}y}{(1+2y\cos{(a)}+y^{2})^2}= \frac{\ln{(2\pi)}}{2}+\frac{1}{4}\psi{(\frac{1}{2}+\frac{a}{2\pi})}+\frac{1}{4}\psi{(\frac{1}{2}-\frac{a}{2\pi})}$$ and if we put $a=\frac{\pi}{2}$ and use the known values (or just calculate them from the integral expression for $psi{(x)}$) $$\psi{(\frac{1}{4})}=-\gamma-\frac{\pi}{2}-\ln{8}$$ and $$\psi{(\frac{3}{4})}=-\gamma+\frac{\pi}{2}-\ln{8}$$ we get this formula $$\int_0^{1}\frac{y\ln{\ln{(\frac{1}{y})}}{\mathrm{d}}y}{(1+y)^2}=\frac{1}{2}\ln{\sqrt{\pi}}-\frac{\ln{4}}{4}-\frac{\gamma}{4}$$ which formula is a special case of a more general one, given by Adamchik in propostion 5 in his paper [@18].\
§25 The first three integrals were all given by Vardi [@7], the first two also by Malmstèn [@3], including Vardi’s integral. And from his formulas there follow many others.\
§26 It is now easyly seen, that we provided everything again, to get many logarithmic integrals. The greatest problem consists in the evaluation of certain sums, involving logarithms, which - as we saw - can be expressed in finite terms with the given Fourier series expansion.\
We could also evaluate integrals as this one $$\int_0^{1}\frac{y\ln{\ln{(\frac{1}{y})}}{\mathrm{d}}y}{(1-y+y^2)^2}=-\frac{\gamma}{3}-\ln{(\frac{6\sqrt{3}}{\pi})}+\frac{\pi \sqrt{3}}{27}[5\ln{(2\pi)}-6\ln{\Gamma(\frac{1}{6})}]$$ if we allowed the use of divergent series or at least the principle of analytic continuation, because we would be lead to certain divergent series, which correspond to certain sums and quotients of geometric series and their derivatives at the point $x=1$.\
§27 But having shown at least a little bit about these integrals and having proved the functional equations for some Dirichlet series, we will put aside more concrete evaluations, because they are well-presented by Medina and Moll [@17] and Adamchik [@18] and we would need divergent series, which require a deeper theory, that cannot be regarded as elementary anymore and that is not as straight-forward as the calculations, we did and explained, in this memoir. We will give more evaluations on another occasion and use Euler’s definition [@8] of a divergent series and its sum and will see, whether it leads to the same results.\
[**Acknowledgement:**]{} I would like to thank Sebastian Koch and Arseny Skryagin for showing interest in this little paper and making many useful suggestions. I would also like to thank Artur Diener for helping a lot, writing this paper.
[99]{} A.M. Legendre, [*Exercises du calcul integral*]{} B. Riemann, [*über die Anzahl der Primzahlen unter einer gegebenen Grösse*]{}, Monatsberichte der Berliner Academie, (1859). C. J. Malmstèn, [*“De integralibus quibusdam definitis seriebusque infinitis”*]{}, Journal für Mathematik, Bd. [**38**]{}, 1-39, (1849). E. E. Kummer, [*Beitrag zur Theorie der Funktion $\Gamma (x)$*]{}, Journal für Mathematik, Bd. 35, 1-4, (1847). E. Sandifer, [*Partial fractions*]{}, How Euler did it, June 2007 H. Hardy, [*Divergent Series*]{}, AMS Chelsea Publishing Vardi, [*Integrals, an Introduction to Analytic Number Theory, American Mathematical Monthly Volume 95 Issue 4, April 1988 Pages 308 - 315*]{} L. Euler, [*De seriebus divergentibus*]{}, Opera Omnia: Series 1, Volume 14, pp. 585 - 617 L. Euler, [*De seriebus De summis serierum reciprocarum ex potestatibus numerorum naturalium ortarum dissertatio altera, in qua eaedem summationes ex fonte maxime diverso derivantur*]{}, Opera Omnia: Series 1, Volume 14, pp. 138 - 155 L. Euler, [*De resolutione fractionum transcendentium in infinitas fractiones simplices*]{}, Opera Omnia: Series 1, Volume 15, pp. 621 - 660 L. Euler, [*De valore formulae integralis $\int\frac{x^{a-1}{\mathrm{d}}x}{\ln{x}}\cdot\frac{(1-x^b)(1-x^c)}{1-x^n}$ a termino $x = 0$ usque ad $x = 1$ extensae*]{}, Opera Omnia: Series 1, Volume 18, pp. 51 - 68 L. Euler, [*De valoribus integralium a termino variabilis $x = 0$ usque ad $x = \infty$ extensorum*]{}, Opera Omnia: Series 1,Volume 19, pp. 217 - 227 L. Euler, [*Evolutio formulae integralis $\int x^{f-1}{\mathrm{d}}x\ln^{\frac{m}{n}}{(x)}$ integratione a valore $x = 0$ ad $x = 1 extensa$*]{}, Opera Omnia: Series 1,Volume 17, pp. 316 - 357 L. Euler, [*Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum*]{}, Opera Omnia: Series 1, Volume 10 L. Euler, [*Remarques sur un beau rapport entre les series des puissances tant directes que reciproques*]{}, Opera Omnia: Series 1, Volume 15, pp. 70 - 90 L. Euler, [*Summatio progressionum $\sin^{\lambda}{(\phi)}+\sin^{\lambda}{(2\phi)}+\sin^{\lambda}{(3\phi)}\cdots+\sin^{\lambda}{(n\phi)}$; $\cos^{\lambda}{(\phi)}+\cos^{\lambda}{(2\phi)}+\cos^{\lambda}{(3\phi)}\cdots+\cos^{\lambda}{(n\phi)}$*]{}, Opera Omnia: Series 1,Volume 15, pp. 168 - 184 L. Medina and V.Moll, [*A class of logarithmic integrals*]{}, Ramanujan Journal, 20, 2009, pp. 91-126 V. Adamchik, [*A class of logarithmic integrals*]{} W. Ford, [*Studies on divergent series and summability and the asymptotic development of functions defined by Maclaurin series*]{}, Chelsea Publishing Company, New York
|
---
author:
- 'M. Sasaki'
- 'T. F. X. Stadlbauer'
- 'F. Haberl'
- 'M. D. Filipović'
- 'and P. J. Bennie'
date: 'Received September 30, 2000; accepted November 2, 2000'
title: 'XMM-Newton EPIC Observation of SMC SNR0102$-$72.3[^1]'
---
Introduction
============
is one of the brightest X-ray sources in the Small Magellanic Cloud (SMC) and was discovered by the IPC instrument onboard the Einstein Observatory (Seward & Mitchell [@SM81]). The Einstein HRI resolved a shell-like supernova remnant (SNR), and optical emission is seen from a variety of filaments rich in oxygen and neon (Dopita & Tuohy [@DT84]) arranged in an incomplete clumpy ring with radius of about 12. Enrichment in the element oxygen is indicative for remnants of a type II SN explosion. The maximum velocity of the optical filaments \[\], not visible in H$\alpha$ (Dopita [et al.]{} [@D81]), was found to be up to 6500km/s (Tuohy & Dopita [@TD83]). The bright knots, assumed to be dense ejecta clumps or Rayleigh-Taylor fingers of ejecta (density $\sim$1cm$^{-3}$), correspond in some parts well with bright X-ray features (Gaetz [et al.]{} [@G00]) but in other parts not. The estimated ionization timescale of $\mathrm{log}(nt)<12$ adds to the origin of the X-ray emission being a shock heated X-ray plasma in a state of non-equilibrium ionization (NEI). Radio observations of with 3 resolution in the 6cm wavelength band by Amy & Ball ([@AB93]) unveiled an outer radio shell diameter of 40$\pm$5. The radio emission is found to lie predominantly outside the bright X-ray emission but within an outer faint X-ray rim of $\sim$44 (Gaetz [et al.]{} [@G00]). Hayashi [et al.]{} ([@H94]) reported on a broadband spectrum of gained from the ASCA SIS instrument showing most of the emission from lying in the 0.5 – 2.3keV energy band. The most prominent line features were interpreted as He-like K$\alpha$, H-like K$\alpha$ and K$\beta$ emission from ionized atoms of the elements oxygen, neon and magnesium. Their difficulties to describe the overall spectrum with simple NEI models was interpreted as an inhomogeneous abundance pattern within the emitting plasma and possibly two components, namely forward shock in the interstellar medium (ISM) and reverse shock in the ejecta, contributing to the overall X-ray emission. XMM-Newton with its high sensitivity now enables spatially resolved spectroscopy on various parts of . The following section describes the details of the observation and the data. Spectra from the selected regions are presented and fitted in Sec.\[xrayspec\]. Finally, the results are discussed in Sec.\[discuss\].
Data
====
------- ------------ ------------------------- ------------------------- ------------- --------------- --------
Orbit Obs. ID Start Time End Time Filter
RA Dec
65 0123110201 16. Apr. 2000, 19:06:50 17. Apr. 2000, 01:27:02 01 03 50.00 $-$72 01 55.0 thin
65 0123110301 17. Apr. 2000, 03:41:01 17. Apr. 2000, 09:44:33 01 03 50.00 $-$72 01 55.0 medium
------- ------------ ------------------------- ------------------------- ------------- --------------- --------
was observed by the XMM-Newton (Jansen [et al.]{} [@J01]) in April 2000 during its calibration phase (see Table \[obslist\]). There were two observations with different filters, thin and medium. In order to study the morphologies of the SNR, XMM-Newton EPIC-MOS (Turner [et al.]{} [@T01]) data were used. The MOS camera collected the emission of the complete SNR, whereas in the EPIC-PN (Strüder [et al.]{} [@S01]) observations the SNR was split into two parts by CCD chip borders. The EPIC-MOS data was processed through the XMM-Newton Science Analysis System (XMM-SAS) and events in the energy band of 0.2 – 3.0keV were selected. Fig.\[mosradio\] shows an overlay of the image created from the EPIC-MOS thin filter observation and contours of a radio image of the 4790MHz observations at the Australia Telescope Compact Array (ATCA).
The region centered on was observed as part of the ATCA mosaic observations of the SMC with a baseline of 375m at frequencies of 1420 and 2370MHz with a corresponding angular resolution of $\sim$90 and 45. ATCA observations in snap-shot mode at 4800 and 8640MHz were undertaken for specific regions of interest including SNR 0102$-$72.3. The baseline of these observations were as well 375m and we achieved resolution of $\sim$30 and $\sim$15, respectively (Filipović & Staveley-Smith [@FS98]). Amy & Ball ([@AB93]) studied the at 4790Hz, but with higher resolution (3). We made use of their data and merged it with our observations (Filipović [et al.]{} [@F98] and references therein) to compensate for possible short space missing flux.
As can be seen in Fig.\[mosradio\], the radio emission of the SNR is highest in the north whereas the brightest knots in X-rays are found in the south. The emission from non-thermal electrons which forms the radio ring, is located outside the bright X-ray ring, especially in the north. This is in good agreement with the comparison of Chandra ACIS data with the radio image of Amy & Ball ([@AB93]) by Gaetz [et al.]{} ([@G00]). Due to higher spatial resolution of Chandra, the ACIS image shows a detailed structure of the SNR not available to XMM, and it was found that the radio emission is mainly located between the bright X-ray ring and the rim of faint X-ray emission.
For spectral studies we analyzed the EPIC-PN data. The effective exposure times were 15.5ksec and 10.9ksec, respectively. We used the standard processed data from XMM-SAS and selected single pattern events only. On the EPIC-PN detector the remnant was located in the CCD chips No. 4 and 7, unfortunately divided into two parts, the larger (eastern) part lying on chip No. 7. For spectral analysis of the SNR we selected a circular region of radius 40 around the center of the X-ray ring. Events in a ring with inner radius of 38 and outer radius of 88 on the chips No. 4 and 7, excluding out of time events, were used to estimate the background. In addition we extracted two regions of the X-ray emission with inner radius of 8 and outer radius of 25 which is thought to arise mainly from the hot ejecta. The first region (hereafter REGION01) matches the northern part on the chip No. 7 starting from 10 to 80 counterclockwise from the north, and covers the faintest part of the X-ray ring. REGION02 is selected from 80 to 185 including the southeastern bright emission of the SNR.
X-ray spectrum {#xrayspec}
==============
For each selected region spectra were created from both the thin and medium filter data separately. In order to combine both observations, the two different spectra were fitted in XSPEC simultaneously with either model. In Fig.\[specsnr\] the thin filter spectrum of is shown. Additionally the best double [VGNEI]{} fit (for details see below) is plotted and the positions of the He-like emission lines of , , , , and as well as the H-like lines of and are marked. These lines have been confirmed in the XMM-Newton RGS spectrum of the same pointings (Rasmussen [et al.]{} [@R01]).
Simply looking at the spectra of the northeastern and southeastern parts (Fig.\[specej1\] and Fig.\[specej2\]) makes a more prominent line emission of the He-like ions of oxygen and neon compared to the line emission of the H-like ions in the southeastern spectrum obvious. In the spectrum of REGION01 the higher ionized and clearly are brighter. Even for the ions of magnesium this difference is visible. The spectrum of REGION02 shows a striking line feature of around 1.34keV with a steep fall-off at higher energies. REGION01 however gives a spectrum with a less prominent line feature and a less steep fall-off towards the higher energy end, where the location of the H-like is expected ($\sim$1.47keV).
Due to these He-like and H-like emission lines of oxygen and neon, as well as other He-like lines, the spectra could not be modeled by one non-equilibrium ionization (NEI) model. Thus we used two generalized NEI components with time varying temperature ([VGNEI]{}, Borkowski [@B00]). The model [VGNEI]{} describes a homogeneous neutral gas which was initially cold, but is heated spontaneously. The ionized plasma is not in thermal equilibrium for small ionization timescales $\tau = n_\mathrm{e}t
< 10^{12}$ s cm$^{-3}$, i. e. the electron temperature is lower than the ion temperature. The temperatures vary with time. Beside the parameters temperature $kT$ (keV), heavy-element abundances, and the ionization timescale $\tau$, the ionization timescale averaged temperature $\langle kT\rangle$ (keV) is determined. For the two [VGNEI]{} components, the abundances were tied together, while the temperatures and the ionization timescales remained free parameters.
Furthermore we applied a plane-parallel shock model ([VNPSHOCK]{}) which takes different NEI states into account. Essential parameters of the model are the mean shock temperature $kT_\mathrm{s}$ (keV), postshock electron temperature $kT_\mathrm{e}$ (keV) immediately behind the shock front, and the ionization timescale $\tau$. In collisionless shocks in SNRs electrons and ions are presumably not in thermal equilibrium. The electrons are heated by Coulomb collisions with ions which have higher temperatures in the postshock plasma. Far behind the shock front, electron temperature becomes equal to ion temperature for higher $\tau$.
[lccccccl]{} & & & &\
& value & (90% c.r.) & value & (90% c.r.) & value & (90% c.r.) &\
& & & & &\
SMC [$N_\mathrm{H}$]{}& 8.74 & (8.01 – 9.31) & 8.13 & (1.49 – 24.9) & 6.93 & (5.05 – 9.57) & [10$^{20}$ cm$^{-2}$]{}\
& & & & & & &\
$kT$ & 1.14 & (1.10 – 1.26) & 1.06 & (0.46 – 1.58) & 0.80 & (0.73 – 0.91) & keV\
$\tau_{1}$ & 0.81 & (0.76 – 0.83) & 1.2 & (0.30 – 13) & 3.9 & (3.4 – 5.0) & 10$^{11}$ s cm$^{-3}$\
$\langle kT\rangle$ & 2.44 & (2.34 – 2.57) & 0.91 & (0.56 – 1.99) & 0.71 & (0.63 – 0.75) & keV\
EM$_{1}$ (frac.) & 0.58 (0.68) & & 1.7 (0.76) & & 6.2 (0.84)& & 10$^{58}$ cm$^{-3}$\
& & & & & & &\
$kT$ & 4.52 & (3.25 – 5.47) & 1.04 & (0.30 – 1.13) & 0.79 & (0.62 – 0.92) & keV\
$\tau_{2}$ & 0.11 & (0.10 – 0.12) & 0.16 & (0.01 – 1.0) & 0.18 & (0.07 – 0.34) & 10$^{11}$ s cm$^{-3}$\
$\langle kT\rangle$ & 3.13 & (2.68 – 3.22) & 0.74 & (0.72 – 0.76) & 0.81 & (0.26 – 1.88) & keV\
EM$_{2}$ (frac.) & 0.27 (0.32) & & 0.54 (0.24) & & 1.2 (0.16) & & 10$^{58}$ cm$^{-3}$\
Oxygen abundance & 1.6 & (1.3 – 1.7) & 3.9 & (3.3 – 4.1) & 4.7 & (4.4 – 9.3) &\
Neon & 4.2 & (4.0 – 4.6) & 5.9 & (5.3 – 6.2) & 7.1 & (5.9 – 13.5) &\
Magnesium & 2.3 & (2.1 – 2.5) & 3.2 & (2.9 – 3.5) & 3.0 & (2.7 – 3.3) &\
Silicon & 1.0 & (0.8 – 1.3) & 0.9 & (0.7 – 1.2) & 0.8 & (0.6 – 0.9) &\
Iron & 0.6 & (0.5 – 0.7) & 0.7 & (0.5 – 0.8) & 0.5 & (0.4 – 0.7) &\
red. $\chi^{2}$ & 1.10 & & 1.63 & & 1.76 & &\
& & & & &\
SMC [$N_\mathrm{H}$]{}& 0.99 & (0.00 – 1.29) & 1.95 & (0.99 – 2.19) & 0.37 & (0.00 – 0.69) & [10$^{20}$ cm$^{-2}$]{}\
$kT_\mathrm{e}$ & 1.15 & (0.86 – 1.19) & 0.08 & (0.01 – 0.18) & 0.08 & (0.01 – 0.20) & keV\
$kT_\mathrm{s}$ & 5.65 & (5.64 – 5.66) & 3.76 & (3.54 – 3.82) & 4.20 & (4.10 – 4.30) & keV\
$\tau$ & 1.1 & (1.0 – 1.2) & 1.0 & (0.9 – 1.1) & 1.1 & (1.0 – 1.2) & 10$^{11}$ s cm$^{-3}$\
Oxygen abundance & 0.8 & (0.7 – 0.9) & 1.1 & (1.0 – 1.2) & 1.1 & (1.0 – 1.2) &\
Neon & 1.4 & (1.3 – 1.5) & 2.4 & (2.3 – 2.5) & 2.6 & (2.5 – 2.7) &\
Magnesium & 0.9 & (0.8 – 1.0) & 1.3 & (1.2 – 1.4) & 1.5 & (1.4 – 1.6) &\
Silicon & 0.3 & (0.2 – 0.5) & 0.3 & (0.2 – 0.4) & 0.4 & (0.3 – 0.5) &\
Iron & 0.3 & (0.2 – 0.4) & 0.2 & (0.2 – 0.3) & 0.3 & (0.2 – 0.4) &\
red. $\chi^{2}$ & 1.21 & & 1.84 & & 2.52 & &\
Notes:
EM$_{1}$ and EM$_{2}$ are emission measures ${\rm EM} = \int n_\mathrm{e} n_\mathrm{H}~\rm{d}V$ of the two [VGNEI]{} components. Fractional values are given in brackets. For the entire SNR, EM$_{1}$ and EM$_{2}$ from the EPIC-PN spectra do not correspond to the total values of the SNR, because a part of the emission was not detected due to the CCD gap.
Abundances are relative to solar values.
The 90% confidence range for temperature, ionization timescale and abundances are given in brackets. The confidence range for abundances are calculated by fixing $kT$ and $\tau$ at best fit values.
The absorption consists of fixed foreground absorption with galactic [$N_\mathrm{H}$]{} of $5.36\times 10^{20}\,\mathrm{cm}^{-2}$ (Dickey & Lockman [@DL90]) and an additional absorption column density which is a free fit parameter with fixed abundances of 0.2 typical for the interstellar gas in the SMC (Russell & Dopita [@RD92]). The resulting parameters of the simultaneous two-observation-fit for the entire SNR and the two regions are given in Table \[parlist\].
In all selected regions fitting the spectrum results in an overabundance of oxygen, neon, and magnesium (see Tab.\[parlist\]). Using the double [VGNEI]{} model, two ionization timescales were determined differing in one order of magnitude, $\tau \simeq 10^{11}$s cm$^{-3}$ and $\tau \simeq 10^{10}$s cm$^{-3}$. The lower $\tau$ component dominates in the softer part of the spectrum below 0.8keV. The higher ionization states are reproduced by the higher $\tau$ component which is prominent in the higher energy end of the spectrum.
In REGION01 the determined temperatures are the highest both in the double [VGNEI]{} model and the [VNPSHOCK]{} model with temperatures higher than 1keV. This is the part with the lowest emission along the X-ray ring. Both the shape of the spectrum and the temperatures indicate that this region is more highly ionized than in the south.
In REGION02 and the entire SNR spectrum the temperatures are lower, but the abundances higher than in REGION01. REGION02 includes the brightest part of the X-ray ring visible in the analyzed EPIC-PN observation and the abundances of oxygen, neon, and magnesium are higher than in the northeast with 3.9, 5.9, and 3.2 relative to solar in the double [VGNEI]{} model, respectively. The comparison of the spectra of the entire SNR and REGION02 makes clear that the temperatures and the abundances are almost the same. Although there is a slight difference in the ionization timescale $\tau_{1}$ of the higher $\tau$ component of the double [VGNEI]{}-model, which is higher in the entire spectrum, the observed X-ray emission of the SNR is dominated by the emission from REGION02. Furthermore the He-like emission lines of could be identified around 2.45keV in the overall spectrum, since the spectrum extends up to 6keV thanks to good photon statistics. For the whole SNR the single plane-parallel shock model yields an unsatisfying fit. This corroborates the results of ASCA observations that the X-ray emission of the SNR originates from at least two different thermal plasma states (Hayashi [et al.]{} [@H94]).
Discussion {#discuss}
==========
is a young SNR with an estimated age of about 1000yr (Tuohy & Dopita [@TD83]), which is no longer in the free expansion phase. Most of the X-ray emission originates from a bright ring with radius $\sim$14 and a mean FWHM of 5 caused by a reverse shock propagating through the ejecta (Gaetz [et al.]{} [@G00]; Hughes [et al.]{} [@H00]). The XMM-Newton EPIC observations confirm this overall picture.
The spectra of the northeastern and southeastern regions of the SNR can be fitted with a two component NEI model, each component with a characteristic single ionization timescale. The emission lines of the two highest ionization stages of oxygen and neon, as well as the two values for $\tau$ point out, that there is an ongoing shock ionization. This effect differs in the northeastern and the southeastern part of the SNR which can be seen in the unequal spectra of these regions (Fig.\[specej1\] and Fig.\[specej2\]). The plasma temperature for the northeastern part is up to 4 times higher than in the rest of the SNR, indicating that the shock velocity is higher in this region ($v_\mathrm{s} \sim
T_\mathrm{s}^{0.5}$). Though most of the X-ray emission of the SNR arises from the hot ejecta, in the northeastern part the forward shock of the blast wave propagating into the ISM becomes important, verified in radio observations showing synchrotron emission right behind the blast wave which can be seen in Fig.\[mosradio\]. The X-ray emission of the entire SNR is dominated by the ejecta emission, which can be seen in the significant similarity between the overall spectrum and the spectrum of the southeastern region.
All the spectra extracted from EPIC-PN data were better reproduced by the two component NEI model than in a single plane-parallel shock model. The two NEI components differ not only in temperatures, but much more significantly in the ionization timescale $\tau$. While the lower $\tau_{2}$ component values themselves are similar for both selected regions as well as for the overall spectrum, the higher $\tau_{1}$ value is the lowest in the northeastern region. In this part of the SNR, also the fractional emission measure of the higher $\tau_{1}$ component is smaller than in the southeastern region or the whole SNR. Since in the southeastern (and the entire SNR) the ratio of the bright X-ray emission originating from the inner parts of the SNR (i.e. the bright X-ray ring) to the X-rays from the outer parts is higher (see Fig.\[mosradio\]), the higher $\tau_{1}$ component can be assigned to the bright X-ray ring, outlining regions with higher densities. The implied density distribution around the SNR ring is also supported by the extended overall shape along the southwest to northeast axis, which was already reported by Gaetz [et al.]{} ([@G00]) and can be verified in Fig.\[mosradio\]. This is indicative of a less decelerated expansion in that direction. Chandra observations have shown that there is evidence for a spatially varying distribution of the ionization stages and the existence of a reverse shock in the ejecta (Gaetz [et al.]{} [@G00]; Flanagan [et al.]{} [@F01]). The results obtained from the spectral analysis of the XMM-Newton EPIC-PN data show that there are differences in the plasma states between various parts of the SNR with spatial temperature and ionization stage variations, contributing to its complicated structure.
The authors wish to thank Andrew Rasmussen for the valuable referee report. The XMM-Newton project is supported by the Bundesministerium für Bildung und Forschung / Deutsches Zentrum für Luft- und Raumfahrt (BMBF/DLR), the Max-Planck Society and the Heidenhain-Stiftung.
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[^1]: Based on observations with XMM-Newton, an ESA Science Mission with instruments and contributions directly funded by ESA Member states and the USA (NASA).
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---
abstract: 'An intense laser field in the high-frequency regime drives carriers in graphene nanoribbons (GNRs) out of equilibrium and creates topologically-protected edge states. Using Floquet theory on driven GNRs, we calculate the time evolution of local excitations of these edge states and show that they exhibit a robust dynamics also in the presence of very localized lattice defects (atomic vacancies), which is characteristic of topologically non-trivial behavior. We show how it is possible to control them by a modulated electrostatic potential: They can be fully transmitted on the same edge, reflected on the opposite one, or can be split between the two edges, in analogy with Hall edge states, making them promising candidates for flying-qubit architectures.'
author:
- 'M. Puviani'
- 'F. Manghi'
- 'A. Bertoni'
title: 'Dynamics and Control of Edge States in Laser-driven Graphene Nanoribbons'
---
Introduction
============
The dynamics of quantum systems under the influence of time-periodic modulations has recently attracted growing attention for the possibility to realize unconventional phases of matter, including topological phases. After theoretical predictions, [@Kitagawa2011; @Lindner2011; @Inoue2010; @Oka2009] the generation and manipulation of topological states through the application of a time-periodic perturbation has been experimentally demonstrated in different systems such as ultra-cold gases in time-dependent optical lattices [@Goldman2016], periodically driven photonic waveguide lattices [@Maczewsky2017; @Rechtsman2013], acoustic systems [@He2016] and topological insulators under circularly polarized light [@W.2012]. The presence of topologically protected edge states responsible for robust one-way edge transport is the common thread between all these diverse systems.
On the theoretical side, a complete topological characterization is provided by the emergence of non-vanishing topological invariants that ensure the presence of gapless edge states and their robustness against disorder. Topological invariants have been defined for systems in static conditions [@Xu2006; @Hasan2010; @Grandi2015; @Grandi2015NJP] and only more recently extended to the periodically driven case [@Rudner2013; @Foa2015; @Fenner2017] where new types of edge modes have been identified which cannot be accounted for using the invariants developed for the static case. [@Kitagawa2012; @Maczewsky2017]
The topological properties of Floquet systems have also been studied in connections with transport properties: The conductance and quantum Hall response of irradiated graphene nanoribbons (GNRs) [@PhysRevLett.113.236803; @PhysRevLett.113.266801; @PhysRevB.89.121401] and of quantum well heterostructures [@PhysRevLett.115.106403] have been calculated by analytical and numerical methods showing distinctive characteristics associated with the presence of chiral edge states. The influence of disorder [@PhysRevLett.113.236803; @PhysRevLett.115.106403] and of coupling with phonons [@PhysRevB.90.195429] has been used as an hallmark of a topologically protected phase. Different kinds of defects in 2D graphene have been shown to host Floquet bound states and their chiral nature has been identified by calculating the probability current around them [@PhysRevB.93.245434].
In this article, we look for another explicit signature of the robustness of edge states in a zig-zag terminated GNR driven by a circularly polarized intense laser field. We explore the real-time evolution of a particle initialized in one of these states and study in particular how time evolution is affected by local defects and potential barriers. Our analysis is based on the Floquet formalism, a Bloch theory in time domain which exploits time-periodicity to solve the time-dependent Schrödinger equation, factorizing the stroboscopic time-dependence of the quasiparticle from the intrinsic periodic one.
Floquet theory and time-dependent velocity
==========================================
The Hamiltonian for a lattice driven by a time-periodic electromagnetic field can be written using the minimal coupling [@PhysRevLett.108.225303]: $$\begin{aligned}
\hat{H}_{\vec{k}} (t) = \sum_{i,i'} \sum_{l, l'} \ J_{i l, i' l'} \ e^{i \left( \vec{k} + \vec{A} (t) \right)}
\, e^{(\vec{R}_l + \vec{\tau}_i - \vec{R}_{l'} + \vec{\tau}_{i'})}
\nonumber \\ \hat{c}^{\dagger}_{i,l}(t) \ \hat{c}_{i',l'}(t) \ ,\end{aligned}$$ $i,i'$ ($l,l'$) being site (cell) indices, with $\hat{c}^{\dagger}_{i,l}(t)$, $\hat{c}_{i',l'}(t) $ the corresponding time-dependent creation and annihilation operators, and $J_{i l, i' l'}$ being the nearest-neighbor hopping term. The vector potential associated with the electromagnetic field is $\vec{A}(t) = A_0 \left( \sin( \Omega t ) \ \hat{x} + \ \cos(\Omega t ) \ \hat{y} \right)$ for clockwise circular polarization, $\Omega = 2 \pi / \text{T}$ the frequency of the driving. Owing to the Floquet theorem, the solution of the time-dependent Schrödinger equation $$\label{ShrTD}
\hat{H} (t) \ | \Psi (t) \rangle = i \partial_t \ | \Psi (t) \rangle$$ can be written in a factorized form as $$\label{TDsol}
| \Psi_{\alpha} (t) \rangle = e^{- i \varepsilon_{\alpha} t} | \Phi_{\alpha}(t) \rangle \ ,$$ with $ | \Phi_{\alpha} (t+T) \rangle = | \Phi_{\alpha}(t) \rangle $. Defining the Floquet operator as $$\hat{H}_F \equiv \hat{H} (t) - i \partial_t \ ,$$ and substituting Eq. (\[TDsol\]) in Eq. (\[ShrTD\]), we obtain a time-independent eigenvalue problem for the Floquet states, with Floquet quasienergies $\varepsilon_{\alpha}$ constant in time: $$\label{EigenFloq}
\hat{H}_F \ | \Phi_{\alpha}(t) \rangle = \varepsilon_{\alpha} \ | \Phi_{\alpha}(t) \rangle \quad .$$ Since $| \Phi_{\alpha}(t) \rangle $ is periodic in time, it can be expanded in Fourier series $$| \Phi_{\alpha}(t) \rangle = \sum_{n = - \infty}^{+ \infty} e^{- i n \Omega t} \ | \phi_{\alpha,n} \rangle \quad .$$ In practice, the Fourier expansion is truncated to include a finite number of modes, up to a cutoff $n_{max}$. This allows one to formulate the eigenvalue problem in Eq. (\[EigenFloq\]) in a standard matrix form whose eigenvalues turn out to be replicas of the static band structure with gaps opening at their crossing points. Due to this truncation we have access to the time evolution at stroboscopic times $t=nT/n_{max}$ only. However, by extending $n_{max}$ it is possible to verify the accuracy of the dynamics at intermediate times.
The solution of the time-dependent Schrödinger equation for a particle in a lattice written in the local basis $\varphi$ is therefore $$\begin{aligned}
\label{SoluT}
\Psi_{\alpha}(r,t) = e^{- i \varepsilon_{\alpha} t } \sum_{i=1}^{N} \sum_{l=1}^{M} \sum_{n = -n_{max}}^{+ n_{max}} \sum_{\vec{k}} c_{i,n}^{\alpha} (\vec{k}) \ e^{- i n \Omega t} \nonumber \\
e^{-i \vec{k} \cdot (\vec{R}_l + \vec{\tau}_i) } \ \varphi (\vec{r}- \vec{R}_l - \vec{\tau}_i) \ ,\end{aligned}$$ where $N$ is the number of sites per cell, $M$ the number of cells, while $n$ are the Floquet indices. The time-dependent expectation values of observables involve these states.[@Manghi2017]\
The time-dependent expectation value of the velocity operator $\hat{\vec{v}}(t) = - i [ \hat{\vec{r}} , \hat{H}(t) ]$ can be analyzed at fixed Floquet eigenvectors obtaining a velocity vector field in real space $$\label{velocity}
\vec{v}_{\alpha} (\vec{r}_i, t) = \langle \Psi_{\alpha} (r,t) | \hat{\vec{v}}(t)| \Psi_{\alpha} (r,t) \rangle.$$ By using the elements of operators $\hat{H}$ and $\hat{\vec{r}}$ in the localized basis, namely $$\langle \varphi_{i,l} | \ \hat{H}(t) \ | \varphi_{i', l'} \rangle = J_{i i', l l'} \ e^{i \vec{A}(t) \cdot \left( \vec{R}_l + \vec{\tau}_i - \vec{R}_{l'} - \vec{\tau}_{i'} \right)}$$ and [@Selloni1984] $$\langle \varphi (\vec{r}- \vec{R}_l - \vec{\tau}_i)|\vec{r}|\varphi (\vec{r}- \vec{R}_{l'} - \vec{\tau}_{i'} ) \rangle = \delta_{i i'} \delta_{l l'} (\vec{R}_l + \vec{\tau}_i) \ ,$$ the velocity vector field turns out to be $$\begin{aligned}
\vec{v}_{\alpha} (\vec{r}_i, t) = \sum_{\substack{i' \\ l, l'}} \sum_{n, m} \sum_{\vec{k}} \ \left( c^{\alpha}_{i',m} (\vec{k}) \right)^* c^{\alpha}_{i,n} (\vec{k}) \ e^{-i (n-m) \Omega t} \nonumber \\
e^{- i \vec{k} \cdot \left( \vec{R}_l + \vec{\tau}_i - \vec{R}_{l'} - \vec{\tau}_{i'} \right) } \ ( \vec{R}_l + \vec{\tau}_i - \vec{R}_{l'} - \vec{\tau}_{i'} ) \nonumber \\
e^{- i \vec{A}(t) \cdot \left( \vec{R}_l + \vec{\tau}_i - \vec{R}_{l'} - \vec{\tau}_{i'} \right) } \ J_{i l , i' l'} \ .\end{aligned}$$
Dynamics of Floquet states in real space
========================================
The probability for an electron in a given state to move from a site $i$ at time $t_0$ to a site $j$ at time $t$ is $| U_{j,i} (t,t_0) |^2 $, where the time-evolution operator $U_{j,i}$ can be expressed interms of Floquet components as follows [@Shirley1965] $$\begin{aligned}
\label{timeEVO}
U_{j,i} (t,t_0) = \sum_{\alpha, n,m} e^{- i \varepsilon_{\alpha} (t-t_0)} \ e^{- i m \Omega t} \ \langle j,m | \varphi_{\alpha,m} \rangle \nonumber \\
\langle \varphi_{\alpha,n} | i,n \rangle \ e^{i n \Omega t_0} \quad .\end{aligned}$$ In order to have a result directly comparable with a real experiment, it is necessary to average the transition probability over the initial time $t_0$ keeping fixed the interval $\Delta_t=(t-t_0)$: $$\label{probab}
P_{j \leftarrow i} (\Delta_t) = \sum_m \bigg\vert \sum_{\alpha, \vec{k}} c^{\alpha}_{j,m} (\vec{k}) \left( c^{\alpha}_{i,0} (\vec{k}) \right)^* e^{-i \varepsilon_{\alpha, \vec{k}} \Delta_t} \bigg\vert ^2 \ .$$ This allows us to determine the electron motion in real space and time, without solving any time-integral: it is noticeable in fact that the system has no memory. Therefore, this quantity can be calculated for any $t$ as large as desired, without knowing anything regarding times before $t$.
Dynamics and robustness of edge states in driven GNR
====================================================
We study an ac-driven zigzag GNR extended along the $x$ axis and we consider the case of $\Omega \gg J $. With $n_{max}=3$ we get already time steps in the attoseconds regime. We verified that the dynamics at intermediate times does not substantially deviate from what is obtained with larger cutoffs.
In the presence of circularly-polarized light, edge states are localized either at the upper or at the lower GNR edge, and are characterized by unidirectional and opposite values of the velocity vector field. This is shown in Fig. \[fig1:edge\_velocity\] where we report the time-averaged velocity $\overline{\vec{v}_{\alpha, \vec{k}} (\vec{r}_i)} = \dfrac{1}{T} \int_{0}^{T} \vec{v}_{\alpha, \vec{k}} (\vec{r}_i, t) \ dt $ calculated over the two edge states.
![\[fig1:edge\_velocity\] (color online) Time-averaged velocity plot in real space for the edge states of the GNR. The arrows represent the velocity averaged over a period of the external field for each site, while the color scale is associated with the projection of such vector along the $x$ direction of the nanoribbon. The inset shows a portion of the GNR lattice, and the curved arrow indicates the circular laser polarization. ](fig1.jpg){width="9cm"}
Now, we want to include a defect at one edge. To this aim, we extend the ribbon unit cell and build a supercell of $12 \times 27$ atoms (the ribbon is 12-atoms wide), with periodic boundary conditions along the $x$ axis. We selectively remove atoms at one edge, as shown in the inset of Fig. \[fig2:Low1Curr\], creating a multi-atom vacancy. Gapless edge states characteristic of driven GNRs [@Manghi2017] persist in the presence of this rather strong perturbation and, interestingly, they give rise to the peculiar time-averaged velocity vector field reported in Fig. \[fig2:Low1Curr\]. Indeed, the velocity field does not seem to be globally affected, although the localized multi-vacancy induces a very steep perturbation and the time averaged vector velocity calculated in the state localized at the lower edge perfectly bypasses the defect. This behavior is distinctive of edge states, whereas the velocity vector field calculated in any other (not edge-localized) state is rather different, as shown in Fig. \[fig3:bulkCurr\].
![\[fig2:Low1Curr\] (color online) Time-averaged velocity in real space for an edge state localized in the lower part of the nanoribbon and propagating rightward. A six-atom vacancy is present at the lower edge as shown in the inset, where the bold dots indicate atoms not included in the calculation and where the curved arrow indicates the circular laser polarization. The arrows represent the velocity averaged over a period of the external field for each site, while the color scale is associated with the projection of these vectors on the $x$ axis.](fig2.jpg){width="9cm"}
![\[fig3:bulkCurr\] (color online) Same as Fig. \[fig2:Low1Curr\], but for a state which is not localized at the edge. ](fig3.jpg){width="9cm"}
The real-time dynamics of the same edge state described in terms of hopping probability as given in Eq. (\[probab\]) is shown in Fig. \[fig4:Evo1\] and illustrates even more clearly how an electron injected in this state travels unaffected after circumventing the defect.
![\[fig4:Evo1\] (color online) Snapshots at three different times (from top to bottom: $t = 40 \ T$, $t = 72 \ T$, $t = 97 \ T$) of the evolution of a state localized at the GNR lower edge in the presence of a multi-atom vacancy. The initial state is taken as a superposition of equally weighted edge states with $k \in [1.1,1.3] \ \pi$. The color scale represents the probability density to find the electron on each site. ](fig4.jpg){width="9cm"}
Topologically protected edge states have been proposed for the realization of quantum computing architectures based on the flying-qubit paradigm [@Stace2004; @Giovannetti2008], where electrons in chiral edge channels in the integer quantum Hall effect host and process quantum information. In the above case, a magnetic field orthogonal to the plane of a quantum well, hosting a low-density electron gas, drives the 2D system in the quantum Hall regime. A pattern of split gates creates the path of Hall edge states that can eventually lead to inter-channel interaction and coherent channel mixing [@Neder2007; @Beggi2015]. This can be obtained with local magnetic fields [@Karmakar2011] and a single edge channel can be split by a proper quantum point contact [@Roddaro2003; @Paradiso2011]. Our proposal addresses the formation and control of edge states in GNRs with a high-frequency laser field. Such edge states represent an alternative to the ones induced by a transverse magnetic field in a 2D semiconductor material through Landau quantization. Indeed, a remarkable difference between the two systems is the robustness of the former against local few-atoms defects. As it can be gathered from Fig. \[fig4:Evo1\], the GNR edge state bypasses the perturbation preserving its original character, thus the localized excitation is neither scattered back nor excited to higher energy states. The reason can be traced to the presence of a single edge state in a given edge and to the large separation between the energy of our edge carrier and other available states with the same k vector (about $1$ eV in the system we simulated). On the contrary, Hall edge states in a mesoscopic slab of a semiconductor, e.g. GaAs, are sensitive to sharp potential discontinuities, that may mix different edge channels [@PhysRevB.45.9059] (with different Landau index, though propagating in the same direction) whose energy difference is of the order of $\hbar\omega_c$, i.e. as small as $10$ meV for GaAs in a $5$ T magnetic field.
Dynamics of edge states in driven GNR with potential barrier
============================================================
We have shown that edge states of an ac-driven GNR are indeed robust against lattice defects. We now show how their real-space pattern can be controlled by adding a modulated electrostatic potential. For simplicity, we consider a constant potential barrier crossing the whole width of the GNR and compute the edge-state dynamics in real time and real space for different heights of the barrier. Notice that the inclusion of the potential does not jeopardize the formation of edge states.
{width="18cm"}
As shown in Fig. \[fig5:Vstep1\], for a potential barrier with on-site energy $V = 0.2 \ J$, i.e. much lower than the inter-site hopping parameter $J$, the particle is perfectly transmitted, fully maintaining its edge character. On the contrary, for a potential barrier substantially higher than the hopping term ($V = 2 J$) the electron localized on one edge is reflected onto the other GNR edge. Although such a steep variation in the value of $V$ is beyond current technology, typically based on external gates leading to a smooth potential modulation, the tailoring of the edge channel path induced by space-dependent on-site energies is remarkable. The same effect can be obtained e.g. by removing a short part of the GNR, and it is also expected to be present for smoother barriers. Simulations of a realistic case of the latter type would require an exceedingly large supercell and are beyond the scope of the present work. The possibility for a particle to be reflected from the lower to the upper edge following the barrier boundary is related again to the robustness of these topologically protected edge states. In particular, a systematic calculation of reflectivity has been performed as a function of the height of the potential barrier, for three different widths (see Fig. \[fig6:Refl\]). For all the widths considered, a peak in reflectivity around $V=0.8\ J$ appears, then a minimum occurs at around $V=1.2\ J$ before perfect reflection is reached for $V \geq 2\ J$. This non-monotonic behavior is similar to the transmission of a one-dimensional barrier, with a notable difference. On the one hand, in the trivial quantum mechanical problem of a square barrier in one dimension, resonance dips occur for particle energies slightly higher than the barrier ($E \gtrsim $ V), i.e. when the wave function has a plane-wave character in the barrier region. On the other hand, in our system, we have a dip in the reflection coefficient when the potential barrier is higher than the hopping energy, i.e. when the wave function decays exponentially inside the barrier along $x$. The observed drop in reflectivity associated with evanescent waves inside the barrier is evocative of a Klein tunneling mechanism [@Katsnelson2006; @Allain2011].
![\[fig6:Refl\] (color online) Reflectivity as a function of the ratio between the height of the potential barrier $V$ and the hopping energy $J$ for different widths [w]{} of the barrier.](fig6.jpg){width="9cm"}
All in all, a greater control shows up in the opportunity of having either transmitted particles on the same edge and/or reflected ones on the opposite edge of the nanoribbon. It is also important to notice that this behavior is persistent for longer potential steps or wider GNR’s.
Conclusion and outlook
======================
In conclusion, an efficient scheme to calculate the time evolution of Floquet states in real time and real space has been developed and applied to demonstrate the topological robustness of edge states in ac-driven graphene nanoribbons. Edge states evolve in time bypassing edge defects undisturbed. Performing the evolution of these states in the presence of a potential barrier, we have shown that it is possible to control the edge carrier dynamics by tailoring the barrier height, in order to switch from a perfect transmission condition to a perfect reflection on the opposite edge of the GNR. This is nontrivial and more intriguing compared to the analogous one-dimensional quantum mechanical problem. This control mechanism can be exploited to realize topologically-protected flying qubits that might parallel similar proposals based on the quantum Hall effect.
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---
abstract: 'Radon transform is widely used in physical and life sciences and one of its major applications is the X-ray computed tomography (X-ray CT), which is significant in modern health examination. The Radon inversion or image reconstruction is challenging due to the potentially defective radon projections. Conventionally, the reconstruction process contains several *ad hoc* stages to approximate the corresponding Radon inversion. Each of the stages is highly dependent on the results of the previous stage. In this paper, we propose a novel unified framework for Radon inversion via deep learning (DL). The Radon inversion can be approximated by the proposed framework with an end-to-end fashion instead of processing step-by-step with multiple stages. For simplicity, the proposed framework is short as iRadonMap (**i**nverse **Radon** transfor**m** **ap**proximation). Specifically, we implement the iRadonMap as an appropriative neural network, of which the architecture can be divided into two segments. In the first segment, a learnable fully-connected filtering layer is used to filter the radon projections along the view-angle direction, which is followed by a learnable sinusoidal back-projection layer to transfer the filtered radon projections into an image. The second segment is a common neural network architecture to further improve the reconstruction performance in the image domain. The iRadonMap is overall optimized by training a large number of generic images from ImageNet database. To evaluate the performance of the iRadonMap, clinical patient data is used. Qualitative results show promising reconstruction performance of the iRadonMap.'
author:
- Ji He
- Jianhua Ma
bibliography:
- 'report.bib'
title: Radon Inversion via Deep Learning
---
Introduction
============
In physical and life sciences, the reconstruction problem, which is to determine the internal structure or some property of the internal structure of an object without having to macroscopically damage the object, is essential. In 1917, Johann Radon of Austria presented a solution to the reconstruction problem with Radon transform and the corresponding inversion formula[@radon20051]. Up to now, Radon transform has been used in various applications, including the removal of multiple reflections[@thorson1985velocity][@sacchi1995high], regional and global seismology[@gorman1999wave][@gu2009mantle], astrophysics[@bracewell1956strip], and computed tomography (CT)[@cormack1963representation]. Among these applications, X-ray CT is one of the most important branches of Radon transform. The X-ray CT, which has great advantages in various pathological diagnoses, is an indispensible imaging modality in modern hospitals and clinics.
Radon inversion or the so-called image reconstruction in X-ray CT is challenging because the radon projections acquired with physical sensors or detectors are probably defective and with noise. Conventionally, the reconstruction process contains multiple *ad hoc* stages to approximate the corresponding Radon inversion. Each of these stages highly depends on the processing results of its previous stage. The processing chain of X-ray CT reconstruction includes the logarithm transformation, scatter correction, beam hardening correction, partial volume effect correction, image reconstruction, and image postprocessing. It is expected that the reconstruction performance will be largely affected if any one of these stages suffer from slight compromise.
In order to obtain promising reconstruction performance for X-ray CT, various advanced algorithms separately addressing the multiple *ad hoc* stages have been proposed. Among these stages, the image reconstruction is one of the most vigour research fields. Regarding the image reconstruction, the most popular method in commercial X-ray CT scanner is filtered back-projection (FBP) algorithm. With a simple filtering operation and back-projection, the FBP algorithm can fleetly reconstruct CT images. The FBP algorithm can achieve promising reconstruction performance if high-quality radon projections could be obtained with adequate X-ray radiation exposure. However, in low-dose X-ray CT (LdCT) imaging, the reconstruction results of FBP algorithm will suffer from severe noise-induced artifacts. To obtain promising reconstruction performance, one can design more elaborate filtering operations for FBP algorithm, which is often not an easy task. In addition, the off-the-shelf denoising algorithms can be adopted to postprocess the reconstructed CT images. These denoising algorithms can improve the image quality to some extent, but the severe streak-like artifacts might be preserved as intrinsic textures.
Currently, model-based iterative reconstruction (MBIR) algorithms are widely studied for LdCT imaging. Usually, the MBIR algorithms simultaneously model both the noise properties of the radon projections and some prior knowledge of the radon projections and/or objective image, which results in a cost function comprised of two terms, namely, a data-fidelity term and a penalty or prior term. Most of the MBIR algorithms assume that two sources of noise, namely, the intrinsic quanta noise and electronic noise, are responsible to the degradation of the radon projections. This results in a similar data-fidelity term for most of the MBIR algorithms. By designing different prior knowledge for the penalty term, these MBIR algorithms can obtain promising reconstruction performances to different extent. Nevertheless, three drawbacks are accompanied with these MBIR algorithms. First, proper distributions have to be assumed to model the noise in the radon projections. Currently, the Poisson and Gaussian distributions are widely used for the intrinsic quanta noise and electronic noise, respectively. However, the real noise distribution is much more complicate than the simple Poisson and Gaussian distributions. Second, in order to obtain promising reconstruction performance, proper prior knowledge has to be designed to constrain solution space of the MBIR algorithms, which is often nontrivial. Third, these MBIR algorithms often involve several projections and back-projections during the optimizations, which will be more time-consuming than the FBP algorithm.
In addition to the respective drawbacks of the FBP and MBIR algorithms, the commonality of these two kinds of algorithms is that they both rely on a priorly calculated projection operator, which largely determines the precision of the final reconstruction. Recently, data-driven image reconstruction is receiving more attention. In [@zhu2018image], Zhu *et al*. proposed a data-driven supervised learning for image reconstruction, which is named as automated transform by manifold approximation (AUTOMAP). Without incorporating a priorly calculated projection operator, the AUTOMAP directly learns a mapping between the sensor and the image domain, which is emerged from an appropriate corpus of training data. Compared to FBP and MBIR algorithms, the AUTOMAP is a unified framework for image reconstruction, which can simultaneously consider the separate *ad hoc* stages in image reconstruction to optimize the final reconstruction performance. However, the AUTOMAP might be unsuitable in clinical CT applications with normal dimension, i.e., $512\times 512$, because of inherence defect of the network architecture of AUTOMAP. The fully-connected layers adopted in the AUTOMAP require a huge amount of computing resource in order to train a model with reasonable reconstruction performance.
Inspired by the concept of unified image reconstruction of the AUTOMAP, in this work we propose a unified framework for Radon inversion that overcomes the drawbacks of the AUTOMAP in reconstructing images from radon projections with large dimensions. For simplicity, the proposed framework is short as RAINAP (Radon inversion approximation). The RAINAP is implemented with an appropriative neural network that based on the corresponding Radon transform. Specifically, the RAINAP can be divided into two segments. In the first segment, we parameterize the filtering operation and back-projection of the FBP algorithm with two learnable appropriative network layers to perform the domain transform, i.e., from radon projections to image domain. In the second segment, we use a convolutional neural network (CNN) to further refine the reconstruction performance. The two segments are overall optimized using a large number of training data to guarantee a promising reconstruction performance.
METHODOLOGY
===========
Theoretical Foundation
----------------------
In this section, we present the theoretical foundation for the network architecture of the iRadonMap. Without loss of generality, the Radon transform discussed in this work is with a parallel-beam X-ray CT imaging geometry. The network architecture of the iRadonMap for X-ray CT imaging geometries of fan-beam and cone-beam can be deduced similarly.
For an arbitrary 2D object $f(x,y)$, the corresponding Radon transform with a parallel-beam X-ray CT imaging geometry can be written as follows: $$\small
\label{Eq_1}
p(s,\theta)=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}{f(x,y)\delta(xcos\theta+ysin\theta-s)dxdy}.$$ Here, $p(s,\theta)$ denotes a radon projection of $f(x,y)$ at a certain view-angle $\theta$. $\delta(\cdot)$ is a Dirac function. $s$ is the position of a detector unit relative to the geometry center of the X-ray imaging system. Thus, $f(x,y)\delta(xcos\theta+ysin\theta-s)$ presents the intersection of an X-ray beam with $f(x, y)$. The radon projections are usually collected within rotation interval of 180 degree, namely, $0\le\theta\le \pi$.
To reconstruct $f(x, y)$, one can use FBP algorithm. Let’s denote the two-dimensional Fourier transform of $f(x,y)$ as $F(\omega,\theta)$ and the one-dimensional Fourier transform of $p(s,\theta)$ as $P(\omega,\theta)$. According to Inverse Fourier transform and Central slice theorem, $f(x,y)$ can be expressed as follows: $$\small
\label{Eq_2}
\begin{array}{l}
f(x,y)=\int\limits_{0}^{2\pi}\int\limits_{0}^{\infty}{F(\omega,\theta)e^{2{\pi}i\omega(xcos\theta+ysin\theta)}{\omega}d{\omega}d\theta}
\\[4mm]
\qquad\quad\,=\int\limits_{0}^{\pi}\int\limits_{-\infty}^{\infty}{P(\omega,\theta)|{\omega}|e^{2{\pi}i\omega(xcos\theta+ysin\theta)}d{\omega}d\theta}.
\end{array}$$ Here, $|{\omega}|$ is the transfer function of the ramp filter. Introducing $Q(\omega,\theta)=|{\omega}|P(\omega,\theta)$ and denoting the Inverse Fourier transform of $Q(\omega,\theta)$ as $q(s, \theta)$, the reconstruction of an arbitrary 2D object $f(x,y)$ via FBP algorithm can be obtained with the following two steps:
1\) apply a ramp filtering operation to $p(s,\theta)$ with respect to the variable $s$ in the Fourier domain, as follows: $$\small
\label{Eq_3}
q(s, \theta)=FT^{-1}\{|{\omega}|{\cdot}FT\{p(s,\theta)\}\};$$ 2) back-project $q(s, \theta)$ to obtain the reconstruction, as follows: $$\small
\label{Eq_4}
f(x,y)=\int\limits_{0}^{\pi}{q(s, \theta)|_{s=xcos\theta+ysin\theta}d\theta}.$$ Here, $s=xcos\theta+ysin\theta$ denotes a sinusoidal track, from which the radon projection points are related to the reconstructed point $(x,y)$. The reconstruction by the FBP algorithm might suffer from noise-induced artifacts due to the degradation of the radon projections. In order to obtain a promising reconstruction, one can apply some off-the-shelf restoration algorithms in the radon projections and/or image domain to further improve the FBP results. This indicates that the Radon inversion can be approximated by several successive operations, each of which is highly dependent on the results of the previous operation. Recently, Zhu *et al*. proposed a unified framework for image reconstruction, namely, AUTOMAP[@zhu2018image], which is also suitable for Radon inversion. With it, one can reconstruct an image from the radon projections in an end-to-end fashion instead of step-by-step with multiple stages. However, it can be difficult to implement for large-size CT images (e.g., $512\times 512$), due to the stack of fully-connected layers in the AUTOMAP, which might cost huge amount of storages and computations. In order to address this issue, in this work we propose a novel unified framework (iRadonMap) to approximate the Radon inversion of large-size CT image with far less parameters than the AUTOMAP.
\
Network Architecture
--------------------
As shown in Fig. \[Fig\_1\], the proposed iRadonMap can be divided into two segments. In the first segment, we use two learnable appropriative network layers to simulate the filtering operation and back-projection in FBP algorithm, respectively. The ramp filtering operation in the FBP algorithm is applied to the radon projections along the view-angle direction, as shown in Eq. (\[Eq\_3\]). In the proposed iRadonMap, we use a learnable fully-connected layer to simulate the filtering operation, which is also applied to the radon projections along the view-angle direction. Regarding the back-projection, we could simply simulate it with another fully-connected operation[@zhu2018image]. However, the fully-connected operation is difficult to implement for Radon inversion of large-size CT images, which involves huge number of parameters and may consume huge amount of computing resources. To address this problem, we use an appropriative network layer to simulate the back-projection. By analysing the back-projection of the FBP algorithm in Eq. (\[Eq\_4\]), it can be observed that a certain reconstruction point only relates to a few radon projected points, which constitute a sinusoid in radon projection domain. This allows us to construct an appropriative learnable operation for back-projection with far fewer parameters than the fully-connected operation. This is denoted as a sinusoidal back-projection layer.
In the second segment, we use a residual CNN to strengthen the reconstruction of the iRadonMap. The residual CNN used in the second segment of the iRadonMap is a fully convolutional network, which adopts the shortcut technique in the ResNet proposed by He *et al*. [@He2015Deep]. Unlike the ResNet, we do not use batch normalization. It is noted that the CNN architecture used in this work for the second segment is not the only possible implementation. Other well-defined architectures, such as autoencoders, U-net can also be adopted.
Network Training
----------------
To obtain promising reconstruction performance, the iRadonMap is overall optimized by minimizing the mean square error (MSE), which is defined as follows: $$\label{Eq_5}
E(\Theta)=\frac{1}{N}\sum\limits_{i=1}^{N}\|\bar x_{i}(\Theta)-x_{i}^{ref}\|_{2}^{2}.$$ Here, $\bar x$ is the final output of the iRadonMap and $x^{ref}$ is the reference image. $N$ is the number of image pairs used for training. $\Theta$ represents the learnable parameters in the iRadonMap. This minimization problem can be solved with various off-the-shelf algorithms, and in this work the RMSProp algorithm (see *http://www.cs.toronto.edu/$\sim$tijmen/csc321/slides/lecture\_slides\_lec6.pdf*) is adopted. The corresponding minibatch size, learning rate, momentum, and weight decay are set to 2, 0.00002, 0.9, and 0.0, respectively. The iRadonMap is implemented on PyTorch deep learning framework [@paszke2017automatic]. The iRadonMap is trained for one week using two NVIDIA Tesla P40 graphics processing units (GPUs) with 24 GB memory capacity each.
\
Datasets
--------
The training dataset of the iRadonMap is consisted of a large number of generic images from ImageNet[@deng2009imagenet]. In this work, we collected 62,899 RGB color images. Similar to [@zhu2018image], the Y-channel luminance of the RGB color images were extracted to form greyscale intensity images. Then, the greyscale intensity images were cropped to the central $512\times 512$ pixels. The radon projections with size of $1160\times 736$ (i.e., 1160 views with 736 detector bins each) of each image is generated with the parallel geometry in the ASTRA Toolbox[@Van2016Fast]. To summarize, in this work a radon projection map with size of $1160\times 736$ is feed into the iRadonMap, which outputs a reconstructed image with size of $512\times 512$. The testing dataset of the iRadonMap is consisted of real clinical patient data, which are provided and authorized by the Mayo Clinic. To demonstrate generalizability of the iRadonMap, no clinical patient data was involved in the training phase. We only test network performance with the clinical patient data.
RESULTS
=======
In practice, different iRadonMap models should be trained for Radon transforms with different imaging geometries. In the preliminary experiments, we study the same parallel imaging geometry but with different number of view-angles, namely, 1160, 580, 290, 145, 72 views. The corresponding results are presented in Fig. \[Fig\_2\]. Specifically, the results of different views are showed in rows from up to down, respectively. The columns from left to right show the Reference, FBP, iRadonMap, different image between the Reference and FBP, and different image between the Reference and iRadonMap, respectively. By comparing the reconstruction results of the iRadonMap and FBP in row 1 to row 3, we can see that the performance of iRadonMap is similar to FBP, which indicates a robust Radon inversion performance of the proposed iRadonMap. Moreover, by comparing the different images in columns 4 and 5, it can be observed that the iRadonMap can preserve more edge details than the FBP. In row 4 and 5, it can be observed that the FBP algorithm suffers from severe streak artifacts with sparse view-angles radon projection data. On the contrary, the reconstruction performance of the iRadonMap is promising with less streak artifacts, compared to that of the FBP.
CONCLUSION
==========
In this paper, we propose a novel framework for Radon inversion via deep learning, namely, iRadonMap. The iRadonMap is an appropriative neural network that consists of two segments. The first segment contains a learnable fully-connected filtering layer and a learnable sinusoidal back-projection layer. The second segment is a common CNN architecture. The preliminary results with clinical data show that the iRadonMap can achieve promising performance in terms of qualitative measurements. More experiments are undergoing.
|
---
abstract: 'Arecibo spectra of the mainline OH maser emission from U Her over more than a decade show variations of the OH emission over these time scales. These observations are combined with high spatial resolution VLBA maps to investigate the changes in the velocities of the maser components. Global properties of the dust shell, such as accelerations, variations in the pump and shell-wide magnetic field changes are examined as possibilities, and eliminated. A solution to the problem involving plasma turbulence and the local magnetic field is introduced, and the relevant time scales of the turbulence are calculated. The turbulent velocity field causes variations on time scales that are too long (of order centuries), while the turbulent magnetic field causes variations on appropriate time scales of a few years. A line-of-sight model of the turbulence is developed and investigated. The complete exploration of this solution requires extensive theoretical and observational work. Possible avenues of investigation of the plasma turbulence model are presented.'
author:
- 'Stacy Palen, John D. Fix'
title: Models of OH Maser Variations in U Her
---
Introduction
============
Radio observations of the OH maser emission from OH/IR stars have often been used to investigate properties of the dust shell, such as its shape (Chapman, Cohen and Saikia 1991, Alcock and Ross 1986), its density structure (MacLow 1996), and its history (Chapman, Cohen and Saikia 1991). With a few exceptions, the maser emission from these stars has not been monitored over decade time scales, in part because the study of maser emission itself is only a few decades old. U Orionis has been studied extensively by many groups since 1972, when an unusual flare in the 1612 emission occurred. In particular, in 1991, Chapman, Cohen and Saikia reported results of a monitoring program of U Ori lasting 6 years. Observing programs spanning decades should prove interesting, because the time scale for gas to cross the maser emitting shell is of order 10 years. U Herculis is a particularly good candidate for maser studies, since it is relatively close ( 385 pc (Chapman, et al. 1994)) and has strong OH maser emission, with a peak maser flux that varies between 3 and 20 Jy in the 1665 and 1667 MHz lines. In 1977, Fix (1999) began a monitoring program of U Her and 11 other OH/IR stars at the Arecibo Radio Telescope. His U Her data was indispensible to this project.
Section \[observations\] describes the Arecibo Radio Telescope (Arecibo) [^1] and Very Long Baseline Array (VLBA) [^2] observations of U Her, as well as the data reduction and analysis carried out prior to modeling. Section 3 explains the need for and the construction of simulated sets of maser components. We explore several explanations for the observed variations in Section \[bulk\], and find that overall changes in the properties of the maser emitting shell do not fully explain the observed variations in the maser velocities. A plasma turbulence model is presented in Section \[plasmaturb\], and shown to be a promising line of inquiry. Section \[discussion\] contains a summary and discussion of these results, as well as suggestions for future work.
Observations, Analysis, and Interpretations {#observations}
===========================================
Observations
------------
Observations of U Her were obtained in both maser main lines (1665 and 1667 MHz), and both right circular polarization (RCP) and left circular polarization (LCP). The total bandpass of all observations was 25 km/s ($\approx$ 488 kHz), although the spectral resolution varied with instrument as described below.
Spectra of the main line maser emission from U Her were obtained at Arecibo at four epochs: 1977.866 (henceforth referred to as 1977), 1979.151 (1979a), 1979.953 (1979b), and 1992.384 (1992). The spectral resolution at most epochs was 0.05 km/s. At 1667 MHz, the 1979a data were of insufficient spectral resolution, and so are not included in the full discussion, but reserved until section 4.1.
On 1995.512 (1995), we used the VLBA, supplemented by one antenna of the Very Large Array (VLA) to map the OH emission. Approximately 3 hours of data were obtained on the target source. The data were correlated using the VLBA correlator in Socorro, NM. The spectral resolution was 0.1 km/s. The beam size was approximately 10$\times$4 mas. 3C273 was used as a flux calibrator, and 3C345 was chosen as a phase calibrator. The *a priori* calibration of these VLBA data was carried out using the spectral-line calibration methods described in the AIPS Cookbook in Chapters 4, 8 and 9. Remaining errors were removed using a loop of self-calibration and CLEAN algorithms.
The red-shifted portion of the 1667 MHz line (from the back of the shell) was too faint to be mapped with the VLBA. Also in 1667 MHz, the VLA antenna recorded interference. The consequent loss of short telescope spacings reduced the sensitivity of the 1667 MHz maps to diffuse emission at angular scales longer than 40 mas. Spectra were created from the VLBA single antenna data. These spectra were significantly noisier than the Arecibo data, since the sensitivity of the single antenna can not compare to Arecibo. This is evident in Figure \[spectra\].
Analysis
--------
Each spectrum was deconstructed into a series of Gaussian components using a least-squares fitting routine. Between 19 and 30 components were required to adequately characterize each spectrum.
The overall maser emitting region was resolved into a number of small maser components as shown in Figure \[maps\]. This “spotty” appearance was evident in all the maps, in both polarizations, and both main lines. The AIPS task SAD was used to find the maser components in the individual maps. The integrated amplitudes of these maser components were then grouped by location across the channels to create a spectrum for each component. This resulted in a set of spatially resolved components (“spatial components”) each with an associated spectrum.
Interpretations
---------------
Figure \[spectra\] shows the dramatic changes in the Arecibo spectra between 1977 and 1992. The changes in the spectra can not be described as simple amplitude variations of the entire line, as would be produced in response to variations in the intensity of the central star.
Only 7 of the 80 maser spots in the maps were spatially coincident, indicating that the individual maser spots are highly polarized, possibly via the Cook mechanism (Cook 1975). The Cook mechanism balances velocity gradients and magnetic field gradients so that only one polarization is amplified. In contrast, the spectra are not highly polarized, implying that an annular region of approximately constant projected outflow velocity may have several components. These two observations together indicate that it is likely that an apparent Zeeman pair in a spectrum is actually a pair of spatially distinct Cook components.
The spectra of the spatial components were compared with the Gaussian components derived directly from fitting Gaussian components to the VLBA spectra. In the case of the 1665 MHz data, 87% of the Gaussian components could be identified with a particular spatial component. 8% of the Gaussian components could not be identified in the maps. The amplitudes of these components were small. 5% had more than one spectral component that could be identified with one spatial location. For the 1667 MHz observations, only 54% of the Gaussian components could be identified with the spatial components. This is probably due to the decreased flux in 1667 MHz, and the decreased sensitivity at larger size scales. These results imply that a component by component analysis of the Arecibo spectra yields variations of real, physically distinct components.
Simulated Data
==============
In order to test whether model results are meaningful, 25 sets of simulated components were created. Each set of simulated components consists of a number of velocities and associated errors. Note that a spectrum was *not* produced by this process. The components have neither width nor amplitude, only velocity. This is the primary parameter of interest in the models below.
All of the relevant information was created using the pseudo-random number generator distributed with ANSI-C. The generator was seeded with a number between 1 and 100. The generator creates a long list of random numbers, from which we created velocities, errors, and numbers of components.
The first step was to choose the number of components in a data set. Since the real data sets contained between 19 and 30 components each, the first 25 random numbers were transformed to values between 19 and 30.
The velocities of the components in a data set were found by transforming the next 25 (say) random numbers into velocities ranging between and , to simulate the range of velocities found in the real spectra. Each simulated data set was created from different lists of random numbers.
We also created “errors” in the simulated components. The errors in the real components ranged between 0.001 and 0.02 km/s, depending roughly on the amplitude of the component, but independent of the velocity. The “errors” in the simulated components were created by transforming the random numbers to numbers between 0.001 and 0.02 km/s. The lists of simulated velocities and simulated errors were collated so that for each simulated set of components, there were between 19 and 30 velocities and associated errors.
Models Involving Overall Properties of the Shell {#bulk}
================================================
There are at least three ways in which the maser components may vary due to changes in the overall properties of the maser shell. The first is that the masers may be moving outward at constant velocity, through regions of varying pump intensity. In this case, components will remain at the same velocity, while changing in amplitude. The second possibility is an acceleration of the shell, either radial or rotational, which changes the velocities of the maser components. The third possibility is a change in the overall magnetic field, which can be observed in changes in Zeeman split components, in which the members of an LCP/RCP pair will change velocity in opposite directions, either moving closer together along the velocity axis, or farther apart.
All of the models involving bulk properties of the shell were investigated using the following algorithm. An initial epoch was chosen, then the model was propagated forward in time, modifying each component of the initial epoch accordingly. For example, if the model predicts that between 1977 and 1979, the components will shift redward in velocity by 0.25 km/s, then 0.25 was subtracted from the velocity of each 1977 component. These modified components were then compared to the 1979 components to look for matches. A match was found when the difference in the predicted and actual velocities was less than or equal to twice the sum of their error bars.
In order to understand the meaning of the statistics, the model was also applied to pairs of simulated data, to compare with the number of matches expected purely by chance. In order for a model to be considered successful, it must accurately predict the changes in the central velocities of the components from one epoch to the next. The percentage of components matched in the real data must be significantly larger than the percentage of components matched in the simulated data. In general, there were 24 comparisons of actual components (2 frequencies, 2 polarizations, and 4 epochs, combined pairwise by epoch), and 51 comparisons of simulated components.
The parameter space for each of the models was completely explored. Each parameter in each model was varied from a theoretical lower bound to a theoretical upper bound, in a series of incremental steps. For each value of these parameters, the number of matches between epochs and between simulated component sets was recalculated. The step size was always carefully chosen so that the model predictions would be adequately sampled. This is explained further for each model below. The number of parameters in each model was small enough that this method of investigation was not overly costly in terms of CPU time.
Changes in the Pump
-------------------
If the masing components move outward at constant velocity, then each individual component remains at the same velocity (projected along the line of sight) while increasing or decreasing in amplitude. Spectral components with have the same velocity in more than one epoch. The results of this model are shown in Figure \[histogram\], and expressed as a percentage of the total number of matching components that might have been found. These two binned distributions yield a reduced chi-square of 1.4, indicating that the distributions are statistically identical. This model fails to fit real data better than it fits simulated data, implying that the matches in the real data are consistent with chance.
Radial Acceleration
-------------------
A radial acceleration changes the velocity of the material as it moves out through the shell. Because of projection effects, the radial acceleration will be most apparent at the front and the rear of the shell, where the expansion is most nearly along the line of sight. A radial acceleration at the limb of the shell will not cause an observable change in the velocity of a maser component located there. The emission from the front and back of the shell is located at the outside of the line profile, and so we expect to see the greatest changes due to radial acceleration in these regions of the line.
The change in the projected velocity due to a radial acceleration is given by $$\Delta v_{pr}=\Delta v_{exp} \cos\theta$$ where $\Delta v_{exp}$ is the total change in the expansion velocity due to acceleration, and $\theta$ is the angle between the maser and the projected center of the shell. The cosine of the angle is equal to the ratio of the observed velocity to the expansion velocity, $v_{obs}/v_{exp}$, so that equation (1) becomes $$\Delta v_{pr}=\Delta v_{exp}\frac{v_{obs}}{v_{exp}}$$ The component will be shifted towards the outside of the line for a positive radial acceleration, and towards the center of the line for a negative acceleration. Red-shifted and blue-shifted lines will move in opposite directions, but both will move towards or away from the center if they are affected by the same acceleration.
The projected velocity at a later epoch, $v_{r,b}$, is related to the projected velocity at an earlier epoch, $V_{obs}$ by $$v_{r,b}=v_{obs}(1\pm\frac{\Delta v_{exp}}{v_{exp}})$$ where $v_{r,b}$ indicates the red- or blue-shifted velocity, and the direction of the acceleration is given by the sign of $\Delta v_{exp}$.
From the largest and smallest velocities of the maser components of U Her, the expansion velocity is between 6 and 8 km/s. This expansion velocity is expected to remain approximately constant since the dust has already condensed (Fix and Cobb 1987). With this range of values for the expansion velocity, the theoretically expected values for $\Delta v_{exp}/v_{exp}$ are less than 0.2 since the expansion velocity changes by at most a few tenths of a km/s per decade. Varying the range of $\Delta v_{exp}/v_{exp}$ between -2 and 2 easily covers the possible range of values, and is in fact much larger than necessary, as it allows even the innermost components to be completely shifted out of the line profile. A step size of 0.0005 was used. The errors in the centroids of the components were greater than 0.001. This step size is approximately 1/8 of twice the sum of the error bars for a pair of components. This is the criterion for finding components at the same velocity, and so this step size should adequately sample the radial acceleration model.
The average percentage of components which could be identified between epochs was 39% for the real data, and 35% for the simulated data. The reduced chi-square of the two distributions (real and simulated data) of these percentages is 1.9, consistent with identical distributions for these small number statistics.
One of the assumptions of this model is that the radial acceleration is spherically symmetric (although the shell need not be). A consequence of this assumption is that the front and the rear of the shell undergo the same acceleration. This constraint was relaxed by allowing the front and back of the shell to have different accelerations. In this case, the reduced chi-square of the distributions of percentages of matched components for real and simulated data was found to be 0.46. This is still consistent with all of the matches in the actual data being found purely by chance.
A second consequence of this assumption is that many components undergo the same acceleration. If this is not the case, this model does not apply, and we must try to model the components individually, and look for changes in the velocities which are linear in time (see section 4.5).
Magnetic Field
--------------
The maser emission from OH/IR stars is often polarized. This indicates that magnetic fields are present which are strong enough to Zeeman split the OH lines. In order to investigate changes in the global magnetic field, we must first look for Zeeman pairs.
The VLBA maps provide the most compelling evidence of magnetic fields strong enough to Zeeman split the lines. In the 1665 MHz emission, only 7 of 47 maser spots appeared in the same location in both RCP and LCP emission. In 1667 MHz, none of the 33 maser components appeared in both polarizations. This implies that at least 80% of the components are polarized, possibly via the Cook mechanism (Cook 1975). As mentioned above, this implies that we are unlikely to find a pair of Zeeman components in the spectra which are real. The probability that an apparent Zeeman pair is actually a pair of spatially distinct Cook components is high.
The importance of the Cook mechanism can be verified directly from the spectral data. The main-line maser emission is thought to arise in a narrow region of the shell (see for example, Collison and Nedoluha, 1994). If the magnetic field is constant (or approximately so) throughout the region, then the Zeeman splitting will also remain constant throughout the shell. In this case, the Cook mechanism may be ignored, since there is no magnetic field gradient. There should be a constant splitting of each of the LCP/RCP pairs, since each component experiences the same magnetic field. If the magnetic field gradient is important, few pairs of LCP and RCP components will be found with the same Zeeman splitting.
The number of Zeeman pairs was calculated by choosing a component in one polarization, then searching in the other polarization for a component offset by a prescribed Zeeman splitting. The range of the splitting was -2 to 2 km/s, and the step size was 0.001 km/s. These parameters were chosen to encompass the entire range of possible values, and to provide a step size smaller than the errors in the velocities of the components. The average percentage of matched components for the actual data was 34.5%, while the average percentage of matched components for the simulated data was 32.5%. When the 1995 data, with its larger spectral resolution, was removed from consideration, the average percentage of matched components in the actual data dropped to 27%. Both of these results are consistent with the results from the simulated data.
Linear and Quadratic Changes in Time
------------------------------------
The investigations described above lead us to conclude that we cannot model the changes in the velocities of the components as global changes caused by constant velocity motions, radial or rotational accelerations, or global magnetic fields. However, it may be possible that all components, while subject to varying physical conditions, are still constrained to change in the same way over time. For example, all components could be subject to an acceleration that varies between one component and another. Models of this type will not give any detailed information about the mechanism of the change, but they will tell us whether the same mechanism operates on each maser feature throughout the observed time period. We note that if the variations are caused by a combination of the above scenarios, then this model should be able to characterize those variations.
The number of 1977 components was in some cases (e.g. 1665 RCP) greater than the number of components in the 1979b or 1992 data. There may be redundancies in the alignment of components, so that two or more 1977 components may be aligned with one of the 1979b or 1992 components. Since it is not possible to determine which of these is the “correct” pairing, we simply counted the number of 1977 data points for which a match may be made. Because of this, the number of successful projections was occasionally higher than the smallest number of components in the three or four epochs.
The four epoch case was completely consistent with the simulated data, implying that the matched components are likely to have been found by chance. The three epoch case is less tightly constrained, and more matching components were found. However, the largest percentage of actual matched components was no more than 5% higher than the highest percentage of simulated matched components. These results are not inconsistent with the results expected purely by chance.
For completeness, we investigated an acceleration which is variable in time. An acceleration which changes over time will produce a velocity that varies quadratically in time. For example, if the radiation pressure changes, the force on the gas will change, and the acceleration of the gas particles will change. The possibility of a quadratic variation of the velocity was investigated, and it was found that the number of free parameters in this model is so large that a projected component can be found in the third or fourth epoch nearly 100% of the time for both real and simulated data.
Plasma Turbulence {#plasmaturb}
=================
We have now shown that the changes in the velocity and polarization structure of the OH main line maser emission from U Her over decade time scales can not be completely explained by overall properties of the shell, such as accelerations, or global magnetic field changes. We are driven to consider explanations which do not involve properties of the shell as a whole, but rather can affect each maser component differently. We know that the magnetic field must be important in these masers because nearly all of the spatial and spectral components are polarized. The masers arise in regions with charge-carrying dust grains moving at a drift velocity which is of order tens of km/s (Collison and Fix 1992). This is about 100 times the thermal speed of the gas derived from the full widths of the spectral components. There is no direct evidence that the dust grains carry charge, although it has long been assumed that they do. It seems plausible that the grains are collisionally charged by interacting with the ionized gas and its electrons, and theoretical models including charged dust match well with infrared observations (Zubko 1998). Plasma turbulence may arise, perhaps via an instability such as those described in plasma kinetic theory. We investigate whether this turbulence could produce changes on the observed time scales.
Time Scales of Turbulence
-------------------------
In the data considered so far, the shortest time scale investigated was a bit more than two years. A better constraint on the time scales may be found by comparing observations closer in time. A second set of observations of U Her were taken on 1979.151 (1979a), about ten months before the 1979.953 (1979b) data set used to investigate the overall variations of the shell. Only the 1665 MHz observations were of high enough spectral resolution to compare with the rest of the data, and so the 1979a observations were not included in the larger study. The 1979a data was subjected to the models described in section 4. These models tended to fit the 1979a and 1979b combination of epochs better than any other pair of epochs, but still, the percentage of components which could be identified across epochs was consistent with, or only marginally better (1-2%) than in the simulated components. Interestingly, even the constant velocity model failed to fit these epochs which were separated by less than one year. This implies that the time scales of the variations of the individual components may be of order months.
The time scale for a turbulent eddy of size L to rearrange is $$\tau=\frac{L}{\delta v_t}$$ where $\delta v_t$ is the “turbulence velocity”, which is usually less than or of the order of the thermal velocity, or the Alfvén speed. In the case of U Her, we can calculate an upper bound to this length scale since the masers are partially resolved. From the period-luminosity relation for Mira variables, the distance to U Her is estimated to be 385 pc (Chapman et al 1994). Feast (1989) quotes an error in the derived magnitude using this method of 0.14. This yields an error in the distance of 26 pc (7%). From the maps, the average angular size of the maser emission regions is about 20 mas. This gives a linear size of $\approx 1 \times 10^{14}$ cm for the regions which produce the maser emission. If the turbulence velocity is the thermal velocity of the gas ($\approx 0.2$km/s=$2 \times 10^{4}$cm/s), then the time scale for complete change of the turbulent medium is about 200 years. This is much longer than the time scale of the variations. The simple turbulent motions of the gas cannot explain variations of the maser components on the time scales observed.
Suppose instead that the magnetic field variations drive the appearance and disappearance of the maser spots. This possibility is supported by the observed importance of the Cook mechanism. If the magnetic field is tied to the charged dust particles, then turbulent features will cross the maser emitting region at the dust drift velocity, given by Collison and Fix (1992) as $\approx20$ km/s. This leads to a time scale of a bit less than two years. While it is not certain whether the dust grains and the magnetic fields travel together, this seems like a reasonable assumption, since the $\approx 1$ mGauss fields inferred from the Zeeman splitting cannot be generated at the central star, and thus must be generated [*in situ*]{} by the motion of the charged particles. Note that the magnetic field gradient is the important quantity in the Cook mechanism, not the magnitude of the magnetic field, and so this calculation is an upper bound on the time scale, since the turbulent magnetic field may have large enough gradients for the Cook mechanism to be rendered ineffective on length and time scales much smaller than the ones calculated here.
Plasma Turbulence Along the Line of Sight: Theory and Model Description
-----------------------------------------------------------------------
One way to investigate the validity of a plasma turbulence model is to investigate whether plasma turbulence can cause bright, polarized components with the same probability as that inferred from the filling factor of the masers in the radiation shell. For example, if masers are produced in 20% of the shell, then we would like to know if plasma turbulence produces bright, polarized components in 20% of simulated lines of sight.
The filling factor of the observed maser emitting regions was found by summing the areas of the maser components, and dividing by the area of the shell projected on the sky. The projected shell was assumed circular, with a diameter of $\approx 0.5$ arcseconds, as indicated by the distribution of the maser emission in the maps. For the 1665 MHz maps, the filling factor of all the components at all velocities is 0.11. For the 1667 MHz maps, the filling factor is 0.05. However, since the back portion of the shell was too faint to map in 1667 MHz emission, the filling factor for the 1667 MHz emission may be as much as 0.1. The filling factor to which the following model results were compared was 0.1.
We can reduce the plasma turbulence problem to one dimension by considering the maser as a line integral along the line of sight. In each incremental path length $ds$, a given velocity $v$, and magnetic field $b$ are present. These values of $v$ and $b$ are the small-scale turbulent velocities and magnetic fields, not the large-scale expansion velocity, or overall magnetic field. We can calculate an effective velocity of emission from this incremental path length from the Doppler-shifted frequencies and the Zeeman splitting of the line due to the magnetic field $$v_{eff}=v \pm \frac{\gamma c}{f}B \label{eq:effvel}$$ where $c$ is the speed of light, $f$ is the frequency in MHz of the transition at rest (1665 or 1667), and $\gamma$ is the frequency splitting due to the Zeeman effect. The sign indicates RCP or LCP shifts. In order for amplification to occur, we must have a large number of path lengths $ds$ with the same effective velocity. Collecting the effective velocities of the path lengths into bins and making a histogram shows the velocity coherence along the line of sight.
In general, each path length will not be saturated. If one of the bins contains more than the number of path lengths required for the maser to become saturated, then the maser will be amplified. A component is considered bright and polarized if the ratio of the maximum bin in RCP to the maximum bin in LCP is greater than 1.3 or less than 0.7. This criterion means simply that the maser component must be at least 30% polarized in order to resemble a Cook polarized component.
MHD turbulence theory shows that the power spectrum of turbulent velocities and magnetic fields is a power-law. In a one dimensional case, the power spectrum is given by $$P(k)=\tilde{A}^2(k)\propto k^{-a},$$ where $\tilde{A}$ is the Fourier transform of a function in velocity space (e.g. the velocity, $v(x)$), and $k$ is the wave number in the Fourier transform space. In the case of Kolmogorov turbulence, the exponent indicating the steepness of the power law, $a$, is 5/3. We adopt this special case as a first approximation of the problem. In our case, however, we may have turbulence on size scales larger than the size of the emitting region, $l_{0}$. These will enter into the power spectrum as a flattening of the slope to a constant near $k$=0. The point of turnover is given by $k_{0}=2\pi/l_{0}$. We can express this modified Kolmogorov power spectrum in functional form as $$P(k) \propto{\left( 1+ \left( \frac{k}{k_0} \right)^2 \right)^{-5/6}}.$$
In order to generate the velocity and magnetic field distribution, we must randomize the power spectrum given by MHD turbulence theory. This guarantees that the general behavior of the velocities and magnetic fields agrees with what is expected from theory. Typically, the power spectrum is randomized by multiplying by a random, complex number $$\tilde{A} (k) = \sqrt{\frac{P(k)}{2}}(D+iF).$$ $D$ and $F$ are both zero mean, Gaussian distributed, unit standard deviation numbers (Spangler 1998). This equation is additionally constrained by the requirement that $A(x)$ (the velocity or magnetic field vector) is real. This requirement will be met when $\tilde{A}^{*}(k)=\tilde{A}(-k)$. By even-odd symmetry, this condition constrains $F$ to be 0 when $\tilde{A}(k)$ is the modified Kolmogorov spectrum being considered here. $A(x)$ is the inverse Fourier transform of $\tilde{A}(k)$ $$A(x)=\int_{-\infty}^{\infty} dk \exp (-2 \pi i k \cdot x) \tilde{A}(k).$$ The velocity is given by $A(x)$. The magnetic field, $b(x)$, was calculated as a linear combination of the velocity, $v(x)$, and a statistically independent function $b_{i}(x)$. The function $b_{i}(x)$ was calculated in the same manner as $v(x)$, using a different amplitude, and a different set of random numbers $D$ in the randomization of the power spectrum. The magnetic field is then given by $$b(x)=\alpha v(x)+(1-\alpha)b_{i}(x),$$ where $\alpha$ is an adjustable parameter, with values between 0 and 1, which indicates the degree of correlation between the magnetic and velocity fields. Plots of the randomized quantities $v(x)$ and $b(x)$ are shown in Figure \[fields\].
Once the velocity and magnetic fields were calculated, the effective velocity, Equation \[eq:effvel\], was calculated, and path lengths grouped in bins by this value. The bin width was chosen to be the spectral resolution of the Arecibo observations (0.05 km/s). The resulting distribution was compared to the criterion for a bright, polarized component. The model was recalculated for many different sets of the random variable $D$, which is equivalent to calculating the model along many lines of sight. The probability of the production of a bright, polarized component was calculated, and compared to the filling factor of the maser components. This probability was calculated for various values of the free parameters $k_{0}$, $\alpha$, $C_{vel}$, and $C_{mag}$.
A subset of the results of this model are shown in Figure \[turbmod\]. The strongest dependence in the model was on the correlation parameter $\alpha$. Cases in which the magnetic and velocity fields were very highly correlated yielded the correct filling factor in only 11% of trials. Generally, the highly correlated case led to too many components, or none, so that either 20-40% of the shell was filled, or there was no maser action. A partial correlation ($\alpha = 0.5$) also gave the correct filling factor in only 11% of trials. For a nearly uncorrelated case, ($\alpha=0.01$), the correct filling factor was obtained 1/3 of the time. This implies that the proper filling factor can be achieved through plasma turbulence, and that the velocity and magnetic fields are most likely uncorrelated. None of the other parameters showed a trend as clear as the dependence on $\alpha$. Overall, 19% of the values of the parameters yielded filling factors that matched the observations.
Discussion
==========
In this study, we have investigated three possible global explanations of the variations of the OH maser emission from U Her. The first possibility was a movement of the maser, at constant velocity, through regions where the pump altered in strength. The second possibility was an acceleration, either radial or rotational, of the shell. The third was a change in the global magnetic field which causes the polarization of the maser components. All three of these possibilities were investigated as bulk properties of the shell, and as properties which changed within the shell, and were peculiar to each maser. None of these possibilities completely characterized the behavior of the main-line maser emission from U Her.
Since the variations cannot be described by overall properties of the shell, and cannot even be modeled to vary in the same manner over time everywhere in the shell, we were driven to seek other explanations. One promising explanation is that plasma turbulence effects alter the magnetic field in the emission region enough to remove the amplification of the masers by the Cook mechanism. If the turbulent magnetic fields are carried by the dust grains, the plasma turbulence model produces changes on time scales of less than one year, which is in agreement with observed time scales of variation. Also, it is possible to create shells with the correct filling factors using this model.
This is a radical departure from the usual school of thought for at least two reasons. The first is that the determining factor in maser emission seems to be the magnetic field gradient, rather than the velocity gradient along the line of sight, since it is the turbulent magnetic field which varies on the observed time scales. The second is that the coherent turbulent “bits” along the line of sight which add up to an amplified, polarized maser component are not necessarily physically contiguous. There may be larger regions of incoherent plasma between the coherent regions, and still the masers will be amplified and polarized. Thus, these masers can not be thought of as entities, or even preferred lines of sight, since the whole line of sight through the dust shell is not necessarily involved in the amplification.
Alcock, C. and R. R. Ross. 1986, , 310, 838 Chapman, Jessica M., R. J. Cohen, and D. J. Saikia. 1991, , 249, 227 Chapman, Jessica M., et al. 1994, , 268, 475 Collison, Alan J. and John D. Fix. 1992, , 390, 191 Collison, Alan J. and Nedoluha, Gerald E. 1994, , 422, 193 Cook, A. H., 1975, , 171, 605 Fix, J. D. and Cobb, Michael L., 1987, , 312, 290 Fix, J. D., 1999, , submitted Mac-Low, Mordecai-Mark. 1996, , 316, 133 Sahai, et al. 1998, , 493, 301 Spangler, S. R. 1998, Physics of Plasmas, preprint Zubko, V. G. 1998, , 295, 109
[^1]: The Arecibo Observatory is part of the National Astronomy and Ionospheric Center, which is operated by Cornell University under a cooperative agreement with the National Science Foundation.
[^2]: The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
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abstract: 'After reviewing the problematic behavior of some previously suggested finite interval spatial operators of the symmetric Riesz type, we create a wish list leading toward a new spatial operator suitable to use in the space-time fractional differential equation of anomalous diffusion when the transport of material is strictly restricted to a bounded domain. Based on recent studies of wall effects, we introduce a new definition of the spatial operator and illustrate its favorable characteristics. We provide two numerical methods to solve the modified space-time fractional differential equation and show particular results illustrating compliance to our established list of requirements, most important to the conservation principle and the second law of thermodynamics.'
address:
- 'Texas A&M University'
- Department of Petroleum Engineering
- Department of Physics
author:
- 'P. P. Valkó'
- 'X. H. Zhang'
title: Finite Domain Anomalous Spreading Consistent with First and Second Law
---
Space-time fractional differential equation ,Caputo derivative ,Riesz derivative ,Laplace transform ,collocation ,finite differences 05.40.-a ,05.60.k
Introduction {#sec1}
============
In practical sense anomalous diffusion can be detected by heavy tails of the resulting density distribution (at a given time) and by the departure from linearly evolving mean-square displacement for an initially concentrated plume [@Schneider89; @Shlesinger93; @Mainardi97]. Such a behavior is well documented for instance for the spreading of contaminants in heterogeneous porous media, where shortcut pathways may be present between two points in space (causing a departure from Fick’s law) and/or particles can be trapped, hindered at various locations (causing memory effects)[@Berkowitz95; @Benson00a; @Benson00b; @Metzler00; @Metzler04]. With a common term, the behavior may be non-local both in space and time [@Zaslavsky02; @Chechkin06; @Zhang09].
Continuous time random walks (CTRW) serve as a small-scale conceptual model for describing anomalous diffusion. Of practical interest is the evolution of the density of a cloud of walkers on the macroscopic scale that is ultimately determined by the statistical characteristics of the jump lengths and waiting times on the microscopic level. The so called Lévy flights are dominated by rare but large jumps and can reproduce the power-law tails of the spatial distributions at a given time. Allowing rare but long waiting times can also lead to marked departure from the scaling law of diffusion. The flux computed on the scale of particle motion, depends on the parameters of the random walk and on the density at the considered time (in the Markovian case) or on the complete density history (in the case of memory effects included). The passage from microscopic to macroscopic scale is performed by letting the characteristic length and time of the particle motion tend to zero [@Montroll65; @Cushman91; @Barkai02]. On unbounded domain, the resulting macroscopic behavior is conveniently described by the space-time fractional diffusion equation [@Gorenflo02]. The passage also results in the generalization of Fick’s law [@Paradisi01; @Neel07].
Brief summary of known results for the unbounded domain case
------------------------------------------------------------
The one-dimensional space-time fractional diffusion equation is written as $$\begin{aligned}
\label{equ1}
\frac{\partial ^\beta}{\partial t^\beta} u(x,t) = a \frac{\partial ^\alpha}{\partial |x|^\alpha} u(x,t)\;\;\; - \infty < x < \infty, \; t > 0\end{aligned}$$ where $a$ is positive constant with dimension $[L^\alpha / T^{\beta}]$. (The dimension of $u$ is $[1/L]$ because it is understood as one dimensional density of a countable quantity.)
The fractional time derivative is taken in the Caputo sense [@Caputo67]. The notation for the operator $\frac{\partial ^\alpha}{\partial |x|^\alpha} $ was introduced by Saichev and Zaslavsky [@Saichev97]. It is understood as the application of the (symmetric) Riesz derivative $\frac{d ^\alpha}{d |x|^\alpha} $ operator with respect to the space variable $x$. The Riesz derivative is defined through the Liouville-Weyl fractional derivatives: $$\begin{aligned}
\frac{d ^\alpha}{d |x|^\alpha} f(x) &=&
\begin{Cases}
-\frac{1}{2 \cos(\pi \alpha /2 )} [ D^\alpha_+ f + D^\alpha_- f ],\;\;\; 0 < \alpha \leq 2,\;\alpha \neq 1\\
-\frac {d}{dx}\textbf{H} (f ; x),\;\;\;\alpha=1\\
- f,\;\;\; \alpha=0
\end{Cases}\end{aligned}$$ where $D^\alpha_\pm$ are called the left- and right Liouville-Weyl derivatives: $$\begin{aligned}
\label{LW1}
D^\alpha_+ &=& \frac{1}{\Gamma(m-\alpha)} \frac{d^m}{dx^m} \int_{-\infty}^{x} \frac{f(\xi)d\xi}{(x-\xi)^{\alpha-m+1}},\;\;\; m = \lceil \alpha \rceil\\
\label{LW2}
D^\alpha_- &=& \frac{(-1)^m}{\Gamma(m-\alpha)} \frac{d^m}{dx^m} \int_{x}^{\infty} \frac{f(\xi)d\xi} {(\xi-x)^{\alpha-m+1}}.\end{aligned}$$ and **H** denotes Hilbert transform. (See a more detailed discussion, for instance, by Chechkin *at al.* [@Chechkin08] )
For the unbounded case the following Cauchy problem can be stated: solve (\[equ1\]) for a given parameter set $\{ 0<\alpha\leq2,\; 0<\beta\leq1,\; a > 0 \}$ augmented with the initial condition $u(x,0)=f_i(x)$, where $f_i$ is a probability density function. The solution of the Cauchy problem can be obtained by the Laplace-Fourier approach, probably first used in this context by Montroll and Weiss [@Montrol73]. The transforms serve double purpose: they provide a better understanding of the operators involved and also lead to the solution for particular cases. In fact, the $\beta$-order Caputo derivative ($0<\beta\leq 1$) is the generalization of the *first derivative* via Laplace transform: $$\begin{aligned}
\textbf{L}\left( \frac{d ^\beta f }{d t^\beta} ; s \right) = s^{\beta-1} \left[ s \textbf{L}(f; s) -f(0)\right]\end{aligned}$$ and the $\alpha$ order Riesz derivative ($0\leq\alpha\leq 2$) is the generalization of the *second derivative* via Fourier transform: $$\begin{aligned}
\textbf{F}\left( \frac{d ^\alpha f }{d |x|^\alpha} ;\omega \right) = -|\omega|^\alpha \textbf{F}(f ;\omega)\end{aligned}$$
The fundamental solution (spreading of an initial Dirac delta) can be obtained by the Laplace-Fourier method and can be given in terms of well investigated special functions. Though various representations are available, their equivalence has been well established [@Mainardi01; @Kilbas06] . The remarkable scaling property of the fundamental solution can be stated as: $$\begin{aligned}
\label{fundsol}
u(x,t) = (a t)^{-\frac{\beta}{\alpha}}\textit{M}\left( \frac{|x|}{(a t)^\frac{\beta}{\alpha} } ; \frac{\beta}{\alpha} \right)\end{aligned}$$ where $\textit{M}$ denotes the Mainardi (or $\textit{M}$) -function given by $$\begin{aligned}
\textit{M}(\xi; \mu)= \frac{1}{2} \sum_{n=0}^\infty \frac{\left(-\xi\right)^n}{n! \; \Gamma\left[-\mu n +(1-\mu)\right]}\end{aligned}$$ (By including the factor $1/2$ we ensure that the integral of (\[fundsol\]) over the $x$-axis is unity.) While computability of the $\textit{M}$-function is far from trivial, it has been basically resolved. For instance, *Mathematica* can calculate it *with any desired accuracy* from its definition above, provided the $\mu$ parameter is passed as a rational fraction. Some known special cases, see e.g. [@Piryatinska05] can be reproduced symbolically with $Mathematica$:
$$\begin{aligned}
\textit{M}(\xi ; 1/2) &=& \frac{1}{2 \sqrt{\pi}} \exp(-\xi^2 /4)\\
\label{M13}
\textit{M}(\xi ; 1/3) &=& \frac{3^{2/3}}{2} Ai(\frac{\xi}{\sqrt[3]{3}}) \\
\textit{M}(\xi ; 2/3) &=& \frac{1}{2 \; 3^{2/3}} \exp(-\frac{2 \xi^3}{27}) \left[
\sqrt[3]{3} \; \xi \; Ai(\frac{ \xi^2}{3 \; 3^{1/3}}) - 3 Ai'(\frac{ \xi^2}{3 \; 3^{1/3}})\right]\end{aligned}$$
where $Ai$ stands for the Airy function and and $Ai'$ for the Airy-prime function.
Anomalous spreading on a finite interval
----------------------------------------
For Fickian diffusion the finite domain model consistent with the first law (conservation principle) is obtained by requiring zero flux at the two endpoints of the considered interval. This translates to the well-known homogenous Neumann boundary conditions for the traditional $\{\beta=1, \; \alpha=2\}$ diffusion equation. Mapping the same physical requirement to mathematically treatable objects for the space-time fractional partial differential equation has turned out to be extremely challenging.
Numerical experimentation followed two complementary approaches: Monte-Carlo (Langevin) simulation of random walk and finite difference approximation to the solution of a space-time fractional differential equation on bounded domain. The first is easier to conduct and always results in physically meaningful results (including conservation, if reflective walls are applied), but those results are difficult to use in a practical sense. The second approach has grown mature during the years. Here we will rely on the well documented “matrix approach” suite by Podlubny *et al.* [@Podlubny09] that provides a general framework to the numerical solution of partial fractional differential equations.
However, the problem is deeper than purely finding an adequate numerical method. In the presence of wall(s) of various properties the spatial operator itself needs to be modified, see e.g. [@Chechkin03; @Metzler07].
In this work we are looking for the solution of the Cauchy problem for the fractional partial differential equation $$\begin{aligned}
\label{problem}
\frac{\partial ^\beta}{\partial t^\beta} u(x,t) = a \frac{\partial ^\alpha}{\partial_{mod} |x|^\alpha} u(x,t)\;,\;\;\;\; 0 \leq x \leq 1 , \;\;\;\; t > 0\end{aligned}$$ when $a > 0$ and the initial condition $u(x,0) = f_i(x), \;\; 0 \leq x \leq 1 $ is a probability density function. The fractional time derivative of order ($0<\beta\leq1$) is in the Caputo sense. We added a subscript “mod” to the spatial operator of order ($0<\alpha\leq2$), because the definitions (\[LW1\]-\[LW2\]) require the extension of the $u(x,t)$ function defined on the interval $[0,1]$ to a left function $f_l(x)$ defined on $(-\infty,1]$ (or, strictly speaking, at least on $(-\infty,x]$,) and to a right function $f_r(x)$ defined on $[0,\infty)$. Notice that these auxiliary functions need not be probability distributions. For brevity, we will call the set of choices we make in creating these extensions a “prescription”. Various prescriptions will give rise to various finite domain Riesz operators and will ultimately define the characteristics of the solution of (\[problem\]).
In the following section (\[sec2\]) we illustrate the problematic behavior of some previously suggested prescriptions and create a “wish list” leading toward a new spatial operator. The subsequent section (\[sec3\]) introduces the new prescription suitable to treat anomalous spreading on a bounded domain and describes its main characteristics. The (\[sec4\]) provides two numerical methods to solve the modified space-time fractional differential equation and shows particular results illustrating the key issues. We finish the paper with summary and conclusions.
Finite domain approaches {#sec2}
========================
The mathematically straightforward prescription to create $f_l(x)$ and $f_r(x)$ is padding the function with zero from both sides: $$\begin{aligned}
\label{prescription 1}
\nonumber
f_l(x)&=& \begin{Cases}
u(x,t),\;\;\; 0 \leq x \leq 1 \\
0,\;\;\; -\infty < x < 0
\end{Cases} \\
f_r(x)&=& \begin{Cases}
u(x,t),\;\;\; 0 \leq x \leq 1 \\
0,\;\;\; 1 < x < \infty
\end{Cases}\end{aligned}$$ With such a prescription, the operator $D^\alpha_+$ will yield the left finite Riemann-Liouville fractional derivative $ _0D^\alpha_x u(x,t)$ and the operator $D^\alpha_-$ will yield the right finite Riemann-Liouville fractional derivative $ _xD^\alpha_1 u(x,t)$, where the integrals are defined already only over the appropriate part of the interval $[0,1]$, see [@Podlubny99]. In most numerical calculations published so far such a prescription has been used. It is also the default in the matrix approach. Starting from an $f_i(x)$ probability distribution obeying $f_i(0)=f_i(1)=0$, it seems reasonable to augment the problem with the two Dirichlet boundary conditions $u(0,t)=u(1,t)=0$ and solve it numerically by finite differences.
It is well known however, that the solution with Dirichlet conditions will not satisfy the first law, even in the case of $\beta=1$ and $\alpha=2$. Fig 1. shows the solution of a well documented problem with data $\{\beta=1/2,\;\alpha=3/2,\;a=1,\; f_i(x) = 6 x (1-x)\}$ obtained via the matrix approach suite. (Notice that the actual a parameter passed on should be $a' = \sqrt2$ instead of $a=1$ because in that suite the Riesz derivative is understood without the factor $\cos(\pi \alpha /2 )$.) In the illustration we also show the evolution of the first integral of the spatial density that is the fraction of substance remaining in the finite domain. As it is obvious, material is lost during the process. Replacing the two boundary conditions by “fractional derivative of order $\alpha -1$ equal to zero” condition – motivated by some interpretation of Fick’s law – does not help either. The problematic behavior of prescription (\[prescription 1\]) has been repeatedly discussed, for instance, in the groundwater literature [@Zhang06].
Here we show, that no boundary condition can be found to reconcile the non-physical nature of prescription (\[prescription 1\]). To this end we introduce a small change into the problem specification. Instead of requiring something at the two boundaries, we require that the numerical solution preserve the two important characteristics of the initial distribution: the first integral over $[0,1]$ is equal to unity and it is symmetric. To satisfy these conditions instead of the two Dirichlet boundary conditions we require $\int^1_0 u(x,t) dx = 1$ and $u(0,t)=u(1,t)$. (The second one is obviously necessary, but it turns out to be sufficient as well.) The two conditions are easily passed on to any finite-difference method, in this case to the matrix approach suite [@Podlubny09]. Summarized in Fig. 2 are the results for $f_i(x)= 6 x (1-x)$. Forcing the model to satisfy the first law, we lost compliance to the second law. The normalized entropy ($\sum_{i=1}^n{u_i \ln{u_i}}/\ln{\frac{1}{n}}$, where $n$ is the number of spatial mesh points) is *decreasing* with time.
Turning to Caputo’s idea
------------------------
Similar experiences known for practitioners have led to various ideas. For instance, del-Castillo-Negrete *et al.* [@Del-Castillo-Negrete08] suggested to use a modification of the Riesz derivative operator, following the recipe of Caputo being so successful for the time derivative. In our terminology, the prescription (explicitly given here only for $ 1 < \alpha \leq 2$) $$\begin{aligned}
\label{prescription 2}
\nonumber
f_l(x)&=& \begin{Cases}
u(x,t) -u(0,t)- x \: \left[\frac{\partial}{\partial x}u(x,t)\right]_{\& \; x=0},\;\;\; 0 \leq x \leq 1 \\
0,\;\;\; -\infty < x < 0
\end{Cases} \\
f_r(x)&=& \begin{Cases}
u(x,t) -u(1,t) + (1-x) \: \left[\frac{\partial}{\partial x}u(x,t)\right]_{\& \; x=1},\;\;\; 0 \leq x \leq 1 \\
0,\;\;\; 1 < x < \infty
\end{Cases}\end{aligned}$$ results in the Caputo form of the finite Riesz derivative. (Notice that the prescription introduces a jump discontinuity between the two functions $f_l$ and $f_r$.) Initiating the spreading process from the uniform distribution will leave the initial state at rest, since the right-hand side of (\[problem\]) will be identically zero. Thus the prescription resolved a contradiction, but now we have to face another one: starting the process from a triangular distribution will also leave the system at rest.
Constructing the flux expression (Fick’s law) directly with the Caputo derivative, Zhang *et al.* [@Zhang07] introduced another variant without explicit use of the first spatial derivative. In our notation, the prescription will take the form: $$\begin{aligned}
\label{prescription Zhang}
\nonumber
f_l(x)&=& \begin{Cases}
u(x,t) -u(0,t),\;\;\; 0 \leq x \leq 1 \\
0,\;\;\; -\infty < x < 0
\end{Cases} \\
f_r(x)&=& \begin{Cases}
u(x,t) -u(1,t),\;\;\; 0 \leq x \leq 1 \\
0,\;\;\; 1 < x < \infty
\end{Cases}\end{aligned}$$ The advantage of prescription (\[prescription Zhang\]) is that an initial triangular distribution will not be a steady-state solution of problem (\[problem\]) any more. However, it is still easy to find an initial condition (e.g. a box function non-zero only over a part of the \[0;1\] interval) that will also be a steady-state solution – in contrast to physical intuition. Fig 3 illustrates the problematic behavior of prescriptions (\[prescription 1\]), (\[prescription 2\]), and (\[prescription Zhang\]).
Summary of desired characteristics for a finite domain model
------------------------------------------------------------
From the introductory numerical experiments we glean a wish list. Sought is a macroscopic model in the form of fractional partial differential equation (\[problem\]) with the “hard” requirements (1-5) and “soft” ones (6-10):
i) If the initial state is a probability distribution (non-negative and with unit area under the curve), this property should be preserved for any time;
ii) If the initial probability distribution is symmetric around $x=0.5$, this property should be preserved for any time;
iii) The only stable steady state should be the uniform distribution;
iv) For any non-uniform initial distribution, the entropy should monotonically increase with time;
v) For integer orders, the model should reduce to known results;
vi) Starting from a Dirac delta distribution, the solution should follow the unbounded fundamental solution for short times;
vii) We still want to preserve the deep correspondence with Caputo fractional derivative in time and LW fractional derivative in space (allowing, however, some liberty in the selection of the $f_l$ and $f_r$ functions);
viii) Motivation should stem from a microscopic CTRW concept;
ix) It is desirable to have analytic solution for special cases;
x) It is desirable to have a numerical solution method within the general framework of discrete fractional calculus [@Podlubny09] .
Notice that *item i)* is sometimes stated as the probability preserving property, or the conservation principle. In this work we take the liberty to refer to it as “first law”. *Item ii)* expresses the invariance with respect to directing the coordinate axis, that is we consider only the symmetric Riesz derivative. *Items iii-iv* are obviously related to the second law (of thermodynamics). In some applications (see e.g. stock prices, [@Scalas06]) the listed requirements can be relaxed, but for description of spreading of material they are obviously necessary.
The effects of walls {#sec3}
====================
A prescription by Krepysheva *et al.*
-------------------------------------
Our starting point is the work of Krepysheva *et al.* [@Krepysheva06a] who visualized a “reflecting” wall at location $x=0$ and showed that, due to its non-local character, the kernel of the fractional space derivative has to be modified. The rule for the hopping particle was that if its jump interacts with the wall, it would continue to move in the mirror direction, preserving the overall length of the “initially intended” jump. In the macroscopic limit, the modified Riesz kernel turned out to be markedly different from the standard one based on finite interval left and right Riemann-Luiville derivatives. In our terminology, the works Krepysheva [@Krepysheva06a; @Krepysheva06b] derived the following specific prescription: Starting from an $u(x,t)$ function available over the non-negative $x$-axis, construct: $$\begin{aligned}
\label{prescription 3}
\nonumber
f_l(x)&=& \begin{Cases}
u(x,t),\;\;\; 0 \leq x \leq 1 \\
u(-x,t),\;\;\; -\infty < x < 0 \\
\end{Cases} \\
f_r(x)&=&u(x,t), \;\;\; 0 \leq x < \infty\end{aligned}$$ Calvo *at al.* [@Calvo07] have developed the idea further, for a rather specific geometric situation, when the random walk is along the perimeter of a circle. Building on these results, van Milligen *at al.* [@vanMilligen08] recently suggested a modification of the spatial operator for our problem (\[problem\]) that involves the Hurwitz zeta function. Another extension of the idea – based on the so-called Kolwankar-Gangal derivative – was proposed by Néel *et al.* [@Neel07]. Recently, Zoia *et al.* [@Zoia07] also discussed the effect of walls, although not from a first law point of view.
A new prescription
------------------
This work suggests another turn in the development for the isolating two-wall case. We recall that the hydrodynamic limit procedure makes use of the fact that in general, measurements correspond to time and length scales much larger than those of particle motions. In our opinion, it follows that even the “extremely long” jumps of a particle must be shorter than the domain size. Our main idea is therefore to limit the jump size to one, and hence allow zero or one particle-wall interaction, but never more than one. This suggests a new prescription: Starting from an $u(x,t)$ function available on $0 \leq x \leq 1$, construct $$\begin{aligned}
\nonumber
f_l(\xi) &=& \begin{Cases}
u(\xi,t),\;\;\; 0 \leq \xi \leq x \\
u(-\xi,t),\;\;\; x-1 \leq \xi \leq 0 \\
0,\;\;\; -\infty < \xi < 0
\end{Cases}
\\
\label{prescription new}
f_r(\xi)&=& \begin{Cases}
u(\xi,t),\;\;\; x \leq \xi \leq 1\\
u(2-\xi,t),\;\;\; 1 < \xi \leq 1 +x\\
0,\;\;\; 1+x < \xi < \infty
\end{Cases}\end{aligned}$$ The total support of both $f_l$ and $f_r$ is always two units long but it moves with the location of $x$. The $f_l$ and $f_r$ functions combined contain every function value from the original $u(x,t)$ twice (one corresponding to direct jump and the other bumped from the wall.) Fig. 4 shows the construction of the extensions $f_l$ and $f_r$ for two specific functions and a *specific location* $x=1/3$.
The modified Riemann-Luiville-Riesz derivative
----------------------------------------------
For comparison purposes, we can cast prescription (\[prescription new\]) into a more familiar form, using the concept of modified left and right finite-interval Riemann-Luiville derivatives: $$\begin{aligned}
_{x-1}D^\alpha_{x,mod} f &=& \frac{1}{\Gamma(m-\alpha)} \frac{d^m}{dx^m} \int_{x-1}^{x} \frac{f_{mod}(\xi)d\xi}{(x-\xi)^{\alpha-m+1}}, \;\;\; m = \lceil \alpha \rceil\\
_{x}D^\alpha_{x+1,mod} f &=& \frac{(-1)^m}{\Gamma(m-\alpha)} \frac{d^m}{dx^m} \int_{x}^{x+1} \frac{f_{mod}(\xi)d\xi} {(\xi-x)^{\alpha-m+1}}.\end{aligned}$$ where $$\begin{aligned}
f_{mod}(\xi) &=& f(2-\xi) \Pi \left(\xi-3/2\right)+f(\xi) \Pi \left(\xi-1/2\right)+
f(-\xi) \Pi \left(\xi+1/2\right) \;\;\;\;\;\;\;\;\;\;\end{aligned}$$ with the Heaviside box function defined as $\Pi \left(\xi\right)=1$ for $|\xi| \leq 1/2$ and zero otherwise. These finite Riemann-Luiville derivatives are based on an interval of total length 2, always centered at the location $x$ where we are interested in the derivative. Therefore, for non-integer $\alpha$ the modified finite-interval Riemann-Luiville-Riesz derivative takes the form $$\begin{aligned}
\label{modRLRiesz}
\frac{d ^\alpha}{d_{mod} |x|^\alpha} f(x) =
-\frac{1}{2 \cos(\pi \alpha /2 )} [ _{x-1}D^\alpha_{x,mod} f +\; _xD^\alpha_{x+1,mod} f]\end{aligned}$$ Definition (\[modRLRiesz\]) and prescription (\[prescription new\]) are equivalent.
It is illuminating to compare the regularly used spatial operator (\[prescription 1\]) and the modified one (\[modRLRiesz\]) for some simple functions defined over $[0;1]$. Fig 5 shows the comparison for $f(x)=x-0.5$, $f(x)=(x-0.5)^2$, and $f(x)=\sin(\pi x)$. Somewhat disappointingly, the modified Riesz derivative (\[modRLRiesz\]) does not eliminate singularity at the end points of the interval for these functions and – by and large – behaves similarly to the commonly used definition (\[prescription 1\]). However, for one function family investigated in the next sub-section, the difference is dramatic.
Eigenfunctions and eigenvectors
-------------------------------
Of particular interest is the the family of functions $\cos( j \pi x)$, $j=0,1, \ldots $. Also shown in Fig. 5 the comparison for $f(x)=\cos(3 \pi x)$. We see that prescription (\[prescription 1\]) leads to singular behavior at the endpoints – as usual. On the other hand, (\[modRLRiesz\]) is not only non-singular, but it *almost* coincides with the Fourier-Riesz derivative of the entire $\cos(3 \pi x)$ function over the interval $[0,1]$, differing only in a constant factor $1.01328\ldots$. Using $Mathematica$ one can show that $$\begin{aligned}
\label{eigenfunctions}
\frac{d ^\alpha}{d_{mod} |x|^\alpha} \cos( j \pi x) &=& c_{\alpha,j} \cos( j \pi x) \;\;\; j=0,1,\ldots\end{aligned}$$ where $c_{\alpha,j}$ is constant. In other words, $\cos( j \pi x)$ is an eigenfunction of the operator (\[modRLRiesz\]) with eigenvalue $c_{\alpha,j}$. Moreover, $c_{\alpha,0}=0$, implying that the uniform distribution has a modified Riesz derivative equal to zero. This property will ensure that the uniform distribution will be a steady state solution of (\[problem\]). We managed to obtain closed form expression for specific $\alpha$ parameters (with repeated help from $Mathematica$) as follows: $$\begin{aligned}
\label{eigenvalues}
c_{2,j} &=& - {(j \pi)}^2, \;\;\; j=1,2, \ldots \\
c_{3/2,j} &=& -2 {(j \pi)}^{3/2} C_F(\sqrt{2 j}), \;\;\; j=1,2, \ldots \\
c_{1/2,j} &=& -2 \sqrt{j \pi } S_F(\sqrt{2 j} ), \;\;\; j=1,2, \ldots\end{aligned}$$ where $C_F$ and $S_F$ denote the $\cos$ and $\sin$ Fresnel integrals, respectively. For other $\alpha$ values we could not obtain an explicit expression, but could still develop a simple code in $Mathematica$ that calculates the eigenvalue *with any required number of digit* accuracy for an $\alpha$ given as a rational fraction (see Appendix).
The solution of the space-time fractional differential equation on the interval \[0,1\] {#sec4}
=======================================================================================
We introduce two approaches. The first one uses Laplace transform in time and collocation in space. The second one is based on finite differences.
The Laplace Transform – Collocation method: an example
------------------------------------------------------
We illustrate this method on a simple example: $\{\beta = 1/2,\;\alpha = 3/2,\; a = 1,\; f_i(x)=1 - \cos(2 \pi x)\}$. Using 6 collocation points: $\textbf{x}_c = \{0, \frac15,\frac25,\frac35,\frac45,1\}$ we seek the Laplace transform of the solution at an arbitrary point, $x$. Introducing the vector notation $$\begin{aligned}
\textbf{v}(s)&=&\{ c_0(s),c_1(s),c_2(s),c_3(s),c_4(s),c_5(s)\} \\
\textbf{g}(x)&=&\{1,\cos (\pi x),\cos (2 \pi x),\cos (3 \pi x),\cos (4 \pi x),\cos (5 \pi x)\}\\
\nonumber
\textbf{e}&=&
\{
0,-2 \pi^{3/2} C_F(\sqrt{2}),-4 \sqrt{2} \pi^{3/2} C_F(2),-6 \sqrt{3} \pi ^{3/2} C_F(\sqrt{6}), \\
&&-16\pi^{3/2} C_F(2\sqrt{2}), -10\sqrt {5}\pi^{3/2} C_F(\sqrt{10} ) \}\end{aligned}$$ we represent the Laplace space solution at $x$ as $$\label{Uxs}
U(x,s) =\textbf{v}(s) . \textbf{g}(x)$$ Then its modified Riesz derivative of order $3/2$ takes the form $$\frac{\partial ^{3/2}}{\partial_{mod} |x|^{3/2}} U(x,s) = \textbf{v}(s).\left[ \textbf{e} \, \textbf{g}(x)\right]$$ (with component by component multiplication between $\textbf{e}$ and $\textbf{g}$) and its Caputo derivative of order $1/2$ is written as $$\textbf{L} \left( \frac{\partial ^{1/2}}{\partial t^{1/2}} u(x,t),\;s \right) = s^{-1/2} \left[s \left( \textbf{v}(s) . \textbf{g}(x)\right) - f_i(x)\right]$$ Writing the partial differential equation $$\label{collocation}
s^{-1/2} \left[s \left( \textbf{v}(s) . \textbf{g}(x)\right) - f_i(x)\right] \;-\; \textbf{v}(s).\left[ \textbf{e} \, \textbf{g}(x)\right] = 0$$ at the 6 collocation points $\textbf{x}_c$, we obtain a system of linear equations in the 6 unknown variables, $\textbf{v}(s)$. The solution turns out to be $$\textbf{v}(s)=\{\frac{1}{s},0,-\frac{1}{4 \sqrt{2} \pi ^{3/2} C_F(2) \sqrt{s}+s},0,0,0 \}$$ and hence we obtain $$U(x,s) = \frac{1}{s}-\frac{\cos (2 \pi x)}{4 \sqrt{2} \pi ^{3/2} C_F(2) \sqrt{s}+s}$$ This can be inverted on $s$ resulting in $$\label{inverted}
u(x,t) = 1 + \exp\left({32 \pi ^3 C_F(2)^2 t}\right) \left[ \textrm{erf} \left(4 \sqrt{2} \pi ^{3/2} C_F(2) \sqrt{t} \right)-1 \right]\cos (2 \pi x)$$ Increasing the number of collocation points does not change the solution (\[inverted\]), it is already exact.
The fundamental solution for $\beta = 1/2,\;\alpha = 3/2$
---------------------------------------------------------
We can repeat the above procedure for $f_i(x)=\delta(x-\frac12) = 1-2 \cos(2 \pi x) + 2 \cos(4 \pi x)-2 \cos(6 \pi x)+ \; \ldots$ and obtain the fundamental solution: $$\begin{aligned}
\nonumber
u(x,t)&=&
1+2 \exp\left({32 \pi ^3 C_F(2)^2 t}\right) \cos (2 \pi x) \left[\textrm{erf}\left(4 \sqrt{2} \pi ^{3/2} C_F(2) \sqrt{t}\right)-1\right]-\\
\nonumber
&&2 \exp\left({256 \pi ^3 C_F(2 \sqrt{2})^2 t}\right) \cos (4 \pi x) \left[\textrm{erf}\left(16 \pi ^{3/2} C_F(2 \sqrt{2}) \sqrt{t}\right)-1\right]+\\\nonumber
&&2 \exp\left({864 \pi ^3 C_F(2 \sqrt{3})^2 t}\right) \cos (6 \pi x) \left[\textrm{erf}\left(12 \sqrt{6} \pi ^{3/2} C_F(2 \sqrt{3}) \sqrt{t}\right)-1\right]-\\
\label{fundsol3}
&&\ldots\end{aligned}$$ Looking back at our wish list, we would like to check *item vi)* requiring the correspondence of the unbounded and bounded solutions at early times. The analytic solution of the problem stated on unbounded domain would take the form: $$\begin{aligned}
\label{fundsol2}
u_\infty(x,t) = t^{-\frac13}\textit{M}\left( |x|\, t^{-1/3} ; 1/3 \right)\end{aligned}$$ where $\textit{M}(\xi ; 1/3)$ is given by (\[M13\]). Fig. 6 compares, at an early time, $t=10^{-3}$, the new fundamental solution (\[fundsol3\]) computed with 50 terms and the well-know unbounded solution (\[fundsol2\]). We see that the effect of the boundary has just started to show. (Unfortunately, the numerical evaluation of the fundamental solution is not trivial – even with $Mathematica$ – because of the extremely large exponents and hence we could not increase the number of terms to get rid of the small oscillations of the curve.)
Pursuing the “analytic” solution further has other drawbacks too. The fundamental solution would contain Mittag-Leffler functions (in addition to the eigenvalues, $c_{\alpha,j}$ and eigenfunctions $\cos(j\: \pi \:x)$ ) in the case of a general $\alpha$ and ${\beta}$. Moreover, any other (non Dirac delta) initial condition would necessitate further numerical convolution.
One can, however, easily construct the system of linear equations for a given set of $\alpha$, $\beta$, $a$ and $f_i(.)$ at any selected value of the Laplace variable $s$, solve the system numerically by Gauss elimination and substitute $\textbf{v}(s)$ into (\[Uxs\]). Therefore, we have a way to calculate (at any specified $x$) the Laplace transform of the solution numerically, and hence we can use a numerical inversion technique [@Valko04]. The procedure is robust, if care is taken to do the Gauss elimination with multiple precision – large enough with respect to the number of terms in the GWR algorithm [@Abate04]. We will call the method LT–collocation with numerical inversion. A $Mathematica$ realization of the algorithm is provided in the Appendix.
We emphasize that the procedure does not require any boundary conditions, rather $x=0$ and $x=1$ are included in the set of collocation points. (The physical “boundary conditions” are taken care of within the spatial operator itself.)
Shown in Fig. 7 is the summary of the results of the procedure for $\{\beta = 1/2,\;\alpha = 3/2,\; a = 1,\; f_i(x)=\delta (x-\frac12)\}$, using $51$ collocation points. Since the LT–collocation method with numerical inversion cannot be started from the Dirac delta “function”, we pass on the solution of the unbounded problem (\[fundsol2\]) at a very early time, $t = 0.00001$ as a new “initial” condition. Then we do the numerical inversion for $t-0.00001$ where $t$ is the time we are interested in. The “fraction of substance still in the domain” is not shown in Fig. 7, because at the “initial” state it is unity (for all practical purposes) and hence it remains unity during all times.
Next we compare traditional $\{\beta = 1,\;\alpha = 2,\; a = 1\}$ and space-time fractional spreading $\{\beta = 1/2,\;\alpha = 3/2,\; a = 1\}$ starting from a non-symmetric initial distribution $f_i(x) = 12 x(1-x)^2$, as illustrated by Figs 8 and 9. The number of collocation points is kept at $51$. We find that the spreading is initially faster for the space-time fractional case, but at late time traditional dispersion becomes faster. There is also a remarkable difference in the “overshoot” of the density at $x=0$ – at least for the studied time points. (We note, however, that such comparisons are of limited value, because the parameter $a$ is not the same in the two cases, in spite of looking the same.)
Solution by the method of finite differences
--------------------------------------------
While the LT–collocation with numerical inversion works effectively, it is still illuminating to solve the problem within the framework of discrete fractional calculus. The form of the modified Riesz operator (\[modRLRiesz\]) suggests that a relatively small modification to the established matrix approach will suffice. Indeed, one has to use prescription (\[prescription new\]) to pad the list of available function values and add a small correction to assure that for a constant function the modified Riesz derivatives yield zero.
We introduced this modification into the matrix approach suite (see Appendix for some details). When working with the suite, we do not use the concept of “mathematical boundary conditions” at all, rather we write the equations for the endpoints $x=0$ and $x=1$ as well. Therefore, the total number of equations remains the same as the number of unknowns. Shown in Fig. 10 is the summary of results for our previous example $\{\beta = 1/2,\;\alpha = 3/2,\; a = 1\}$ (that is $a' = \sqrt2$) when the initial condition is $f_i(x) = 12 x(1-x)^2$ and the finite difference step sizes are $\Delta x=0.02$ and $\Delta t=0.01$. (Notice that the number of unknowns in the matrix approach was 5000 and the system matrix had $2.5$ million elements, hence we were limited by computer memory.) Regarding accuracy, the results are still behind the ones depicted in Fig. 9, but the overall correspondence is remarkable. In particular, the fraction of substance still in the domain has less than 2 % error. Not only the modified Riesz derivative is “probability preserving” but also is our finite difference representation of it.
Summary and conclusions
=======================
We have introduced a new version of the finite Riesz derivative. The new spatial operator – combined with the Caputo derivative in time – results in a space-time fractional differential equation that is well suited to describe anomalous spreading of substance in a finite domain. The space-time fractional differential equation satisfies our postulated requirements. If the initial state is a probability distribution (non-negative and with unit area under the curve), this property is preserved for any later time; if the initial probability density is symmetric around $x=0.5$, this property is also preserved. We could not prove rigorously that the only stable steady state is the uniform distribution, but all specific analytic formulaes and numerical examples indicated so. In all our examples, starting from a non-uniform initial distribution, the entropy monotonically increased with time. For integer orders, the model reproduces the known results of Fickian-Markovian diffusion over a finite domain. Starting from a Dirac delta distribution, the solution follows the unbounded fundamental solution of the space-time fractional differential equation for short times. We could preserve the deep correspondence with Liouville-Weyl fractional derivative in space by creating the appropriate prescription, however we are not sure that this is the only (or even the best) way to do it. The new approach arose from a microscopic CTRW concept and we could manage to provide analytic solution for special cases. While our preferred solution method is the LT–collocation with numerical inversion, we could also extend the matrix approach and hence fit our operator into the mainstream framework of discrete fractional calculus. In this work we focused on “pure” diffusion but we do not envisage any difficulty in considering simultaneous advection or other external potential field.
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$Mathematica$ v7.0 $\textrm{http://www.wolfram.com/}$ (Last visited: Aug. 15, 2009)
Appendix
========
Calculations were done in $Mathematica$. Here we show some code snippets and results in order to ease reproduction of our results.
The code snippet for the LT-collocation method with numerical inversion is the following:
$$\begin{array}{lll}
\textsf{vari}[\textsf{s$\_$}]&=&\textsf{Table}[c[j][s],\{j,0,\textsf{nm}\}];\\
\textsf{cosi}[\textsf{x$\_$}]&=&\textsf{Table}[\textsf{Cos}[ (j \pi )\textsf{ }x],\{j,0,\textsf{nm}\}];\\
\textsf{symi}[\textsf{x$\_$}]&=&\textsf{Table}[\textsf{scos}[\alpha ][j][x],\{j,0,\textsf{nm}\}];\\
\textsf{equl}[\textsf{s$\_$}][\textsf{x$\_$}]&=&s{}^{\wedge} {(\textsf{$\beta $}-1)} \; (s\textsf{ }\textsf{vari}[s].\textsf{cosi}[x]\textsf{ }-\textsf{fi}[x]) - a \; \textsf{vari}[s].\textsf{symi}[x] ;\\
\textsf{xc}&=&N[\textsf{Range}[0,1,1/\textsf{nm}],\textsf{maxprec}];\\
\{\textsf{mb}[\textsf{s$\_$}],\textsf{As}[\textsf{s$\_$}]\}&=&\textsf{Map}[\textsf{Normal},
\textsf{CoefficientArrays}[\textsf{equl}[s][\textsf{xc}],\textsf{vari}[s]]];\\
\textsf{sols}[\textsf{s$\_$}]&:=&\textsf{sols}[s]=\textsf{LinearSolve}[\textsf{As}[s],-\textsf{mb}[s]];\\
\textsf{Uxs}[\textsf{x$\_$}][\textsf{s$\_$}]&:=&\textsf{sols}[s].\textsf{cosi}[x];\\
\textsf{uxt}[\textsf{M$\_$}][\textsf{x$\_$}][\textsf{t$\_$}]&:=&\textsf{GWR}[\textsf{Uxs}[x],t,\textsf{M}];\\
\end{array}$$
The user has to provide $\beta, \alpha, a$, the function $fi[x\_]$ in addition to the integer $nm$ (the number of collocation points minus one.) The number of terms in the GWR algorithm $M$ [@Abate04] and the maximum used precision $maxprec$ (we used 200) are also required. Once defined, the $\textsf{uxt}[\textsf{M}][\textsf{x}][\textsf{t}]$ function can be used to calculate the solution at a specific $x$ and $t$. The $\textsf{scos}[\alpha ][j][x]$ expression should evaluate to $-2 j^{3/2} \pi^{3/2} FresnelC[\sqrt{2 j}] Cos[j \pi x]$ when $\alpha =3/2$. In general, it will be the product of an eigenvalue $cei[\alpha][j]$ and the appropriate $Cos[j \pi x]$. While no general formula is currently available for the eigenvalue, we can calculate it by the following code snippet for a given rational fraction $\alpha$ and positive integer $j$:
$$\begin{array}{lcl}
\textsf{eug}=\textsf{EulerGamma};\\
h[j][x\_]=\textsf{Cos}[j \: \pi x];
\\
\textsf{cei}[\alpha][j]=N[ \textsf{symRmod}[h[j],\textsf{eug},\alpha]/\textsf{h}[j][\textsf{eug}], \textsf{maxprec}];
\end{array}$$ where
$$\textsf{symRmod}[f\_,x\_,\alpha\_]:=-\frac{1}{2 Cos[\alpha \pi /2]} (\textsf{lRmod}[f,x,\alpha]+
\textsf{rRmod}[f,x,\alpha]);$$
In the above code the left modified Rieman-Luiville-Riesz derivative is calculated from
$$\begin{array}{l}
\textsf{lRmod}[\textsf{f$\_$},\textsf{t$\_$},\alpha \_] := \textsf{Module}[\{\textsf{fext},m=\textsf{Ceiling}[\alpha ]\},
\\
\textsf{fext}[\textsf{x$\_$}]=f[x]\;\textsf{UnitBox}[x-1/2]+f[2-x]\;\textsf{UnitBox}[x-3/2]+
f[-x]\;\textsf{UnitBox}[x+1/2];\;\;\;\;
\\
\textsf{If} [ \textsf{IntegerQ}[\alpha ],\;
D[f[t],\{t,\alpha \}],
\\
\frac{1}{
\textsf{Gamma[m - $\alpha$ ]}
}\;
D[(\textsf{Integrate}[\frac{\textsf{fext}[\tau ]}{(\textsf{eug} -\tau ){}^{\wedge}(\alpha +1-m)},
\{\tau ,\textsf{eug}-1,\textsf{eug}\}])\textsf{/.}\textsf{eug} \rightarrow t,
\{t,m \}]]]
\end{array}$$ and the right modified finite-interval Rieman-Luiville-Riesz derivative is calculated from $$\begin{array}{l}
\textsf{rRmod}[\textsf{f$\_$},\textsf{t$\_$},\alpha \_] := \textsf{Module}[\{\textsf{fext},m=\textsf{Ceiling}[\alpha ]\},
\\
\textsf{fext}[\textsf{x$\_$}]=f[x]\;\textsf{UnitBox}[x-1/2]+f[2-x]\;\textsf{UnitBox}[x-3/2]+
f[-x]\;\textsf{UnitBox}[x+1/2];\;\;\;\;
\\
\textsf{If} [ \textsf{IntegerQ}[\alpha ],\;
D[f[t],\{t,\alpha \}],
\\
\frac{(-1)^m}{
\textsf{Gamma[m - $\alpha$ ]}
}\;
D[(\textsf{Integrate}[\frac{\textsf{fext}[\tau ]}{(\tau-\textsf{eug} ){}^{\wedge}(\alpha +1-m)},
\{\tau ,\textsf{eug},\textsf{eug}+1 \}])\textsf{/.}\textsf{eug} \rightarrow t,
\{t,m \}]]]
\end{array}$$ (The extensive use of “EulerGamma” is somewhat arbitrary but proved useful in the context of the current version – v7.0 – of $Mathematica$ [@MMA].)
Without going into details of the derivation, here we illustrate the concept of extending the matrix approach suite of Podlubny *et al.* [@Podlubny09]. For $1<\alpha\leq2$, the current symmetric Riesz function of the suit (called $ransym$) would hypothetically provide the following array $SR$ of symmetric Riesz derivatives at points $\{0,1h,2h,3h,4h\}$ where the function values $\{y_0,y_1,y_2,y_3,y_4\}$ are known:
$$\begin{array}{lll}
SR_0&=&
2^{-1+2 \alpha } (1-\alpha ) y_0+\\&&2^{-1+2 \alpha } y_1 \\
SR_1&=& \frac{1}{3} 4^{-2+\alpha } (24+12 (-1+\alpha ) \alpha ) y_0-\\&&4^{\alpha } \alpha y_1+\\
&&
\frac{1}{3} 4^{-2+\alpha } (24+12 (-1+\alpha ) \alpha ) y_2-\\&&\frac{1}{3} 4^{-1+\alpha } (-2+\alpha ) (-1+\alpha ) \alpha y_3+\\
&&
\frac{1}{3} 4^{-2+\alpha } (-3+\alpha ) (-2+\alpha ) (-1+\alpha ) \alpha y_4 \\
SR_2&=& \frac{1}{3} 4^{-1+\alpha } (2-\alpha ) (-1+\alpha ) \alpha y_0+\\&&\frac{1}{3} 4^{-1+\alpha } (6+3 (-1+\alpha ) \alpha ) y_1-\\
&&4^{\alpha } \alpha y_2+\\&&
\frac{1}{3} 4^{-1+\alpha } (6+3 (-1+\alpha ) \alpha ) y_3+\\
&&\frac{1}{3} 4^{-1+\alpha } (2-\alpha ) (-1+\alpha ) \alpha y_4 \\
SR_3&=& \frac{1}{3} 4^{-2+\alpha } \alpha \big(-6+11 \alpha -6 \alpha ^2+\alpha ^3\big) y_0-\\&&\frac{1}{3} 4^{-1+\alpha } \alpha \big(2-3 \alpha +\alpha ^2\big) y_1+\\
&&4^{-1+\alpha } \big(2-\alpha +\alpha ^2\big) y_2-\\&&
4^{\alpha } \alpha y_3+\\
&&4^{-1+\alpha } \big(2-\alpha +\alpha ^2\big) y_4 \\
SR_4&=& 2^{-1+2 \alpha } y_3+\\&&2^{-1+2 \alpha } (1-\alpha ) y_4\\
\end{array}$$
A symmetric Riesz function modified according to prescription (\[prescription new\]) would hypothetically provide the following array $SRM$ for the same input:
$$\begin{array}{lll}
SRM_0&=&
-2^{2 \alpha } \alpha y_0+\\&&2^{-1+2 \alpha } \big(2-\alpha +\alpha ^2\big) y_1-\\&&
\frac{1}{3} 2^{-1+2 \alpha } \alpha \big(2-3 \alpha +\alpha ^2\big) y_2+\\&&\frac{1}{3} 2^{-3+2 \alpha } (-3+\alpha ) \alpha \big(2-3 \alpha +\alpha ^2\big) y_3+\\&&
\frac{1}{3} 2^{-3+2 \alpha } (4-\alpha ) (-3+\alpha ) \big(2-3 \alpha +\alpha ^2\big) y_4 \\
SRM_0&=&
4^{-1+\alpha } \big(2-\alpha +\alpha ^2\big) y_0-\\&&\frac{1}{3} 4^{-1+\alpha } \alpha \big(14-3 \alpha +\alpha ^2\big) y_1+\\&&
\frac{1}{3} 4^{-2+\alpha } \big(24-18 \alpha +23 \alpha ^2-6 \alpha ^3+\alpha ^4\big) y_2+\\&&\frac{1}{3} 4^{-2+\alpha } \big(-48+92 \alpha -58 \alpha ^2+16 \alpha ^3-2 \alpha ^4\big) y_3+\\&&
\frac{1}{3} 4^{-2+\alpha } \big(-6 \alpha +11 \alpha ^2-6 \alpha ^3+\alpha ^4\big) y_4 \\
SRM_0&=&
-\frac{1}{3} 4^{-1+\alpha } \alpha \big(2-3 \alpha +\alpha ^2\big) y_0+\\&&\frac{1}{3} 4^{-2+\alpha } \big(24-18 \alpha +23 \alpha ^2-6 \alpha ^3+\alpha ^4\big) y_1+\\&&
\frac{1}{3} 4^{-2+\alpha } \big(-48+52 \alpha -70 \alpha ^2+20 \alpha ^3-2 \alpha ^4\big) y_2+\\&&\frac{1}{3} 4^{-2+\alpha } \big(24-18 \alpha +23 \alpha ^2-6 \alpha ^3+\alpha ^4\big) y_3+\\&&
\frac{1}{3} 4^{-2+\alpha } \big(-8 \alpha +12 \alpha ^2-4 \alpha ^3\big) y_4 \\
SRM_0&=&
\frac{1}{3} 4^{-2+\alpha } \alpha \big(-6+11 \alpha -6 \alpha ^2+\alpha ^3\big) y_0-\\&&\frac{1}{3} 2^{-3+2 \alpha } \big(24-46 \alpha +29 \alpha ^2-8 \alpha ^3+\alpha ^4\big) y_1+\\&&
\frac{1}{3} 4^{-2+\alpha } \big(24-18 \alpha +23 \alpha ^2-6 \alpha ^3+\alpha ^4\big) y_2+\\&&\frac{1}{3} 4^{-2+\alpha } \big(-56 \alpha +12 \alpha ^2-4 \alpha ^3\big) y_3+\\&&
\frac{1}{3} 4^{-2+\alpha } \big(24-12 \alpha +12 \alpha ^2\big) y_4 \\
SRM_0&=&
\frac{1}{3} 2^{-3+2 \alpha } \big(-24+50 \alpha -35 \alpha ^2+10 \alpha ^3-\alpha ^4\big) y_0+\\&&\frac{1}{3} 2^{-3+2 \alpha } \alpha \big(-6+11 \alpha -6 \alpha ^2+\alpha ^3\big) y_1\\&&
-\frac{1}{3} 2^{-1+2 \alpha } \alpha \big(2-3 \alpha +\alpha ^2\big) y_2+\\&&2^{-1+2 \alpha } \big(2-\alpha +\alpha ^2\big) y_3\\&&
-2^{2 \alpha } \alpha y_4
\end{array}$$
One can easily check by substitution, that when $y_0=y_1=y_2=y_3=y_4$, each derivative is zero $SRM_0=SRM_1=SRM_2=SRM_3=SRM_4=0$, and when $y_0=1, y_1=0.5, y_2=0,y_3=-0.5,y_4=-1$, the sum is zero $SRM_0+SRM_1+SRM_2+SRM_3+SRM_4\;=\;0$.
Figure captions
===============
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|
---
abstract: 'We determine the behavior under Weil restriction of the group of connected components of the special fiber of an arbitrary smooth group scheme (whose Weil restriction exists) over an arbitrary (commutative and unital) local ring. Applications to Néron models are given.'
address:
- 'Università degli Studi di Padova, Dipartimento di Matematica, via Trieste 63, I-35121 Padova'
- 'Departamento de Matemáticas, Universidad de La Serena, Cisternas 1200, La Serena 1700000, Chile'
author:
- Alessandra Bertapelle
- 'Cristian D. González-Avilés'
title: Groups of Components and Weil Restriction
---
[^1]
[^2]
Introduction
============
Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$ and let ${k^{\le\rm s}}$ be a fixed separable algebraic closure of $k$. Let $A_{K}$ be an abelian variety over $K$ with Néron model $A$ over $R$ and let $\pi_{0}(\lbe A_{\le\rm s}\lbe)$ denote the $k$-group scheme of connected components of the special fiber $A_{\le\rm s}$ of $A$. Let $A^{\vee}_{K}$ be the dual (i.e., Picard) variety of $A_{K}$ and write $A^{\vee}$ and $\pi_{0}(\lbe A^{\vee}_{\le\rm s}\lbe)$ for the corresponding objects associated to the abelian variety $A^{\vee}_{K}$. When $k$ is perfect, Grothendieck’s pairing $\pi_{0}(\lbe A_{\le\rm s}\lbe )({k^{\le\rm s}})\times\pi_{0}(\lbe A^{\vee}_{\le\rm s})({k^{\le\rm s}})\to{{\mathbb Q}}/{{\mathbb Z}}$ [@sga7 IX, (1.2.2)] has been verified to be perfect in all cases except when $K$ has positive characteristic and $k$ is infinite; see [@bb §1] for further comments. More recently a general proof of the perfectness of Grothendieck’s pairing has been announced in [@s]. However, when $k$ is [*imperfect*]{}, there exist examples that show that the above pairing is no longer perfect. The first such examples were constructed in [@bb comment after Corollary 2.5] using the Weil restriction functor. It is for this reason, at least, that a full understanding of the behavior of the groups of connected components of abelian (or more general smooth group) varieties under Weil restriction is desirable. Previously, this has been achieved only in the case of abelian varieties under various restrictions. See Remark \[rem\]. In this paper we completely determine the behavior under Weil restriction of the group of connected components of the special fiber of an arbitrary (possibly non commutative) smooth group scheme over an arbitrary (possibly non noetherian) local ring. Our main result is the following
\[main\] Let $R^{\e\prime}/R$ be a finite flat extension of local rings with associated residue field extension $k^{\le\prime}\be/k$. Let $G^{\le\prime}$ be a smooth $R^{\e\prime}$-group scheme such that the Weil restriction $\Re_{\le R^{\e\prime}\be/\be R}(G^{\e\prime}\le)$ exists. Then $\Re_{\le k^{\le\prime}\be/k}(\pi_{0}(G^{\e\prime}_{\lbe{\rm{s}}}\le))$ exists and there exists a canonical isomorphism of étale $k$-group schemes $$\pi_{0}(\Re_{\le R^{\e\prime}\be/\be R}(G^{\e\prime}\le)_{{\rm{s}}})\simeq\Re_{\le k^{\le\prime}\be/k}(\pi_{0}(G^{\e\prime}_{\lbe{\rm{s}}}\le)).$$
The theorem applies, in particular, to smooth and quasi-projective $R^{\e\prime}$-group schemes since $\Re_{\le R^{\e\prime}\be/\be R}(G^{\e\prime}\le)$ is well-known to exist in this case. More general conditions that guarantee the existence of $\Re_{\le R^{\e\prime}\be/\be R}(G^{\e\prime}\le)$ can be found in [@blr §7.6, Theorem 4, p. 194] and [@bga Theorem 2.15].
The proof of the theorem, which is simpler than the proofs of the particular cases alluded to above, essentially relies only on basic properties of the Weil restriction functor and the following key fact: if $k$ is a field, $R^{\e\prime}$ is a finite local $k$-algebra and $G^{\e\prime}$ is a smooth $R^{\e\prime}$-group scheme with geometrically connected special fiber, then $\Re_{\le R^{\e\prime}\be/\lbe k}(G^{\e\prime}\le)$ is geometrically connected.
Some applications of Theorem \[main\] are discussed at the end of the paper.
We thank Brian Conrad for helping us shorten our original proof of the main theorem. We also thank Bas Edixhoven and Dino Lorenzini for helpful comments.
Preliminaries
=============
All rings considered in this paper are commutative and unital. If $X$ is a scheme, the topological space underlying $X$ will be denoted by $|X|$. The identity morphism of $X$ will be denoted by $1_{\be X}$.
Let $S$ be a scheme, let $X$ be an $S$-scheme and let $S^{\e\prime\prime}\to S^{\le\prime}\to S$ be morphisms of schemes. We will make the identification $$\label{id1}
(X\times_{S}S^{\e\prime}\e)\times_{S^{\le\prime}}S^{\e\prime\prime}=X\times_{S}S^{\e\prime\prime}.$$ If $A\to B$ is a ring homomorphism and $X$ is an $A$-scheme, $X_{\be B}$ will denote $X\times_{{\mathrm{ Spec}\,}A}{\mathrm{ Spec}\,}B$.
Let $W\to S^{\e\prime}\to S$ be morphisms of schemes. If $Z$ is an $S$-scheme and $\textrm{pr}_{\le 1}\colon Z\be\times_{S}\lbe S^{\e\prime}\to Z$ is the first projection then, by the universal property of the fiber product, the map $$\label{bc}
{ \mathrm{Hom}}_{S^{\prime}}(W,Z\be\times_{S}\be S^{\e\prime}\e)\overset{\be\!\sim}{\to}{ \mathrm{Hom}}_{\le S}(W,Z\le),\quad g\mapsto \textrm{pr}_{1}\circ g,$$ is a bijection.
Let $A$ be an artinian local ring with residue field $k$ and let $G$ be an $A$-group scheme [*locally of finite type*]{}. We will identify $G$ with the functor on $({\mathrm }{Sch}/A)$ represented by $G$. Let $G_{\lbe \textrm{s}}:=G\!\times_{{\mathrm{ Spec}\,}A}\be{\mathrm{ Spec}\,}k$ be the special fiber of $G\e$[^3]. Now let $|G|^{\le 0}\subset |G|$ denote the connected component of the identity $1\in G$ and define a subfunctor $G^{\e 0}$ of $G$ by $$\label{idcomp1}
G^{\e 0}(T)=\{u\in G(T)\colon u(|T|)\subseteq |G|^{\le 0}\},$$ where $T$ is any $A$-scheme. Since $G$ is locally of finite type over $A$, is represented by an open and normal subgroup scheme $G^{\e 0}$ of $G$ [@sga3 $\text{VI}_{\text{A}}$, Proposition 2.3.1]. Note also that, since every subgroup scheme of $G$ is closed [@sga3 $\text{VI}_{\text{A}}$, Corollary 0.5.2], $G^{\e 0}$ is open and closed in $G$. Assume now that $G$ is locally of finite type and [*flat*]{} over $A$. Then, by [@sga3 $\text{VI}_{\text{A}}$, Proposition 5.5.1(i) and Theorem 3.2(vi)], the fppf quotient sheaf of groups $$\pi_{0}(G):=G^{\e 0}\e\backslash G$$ is represented by an étale $A$-group scheme and the canonical morphism $$\label{pim}
p_{\le G}\colon G\to \pi_{0}(G)$$ is faithfully flat and locally of finite presentation. Further, by [@sga3 $\text{VI}_{\text{B}}$, Proposition 3.3], there exists a canonical isomorphism of $k$-group schemes $(G_{\rm s}\lbe)^{0}\simeq (G^{\e 0})_{\rm s}$ and therefore $\pi_{0}(G_{\rm s}\le)\simeq\pi_{0}(G\le)_{\rm s}$.
Note that, if $G$ is [*smooth*]{} over $A$, then $G^{\e 0}$ is smooth as well and therefore is a smooth morphism by [@ega $\text{IV}_{4}$, Proposition 17.5.1] and [@sga3 ${\rm{VI}}_{\rm B}$, Proposition 1.3].
Let $f\colon S^{\e\prime}\to S$ be a morphism of schemes and let $X^{\prime}$ be an $S^{\le\prime}$-scheme. We will say that [*the Weil restriction of $X^{\prime}$ along $f$ exists*]{} (or, more concisely, that [*$\Re_{S^{\le\prime}\be/S}(X^{\prime})$ exists*]{}) if the contravariant functor $(\mathrm{Sch}/S)\to(\mathrm{Sets}), T\mapsto{ \mathrm{Hom}}_{
S^{\le\prime}}(T\times_{S}S^{\le\prime},X^{\le\prime}\le)$, is representable, i.e., if there exists a pair $(\Re_{S^{\le\prime}\be/S}(X^{\prime}\e), q_{\le X^{\prime}\!,\e S^{\le\prime}\be/S})$, where $\Re_{S^{\le\prime}\be/S}(X^{\prime}\e)$ is an $S$-scheme and $q_{\le X^{\prime}\be,\e S^{\le\prime}\be/S}\colon \Re_{S^{\le\prime}\be/S}(X^{\prime}\e)_{S^{\le\prime}}\to X^{\prime}$ is an $S^{\prime}$-morphism of schemes, such that the map $$\label{wr}
{ \mathrm{Hom}}_{\le S}\e(T,\Re_{S^{\le\prime}\be/S}(X^{\le\prime}\e))\overset{\!\sim}{\to}{ \mathrm{Hom}}_{
S^{\le\prime}}(T\!\times_{S}\!S^{\e\prime},X^{\le\prime}\e), \quad g\mapsto q_{\le X^{\prime}\be,\e S^{\le\prime}\be/S}\circ g_{\le S^{\le\prime}}$$ is a bijection [@ega1 (1.1.8), p. 22]. The pair $(\Re_{S^{\le\prime}\be/S}(X^{\prime}\e), q_{\le X^{\prime}\!,\e S^{\le\prime}\be/S})$ (or, more concisely, the scheme $\Re_{S^{\le\prime}\be/S}(X^{\prime}\e)$) is called the [*Weil restriction of $X^{\prime}$ along $f$*]{}. If $S^{\le\prime}={\mathrm{ Spec}\,}A^{\le\prime}$ and $S={\mathrm{ Spec}\,}A$ are affine, we will write $(\Re_{A^{\le\prime}/A}(X^{\prime}\le),q_{\le X^{\prime}\!,\e A^{\le\prime}\lbe/A})$ for $(\Re_{S^{\le\prime}\be/S}(X^{\prime}\le),q_{\le X^{\prime}\!,\e S^{\le\prime}\lbe/S})$ and refer to $\Re_{A^{\le\prime}/A}(X^{\prime}\le)$ as the [*Weil restriction of $X^{\prime}$ along $A^{\le\prime}/A$*]{}.
If $X^{\prime}$ is an $S^{\e\prime}$-scheme such that $\Re_{S^{\le\prime}\be/S}(X^{\prime}\le)$ exists and $T\to S$ is any morphism of schemes, then $\Re_{\le S^{\le\prime}_{T}\lbe/ T}\le(\e X^{\prime}\!\times_{S^{\prime}}\! S^{\e\prime}_{\le T})$ exists as well and , and yield a canonical isomorphism of $T$-schemes $$\label{wrbc}
\Re_{S^{\le\prime}\be/S}(X^{\prime}\le)\times_{S}T\overset{\!\sim}{\to}\Re_{\le S^{\le\prime}_{T}\lbe/ T}\le(\e X^{\prime}\!\times_{S^{\prime}}\! S^{\e\prime}_{\le T}).$$ Further, if $S^{\le\prime\prime}\to S^{\le\prime}\to S$ are morphisms of schemes and $X^{\prime\prime}$ is an $S^{\le\prime\prime}$-scheme such that $\Re_{\e S^{\prime\prime}\be/S^{\le\prime}\be}\e(X^{\prime\prime})$ exists, then implies that $\Re_{S^{\prime}\be/S}(\Re_{\e S^{\prime\prime}\be/S^{\le\prime}\be}\e(X^{\prime\prime}))$ exists if, and only if, $\Re_{\e S^{\prime\prime}\!/S} (X^{\prime\prime})$ exists. If this is the case, then there exists a canonical isomorphism of $S$-schemes $$\label{wrcomp}
\Re_{S^{\prime}\be/S}(\Re_{\e S^{\prime\prime}\be/S^{\le\prime}\be}\e(X^{\prime\prime}))\overset{\!\lbe\sim}{\to}\Re_{\e S^{\prime\prime}\be/S} (X^{\prime\prime}) .$$
Let $f\colon S^{\e\prime}\to S$ be a finite and locally free (i.e., flat and of finite presentation) morphism of schemes and let $X^{\e\prime}$ be an $S^{\e\prime}$-scheme. By [@blr §7.6, Theorem 4, p. 194], $\Re_{S^{\le\prime}\be/S}(X^{\prime}\e)$ exists if, for each point $s\in S$, every finite set of points of $X^{\le\prime}\times_{S}{\mathrm{ Spec}\,}k(s)$ is contained in an open affine subscheme of $X^{\prime}$. This condition is satisfied if $X^{\e\prime}$ is quasi-projective over $S^{\e\prime}$ [@ega II, Definition 5.3.1 and Corollary 4.5.4]. A weaker condition for the existence of $\Re_{S^{\le\prime}\be/S}(X^{\prime}\e)$ is given in [@bga Theorem 2.15].
\[et-eq\] Let $k$ be a field and let $A$ be a finite local $k$-algebra with residue field $k^{\e\prime}$. Let $X$ be an étale $A$-scheme. Then the following hold.
- $\Re_{k^{\le\prime}\be/A}(X_{\rm s})$ exists and is canonically isomorphic to $X$.
- $\Re_{A/k}(X)$ and $\Re_{\le k^{\le\prime}\be/k}(X_{\rm s}\lbe)$ exist and are canonically isomorphic.
Let $T$ be any $A$-scheme. Since $T_{\rm s}\to T$ is a nilpotent immersion and $X$ is formally étale over $A$, the canonical map ${ \mathrm{Hom}}_{A}\e(T,X\le)\to{ \mathrm{Hom}}_{\e k^{\prime}}\e(T_{\rm s},X_{\rm s}\le),\, g\mapsto g_{\e\rm s},$ is a bijection [@ega $\text{IV}_{4}$, Remark 17.1.2(iv)]. Thus $(\Re_{k^{\le\prime}\be/A}(X_{\rm s}), q_{\le X_{\rm s}\lbe,\e k^{\le\prime}\be/A})=(X,1_{\be X_{\rm s}})$ is the Weil restriction of $X_{\rm s}$ along $k^{\e\prime}/A$. Assertion (i) is now clear. Now, by (i), $\Re_{\le A\lbe/k}(X)$ exists if, and only if, $\Re_{\le A\lbe/k}(\Re_{\e k^{\prime}\be/A}(X_{\rm s}))\simeq \Re_{\e k^{\prime}\be/k}(X_{\rm s})$ exists and, if this is the case, then $\Re_{\le A\lbe/k}(X)$ and $\Re_{\le k^{\le\prime}\be/k}(X_{\rm s}\lbe)$ are canonically isomorphic étale $k$-schemes. Thus it remains to show that $\Re_{\e k^{\prime}\be/k}(X_{\rm s})$ exists. By [@blr §7.6, Theorem 4, p. 194], it suffices to check that any finite set of points of $X_{\rm s}$ is contained is an affine open subscheme of $X_{\rm s}$. Since $X_{\rm s}$ is an étale $k^{\e\prime}$-scheme, $X_{\rm s}$ is isomorphic to a sum $\coprod_{i\in I} {\mathrm{ Spec}\,}k^{\e\prime}_{i}$, where each $k^{\e\prime}_{i}$ is a finite and separable extension of $k^{\e\prime}$ [@ega $\text{IV}_{4}$, Corollary 17.4.2$({\text d}^{\prime}\e)$]. It is now clear that the stated condition holds.
Proof of the main theorem
=========================
An [*extension of local rings*]{} is a flat homomorphism $R\to R^{\e\prime}$ of local rings. Then $R\to R^{\e\prime}$ is faithfully flat and therefore injective [@mat Chapter 2, (4.A), Corollary, p. 27, and (4.C)(i), p. 28]. Let ${\mathfrak m}$ and ${\mathfrak m}^{\e\prime}$ denote the maximal ideals of $R$ and $ R^{\e\prime}$, respectively. Then $R\to R^{\e\prime}$ induces an extension of residue fields $k=R/{\mathfrak m}\hookrightarrow k^{\e\prime}=R^{\e\prime}/{\mathfrak m}^{\e\prime}$.
\[op4\] Let $k$ be a field and let $A$ be a finite local $k$-algebra with residue field $k^{\e\prime}$. Let $G$ be a smooth $A$-group scheme such that $\Re_{\le A/k}(G\le)$ exists. Then $\Re_{\e k^{\le\prime}\be/k}(\pi_{0}(G_{{\rm{s}}}\le))$ exists and there exists a canonical isomorphism of étale $k$-group schemes $\pi_{0}(\Re_{\le A/k}(G\le))\simeq\Re_{\e k^{\le\prime}\be/k}(\pi_{0}(G_{{\rm{s}}}\le))$.
The existence assertion follows from Proposition \[et-eq\](ii). Now, since $G^{\e 0}$ is an open and closed subgroup scheme of $G$, $\Re_{\le A/k}(G^{\e 0})$ is an open and closed subgroup scheme of $\Re_{\le A/k}(G\le)$ by [@blr §7.6, Proposition 2, p. 192]. Further, since $(G^{\e 0})_{\rm s}$ is geometrically connected [@sga3 $\text{VI}_{\text{A}}$, Proposition 2.4(i)], $\Re_{\le A/k}(G^{\e 0})$ is geometrically connected as well by [@cgp Proposition A.5.9][^4]. It follows that $\Re_{\le A/k}(G^{\e 0})=\Re_{\le A/k}(G\le)^{0}$. Now, since $p_{\le G}\colon G\to\pi_{0}(G\le)$ is smooth and surjective, the induced smooth morphism $\Re_{\le A/k}(G\le)\to \Re_{\le A/k}(\pi_{0}(G\le))$ is surjective by [@cgp Corollary A.5.4(1)]. Thus there exists an exact and commutative diagram of sheaves of groups on $(\textrm{Sch}/k)_{{\mathrm{fppf}}}$ $$\begin{CD}
1@>>>\Re_{\le A\be/k}(G\le)^{0}@>>>\Re_{\le A\be/k}(G\le)@>>>\pi_{0}(\Re_{\le A\be/k}(G\le)) @>>>1
\\
@. @| @| @VV\wr V\\
1@>>>\Re_{\le A\be/k}(G^{\e 0})@>>> \Re_{\le A/k}(G\le)@>>>
\Re_{\le A/k}(\pi_{0}(G\le))@>>>1,
\end{CD}$$ which yields a canonical isomorphism of étale $k$-group schemes $ \pi_{0}(\Re_{\le A\be/k}(G\le))\simeq \Re_{\le A/k}(\pi_{0}(G\le))$. Now Proposition \[et-eq\](ii) completes the proof.
[*We may now prove Theorem \[main\].*]{}
The existence assertion follows from Proposition \[et-eq\](ii). Let $A:=R^{\e\prime}\otimes_{R}k=R^{\e\prime}/{\mathfrak m}R^{\e\prime}$, which is a finite local $k$-algebra with residue field $k^{\e\prime}$ and let $S^\prime={\mathrm{ Spec}\,}R^\prime$. Then $G^{\e\prime}_{\! A}:=G^{\e\prime}\!\times_{S^{\le\prime}}\be{\mathrm{ Spec}\,}A$ is a smooth $A$-group scheme. Now, since $\Re_{\le R^{\e\prime}\be/\be R}(G^{\e\prime}\le)$ exists by hypothesis, $\Re_{A/k}(G^{\e\prime}_{\! A}\le)$ exists as well and the map for $(X^{\prime}\be,\lbe S^{\e\prime}\be/S,T\e)=(G^{\e\prime}\be,\lbe R^{\e\prime}\be/R, k\le)$ is an isomorphism of smooth $k$-group schemes $\Re_{\le R^{\e\prime}\be/\be R}(G^{\e\prime}\le)_{{\rm{s}}}\simeq
\Re_{A/k}(G^{\e\prime}_{\! A}\le)$. On the other hand, Proposition \[op4\] yields a canonical isomorphism of étale $k$-group schemes $\pi_{0}(\Re_{A/k}(G^{\e\prime}_{\! A})\e)\simeq\Re_{\le k^{\le\prime}\be/k}(\pi_{0}(G^{\e\prime}_{{\rm{s}}}))$, whence the theorem follows.
Some applications
=================
We present the following consequences of Theorem \[main\].
\[cor2\] Let $R^{\e\prime}\be/R$ be a finite extension of local rings such that the associated extension of residue fields $k^{\e\prime}\be/k$ is [purely inseparable]{}. Let $G^{\e\prime}$ be a smooth $R^{\e\prime}$-group scheme such that $\Re_{\le R^{\e\prime}\be/\be R}(G^{\e\prime}\le)$ exists. Then there exists a canonical isomorphism of étale $k^{\e\prime}$-group schemes $\pi_{0}(\Re_{\le R^{\e\prime}\be/\be R}(G^{\e\prime}\le)_{{\rm{s}}})_{k^{\le\prime}}\simeq\pi_{0}(G^{\e\prime}_{{\rm{s}}}\le)$.
If $\chi\colon\pi_{0}(\Re_{\le R^{\e\prime}\be/\be R}(G^{\e\prime}\le)_{{\rm{s}}})\overset{\!\sim}{\to}\Re_{\le k^{\le\prime}\be/k}(\pi_{0}(G^{\e\prime}_{{\rm{s}}}\le))$ is the isomorphism of Theorem \[main\], then the diagram of étale $k^{\le\prime}$-group schemes $$\begin{CD}
\pi_{0}(\Re_{\le R^{\e\prime}\be/\be R}(G^{\e\prime}\le)_{{\rm{s}}})_{k^{\le\prime}} @>\chi_{ k^{\le\prime}}>>\Re_{\le k^{\le\prime}\be/k}(\pi_{0}(G^{\e\prime}_{{\rm{s}}}\le))_{k^{\le\prime}}\\
@VV\pi_{0}((q_{\le G^{\le\prime}\be,\le R^{\le\prime}\be/\be R}\lbe)_{k^{\le\prime}})V @VV q_{\e\pi_{0}(G^{\le\prime}_{\rm{s}}\le)\lbe,\, k^{\le\prime}/k}V\\
\pi_{0}(G^{\e\prime}_{\textrm{s}}\le) @= \pi_{0}(G^{\e\prime}_{\textrm{s}}\le)
\end{CD}$$ commutes. Further, since ${\mathrm{ Spec}\,}k^{\e\prime}\to{\mathrm{ Spec}\,}k$ is a finite and locally free universal homeomorphism, the right-hand vertical map in the above diagram is an isomorphism[^5]. Thus the left-hand vertical map is an isomorphism as well, which completes the proof.
\[cor1\] Let $R^{\e\prime}\be/R$ be a finite extension of discrete valuation rings with associated residue and fraction field extensions $k^{\le\prime}\be/k$ and $K^{\le\prime}\be/K$. Let $G^{\e\prime}$ be a smooth, separated and commutative $R^{\e\prime}$-group scheme which is a Néron model[^6] of its generic fiber $G^{\e\prime}_{\lbe\eta}$. Then $\Re_{\le R^{\e\prime}\be/\be R}(G^{\e\prime}\le)$ exists, is a Néron model of $\Re_{\le K^{\prime}\be/\be K}\lbe(G^{\e\prime}_{\lbe\eta}\lbe)$ and there exists a canonical isomorphism of étale $k$-group schemes $\pi_{0}(\Re_{\le R^{\e\prime}\be/\be R}(G^{\e\prime}\le)_{{\rm{s}}})\simeq\Re_{\le k^{\le\prime}\be/k}(\pi_{0}(G^{\e\prime}_{\lbe \rm s}\le))$.
The existence of $\Re_{\le R^{\e\prime}\be/\be R}(G^{\e\prime}\le)$, as well as the fact that it is a Néron model of $\Re_{\le K^{\prime}\!/\be K}\lbe(G^{\e\prime}_{\lbe \eta}\lbe)$, follow from [@blr §10.1, proof of Proposition 4, p. 291]. The corollary is now immediate from the theorem.
\[rem\] In the case where $R^{\e\prime}\be/R$ is a finite extension of discrete valuation rings, the isomorphism of Corollary \[cor1\] was discussed in [@lor2 Proposition 3.19] when the associated extension of fraction fields $K^{\le\prime}\be/K$ is Galois, $k$ is perfect and $G^{\e\prime}$ is the Néron model of an abelian variety over $K^{\prime}$. Corollary \[cor2\] is proved in [@bb Proposition 1.1] (see also [@ell proof of Theorem 1]) when $K^{\le\prime}\be/K$ is separable, $G^{\e\prime}$ is as above and $R$ is henselian. The proofs of both results make use of a filtration introduced by B. Edixhoven.
Let $R^{\e\prime}\be/R$ be a finite extension of discrete valuation rings with associated fraction field extension $K^{\le\prime}\be/K$ and residue field extension $k^{\e\prime}\be/k$. Let ${\overline{k}}$ be a fixed algebraic closure of $k$ containing $k^{\e\prime}$ and let ${k^{\le\rm s}}$ and $(k^{\e\prime})^{\rm s}$ denote, respectively, the separable closures of $k$ and $k^{\e\prime}$ inside ${\overline{k}}$. Now let $A_{K^{\le\prime}}$ be an abelian variety over $K^{\le\prime}$ with Néron model $A$ over $R^{\e\prime}$ and recall from [@sga7 IX, (1.2.2)] Grothendieck’s pairing $$\label{gpair}
\pi_{0}(\lbe A_{\le\textrm{s}}\lbe )((k^{\e\prime})^{\rm s})\times\pi_{0}(\lbe A^{\vee}_{\le\textrm{s}})((k^{\e\prime})^{\rm s})\to {{\mathbb Q}}/{{\mathbb Z}}\e,$$ where $A^{\vee}$ is the Néron model of the abelian variety $A^{\vee}_{K^{\le\prime}}$ dual to $A_{K^{\le\prime}}$. When the preceding pairing is perfect, $\pi_{0}(\lbe A_{\le\textrm{s}}\lbe )((k^{\e\prime})^{\rm s})$ and $\pi_{0}(\lbe A^{\vee}_{\le\textrm{s}})((k^{\e\prime})^{\rm s})$ are isomorphic as abelian groups. As noted by D. Lorenzini, it is not known, in general, whether $\pi_{0}(\lbe A_{\le\textrm{s}}\lbe )((k^{\e\prime})^{\rm s})$ and $\pi_{0}(\lbe A^{\vee}_{\le\textrm{s}})((k^{\e\prime})^{\rm s})$ are always isomorphic as abelian groups, i.e., even when is not perfect. See [@lor p. 1, foonote] and [@lor2 p. 2032]. Regarding this question, we observe the following consequence of the previous corollary.
\[cor3\] Let $R^{\e\prime}\be/R$ be a finite extension of discrete valuation rings with associated fraction field extension $K^{\e\prime}\be/K$. Let $A_{K^{\prime}}$ be an abelian variety over $K^{\e\prime}$ with dual abelian variety $A^{\vee}_{K^{\le\prime}}$ and let $A$ and $A^{\vee}$ denote, respectively, the Néron models of $A_{K^{\prime}}$ and $A^{\vee}_{K^{\le\prime}}$ over $R^{\e\prime}$. Assume that Grothendieck’s pairing is perfect. Then there exists an isomorphism of abelian groups $$\pi_{0}(\Re_{\le R^{\e\prime}\be/\be R}(A\le)_{{\rm{s}}})({k^{\le\rm s}})\simeq \pi_{0}(\Re_{\le R^{\e\prime}\be/\be R}(A^{\vee}\le)_{{\rm{s}}}\lbe)({k^{\le\rm s}}).$$
By Corollary \[cor1\], we have $$\pi_{0}(\Re_{\le R^{\e\prime}\be/\be R}(A\le)_{{\rm{s}}})({k^{\le\rm s}})\simeq\Re_{\le k^{\le\prime}\be/k}(\pi_{0}(A_{\le\rm s}\le))({k^{\le\rm s}})=\pi_{0}(A_{\le \rm s}\le)(k^{\le\prime}\be\otimes_{k}\be{k^{\le\rm s}}).$$ Now, by [@bou V, §6, no. 7, Theorem 4, p. A.V.34] and an inductive limit argument, $k^{\le\prime}\otimes_{k}{k^{\le\rm s}}$ is a finite reduced ${k^{\le\rm s}}$-algebra. Consequently (see, e.g., [@am Theorem 8.7, p. 90]), $k^{\le\prime}\otimes_{k}{k^{\le\rm s}}\simeq\prod_{i\in I}k_{i}$ for some finite set $I$, where each $k_{i}$ is a finite field extension of ${k^{\le\rm s}}$. Thus $\pi_{0}(A_{\le \rm s}\le)(k^{\le\prime}\lbe\otimes_{k}\lbe{k^{\le\rm s}})\simeq\prod_{i\in I}
\pi_{0}(A_{\le \rm s}\le)(k_{i})$. Now, since each $k_{i}/{k^{\le\rm s}}$ is purely inseparable, the associated morphism ${\mathrm{ Spec}\,}k_{i}\to{\mathrm{ Spec}\,}{k^{\le\rm s}}$ is a universal homeomorphism. Thus, by [@sga4 VIII, Corollary 1.2], the canonical homomorphism $\pi_{0}(A_{\le \rm s}\le)({k^{\le\rm s}})\to \pi_{0}(A_{\le \rm s}\le)(k_{i})$ is an isomorphism for each $i\in I$. Further, since $(k^{\e\prime})^{\rm s}/{k^{\le\rm s}}$ is also purely inseparable, the canonical homomorphism $\pi_{0}(A_{\le \rm s}\le)({k^{\le\rm s}})\to \pi_{0}(A_{\le \rm s}\le)((k^{\e\prime})^{\rm s})$ is an isomorphism as well. We conclude that $\pi_{0}(A_{\le \rm s}\le)(k^{\le\prime}\!\otimes_{k}\!{k^{\le\rm s}})\simeq \pi_{0}(A_{\le \rm s}\le)((k^{\e\prime})^{\rm s})^{I}$. Similarly, $\pi_{0}(A^{\vee}_{\le \rm s}\le)(k^{\le\prime}\!\otimes_{k}\!{k^{\le\rm s}})\simeq \pi_{0}(A^{\vee}_{\le \rm s}\le)((k^{\e\prime})^{\rm s})^{I}$. Since $\pi_{0}(\lbe A_{\le\textrm{s}}\lbe )((k^{\e\prime})^{\rm s})$ and $\pi_{0}(\lbe A^{\vee}_{\le\textrm{s}})((k^{\e\prime})^{\rm s})$ are isomorphic by the perfectness of , the corollary follows.
The interest of the above corollary is that, when $K^{\le\prime}\!/K$ is separable so that $\Re_{\le K^{\prime}\!/\lbe K}(A_{K^{\prime}}\be)$ is an abelian variety (see [@ed Proposition 4.1]), Grothendieck’s pairing $$\label{gpair2}
\pi_{0}(\Re_{\le R^{\e\prime}\be/\lbe R}(A\le)_{{\rm{s}}})({k^{\le\rm s}})\times\pi_{0}(\Re_{\le R^{\e\prime}\be/\lbe R}(A^{\vee}\le)_{{\rm{s}}}\lbe)({k^{\le\rm s}})\to{{\mathbb Q}}/{{\mathbb Z}}$$ may not be perfect [@bb comment after Corollary 2.2], although, by the corollary, the groups involved are isomorphic when is perfect. In other words, Corollary \[cor3\] suggests that the answer to Lorenzini’s question may well be positive.
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[^1]: A. B. was partially supported by PRAT 2013 ”Arithmetic of Varieties over Number Fields" CPDA 135371/13
[^2]: C.G.-A. was partially supported by Fondecyt grant 1120003.
[^3]: \[foot\] Here the subscript “$\rm s$" indicates “special fiber” rather than the closed point of ${\mathrm{ Spec}\,}A$. We use this notation because it remains unchanged by a change of rings, i.e., if $A^{\prime}$ is another artinian local ring with residue field $k^{\e\prime}$ and $G^{\e\prime}$ is an $A^{\prime}$-group scheme, then it should be clear that $G^{\e\prime}_{\lbe \textrm{s}}$ means $G^{\e\prime}\!\times_{{\mathrm{ Spec}\,}A^{\prime}}\be{\mathrm{ Spec}\,}k^{\e\prime}$.
[^4]: The proof in \[loc.cit.\] remains valid if the quasi-projectivity assumption made there is replaced by the condition that the Weil restrictions involved exist.
[^5]: Proofs of these facts, currently available at <http://arxiv.org/abs/1501.05621v3>, will appear in [@bga]
[^6]: Only assumed to be locally of finite type [@blr §10.1. Definition 1, p. 289].
|
---
abstract: 'Motivated by the STM experimental data on Bi$_{2}$Sr$_{2}$CaCU$_{2}$O$_{8+x}$ which indicate the tunneling conductance asymmetry $\sigma(-V)\neq\sigma(V)$, we report that such a behavior can be explained in terms of the boson fermion model. It has been shown in the recent studies, based on various selfconsistent techniques to capture the many-body effects, that the low energy spectrum of the boson fermion model is featured by an appearance of the pseudogap at $T^{*}>T_{c}$. We argue that the pseudogap structure has to exhibit a particle-hole asymmetry. This asymmetry may eventually depend on the boson concentration.'
address:
- 'Institute of Physics, Maria Curie Skłodowska University, 20-031 Lublin, Poland'
- 'Centre de Recherches sur les Très Basses Températures CNRS, 38-042 Grenoble Cedex 9, France'
author:
- 'T. Domański and J. Ranninger'
title: |
Thermodynamics and tunneling spectroscopy in the pseudogap regime\
of the boson fermion model
---
Introduction
============
Recently there was a considerable amount of studies of the pseudogap phenomenon observed in a variety of experiments on the high temperature superconductors (HTSC) [@Timusk-99]. There are two main theoretical interpretations which are presently widely considered in the literature: (i) the pseudogap as a precursor of the emerging pairing fluctuations, and (ii) the pseudogap understood in terms of some new (hidden) ordering taking place in a vicinity of the superconducting phase. Some selective overview can be found e.g. in the recent monograph [@Carlson-02].
Among theoretical attempts to explain the pseudogap effect of HTSC materials there is a model of itinerant electrons or holes which coexist and interact with the local bound pairs (hard-core bosons) [@Ranninger-85]. It is worth mentioning, that the pseudogap has been foreseen within this boson fermion (BF) model a long time before the convincing experimental data became available (see the last paragraph in section IV of the Ref. [@Micnas-90]).
Pseudogap phase of the BF model is a manifestation of the pairing-wise correlations which start to appear in a system when the transition temperature $T_{c}$ is approached from above (the precursor type interpretation). On a microscopic basis it means that below a certain temperature $T^{*}$ fermions start to couple into the pairs. These are, however, weakly ordered in phase due to the small superfluid stiffness. The phase coherence sets in at a sufficiently low temperature $T_{c} \leq T^{*}$ and then the true superconducting transition occurs. Presence of the incoherent ($T^{*}<T<T_{c}$) or the coherent ($T_{c} \geq T$) fermion pairs is accompanied by either the pseudogap or the true superconducting gap formed around the Fermi energy $\varepsilon_{F}$.
In general, some advanced methods of the many body theory are required to explore a pseudogap phase within any model. It is because the single-particle and the two-particle correlations are then of equal importance. They should be properly treated taking account of possible feedback effects between both channels in a controlled way. In a context of the BF model such requirements were obtained so far via: a) the selfconsistent perturbative investigation [@perturbative; @Ren-98], b) the dynamical mean field theory [@DMFT], and c) the flow equation study [@Domanski-01]. Some alternative way was studied by Micnas [*et al*]{} [@Micnas-01] who considered the Kosterlitz-Thouless criterion for determination of the superconducting transition $T_{c}=T_{KT}$ and identified $T^{*}$ with the mean field estimate $T_{c}^{(MF)}$. Authors were able to reproduce qualitatively the Uemura type plots $T_{c}$ versus $\rho_{s}$.
In this short paper we extend our previous study [@Domanski-01] to analyze the pseudogap shape and its variation with temperature. Direct consequences of the particle-hole asymmetric pseudogap are illustrated on an example of the STM current conductance.
Model and the origin of correlations
====================================
We consider the BF model which is described by the following Hamiltonian [@Ranninger-85; @Micnas-90] $$\begin{aligned}
H^{BF} & = & \sum_{i,j,\sigma} \left( t_{ij} - \mu
\delta_{i,j} \right) c_{i\sigma}^{\dagger} c_{j\sigma}
+ \sum_{i} E_{0} b_{i}^{\dagger} b_{i}
\nonumber \\ & + &
v \sum_{i} \left( b_{i}^{\dagger} c_{i\downarrow}
c_{i\uparrow} + b_{i} c_{i\uparrow}^{\dagger}
c_{i\downarrow}^{\dagger} \right)
\label{BF}\end{aligned}$$ where $E_{0}=\Delta_{B}-2\mu$ and $\Delta_{B}$ is the boson energy, $\mu$ is the chemical potential. The second quantization operators $c_{i\sigma}^{(\dagger)}$ refer to the conduction band particles (electrons or holes) and $b_{i}^{(\dagger)}$ to the composite hard-core bosons (for instance they can represent the trapped electron pairs $b_{i}=d_{i\downarrow}d_{i\uparrow}$). Itinerant fermions propagate between the lattice sites $i$ and $j$ via the hopping integral $t_{ij}$ whereas bosons are assumed to be infinitely heavy. The hard-core property of bosons means that either $0$ or $1$ boson is allowed to occupy a given lattice site. This local constraint can be formally expressed [@Micnas-90] through the following semi-bosonic commutation relations $\left[ b_{i},b_{j}^{\dagger}\right] = \delta_{i,j}
\left(1-2b_{i}^{\dagger}b_{i}\right)$ and $\left[b_{i},b_{j}\right]
=0=\left[b_{i}^{\dagger},b_{j}^{\dagger}\right]$.
Mechanism of superconductivity and all the other forms of correlations in the BF model (\[BF\]) are caused by the boson fermion charge exchange potential $v$. By decaying into the fermion pairs, bosons gain effectively some mobility. If temperature decreases below the critical value $T_{c}$ then (for $dim>2$) some fraction of bosons gets “frozen” into the BE condensate $n_{0}(T) = \frac{1}{N} \left< b_{{\bf q}={\bf 0}}^{\dagger}
b_{{\bf q}={\bf 0}} \right>$. For $n_{0}(T)\neq 0$, fermions are simultaneously driven into the broken symmetry superconducting state. It can be shown [@Micnas-90; @Domanski-01] that the energy gap in the superconducting fermion subsystem is $v \sqrt{n_{0}(T)}$.
For temperatures slightly higher than $T_{c}$ there exist many bosons which occupy the small momenta ${\bf q} \sim {\bf 0}$ states. Because of the interaction $v \sum_{{\bf k},{\bf q}}
\left( b_{\bf q}^{\dagger} c_{{\bf k}+{\bf q}/2\downarrow}
c_{-{\bf k}+{\bf q}/2\uparrow} + h.c. \right)$, these boson states $\left| n_{\bf q} \right>_{B}$ are strongly mixed with the fermion states $\left| n_{{\bf k}+{\bf q}/2\downarrow}
\right>_{F}$ $\left| n_{-{\bf k}+{\bf q}/2\uparrow} \right>_{F}$ and thereby the life time of fermions might be reduced, especially for $|{\bf k}| \sim k_{F}$. In consequence, we expect that the fermion density of states might be suppressed near $\varepsilon_{F}$.
With the on-site boson fermion interaction given in (\[BF\]) one can generate only the isotropic gap/pseudogap. Of course, the HTSC materials are characterized by the anisotropic order parameters of the $d$-wave symmetry with a possible admixture of the $s$-wave component [@mixed_symmetry]. To capture this aspect it is enough to introduce the intersite coupling $v_{i,j} b_{i}^{\dagger} \left( c_{i\downarrow} c_{j\uparrow}
+ c_{j\downarrow}c_{i\uparrow} \right) + h.c.$ when both, the superconducting gap [@Micnas-01; @Domanski-02; @anisotropicBF] and the pseudogap [@Ren-98] become anisotropic. Here we only discuss the results for the isotropic case but, at a price of more difficult numerical computations, the same procedure can be easily extended to the anisotropic pairing.
The effective spectra
=====================
In order to determine the effective fermion and boson spectra of the model (\[BF\]) we utilize the flow equation technique proposed by Wegner [@Wegner-94]. The main idea behind is to disentangle the coupled boson and fermion subsystems via a sequence of canonical transformations $H(l)=e^{-S(l)}He^{S(l)}$, where $l$ is a continuous parameter. We start at $l=0$ by putting $H^{BF} \equiv H(0)$, and proceed till $l=\infty$, when we want to obtain $H(\infty)=H^{F}_{eff}+H^{B}_{eff}$. All the way, from $l=0$ to $l=\infty$, we adjust the operator $S(l)$ according to the Wegner’s prescription [@Wegner-94]. In practice, disentangling of fermion from boson subsystem can be done within an accuracy of the order $v^{3}$ [@Domanski-01]. To simplify the matters we neglect the hard-core constraint and use the pure bosonic relations $\left[ b_{i},b_{j}^{\dagger}
\right] \simeq \delta_{i,j}$ which should be valid for small boson concentrations $n_{B}=\left< b_{i}^{\dagger} b_{i}\right>$.
After the disentangling procedure is finished we obtain the following structure for the boson contribution to the effective Hamiltonian $H^{B}_{eff}=\sum_{\bf q}
\left( E_{\bf q}-2\mu\right)$. The initial boson energy $\Delta_{B}$ is thus transformed into the dispersion $E_{\bf q}$ which is characterized by the width proportional to $v^{2}$ and the effective boson mass comparable with the mass of fermions [@perturbative; @Domanski-01]. $H^{F}_{eff}$ part, on the other hand, is given as $H^{F}_{eff}=\sum_{{\bf k},\sigma} \left(\varepsilon_{\bf k}-\mu
\right) c_{{\bf k}\sigma}^{\dagger}c_{{\bf k}\sigma} + \frac{1}{N}
\sum_{{\bf k},{\bf p},{\bf q}} U_{{\bf k},{\bf p},{\bf q}}
c_{{\bf k}\uparrow}^{\dagger} c_{{\bf p}\downarrow}^{\dagger}
c_{{\bf q}\downarrow}c_{{\bf k}+{\bf p}-{\bf q}\uparrow}$. Renormalization of $\varepsilon_{\bf k}$ with respect to the initial dispersion $\varepsilon_{\bf k}^{0}$ takes place mainly around ${\bf k}_{F}$. There is also induced the long range fermion-fermion interaction $U_{{\bf k},{\bf p},{\bf q}}$ which has somewhat unusual resonant-like character as shown in Figs 7 and 8 of [@Domanski-01] for the BCS $U_{{\bf k},-{\bf k},{\bf q}}$ and for the density-density $U_{{\bf k},{\bf q},{\bf q}}$ channels.
Previously [@Domanski-01] we discussed the fermion spectrum only on a basis of the quasiparticle energy $\varepsilon_{\bf k}$. However, in some cases, a considerable influence may also arise from the fermion-fermion interactions. These interactions are in principle small, $|U_{{\bf k},{\bf p},{\bf q}}|$ is of the order $v^{2}$, so we can treat them perturbatively. The effective dispersion $\overline{\varepsilon}_{{\bf k}\sigma}(l)$ is for $T>T_{c}$ given by $$\begin{aligned}
\overline{\varepsilon}_{{\bf k}\uparrow} = \varepsilon_{\bf k}
+ \frac{1}{N}\sum_{\bf q} U_{{\bf k},{\bf q},{\bf q}}
\left< c_{{\bf q}\downarrow}^{\dagger} c_{{\bf q}\downarrow}
\right> .
\label{correction}\end{aligned}$$ At $T<T_{c}$ one should also consider the other contribution coming from the BCS channel $U_{{\bf k},-{\bf k},{\bf q}}$. We restrict our attention only to the normal phase ($T>T_{c}$).
In the left h.s. panel of figure \[figure1\] we show the density of fermion states $\rho(\omega) \equiv
\frac{1}{N}\sum_{\bf k} \delta \left( \omega - \overline
{\varepsilon}_{\bf k} \right)$. Note, that there appears a pseudogap which deepens with a decreasing temperature $T$. The pseudogap structure has a clear particle-hole asymmetry at all the temperatures. Asymmetry finally disappears at very high temperatures $T \simeq 0.1$ (not shown here). The boson density of states $\frac{1}{N} \sum_{\bf q} \delta
\left( \Omega - E_{\bf q} \right)$ is much less affected by a varying temperature (see Fig. 4 in [@Domanski-01]). However, upon decreasing $T$ we observe (see the right h.s.panel of figure \[figure1\]) a considerable redistribution of boson occupancy $N_{B}(\Omega)=\frac{1}{N} \sum_{\bf q}
\delta \left(\Omega - E_{\bf q} \right) f_{BE} \left(E_{\bf q}
-2\mu,T\right)$, here $f_{BE}$ is the Bose Einstein distribution. By comparing both the panels of figure \[figure1\] we notice that the pseudogap builds up when bosons start populating the low energy states $E_{{\bf q} \simeq {\bf 0}}$.
Asymmetry of the pseudogap structure is mainly controlled by the boson concentration $n_{B}$. Here is a simple argumentation. If the boson energy is located in a center of the fermion band ($\Delta_{B}=0$), then for the exactly half-filled fermion and boson subsystems they both must have symmetric spectra. In particular, the pseudogap would then become symmetric too. For the situation presented in Fig. \[figure1\] we have $n_{F} \simeq 1$, so it can only be the boson concentration $n_{B}$ responsible for the asymmetry of $\rho(\omega)$. A more detailed analysis will be presented in the future publication.
Single particle spectroscopy
============================
Our results, in particular effects of the particle-hole asymmetry, can be well illustrated by calculating the single particle tunneling current $J$. The differential conductance $\sigma(V)=dJ/dV$ as a function of bias voltage $V$ is a direct probe of the density of states below and above the Fermi energy. We use the following expression for the STM current $$\begin{aligned}
J(V) = const \int_{-\infty}^{\infty} d \omega \; \rho(\omega)
\; \left[ f_{FD}(\omega,T) \right.
\nonumber \\
\left. - f_{FD}(\omega-eV,T) \right] ,
\label{J}\end{aligned}$$ where $f_{FD}$ stands for the Fermi Dirac distribution. As usually, we neglect the energy $\omega$ and ${\bf k}$-dependence of the tunneling matrix [@Norman-00].
In figure \[figure2\] we show the conductance $\sigma(V)$ of the STM current (\[J\]) obtained for the same set of parameters as in Fig. \[figure1\]. We obtain the negative-positive asymmetric characteristics because, at low T, the conductance is roughly proportional to the density of states $\rho(\omega)$. Our results agree very well with the experimental data reported by Renner [*et al*]{} [@Renner-96]. Unfortunately, we are unable to pass through $T_{c}$ (we solve the flow equations using the one dimensional tight binding dispersion $\varepsilon_{\bf k}^{0}$ [@Domanski-01] when $T_{c}^{dim=1}=0$). However, for the realistic $dim>2$, we expect the asymmetry to survive even at $T<T_{c}$ as seen experimentally [@Renner-96].
Let us point out the main features of the pseudogap probed experimentally by the STM conductance [@Renner-96]: (i) it is asymmetric, (ii) its magnitude (the peak to peak distance) is almost temperature independent, and (iii) the pseudogap deepens with a decreasing temperature while the coherence peaks gradually start to appear. The BF model is capable to reproduce all these features (i)-(iii). Some other theoretical concepts discussed in the literature to explain $\sigma(V)$, e.g. [@Norman-00] and references cited therein, are dealing with the physics which microscopically is very close to the BF model (\[BF\]).
[**Acknowledgment**]{} T.D. kindly acknowledges hospitality of the J. Fourier University and Centre de Recherches sur les Très Basses Temperatures in Grenoble, where this study was carried out. The work was partly supported by the Polish State Committee for Scientific Research, grant No. 2P03B 106 18.
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|
The development of integrated photonic circuits requires all-optical elements for high-speed processing of light signals. The optical diode (OD) is one of such indispensable elements. Similar to electronic diodes, it allows the flow of light only in one direction and can reduce problems caused by unwanted reflections or interference effects. The successful design of an OD relies on the breaking of time-reversal symmetry [@Potton2004]. The Faraday effect in magneto-optical media is mostly used for this purpose. There has been a considerable progress during recent years in making such devices suitable for integration into waveguides on a chip [@Doetsch2005; @Takeda2008; @Jalas2010]. However, the need to apply a strong external magnetic field still remains a major limiting factor for their development. Alternative approaches to fabricate ODs are based on chiral media such as cholesteric liquid crystals [@Hwang2005] or metamaterials [@Menzel2010; @Zhukovsky2009]. Although their figures of merit are significantly lower, they can be produced with very small sizes and do not require any external fields.
Nonlinear effects attracted a lot of attention as a means of achieving unidirectional transmission. They can be subdivided into two large groups. The first one uses the nonreciprocal conversion of waveguide modes due to asymmetrically placed defects in directional couplers [@Alberucci2008] and quasi-phase-matched gratings [@Gallo2001; @Yu2010]. As an option, the asymmetry can be created by launching additional signals that modulate the refractive index dynamically, leading to indirect photonic transitions [@Yu2009]. The second group uses the folding of resonances in microcavities (MCs) possessing the Kerr nonlinearity and can be implemented in many different configurations [@Chremmos2010]: from photonic crystals with embedded defects of various form and dimensionality [@Soljacic2003; @Zhao2006; @Cai2008] to quasiperiodic [@Biancalana2008; @Grigoriev2010] and aperiodic [@Shadrivov2010; @Zhukovsky2011] structures.
The resonant properties of the MCs allow ODs of smaller sizes to be constructed while keeping a large contrast ratio between the transmission in the forward and backward directions. However, a common problem in the designs based on the usage of MCs is that the high contrast ratio is often achieved at the expense of lowering the maximal transmission in the forward direction. In this Letter, we show how to solve this problem and to create nonlinear ODs with a negligible insertion loss.
It is known that the mirror symmetric MCs demonstrate resonances of perfect transmission. They can be used as a building block to construct more complex structures, and if these MCs are designed to have a common resonance of perfect transmission, the resulting structure will also show it in the linear spectrum [@Zhukovsky2010]. To create asymmetry, it is sufficient to use two such MCs and to modify their sensitivity to the intensity of incident waves. Without any loss of generality, we can consider a multilayered geometry shown in Fig. \[fig1\](a). In this case, the nonlinear response of each MC can be changed either directly by varying the thickness of the nonlinear defect region or by modifying the localization strength via the thickness of the Bragg mirrors. These two degrees of freedom can be rigorously taken into account in the framework of coupled-mode theory [@Bravo-Abad2007; @Haus1984], and we will investigate how to use them efficiently to preserve the perfect transmission in the nonlinear case.
{width="80mm"}
The coupled-mode equations for the mirror symmetric MC can be written as $$\label{eqA}
\frac {dA} {dt} =
- \left[ i
\left(
\omega_0 - \gamma_{ \rm{A} } \frac {|A|^2} {I_{ \rm{A} }}
\right)
+ \gamma_{ \rm{A} }
\right] A
+ \gamma_{ \rm{A} } (u_+ \pm s_- ),$$ $$\label{eqUV}
\left( \begin{array}{c}
u_- \\ s_+
\end{array} \right)
= \mp
\left( \begin{array}{c}
u_+ \\ s_-
\end{array} \right)
+ A
\left( \begin{array}{c}
\pm1 \\ 1
\end{array} \right),$$ where $A(t)$ is the amplitude of the mode $E_{ \rm{A} }(x)$ with the resonant frequency $\omega_0$ and damping constant $\gamma_{ \rm{A} }$, $u_\pm$($s_\pm$) are the amplitudes of the forward and backward propagating plane waves on the left (right) side of the MC \[Fig. \[fig1\](b)\]. The Kerr nonlinearity enters via the characteristic intensity $I_{ \rm{A} }$ and is responsible for the nonlinear shift of resonant frequencies. If the mode $E_{ \rm{A} }(x)$ is even (odd), the lower (upper) signs should be used in these equations.
It is convenient to go into the frequency domain and to derive the transfer matrix ${\bf{M}}_{\rm{A}}$ which relates the plane waves on the opposite sides of the MC as $[u_+, \; u_- ]^{\rm{T}} = {\bf{M}}_{\rm{A}} [s_+, \; s_- ]^{\rm{T}}$ where $$\label{eqTmatrix}
{\bf{M}}_{\rm{A}} =
{\bf{I}} - i
\left(
\frac {\omega - \omega_0} {\gamma_{ \rm{A} }}
+ \frac {
\left|
s_+ \pm s_-
\right|
^2 } {I_{ \rm{A} }}
\right)
\left[
{\begin{array}{cc}
1 & \pm 1 \\
\mp 1 & -1 \\
\end{array}}
\right]$$ and $\bf{I}$ is the identity matrix. Regardless of the nonlinear term, the form of the transfer matrix (\[eqTmatrix\]) satisfies such fundamental physical properties as the time-reversal symmetry (${\bf{M}}_{22} = {\bf{M}}_{11}^*$, ${\bf{M}}_{12} = {\bf{M}}_{21}^* $), the conservation of energy flow ($\det {\bf{M}} = 1$) and in addition the spatial mirror symmetry (${\bf{M}}_{12} = - {\bf{M}}_{21}$).
The transmission spectrum for waves incident from the left can be found as $T_{\rm{A}} = |s_+ / u_+ |^2 = |({\bf{M}}_{\rm{A}})_{11}|^{-2}$ $$\label{eqTransmission}
T_{ \rm{A} } = \frac {I_{ \rm{out} }} {I_{ \rm{in} }} =
\left[
1 + \left(
\frac {\omega - \omega_0} {\gamma_{ \rm{A} }} + \frac {I_{ \rm{out} }} {I_{ \rm{A} }}
\right) ^ 2
\right] ^ {-1}.$$ It follows from the formula (\[eqTransmission\]) that the condition of perfect transmission can be written as $I_{\rm{in}} / I_{\rm{A}} = - \Delta \omega / \gamma_{\rm{A}}$, where $\Delta \omega = \omega - \omega_0$ is the frequency detuning. This condition can be generalized to the case of two or more MCs which have the same resonant frequency but differ in their characteristic intensities and quality factors $Q = \omega_0 / (2 \gamma )$ $$\label{eqCondition}
Q_{\rm{A}} I_{\rm{A}} = Q_{\rm{B}} I_{\rm{B}}.$$ It is useful to introduce the parameter $f > 1$ as a measure of asymmetry between these MCs $f = Q_{\rm{A}} / Q_{\rm{B}} = I_{\rm{B}} / I_{\rm{A}}$. Since larger quality factors generally imply smaller characteristic intensities, the condition (\[eqCondition\]) is not difficult to fulfill.
The hysteresis of transmission for single MCs ’A’ and ’B’ can be computed by using Eq. (\[eqTransmission\]) and is shown in Fig. \[fig2\](a). For any value of the parameter $f$ and the frequency detuning $\Delta \omega$, the resonances of perfect transmission are equally shifted by the Kerr nonlinearity and coincide in both MCs. However, the switching thresholds can be different, and a specific range of frequency detunings exists when only one of the MCs shows the bistable behavior.
{width="80mm"}
The hysteresis of transmission for coupled MCs ’A’ and ’B’ can be found similar to the linear transfer matrix method. After choosing some output intensity, the nonlinear transfer matrices $\bf{M}_{\rm{A}}$ and $\bf{M}_{\rm{B}}$ defined according to Eq. (\[eqTmatrix\]) can be computed in a reversed order, giving the total transfer matrix of the structure and the input intensity as a result. It turns out however that the matrices do not commute in the nonlinear case, and their multiplication leads to a different answer depending on the direction of incidence \[Fig. \[fig2\](b)\]. The effect is maximized when the MCs are of the opposite parity and can be explained in the following way. After reaching the second MC in the system, a part of the signal is reflected and increases the field in the first MC due to constructive interference. When ’A’ is used as the first MC, the small increase causes the switching inside it, and the transmission through both MCs is greatly enhanced. On the contrary, when ’B’ is used as the first MC, the small increase draws it away from the resonance so that the switching to a higher transmission state is strongly suppressed. To confirm these considerations, we performed a time domain simulation \[Fig. \[fig3\]\].
{width="80mm"}
In comparison to other approaches based on the usage of MCs, the proposed design of OD offers several important advantages. First of all, it shows not only a low insertion loss, but also it is stable for small variations of the input intensity and frequency detuning. This is mostly because the resonance of perfect transmission changes its shape and flattens around the optimal set of parameters. Secondly, the switching thresholds for the opposite propagation directions are separated considerably. This prevents the excitation of a higher transmission branch on the hysteresis curve by waves moving in the blocked direction. At the same time, the waves moving in the allowed direction can cause an immediate switching to a perfect transmission even if they have relatively small intensities.
We found that the above properties are more pronounced when the parameter $f$ is in the following range $2 < f < 3$ and the typical contrast ratio between the transmission in the opposite directions varies from $10$ to $20$. Although this number is not very large, the low insertion loss makes a series connection of such diodes to be very favorable so that arbitrary high transmission contrasts can be achieved in principle.
The OD can be implemented in the form of a multilayered structure \[cf. Fig. \[fig1\](a)\]. Using layers of the quarter-wave optical thickness, the corresponding symbolic formula can be written as $$\label{eqSymbolic}
%\underbrace {
({ \rm{HL} }) ^ {p_{ \rm{A} }}
{ \rm{D} } ^ {2m_{ \rm{A} }}
({ \rm{LH} }) ^ {p_{ \rm{A} }}
%}_{ \rm{Cavity \; A} }
\;
%\underbrace {
({ \rm{LH} }) ^ {p_{ \rm{B} }}
{ \rm{D} } ^ {2m_{ \rm{B} }}
({ \rm{HL} }) ^ {p_{ \rm{B} }}
,%}_{ \rm{Cavity \; B} },$$ where ’H’ (’L’) denotes layers with higher (lower) linear refractive index, ’D’ is a defect layer made from a Kerr-nonlinear material, $p_{\rm{A}(\rm{B})}$ and $m_{\rm{A}(\rm{B})}$ are integers. The wavelength used for the quarter wave condition determines the resonance in the vicinity of which the nonreciprocal behavior can be observed. For the optimal performance, the linear refractive indices should satisfy the following relation $(n_{\rm{H}} / n_{\rm{L}}) ^ {p_{\rm{B}}- p_{\rm{A}}} = n_{\rm{D}}$. In this case, the characteristic intensities will be inversely proportional to the width of defect regions, and the parameter $f$ can be found as $f = m_{\rm{A}} / m_{\rm{B}}$.
In conclusion, we proposed how to create an efficient OD by combining two nonlinear MCs of opposite parity. Our design shows a strong nonreciprocal behavior together with a negligible insertion loss and can be fabricated in a variety of configurations such as multilayered structures or photonic crystals with embedded defects.
This work was supported by the German Max Planck Society for the Advancement of Science (MPG).
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|
Heavy fermion superconductors have surprising properties on both the microscopic and the macroscopic level. Charge carrier’s pairing mechanism is unconventional. The vortex state is also rather different from that of s-wave superconductors: there exist unusual asymmetric vortices, and phase transitions between numerous vortex lattices take place[@Sauls1]. For the best studied material UPt$_{3}$ several phenomenological theories have been put forward [@Machida1; @Sauls2; @Joynt] which utilize a multicomponent order parameter. In particular, great effort has been made to qualitatively and quantitatively map the intricate $H-T$ phase diagram. Most attention has been devoted to the region of magnetic fields near $H_{c2}$.
Magnetization curves of UPt$_{3}$ near $H_{c1}$ are also rather unusual (see Fig.1). Theoretically, if the magnetization is due to penetration of vortices into a superconducting sample then one expects $-4\pi M$ to drop with an infinite derivative at $H_{c1}$ (dotted line). On the other hand experimentally $-4\pi M$ continues to increase smoothly (squares and triangles represent rescaled data taken from refs. [@Maple] and [@Zhao] respectively). Such a behavior was attributed to strong flux pinning or surface effects [@Maple]. However both experimental curves in Fig.1, as well as the other ones found in literature, are close to each other if plotted in units of $H_{c1}$. There might be a more fundamental explanation of the universal smooth magnetization curve near $H_{c1}$. If one assumes that fluxons are of unconventional type for which interaction is long range then precisely this type of magnetization curve is obtained.
Magnetization near $H_{c1}$ due to fluxons carrying $N$ units of flux $\Phi
_{0}\equiv hc/2e$, with line energy $\varepsilon $ and mutual interaction $%
V(r)$, is found by minimizing the Gibbs energy of a very sparse triangular lattice: $$G(B)=\frac{B}{N\Phi _{0}}\left[ \varepsilon +3V(a)\right] -\frac{BH}{4\pi },
\label{Gibbs-energy}$$ where $a=(\Phi _{0}/B\sqrt{3})^{\frac{1}{2}}$ is lattice spacing. When $%
V(r)\sim \exp [-\lambda r],$ the magnetic induction has the conventional behavior $B\sim \left[ \log \left( H-H_{c1}\right) \right] ^{-2}$[@Tinkham], while if it is long range, $V(r)\sim 1/r^{n},$ then one finds $%
B\sim \left( H-H_{c1}\right) ^{n+1}$. The physical reason for this different behavior is very clear. For a short range repulsion, if one fluxon penetrated the sample, many more can penetrate almost with no additional cost of energy. This leads to the infinite derivative of magnetization. On the other hand for a long range interaction making a place for each additional fluxon becomes energy consuming. Derivative of magnetization thus becomes finite.
It is generally assumed that although vortices in UPt$_{3}$ differ from the usual Abrikosov vortices in many details [@Tokuyasu1] two important characteristics are preserved. First, their size $\lambda $ is well defined: magnetic field and interactions between vortices vanish exponentially beyond this length. Second, their energy is proportional to $\log \kappa $. However, in this note we show on the basis of topological analysis of a model by Machida [*et al.* ]{}[@Machida2] that there exists an additional class of fluxons which we call magnetic skyrmions. They carry two units of magnetic flux $N=2$ and do not have a singular core, similar to ATC texture in superfluid $^{3}$He [@Salomaa]. We show that their line energy $\varepsilon \approx 2\varepsilon _{0}$, $\varepsilon _{0}\equiv
\left( \Phi _{0}/4\pi \lambda \right) ^{2}$ is independent $\kappa $ and is smaller than that of Abrikosov vortices for strongly type II superconductors like UPt$_{3}$ ($\kappa \sim 50$). Magnetic skyrmion lattice becomes the ground state at low magnetic fields $(H-H_{c1})/H_{c1}\ll 1$. We further find that the repulsion of magnetic skyrmions is, in fact, long range: $%
V(r)\sim 1/r$. This allows us to produce a nice fit to the magnetization curve (solid line in Fig.1) which is universal (independent of $\kappa $).
The order parameter in the weak spin–orbit coupling model of UPt$_{3}$ is a three dimensional complex vector: $\psi _{i}(\vec{r})$[@Machida1]. The Ginzburg–Landau free energy reads: $$F=F_{sym}+\Delta F, \label{Tot-energy}$$ $$\begin{aligned}
&F_{sym}=-\alpha \psi _{i}\psi _{i}^{\ast }+\frac{\beta _{1}}{2}(\psi
_{i}\psi _{i}^{\ast })^{2}+\frac{\beta _{2}}{2}|\psi _{i}\psi _{i}|^{2}
\label{Pot-energy} \\
&+K_{1}\left( |{\cal D}_{x}\psi _{i}|^{2}+|{\cal D}_{y}\psi
_{i}|^{2}\right) +K_{2}\left| {\cal D}_{z}\psi _{i}\right| ^{2}+\frac{1}{%
8\pi }B_{j}^{2}, \label{Grad-energy} \\
&\Delta F=-\gamma |\psi _{x}|^{2}-\lambda |\psi _{z}|^{2}+\frac{\Delta \chi
}{2}|\psi _{i}B_{i}|^{2} \label{Sym-break-pot} \\
&+\sum_{i=x,y}\left[ k_{1}^{i}\left( |{\cal D}_{x}\psi _{i}|^{2}+|{\cal D}%
_{y}\psi _{i}|^{2}\right) +k_{2}^{i}\left| {\cal D}_{z}\psi _{i}\right| ^{2}%
\right] , \label{Sym-break-grad}\end{aligned}$$ where ${\cal D}_{j}\equiv \partial _{j}-i(2{e}/{\hbar c})A_{j}$ and $%
B_{j}=(\nabla \times \vec{A})_{j}$. We separated eq.(\[Tot-energy\]) into a symmetric part $F_{sym}$ which is invariant under the spin rotation group $%
\ $SO(3)$_{spin}$ acting on the index $i$ of the order parameter, and into terms breaking the SO(3)$_{spin}$ symmetry (anisotropy, coupling to antiferromagnetic spin fluctuations and spin-orbit coupling). Although $%
\Delta F$ is crucial in explaining the double superconducting phase transition in UPt$_{3}$ at zero external magnetic field and the shape of $%
H_{c2}(T)$ curve on the $H-T$ phase diagram, it can be considered as a small perturbation in the low temperature superconducting phase (phase B) well below its critical temperature $T\ll T_{c}^{-}\simeq .45$ K and at low magnetic fields $H\simeq H_{c1}$. Indeed, estimation of coefficients of $%
\Delta F$ at $T=T_{c}^{-}/2$ yields: $\gamma /\alpha \simeq .2,\lambda
/\alpha \simeq .05$ and $\left( \frac{\Delta \chi }{2}H_{c1}^{2}\right)
/\left( \frac{\alpha ^{2}}{2\beta _{1}}\right) \simeq 10^{-6}$, and also $%
k\ll K$ [@Machida1]. Therefore, in a certain range of magnetic fields and temperatures there is an approximate O(3) symmetry and we first turn to minimize $F_{sym}$.
In the vacuum of phase B the order parameter is $\vec{\psi}=\psi _{0}(\vec{n}%
+i\vec{m})/\sqrt{2}$,$\psi _{0}^{2}\equiv \alpha /\beta _{1}$, $\;\vec{n}%
\perp \vec{m}$, $\vec{n}^{2}=\vec{m}^{2}=1.$ Stability requires: $\alpha >0$, $\beta _{1}>0$ and $\beta _{2}>-\beta _{1}.$ The symmetry breaking pattern of phase B is as follows. Both the spin rotations SO(3)$_{spin}$ and the U(1) gauge symmetries are spontaneously broken, but a diagonal subgroup U(1) survives. The subgroup consists of combined transformations: rotations by angle $\vartheta $ around the axis $\vec{l}$ $\equiv \vec{n}\times \vec{m}$ accompanied by gauge transformations $e^{i\vartheta }$. Each vacuum state is specified by orientation of a triad of orthonormal vectors $\vec{n},$ $\vec{m%
},$ $\vec{l}$. The vacuum manifold is therefore isomorphic to SO(3). Topological defects might be of two kinds: regular and ”singular”. To find regular topological line defects, it is enough to consider the London approximation [@Salomaa], i.e. to assume that the order parameter gradually changes in space from one vacuum to another. The triad $\vec{n},$ $%
\vec{m},$ $\vec{l}$ then becomes a field. The Abrikosov vortex is a singular defect: it has a core where the modulus of the order parameter vanishes and energy diverges logarithmically. Accordingly, a cutoff parameter – the correlation length – should be introduced and one obtains $\log \kappa $ dependence for the vortex line energy. This fact alone means that if there exists a regular solution it is bound to become energetically favorable for large enough $\kappa $. Below we consider a situation when the external magnetic field is oriented along $z$ axis and all configurations are translationally invariant in this direction. We use dimensionless units: $%
r\equiv \lambda \tilde{r}$, $A\equiv \left( \Phi _{0}/2\pi \lambda \right)
\tilde{A}$, $B\equiv \left( \Phi _{0}/2\pi \lambda ^{2}\right) \tilde{B}$ and $F=(\varepsilon _{0}/2\pi \lambda ^{2})\tilde{F},$ where $\lambda \equiv
(\Phi _{0}/2\pi )\sqrt{\beta _{1}/4\pi \alpha K_{1}}$ (the tilde will be dropped hereafter). The free energy takes form $$F_{L}=1/2\left( \partial _{k}\vec{l}\right) ^{2}+\left( \vec{n}\partial _{k}%
\vec{m}-A_{k}\right) ^{2}+B_{k}^{2} \label{main functional}$$ and the field equations are $$\begin{aligned}
&n_{p}\vec{\nabla}m_{p}\!\!-\vec{A} =\vec{\nabla}\times \left( \vec{\nabla}%
\times \vec{A}\right) =\vec{j}, \label{current-eq} \\
&\Delta \vec{l}-\vec{l}(\vec{l}\cdot \Delta \vec{l})+2j_{k}(\vec{l}\times
\partial _{k}\vec{l}) =0. \label{OP-eq}\end{aligned}$$ Eq.(\[current-eq\]) shows that the superconducting velocity is given by $%
n_{p}\vec{\nabla}m_{p}=-\vec{\nabla}\vartheta $, where the angle $\vartheta $ specifies the orientation of vector $\vec{n}$ or $\vec{m}$ in the plane perpendicular to $\vec{l}$ (see insert in Fig.2). Thus, $\vartheta $ is the superconducting phase.
Now we proceed to classify the boundary conditions. Magnetic field vanishes at infinity, while topology of the orientation of the triad $\vec{n},$ $\vec{%
m},$ $\vec{l}$ at different distant points is described by the first homotopy group of vacuum manifold: $\pi _{1}$(SO(3))$=$Z$_{2}$ [@Salomaa]. It yields a classification of solutions into two topologically distinct classes (”odd” and ”even”). This classification is too weak, however, for our purposes because it does not guarantee nontrivial flux penetrating the plane. We will see that configurations having both ”parities” are of interest. In the presence of the magnetic flux possible configurations are further constrained by the flux quantization condition. The vacuum manifold is naturally factored into SO(3)$\rightarrow $SO(2)$\otimes $S$_{2}$ where the S$_{2}$ is set of directions of $\vec{l}$ and the SO(2) is the superconducting phase $\vartheta $. For a given number of flux quanta $N$, the phase $\vartheta $ makes $N$ windings at infinity, see Fig. 2. The first homotopy group of this part is therefore fixed: $\pi _{1}$(SO(2))$=$Z$.$ If, in addition, $\vec{l}$ is constant, there is no way to avoid singularity in the phase $\vartheta $ where $|\vec{\psi}|=0$. However, general requirement that a solution has finite energy is much weaker. It tells us that the direction of $\vec{l}$ should be fixed only at infinity [@footnote]. The relevant homotopy group is nontrivial: $\pi _{2}$(S$_{2} $)$=$Z. The second homotopy group appears because fixing $\vec{l}$ at infinity (say, up) effectively ”compactifies” two dimensional physical space into S$_{2}.$ Unit vector $\vec{l}$ winds towards the center of the texture. The new topological number is $Q=(1/8\pi )\int \varepsilon _{ij}\,\vec{l}\,\left(
\partial _{i}\vec{l}\times \partial _{j}\vec{l}\right) d^{2}r $. Therefore, all configurations fall into classes characterized by the two integers $N$ and $Q$. For regular solutions, however, these two numbers are not independent. Upon integrating the supercurrent equation, eq.(\[current-eq\]) along a remote contour and making use of the identity $%
\varepsilon _{pqs}l_{p}(\partial _{i}l_{q})(\partial _{j}l_{s})=(\partial
_{i}n_{p})(\partial _{j}m_{p})-(\partial _{i}m_{p})(\partial _{j}n_{p}),$ we obtain: $Q=N/2.$ We call these regular solutions magnetic skyrmions.
The lowest energy solution within the London approximation corresponds to $%
N/2=Q=-1$ (or $N/2=Q=+1$). We analyze a cylindrically symmetric situation and choose the triad $\vec{n},$ $\vec{m},$ $\vec{l}$ in the form: $$\begin{aligned}
\vec{l} &=&\vec{e}_{z}\cos \Theta (\rho )+\vec{e}_{\rho }\sin \Theta (\rho ),
\nonumber \\
\vec{n} &=&\left( \vec{e}_{z}\sin \Theta (\rho )-\vec{e}_{\rho }\cos \Theta
(\rho )\right) \sin \varphi +\vec{e}_{\varphi }\cos \varphi , \label{anzatz}
\\
\vec{m} &=&\left( \vec{e}_{z}\sin \Theta (\rho )-\vec{e}_{\rho }\cos \Theta
(\rho )\right) \,\cos \varphi -\vec{e}_{\varphi }\sin \varphi , \nonumber\end{aligned}$$ where $\rho $ and $\varphi $ are polar coordinates and $\Theta =\widehat{%
\vec{e}_{z}\vec{l}}$. Boundary conditions are: $\Theta (0)=\pi $ and $\Theta
(\infty )=0$. The vector potential is given by $\vec{A}=A(\rho )\vec{e}%
_{\varphi }$. The general form of such a configuration is shown in Fig.2. The unit vector $\vec{l}$ (solid arrows) flips its direction from up to down as it moves from infinity toward the origin. The phase $\vartheta $ (arrow inside small circles in Fig.2) winds twice while completing an ”infinitely remote” circle. If in eq. (\[main functional\]) only the first term were present we would deal with a standard $SO(3)$ invariant nonlinear $\sigma $-model [@Rajaraman]. Being scale invariant, it possesses infinitely many pure skyrmion solutions $\Theta _{s}(\rho ;\delta )=2\arctan (\delta /\rho ),
$ which have the same energy equal to $2$ (in units of $\varepsilon _{0}$) for any size $\delta $ of a skyrmion. However, in the present case the structure of the order parameter is more complex and the above degeneracy is lifted by the second and third terms of eq. (\[main functional\]).
Below we make use of the functions $\Theta _{s}(\rho ;\delta )$ to explicitly construct the variational configurations. We show that as size of these configurations increases the energy is reduced to a value arbitrarily close to the absolute minimum of $\varepsilon _{ms}=2$. Substituting eq.(\[anzatz\]) into eq.(\[main functional\]) and integrating over the $x-y$ plane we obtain the energy of the magnetic skyrmion in the form: $%
\varepsilon _{ms}=\varepsilon _{s}+\varepsilon _{cur}+\varepsilon _{mag}$, where $\varepsilon _{s}\equiv \int \rho d\rho \left( \Theta ^{\prime
2}/2+\sin ^{2}\Theta /2\rho ^{2}\right) $, $\varepsilon _{cur}\equiv \int
\rho d\rho \left[ (1+\cos \Theta )/\rho +A\right] ^{2}$ and $\varepsilon
_{mag}\equiv \int \rho d\rho \left( A/\rho +A^{\prime }\right) ^{2}$. The first term $\varepsilon _{s}$ is the same as in nonlinear $\sigma $-model without magnetic field. It is bound from below by $2$, the energy of a pure skyrmion. The second term $\varepsilon _{cur}$, the ”supercurrent” contribution, is positive definite. One still can maintain zero value of this term when the field $\Theta (\rho )$ is a pure skyrmion $\Theta
_{s}(\rho ;\delta )$ of certain size $\delta $. Assuming this one gets: $%
A(\rho )=-(1+\cos \Theta )/\rho =-2\rho /(\rho ^{2}+\delta ^{2})$. The third term, the magnetic field contribution (which is also positive definite), becomes $\varepsilon _{mag}=8/3\delta ^{2}$. It is clear that when $\delta
\rightarrow \infty $ we obtain energy arbitrarily close to the lower bound: $%
\varepsilon _{ms}\leq 2+8/3\delta ^{2}\rightarrow 2$. Single magnetic skyrmion therefore blows up.
If many magnetic skyrmions are present, then their interactions can stabilize the system. They repel each other, as we will see shortly, and therefore form a lattice. Since they are axially symmetric, the interaction is axially symmetric and thus a triangular lattice is expected. Assume that the lattice spacing is $a$. At the boundaries of the hexagonal unit cells the angle $\Theta $ is zero, while at the centers it is $\pi $. The magnetic field $B$ is continuous on the boundaries. Therefore, to analyze a magnetic skyrmion lattice we should solve eqs.(\[current-eq\])–(\[OP-eq\]) on the unit cell with these boundary conditions demanding that two units of flux pass through the cell (by adjusting the value of magnetic field on the boundary). We have approximated the hexagonal unit cell by a circle of radius $R=3^{3/4}a/\sqrt{\pi }$ and the same area, and performed numerical integration of the equations
$$\begin{aligned}
A^{\prime \prime }+\frac{A^{\prime }}{\rho }-\frac{A}{\rho ^{2}}-A-\frac{%
1+\cos \Theta }{\rho } &=&0, \label{rot-eq1} \\
\Theta ^{\prime \prime }+\frac{\Theta ^{\prime }}{\rho }+\frac{\sin \Theta }{%
\rho }\left( \frac{2+\cos \Theta }{\rho }+2A\right) &=&0, \label{rot-eq2}\end{aligned}$$
which follow from the cylindrically symmetric Ansatz of eq.(\[anzatz\]). Calculations for $R$ from $R=5$ till $R=600$ were done by means of a finite element method. The energy per unit cell in a wide range of $R$ is satisfactory described (deviation at $R=10$ is $1\%$) by the function $%
\varepsilon _{cell}\simeq 2+5.62/R$. Note that in the limit $R\rightarrow
\infty $ we recover our previous variational estimate: $\varepsilon
_{cell}\rightarrow \varepsilon _{ms}=2.$ The dominant contribution to magnetic skyrmion energy at large $\ R$ comes from the first term $%
\varepsilon _{s}$, similar to the analytical variational state described above. The contribution to $\varepsilon _{cell}$ from magnetic field, $%
\varepsilon _{mag}$, is small for large $R$ but becomes significant in denser lattices. The most interesting feature of the solution is that the supercurrent contribution $\varepsilon _{cur}$ to the energy of magnetic skyrmion is negligibly small for all considered values of $R$. This is to be compared with the usual Abrikosov vortex where at high $\kappa $ the total energy is dominated by magnetic and supercurrent contributions which are of the same order of magnitude. Most of the flux goes through the region where the vector $\vec{l}$ is oriented upwards. In other words, the magnetic field is concentrated close to the center of a magnetic skyrmion.
Line energy of Abrikosov vortices $\varepsilon _{v}$ for the present model was calculated numerically (beyond London approximation) in [@our]. For $%
\kappa =20$ and $50$ we obtain $2\varepsilon _{v}/\varepsilon _{ms}\approx
3.5$ and $4.4$ respectively. Therefore we expect that the lower critical field of UPt$_{3}$ is determined by magnetic skyrmions: $h_{c1}=\varepsilon
_{ms}/2N$. Returning to physical units, $$H_{c1}=\Phi _{0}/4\pi \lambda ^{2}. \label{Hc1}$$ To find magnetization, we now utilize eq.(\[Gibbs-energy\]). Interactions among magnetic skyrmions follows easily from the energy of a unit cell of the hexagonal lattice: $V(r)=2(\varepsilon _{cell}-2)/6\simeq 1.87/r$. The resulting averaged magnetic induction, in units of $\Phi _{0}/2\pi \lambda
^{2}$, reads $$B\simeq 0.25\left( H/H_{c1}-1\right) ^{2}.\;\; \label{magnetization}$$ This agrees very well with the experimental results, see Fig.1. For fields higher then several $H_{c1}$ London approximation is not valid anymore since magnetic skyrmions will start to overlap. In this case one expects that ordinary Abrikosov vortices, which carry one unit of magnetic flux, become energetically favorable. The usual vortex picture has indeed been observed at high fields by Yaron[* et al*]{}. [@Yaron] Curiously, our result is similar to conclusions of Burlachkov [*et al.*]{}[@Burlachkov] who investigated stripe-like (quasi one dimensional) spin textures in triplet superconductors. Having established the magnetic skyrmion solution of $%
F_{sym}$ we next estimated how it is influenced by various terms of $\Delta F
$ eqs.(\[Sym-break-pot\])–(\[Sym-break-grad\]). It was found that these perturbations do not lead to destabilization of a magnetic skyrmion.
In conclusion, we have performed a topological classification of the solutions in SO(3)$_{spin}$ symmetric GL free energy. This model, with addition of very small symmetry breaking terms, describes heavy fermion superconductor UPt$_{3}$ and possibly other p-wave superconductors. A new class of topological solutions in weak magnetic field was identified. These solutions, magnetic skyrmions, do not have normal core. At small magnetic fields the magnetic skyrmions are lighter then Abrikosov vortices and therefore dominate the physics. Magnetic skyrmions repel each other as $1/r$ at distances much larger then magnetic penetration depth forming a relatively robust triangular lattice. $H_{c1}$ is reduced by a factor $\log
\kappa $ as compared to that determined by usual Abrikosov vortex (see eq.(\[Hc1\])).
The following characteristic features, in addition to the slope of the magnetization curve, can allow experimental identification of a magnetic skyrmions lattice.
1\. Unit of flux quantization is $2\Phi _{0}$.
2\. Superfluid density $|\vec{\psi}|^{2}$ is almost constant throughout the mixed state. This can be tested using STM techniques.
3\. Due to the fact that there is no normal core, in which usually dissipation and pinning take place, one expects that pinning effects are reduced.
It is interesting to note that our results are actually applicable to another model of UPt$_{3}$ with accidentally degenerate AE representations [@Zhitom1]. Although this model adopts the strong spin–orbit coupling scheme, it has a structure closely related to $F_{sym}$ of eqs.(\[Pot-energy\])–(\[Grad-energy\]) at low temperatures where both order parameters become of equal importance and can be viewed as a single three dimensional order parameter.
The authors are grateful to B. Maple for discussion of results of Ref. 5, to L. Bulaevskii, T.K. Lee and J. Sauls for discussions and to A. Balatsky for hospitality in Los Alamos. The work is supported by NSC, Republic of China, through contract \#NSC86-2112-M009-034T.
J.A. Sauls, Adv. Phys., [**43**]{}, 113 (1994); M. Sigrist, K. Ueda, Rev. Mod. Phys. [**63**]{}, 239 (1991); I.A. Luk’yanchuk, M.E. Zhitomirsky, Supercond. Rev. [**1**]{}, 207 (1995).
K.Machida, M.Ozaki, Phys. Rev. Lett. [**66**]{}, 3293 (1991); K.Machida, T.Ohmi, M.Ozaki, J. Phys. Soc. Jpn. [**62**]{}, 3216 (1993); T.Ohmi, K.Machida; [*ibid*]{} [**65**]{}, 4018 (1996).
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Z.Zhao *et al.*, Phys. Rev. Lett. [**43**]{}, 13720 (1991).
M.Tinkham, [*Introduction to superconductivity*]{} (McGraw–Hill, 1996).
T.A.Tokuyasu, D.W.Hess, J.A.Sauls, Phys. Rev. [**B41**]{}, 8891 (1990); T.A.Tokuyasu, J.A.Sauls, Physica B [**165 & 166**]{}, 347 (1990); Yu.S.Barash, A.S.Mel’nikov, Sov. Phys. JETP [**73**]{} (1), 170 (1991); K.Machida, T.Fujita, T.Ohmi, J. Phys. Soc. Jpn. [**62**]{}, 680 (1993).
K.Machida, T.Ohmi, J. Phys. Soc. Jpn. [**67**]{}, 1122 (1998).
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This follows from the presence of $(\partial _{i}\vec{l}%
)^{2}$ term in $F_{sym}$ of eq. (\[main functional\]), which cannot be ”gauged away” similar to the corresponding term for the SO(2) part.
R.Rajaraman, [*Solitons and instantons* ]{}(North-Holland, 1982).
A.Knigavko, B.Rosenstein, Phys. Rev. [**B58**]{}, 9354 (1998).
U.Yaron [*et al.,*]{} Phys. Rev. Lett. [**78**]{}, 3185 (1998).
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\#1
[**DYNAMICS OF $^1S_0$ DIPROTON FORMATION IN THE REACTIONS $pp\to \{pp\}_s\pi^0$ AND $pp\to \{pp\}_s\gamma $ IN THE GEV REGION** ]{}
Yu.N. Uzikov$^{1 \dag}$
[(1) [*Joint Institute for Nuclear Researches, LNP, Dubna, Moscow reg. Russian Federation* ]{}\
$\dag$ [*E-mail: [email protected]* ]{}]{}
Introduction
============
Quasi-binary reactions $AB\to \{pp\}_sC$ with formation of a proton pair at small excitation energy $E_{pp}=0-3$ MeV, i.e. the $^1S_0$ diproton $\{pp\}_s$, are of great interest at high transferred momenta since transition amplitudes of these reactions require high momentum components of the pp-wave function. In comparison to very similar (in kinematics) reactions $AB\to dC$ with the final deuteron $d$, the reactions with the diproton are expected to give more definite information on short-range NN-dynamics. The reason is that the contribution of non-short range mechanisms related to excitation of the $\Delta$-isobars in intermediate states is expected to be strongly suppressed for the $AB\to \{pp\}_sC$ reactions as compared to the $AB\to dC$ due to isospin symmetry and conservation of angular momentum and parity. So, in the reaction $pd\to \{pp\}_sn$ this suppression is given by the factor $\frac{1}{9}$ [@uz2002]. Furthermore, in the reaction $pp\to \{pp\}_s\pi^0$ the intermediate S-wave $\Delta N$ state is completely forbidden [@niskanen]. Similarly, in the $pp\to \{pp\}_s\gamma$ reaction direct excitation of the $\Delta-$isobar, dominating the $\gamma d\to pn$ reaction via $M1$ transition is also forbidden.
Contrary to those expectations, the cross section of the reactions $pp\to \{pp\}_s\pi^0$ [@kurbatov] and $pp\to \{pp\}_s\gamma $ [@komar08] recently measured in forward direction for beam energies $0.5 - 2.0$ GeV and $0.35 - 0.55$ GeV, respectively, demonstrate prominent peaks in the $\Delta(1232)$-isobar region. In the deuteron breakup reaction $pd\to \{pp\}_sn$ measured in Ref.[@komar2003] the $\Delta(1232)$ peak is non-visible in the energy dependence of the cross section for the backward scattered neutron, however, theoretical analyses [@jhuz2003; @uzjhcw] suggest, that the $\Delta$ contribution dominates in this reaction at $0.5 - 1.3$ GeV.
Observation of the $\Delta$ peaks in the data on the reactions $pp\to \{pp\}_s\pi^0$ [@kurbatov] and $pp\to \{pp\}_s\gamma $ [@komar08] would mean that the high momentum component of the NN-wave function, which might be hidden by the $\Delta$- contribution in the corresponding reactions with the deuteron, is actually rather week. In other words, new data [@kurbatov; @komar08], most likely, confirm the result of the previous analysis of the reaction $pd\to\{pp\}_sn$ [@jhuz2003], which suggests softness of the NN-interaction potential at short distances. To study this conjecture theoretical analysis is required.
Here we present the results of calculations of the differential cross sections of the reactions $pp\to \{pp\}_s\pi^0$ and $pp\to \{pp\}_s\gamma$ at $\theta_{cm}=0^\circ$ within the one-pion exchange model, which includes the subprocesses $\pi^0 p\to \pi^0 p$ and $\pi^0 p\to \gamma p$, respectively. A similar model with the subprocess $\pi^0 d\to np$ was applied earlier to the reaction $pd\to \{pp\}_sn$ [@uzjhcw].
The deuteron breakup reaction $pd\to \{pp\}_sn$
===============================================
The deuteron breakup reaction $pd\to \{pp\}_sn$ was studied at COSY [@komar2003]. A theoretical analisys [@jhuz2003] was performed within the sum of the following mechanisms: one-nucleon-exchange (ONE) with initial and final state interaction included, $\Delta$-isobar excitation ($\Delta$) and single-scattering (SS).
![ The $pd\to \{pp\}_sn$ data [@komar2003] ($\bullet$) at $\theta_{cm}^n=180^\circ$ verus the beam energy in comparison with the ONE (1, 2) and $\Delta$ mechanisms (3,4) calculations from Ref. [@jhuz2003]. Replacement of the Paris NN potential (1,3) by the CD Bonn one (2,4), decreases the ONE contribution, but increases the $\Delta$-contribution. []{data-label="figpd"}](cdparoned.ps){width="100mm"}
The analysis shows (see Fig. \[figpd\]) that at 0.8 GeV the ONE mechanism has a minimum due to repulsive core in the NN-interaction at $r_{NN}\sim 0.5
$fm, but the $\Delta$ contribution has a maximim at 0.6 GeV and completely dominates this reaction. This $\Delta$-maximum is not visible as a bump in the cross section due to a large ONE contribution below the ONE-node. However, only rather soft NN-interaction potential like the CD Bonn one [@Machleidt] provides agreement with the data. When replacing a hard NN-interaction potential (RSC [@Reid], Paris [@paris]) by the soft one (CD Bonn) the ONE contribution decreases, whereas the $\Delta$ contribution increases providing agreement with the COSY data. On the other hand, more hard NN-models like Paris and especially RSC provide too large magnitude of the high momentum components of the NN-wave function and, therefore, lead to strong contradiction with the data especially above 1 GeV (see Ref. [@jhuz2003]). Further analysis [@uzjhcw] within the OPE model with the subprocess $\pi ^0d\to pn$ provided an independent confirmation of the dominant contribution of the $\Delta(1232)$-isobar in this reaction at $0.5 - 1$ GeV and suggested sizable admixture of the ONE mechanism compatible with the CD Bonn model.
The reaction $pp\to \{pp\}_s\pi^0$
==================================
The reaction $pp\to \{pp\}_s\pi^0$ is the simplest inelastic process in the pp-collision, which can reveal underlying dynamics of NN interaction. Restriction to only one pp-partial wave (s-wave) in the final state considerably simplifies a comparison with theory. The reaction $pp\to \{pp\}_s\pi^0$ is very similar kinematically to the reaction $pp\to d\pi^+$, but its dynamics can be essentially different. In fact, quantum numbers of the diproton state ($J^\pi=0^+,\,I=1,\, S=0, \, L=0$) differ from these for the deuteron ($J^\pi=0^+, I=0,\, S=1, L=0,2$). Therefore, transition matrix elements for these two reactions are also different. Due to the generalized Pauli principle and angular momentum and P-pariry conservation only negative parity states are allowed in the reaction $pp\to \{pp\}_s\pi^0$. Thus, for the intermediate $\Delta N$ state only odd partial waves are allowed. In contrast, in the $pp\to d\pi^+$ reaction both negative and positive parity states are allowed and formation of the intermediate S-wave $\Delta N$ state with $J^P=2^+$ leads to a perfect resonance looping in the $^1D_2$ $pp$-partial wave in the respective Argand diagram [@arndt]. Therefore, the relative contribution of the $\Delta$-mechanism to the reaction $pp\to \{pp\}_s\pi^0$ is expected to be suppressed as compared to the reaction $pp\to d\pi^+$. This argument was applied in Ref. [@uzwilk2001] to explain a very small ratio (less of few percents) of the spin-singlet to spin-triplet pn-pairs observed in the LAMPF data [@HGabitch] in the final state interaction region of the reaction $pp\to pn\pi^+$ at proton beam energy 0.8 GeV. Furthermore, since $\Delta-$type mechanisms are of long-range type, reduction of their contribution would mean that other mechanisms, like $N^*$-exchanges [@sharmamitra] which are more sensitive to short-range NN-dynamics, could be more pronounced in the reaction $pp\to \{pp\}_s\pi^0$ as compared to the $pp\to d\pi^+$ reaction [@ponting].
The cross section of the reaction $pp\to \{pp\}_s\pi^0$ was measured recently at energy 0.8 GeV in Ref.[@dymov06] and at beam energies $0.5 - 2.0$ GeV in Ref. [@kurbatov]. At zero angle, the data [@kurbatov] show a broad maximum in the energy dependence of the cross section at $0.5 - 1.4$ GeV. This maximum is similar in shape and position to the well known $\Delta-$ maximum in the reaction $pp\to d\pi^+$. However, a comparison with the microscopical model calculation [@niskanen], which includes $\Delta(1232)$-isobar excitation and s-wave $\pi N$-rescattering, reveals very strong disagreement between the model and the data [@kurbatov] at energies $0.5 - 0.9$ GeV both in the absolute value and shape of energy dependence of the cross section.
![The differential cross section of the reaction $pp\to \{pp\}_s\pi^0$ versus the beam energy at $\theta_{cm}=0^\circ$. The OPE model (full line) is compared with the data: $\bullet$ – [@kurbatov], triangles – [@bilger]. The dashed curve shows the OPE result obtained without the isospin $3/2$ contribution to the amplitude $\pi^0 p\to \pi^0 p$. The calculated cross sections are scaled by the factor 1/6 (see text). []{data-label="figpi0"}](pppppi0s.ps){width="100mm"}
Here [@uzpi02008] we analyse these data employing a simpler model, which includes the subprocess $\pi ^0 p\to \pi^0 p$ and the final state pp$(^1S_0)$-interaction (Fig.\[fig1\]). The formalism is very similar to that developed for the $pd\to \{pp\}_sn$ reaction [@uzjhcw]. We use the impulse approximation, i.e. the amplitude of the reaction $\pi^0p \to \pi^0p$ is taken off the loop inegral sing. Therefore, the cross section of the reaction $pp\to \{pp\}_s\pi^0$ in forward direction is proportional to the forward cross section of the reaction $\pi^0p \to \pi^0p$ taken from the data [@arndt]. The structure formfactor is calculated using the CD Bonn model for pp-interaction [@Machleidt]. The cutoff parameter for the monopole formfactor in the $\pi NN$ vertex is taken as $\Lambda=0.65$ GeV/c. The results presented in Fig.\[figpi0\] by full line show that the observed shape of the peak is in agreement with the dominance of the $\Delta(1232)$-isobar contribution. Indeed, exclusion of the the isospin $3/2$ contribution from the amplitude of reaction $\pi^0p \to \pi^0p$ (dashed curve) leads to strong disagreement with the data. In absolute value the OPE cross section overestimates the data by factor of 6. The main part of this factor can be explained by the employed impulse approximation. Indeed, within the impulse approximation one cannot exclude intermediate $\Delta N$-states of positive parity, which are forbidden in this reaction. In order to exclude these states one needs to consider the $\Delta$-isobar ecxitation explicitely.
The reaction $pp\to \{pp\}_s\gamma$
===================================
Another simplest process which allows to probe fundamental properties of NN system is photoabsorption on two nucleon systems. The deuteron photodisintegration reaction $\gamma d\to pn$ is widely used as a testing ground for different theoretical models of the NN-interaction, however, much less is known on the photodisintegration of the diproton, $\gamma \{pp\}_s\to pp$, or the inverse process of the photoproduction $pp\to \{pp\}_s\gamma$. Whereas in the photodisinegration of the deuteron the M1 magnetic dipole transition dominates at several hundred MeV through the excitation of the $\Delta(1232)$ isobar, in the reaction with the $^1S_0$ diproton M-odd multipoles are forbidden due to angular momentum and parity conservation. Therefore, there is no direct contribution of the intermediate S-wave $\Delta N$ states in the reaction $pp\to \gamma \{pp\}_s$. Non-direct excitation of the $^5S_2$ $\Delta N$ state is possible via the E2 transition [@WNA], but this contribution is expected to be less important than the M1-transition. The OPE model of the reaction $pp\to \{pp\}_s\gamma$ allows to account for the $\Delta$ contributions via the subprocess $\pi^0 p\to p\gamma$. The corresponding OPE diagram is similar to those in Fig. \[fig1\], but with the subproscess $\pi^0 p\to p\gamma$ in the down vertex. The result of the OPE calculations are shown in Fig.\[figgamma\]. One can see that this model explaines the observed in Ref. [@komar08] rise of the cross section almost quantatively. The second bump at 1.6 GeV is caused by the energy dependence of the $\pi^0 p\to p\gamma$ cross section [@arndt] and related to excitation of more heavy nucleon isobars.
![The forward differential cross section $pp\to \{pp\}_s\gamma$ in comparison with the OPE model (curve scaled by the factor 0.8). Data ($\bullet$) are taken from Ref. [@komar08]. []{data-label="figgamma"}](ppppgams.ps){width="90mm"}
Conclusion
==========
Parity and angular momentum conservation exclude the S-wave $\Delta N$-intermediate state from the reaction $pp\to \{pp\}_s\pi^0$. In similar way, the M1 transition, dominating in the $\gamma d\to pn$ reaction at several hundred MeV via excitation of the $\Delta$-isobar, is forbidden in the reaction $pp\to \{pp\}_s\gamma$. This suppression is similar to that in the reaction $pd\to \{pp\}_sn$ [@uz2002; @jhuz2003] as compared to the $pd\to dp$ process. Therefore, one could expect that some features of the short-range dynamics, which, perhaps, are not visible in the reactions with deuteron, $pp\to d\pi^+$ and $\gamma d \to pn$, may reveal themselves in the corresponding reactions with the diproton. The OPE calculations, in agreement with the data show, however, that the $\Delta$-contribuition is still significant in the reactions $pp\to \{pp\}_s\pi^0$ and $pp\to \{pp\}_s\gamma$. It would mean that short-range effect is rather minor itself in these reactions in the considered region.
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abstract: 'Recently, Chen, Hou and Jin used both Abel’s lemma on summation by parts and Zeilberger’s algorithm to generate recurrence relations for definite summations. Meanwhile, they proposed the Abel-Gosper method to evaluate some indefinite sums involving harmonic numbers. In this paper, we use the Abel-Gosper method to prove an identity involving the generalized harmonic numbers. Special cases of this result reduce to many famous identities. In addition, we use both Abel’s lemma and the WZ method to verify and to discover identities involving harmonic numbers. Many interesting examples are also presented.'
author:
- |
Hai-Tao Jin$^1$ and Daniel K. Du$^2$\
$^1$School of Science,\
Tianjin University of Technology and Education,\
Tianjin 300222, P. R. China\
$^2$Center for Applied Mathematics\
Tianjin University, Tianjin 300072, P. R. China\
$^[email protected], $^[email protected]
title: '**Abel’s Lemma and Identities on Harmonic Numbers**'
---
[*Keywords*]{}: harmonic number, Abel’s lemma, Able-Gosper method, Abel-WZ method
[*AMS Classification*]{}: 05A19, 33F10, 11B99
Introduction
============
The objective of this paper is to employ Abel’s lemma on summation by parts and hypergeometric summation algorithms to verify and to discover identities on the harmonic as well as generalized harmonic numbers.
Recall that for a positive integer $n$ and an integer $r$, the generalized harmonic numbers in power $r$ are given by $$H_n^{(r)}=\sum_{k=1}^{n}\frac{1}{k^r}.$$ For convenience, we set $H_n^{(r)}=0$ for $n\leq 0$. As usual, $H_n =H_n^{(1)}$ are the classical harmonic numbers. We also define (see [@Chu2005]) $$H_n(x)=\sum_{k=1}^{n}\frac{1}{k+x}, \quad x\neq -1,-2,\ldots$$ for $n\geq 1$ and $H_n(x)=0$ when $n\leq 0$. Identities involving these numbers have been extensively studied and applied in the literature, see, for example, [@chosri2011; @Chu2005; @GKP1994; @Paule-Schneider2003; @Spies1990]. Also recall that Abel’s lemma [@Abel] on summation by parts is stated as follows.
For two arbitrary sequences $\{a_k\}$ and $\{b_k\}$, we have $$\sum_{k=m}^{n-1} (a_{k+1}-a_k)b_k =\sum_{k=m}^{n-1}a_{k+1}(b_k-b_{k+1})+a_{n}b_{n}-a_{m}b_{m}.$$
For a sequence $\{\tau_k\}$, define the forward difference operator $\Delta$ by $$\Delta \tau_k = \tau_{k+1} - \tau_k.$$ Then Abel’s lemma can be written as $$\label{Abel'lemma}
\sum_{k=m}^{n-1}b_k\Delta a_k=-\sum_{k=m}^{n-1}a_{k+1}\Delta b_k+a_{n}b_{n}-a_{m}b_{m}.$$
Graham, Knuth and Patashnik [@GKP1994] reformulated Abel’s lemma in terms of finite calculus to evaluate several sums on harmonic numbers. Recently, Chen, Hou and Jin [@Chen-Hou-Jin] proposed *the Abel-Gosper method* and derived some identities on harmonic numbers. The idea can be explained as follows. Let $f_k$ be a hypergeometric term, i.e., $f_{k+1}/f_k$ is a rational function of $k$. First, we use Gosper’s algorithm [@PWZ1996] to find a hypergeometric term $a_k$ (if it exists) satisfying $\Delta a_k = f_k$. Then, by Abel’s lemma, we have $$\label{H-trans}
\sum_{k=m}^{n-1} f_k H_k = \sum_{k=m}^{n-1} H_k \Delta a_k = -\sum_{k=m}^{n-1} \frac{a_{k+1}}{k+1}
+ a_n H_n - a_m H_m.$$ Hence we can transform a summation involving harmonic numbers into a hypergeometric summation. For example, let $$S(n) = \sum_{k=1}^{n} H_k.$$ We have $$S(n) = \sum_{k=1}^n H_k \Delta k = - \sum_{k=1}^n (k+1) \Delta H_k + (n+1)H_{n+1} - H_1 = (n+1) H_n - n.$$ In this framework, they combine both Abel’s lemma and Zeilberger’s algorithm to find recurrence relations for definite summations involving non-hypergeometric terms. For example, they can prove the Paule-Schneider identity [@Paule-Schneider2003] $$\sum_{k=0}^n\left(1+3(n-2k)H_k\right){n\choose k}^3 = (-1)^n,$$ and Calkin’s identity [@Calkin1994] $$\sum_{k=0}^n\left(\sum_{j=0}^k{n\choose j}\right)^3 = n 2^{3n-1} +2^{3n} -3n2^{n-2}{2n\choose n}.$$
In this paper, we use the Abel-Gosper method to generalize the following well-known inversion formula (see, for example [@Gould1972 (1.46)]) $$\label{Binom1-eq-1}
\sum_{k} (-1)^{k-1}\binom{n}{k} H_k=\frac{1}{n}.$$ To be specific, we have
\[Main-1\] Let $m,s,p,n$ are nonnegative integers with $n\geq p$ and $m\geq 1$, then $$\label{Main-1-eq}
\sum_{k=p}^{n} (-1)^{k-1}\binom{n}{k} \binom{k}{p} H_{mk+s}(x)= \left\{
\begin{array}{cc}
\frac{(-1)^{p}m^{n-p-1}n!}{(n-p)p!}\sum\limits_{i=1}^{m}\frac{1}{\prod_{u=p}^{n-1}(mu+s+x+i)},&
n>p,\\[7pt]
(-1)^{p-1}H_{mp+s}(x),& n=p.
\end{array}
\right.
$$
It is readily to see that identity reduce to inversion formula by setting $p=0, m=1, s=0$ and $x=0$. More interesting special cases of can be found in Section 2.
In addition, by combining Abel’s lemma with the WZ method, we establish the *Abel-WZ method* to construct identities on harmonic numbers from known hypergeometric identities. For example, we shall reestablish the following identity due to Prodinger [@Prodinger2008]. $$\label{Prodinger2008}
\sum_{k=0}^{n} (-1)^{n-k}\binom{n}{k}\binom{n+k}{k} H_{k}^{(2)}=2\sum_{k=1}^{n}\frac{(-1)^{k-1}}{k^2}.$$
The paper is organized as follows. In Section 2, we shall give a proof of Theorem \[Main-1\] by the Abel-Gosper method. Special cases of Theorem \[Main-1\] and more examples are also displayed. In Section 3, we introduce the Abel-WZ method and then construct many interesting identities on harmonic numbers from hypergeometric identities.
The Abel-Gosper Method {#Sec-Abel-Gosper}
======================
We first make use of the Abel-Gosper method to prove Theorem \[Main-1\].
[[*Proof of Theorem \[Main-1\].*]{}]{} Let $$S_{m,s,p}(n,x)=\sum_{k=p}^{n} (-1)^{k-1}\binom{n}{k} \binom{k}{p} H_{mk+s}(x),$$ and $$F(n,k)=(-1)^{k-1}\binom{n}{k} \binom{k}{p}.$$ By Gosper’s algorithm, we have $$F(n,k)=\Delta_k G(n,k),$$ where $$G(n,k)=\frac{(-1)^k(k-p)}{n-p}\binom{n}{k}\binom{k}{p}.$$ Thus it follows that $$S_{m,s,p}(n,x)=\sum_{k} \Delta_k G(n,k) H_{mk+s}(x).$$ Employing Abel’s lemma and noticing the boundary values, we find that $$S_{m,s,p}(n,x)=-\frac{1}{n-p}\sum_k
(-1)^{k-1}(k+1-p)\binom{n}{k+1}\binom{k+1}{p}\sum_{i=1}^{m}\frac{1}{mk+s+i+x}.$$ For $1\leq i \leq m$, set $$S_i(n)=\sum_k
(-1)^{k-1}(k+1-p)\binom{n}{k+1}\binom{k+1}{p}\frac{1}{mk+s+i+x}.$$ Then Zeilberger’s algorithm (see [@PWZ1996]) returns the recurrence equation $$(mn+s+i+x)S_i(n+1)-m(n+1)S_i(n)=0.$$ By the initial value $$S_i(p+1)=\frac{(-1)^{p+1}(p+1)}{mp+s+i+x},$$ we obtain $$S_i(n)=(-1)^{p+1}\frac{m^{n-p-1}n!}{p!\prod_{u=p}^{n-1}(mu+s+i+x)}.$$ Equation is then established by noticing that $$S_{m,s,p}(n,x)=-\frac{1}{n-p}\sum_{i=1}^{m}S_i(n),\quad n>p$$ and $S_{m,s,p}(p)=(-1)^{p-1}H_{mp+s}(x)$.
Now let us show some special cases of Theorem \[Main-1\]. By setting $m=1$ and $x=0$, reduces to the following identity.
\[Main-1-sp1\] Let $n,p$, $s$ be nonnegative integers and $n>p$, then we have $$\label{Main-1-sp1-eq}
\sum_{k=p}^{n}(-1)^{k-1}\binom{n}{k}\binom{k}{p}H_{k+s}=\frac{(-1)^p \binom{p+s}{s}}{(n-p)\binom{n+s}{s}}.$$
The special cases $p=0$ and $s=0$ of are given in [@Spies1990; @WangWeiPing2010]. By setting $m=2,s=0$ and $x=0$ in , we are led to the following identity .
\[Main-1-sp2\] Let $n, p$ be nonnegative integers and $n>p$, then we have $$\label{Main-1-sp2-eq}
\sum_{k=p}^{n} (-1)^{k-1}\binom{n}{k} \binom{k}{p}
H_{2k}=\frac{(-1)^p}{(n-p)}\left(\frac{1}{2}+\frac{
2^{2n-2p-2}\binom{2p}{p}}{\binom{2n-1}{n-1}}\right).$$
Using the relation $k^2=2\binom{k}{2}+\binom{k}{1}$ and the cases $p=1,2$ of , we arrive at an identity due to Sofo [@Sofo2009]. $$\label{Main-1-sp2-eq1}
\sum_{k} (-1)^{k-1}\binom{n}{k} k^2
H_{2k}=\frac{n}{2(n-1)(n-2)}+\frac{2^{2n-4}}{(n+2)\binom{2n-1}{n-3}}, \quad n>2.$$
Note that we can also derive identities involving the generalized harmonic numbers $H_n^{(r)}$ from Theorem \[Main-1\]. To this end, we need the operators $\L$ and $\D$ which are defined by $\L f(x)=f(0)$ and $\D f(x)=f'(x)$. It is readily to see that $$\L {\D}^m H_n(x)=(-1)^{m}m!H^{(m+1)}_n.$$
By setting $m=1$ and $p=0$ in , we get the following result (see [@Larcombe2005]).
$$\label{Main-2-sp-eq}
\sum_{k=0}^{n} (-1)^{k-1}\binom{n}{k}H_{k+s}(x)=\frac{n!}{n
(s+x+1)_n}.$$
Then applying the operator $\L \D$ to both sides of , we obtain a formula given in [@WangWeiPing2010] $$\label{Main-2-sp-eq-1}
\sum_{k=0}^{n}(-1)^{k-1}\binom{n}{k}H^{(2)}_{k+s}=
-\frac{1}{n}(H_s-H_{n+s})\binom{n+s}{s}^{-1}.$$ Furthermore, applying the operator $\L \D^2$ to both sides of gives $$\sum_{k=0}^{n}(-1)^{k-1}\binom{n}{k}H^{(3)}_{k+s}=
\frac{1}{2n}\Big((H_{n+s}-H_s)^2+H^{(2)}_{n+s}-H^{(2)}_s\Big)\binom{n+s}{s}^{-1}.$$
More generally, leads to the following inversion formula by applying the operator $\L \D^m$ to its both sides.
For positive integers $n$ and $m$, we have $$\label{Main-2-sp-eq-2}
\sum_{k=1}^{n}(-1)^{k-1}\binom{n}{k}H^{(m+1)}_{k}=\frac{1}{n}\sum_{1\leq
j_1\leq j_2\leq\cdots\leq j_m\leq n}\frac{1}{j_1 j_2\cdots j_m}.$$
Setting $s=0$ in and applying the operator $\L {\D}^m$ to its both sides, we have $$(-1)^{m}m! \sum_{k} (-1)^{k-1}\binom{n}{k}
H^{(m+1)}_k=\frac{n!}{n}LD^m\frac{1}{(x+1)_n}.$$ By the partial fraction decomposition $$\frac{1}{(x+1)_n}=\sum_{k=1}^{n}\frac{1}{(x+k)\prod\limits_{1\leq
j\neq k\leq n}(j-k)},$$ we find $$\frac{n!}{n}\L {\D}^m\frac{1}{(x+1)_n}=(-1)^m
m!\frac{1}{n}\sum_{k=1}^{n}\binom{n}{k}\frac{(-1)^{k-1}}{k^m}.$$ Finally, using Dilcher’s formula [@Dilcher1995] $$\sum_{k=1}^{n}\binom{n}{k}\frac{(-1)^{k-1}}{k^m}=\sum_{1\leq j_1\leq
j_2\leq\cdots\leq j_m\leq n}\frac{1}{j_1 j_2\cdots j_m},$$ we arrive at .
Similarly, we can use the Abel-Gosper method to find many other identities. Here are some examples.
For $n\in \mathbb{N}$ and $x\in \mathbb{C}\setminus \{-1,-2,\ldots\}$, we have $$\begin{aligned}
\sum_{k=0}^{n}\frac{(x+1)_k}{k!}H_k&=\frac{1}{x+1}\left(1+\frac{(x+1)_{n+1}}{n!}\left(H_n-\frac{1}{x+1}\right)\right),\\[6pt]
\sum_{k=0}^{n}\frac{k!}{(x+1)_{k}}H_k&=\left\{
\begin{array}{ll}
\frac{1}{(x-1)^2}\left(x-\frac{n!}{(x+1)_{n}}\left((x-1)(n+1)H_{n}+n+x\right)\right), &\mbox{if}\quad x\neq 1,\\[6pt]
\frac{H^2_{n+1}-H^{(2)}_{n+1}}{2},&\mbox{if}\quad x= 1.
\end{array}
\right.\end{aligned}$$
We remark that the second identity also holds when $x$ is a negative integer. In this case, it it equivalent to the following formula (see [@GKP1994 Exercise 6.53]). $$\sum_{k=0}^{n}\frac{(-1)^k}{\binom{m}{k}}H_{k}=
\frac{(-1)^n}{\binom{m}{n}}\left[\frac{n+1}{m+2}H_{n}+\frac{m+1-n}{(m+2)^2}\right]-\frac{m+1}{(m+2)^2},$$ where $m,n\in \mathbb{N}$ and $n\leq m$.
For $n,m,p\in \mathbb{N}$, we have the following three identities. $$\begin{aligned}
&\sum_{k=0}^{n}(-1)^{k-1}\frac{\binom{n}{k}}{\binom{k+p}{p}}H_{k+p}=\frac{n-p(n+p)H_p}{(n+p)^2},\\[6pt]
&\sum_{k=0}^{n}(-1)^{k-1}\frac{k \binom{n}{k}}{\binom{k+p}{p}}H_{k}=\frac{pn(1+H_{p-1}-H_{n+p-2})}{(n+p)(n+p-1)},\quad p\geq 2,\\[6pt]
&\sum_{k=0}^{n}(-1)^{k-1}\frac{k^2 \binom{n}{k}}{\binom{k+p}{p}}H_{k}=\frac{pn((n-p)(H_{n+p-3}-H_{p-1})-(2n-p))}{(n+p)(n+p-1)(n+p-2)},\quad p\geq 3.\end{aligned}$$
We remark that the first formula is due to Sofo [@Sofo2008] and the remaining two are obtained by Chu [@Chu2012].
Using the Abel-Gosper method iteratively, we can prove the following identity.
For $n,p\in \mathbb{N}$ and $n>p$, we have $$\label{new-2}
\sum_{k} (-1)^{k-1}\binom{n}{k} \binom{k}{p}H_k^{2}=\frac{(-1)^p}{n-p}(H_n-2H_{n-p-1}+H_p).$$
The Abel-WZ Method
==================
In this section, we shall illustrate how to combine Abel’s lemma with the WZ method to derive identities on harmonic numbers.
Recall that a pair of hypergeometric functions $(F(n,k),G(n,k))$ is called a WZ pair if the following WZ equation holds $$F(n+1,k)-F(n,k)=G(n,k+1)-G(n,k).$$ For a given $F(n,k)$, the WZ method will give such $G(n,k)$ if it exists, see for example [@PWZ1996]. Now we are ready to describe the Abel-WZ method. Assuming that we have the following hypergeometric identity. $$\sum_k F(n,k)=f(n).$$ In most cases, we can obtain a WZ pair $$\left(\frac{F(n,k)}{f(n)},G(n,k)\right).$$ Then for the sum $S(n)=\sum_{k\geq 0} F(n,k)b_k$, where $b_k$ is harmonic number, we have $$\frac{S(n+1)}{f(n+1)}-\frac{S(n)}{f(n)}=\sum_k (G(n,k+1)-G(n,k))b_k.$$ Denote by $U(n)=\sum_k (G(n,k+1)-G(n,k))b_k$. Then by Abel’s lemma, we have (here we omit the boundary values) $$U(n)=-\sum_k G(n,k+1)\Delta_k b_k.$$ Again, if $\Delta_k b_k$ is hypergeometric, $U(n)$ can be treated by Zeilberger’s algorithm. Moreover, if $U(n)$ can be expressed in closed form, we then establish an identity of the form $$S(n)=f(n)\sum_{k\leq n-1} U(k).$$
We begin by an identity due to Prodinger [@Prodinger2008].
For $n\in \mathbb{N}$, we have $$\label{Prodinger2008}
\sum_{k=0}^{n} (-1)^{n-k}\binom{n}{k}\binom{n+k}{k} H_{k}^{(2)}=2\sum_{k=1}^{n}\frac{(-1)^{k-1}}{k^2}.$$
Denote the left side of by $S(n)$. For $F(n,k)=(-1)^{n-k}\binom{n}{k}\binom{n+k}{k}$, the WZ method gives $$F(n+1,k)-F(n,k)=G(n,k+1)-G(n,k),$$ where $$G(n,k)=\frac{2(-1)^{n-k}k^2{n\choose k}{n+k\choose k}}{(n-k+1)(n+1)}.$$ Multiplying both sides of the WZ equation by $H_{k}^{(2)}$ and summing over $k$ gives $$S(n+1)-S(n)=\sum_{k}(G(n,k+1)-G(n,k))H_{k}^{(2)}.$$ Then applying Abel’s lemma to the right hand side of the above identity and noting the boundary values, we have $$\begin{aligned}
S(n+1)-S(n)&=\sum_k \frac{-G(n,k+1)}{(k+1)^2}\\
&=\sum_{k\geq 0} \big( T(k+1)-T(k) \big) \\[5pt]
&=-T(0)=2\frac{(-1)^n}{(n+1)^2},\end{aligned}$$ where $$T(k)=\frac{2(-1)^{n-k-1}(k+1)^2{n\choose k+1}{n+k+1\choose k+1}}{(n-k)(n+1)^3}.$$ Thus we have $$S(n)=S(0)+2\sum_{k=1}^{n}\frac{(-1)^{k-1}}{k^2}.$$ By the initial value $S(0)=0$, we complete the proof.
The underlying hypergeometric identity of the above theorem is the special case $p=0$ of $$\sum_{k=0}^{n}(-1)^k\binom{n}{k}\binom{n+k}{k}\binom{k}{p}=(-1)^n\binom{n+p}{p}\binom{n}{p},$$ which enables us to establish the following identities.
For $n,p\in \mathbb{N}$ and $n\geq p$, we have $$\begin{aligned}
\sum_{k=0}^{n}(-1)^{n-k}{n\choose k}{n+k\choose k} H_{2k}&=3H_{n}-H_{\lfloor \frac{n}{2}\rfloor}, \\[6pt]
\sum_{k=0}^{n}(-1)^k\binom{n}{k}\binom{n+k}{k}\binom{k}{p}H_k&=(-1)^n\binom{n+p}{p}\binom{n}{p}(2H_n-H_p),\\[6pt]
\sum_{k=0}^{n}(-1)^k\binom{n}{k}\binom{n+k}{k}\binom{k}{p}H_{n+k}&=(-1)^n\binom{n+p}{p}\binom{n}{p}(H_{n+p}+H_n-H_p).\end{aligned}$$ The cases $p=0,1$ of the last two formulas can be found in [@Prodinger2008] and [@Osburn] respectively.
The following contents of this section consist of several selected examples.
From the binomial theorem $\sum_{k}\binom{n}{k}\lambda^{n-k}\mu^k=(\lambda+\mu)^n$, we can derive the following formula due to Boyadzhiev [@Boyadzhiev2009]. $$\label{Boyadzhiev}
\sum_{k=1}^{n}\binom{n}{k}H_k \lambda^{n-k}\mu^k=(\lambda+\mu)^n H_n-
\left(\lambda (\lambda+\mu)^{n-1}+\frac{\lambda^2}{2}(\lambda+\mu)^{n-2}+\cdots+\frac{\lambda^n}{n} \right).$$
From identity $$\sum_{k=p}^{n}{n\choose k}^2{k\choose p}={2n-p\choose n}{n\choose p},$$ we can derive $$\label{eq-wz-hk}
\sum_{k=p}^{n}{n\choose k}^2{k\choose p}H_k={2n-p\choose n}{n\choose p}(2H_n-H_{2n-p}).$$ The special cases $p=0$ and $p=1$ are due to Paule and Schneider [@Paule-Schneider2003].
From the identities $$\sum_{k=0}^{2n}(-1)^k{2n\choose k}^2=(-1)^n{2n\choose n}$$ and $$\sum_{k=0}^{2n}(-1)^k{2n\choose k}^3=(-1)^n\frac{(3n)!}{n!^3},$$ we have $$\begin{aligned}
\sum_{k=0}^{2n}(-1)^k{2n\choose k}^2 H_k&=&(-1)^n{2n\choose n}\frac{H_n+H_{2n}}{2},\label{eq-wz-41}\\ [6pt]
\sum_{k=0}^{2n}(-1)^k{2n\choose k}^3 H_k&=&(-1)^n\frac{(3n)!}{n!^3}\frac{H_n+2H_{2n}-H_{3n}}{2},\label{eq-wz-42}\\ [6pt]
\sum_{k=0}^{2n}(-1)^k{2n\choose k}^3 H_{k}^{(2)}&=&(-1)^n\frac{(3n)!}{n!^3}\frac{H_{n}^{(2)}+H_{2n}^{(2)}}{2}.\label{eq-wz-43}\end{aligned}$$ The last two formulas can be found in [@Driver-Schneider20062] and [@ChuFu2009] respectively.
[**Acknowledgments.**]{} We wish to thank Qing-Hu Hou for helpful comments and discussions. This work was supported by the National Science Foundation of China (Tianyuan Fund for Mathematics) and the Project Sponsored by the Scientific Research Foundation of Tianjin University of Technology and Education.
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abstract: 'We define the 2-groupoid of descent data assigned to a cosimplicial 2-groupoid and present it as the homotopy limit of the cosimplicial space gotten after applying the 2-nerve in each cosimplicial degree. This can be applied also to the case of $n$-groupoids thus providing an analogous presentation of “descent data" in higher dimensions.'
author:
- Matan Prezma
title: Higher Descent Data as a Homotopy Limit
---
[Introduction]{} In this note we reinterpret algebro-geometric information, namely descent data, in a homotopically-invariant way. Given a cosimplicial 2-groupoid $\G^\bullet$, its descent data is (the 2-nerve of) a 2-groupoid $\mb{Desc}(\G^\bullet):=Tot_r(\N\G^\bullet)$ (where $Tot_r$ means “totalization without degeneracies") whose path components coincide with the set of descent data modulo the gauge equivalence relation (see [@BGNT] and also [@Ye1 Definitions 1.4, 1.5]). We show that this 2-groupoid is (canonically equivalent to) the homotopy limit $\operatorname{holim}_{\Sim}\N\G^\bullet$ where $\N$ is the 2-nerve applied on each level. Thus, given a weak equivalence of cosimplicial 2-groupoids $\G^\bullet\rar \H^\bullet$, the map $\mb{Desc}(\G^\bullet)\rar \mb{Desc}(\H^\bullet)$ is a weak equivalence of 2-groupoids; this generalizes [@Ye1 Theorem 0.1]. We know of two situations in which this setup can arise.
The first concerns Maurer-Cartan equations. Consider a cosimplicial DGLA, which shows up for instance as the Čech construction for a sheaf of nilpotent parameter DGLAs. Taking the Deligne 2-groupoid (which encodes solutions to Maurer-Cartan equations) of each cosimplicial degree gives rise to a cosimplicial 2-groupoid. As follows from [@Ye2 Theorem 0.4], a quasi-isomorphism of cosimplicial pronilpotent DGLAs of quantum type (i.e. concentrated in degrees $\geq -1$) induces a weak equivalence of cosimplicial 2-groupoids.
The second is in the classification of $\G$-gerbes for a sheaf of groups $\G$ (see [@Br1], [@Br2]). There, the cosimplicial 2-groupoid arises via the Čech construction (with respect to a cover) from the sheaf of 2-groups (or crossed modules) $\G \rar Aut(\G)$ and the descent data approximates isomorphism classes of $\G-$gerbes. In some cases, for example when the cover totally trivializes all $\G$-gerbes, $\pi_0\mb{Desc}(\G)$ will classify all $\G-$gerbes and a refinement will yield a weak equivalence of cosimplicial 2-groups.
Descent data is intimately related to non-abelian cohomology. For this reason, the role of codegeneracies is degenerate and we can consider the restricted totalization (see §\[Tot\]) which simplifies the homotopical framework. This eliminates the difficulty arising from the fact that the cosimplicial simplicial set gotten by taking the 2-nerve of each level of a cosimplicial 2-groupoid need not be Reedy fibrant (see [@Ja Example 9]) and gives an argument which is also valid for the case of $n$-groupoids; this is discussed in §\[n-descent\].
I would like to thank Amnon Yekutieli for introducing and motivating the question at hand and to Yonatan Harpaz for a useful discussion.
[Totalization and Restricted Totalization]{}\[Tot\] Let $\Sim$ be the category whose objects are non-empty finite ordinals $[0],[1],...,[n]$ where $[n]=\{0,1,...,n\}$ and whose morphisms are weakly order preserving functions. Every morphism in $\Sim$ is a composition of face maps $d^i:[n-1]\rar [n]$ and degeneracies $s^i:[n+1]\rar [n]$, $i=0,...,n$. A simplicial set is a functor $X:\Sim^{op}\rar Set$ and we write $X_n:=X([n])$, $d_i:=X(d^i)$, $s_i:=X(s^i)$. Write $\sset$ for the category whose objects are simplicial sets and whose morphisms are natural transformations.\
A *cosimplicial object* in a category $\mathcal{C}$ is a functor $\Sim\rar \cl{C}$. In particular, a *cosimplicial simplicial set* is a cosimplicial object in $\sset$. We write $\csset$ for the category whose objects are cosimplicial simplicial sets and whose morphisms are natural transformations. If $X$ is a cosimplicial object we will denote the object assigned to $[n]$ by $X^n$. The maps $d^i:=X(d^i)$ and $s^i:=X(s^i)$ are called cofaces and codegeneracies respectively. The *cosimplicial standard simplex* $\Delta$ has $\Delta^n$ in its $n$-th cosimplicial degree and cofaces and codegeneracies induced by precomposition. For $X,Y\in\csset$, the product $X\times Y$ is the cosimplicial simplicial set with $(X\times Y)^n:=X^n\times Y^n$ and for $A\in\sset$, we write, by abuse of notation, $A$ for the constant cosimplicial simplicial set with $A^n:=A$ for all $n$ and cofaces and codegeneracies being identities.
The category $\csset$ is enriched over simplicial sets. Given $X,Y\in \csset$, the ‘internal hom’ $\uline{\csset}(X,Y)$ is the simplicial set whose $n$-simplices are $$\uline{\csset}(X, Y)_n=\csset(X\times \Delta^n,Y)$$ Here, $X\times \Delta^n$ is the product of $X$ with the constant cosimplicial simplicial set $\Delta^n$.
With this enrichment, $\csset$ is a simplicial category in the sense of [@GJ II,2.1] or in our terminology, *tensored and cotensored* over $\sset$ (see [@GJ II, 2.5]). For $A,B\in\csset$ we denote the tensor and cotensor functors by $$A\times (-): \sset\rar \csset\;\;and\;\;B^{(-)}:(\sset)^{op} \rar \csset$$ respectively; these are the left adjoints of $\uline{\csset}(A,-)$ and $\uline{\csset}(-,B)$.
The *totalization* $Tot:\csset\rar \sset$ is the simplicial set $Tot(X^\bullet)=\uline{\csset}(\Delta^\bullet,X^\bullet)$.
We let $\Sim_{r}$ denote the subcategory of $\Sim$ with the same objects but only injective maps i.e. compositions of face maps $d^i$. A *restricted cosimplicial object* in a category $\cl{C}$ is a functor $\Sim_{r}\rar \cl{C}$; it is also called a *semi-cosimplicial object* by some authors. In particular, a restricted cosimplicial object in $\sset$ is called a *restricted cosimplicial simplicial set*. There is an obvious ‘restriction’ functor $r:\csset\rar \rcsset$ and in particular we have $r\Delta\in\rcsset$. The category $\rcsset$ is again enriched over simplicial sets so that if $X,Y\in \rcsset$ we denote $\uline{\rcsset}(X,Y)\in \sset$. Its $n$-simplices are $\uline{\rcsset}(X,Y)_n:=\rcsset(X\times r\Delta^n,Y)$ and given $\theta:[m]\rar [n]$ in $\Sim$, the map $\theta^*:\uline{\rcsset}(X,Y)_n\rar \uline{\rcsset}(X,Y)_m$ is induced by composing with the map $\theta_*:r\Delta^m\rar r\Delta^n$. Simplicial identities hold since their opposites hold in $\Delta$. The arguments in [@GJ II,2.5] may be used verbatim to show that $\rcsset$ is tensored and cotensored over $\sset$.
The *restricted totalization* is the functor $Tot_r:\rcsset\rar \sset$ defined by $Tot_r(X^\bullet)=\uline{\rcsset}(r\Delta^\bullet,X^\bullet)$.
More generally we can use ends (see [@Ma IX.5]) to get:
\[abstract tot\] Let $\C$ be a category cotensored over simplicial sets.
1. The *totalization* of $\G^\bullet \in \C^{\Sim}$ is the object of $\C$ is given by the end $$Tot(\G^\bullet):=\int_{[n]\in \Sim}(\G^n)^{\Delta^n}.$$
2. The *restricted totalization* of $\G^\bullet \in \C^{\Sim_r}$ is the object of $\C$ is given by the end $$Tot_r(\G^\bullet):=\int_{[n]\in \Sim_r}(G^n)^{\Delta^n}.$$
[Model structures]{} We assume the reader is familiar with the definition of a model category. Let us shortly spell out the definition of a simplicial model category.
A model category $\cl{M}$ is called *simplicial* if it is enriched with tensor and cotensor over $\sset$ and satisfies the following axiom [@Qu II.2 SM7]: If $f:A\rar B$ is a cofibration in $\cl{M}$ and $i:K\rar L$ is a cofibration in $\sset$ then the map $$q:A\otimes L\coprod_{A\otimes K} B\otimes K\rar B\otimes L$$
1. is a cofibration;
2. is a weak equivalence if either
1. $f$ is a weak equivalence in $\cl{M}$ or
2. $i$ is a weak equivalence in $\sset$.
A category $\cl{R}$ is called a *Reedy category* if it has two subcategories $\cl{R}_+,\cl{R}_-\subseteq \cl{R}$ and a *degree* function $d:ob(\cl{R})\rar \alpha$ where $\alpha$ is an ordinal number such that:
- Every non-identity morphism in $\cl{R}_+$ raises degree;
- Every non-identity morphism in $\cl{R}_-$ lowers degree;
- Every morphism in $\cl{R}$ factors uniquely as a map in $\cl{R}_-$ followed by a map in $\cl{R}_+$.
The category $\Sim$ is a Reedy category with $\Sim_+=\Sim_{inj}\;\;(=\Sim_{r})$, $\Sim_-=\Sim_{surj}$ and the obvious degree function.
Let $\cl{R}$ be a Reedy category and $\cl{C}$ any category. Given a functor $X:\cl{R}\rar \cl{C}$ and an object $n\in\cl{R}$ we set $X^n:=X(n)$ (to relate to the case $\cl{R}=\Delta$), and define the $n$-th *latching object* to be
$$L^nX=colim_{\mathbb{L}(\cl{R})}X^s$$ where $\mathbb{L}(\cl{R})$ is the full subcategory of the over category ${\cl{R}}_+/n$ containing all objects except the identity $id_n$.
Dually, define the $n$-th *matching object* to be
$$M^nX=lim_{\mathbb{M}(\cl{R})}X^s$$ where $\mathbb{M}(\cl{R})$ is the full subcategory of the under category $n/\cl{R}_-$ containing all objects except $id_n$. We have natural morphisms $$L^nX\rar X^n\rar M^nX.$$ The importance of a Reedy structure on $\cl{R}$ is due to the following:
[@Re] Let $\cl{R}$ be a Reedy category and $\cl{M}$ a model category. The functor category $\cl{M}^{\cl{R}}$ admits a structure of a model category, called *Reedy model structure* in which a map $X\rar Y$ is a
- Weak equivalence iff $X^n\rar Y^n$ is a weak equivalence in $\cl{M}$\
for every $n$.
- Cofibration iff the map $L^nY\coprod_{L^nX}X^n\rar Y^n$ is a cofibration in $\cl{M}$\
for every $n$.
- Fibration iff the map $X^n\rar M^nX\times_{M^nY}Y^n$ is a fibration in $\cl{M}$\
for every $n$.
In particular, an object $X$ is
- Fibrant iff $X^n\rar M^n X$ is a fibration in $\cl{M}$ for every $n$.
- Cofibrant iff $L^nX\rar X^n$ is a cofibration in $\cl{M}$ for every $n$.
Moreover [@An Theorem 4.7], if the model structure on $\cl{M}$ is simplicial, so is the Reedy model structure on $\cl{M}^\cl{R}$.
The Kan-Quillen model structure $\sset_{K-Q}$ and the Reedy structure on $\Sim$ (respectively $\Sim_{r}$) induce a simplicial model structure on $\csset$ (respectively $\rcsset$).
\[simplex\] The object $X=\Delta^\bullet\in \csset$ is Reedy cofibrant. The map $L^nX\rar X^n$ is the inclusion $\partial \Delta^n \hrar \Delta^n$ which is a cofibration of simplicial sets.
Next, we recall another model structure on $\rcsset$.
The simplicial enrichment of $\rcsset$ can be extended to a simplicial model structure, called the *projective model structure*, in which a map $X\rar Y$ is a
- weak equivalence if for each $n$, $X^n\rar Y^n$ is a weak equivalence.
- fibration if for each $n$, $X^n\rar Y^n$ is a Kan fibration.
- cofibration if it has the left lifting property with respect to trivial fibrations.
In particular, $X$ is a fibrant object iff $X^n$ is a Kan complex for every $n$.
Suppose $\cl{R}$ is a Reedy category and $\cl{M}$ is a model category. In general, if the projective model structure on $\cl{M}^\cl{R}$ exists (e.g. when $\cl{M}$ is sufficiently nice) it will be very different than the Reedy model structure. However, in special cases the two may coincide.
If $\cl{R}=\cl{R}_+$ the projective and Reedy model structures on $\cl{M}^\cl{R}$ coincide.
In this case, for every $X\in \cl{M}$ the $n$-th matching object $M^nX$ equal the terminal object, being the limit over the empty diagram, so that a map $X\rar Y$ is a Reedy fibration iff $X^n\rar Y^n$ is a fibration in $\cl{M}$. This means that the two model structures have the same classes of weak equivalences and fibrations, and hence coincide.
For $\cl{R}=\Sim_{r}$ we obtain:
\[restricted simplex\] The Reedy and projective model structures on $\rcsset$ coincide. Thus, an object $X\in \rcsset$ is Reedy fibrant iff $X^n$ is a Kan complex for each $n$.
\[rDelta is cofibrant\] By example \[simplex\], $\Delta^\bullet$ is Reedy cofibrant in $\csset$ and since the indexing category defining $L^n\Delta^\bullet$ depends only on $\Sim_+=\Sim_{r}$, we have $L^n\Delta^\bullet =L^n r\Delta^\bullet$. Thus, the map $L^n r\Delta^n\rar r\Delta^n$ is again the inclusion $\partial \Delta^n \hrar \Delta^n$ so that $r\Delta^\bullet$ is Reedy cofibrant in $\rcsset$.
[2-Groupoids]{}\[2-groupoids\]
A (strict) *2-groupoid* is a groupoid-enriched (small) category in which all morphisms are invertible.
Explicitly, a 2-groupoid consists of:
- a set of *objects*;
- for every pair of objects $x,y$, a set of *1-morphisms*, written as $f:x\rar y$; and, for every object $x$, a distinguished 1-morphism $1_x:x\rightarrow x$;
- for every pair of 1-morphisms $f,g:x\rightarrow y$ a set of *2-morphisms*, written as $a:f\Rightarrow g$; and, for every 1-morphism $f$, a distinguished 2-morphism $1_f:f\Rightarrow f$
together with a composition law for 1-morphisms and vertical and horizontal composition laws for 2-morphisms (denoted by $*$ and $\circ$ respectively) subject to three axioms, expressing associativity of composition and left and right unit laws and in addition satisfy the ‘interchange law’: $$(b *a)\circ (b' *a')=(b'\circ b)*(a'\circ a).$$ All morphisms are invertible with respect to these composition laws.
There are 2-categorical analogues for the notions of a functor and natural transformation. However, since 2-categories have 2-morphisms, an additional ‘level of arrows’ reveals itself, namely, the one of modifications. There is some ambiguity regarding these notions, since one can consider also their weak versions. For the sake of clarity, we spell out the definitions we use, which are taken from [@Gr I,2.2;I,2.3].
\[cartesian closed structure\] Let $\G,\H$ be a pair of 2-groupoids.
(I) A (strict) *2-functor* $\Phi:\G\rar \H$ is a groupoid-enriched functor between the underlying groupoids of $\G$ and $\H$. Explicitly, $\Phi$ assigns:
- to each object $x\in \G$, an object $\Phi x\in \H$,
- to each 1-morphism $f:x\rar y\in \G$, a 1-morphism $\Phi f:\Phi x\rar \Phi y\in \H$,
- to each 2-morphism $a:f\Rightarrow g\in \G$ a 2-morphism $\Phi a:\Phi f\Rightarrow \Phi g\in \H$
and this assignment respects all compositions and units.
(II) Given a pair of 2-functors $\Phi,\Psi:\G\rar \H$ between 2-groupoids, a (strict) *2-natural transformation* $\Theta:\Phi\Rightarrow \Psi$ consists of a 1-morphism $\eta_x:\Phi x\Rightarrow \Psi x$ for every object $x\in \G$ which is natural in the sense that for every 2-morphism $a:f\Rightarrow g$ in $\G$, the diagram $$\xymatrix{\Phi(x)\ar[d]_{\eta_x}\rtwocell_{\Phi f}^{\Phi g}{\Phi a} & \Phi(y)\ar[d]^{\eta_y} \\ \Psi(x)\rtwocell^{\Psi f}_{\Psi g}{\Psi a} & \Psi(y)}$$ is commutative in that $1_{\eta_y}\circ \Phi a= \Psi a \circ \eta_x$ as 2-morphisms in $\H$.
(III) Given a pair of 2-natural transformations $\eta,\theta:\Phi\Rightarrow \Psi$, a (strict) *modification* $\mu:\eta \Rrightarrow \theta$ consists of a 2-morphism $\mu_x:\eta_x\Rightarrow \theta_x$ in $\H$ for every object $x\in \G$ such that for every 1-morphism $f:x\rar y$ in $\G$, the diagram $$\xymatrix{\Phi(x)\ar[d]_{\Phi f}\rtwocell_{\eta_x}^{\theta_x}{\mu_x} & \Psi(x)\ar[d]^{\Psi f} \\ \Phi(y)\rtwocell^{\eta_y}_{\theta_y}{\mu y} & \Psi(y)}$$ is commutative in the sense of (II).
We denote by ${\kern0.05ex\vcenter{\hbox{\footnotesize\ensuremath{2}}}\kern0.05ex\mathcal{G}pd}$ the category of 2-groupoids and strict 2-functors between them. The collection of 2-functors from $\G$ to $\H$, their 2-natural transformations and their modifications is naturally a 2-category (see [@Gr 2.3]) which is in fact a 2-groupoid because of invertibility of 1-and 2-morphisms in the codomain $\H$. We denote this $2$-groupoid by $\uline{{\kern0.05ex\vcenter{\hbox{\footnotesize\ensuremath{2}}}\kern0.05ex\mathcal{G}pd}}(\G,\H)$.
\[cartesian closeness\][(cf. [@Gr 2.3])]{} The category ${\kern0.05ex\vcenter{\hbox{\footnotesize\ensuremath{2}}}\kern0.05ex\mathcal{G}pd}$ is cartesian closed with respect to $\uline{{\kern0.05ex\vcenter{\hbox{\footnotesize\ensuremath{2}}}\kern0.05ex\mathcal{G}pd}}(\G,\H)$.
Let $\Sim_{\leq n}$ be the full subcategory of $\Sim$ with objects $[0],...,[n]$ and let $\sset_{\leq n}$ be the category of functors $(\Sim_{\leq n})^{op}\rar \mathcal{S}et$. Objects of $\sset_{\leq n}$ are called *$n$-truncated simplicial sets*. The inclusion $\Sim_{\leq n}\rar \Sim$ induces a ‘truncation functor’ $tr_n:\sset\rar \sset_{\leq n}$ which admits right and left adjoints $cosk_n:\sset_{\leq n}\rar \sset$ and $sk_n:\sset_{\leq n}\rar \sset$ respectively. We denote by $Cosk_n:\sset\rar \sset$ the composition $cosk_n\circ tr_n$ and by $Sk_n$ the composition $sk_n\circ tr_n$. The functor $Sk_n$ takes a simplicial set and creates a new simplicial set from its $n$-truncation by adding degenerate simplices in all levels above $n$; it is the simplicial analogue of the $n$-skeleton of a $CW$ complex. The functor $Cosk_n$ has a more involved simplicial description; it is the simplicial analogue of the $(n-1)$th Postnikov piece $P_{n-1}$.\
By abstract considerations, one can show that $Cosk_n$ is right adjoint to $Sk_n$. Thus, a map $X\rar Cosk_n Y$ correspond precisely to a map $Sk_n X\rar Y$.
A simplicial set $X$ is called *$n$-coskeletal* if the canonical map $X\rar Cosk_n X$ is an isomorphism.
In particular, given an $n$-truncated simplicial set $X$, $cosk_n X$ is an $n$-coskeletal simplicial set. Thus, in order to define an $n$-coskeletal simplicial set it is enough to define its $n$-truncation.
In §\[vertices and homotopies\] we intend to interpret the definition of descent data in terms of the 2-nerve. In order to improve readability, we now rewrite the definition of [@MS §2] with the notations relevant for our formulae.
\[2-nerve def\] The *2-nerve* is the functor $\N:{\kern0.05ex\vcenter{\hbox{\footnotesize\ensuremath{2}}}\kern0.05ex\mathcal{G}pd}\rar \sset$ which takes a 2-groupoid $\G$ to the 3-coskeletal simplicial set $\N\G$ whose
- 0-simplices are the objects of $\G$;
- 1-simplices are the morphisms of $\G$;
- 2-simplices are triangles of the form $$\xymatrix @=0.65pc{& & x_1\ar[ddrr]^{g_{12}} & &\\& & & & \\ x_0\ar@{}[uu]_(0.4){\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\kern0.05ex\vcenter{\hbox{\LARGE\ensuremath{\Uparrow}}}\kern0.05ex}{a}_{012}}\ar[uurr]^{g_{02}}\ar[rrrr]_{g_{01}} & & & & x_2 }$$
where $g_{ij}:x_i\rightarrow x_j$ and $\alpha:g_{02}\Rightarrow g_{12}\circ g_{01}$ are 1-and 2-morphisms (respectively) in $\G$;
- 3-simplices are commutative tetrahedra of the form
$\xymatrix@!=2pc{ & x_3 & \\ & x_1\ar@{}[u]^(0.2){\;\;\;\;\overset{{\kern0.05ex\vcenter{\hbox{\Large\ensuremath{\Rightarrow}}}\kern0.05ex}}{a_{013}}}_(0.2){\overset{{\kern0.05ex\vcenter{\hbox{\Large\ensuremath{\Rightarrow}}}\kern0.05ex}}{a_{123}}}\ar[u]|{g_{13}}\ar[dr]|(0.3){g_{12}} & \\ x_0\ar@{}[ur]|{\overset{{\kern0.05ex\vcenter{\hbox{\LARGE\ensuremath{\Rightarrow}}}\kern0.05ex}}{a_{023}}}\ar[uur]^{g_{03}}\ar@{.>}[ur]|(0.7){g_{01}}\ar[rr]|{g_{02}}\ar@{}[u]_(0.5){\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\kern0.05ex\vcenter{\hbox{\LARGE\ensuremath{\Uparrow}}}\kern0.05ex}a_{012}} & & x_2\ar[uul]_{g_{23}}}$ $\xymatrix{\\a_{ijk}:g_{ik}\Rightarrow g_{jk}\circ g_{ij}.}$
Commutativity of this tetrahedron means that the diagram of 2-morphisms
$$\label{tetrahedron commutativity}
\xymatrix@=1.5pc{g_{03}\ar[rr]^{a_{023}}\ar[d]_{a_{013}} && g_{23}\circ g_{02}\ar[d]^{1_{g_{23}}\circ a_{012}} \\ g_{13}\circ g_{01}\ar[rr]_{a_{123}\circ 1_{g_{01}}} && g_{23}\circ g_{12}\circ g_{01}}$$
commutes.
We will need four well-known properties of the 2-nerve:
[@MS]\[2-nerve\]
1. $\N$ preserves products.
2. For every 2-groupoid $\G$, $\N\G$ is a Kan complex.
3. A map of 2-groupoids $\G\rar \H$ is a weak equivalence iff $\N\G\rar \N\H$ is a weak equivalence of simplicial sets.
4. $\N$ admits a left adjoint $\W:\sset\rar {\kern0.05ex\vcenter{\hbox{\footnotesize\ensuremath{2}}}\kern0.05ex\mathcal{G}pd}$, called the *Whitehead 2-groupoid*. \[W\]
The category ${\kern0.05ex\vcenter{\hbox{\footnotesize\ensuremath{2}}}\kern0.05ex\mathcal{G}pd}$ admits a natural simplicial enrichment via $\N\uline{{\kern0.05ex\vcenter{\hbox{\footnotesize\ensuremath{2}}}\kern0.05ex\mathcal{G}pd}}(-,-)$. This enrichment is nicely behaved in the following sense:
\[2gpd is t-c\] The simplicially-enriched category ${\kern0.05ex\vcenter{\hbox{\footnotesize\ensuremath{2}}}\kern0.05ex\mathcal{G}pd}$ is tensored and cotensored over $\sset$.
We need to verify the conditions of [@GJ II,2.1]. The functor $((-)\times \G)\circ \W$ is a left adjoint to $\N{\kern0.05ex\vcenter{\hbox{\footnotesize\ensuremath{2}}}\kern0.05ex\mathcal{G}pd}(\G,-)$ and the functor $\uline{{\kern0.05ex\vcenter{\hbox{\footnotesize\ensuremath{2}}}\kern0.05ex\mathcal{G}pd}}(W(-),\H)$ is a left adjoint to $\N\uline{{\kern0.05ex\vcenter{\hbox{\footnotesize\ensuremath{2}}}\kern0.05ex\mathcal{G}pd}}(-,\H)$.
It is worth notice that [@No 5.1] shows the inner hom described in \[cartesian closed structure\] does not induce a simplicial model category structure on ${\kern0.05ex\vcenter{\hbox{\footnotesize\ensuremath{2}}}\kern0.05ex\mathcal{G}pd}$ via setting the simplicial mapping space to be $\N\uline{{\kern0.05ex\vcenter{\hbox{\footnotesize\ensuremath{2}}}\kern0.05ex\mathcal{G}pd}}(\G,\H)$. However in the current note, the main homotopical part is done in the category of (cosimplicial) simplicial sets so that we do not need a full-fledged homotopy theory of ${\kern0.05ex\vcenter{\hbox{\footnotesize\ensuremath{2}}}\kern0.05ex\mathcal{G}pd}$.
We shall need a slight generalization of proposition \[2gpd is t-c\]:
If $\C$ is tensored and cotensored over $\sset$ and $I$ is any small category, then the functor category $\C^I$ is again tensored and cotensored over $\sset$.
Denote the inner homs and their adjoints by
$\xymatrix{\C\ar@<-0.5ex>[r]_{\uline{\C}(X,-)} & \sset\ar@<-0.5ex>[l]_{X\otimes(-)}}$ and $\xymatrix{\C^{op}\ar@<-0.5ex>[r]_{\uline{\C}(-,Y)} & \sset\ar@<-0.5ex>[l]_{Y^{(-)}}}$.
One defines for $\undertilde{X}\in \C^I$ and $K\in \sset$, $(\undertilde{X}\otimes K)_\alpha :=\undertilde{X}_\alpha\otimes K$ and $(\undertilde{X}^K)_\alpha:=(\undertilde{X}_\alpha)^K$ for every $\alpha\in I$. Then, $\uline{\C^I}(\undertilde{X},\undertilde{Y})_n=\C^I(\undertilde{X}\otimes\Delta^n, \undertilde{Y})$ with the obvious face and degeneracy maps provides the desired inner hom.
\[cosimplicial 2-groupoids are t-c\] The categories ${\kern0.05ex\vcenter{\hbox{\footnotesize\ensuremath{2}}}\kern0.05ex\mathcal{G}pd}^\Sim$ and ${\kern0.05ex\vcenter{\hbox{\footnotesize\ensuremath{2}}}\kern0.05ex\mathcal{G}pd}^{\Sim_r}$ are tensored and cotensored over simplicial sets.
The last corollary enables us to express the totalization as an end via Definition \[abstract tot\].
By abuse of notations, we denote by $\N,\;\W$ the prolongation of the 2-nerve and Whitehead 2-groupoid functors to the categories ${\kern0.05ex\vcenter{\hbox{\footnotesize\ensuremath{2}}}\kern0.05ex\mathcal{G}pd}^{\Sim},\;\csset$ (respectively). Since (level-wise) coproducts define the tensoring (over $\sset$) in ${\kern0.05ex\vcenter{\hbox{\footnotesize\ensuremath{2}}}\kern0.05ex\mathcal{G}pd}$ and $\csset$ and $\W$ commutes with coproducts, the premisses of [@GJ Lemma 2.9(1)] are satisfied and we have:
\[Enriched adjunction\] There is an enriched adjunction $$\uline{{\kern0.05ex\vcenter{\hbox{\footnotesize\ensuremath{2}}}\kern0.05ex\mathcal{G}pd}^{\Sim}}(\W \G^\bullet,\H^\bullet) \cong \uline{\csset}(\G^\bullet,\N\H^\bullet)$$
[Descent data of cosimplicial 2-groupoids]{}\[vertices and homotopies\] Following [@BGNT], descent data of a cosimplicial crossed groupoid is defined in [@Ye1]. Since crossed groupoids can be viewed precisely as 2-groupoids (e.g. as a special case of [@BH1]), a translation leads to the following:
(cf. [@Ye1 Definition 1.4]) Given a cosimplicial 2-groupoid $\G^\bullet=\{\G^n\}$, a *descent datum* is a triple $(x,g,a)$ in which:
1. $x$ is an object of $\G^0$;
2. $g:d^1x\rightarrow d^0x$ is a 1-morphism in $\G^1$ and
3. $a:d^1g\Rightarrow d^0g\circ d^2g$ is a 2-morphism in $\G^2$.
such that $$\label{tetrahedron}
\tag{twisted 2-cocycle}
(1_{d^1d^0g}\circ d^3 a)*d^1a=(d^0a \circ 1_{d^2d^2g})*d^2a.$$
Let $(x,g,a)$ be a descent datum of $\G^\bullet$. Write $x_i\equiv x_i^{(1)}\; (i=0,1)$ for the object of $G^1$ corresponding to the vertex $(i)$ of $\Delta^1$, i.e. $x_i=d^jx$ where $\{j\}=\{0,1\}\setminus \{i\}$; thus $g:x_0\rightarrow x_1$. Similarly, write $x_i\equiv x_i^{(2)}\; (i=0,1,2)$ and $g_{ij}\equiv g_{ij}^{(2)} \; (0\leq i<j\leq 2)$ for (respectively) the object and 1-morphism of $\G^2$ corresponding to the vertex $(i)$ and edge $(ij)$ of $\Delta^2$. In other words, $x_i =d^kd^jx$ where $\{j<k\}=\{0,1,2\}\setminus \{i\}$ and $g_{ij}=d^kg$ where $\{k\}=\{0,1,2\}\setminus \{i,j\}$; thus $g_{ij}:x_i\rightarrow x_j$ and $a:g_{02}\Rightarrow g_{12}\circ g_{01}$. Finally, write $x_i \equiv x_i^{(3)}\; (i=0,...,3)$, $g_{ij}\equiv g_{ij}^{(3)} \;(i<j)$ and $a_{ijk}\equiv a_{ijk}^{(3)}\;(i<j<k)$ for (respectively) the object, 1-morphism and 2-morphism of $\G^3$ corresponding to the vertex $(i)$, edge $(ij)$ and face $(ijk)$ of $\Delta^2$; thus $g_{ij}:x_i\rightarrow x_j$ and $a_{ijk}:g_{ik}\Rightarrow g_{jk}\circ g_{ij}$. With these notations in mind, one can immediately see that the twisted cocycle condition corresponds precisely to the commutativity of a tetrahedron $t$ in $\G^3$ as in \[2-nerve def\]. Thus, such triples are in 1-1 correspondence with diagrams of simplicial sets of the form
$$\label{vertex}
\xymatrix{\Delta^0\ar[d]_x\ar@<0.5ex>[r]\ar@<-0.5ex>[r] & \Delta^1\ar[d]_g\ar@<1ex>[r]\ar[r]\ar@<-1ex>[r] & \Delta^2\ar[d]_a\ar@<1ex>[r]\ar@<0.35ex>[r]\ar@<-0.35ex>[r]\ar@<-1ex>[r] &\Delta^3\ar[d]_t\\
\N\G^0\ar@<0.5ex>[r]\ar@<-0.5ex>[r] & \N\G^1\ar@<1ex>[r]\ar[r]\ar@<-1ex>[r] & \N\G^2\ar@<1ex>[r]\ar@<0.4ex>[r]\ar@<-0.4ex>[r]\ar@<-1ex>[r] &\N\G^3\;\;.}$$
Since $\N\G^n$ is 3-coskeletal, diagrams as above are in turn the 0-simplices $$Tot_r(r\N\G^\bullet)_0=\uline{\rcsset}(r\Delta^\bullet,r\N\G^\bullet)_0=\rcsset(r\Delta^\bullet,r\N\G^\bullet).$$
\[definition of gauge\](cf. [@Ye1 definition 1.5]) Let $\mb{d}=(x,g,a),\;\mb{d'}=(x',g',a')$ be a pair of descent data of $\G^\bullet$. A *gauge transformation* $\mb{d}\leadsto \mb{d'}$ is a pair $(f,c)$ in which:
1. $f:x\rightarrow x'$ is a 1-morphism in $\G^0$ and
2. $c: d^0f\circ g_{01} \Rightarrow g'_{01}\circ d^1f$ is a 2-morphism in $\G^1$ (see diagram \[c\])
$$\label{c}
\xymatrix{x_0\ar[r]^{g_{01}}\ar@{}|{{\kern0.05ex\vcenter{\hbox{\Large\ensuremath{\Swarrow}}}\kern0.05ex}c}[dr]\ar[d]_{f_0} & x_1
\ar[d]^{f_1\quad (f_0:=d^1f,\;f_1:=d^0f)}
\\ x'_0\ar[r]_{g'_{01}} & x'_1}$$
such that the prism in $\G^2$ $$\label{p}
\xymatrix@=2.5pc
{& x_1\ar@{.>}|{f_1}[ddd]^(0.72){\quad}="0"\ar|{g_{12}}[dr] &\\
x_0\ar|{f_0}[ddd]\ar|{g_{01}}[ur]\ar|{g_{02}}[rr]\ar@{}[dddr]^(0.45){\quad}="1"
&\ar@{=>}@/^.6pc/^{\quad c_{02}}"0";"1" {} ^{} \ar@{}[u]_(.3){{\kern0.05ex\vcenter{\hbox{\Large\ensuremath{\Uparrow}}}\kern0.05ex}a_{012}} & x_2\ar|{f_2}[ddd]^{\quad (c_{ij}: f_j\circ g_{ij}\Rightarrow g'_{ij}\circ f_i)}\\
\ar@{}[r]^{{\kern0.05ex\vcenter{\hbox{\Large\ensuremath{\Swarrow}}}\kern0.05ex}c_{01}} &\ar@{}[r]^{{\kern0.05ex\vcenter{\hbox{\Large\ensuremath{\Swarrow}}}\kern0.05ex}c_{12}} & \\
& x'_1\ar@{.>}|{g'_{12}}[dr] & \\ x'_0\ar@{.>}|{g'_{01}}[ur]\ar|{g'_{02}}[rr] &\ar@{}[u]_(.3){{\kern0.05ex\vcenter{\hbox{\Large\ensuremath{\Uparrow}}}\kern0.05ex}a'_{012}} & x'_2}$$ is commutative in the sense of \[tetrahedron commutativity\].
Let $Desc(\G^\bullet)$ denote the set of descent data of $\G^\bullet$. The relation $\mb{d} R\mb{d'}\Leftrightarrow \exists \mb{d}\leadsto\mb{d'}$ is an equivalence relation on $Desc(\G^\bullet)$ and we denote by $\oline{Desc}(\G^\bullet)$ its quotient (cf. [@Ye1], definition 1.8). We now claim that:
\[translation\] For any cosimplicial 2-groupoid $\G^\bullet$, there is a (natural) isomorphism $$\oline{Desc}(\G^\bullet)\cong\pi_0Tot_r(r\N\G^\bullet)$$
In order to prove Theorem \[translation\] we would like to view a gauge transformation $\mb{d}\leadsto \mb{d'}$ as a path between two vertices of $Tot_r(r\N\G^\bullet)$ but there is a slight problem. Given a pair of descent data, thought of as 4-tuples $\mb{d}=(x,g,a,t)$ and $\mb{d'}=(x',g',a',t')$ of the form \[vertex\], a path between them is an element of $$Tot_r(r\N\G^\bullet)_1=\uline{\rcsset}(r\Delta^\bullet,r\N\G^\bullet)_1=\rcsset(r\Delta^\bullet\times r\Delta^1,r\N\G^\bullet)$$ that restricts to $\mb{d}$ and $\mb{d'}$ via the maps $d^1,d^0:\xymatrix@1{\Delta^0\ar@<0.5ex>[r]\ar@<-0.5ex>[r] & \Delta^1}$. Since $\N\G^n$ is 3-coskeletal, such elements correspond to diagrams of the form $$\label{path}
\xymatrix{\Delta^0\times\Delta^1\ar[d]_f\ar@<0.5ex>[r]\ar@<-0.5ex>[r] & \Delta^1\times\Delta^1\ar[d]_{c'}\ar@<1ex>[r]\ar[r]\ar@<-1ex>[r] & \Delta^2\times\Delta^1\ar[d]_{p'}\\
\N\G^0\ar@<0.5ex>[r]\ar@<-0.5ex>[r] & \N\G^1\ar@<1ex>[r]\ar[r]\ar@<-1ex>[r] & \N\G^2}$$ that restrict to that restrict to $(x,g,a)$ and $(x,g,a)$.
The last diagram carries an automatic ‘triangulation’. The map $c'$ is a diagram in $\G^1$ of the form $$\label{c'}
\xymatrix{x_0\ar[r]^{g_{01}}\ar|{h}[dr]^{{\kern0.05ex\vcenter{\hbox{\Large\ensuremath{\Swarrow}}}\kern0.05ex}}_{{\kern0.05ex\vcenter{\hbox{\Large\ensuremath{\Swarrow}}}\kern0.05ex}}\ar[d]_{f_0} & x_1\ar[d]^{f_1} \\ x'_0\ar[r]_{g'_{01}} & x'_1}$$ which is a triangulation of \[c\]; and similarly, the map $p'$ is a diagram in $\G^2$ which is a triangulation of \[p\].
There are two possible solutions for that. The first (which was suggested by the referee) is to change the framework into crossed complexes, relying on [@BH2 Theorem 2.4] and obtain a description of gauge transformations as maps of crossed complexes. The second, which we will adopt for the sake of simplicity, is to notice the following:
\[path-gauge\] Every gauge transformation $\mb{d}\leadsto \mb{d'}$ gives rise to a canonical path in $Tot_r(\N\G^\bullet)$ between $\mb{d}$ and $\mb{d'}$ and every such path gives rise to a canonical gauge transformation.
Given a path between $\mb{d}$ and $\mb{d'}$, represented by a triple $(f,c',p')$ as in \[path\], one can compose the 2-morphisms appearing in $c'$ and in the squares of $p'$ to obtain a triple $(f,c,p)$ and hence a gauge transformation $\mb{d}\leadsto \mb{d'}$. Conversely, given a gauge transformation $(f,c):\mb{d}\leadsto \mb{d'}$, one obtains, from condition \[p\] of definition \[definition of gauge\] a prism $p$ in $\G^2$. Then, by inserting the 1-morphism $f_1\circ g_{01}$ as the diagonal in \[c\] and $1_{f_1\circ g_{01}}$ in the upper triangle, one obtains a diagram of the form \[c’\] and a similar procedure on \[p\] yields a prism $p'$. The triple $(f,c',p')$ is the resulting path.
Expressing the totalization as an end allow us to reveal its higher structure:
\[2-groupoid structure\] For a cosimplicial 2-groupoid $\G^\bullet$, there are natural isomorphisms
1. $Tot(\N\G^\bullet)\cong \N Tot(\G^\bullet)$;
2. $Tot_r(r\N \G^\bullet)\cong \N Tot_r(r\G^\bullet)$;
(see definition \[abstract tot\]).
We only prove (1) as the proof of (2) is identical. Since $\N$ is a right adjoint, it commutes with limits. Relying on [@Ma IX.5], $$\begin{split}
\N Tot(\G^\bullet)=N\left( \displaystyle\int_{[n]\in \Sim} (\G^n)^{\Delta^n}\right) \cong \displaystyle\int_{[n]\in\Sim}\N\left( (\G^n)^{\Delta^n}\right)\cong \displaystyle\int_{[n]\in\Sim}(\N\G^n)^{\Delta^n}=Tot(\N\G^\bullet)
\end{split}$$ where the last isomorphism comes from [@GJ II, Lemma 2.9(2)] relying on the fact that $\W$ commutes with arbitrary coproducts.
Thus, we define:
\[descent 2-groupoid\] Given a cosimplicial 2-groupoid $\G^\bullet$, its *descent 2-groupoid* is\
$\mb{Desc}(\G^\bullet):=Tot_r(\N\G^\bullet).$
Since $Tot_r(r\N\G^\bullet)$ is a Kan complex (being the 2-nerve of a 2-groupoid), $\pi_0Tot_r(\N\G^\bullet)=Tot_r(\N\G^\bullet)_0/\sim$ where $\mb{d}\sim \mb{d'}$ iff there is a path between them. By lemma \[path-gauge\] this equivalence relation is equal to the gauge equivalence relation.
[Invariance of descent data]{}\[Invariance of descent data\] Theorem \[translation\] enables us to use homotopy-theoretic tools to prove invariance of descent data under weak equivalence. We need one more simple theorem:
\[holim\] For any cosimplicial 2-groupoid $\G^\bullet$, there is a (natural) weak equivalence $Tot_r(\N\G^\bullet)\simeq \operatorname{holim}_{\Sim}\N\G^\bullet$,
In the simplicial model category $\rcsset_{proj}$, the homotopy limit (over $\Sim_{r}$) of a fibrant object can be described as the internal mapping space from a weakly contractible cofibrant object [@Hi Theorem 19.4.6(2)]. In our case, $\N\G^\bullet$ is fibrant and $r\Delta^\bullet$ is (weakly contractible and) cofibrant (see remark \[rDelta is cofibrant\]). Thus, $Tot_r(r\N\G^\bullet)=\uline{\rcsset}(r\Delta^\bullet, \N\G^\bullet)\simeq \operatorname{holim}_{\Sim_{r}}r\N\G^\bullet$. By ([@DF Lemma 3.8]), $\operatorname{holim}_{\Sim_{r}}r\N\G^\bullet\sim \operatorname{holim}_{\Sim}\N\G^\bullet$.
In light of definition \[descent 2-groupoid\] and the previous theorem it now follows that:
A weak equivalence of cosimplicial 2-groupoids $\G^\bullet\rar \H^\bullet$ induces a weak equivalence of 2-groupoids $\mb{Desc}(\G^\bullet)\rar \mb{Desc}(\H^\bullet)$.
In particular, we have:
(cf. [@Ye1 Theorem 2.4]) If $\G^\bullet\rar \H^\bullet$ is a weak equivalence of cosimplicial 2-groupoids, the induced map $\oline{Desc}(\G^\bullet)\rar \oline{Desc}(\H^\bullet)$ is an isomorphism of sets
By Theorems \[translation\] and \[holim\], the map $\oline{Desc}(\G^\bullet)\rar \oline{Desc}(\H^\bullet)$ coincides with $\pi_0(holim_{\Sim}\N\G^\bullet)\rar \pi_0(holim_{\Sim}\N\H^\bullet)$ and $\N$ and $\operatorname{holim}_\Sim$ preserve weak equivalences.
[Descent data of cosimplicial $n$-groupoids]{}\[n-descent\] The techniques of §\[2-groupoids\]–§\[Invariance of descent data\] work equaly well in higher dimensions. Here, we write down the details for the case of (strict) $n$-groupoids and the corresponding $n$-nerve $\N_{(n)}$ in the sense of [@St] but the same arguments work for weaker notions of $n$-groupoids, e.g. Tamsamani $n$-groupoids. We only need two ingredients. The first is that $N_{(n)}$ admits a left adjoint (and hence commutes with limits); this is true since the inclusion $n\mathcal{G}pd\hookrightarrow nCat$ admits a left adjoint $\Pi_n:nCat\rar n\mathcal{G}pd$ and thus the composite $\Pi_n\circ \tau_n$ (where $\tau_n$ is the fundamental $n$-category) is the desired left adjoint. The second ingredient is that $\N_{(n)}\G$ is a Kan complex for every $n$-groupoid $\G$; this goes back to [@Da]. In light of theorem \[translation\], it makes sense to define:
\[n-descent data\] Let $\G^\bullet$ be a cosimplicial $n$-groupoid. Its *$n$-descent data* is the simplicial set $\mb{Desc}_n(\G^\bullet):=Tot_r(\N_{(n)}\G)$.
Definition \[n-descent data\] makes sense formally, but its geometric meaning is unknown to us. Nevertheless, the formal reasoning of proposition \[2-groupoid structure\] implies:
The simplicial set $\mb{Desc}_n(\G^\bullet)$ is the $n$-nerve of an $n$-groupoid.
Moreover, theorem \[holim\] generalizes immediately.
Let $\G^\bullet$ be a cosimplicial (strict) $n$-groupoid. There is a natural weak equivalence $Tot_r(r\N_{(n)}\G^\bullet)\simeq \operatorname{holim}_{\Sim} \N_{(n)}\G^\bullet$.
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abstract: 'Bobkov (J. Theoret. Probab. **18**(2) (2005) 399–412) investigated an approximate de Finetti representation for probability measures, on product measurable spaces, which are symmetric under permutations of coordinates. One of the main results of that paper was an explicit approximation bound for permanents of complex rectangular matrices, which was shown by a somewhat complicated induction argument. In this paper, we indicate how to avoid the induction argument using an (asymptotic) expansion. Our approach makes it possible to give new explicit higher order approximation bounds for such permanents and in turn for the probability measures mentioned above.'
address: 'FB IV – Department of Mathematics, University of Trier, 54286 Trier, Germany.'
author:
- Bero Roos
date: 'December 14, 2013: Revised version'
title: 'On Bobkov’s approximate de Finetti representation via approximation of permanents of complex rectangular matrices'
---
Introduction
============
Suppose that $X:=(X_1,X_2,X_3,\dots)$ is an infinite exchangeable sequence of random variables on a probability space $(\Omega,{\mathcal{A}},P)$ with values in a measurable space $(S,{\mathcal{S}})$, that is, the distribution $P^{X}$ of $X$ on the infinite product measurable space $(S^\infty,{\mathcal{S}}^{\otimes\infty})$ is invariant under permutations of a finite number of coordinates. The de Finetti Theorem says that, under mild assumptions on the space $(S,{\mathcal{S}})$, there is a probability space $(T,{\mathcal{T}},\nu)$ and a Markov kernel $\mu:\,T\times{\mathcal{S}}\longrightarrow[0,1]$, $(t,A)\mapsto\mu_t(A)$ such that $$\begin{aligned}
P^{X}=\int_T(\mu_t)^{\otimes\infty}\,{{\mathrm{d}}}\nu(t).\end{aligned}$$ For instance, it suffices to assume that $(S,{\mathcal{S}})$ is a Borel (or standard) measurable space, i.e. Borel isomorphic to some Borel measurable subset of ${\mathbb{R}}$ (see Hewitt and Savage [@HS55] or Diaconis and Freedman [@DF80]).
For a finite exchangeable sequence, an analogous representation does not generally hold, but there are approximate de Finetti results. In what follows, let $N\in{\mathbb{N}}$, $n\in{\underline{N}}=\{1,\dots,N\}$, and let $Y_N=(X_1,\dots,X_N)$ be an exchangeable family of $S$-valued random variables, that is, the distribution $P^{Y_N}$ of $Y_N$, defined on $(S^N,{\mathcal{S}}^{\otimes N})$, is invariant under permutation of coordinates. Let $Q_1$ be the probability measure on $(S^n,{\mathcal{S}}^{\otimes n})$ defined by $$\begin{aligned}
Q_1(A)=\int\Big(\frac{1}{N}\sum_{j=1}^N
{\delta}_{X_j(\omega)}\Big)^{\otimes n}(A)\,{{\mathrm{d}}}P(\omega)\end{aligned}$$ for $A\in{\mathcal{S}}^{\otimes n}$, where ${\delta}_x$ denotes the Dirac measure at the point $x\in S$. In other words, $Q_1$ is the $P$-expectation of the $n$-th power of an empirical measure on $(S^N,{\mathcal{S}}^{\otimes N})$. The following results can be found in Diaconis and Freedman [@DF80]. They showed that $$\begin{aligned}
\label{eq719}
{d_{\mathrm{TV}}(P^{Y_n},Q_1)}\leq 1-\frac{N!}{(N-n)!N^n}
\leq \frac{n(n-1)}{2N},\end{aligned}$$ where ${d_{\mathrm{TV}}(R,R')}=\sup_{A\in{\mathcal{S}}^{\otimes n}}{\vertR(A)-R'(A)\vert}$ denotes the total variation distance between finite signed measures $R$ and $R'$ on $(S^n,{\mathcal{S}}^{\otimes n})$. Hence, if $\frac{n^2}{N}$ is small then $P^{Y_n}$ has an approximate de Finetti representation $Q_1$. It turned out that, in general, the bound (\[eq719\]) is sharp. However, if $S$ is finite and of cardinality ${|S|}=d\in{\mathbb{N}}$, then the nice inequality $$\begin{aligned}
\label{eq1755}
{d_{\mathrm{TV}}(P^{Y_n},Q_1)}\leq \frac{dn}{N}\end{aligned}$$ is available, which, in the case of finite $S$, is better than (\[eq719\]) if $d$ is sufficiently small compared to $n$.
On the other hand, it is possible to obtain similar good bounds in the general case if the total variation distance is replaced by a weaker metric. Let ${\mathcal{F}}_n$ be the set of all functions $f:\,S^n\longrightarrow {\mathbb{C}}$ such that measurable $f_1,\dots,f_n:\,S\longrightarrow {\mathbb{C}}$ exist with ${\vertf_k(x_k)\vert}\leq1$ for $k\in{\underline{n}}$ and $f(x)=\prod_{k=1}^nf_k(x_k)$ for all $x=(x_1,\dots,x_n)\in S^n$. We write $f=\bigotimes_{k=1}^n f_k$. Furthermore, let ${\underline{N}}_{\neq}^n=\{(j_1,\dots,j_n)\in{\underline{N}}^n\,|\,
j_k\neq j_\ell \mbox{ for all }k,\ell\in{\underline{n}}\mbox{ with }
k\neq \ell\}$.
Bobkov [@bob05] showed in his Theorem 1.1 (see also p. 405 there) the inequality $$\begin{aligned}
\label{eq41887}
\sup_{f\in{\mathcal{F}}_n}{\Big\vert\int f\,{{\mathrm{d}}}(P^{Y_n}-Q_1)\Big\vert} \leq C\frac{n}{N}
\quad \mbox{ with } C=16.\end{aligned}$$ For the proof, he used the representation $$\begin{aligned}
\lefteqn{
\int f\,{{\mathrm{d}}}(P^{Y_n}-Q_1)
=\int\Big(\prod_{k=1}^nf_k(X_k(\omega)) -
\int f\,{{\mathrm{d}}}\Big(\frac{1}{N}\sum_{j=1}^N
{\delta}_{X_j(\omega)}\Big)^{\otimes n}\Big)\,{{\mathrm{d}}}P(\omega)}
\qquad\qquad\nonumber\\
&=\int\Big(\frac{(N-n)!}{N!}\sum_{j\in{\underline{N}}_{\neq}^n}
\prod_{k=1}^nf_k(X_{j_k}(\omega))
-\prod_{k=1}^n\Big(\frac{1}{N}\sum_{j=1}^N
f_k(X_j(\omega))\Big)\Big)\,{{\mathrm{d}}}P(\omega)\label{eq892762}\end{aligned}$$ for $f=\bigotimes_{k=1}^n f_k\in{\mathcal{F}}_n$ and a remarkable approximation result for permanents of complex rectangular matrices (see Theorem A below), which he proved by using a somewhat complicated induction argument. The permanent of a complex rectangular matrix $Z=(z_{j,k})\in{\mathbb{C}}^{N\times n}$ with $N\in{\mathbb{N}}$ and $n\in{\underline{N}}$ is defined by $$\begin{aligned}
{\mathrm{Per}(Z)}:=\sum_{j\in{\underline{N}}_{\neq}^n}\prod_{k=1}^nz_{j_k,k}.\end{aligned}$$ For general properties of permanents, we refer the reader to Minc [@min78] and Cheon and Wanless [@CW05].
**Theorem A.** (Bobkov [@bob05 Theorem 2.1]) *Let $N\in{\mathbb{N}}$, $n\in{\underline{N}}$ and $Z=(z_{j,k})\in{\mathbb{C}}^{N\times n}$. For $j\in{\underline{N}}$ and $k\in{\underline{n}}$, we assume that ${\vertz_{j,k}\vert}\leq1$ and set $\widetilde{z}_k=\frac{1}{N}\sum_{j=1}^Nz_{j,k}$. Then $$\begin{aligned}
\label{eq2645}
{\Big\vert\frac{(N-n)!}{N!}{\mathrm{Per}(Z)}-\prod_{k=1}^n\widetilde{z}_k\Big\vert}\leq
C\frac{n}{N} \quad \mbox{ with }C=16.\end{aligned}$$*
From Proposition 4.1 in Bobkov [@bob04] it follows that (\[eq41887\]) and (\[eq2645\]) hold with the better constant $C=6$ if $z_{j,k}=z_{j,1}$ for all $j\in{\underline{N}}$ and $k\in{\underline{n}}$. However, Theorem \[th12\] below shows that $C$ can always be taken smaller than 3.57.
For two finite signed measures $R$ and $R'$ on $(S^n,{\mathcal{S}}^{\otimes n})$, let $$\begin{aligned}
{d_{\mathrm{PV}}}(R,R')
=\sup_{A_1,\dots,A_n\in{\mathcal{S}}}{\vertR(A_1\times\ldots\times A_n)
-R'(A_1\times\ldots\times A_n)\vert}\end{aligned}$$ denote the so-called product variation between $R$ and $R'$. Obviously ${d_{\mathrm{PV}}}$ is a metric on the set of all finite signed measures on $(S^n,{\mathcal{S}}^{\otimes n})$. Furthermore, $$\begin{aligned}
{d_{\mathrm{PV}}}(R,R')\leq\sup_{f\in{\mathcal{F}}_n}{\Big\vert\int f\,{{\mathrm{d}}}(R-R')\Big\vert}. \end{aligned}$$ Therefore (\[eq41887\]) and the inequalities of Theorem \[th363\] below imply bounds for ${d_{\mathrm{PV}}}$.
In the next section, we present refinements of (\[eq2645\]), see Theorems \[th2994\], \[th12\] and Corollary \[cor387\]. The latter together with (\[eq892762\]) and a similar representation implies Theorem \[th363\] below, the first part of which is better than (\[eq41887\]) with $C=3.57$ if $\frac{n}{N}\leq\frac{1}{2}$. The second part shows that, if $n\geq 2$ and in turn $N\geq 2$, a more accurate approximation of $P^{Y_n}$ by a finite signed measure $Q_2$ on $(S^n,{\mathcal{S}}^{\otimes n})$ is possible, where $$\begin{aligned}
Q_2(A)
&=Q_1(A)-\frac{1}{N(N-1)}\sum_{K\subseteq{\underline{n}}:\,{|K|}=2}
\sum_{j=1}^N\int\Big(\bigotimes_{k\in {\underline{n}}}
R_{j,k,K}(\omega)\Big)(A)\,{{\mathrm{d}}}P(\omega) \end{aligned}$$ for $A\in{\mathcal{S}}^{\otimes n}$ and $$\begin{aligned}
R_{j,k,K}(\omega)=\left\{
\begin{array}{ll}
{\delta}_{X_j(\omega)}-\frac{1}{N}
\sum_{\ell=1}^N{\delta}_{X_\ell(\omega)},&
\mbox{ if } k\in K,\\
\frac{1}{N}\sum_{\ell=1}^N{\delta}_{X_\ell(\omega)},&
\mbox{ if } k\in{\underline{n}}\setminus K.
\end{array}
\right.\end{aligned}$$
\[th363\] Under the assumptions above and if $\frac{n}{N}<1$, we have $$\begin{aligned}
\sup_{f\in{\mathcal{F}}_n}
{\Big\vert\int f\,{{\mathrm{d}}}(P^{Y_n}-Q_1)\Big\vert}
&\leq \frac{n}{N}
+2.12\,\frac{(\frac{n}{N})^{3/2}}{(1-\frac{n}{N})^{3/4}},
\label{eq71665}\\
\sup_{f\in{\mathcal{F}}_n}
{\Big\vert\int f\,{{\mathrm{d}}}(P^{Y_n}-Q_2)\Big\vert}
&\leq\sqrt{3}\Big(\frac{n}{N}\Big)^{3/2}
+2.27\,\frac{(\frac{n}{N})^2}{(1-\frac{n}{N})^{3/4}}, \quad
\mbox{ if }n\geq 2.\label{eq71666}\end{aligned}$$
Higher order results are also possible using Theorem \[th2994\] or Theorem \[th12\] below. We omit the details.
Approximation of permanents
===========================
For $n\in{\mathbb{N}}$, the indeterminate $x=(x_1,\dots,x_n)$ and $r\in{{\mathbb{Z}}_+}^n=\{0,1,2,\dots\}^n$, we set $x^r=\prod_{k\in{\underline{n}}}x_k^{r_k}$ and write $a_r={\mathrm{Coeff}(x^r;\,\sum_{s\in{{\mathbb{Z}}_+}^n}a_sx^s)}$ for the coefficient of $x^r$ in the formal power series $\sum_{s\in{{\mathbb{Z}}_+}^n}a_sx^s$, $(a_s\in{\mathbb{C}})$. Sometimes $y$ will be our indeterminate. However, the symbols $x$ and $y$ may have other meanings as indicated below. In what follows, we use the simple fact that, for $N\in{\mathbb{N}}$, $n\in{\underline{N}}$ and $Z=(z_{j,k})\in{\mathbb{C}}^{N\times n}$, $$\begin{aligned}
\label{eq1326}
{\mathrm{Per}(Z)}=
{\mathrm{Coeff}\Big(x_1\cdots x_n;\,\prod_{j=1}^N
\Big(1+\sum_{k=1}^nz_{j,k}x_k\Big)\Big)}.\end{aligned}$$ Furthermore, if additionally $Z$ has identical columns, i.e. $z_{j,k}=z_{j,1}$ for all $j\in{\underline{N}}$ and $k\in{\underline{n}}$, then $$\begin{aligned}
\label{eq1326b}
{\mathrm{Per}(Z)}=n!\,{\mathrm{Coeff}\Big(y^n;\,\prod_{j=1}^N(1+z_{j,1}y)\Big)}.\end{aligned}$$
The main result of this section is Theorem \[th2994\] below and requires the following lemmas, the first of which plays a prominent role in the theory of polynomials over infinite dimensional spaces. Its proof is due to Hörmander [@hoe54 Theorem 4]; see also Harris [@har96] and Dineen [@din99 Proposition 1.44 and the notes on page 79]. However, first versions for real spaces were already shown in Kellogg [@kel28] and Banach [@ban38].
\[la61966\] Let $n\in{\mathbb{N}}$, $E$ be a complex Hilbert space, $F$ be a complex Banach space, $g:\,E^n\longrightarrow F$ be $n$-linear (i.e. linear in each component), continuous and symmetric in its arguments. Let $\widehat{g}(x)=g(x,\dots,x)$ for $x\in E$ and $$\begin{aligned}
\|g\|&=\sup\{\|g(x_1,\dots,x_n)\|\,|\,x_i\in E,\|x_i\|\leq1
\mbox{ for each }i\in{\underline{n}}\}\\
\|\widehat{g}\|&=\sup\{\|\widehat{g}(x)\|\,|\,x\in E,\,\|x\|\leq 1\}.\end{aligned}$$ Then $\|g\|=\|\widehat{g}\|$.
The proof of the next lemma uses Lemma \[la61966\] and the Cauchy integral formula. We note that the more complicated Lemma 3 in [@roo01] only yields a weaker result under the assumptions used here. We always set $0^0=1$.
\[la2655\] Let $N\in{\mathbb{N}}$, $n\in{\underline{N}}$ and $A=(a_{j,k})\in{\mathbb{C}}^{N\times n}$. For each $k\in{\underline{n}}$, we assume that $\sum_{j=1}^Na_{j,k}=0$ and set $\alpha_k=\frac{1}{N}\sum_{j=1}^N{\verta_{j,k}\vert}^2$. Then we have $$\begin{aligned}
\label{eq45}
{\vert{\mathrm{Per}(A)}\vert}
&\leq\frac{n!\,N^{N/2}}{(N-n)^{(N-n)/2}n^{n/2}}
\prod_{k=1}^n\sqrt{\alpha_k}.\end{aligned}$$
We may assume that $\alpha_k\neq0$ for each $k\in{\underline{n}}$. Let $E=\{x={^{{{\mathrm{t}}}}}(x_1,\dots,x_N)\in{\mathbb{C}}^{N\times 1}\,|\,
\sum_{j=1}^Nx_j=0\}$ be equipped with the standard inner product and consider $F={\mathbb{C}}$, $g:\,E^n\longrightarrow F$, $g(x^{(1)},\dots,x^{(n)})
={\mathrm{Per}(x^{(1)},\dots,x^{(n)})}$ for $x^{(1)},\dots,x^{(n)}\in E$, where “${{\mathrm{t}}}$” denotes transposition. It is easily seen that Lemma \[la61966\] can be applied, which gives ${\vert{\mathrm{Per}(A)}\vert}\leq \|\widehat{g}\|N^{n/2}\prod_{k=1}^n\sqrt{\alpha_k}$. Using (\[eq1326b\]), we obtain for $x\in E$ with $\|x\|\leq 1$ and arbitrary $r\in(0,\infty)$ that $$\begin{aligned}
{\vert\widehat{g}(x)\vert}
&=\frac{n!}{2{{\mathrm{\pi}}}r^n}{\Big\vert\int_{-{{\mathrm{\pi}}}}^{{\mathrm{\pi}}}{{\mathrm{e}}}^{-{{\mathrm{i}}}nt}
\prod_{j=1}^N\Big(1+x_jr{{\mathrm{e}}}^{{{\mathrm{i}}}t}\Big){{\mathrm{d}}}t\Big\vert}\\
&\leq\frac{n!}{r^n}\sup_{t\in[-{{\mathrm{\pi}}},{{\mathrm{\pi}}}]}
\prod_{j=1}^N{\vert1+x_{j}r{{\mathrm{e}}}^{{{\mathrm{i}}}t}\vert}
\leq\frac{n!}{r^n} \Big(1+\frac{r^2}{N}\Big)^{N/2};\end{aligned}$$ the last inequality follows from the inequality between arithmetic and geometric means. Indeed, for $w\in E$, we have $$\begin{aligned}
\prod_{j=1}^N{\vert1+w_j\vert}
&=\Big(\prod_{j=1}^N(1+2{\mathrm{Re}}(w_j)+{\vertw_j\vert}^2)\Big)^{1/2}
\leq\Big(1+\frac{1}{N}\sum_{j=1}^N{\vertw_j\vert}^2\Big)^{N/2},\end{aligned}$$ where ${\mathrm{Re}}(w_j)$ denotes the real part of $w_j$. Let $\varepsilon\in(0,\infty)$ and $$r=\Big(\frac{nN}{N-n+\varepsilon}\Big)^{1/2}.$$ Letting $\varepsilon\to0$ yields $\|\widehat{g}\|\leq\frac{n!N^{(N-n)/2}}{(N-n)^{(N-n)/2}n^{n/2}}$ and the result is shown.
Inequality (\[eq45\]) can be viewed as a Hadamard type inequality for permanents of matrices with zero column sums. Another inequality of this type is $$\label{eq617}
{\vert{\mathrm{Per}(Z)}\vert}
\leq N!\prod_{k=1}^N\Big(\frac{1}{N}
\sum_{j=1}^N{\vertz_{j,k}\vert}^2\Big)^{1/2},$$ which is valid for general quadratic matrices $Z=(z_{j,k})\in{\mathbb{C}}^{N\times N}$ with $N\in{\mathbb{N}}$. Carlen *et al.* [@CLL06] gave two proofs of (\[eq617\]), which, however, also follows directly from Lemma \[la61966\] together with the inequality between arithmetic and geometric means.
Inequality (\[eq617\]) can be used to derive an alternative bound for the left-hand side of (\[eq45\]) as follows. Consider the assumptions of Lemma \[la2655\] and define $Z=(z_{j,k})\in{\mathbb{C}}^{N\times N}$ with $z_{j,k}=a_{j,k}$ for $j\in{\underline{N}}$, $k\in{\underline{n}}$ and $z_{j,k}=1$ for $j\in{\underline{N}}$, $k\in{\underline{N}}\setminus{\underline{n}}$. Then $$\label{eq618}
{\vert{\mathrm{Per}(A)}\vert}=\frac{{\vert{\mathrm{Per}(Z)}\vert}}{(N-n)!}
\leq\frac{N!}{(N-n)!}\prod_{k=1}^n\sqrt{\alpha_k}.$$ However, it turns out that (\[eq45\]) is always better than the inequality in (\[eq618\]), since $$\frac{N^{N}}{(N-n)^{N-n}n^{n}}
\leq \Big(\prod_{m=1}^{N-n}\frac{N-m+1}{N-n-m+1}\Big)
\Big(\prod_{m=1}^n\frac{N-m+1}{n-m+1}\Big)
={\genfrac{(}{)}{0pt}{}{N}{n}}^2.$$
\[la124376\] Let $n\in{\mathbb{N}}$, $m\in{\underline{n}_0}=\{0,\dots,n\}$ and $w_{1,k},w_{2,k}\in{\mathbb{C}}$ for $k\in{\underline{n}}$. Then $${\Big\vert{\mathrm{Coeff}\Big(y^m;\,\prod_{k=1}^n(w_{1,k}+w_{2,k}y)\Big)}\Big\vert}
\leq {\genfrac{(}{)}{0pt}{}{n}{m}}
\Big(\frac{1}{n}\sum_{k=1}^n {\vertw_{2,k}\vert}^2\Big)^{m/2}
\Big(\frac{1}{n}\sum_{k=1}^n{\vertw_{1,k}\vert}^2\Big)^{(n-m)/2}.$$
Using Cauchy’s inequality, we obtain $$\begin{aligned}
\lefteqn{{\Big\vert{\mathrm{Coeff}\Big(y^m;\,\prod_{k=1}^n(w_{1,k}+w_{2,k}y)\Big)}\Big\vert}
={\Big\vert
\sum_{K\subseteq{\underline{n}}:\,{|K|}=m}
\Big(\prod_{k\in K}w_{2,k}\Big)\prod_{k\in{\underline{n}}\setminus K}
w_{1,k}\Big\vert}}\\
&\leq\Big(\sum_{K\subseteq{\underline{n}}:\,{|K|}=m}
\prod_{k\in K}{\vertw_{2,k}\vert}^2\Big)^{1/2}
\Big(\sum_{K\subseteq{\underline{n}}:\,{|K|}=m}
\prod_{k\in{\underline{n}}\setminus K}
{\vertw_{1,k}\vert}^2\Big)^{1/2}\\
&= {\mathrm{Coeff}\Big(y^m;\,\prod_{k=1}^n(1+
{\vertw_{2,k}\vert}^2 y)\Big)}^{1/2}
{\mathrm{Coeff}\Big(y^{n-m};\,\prod_{k=1}^n(1+
{\vertw_{1,k}\vert}^2 y)\Big)}^{1/2}.\end{aligned}$$ The assertion now follows from a result due to Maclaurin, which says that if $g_1,\dots,g_n\in[0,\infty)$, then $\big(\frac{1}{{\genfrac{(}{)}{0pt}{}{n}{\ell}}}{\mathrm{Coeff}(y^\ell;\,
\prod_{k=1}^n(1+g_ky))}\big)^{1/\ell}$ is non-increasing in $\ell\in{\underline{n}}$, see Hardy *et al.* [@HLP52 Theorem 52, page 52].
\[la31765\] Let $n,N\in{\mathbb{N}}$, $m\in{{\mathbb{Z}}_+}$ with $m\leq \min\{n,N\}$, $(a_{j,k})\in{\mathbb{C}}^{N\times n}$ with $\sum_{j=1}^Na_{j,k}=0$ for all $k\in{\underline{n}}$, $b\in{\mathbb{C}}^n$, $\alpha=\frac{1}{nN}\sum_{j=1}^N\sum_{k=1}^n {\verta_{j,k}\vert}^2$, $\beta=\frac{1}{n}\sum_{k=1}^n{\vertb_k\vert}^2$. Then $$\begin{aligned}
{\Big\vert{\mathrm{Coeff}\Big(x_1\cdots x_n;\,\Big(\sum_{k=1}^nb_kx_k\Big)^{n-m}
\prod_{j=1}^N\Big(1+\sum_{k=1}^n a_{j,k}x_k\Big)\Big)}\Big\vert}
\leq \frac{n!N^{N/2}\alpha^{m/2}\beta^{(n-m)/2}}{
(N-m)^{(N-m)/2}m^{m/2}}.\end{aligned}$$
Let $\alpha_k=\frac{1}{N}\sum_{j=1}^N{\verta_{j,k}\vert}^2$, $(k\in{\underline{n}})$. An application of Lemma \[la2655\] gives $$\begin{aligned}
\lefteqn{\frac{1}{(n-m)!}
{\Big\vert{\mathrm{Coeff}\Big(x_1\cdots x_n;\,
\Big(\sum_{k=1}^nb_kx_k\Big)^{n-m}
\prod_{j=1}^N\Big(1+\sum_{k=1}^n a_{j,k}x_k\Big)\Big)}\Big\vert}} \\
&={\Big\vert\sum_{K\subseteq{\underline{n}}:\;{|K|}=n-m}
{\mathrm{Coeff}\Big(x_1\cdots x_n;\,\Big(\prod_{k\in K}(b_kx_k)\Big)
\prod_{j=1}^N\Big(1+\sum_{k=1}^n a_{j,k}x_k\Big)\Big)}\Big\vert} \\
&={\Big\vert\sum_{K\subseteq{\underline{n}}:\;{|K|}=n-m}
{\mathrm{Coeff}\Big(\prod_{k\in{\underline{n}}\setminus K}x_k;\,
\prod_{j=1}^N\Big(1+\sum_{k\in{\underline{n}}\setminus K}
a_{j,k}x_k\Big)\Big)}\prod_{k\in K}b_k\Big\vert} \\
&\leq\frac{m!N^{N/2}}{(N-m)^{(N-m)/2}m^{m/2}}
\sum_{K\subseteq{\underline{n}}:\;{|K|}=n-m}
\Big(\prod_{k\in{\underline{n}}\setminus K} \sqrt{\alpha_k}\Big)
\prod_{k\in K}{\vertb_k\vert}\nonumber\\
&=\frac{m!N^{N/2}}{(N-m)^{(N-m)/2}m^{m/2}}
{\mathrm{Coeff}\Big(y^m;\,\prod_{k=1}^n({\vertb_k\vert}+\sqrt{\alpha_k}y)\Big)}.\end{aligned}$$ The proof is easily completed using Lemma \[la124376\].
\[la287765\] For $r\in{{\mathbb{Z}}_+}$, $t,x\in[0,1]$, we have $\sum_{m=0}^r(m+1)^tx^m\leq(\frac{1-x^{r+1}}{1-x})^{1+t}$.
This follows from $\sum_{m=0}^r(m+1)x^m
=\frac{1-(r+2)x^{r+1}+(r+1)x^{r+2}}{(1-x)^2}
\leq(\frac{1-x^{r+1}}{1-x})^{2}$ and Hölder’s inequality, i.e. $\sum_{m=0}^r(m+1)^tx^m\leq
(\sum_{m=0}^r(m+1)x^m)^{t}(\sum_{m=0}^rx^m)^{1-t}$.
The following lemma is more precise than Lemma 3 in [@roo00].
\[la4866\] Let $\ell,m,N\in{\mathbb{N}}$, $\ell\leq m\leq N$ and $C_\ell= \big(\frac{{{\mathrm{e}}}^\ell\ell!}{\ell^{\ell+1/2}}\big)^{1/2}$. Then $$\begin{aligned}
\label{eq1987}
\frac{N^{N/2}}{(N-m)^{(N-m)/2}m^{m/2+1/4}{\genfrac{(}{)}{0pt}{}{N}{m}}^{1/2}}
\leq C_\ell.\end{aligned}$$
Let $p(m,N)=\frac{N^{N}}{(N-m)^{N-m}{\genfrac{(}{)}{0pt}{}{N}{m}}}$. Since $q(k):=(\frac{k}{k+1})^k$ is decreasing in $k\in{{\mathbb{Z}}_+}$, we have $$\begin{aligned}
\frac{p(m,N)}{p(m,N+1)}=\frac{N^N(N+1-m)^{N+1-m}(N+1)}{(N+1)^{N+1}
(N-m)^{N-m}(N+1-m)}=\frac{q(N)}{q(N-m)}\leq 1.\end{aligned}$$ Hence $p(m,N)\leq \lim_{\widetilde{N}\to\infty}p(m,\widetilde{N})={{\mathrm{e}}}^m m!$ and therefore the left-hand side of (\[eq1987\]) is bounded by $(\frac{{{\mathrm{e}}}^mm!}{m^{m+1/2}})^{1/2}$. Since this is decreasing in $m$ (cf. Mitrinović [@mit70 p. 183]), the assertion follows.
We now present our first main result, which generalizes Theorem A. Indeed, it will turn out that $\gamma\leq\frac{n}{N}$ and, for $\ell=1$, $H_\ell(Z)=\prod_{k=1}^n\widetilde{z}_k$, see the Remarks \[rem474\] and \[rem76345\] below. A further advantage of $\gamma$ is that it can be equal to zero, namely in the case $z_{j,k}=\widetilde{z}_k$ for all $j\in{\underline{N}}$ and $k\in{\underline{n}}$. We note that the singularity in (\[eq7435\]) can be removed, see Theorem \[th12\] below.
\[th2994\] Let $N\in{\mathbb{N}}$, $n\in{\underline{N}}$, $\ell\in{\underline{n}}$ and $Z=(z_{j,k})\in{\mathbb{C}}^{N\times n}$. For $j\in{\underline{N}}$ and $k\in{\underline{n}}$, we assume that ${\vertz_{j,k}\vert}\leq 1$ and set $\widetilde{z}_k=\frac{1}{N}\sum_{j=1}^Nz_{j,k}$, $a_{j,k}=z_{j,k}-\widetilde{z}_k$, $U_j(x)=\sum_{k=1}^n a_{j,k}x_k$, where $x=(x_1,\dots,x_n)$ is an indeterminate. Further, let $C_\ell$ be as in Lemma \[la4866\], $$\begin{aligned}
\alpha&=\frac{1}{nN}\sum_{j=1}^N\sum_{k=1}^n{\verta_{j,k}\vert}^2
,\quad \beta=\frac{1}{n}\sum_{k=1}^n{\vert\widetilde{z}_k\vert}^2,
\quad \gamma=\frac{n\alpha}{N}\min\Big\{n,\,\frac{1}{1-\beta}\Big\},\\
G_m(Z)&=\frac{(N-m)!}{(n-m)!N!}{\mathrm{Coeff}\Big(x_1\cdots x_n;\,
\Big(\prod_{j=1}^N(1+U_j(x))\Big)
\Big(\sum_{k=1}^n\widetilde{z}_kx_k\Big)^{n-m}\Big)}\end{aligned}$$ for $m\in{\underline{n}_0}$ and set $H_\ell(Z)=\sum_{m=0}^\ell G_m(Z)$. If $\gamma<1$, then $$\begin{aligned}
\label{eq7435}
{\Big\vert\frac{(N-n)!}{N!}{\mathrm{Per}(Z)} -H_\ell(Z)\Big\vert}
\leq (\ell+1)^{1/4}C_{\ell+1}\,
\frac{\gamma^{(\ell+1)/2}}{(1-\gamma)^{3/4}}.\end{aligned}$$
Let $W_m(x)={\mathrm{Coeff}(y^m;\,\prod_{j=1}^N(1+U_j(x)y))}$ for $m\in{\underline{N}_0}$. In view of (\[eq1326\]), $$\begin{aligned}
\lefteqn{\prod_{j=1}^N\Big(1+\sum_{k=1}^nz_{j,k}x_k\Big)
=\prod_{j=1}^N\Big(U_j(x)+1+\sum_{k=1}^n\widetilde{z}_kx_k\Big)}\\
&=\sum_{m=0}^NW_m(x)\Big(1+\sum_{k=1}^n\widetilde{z}_kx_k\Big)^{N-m}
=\sum_{m=0}^N\sum_{r=0}^{N-m}{\genfrac{(}{)}{0pt}{}{N-m}{r}}W_m(x)
\Big(\sum_{k=1}^n\widetilde{z}_kx_k\Big)^{r},\end{aligned}$$ and $$\begin{aligned}
\label{eq3376}
G_m(Z)&=\frac{(N-m)!}{(n-m)!N!}{\mathrm{Coeff}\Big(x_1\cdots x_n;\,W_m(x)
\Big(\sum_{k=1}^n\widetilde{z}_kx_k\Big)^{n-m}\Big)},\end{aligned}$$ we see that $$\begin{aligned}
{\mathrm{Per}(Z)}
&=\sum_{m=0}^n{\genfrac{(}{)}{0pt}{}{N-m}{n-m}}
{\mathrm{Coeff}\Big(x_1\cdots x_n;\,W_m(x)
\Big(\sum_{k=1}^n\widetilde{z}_kx_k\Big)^{n-m}\Big)}\\
&=\frac{N!}{(N-n)!}\sum_{m=0}^nG_m(Z)\end{aligned}$$ and therefore $\frac{(N-n)!}{N!}{\mathrm{Per}(Z)}=H_n(Z)$. Using Lemmas \[la31765\] and \[la4866\] and the simple inequality ${\genfrac{(}{)}{0pt}{}{n}{m}}\leq{\genfrac{(}{)}{0pt}{}{N}{m}}(\frac{n}{N})^m$ for $m\in{\underline{n}_0}$, we obtain $$\begin{aligned}
\lefteqn{
{\Big\vert\frac{(N-n)!}{N!}{\mathrm{Per}(Z)}-H_\ell(Z)\Big\vert}
\leq \sum_{m=\ell+1}^n{\vertG_m(Z)\vert}}\\
&\leq\sum_{m=\ell+1}^n
\frac{N^{N/2}}{(N-m)^{(N-m)/2}m^{m/2}}
\frac{{\genfrac{(}{)}{0pt}{}{n}{m}}}{{\genfrac{(}{)}{0pt}{}{N}{m}}}\alpha^{m/2}\beta^{(n-m)/2}\\
&\leq C_{\ell+1}\sum_{m=\ell+1}^n
\frac{{\genfrac{(}{)}{0pt}{}{n}{m}}}{{\genfrac{(}{)}{0pt}{}{N}{m}}^{1/2}}m^{1/4}
\alpha^{m/2}\beta^{(n-m)/2}\\
&\leq C_{\ell+1} \sum_{m=\ell+1}^n m^{1/4}\gamma^{m/2}
\Big({\genfrac{(}{)}{0pt}{}{n}{m}}\max\Big\{1-\beta, \frac{1}{n} \Big\}^m
\beta^{n-m}\Big)^{1/2},\end{aligned}$$ where we used that $\beta\in[0,1]$. By applying Cauchy’s inequality and the fact that, since $\ell\geq1$, $\sum_{m=\ell+1}^n{\genfrac{(}{)}{0pt}{}{n}{m}}\frac{1}{n^{m}}\leq
(1+\frac{1}{n})^n-2\leq{{\mathrm{e}}}-2< 1$, we obtain $$\begin{aligned}
{\Big\vert\frac{(N-n)!}{N!}{\mathrm{Per}(Z)}-H_\ell(Z)\Big\vert}
&\leq C_{\ell+1} \Big(\sum_{m=\ell+1}^n
\sqrt{m}\gamma^{m}\Big)^{1/2}\\
&\leq (\ell+1)^{1/4}C_{\ell+1}\gamma^{(\ell+1)/2}
\Big(\sum_{m=0}^{n-\ell-1} \sqrt{m+1}\gamma^{m}\Big)^{1/2}.\end{aligned}$$ It remains to use Lemma \[la287765\] with $t=\frac{1}{2}$.
For the rest of the paper, let the notation of Theorem \[th2994\] hold.
\[rem474\] We have $\gamma\leq \frac{n}{N}$, since $$\begin{aligned}
\label{eq217}
\alpha=\frac{1}{nN}\sum_{j=1}^N\sum_{k=1}^n
{\vertz_{j,k}\vert}^2-\beta\leq 1-\beta.\end{aligned}$$ In particular, if ${\vertz_{j,k}\vert}=1$ for all $j\in{\underline{N}}$ and $k\in{\underline{n}}$, then $\alpha=1-\beta$. Indeed, writing $z_{j,k}=u_{j,k}+{{\mathrm{i}}}v_{j,k}$ and $\widetilde{z}_k=\widetilde{u}_k+{{\mathrm{i}}}\widetilde{v}_k$ with $u_{j,k},v_{j,k}\in{\mathbb{R}}$, $\widetilde{u}_k=\frac{1}{N}\sum_{j=1}^Nu_{j,k}$ and $\widetilde{v}_k=\frac{1}{N}\sum_{j=1}^Nv_{j,k}$, we obtain $$\begin{aligned}
\alpha
&=\frac{1}{nN}\sum_{j=1}^N\sum_{k=1}^n((u_{j,k}-\widetilde{u}_k)^2
+(v_{j,k}-\widetilde{v}_k)^2)\\
&=\frac{1}{nN}\sum_{k=1}^n\Big(\sum_{j=1}^N
(u_{j,k}^2+v_{j,k}^2)-N(\widetilde{u}_k^2+\widetilde{v}_k^2)\Big),\end{aligned}$$ from which (\[eq217\]) follows.
Let us now collect some properties of the first few $G_m(Z)$, where we always assume that $m\in{\underline{n}_0}$.
\[rem76345\]
The first few $G_m(Z)$ can be evaluated as follows: $$\label{eq1756}
\begin{split}
G_0(Z)&=\prod_{k=1}^n\widetilde{z}_k,\quad
G_1(Z)=0,\\
G_2(Z)&=- \frac{(N-2)!}{N!}\sum_{K\subseteq{\underline{n}}:\,{|K|}=2}
\Big(\sum_{j=1}^N\prod_{k\in K}a_{j,k}\Big)
\prod_{k\in{\underline{n}}\setminus K}\widetilde{z}_k,
\\
G_3(Z)&=2\frac{(N-3)!}{N!}\sum_{K\subseteq{\underline{n}}:\,{|K|}=3}
\Big(\sum_{j=1}^N\prod_{k\in K}a_{j,k}\Big)
\prod_{k\in{\underline{n}}\setminus K}\widetilde{z}_k.
\end{split}$$ In order to prove this, let $$\begin{aligned}
V_m(x)=\sum_{j=1}^N(-U_j(x))^m,\quad
W_m(x)={\mathrm{Coeff}\Big(y^m;\,\prod_{j=1}^N(1+U_j(x)y)\Big)}\end{aligned}$$ for $m\in{\underline{N}_0}$. We have $$\begin{aligned}
W_m(x)=-\frac{1}{m}\sum_{k=0}^{m-2}W_k(x)
V_{m-k}(x)\quad \mbox{ for }m\in{\underline{N}},\end{aligned}$$ which can be shown in the same way as (10) in [@roo00]. In particular, $$\label{eq21756}
\begin{split}
W_0(x)&=1,\quad W_1(x)=0,\quad W_2(x)=-\frac{1}{2}V_2(x),
\\
W_3(x)&=-\frac{1}{3}V_3(x),\quad W_4(x)=\frac{1}{8}(V_2(x))^2
-\frac{1}{4}V_4(x).
\end{split}$$ In view of (\[eq3376\]), (\[eq21756\]) and $$\begin{aligned}
\lefteqn{{\mathrm{Coeff}\Big(x_1\cdots x_n;\,V_m(x)\Big(\sum_{k=1}^n
\widetilde{z}_kx_k\Big)^{n-m}\Big)}}\nonumber\\
&=(-1)^m\sum_{j=1}^N{\mathrm{Coeff}\Big(x_1\cdots x_n;\,
(U_j(x))^m\Big(\sum_{k=1}^n\widetilde{z}_kx_k\Big)^{n-m}\Big)}\nonumber\\
&=(-1)^m(n-m)!\,m!\sum_{j=1}^N\sum_{K\subseteq{\underline{n}}:\,{|K|}=m}
\Big(\prod_{k\in K}a_{j,k}\Big)
\prod_{k\in{\underline{n}}\setminus K}\widetilde{z}_k,\label{eq17216}\end{aligned}$$ for $m\in{\underline{n}}$, we see that (\[eq1756\]) is true. We note that the representations in (\[eq1756\]) of $G_2(Z)$ and $G_3(Z)$ have a simple form, but the omitted ones of $G_m(Z)$ with $m\geq 4$ are more complicated.
From the above, we obtain that $H_1(Z)=\prod_{k=1}^n\widetilde{z}_k$ and, if $n\geq 2$, $$\label{eq71977}
H_2(Z)=\prod_{k=1}^n\widetilde{z}_k-\frac{1}{N(N-1)}
\sum_{K\subseteq{\underline{n}}:\,{|K|}=2}
\Big(\sum_{j=1}^N\prod_{k\in K}a_{j,k}\Big)
\prod_{k\in{\underline{n}}\setminus K}\widetilde{z}_k.$$
Let us derive some bounds for ${\vertG_2(Z)\vert}$ and ${\vertG_3(Z)\vert}$. From (\[eq17216\]) and Lemma \[la124376\] it follows that, for $m\in{\underline{n}}$, $$\begin{aligned}
\lefteqn{{\Big\vert{\mathrm{Coeff}\Big(x_1\cdots x_n;\,V_m(x)\Big(\sum_{k=1}^n
\widetilde{z}_kx_k\Big)^{n-m}\Big)}\Big\vert}}\\
&\leq(n-m)!\,m!\sum_{j=1}^N
{\Big\vert{\mathrm{Coeff}\Big(y^m;\,\prod_{k=1}^n(\widetilde{z}_k+a_{j,k}y)\Big)}\Big\vert}\\
&\leq n!\sum_{j=1}^N\Big(\frac{1}{n}\sum_{k=1}^n
{\verta_{j,k}\vert}^2\Big)^{m/2}\beta^{(n-m)/2},\end{aligned}$$ which together with (\[eq1756\]) gives $$\begin{aligned}
{\vertG_2(Z)\vert}&\leq \frac{n(n-1)}{2(N-1)}\alpha\beta^{(n-2)/2},\\
{\vertG_3(Z)\vert}&\leq \frac{1}{3}\frac{n!(N-3)!}{(n-3)!N!}
\sum_{j=1}^N\Big(\frac{1}{n}\sum_{k=1}^n
{\verta_{j,k}\vert}^2\Big)^{3/2}\beta^{(n-3)/2}.\end{aligned}$$
The inequalities given above can be used to derive bounds for ${\vertG_2(Z)\vert}$ and ${\vertG_3(Z)\vert}$ depending on $\gamma$. For precise calculations, we use the notation $$\begin{aligned}
\gamma(d)=\frac{n\alpha}{N}\min\Big\{dn,\,\frac{1}{1-\beta}\Big\}\end{aligned}$$ for $d\in(0,\infty)$, giving $\gamma=\gamma(1)$. We have $$\begin{aligned}
{\vertG_2(Z)\vert}&\leq \gamma(1/2)
\max\Big\{\beta^{(n-2)/2},\,\frac{1}{2}n(1-\beta)\beta^{(n-2)/2}
\Big\}\frac{N(n-1)}{(N-1)n}\leq \gamma(1/2),\label{eq4236a}\\
{\vertG_3(Z)\vert}&\leq \sqrt{3}\frac{(n-1)(n-2)N^2}{(N-1)(N-2)n^2}
\sum_{j=1}^N\Big(
\frac{1}{N^2}\sum_{k=1}^n{\verta_{j,k}\vert}^2\min\Big\{\frac{n}{3},
\frac{1}{1-\beta}\Big\}\Big)^{3/2}\nonumber\\
&\quad{}\times\max\Big\{\beta^{(n-3)/2},\,
\Big(\frac{n}{3}\Big)^{3/2}
(1-\beta)^{3/2}\beta^{(n-3)/2}\Big\}\nonumber\\
&\leq \sqrt{3}\sum_{j=1}^N\Big(
\frac{1}{N^2}\sum_{k=1}^n{\verta_{j,k}\vert}^2\min\Big\{\frac{n}{3},
\frac{1}{1-\beta}\Big\}\Big)^{3/2}.\label{eq4236}\end{aligned}$$ We note that (\[eq4236\]) implies that ${\vertG_3(Z)\vert}$ is bounded by $\sqrt{3}(\gamma(1/3))^{3/2}$, which is however of worse order.
The following result is a consequence of Theorem \[th2994\], (\[eq4236a\]) and (\[eq4236\]).
\[cor387\] If $\gamma<1$, then $$\begin{aligned}
{\Big\vert\frac{(N-n)!}{N!}{\mathrm{Per}(Z)} -\prod_{k=1}^n\widetilde{z}_k\Big\vert}
&\leq \gamma(1/2)
+\frac{3^{1/4}C_{3}\,\gamma^{3/2}}{(1-\gamma)^{3/4}},
\label{eq81776}\\
{\Big\vert\frac{(N-n)!}{N!}{\mathrm{Per}(Z)} -H_2(Z)\Big\vert}
&\leq \sqrt{3}\sum_{j=1}^N\Big(
\frac{1}{N^2}\sum_{k=1}^n{\verta_{j,k}\vert}^2\min\Big\{\frac{n}{3},
\frac{1}{1-\beta}\Big\}\Big)^{3/2}\nonumber\\
&\quad{}+\frac{2^{1/2}C_{4}\,\gamma^{2}}{(1-\gamma)^{3/4}},
\label{eq81777}\end{aligned}$$ where the second inequality requires $n\geq 2$.
Inequality (\[eq71665\]) follows from (\[eq81776\]) and (\[eq892762\]), while (\[eq71666\]) can be easily be shown using (\[eq81777\]), (\[eq71977\]) and the representation $$\begin{aligned}
\lefteqn{\int f\,{{\mathrm{d}}}(P^{Y_n}-Q_2)
=\int\Big(\frac{(N-n)!}{N!}\sum_{j\in{\underline{N}}_{\neq}^n}
\prod_{k=1}^nf_k(X_{j_k}(\omega))-\prod_{k=1}^n\zeta_k(\omega)}
\qquad\\
&{}+\frac{1}{N(N-1)}
\sum_{K\subseteq{\underline{n}}:\,{|K|}=2}\sum_{j=1}^N
&\prod_{k\in K}\Big( f_k(X_j(\omega))-\zeta_k(\omega)\Big)
\prod_{k\in{\underline{n}}\setminus K}\zeta_k(\omega)\Big)\,{{\mathrm{d}}}P(\omega) \end{aligned}$$ for $f=\bigotimes_{k=1}^n f_k\in{\mathcal{F}}_n$, where $\zeta_k(\omega)=\frac{1}{N}\sum_{j=1}^Nf_k(X_j(\omega))$.
We now show that the singularity in (\[eq7435\]) can be removed.
\[th12\] For fixed $\ell\in{\underline{n}}$, let $\kappa_\ell$ be the smallest absolute constant such that, without any restrictions on $\gamma$, $$\begin{aligned}
{\Big\vert\frac{(N-n)!}{N!}{\mathrm{Per}(Z)} -H_\ell(Z)\Big\vert}
\leq \kappa_\ell\,\gamma^{(\ell+1)/2}.\end{aligned}$$ Then $\kappa_\ell\leq \frac{(\ell+1)^{1/4}C_{\ell+1}}{(1-x_\ell)^{3/4}}$, where $x_\ell\in(0,1)$ is the unique positive solution of the equation $$\begin{aligned}
\label{eq8546}
2+2^{1/4}C_{2}x\Big(\frac{1-x^{\ell-1}}{1-x}\Big)^{3/4}
=(\ell+1)^{1/4}C_{\ell+1}\,\frac{x^{(\ell+1)/2}}{(1-x)^{3/4}},
\quad(x\in(0,1)).\end{aligned}$$ In particular, $\kappa_1\leq 3.57$, $\kappa_2\leq 5.53$ and $\kappa_3\leq7.08$.
Dividing (\[eq8546\]) by $x^{(\ell+1)/2}$ yields a decreasing left-hand side, whereas the right-hand side remains increasing in $x$. Therefore (\[eq8546\]) has indeed a unique positive solution $x_\ell\in(0,1)$. Similarly as in the proof of Theorem \[th2994\], we have $$\begin{aligned}
\lefteqn{{\Big\vert\frac{(N-n)!}{N!}{\mathrm{Per}(Z)} -H_\ell(Z)\Big\vert}
\leq\frac{(N-n)!}{N!}{\vert{\mathrm{Per}(Z)}\vert}+{\vertH_\ell(Z)\vert}
\leq2+\sum_{m=2}^\ell{\vertG_m(Z)\vert}}\\
&\leq2+2^{1/4}C_{2}\gamma
\Big(\sum_{m=0}^{\ell-2}\sqrt{m+1}\gamma^{m}\Big)^{1/2}
\leq2+2^{1/4}C_{2}\gamma
\Big(\frac{1-\gamma^{\ell-1}}{1-\gamma}\Big)^{3/4}
=:h(\gamma).\end{aligned}$$ If $\gamma\in[0,x_\ell]$, we obtain by Theorem \[th2994\] that $$\begin{aligned}
{\Big\vert\frac{(N-n)!}{N!}{\mathrm{Per}(Z)} -H_\ell(Z)\Big\vert}
&\leq(\ell+1)^{1/4}C_{\ell+1}\,
\frac{\gamma^{(\ell+1)/2}}{(1-\gamma)^{3/4}}
\leq\frac{(\ell+1)^{1/4}C_{\ell+1}}{(1-x_\ell)^{3/4}}\,
\gamma^{(\ell+1)/2}.\end{aligned}$$ If $\gamma\in(x_\ell,\infty)$, then $$\begin{aligned}
{\Big\vert\frac{(N-n)!}{N!}{\mathrm{Per}(Z)} -H_\ell(Z)\Big\vert}
&\leq h(\gamma)
\leq \frac{h(x_\ell)}{x_\ell^{(\ell+1)/2}}\gamma^{(\ell+1)/2}
=\frac{(\ell+1)^{1/4}C_{\ell+1}}{(1-x_\ell)^{3/4}}\,
\gamma^{(\ell+1)/2}.\end{aligned}$$ It remains to use that $x_1\leq 0.5611$, $x_2\leq0.7222$ and $x_3\leq 0.7812$.
Acknowledgment {#acknowledgment .unnumbered}
==============
The author is indebted to an anonymous reviewer for bringing to his attention the result given in Lemma \[la61966\] and for the indication of how it helps to improve the previous version of Lemma \[la2655\]. This led to the significant improvement of constants in several upper bounds. The author also thanks Lutz Mattner for helpful comments.
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---
abstract: 'CAS Journal Ranking, a ranking system of journals based on the bibliometric indicator of citation impact, has been widely used in meso and macro-scale research evaluation in China since its first release in 2004. The ranking’s coverage is journals which contained in the Clarivate’s Journal Citation Reports (JCR). This paper will mainly introduce the upgraded version of the 2019 CAS journal ranking. Aiming at limitations around the indicator and classification system utilized in earlier editions, also the problem of journals’ interdisciplinarity or multidisciplinarity, we will discuss the improvements in the 2019 upgraded version of CAS journal ranking (1) the CWTS paper-level classification system, a more fine-grained system, has been utilized, (2) a new indicator, Field Normalized Citation Success Index (FNCSI), which ia robust against not only extremely highly cited publications, but also the wrongly assigned document type, has been used, and (3) the calculation of the indicator is from a paper-level. In addition, this paper will present a small part of ranking results and an interpretation of the robustness of the new FNCSI indicator. By exploring more sophisticated methods and indicators, like the CWTS paper-level classification system and the new FNCSI indicator, CAS Journal Ranking will continue its original purpose for responsible research evaluation.'
address: 'National Science Library, Chinese Academy of Sciences, Beijing 100190, P. R. China'
author:
- Zhesi Shen
- Sichao Tong
- Fuyou Chen
- Liying Yang
bibliography:
- 'JournalRanking.bib'
title: 'The utilization of paper-level classification system on the evaluation of journal impact'
---
Journal ranking ,Field normalization ,Citation Success Index
Introduction {#sec:introduction}
============
History of CAS Journal Ranking
------------------------------
The CAS journal ranking, an annually released journal ranking by the Center of Scientometrics (CoS), National Science Library of Chinese Academy of Sciences (CAS), is a journal ranking widely used in China. It ranks journals contained in the Clarivate’s Journal Citation Reports (JCR), based on bibliometrics data. We ’ll sketch out its history and mainly introduce the upgraded version of the 2019 CAS journal ranking, which we firstly utilize the CWTS paper-level classification system and a new indicator, the Field Normalized Citation Success Index (FNCSI).
The non-Field Normalized impact factor (JIF), which has been widely used as a journal indicator, performs differently in different research domains. Around the year 2000, in practical administrative work, the CoS research group has gradually identified that the impact factor was in a misused situation in most cases at that time in China. Aiming to compare or analyze journals separately in different scientific domains, the CoS research group released the first edition of CAS journal ranking in 2004, becoming popularly used in China. Journals can be grouped by subject area (major areas developed from degree classification by Degree Office of the Ministry of Education of the People’s Republic of China), subject category (the same specific subject categories developed from the JCR journal subject categories in the Web of Science database).
CAS journal ranking has been applied in many cases, varying from supporting related scientific policy-making of institutions to providing journals’ information to researchers. For the institutional level, they can know the performance of scientific output via drawing their distributions in CAS journal ranking, this information can help them when making related policies. Among the cash-per-publication reward policies in China, CAS journal ranking plays a dominant role. Chinese universities usually reward researchers for scientific output, motivating scientific research. [@RN1071] analyze 168 reward policies in China, and they find that there is an increasing trend of adopting CAS journal ranking in Chinese universities from 2005, after the first edition of CAS journal ranking was released. And there are 99 of reward policies taking CAS journal ranking as the reference by 2016. For researchers, CAS journal ranking can help them know journals of targeted fields, from a relatively comprehensive view, when submitting their research output. Additionally, some journals utilize CAS journal ranking as the source of information, about themselves and other journals.
Limitations of old CAS Journal Ranking
--------------------------------------
A limitation relates to indicator exists in old CAS journal ranking. For a journal, the citation distributions are skewed, and JIF can be vastly affected by the tail of highly-cited papers. We previously utilize a three-year averaged JIF to alleviate such fluctuation. However, it is still not robust enough against occasionally highly-cited papers.
The second limitation is that the journal classification system used in the old CAS journal ranking is not fine-grained. Regarding citation practices, [@RN1087] proposes the citation potential which can be defined as the probability of being cited, perform significantly differently in different fields, and we previously use the JCR journal subject categories in the Web of Science database. However, it is still not fine-grained, differences in citation also exist within fields (e.g., citation performs differ between different areas within a medical field in the study by [@RN1085], based on the subject categories in the WoS database, which we use in old CAS journal ranking).
We plot a science map for journals from all fields (please see Figure \[fig:exp\_jif\]) with each dot representing a journal and the color representing potential citation. The layout of this map is used in an earlier paper [@RN1076] based on journal citation network. Here we use journal’s expected JIF as an indicator for potential citation, the detailed formula can be found in the data and method section. The color of each dot is related to the value of the corresponding journal’s expected JIF: the more red/blue the color is, the larger/smaller the value is. Figure \[fig:exp\_jif\]) indicates a clear distinction between the potential citation between different research fields. We can see the phenomenon of citation performs differ between different areas exist within not only the above studied medical fields but also many other fields, for example, the upper part and the lower part of the Math category obviously perform differently.
![Map of scientific journals with expected JIF.[]{data-label="fig:exp_jif"}](figure/Expected_JIF_label.png){width="80.00000%"}
We then take journals from JCR category: Statistics & Probability as an example. Looking at Figure \[fig:exp\_jif\_stat\], each dot represent a journal, we color journals titled with probability in blue, and in general, most blue dots have smaller expected JIF, indicating that distinction of citation potential probably exists between journals from different topic, within Statistics & Probability category, e.g., Probability related journals perform more weakly in citation potential.
![Correlation of JIF and expected JIF for journals in Statistics and Probability category.[]{data-label="fig:exp_jif_stat"}](figure/JIF_Expected_JIF_stat.pdf){width="50.00000%"}
A third limitation is typically related to journals’ interdisciplinarity or multidisciplinarity. In addition to multidisciplinary scopes included in more journals from a general view, research topic can span across established disciplines [@RN1081], bringing benefits and challenges, especially in journal impact studies. Utilizing a more fine-grained classification system and more sophisticated indicators can partly be a solution to this phenomenon. For example, Nature Communications and Science Advance, two famous open access multidisciplinary journals, having similar Journal Impact Factor (JIF), but their amount and distribution of covered topics are quite different. Similar situations will also happen to specialized journals. Also, some research journals will publish a far greater proportion of reviews than others, usually leading to high JIF, this is a forth limitation.
For these limitations above, we make improvements in the upgraded version of the 2019 CAS journal ranking, which has been firstly released in January 2020 on the official website[^1]. Refinements in this release include the followings:
- The CWTS paper-level classification system, a more fine-grained system, has been utilized to address the above classification system related problem and journals’ interdisciplinarity related problem.
- Instead of JIF, a new indicator, Field Normalized Citation Success Index (FNCSI), has been used in the upgraded version. On the insensitivity side, compared with other citation impact factors, e.g., the three-year average JIF utilized in earlier editions, it excels no merely in the robustness of the occasional ultra-small number of extremely highly cited publications, but also in the robustness against the wrongly assigned document type.
- In addition, from a paper-level instead of journal-level, we calculate the indicator within article/review type papers.
More detailed information about the above refinements will be discussed later in this paper. Data and Methods section will introduce data coverage, CWTS paper-level classification system and the indicator utilized in the upgraded version of the 2019 CAS journal ranking. Results section includes a small part of the CAS journal ranking result and interpretation regarding the advantage of FNCSI. We finally discuss that attention should be paid on how to use CAS journal ranking appropriately for responsible research evaluation. Ongoing work and future plans will also be discussed.
Method and Data {#sec:method}
===============
Journals and citation data
--------------------------
The CAS journal ranking includes the journals which contained in the Clarivate’ Journal Citation Reports (JCR) [@JCR2018]. For journals’ citations data, we use Journal Impact Factor contributing items, which released by Clarivate’ Journal Citation Reports. This contains citations in year Y of each article and review, published in years Y-1 and Y-2, which counted towards the journal’s impact factor.
Paper-level classification data
-------------------------------
The data utilized in the CWTS paper-level classification was collected from Clarivate’ Web of Science database, with the document types article and review, which were published between 2000 and 2018, and this classification system only included publications from the SCI and SSCI database. For the details of constructing the CWTS paper-level classification system, we refer to [@RN1082; @RN1074] for a more detailed introduction of the classification methods from exploring the relatedness of publications to clustering publications into groups. This classification system consists of three levels - macro, meso, and micro levels - according to different granularity. Here we use the micro-level with about 4,000 clusters. It should be noted that, in the released CWTS paper-level classification data, publications from trade journals and several local journals are excluded, i.e., these journals cannot be evaluated. Here we try to include as many journals as possible, thus for these unclassified publications, we retrieve their related records from WoS and put them into corresponding clusters based on the clusters of the retrieved related records using the majority rule. In total, 99% of publications reported in JCR are included for calculation and 98% of journals having more than 90% of their total publications are included.
Journal Ranking Indicators
--------------------------
In CAS Journal Ranking 2019, we follow the idea of Citation Success Index (CSI) and extend it to a field normalized version. The original CSI presented to compare the citation capacity between two journals [@stringer2008effectiveness; @Milojevic2017; @Shen2018], is defined as the probability of a randomly selected paper from one journal having more citations than a randomly selected paper from the other journal. Following the same idea, we propose the Field Normalized Citation Success Index (FNCSI). The FNCSI is defined as the probability that the citation of a paper from journal $A$ is larger than a random paper in the same topics and with the same document type from other journals. For the details please refer to the section below. For comparison, we also consider the Field Normalized Impact Factor (FNIF).
### Field Normalized Citation Success Index (FNCSI)
For journal A, the probability that the citation of a paper from journal A is larger than a random paper in the same topics and document type from other journals, is defined as below: $$S_A = P(c_a > c_o | a\in A, o\in O) =\sum_{t,d}P(A^{t,d})P(c_a>c_o|a\in A^{t,d},o\in O^{t,d})
\label{eq:fncsi}$$ For a specific research topic t, its FNCSI is defined as below: $$S^t_A = \frac{1}{N_{A^t}}\sum_{d}N_{A^{t,d}}\left[\frac{\sum_{a\in A^{t,d},o\in O^{t,d}}1(c_a > c_o) +\sum_{a\in A^{t,d},o\in O^{t,d}}0.5(c_a = c_o)}{N_{A^{t,d}}N_{O^{t,d}}}\right]$$ Journal A usually invloves several research topics from the micro level of the system, then the total FNCSI of Journal A can be sumed from its invloved topics as below: $$S_A = \frac{1}{N_A}\sum_{t}N_{A^{t}}S^t_A$$ where $t\in \{\text{topic}_1,\text{topic}_2,\text{topic}_3,....\}, d\in \{\text{article},\text{review}\},A^{t,d}$ represents the publications clustered in topic $t$ with document type $d$ in journal $A$.
### Field Normalized Impact Factor (FNIF)
Field Normalized Impact Factor (FNIF) use the same classification system as FNCSI but uses the commonly used average citation based normalization approach, i.e., each citation is normalized by the average citation of papers in the same topic cluster and with the same document type. For instance, the FNIF of journal $A$ is defined as: $$F_A = \frac{\sum_{t,d}\sum_{a\in A^{t,d}}c_a/\mu_{t,d}}{N_A}
\label{eq:fnif}$$ where $\mu_{t,d}$ is the average citation of papers in topic $t$ with document type $d$. By comparing the results of FNCSI and FNIF, we can see the advantages of CSI.
### Expected JIF
As we mentioned earlier, for each journal, we use expected JIF as an indicator of potential citation: $$E_A = \frac{\sum_t\mu_tN^t_A}{N_A}
\label{eq:exp_jif}$$ where $\mu_t$ is the average citation of papers in topic $t$.
Results {#sec:result}
=======
Ranking Results
---------------
In this section we present the results of CAS Journal Ranking based on FNCSI and the comparisons with other indicators. Table \[tab:top20\] shows the top 20 ranked journals according to FNCSI. Here we only list journals mainly publishing research articles. The top five journals are well-acknowledged in natural and life science. The rest journals belong to different fields and not concentrate on a single field or narrow fields. If we take a look at the publishers of these journals, we can see that this list is dominated by Nature-titled journals, Lancet-titled journals and Cell-titled journals.
The corresponding rankings based on journals’ FNIF values of these top 20 journals are also presented Table \[tab:top20\]. Among these journals, the rankings of [*Cancer Cell*]{}, [*Nature Neuroscience*]{}, [*Cell Metabolism*]{} and [*Nature Immunology*]{} are boosted most from the FNCSI indicator, they all climb more than 20 positions. Only [*Lancet Oncology*]{} shows a slight drop in position from the FNCSI indicator. Overall, Journals from medical-related categories mostly have a relatively big gap between these indicators. In Appendix Table \[tab:top20\_fncsi\_fnif\] we present the top 20 journals both for FNCSI and FNIF.
The correlation among these journal citation indicators are shown in Figure 3, we can see that FNCSI and FNIF are highly correlated (spearman correlation: 0.98, p-value: 0.0). In the lower part of Figure \[fig:corr\_rankings\], we highlight several journals that having worse rankings in FNCSI compared with FNIF. These journals share a common property that they each have one or several highly cited papers and a majority of poorly cited papers, e.g., Chinese Phys C has one paper cited more than 2000 times but about 70% papers are zero cited [@JCR2018].
![Correlation of rankings based on FNCSI and FNIF.[]{data-label="fig:corr_rankings"}](figure/FNCSI_FNIF_scatter.pdf){width="60.00000%"}
Earlier in this article, we discuss the difference of citation potential exists between journals from different topics, within the Statistics & Probability category. Here in Table \[tab:prob\], we give the top 20 ranked journals (which mainly publishing research articles) according to FNCSI in this category. And to some extent, we can find that journals perform weakly in citation potential have been revealed by FNCSI, such as several well-acknowledged journals like [*Annals of Statistics*]{}, [*Annals of Probability*]{} and [*Biometrika*]{}.
Robustness
----------
### Robust against extremely highly cited publications
The robustness of an indicator represents its sensitivity to changes in the set of publications based on which it is calculated. A robust indicator will not change a lot against the occasional ultra-small number of highly cited publications. To measure the robustness of an indicator we construct several sets of publications for each journal with bootstrapping method and recalculate the indicator and rankings accordingly. For instance, for a journal with N publications, we randomly selected N publications with replacement, calculate these indicators, and get a new ranking for each journal. We simulate this procedure for 100 times and obtain 100 rankings for each journal. Figure \[fig:ranking\_robust\](a) shows the distribution of the obtained rankings of Chinese Physics: C. We can see that the range of ranking from FNCSI varies much less than FNIF. The citation distribution of Chinese Physics: C is highly skewed, with one paper cited about two thousand times and about 70% papers not cited. Thus FNIF depends strongly on whether this highly cited are included in calculation or not.
To get an overview of the indicators’ robustness, we calculate the relative change of rankings for these indicators. The relative change of ranking is defined as: $$\Delta = \frac{1}{N}\sum^N_{j} \frac{\text{max} \{R_j\} - \text{min} \{R_j\}}{\text{avg} \{R_j\}}$$ where $\{R_j\}$ is the rankings of journal j obtained from the above simulation. As shown in Fig. \[fig:ranking\_robust\](b), the relative change of FNCSI is smaller than FNIF implying that FNCSI is more robust than FNIF as FNCSI mainly focus on the central tendency of the citation distribution and is not easily affected by occational highly cited papers.
![(a) Ranking variability of [*Chinese Physics: C*]{} for FNCSI and FNIF. (b) Relative change of rankings based on FNCSI and FNIF[]{data-label="fig:ranking_robust"}](figure/robust_cpc.pdf "fig:"){width="45.00000%"} ![(a) Ranking variability of [*Chinese Physics: C*]{} for FNCSI and FNIF. (b) Relative change of rankings based on FNCSI and FNIF[]{data-label="fig:ranking_robust"}](figure/robust_relative_change.pdf "fig:"){width="45.00000%"}
### Robust against Document Type
Citation patterns are expected to vary a lot across different document types (Price, 1965). When conducting the field normalization, we also consider the document type, thus wrongly assigned document types will affect the journals’ indicators and rankings. To test the sensitivity of indicators against wrongly labeled document types, here we generate a virtual dataset:
- for each journal, we turn its most highly cited paper to the opposite, i.e., Article to Review or Review to Article,
and then we recalculate the journal indicators and obtain the new rankings based on FNCSI and FNIF respectively. The comparison of rankings based on this changed data with the original rankings is shown in Fig. \[fig:robust\_dt\]. We can see that almost all the orange dots (FNCSI-based) locate closely along the diagonal line while the blue squares(FNIF-based) spread much broader which implying that rankings based on FNCSI are more robust against wrongly labeled document type than rankings based on FNIF.
![Robustness against document type for FNCSI and FNIF.[]{data-label="fig:robust_dt"}](figure/robust_DT.pdf){width="60.00000%"}
Conclusion and Discussion {#sec:conclusion}
=========================
In this paper, we briefly describe the CAS Journal Ranking’s history and its practical applications by Chinese universities and institutes in rewarding, promotion and research performance monitoring. We also discuss a number of limitations in earlier editions of the CAS Journal Ranking, and our exploration of solving these problems. To better solve these problems we introduce the new indicator - Field Normalized Citation Success Index - which is used in the CAS Journal Ranking 2019 upgraded version. The FNCSI extends the idea of CSI and uses a fine-grained paper-level classification system to eliminate the citation difference among fields. We also consider the difference citation potential between articles and reviews in normalization. A detailed comparison between FNCSI and FNIF indicating that the ranking result obtained from FNCSI is favorable and is robust against extremely highly-cited publications and wrongly assigned document type.
We need to point out, towards to one of the important issue that evaluating citation performance fairly between different research fields, some contributed work has been done from the source-side, which is originated from [@zitt2008modifying], to solve the field normalized issue, including the source normalized impact per paper (SNIP) indicator [@RN1021], the revised SNIP indicator [@RN1016]. Comparisons and discussions between the source(citing)-side approach and cited-side approach have been done by [@RN817; @waltman2013source; @ruiz-castillo2014the], and still have been inconclusive, here we refer to the overviews of these discussions provided by [@RN37] and [@glanzel2019springer]. We also plan to do an empirical comparison between these indicators. Besides, as previously mentioned about limitations in the earlier editions of CAS journal ranking, with respect to occasionally highly-cited papers, the revised SNIP indicator has the same problem. [@RN802] give an example of the journal Advances in Physics which fluctuates significantly across time based on SNIP and they tried to address this problem by adopting the Elo rating system which takes journals’ historical performance into consideration.
In addition, we have an ongoing exploration of providing journal profiles which will provide more detailed information about journals’ covered topics and facilitate the comparison of journals on a target topic. This journal profile module will be added to the CAS journal ranking in future editions.
Around 1990, China started launching a reward policy to encourage Chinese scholars to join the international research community and publish papers in international journals, mainly the WoS-indexed papers [@RN1083]. Till now, Chinese institutions all have their own reward policy [@RN1071], and these policy which mostly reference CAS Journal Ranking, has indeed succeeded in promoting China’s international scientific publications in the past period. CAS journal ranking truly promotes understanding more about journals for Chinese policymakers and researchers. However at the same time, we are aware of the inappropriate employ that comes along also has a negative impact as indicators’ function may easily be warped in practical evaluation, even becoming a driving force of research [@RN1075; @RN1084]. We here especially notice its misused in evaluating individual research, like in those cash reward policies which have been analyzed in earlier study [@RN1071], most of them take CAS journal ranking, or other bibliometric indicators, as the golden rule instead of as a reference or supporting measures. We here call on any practice of using journal indicators should meet the criteria proposed by [@RN1084]:
- "Justified. Journal indicators should have only a minor and explicitly defined role in assessing the research done by individuals or institutions [@RN1073].
- Contextualized. In addition to numerical statistics, indicators should report statistical distributions (for example, of article citation counts), as has been done in the Journal Citation Reports since 2018 [@RN1072]. Differences across disciplines should be considered.
- Informed. Professional societies and relevant specialists should help to foster literacy and knowledge about indicators. For example, a PhD training course could include a role-playing game to demonstrate the use and abuse of journal indicators in career assessment.
- Responsible. All stakeholders need to be alert to how the use of indicators affects the behaviour of researchers and other stakeholders. Irresponsible uses should be called out."
Following these criteria, we, the CoS research group, will continue our original purpose for responsible research evaluation, exploring more sophisticated methods and indicators, constantly improving the science of CAS Journal Ranking.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank Dr. Nees J van Eck and CWTS for providing the paper-level classification data, and thank Ms. M. Zhu for valuable discussion.
Author contribution {#author-contribution .unnumbered}
===================
Conceptualization: SZ, YL
Data Curation: CF, SZ
Formal analysis: SZ, TS
Methodology: SZ, YL, TS
Writing – original draft: SZ, TS
Writing – review & editing: SZ, YL, SF, CF
Supervision: YL
References {#references .unnumbered}
==========
Appendix {#appendix .unnumbered}
========
Appendix A. Top 20 ranked research journals {#appendix-a.-top-20-ranked-research-journals .unnumbered}
-------------------------------------------
In Table \[tab:top20\_fncsi\_fnif\], we list the top 20 ranked research journals based on FNCSI and FNIF respectively. Compared with the journals of selected according to FNCSI, the top four journals via FNIF are all medical-related.
\[tab:top20\_fncsi\_fnif\]
Appdendix B. Additional results on robust comparison between FNCSI and FNIF {#appdendix-b.-additional-results-on-robust-comparison-between-fncsi-and-fnif .unnumbered}
---------------------------------------------------------------------------
In this section, we present some additional results on the robustness of the proposed journal indicators. In Figure \[fig:ranking\_robust\](b) we have illustrated the relative change of rankings based on FNCSI and FNIF, here we demonstrate some further analysis and results. In Figure \[fig:robust\_up\_low\] we compare the 1st quartile and 3rd quartile rankings obtained from the 100 simulations for each journal. The x-axis is the 1st quartile and the y-axis is the 3rd quartile. We can see for both FNCSI and FNIF, the dots mainly located along the diagonal line implying that the rankings of most journals are stable. When comparing the orange dots(FNCSI) and blue squared(FNIF), we can see the spreading area of orange dots is smaller than the blue squares indicating that rankings based on FNCSI are more stable than rankings based on FNIF when dealing with some special journals.
![Change of rankings based on FNCSI and FNIF.[]{data-label="fig:robust_up_low"}](figure/robust_up_low.pdf){width="60.00000%"}
Journal indicators should also be stable across time as a journal’s reputation and quality will not change dramatically. In Figure \[fig:robust\_time\] we present the evolution of rankings based on JIF, FNIF and FNCSI for the journal [*J Math Sociol*]{}. We can see the rankings of JIF and FNIF show a big jump in the year 2018 compared with its rankings in previous years. However, the ranking of FNCSI only increases a little. Here because of data availability, we only calculated the indicators for 2017 and 2018, we will continue to monitor this journal’s performance in 2019 and forthcoming years.
![Evolution of percentile rank for [*J Math Sociol*]{} based on different indicators. The percentile ranking is calculated within the [*Mathematics, Interdisciplinary applications*]{} category.[]{data-label="fig:robust_time"}](figure/stable_year.pdf){width="60.00000%"}
[^1]: www.fenqubiao.com
|
---
abstract: 'Recent calculations using non-linear relativistic cosmological perturbation theory show biases in the mean luminosity distance and distance modulus at low redshift. We show that these effects may be understood very simply as a non-relativistic, and purely kinematic, Malmquist-like bias, and we describe how the effect changes if one averages over sources that are limited by apparent magnitude. This effect is essentially identical to the distance bias from small-scale random velocities that has previously been considered by astronomers, though we find that the standard formula overestimates the homogeneous bias by a factor 2.'
author:
- |
Nick Kaiser & Michael J. Hudson\
$^1$Institute for Astronomy, University of Hawaii\
$^2$Department of Physics and Astronomy, University of Waterloo
title: Kinematic Bias in Cosmological Distance Measurement
---
Cosmology: theory, observations, distance scale, large-scale structure; galaxies: distances and redshifts
Introduction
============
It is well known that the local rate of expansion $H_0$ is significantly perturbed, at linear order, by peculiar velocities associated with the growth of density perturbations. The impact of this on cosmological parameter estimation is quantified theoretically by calculating the covariance of the 1st order velocity field which is given in terms of the power spectrum of density fluctuations (Hui & Greene 2006; Cooray & Caldwell 2006; Davis et al. 2011; Kaiser & Hudson 2014).
The subject of this paper, in contrast, is the systematic [*bias*]{} in distances, and therefore $H_0$, caused by velocities, and which is a second order effect. This has been studied using 2nd order relativistic cosmological perturbation theory in a number of recent papers (Vanderveld, Flanagan & Wasserman, 2007; Li & Schwarz 2008; Clarkson, Ananda & Larena 2009; Umeh, Larena & Clarkson 2011; Gasperini et al. 2011; Wiegand & Schwarz 2012; Fanizza et al. 2013; Ben-Dayan et al. 2012a, 2012b, 2013a, 2013b, 2014).
These papers all compute the deviation of quantities such as the mean luminosity distance and distance modulus (log distance), averaged over a surface of constant redshift, from that which would apply in a homogeneous universe. Second order perturbation theory is being used in order to explore the regime of non-linear gravitational dynamics. Most of these papers describe the effect as backreaction from the formation of structure, though the term may be being used in a relatively broad sense compared to the narrow definition as the effect of non-commutativity of spatial averaging and time evolution deriving from the non-linearity of Einstein’s equations.
Quantitative predictions in the context of conventional structure formation models are provided in e.g. figure 6 of Ben-Dayan et al. 2013b which shows that the bias falls off inversely as the square of the redshift; that the fractional perturbation to the mean distance $\delta_d \equiv \langle \delta d_L \rangle / d_L$ is positive, and that the perturbation to the mean flux density $\Phi$ is negative with $\delta_\Phi \equiv \langle \delta \Phi \rangle / \Phi \simeq - 0.5 \delta_d$. Further, according to Ben-Dayan et al. 2014 (hereafter BDMS14), for low redshift $z \ll 1$ the mean flux density perturbation is given in terms of $\langle v^2 \rangle$, the total variance of the first order line-of-sight peculiar velocity, by $\delta_\Phi = - \langle v^2 \rangle / c^2 z^2$, and they give the bias in the distance modulus $\mu = 5 \log d_L = (5 / \ln 10) \ln d_L$ as $\langle \delta \mu \rangle = (7.5 / \ln 10) \langle v^2 \rangle / c^2 z^2$.
There are two surprising features of these results if they are assumed to be caused by inhomogeneity affecting the evolution of the averaged universe. First, a cosmological effect would be expected to grow with increasing redshift rather than decrease. Second, one would expect perturbations to distance, distance modulus and flux density to be related by $\langle\delta \mu \rangle = (5 / \ln 10) \langle\delta d_L \rangle / d_L$ and $\langle \delta \Phi \rangle / \Phi = -2 \langle \delta d_L \rangle / d_L$, just as for an individual ‘standard candle’. The relations between these quantities obtained from perturbation theory are quite different, and suggest that the cause of these effects are fluctuations. In that case, the usual relations for a standard candle would not apply, simply because of the non-commutativity of averaging and non-linear transformations; the mean of the square of a fluctuating quantity, for example, is of course not the same as the square of the mean. The effect of fluctuations and the non-linearity of the relationships between $d_L$, $\mu$ and $\Phi$ was discussed by BDMS14 who noted that the bias in $H_0$ depends on the observable used, and by Ben-Dayan et al. 2013a, who argued for using the flux density $\Phi$ in $H_0$ measurements, claiming this to be the least sensitive to fluctuations.
Statistical biases in distance estimation, often associated with the names Eddington (1914) and Malmquist (1920), have been known and widely studied for a long time, in the context of both cosmological parameter estimation and measurements of large-scale peculiar motions or ‘cosmic flows’. Substantial biases may result from the typically $\sim 20$% uncertainty in luminosity distance estimators for galaxies such as are obtained from the Tully-Fisher (TF) relation for spirals (Tully & Fisher 1977) and from the ‘fundamental plane’ (FP) for elliptical galaxies (Djorgovski & Davis, 1987; Dressler et al. 1987). In particular, distance estimates to galaxies may suffer so-called ‘homogeneous Malmquist bias’ in that field galaxies in some range of estimated distance will tend to have true distances that are, on average, systematically enhanced as more galaxies are scattered inward from larger distances than outward from smaller distances (see Lynden-Bell et al. 1988; Willick 1994; and the reviews of Faber et al. 1994 and Strauss & Willick 1995 for more details). Lynden-Bell et al. 1988 showed that with a log-normal model for the distribution of distance errors the mean log-distance in a spatially homogeneous universe would be biased upward by $\delta \ln d = 3 \Delta^2$ where $\Delta^2$ is the fractional distance error variance.
This particular kind of bias may be avoided by considering the mean peculiar [*displacement*]{} in redshift-space (where neighbouring sources have, to a good approximation, the same distance) rather than the peculiar motion in estimated distance space (Schechter 1980). This bias is also not particularly relevant to the calculations above as they effectively assume perfect standard candles. What [*is*]{} relevant is the residual bias that persists after the bias from distance errors has been eliminated. This is driven by small-scale velocity dispersion which causes a scatter in the true distance for objects at the same redshift. This was first considered by Lynden-Bell (1992) who calculated the shift in the mode of the distribution of log-distances for objects of a given recession velocity. Specialising to uniform density and ignoring selection effects and streaming motions gives $\delta \ln d = 3 \sigma_v^2 / c^2 z^2$ where $\sigma_v$ is the velocity dispersion. Willick et al. 1997 also found, under the same simplifying assumptions, that velocity dispersion induces a bias in the apparent magnitude (or distance modulus) of sources of given redshift of $\delta m = 3 \times (5/\ln 10) \times \sigma_v^2 / c^2 z^2$. And both of these are just what one would expect from the Lynden-Bell et al. 1988 formula for the standard Malmquist effect with fractional distance error variance $\Delta^2 = \sigma_v^2 / c^2 z^2$, which seems very reasonable.
Lynden-Bell (1992) and Willick et al. (1997) considered the effect of motions on small-scales that are modelled as spatially incoherent with galaxies behaving like a gas of particles with a Maxwellian velocity distribution. This is very different from the modelling of velocities in perturbation theory, where the motion is like that of a smooth, cold fluid. But otherwise the results are qualitatively the same in that the bias falls off as $1/z^2$ and is proportional to the mean square velocity. This might lead one to suspect that the perturbation theory results are simply the analogue of Malmquist-like bias from small scale motions; which are entirely a consequence of kinematics and statistics. On closer inspection, however, there is a difference in that $\delta m$ is twice as large as the $\delta \mu$ of Ben-Dayan et al. 2014.
In this paper we will explore these biases further. The questions we address are: To what extend can the perturbation theory results be understood in terms of kinematics and statistics? Why does there appear to be a difference between the effects of perturbative flows and small-scale incoherent motions? Is this some subtle relativistic effect? Or might it perhaps derive from some significant difference between the statistical properties of small-scale and large-scale motions? or from the neglect of density perturbations associated with the latter? Another question is why the perturbation theory analysis result for the bias on local measurements of $H_0$ is determined by the total velocity variance, including that from very long wavelength perturbations, when one would expect only relative motions – which for super-survey scale modes are suppressed – to appear.
Malmquist Bias from Large-Scale Coherent Flows {#sec:coherentflows}
==============================================
Here we will calculate the kinematic bias arising from ‘coherent flows’ or ‘streaming motions’; these being the focus of the relativistic perturbation theory calculations. We consider small-scale ‘thermal’ motions later. Since we are interested in the low redshift regime $z \ll 1$ we work in flat, empty space and, we will also ignore special relativistic effects as the effects of interest here are generally of order $\sim (v / cz)^2 \gg (v/c)^2$.
We first consider the bias in the distance and related quantities when averaged over the surface of constant redshift as this is simple, illustrates the key features of the phenomenon, and is what was considered in the relativistic perturbation theory studies. We then generalise the analysis to the more realistic case where we average these quantities over sources.
Area Averaged Bias {#subsec:area_average}
------------------
We imagine an ensemble of realisations of a smooth field of test particles that have a spatially continuous velocity field that consists of a Hubble flow $H \br$ plus a statistically homogeneous random velocity perturbation field, and where one particle is selected at random as the observer and is taken to lie at the origin of spatial coordinate system. Let the velocity with respect to this observer be $\bu(\br)$ and define the peculiar velocity $\bv = \bu - H \br$. Let us further assume, in the spirit of perturbation analysis, that the amplitude and scale length for perturbations in the peculiar velocity are such that there is a unique mapping from velocity (or redshift) space to real space; i.e. all particles in some region of redshift space have the same peculiar velocity. Working in units such that both the speed of light $c$ and the expansion rate $H$ are unity, the distance is $d = |\br| = z - v$ where $v$ is the line-of-sight component of the peculiar velocity.
Consider a cone of infinitesimal solid angle $d \Omega$. In redshift space, the intersection of this cone and a constant-$z$ surface has area $dA_z = z^2 d \Omega$. That two dimensional surface maps to surface element in real space that will lie at a perturbed distance $d = z - v = z (1 - v/z)$ and which will, in general, be slightly tilted relative to the line of sight as there will, in general, be some gradient of $v$ transverse to the line of sight $\bnablaperp v$. The surface element area in real space is then $$dA_r = (1 - v/z)^2 (1 + |\bnablaperp v|^2 / 2) dA_z .$$
The average of the fractional perturbation to the distance $\delta_d = (d-z)/z = -v/z$ over a solid angle $\Delta \Omega$, weighted by real-space area, is then $$\overline{\delta_d}
= \frac{\int d\Omega (1 - v/z)^2 (1 + |\bnablaperp v|^2 / 2)(-v/z)}
{\int d\Omega (1 - v/z)^2 (1 + |\bnablaperp v|^2 / 2)} .
\label{eq:delta_d_bar_area_average}$$
We wish to evaluate $\overline{\delta_d}$ accurate to second order in velocities. Since there is a factor $v/z$ in the numerator, that means we need only keep first order terms in the denominator, and we can completely ignore the transverse derivative terms as they appear only at third order, to give $$\overline{\delta_d} =
\int \frac{d\Omega}{\Delta\Omega} \left\{- \frac{v}{z} + \frac{2v^2}{z^2}
- \frac{v}{z} \int \frac{d\Omega'}{\Delta\Omega} \frac{2 v'}{z} \right \}
\label{eq:delta_d_bar_expansion_area_average}$$ the last factor here allowing for correlation between the numerator and denominator in (\[eq:delta\_d\_bar\_area\_average\]).
The integrals here are evaluated on the surface $z = $ constant i.e. on the perturbed surface in real space $d = z - v$. Working to second order precision, $\overline{\delta_d}$ is given in terms of quantities on the constant distance surface $d = z$ using $v(z - v) = v(d=z) - v dv/dd + \ldots = v - (1/2) d v^2 / dz + \ldots$ (we can ignore the effect on the second order terms above as the change in these is third order in $v$).
Taking the ensemble average, which we will denote by $\langle \ldots \rangle$, the expectation value of the first order term here vanishes as the velocity is equally likely to be positive as negative – this is equally true in real-space and redshift-space since, like $v(d)$, $d v^2 / dz$ is equally likely to be positive or negative – with the result $$\begin{split}
\langle \overline{\delta_d} \rangle & = \frac{2\langle v^2 \rangle}{z^2} -
\frac{2}{z^2} \int \frac{d\Omega}{\Delta\Omega} \int \frac{d\Omega'}{\Delta\Omega} \langle v v' \rangle \\
& \quad\quad = \frac{1}{z^2} \int \frac{d\Omega}{\Delta\Omega} \int \frac{d\Omega'}{\Delta\Omega} \langle (v - v')^2 \rangle .
\end{split}
\label{eq:delta_d_avg}$$
The last expression above makes it clear that $\langle \overline{\delta_d} \rangle > 0$ so the mean distance is biased upwards. It also shows that, for an averaging area that subtends a small solid angle $\Delta \Omega \ll 1$, only velocities caused by density perturbations with scale comparable to or smaller than the averaging region contribute significantly to the bias; for perturbations much larger than the averaging region size the velocity will vary little within the area so $v' \simeq v$ and the bias is strongly suppressed.
If instead of the perturbation to the distance, which is linear in $v$ (for given $z$), we calculate the perturbation to some observable $X$ that is a non-linear function of distance like the flux-density or the distance modulus then we need to include the second order term in the expansion of $X$ expressed as a function of $v/z$. If the perturbation is $\delta X = a v/z + b v^2 / z^2 + \ldots$ then we simply replace the factor $(-v/z)$ in (\[eq:delta\_d\_bar\_area\_average\]) by $a v/z + b v^2 / z^2$ and performing the same expansion – dropping terms that are cubic or higher in the velocity – and ensemble averaging that led to (\[eq:delta\_d\_bar\_expansion\_area\_average\]) and then to (\[eq:delta\_d\_avg\]) now gives $$\langle \overline{\delta X} \rangle = (-2 a + b) \frac{\langle v^2 \rangle}{z^2}
+ \frac{2 a}{z^2} \int \frac{d\Omega}{\Delta\Omega}
\int \frac{d\Omega'}{\Delta\Omega} \langle v v' \rangle .
\label{eq:delta_X_avg}$$
We can use this to give the fractional perturbation to the flux density of standard sources. These have $\Phi(d) \propto 1 / d^2$ so $\Phi(d) = \Phi(z) (1 - v/z)^{-2}$ and $\delta_\Phi \equiv (\Phi(d) - \Phi(z)) / \Phi(z) = (1 - v/z)^{-2} - 1 = 2 v/z + 3 v^2 / z^2 + \ldots$ so the ensemble average of the area averaged flux density perturbation $\delta_\Phi$ is given by (\[eq:delta\_X\_avg\]) with $a = 2$, $b = 3$ or $$\langle \overline{\delta_\Phi} \rangle = - \frac{\langle v^2 \rangle}{z^2} +
\frac{4}{z^2} \int \frac{d\Omega}{\Delta\Omega} \int \frac{d\Omega'}{\Delta\Omega} \langle v v' \rangle .
\label{eq:delta_Phi_avg}$$
Similarly, the perturbation to the distance modulus (DM) $\mu \equiv 5 \log_{10} d$ is $\delta\mu = \alpha \ln (1 - v/z) = - \alpha (v/z + v^2 / 2 z^2 + \ldots)$ with $\alpha \equiv 5 / \ln 10 \simeq 2.17$, so the ensemble average of the area average of $\delta \mu$ is given by (\[eq:delta\_X\_avg\]) with $a = -\alpha$, $b = - \alpha / 2$ or $$\langle \overline{\delta\mu} \rangle = \alpha
\left[ \frac{3 \langle v^2 \rangle}{2z^2} -
\frac{2}{z^2} \int \frac{d\Omega}{\Delta\Omega} \int \frac{d\Omega'}{\Delta\Omega} \langle v v' \rangle \right] .
\label{eq:delta_mu_avg}$$
Note that in both of these cases, in contrast to (\[eq:delta\_d\_avg\]), there is not complete suppression of the effect of perturbations on scales larger than the averaging area.
If we take the averaging area to cover the entire sky, and assume that the redshift is sufficiently large that the distance to this shell is much greater than the coherence scale for the velocity fluctuations then the second term involving $\langle v v' \rangle$ in each of equations \[eq:delta\_d\_avg\], \[eq:delta\_Phi\_avg\] & \[eq:delta\_mu\_avg\] will be much smaller than the first term and we have $$\begin{split}
\langle \overline{\delta_d} \rangle & = 2 \langle v^2 \rangle / z^2 \\
\langle \overline{\delta_\Phi} \rangle & = - \langle v^2 \rangle / z^2 \\
\langle \overline{\delta\mu} \rangle & = (7.5 / \ln(10)) \langle v^2 \rangle / z^2 . \\
\end{split}
\label{eq:delta_bar_collection}$$ These are identical to the low-$z$ limit expressions of BDMS14. So the relatively large low-$z$ effects are not in an essential way a result of non-linearity of gravitational dynamics (relativistic or Newtonian) as they are fully accounted for by kinematics and statistics. We believe, of course, that the velocities we observe are really caused by gravity, and non-linear structure is involved, but our point here is that the same bias would be found if one were observing test particles of negligible mass with peculiar motions caused by non-gravitational forces.
These Malmquist-like biases are easy to understand. The perturbation to the mean distance, for example, comes about because even though the velocity field on a sphere of constant-$z$ is equally likely to be positive or negative, so as many areas (or solid angle elements at the observer) get pushed out as get pushed in in distance-space, those that get pushed out to larger $d$ get pushed in the radial direction and so get expanded in area by a factor $(1 - v/z)^2 \simeq 1 - 2 v/z$ (see figure \[fig:spherefig\]). Similarly those that get displaced inwards get compressed. The result is a rectification of the real-space area averaged distance. The different numerical factors for the other variables comes about simply because they are non-linear functions of the distance.
![Dotted lines are lines of longitude and latitude on the surface of constant redshift. On this surface, peculiar velocities are equally likely to be positive as negative. The cone illustrates how a section of this sphere maps to real space for the case of a negative peculiar velocity. The section is pushed out radially away from the observer – who resides at the centre of the sphere – and consequently is expanded in area. Similarly, for a positive peculiar velocity the section would be compressed. The result of this is that the average of the distance, when weighted by real-space area, is positive. This is the cause of the bias found in the relativistic perturbation theory analyses. More relevant to real observations is the bias in distance averaged over the sources that lie in a shell of given redshift. We consider this in §\[subsec:galaxy\_average\]. There we find that there are some relatively minor differences that arise from the clustering of sources and from the Jacobian involved in transforming volumes from redshift to real space, but the main difference is that the generalisations of (\[eq:delta\_bar\_collection\]) have different numerical pre-factors when the sources are subject to selection based on flux density.[]{data-label="fig:spherefig"}](spherefig.ps){width="85mm"}
BDMS14 noted that the above imply that the bias in $H_0$ obtained from the area-averaged flux density is a factor 3 lower than that obtained from averaging the distance modulus. The above analysis shows that one can do even better by averaging $\Phi^{3/2} \propto (1 - v/z)^{-3}$ since this gives $a = 3$ and $b = 6$ so $-2a + b = 0$ and, in the approximation that the depth is greater than the coherence scale used to obtain (\[eq:delta\_bar\_collection\]), the bias vanishes.
We would emphasise that, according to our analysis, the simple results (\[eq:delta\_bar\_collection\]) are only valid for velocity perturbations with coherence scale less than the distance. But at the same time the effects are really only significant at low redshift because of the $1/z^2$ scaling. For realistic power spectra there is significant contribution to the velocity variance from quite large scales; certainly extending to tens if not hundreds of Mpc, so except for observations at much greater distance – where the effects rapidly become uninterestingly small – one should not use these formulae with the total velocity variance computed in the usual way from the matter power spectrum rather one should use equations \[eq:delta\_d\_avg\], \[eq:delta\_Phi\_avg\] & \[eq:delta\_mu\_avg\] that incorporate the terms involving the velocity correlation function $\langle vv' \rangle$.
It is also important to realise that we have defined the peculiar velocity here such that the velocity of the observer vanishes. Thus the variances and co-variances in these equations are of velocities relative to the observer, which in practice is usually taken to mean relative to the velocity of the local group (LG), since it is the LG peculiar velocity, unlike the motion of the earth or the sun, that is thought to best reflect the gravitational acceleration from large scale structures. This eliminates the effect of perturbations on scales much greater than the survey depth which would otherwise give unphysical effects if the total velocity variance were used.
This is somewhat at odds with BDMS14, and deserves some clarification. Their equations 5,6 give a bias that depends on the total velocity dispersion, including a contribution that comes from modes which are larger than the survey scale spanned by the target objects (in their case $H_{0}$ calibrators). This is the dispersion of one component of the velocity of a galaxy relative to the ‘cosmic-frame’, as is thought to be well approximated by the frame in which the CMB has zero dipole (since any intrinsic dipole is usually thought to be very small). In their discussion of this BDMS14 say that they remove the motion of the observer since the observations are usually quoted in the CMB frame, corresponding to $\bv_0 = 0$, and that a non-vanishing observer velocity would nearly double the effect. This doubling seems to us to be misleading. The observer velocity is not zero in the CMB frame – the LG is moving at about 600 km/s in that frame – but the CMB frame is not of much relevance here as the results should be independent of any frame that the observers choose to refer the observations to. Our formulae, including the correlation function $\langle vv' \rangle$, refer to ensemble averages and, if one had no idea how the LG motion originated, then these should be in the LG frame. Working in the LG frame would indeed increase the co-variance from perturbations on scales smaller than the survey scale, though the effect of motions on larger scales would still be suppressed.
But there is a difference between the variance of the motions of different source regions and our motion, which has a variance in an ensemble sense, but we only sample one realisation of the ensemble (though it is a realisation of all three components of the velocity, not just one). The exact impact of the LG’s motion depends on the depth of the gravitational sources that are responsible for its motion: if these sources are deeper then the $H_{0}$ secondary calibrators themselves, then the $H_{0}$ calibrators and the LG motion share the same bulk velocity and so, by operating in the LG frame, these super-survey modes disappear, as noted above. If on the other hand, the source of the LG’s motion is very local to the LG itself (for example, a very nearby attractor such as Virgo), then, when operating in the LG frame, the LG motion induces a coherent dipole pattern (see Kaiser & Hudson 2014 and references therein). This coherent dipole is different in character to the less-coherent distortion due to the motions of the $H_{0}$ calibrators.
In practice, however, the LG’s motion arises from gravitational sources over a wide range of distances, so the true situation is more complicated than the two scenarios sketched above. Fortunately, by mapping out the distribution of nearby galaxies with an all-sky redshift survey and predicting peculiar velocities via linear perturbation theory, we now have a good idea of the gravitational sources responsible for much of the LG’s motion (e.g. Erdogdu et al. 2006; Lavaux & Hudson 2011, Carrick et al 2014). Consequently, because in practice these surveyed volumes contain within them the secondary calibrators with which one is attempting to measure the local $H_0$, the bias in the local value of $H_0$ could be reduced by working in the frame of the redshift survey itself. In other words, the solution is to use the predicted peculiar velicities to correct for the redshifts of the calibrators (Neill et al. 2007, Riess et al. 2011), leaving only 150-200 km/s of peculiar velocity not well described by linear theory (Carrick et al. 2014).
Galaxy Averaged Bias {#subsec:galaxy_average}
--------------------
We now explore how the bias changes if, as is the case in reality, we average distances over galaxies (or supernovae), rather than perform an area weighted average on the surface of constant redshift, and allow for the fact that such sources are subject to selection bias. This involves weighting by volume elements of a shell that maps to a shell of constant thickness in redshift space, rather than by area on the surface of constant $z$, and this introduces a factor which is the Jacobian of the real- to redshift-space transformation. And there are additional weighting factors coming from the varying real-space density of galaxies arising from structure and from the distant dependent selection function.
Consider a segment of a spherical shell in redshift space at redshift $z$ and thickness $dz$ that subtends a solid angle $d\Omega$ at the observer, and which therefore has volume $dV_z = d\Omega z^2 dz$. This maps to a volume in real-space $$dV_r = (1 - v/z)^2 (1 - dv/dz) dV_z$$ where we see the Jacobian $1 - dv/dz$. Unlike the tilt factor $1 + |\bnablaperp v|^2 / 2$, which was ignorable, this has a first order component.
The expected number of detected galaxies is proportional to the product $dV_r (1 + \delta) \phi(d)$ where $\delta$ is the real-space galaxy density contrast and $\phi$ is the selection function, which we can take to be a function of real distance $d$, since e.g. effects from aberration caused by our motion changing area of galaxies is an order $v/c$ effect and is relatively negligible.
For $v \ll z$ we can make a first order expansion and write the latter as $\phi(d) = \phi(z) (1 - (v/z) d \ln \phi / d \ln z + \ldots)$. As before we shall only need to keep the first order term in the expansion of $(1 - v/z)^2$, so we can use $(1 - v/z)^2 \phi(d) = \phi(z) (1 - (2 + \gamma) v/z + \ldots)$ where $\gamma \equiv d \ln \phi / d \ln z$.
The average of the fractional perturbation to the distance over a solid angle $\Delta \Omega$, weighted by galaxy number, is then $$\overline{\delta_d}
= \frac{\int d\Omega (1 - (2 + \gamma) v/z) (1 - dv/dz) (1 + \delta) (-v/z)}
{\int d\Omega (1 - (2 + \gamma) v/z) (1 - dv/dz) (1 + \delta)} .
\label{eq:delta_d_bar}$$
As before, we wish to evaluate $\overline{\delta_d}$ accurate to second order in perturbed quantities (now including $\delta$ as well as velocity). Expanding and neglecting terms that are cubic or higher yields $$\begin{split}
\overline{\delta_d} & = - \int \frac{d\Omega}{\Delta\Omega} \frac{v}{z} \left\{
1 - \frac{(2 + \gamma) v}{z} - \frac{dv}{dz} + \delta \right. \\
& \quad\quad\quad
\left. + \int \frac{d\Omega'}{\Delta\Omega}
\left(\frac{(2 + \gamma) v'}{z} + \frac{dv'}{dz} - \delta'\right) \right\} .
\end{split}
\label{eq:delta_d_bar_expansion}$$
Again, when we take the ensemble average we will assume that the first order terms vanish by symmetry. As already noted the product of $v$ and $dv/dz$ should average to zero, as does the product of $v$ and $\delta$.
But we have extra 2nd order term involving the product of $v/z$ with $\delta'$ and $dv'/dz$. For a statistically homogeneous random field the expectation of the field and its derivative at two different locations does not, in general, vanish, nor is $\langle v \delta' \rangle = 0$ in general. Generalising to an observable $X$ whose perturbation has the expansion $\delta X = a v/z + b v^2 / z^2 + \ldots$ as before, the analogue of (\[eq:delta\_X\_avg\]) is $$\begin{split}
\langle \overline{\delta X} \rangle & = (- (2 + \gamma) a + b) \frac{\langle v^2 \rangle}{z^2}
+ \frac{a}{z^2} \int \frac{d\Omega}{\Delta\Omega} \int \frac{d\Omega'}{\Delta\Omega} \biggl\{ \\
& \quad\quad
(2 + \gamma) \langle v v' \rangle + z\langle dv'/dz - \delta' \rangle \biggr\} .
\end{split}
\label{eq:delta_X_galaxy_avg}$$
On dimensional grounds, one might expect these new terms appearing in the double integral to have a large contribution (as compared to the term involving $v v'$) from perturbations with wavelength $\lambda \ll z$ since both $\delta'$ and $dv'/dz \sim v / \lambda$. But that is misleading for the following reason. That part of the velocity field which derives from waves in the Fourier spectrum with wave-number $k = 2 \pi / \lambda$ has a coherence scale of order $\lambda$. So pairs of points that have significant correlation are restricted to have separation $\sim \lambda$, and if $\lambda \ll z$ these pairs have a separation whose direction is nearly perpendicular to the line-of-sight. This actually suppresses the contribution to $\langle \overline{\delta_d}\rangle $ from the $v dv'/dz$ term to be smaller than that from the $v v'$ term. The same is true for the term involving $\langle v \delta' \rangle$. Thus the differences introduced by averaging over galaxies, as opposed to the simpler averaging over areas, are small.
As was the case of averaging weighting by area, if we average over the entire sky and assume that this covers many ‘coherence-areas’, then we can ignore the double integral in (\[eq:delta\_X\_galaxy\_avg\]) and we have, in analogy with (\[eq:delta\_bar\_collection\]), $$\begin{split}
\langle \overline{\delta_d} \rangle & = (2 + \gamma) \langle v^2 \rangle / z^2 \\
\langle \overline{\delta_\Phi} \rangle & = - (1 + 2 \gamma) \langle v^2 \rangle / z^2 \\
\langle \overline{\delta\mu} \rangle & = (5 / \ln(10)) (3/2 + \gamma)
\langle v^2 \rangle / z^2 . \\
\end{split}
\label{eq:delta_bar_collection_galaxy_avg}$$
For distances of practical interest, the actual bias involves the additional terms in (\[eq:delta\_X\_galaxy\_avg\]). But the simpler expressions above are potentially useful in a situation where large-scale motions have been modelled and corrected for, as they would then describe any residual bias caused by un-modelled motions on smaller scales.
At any redshift the variable $d_L^n$ is unbiased for $n = -3 - 2 \gamma(z)$. In terms of flux density $\Phi$ (and selection function $\phi$) this is $\Phi^{3/2 + d \ln \phi / d \ln z}$. At the distance at which the number of galaxies per logarithmic interval of distance is maximised – the distance where most of the galaxies reside, in some sense – the selection function is falling as $\phi \propto d^{-3}$ and so the unbiased variable is $d_L^3 \propto \Phi^{-3/2}$ (as compared to the $\Phi^{+3/2}$ that applies if there is no distance dependence selection).
Malmquist Bias from Incoherent Small-Scale Motions {#sec:incoherentflows}
==================================================
The foregoing analysis was somewhat restricted in that it was assumed that at each point in real-space there is a single velocity – i.e. that the galaxies move like a fluid, thus ruling out application to bound virialised systems where there are multiple streams – and yet more restrictive in that it was assumed that there was a unique velocity at each point in redshift space; which rules out e.g. ‘triple valued’ regions in redshift space that exist around clusters. These assumptions are reasonable only for large scale motions.
At the other extreme, a useful and commonly used model for small scale motions within bound structures is that these motions are spatially incoherent with peculiar velocities drawn from a distribution function $P_v(v)dv$. As mentioned in the Introduction, the bias caused by small-scale motions with an assumed Maxwellian distribution (for which the distribution of the line-of-sight velocity is Gaussian) was considered by Lynden-Bell (1992) and by Willick et al. 1997, both of whom found an effect qualitatively similar to, but twice as large as, the bias obtained from perturbation analysis (for motions with coherence scale less than the size of the averaging region). This is puzzling. Why would the result care about whether the coherence scale is just much smaller than the averaging cell size or microscopically small?
We now show, at least in the limit that $v \ll z$, that the result for $\langle \overline{\delta\mu} \rangle$ in (\[eq:delta\_bar\_collection\]) applies also to small-scale incoherent motions, and that the use of the standard formula for the bias with distance errors replaced by velocity errors, while entirely plausible, actually over-predicts the effect (by a factor 2 in the case that selection is ignorable).
At any $z$, $P(d|z) \propto P(d,z) = P(z | d) P(d)$. But $z = d + v$, so $P(z | d) = P_v(z-d | d)$. If we assume that the distribution of peculiar velocities $P_v$ is position independent, then $P(d|z) \propto P_d(d) P_v(z-d)$, from which we can compute expectation values for distance, distance modulus etc..
With $\delta \mu = \alpha \ln (d/z)$ and assuming galaxies are uniformly distributed in angle, but subject to some smoothly varying selection function $\phi(d)$, so $P_d(d) = P_d(z) (1 - (2 + \gamma) v / z + \ldots)$, the mean DM for galaxies at redshift $z$ is $$\langle \delta \mu | z \rangle =
\frac{\int dv \left(1 - (2 + \gamma) \frac{v}{z}\right) P_v(v) \alpha
\left(-\frac{v}{z} - \frac{v^2}{2 z^2} + \ldots\right)}
{\int dv \left(1 - (2 + \gamma) \frac{v}{z}\right) P_v(v)}
\label{eq:delta_DM_int}$$ or, keeping only terms up to second order in velocity in the numerator and only the leading order term in the denominator, $$\langle \delta \mu | z \rangle = \alpha (-\langle v\rangle / z +
(\gamma + 3 / 2) \langle v^2\rangle / z^2) .
\label{eq:delta_DM}$$
For the assumed centred Gaussian distribution, $\langle v\rangle = 0$ and (\[eq:delta\_DM\]) agrees with the the third of (\[eq:delta\_bar\_collection\_galaxy\_avg\]) and, ignoring selection (i.e. setting $\gamma = 0)$, we have $$\langle \delta \mu | z \rangle = (7.5 / \ln(10)) \langle v^2 \rangle / z^2
\label{eq:delta_mu_incoherent}$$ in accord with the third of (\[eq:delta\_bar\_collection\]) but in conflict with equation 15 of Willick et al. 1997 and at odds both with equation 9.17 of Lynden-Bell (1992) and with the seemingly reasonable analogy with Lynden-Bell et al. 1988, all of which would suggest that for a uniform spatial distribution of galaxies $\delta \ln d = 3 \langle v^2 \rangle / z^2$, which is twice as large as what we have here.
The reconciliation with Lynden-Bell (1992) is that the quantity he considers is the [*mode*]{} of $P(\ln(d) | v)$ the distribution of log-distances given an observed recession velocity $v$ and assuming a Gaussian scatter in $v$. That is the most [*probable*]{} log-distance. But what we are interested in here is the [*mean*]{} of the log-distance. The $\ln d$ probability distribution, under these conditions, is asymmetric, and the shift of the mean is half the shift of the mode. Using the shift of the mode, we would argue, overestimates the ‘homogeneous Malmquist bias’ caused by small scale velocity dispersion by a factor two.
Regarding the analogy with Lynden-Bell et al. 1988, what they assumed was a model for FP distance errors in which the probability distribution for the estimated log-distance $l_e$ given a true log-distance $l = \ln d$ was a Gaussian: $$P(l_e | l) = (2 \pi \Delta^2)^{-1/2} \exp(-(l_e - l)^2 / (2 \Delta^2)).
\label{eq:DLB++88model}$$ In the present context redshift $z$ plays the role of estimated distance, with $v$ the distance error. But the model (\[eq:DLB++88model\]) differs from that assumed above (with a Gaussian distribution for velocity errors) in two respects: First, this distribution implies an asymmetric distribution for the peculiar velocity, with a non-zero mean and asymmetric tails. Second, the fractional distance error is independent of distance, so in this model the absolute error grows with distance. This is appropriate for TF or FP distances, but not for errors produced by random motions. As we show in the appendix, the former does not, by itself, resolve the inconsistency; if one uses the moments of $v$ implied by this distribution in (\[eq:delta\_DM\]) this gives $\delta \mu = (5 / \ln 10) \Delta^2$, which does not agree with (\[eq:delta\_mu\_incoherent\]) nor, for that matter, is it in accord with $\delta \ln d = 3 \Delta^2$. The full resolution, again demonstrated in the appendix, is that one needs to modify the above argument to treat the case of distance independent [*fractional*]{} distance errors, and the bias is then given by (\[eq:generalisedHMB\]) which is very similar to (\[eq:delta\_DM\]) but which has the numerical factor $\gamma + 3/2$ replaced by 7/2. Using the first and second velocity moments implied by (\[eq:DLB++88model\]) in (\[eq:generalisedHMB\]) gives $\langle d \mu | z \rangle = (15 / \ln(10)) \langle v^2 \rangle / z^2$ in accord with the usual formula $\delta \ln d = 3 \langle v^2 \rangle / z^2$. But this is not correct for distance errors from velocities, which is what we are considering here, where it is the [*absolute*]{} rather than fractional distance error that is independent of distance, and where the velocity distribution is symmetric.
The above argument is idealised in that it assumes both the density of galaxies and the velocity distribution function to be independent of position. Regarding the homogeneous Malmquist bias the effect of relaxing this is that the expectation of the sky-averaged $\overline{\delta \mu}$ involves the galaxy weighted velocity variance. For large-scale density perturbations there is also an inhomogeneous Malmquist bias term (whose expectation vanishes), just as found by Lynden-Bell (1992). In this regard, we note that the variable $\Phi^{3/2 + d \ln \phi / d \ln z}$ is only unbiased with respect to the homogeneous Malmquist bias and is still affected by the inhomogeneous Malmquist bias.
Summary
=======
We have shown in §\[sec:coherentflows\] that the relatively large low-redshift perturbations to the mean distances, flux densities or distance moduli obtained from relativistic second order perturbation theory can be understood as a purely classical kinematic and statistical Malmquist-like effect and are not, in any essential way, a manifestation of non-linear dynamics. While gravity is involved in generating peculiar velocities, precisely the same bias would be found if one were observing test particles with non-gravitationally generated motions. The relativistic treatment may contain other effects that are essentially gravitational in nature, but as they are apparently extremely small they are of limited interest.
In §\[subsec:galaxy\_average\] we generalised the analysis to obtain the bias when, as in reality, the distance is averaged over sources such as galaxies or supernovae that are subject to selection bias.
Our analysis provides formulae that could, in principle, be used to correct for biases in distance, and hence in the ‘local’ value of $H_0$, from large-scale or small-scale motions. For the former, our results properly account for covariance and suppression of the effect of super-survey modes that is missing from the relativistic perturbation theory papers. But we emphasise that the effects on $H_0$ at least are very small and much smaller than the fluctuations in measurements of $H_0$ that arise in linear theory. We have shown in §\[sec:incoherentflows\] that small-scale incoherent velocities have essentially the same effect. They do not cause a perturbation to log-distance $\delta \ln d = 3 \sigma_v^2 / c^2 z^2$ as has previously been found, and as would seem reasonable by analogy with the commonly used formula for homogeneous Malmquist bias. The effect is a factor two smaller. The reason that the standard formula is not valid for bias from velocity dispersion is in part because the model implies an unrealistic distribution of velocities and in part because it assumes that the distance errors scale linearly with distance whereas errors from motions are distance independent.
We showed that the average of $\Phi^{3/2 + d \ln \phi / d \ln r}$ does not suffer velocity dispersion induced homogeneous Malmquist bias.
We provide in appendix \[sec:generalisedHMB\], a slightly generalised formula for the homogeneous Malmquist bias produced by errors in estimated luminosity distance – as from e.g. Tully-Fisher or fundamental plane techniques – when using the ‘forward’ method. This result is only valid for $\Delta^2 \ll 1$, but makes no assumption about the form for the distribution function for the distance errors.
Acknowledgements
================
The authors thank Ruth Durrer, Dominik Strauss, Giovanni Marozzi and Ido Ben-Dayan for helpful discussions. MH acknowledges support of NSERC. NK gratefully acknowledges the support and stimulation of the Cifar Cosmology and Gravitation program.
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Malmquist Bias from Luminosity Distance Errors {#sec:generalisedHMB}
==============================================
We now obtain the analogue of (\[eq:delta\_DM\]) for the situation where distances are estimated from the source flux density, rather than redshift, and the distance error scales linearly with distance as is appropriate, to a first approximation, for TF or FP distances or for supernovae. Thus we assume sources with real distances $d$ and estimated distances $z$ and distance error $z - d = v$. I.e. we use the same notation as before, but with a different interpretation as the the cause of the errors, and to obtain a distance independent distribution for fractional errors we take $P_v(v|d) dv = f(v/d) dv/d$ where $f(y)$ is some normalised bell-shaped function: $\int dy \; f(y) = 1$.
If we assume the fractional distance errors are small $v \ll d$, we have $$\begin{split}
& P_v(v|d) dv = \frac{z}{z-v} f\left(\frac{v}{z-v}\right) \frac{dv}{z} \\
% & \quad = \left(1 + \frac{v}{z} + \ldots\right) f\left(\frac{v}{z} + \frac{v^2}{z^2} + \ldots\right) \frac{dv}{z} \\
& \quad = \left(1 + \frac{v}{z} + \ldots\right)
\left(f\left(\frac{v}{z}\right) + \frac{v^2}{z^2} f'\left(\frac{v}{z}\right) + \ldots\right) \frac{dv}{z}
\end{split}
\label{eq:TF+FP_Pv}$$ where $f'(y) = df/dy$.
Our goal is to compute $\langle d | z \rangle$ from the conditional distribution of distance $P(d | z) \propto P(z|d) P(d)$. Previously we used $P(d) = d^2 \phi(d)$, but here the estimated distance is not the redshift, it is the inverse square root of the flux density, so a magnitude limit imposes a selection that is a function of the estimated distance ($z$ in our notation). The upshot, as explained by Strauss & Willick (1995), is that the selection function drops out when we compute $\langle d | z \rangle$ or, equivalently, the bias is the same as obtained without any selection.
Using (\[eq:TF+FP\_Pv\]) in (\[eq:delta\_DM\_int\]) we find two extra terms that produce significant contributions to the numerator (when multiplied by $-v/z$ and integrated): $$- z^2 \int \frac{dv}{z} \left(\frac{v^2}{z^2} f\left(\frac{v}{z} \right) + \frac{v^3}{z^3} f'\left(\frac{v}{z}\right) \right)
= 2 \langle v^2 \rangle$$ where we have integrated by parts and assumed that $f$ falls to zero for large argument sufficiently fast that the boundary term is negligible.
As in (\[eq:delta\_DM\]) one finds, in additional to a term proportional to the variance in the distance error, the mean distance error $-\langle v \rangle$. This is not necessarily zero – it is not zero, for instance, if the distance estimator is obtained by minimising residuals in magnitude – but it is reasonable to assume that the strength of any bias in $\langle v \rangle$ is, to order of magnitude, at most proportional to the variance $\langle v^2 \rangle$. Keeping terms up to quadratic order in the distance error $v$ and ignoring the sub-dominant terms in the denominator in (\[eq:delta\_DM\_int\]) yields the general result valid up to linear order in the fractional distance variance $\Delta^2 = \langle v^2 \rangle / d^2$ for the homogeneous Malmquist bias $$\langle d \mu | z \rangle = \alpha ( - \langle v\rangle / z + 7/2 \langle v^2\rangle / z^2)
\label{eq:generalisedHMB}$$ where $v$ is minus the distance error and $z$ can be taken to be either the estimated distance $z$ or the real distance $d = z-v$. One can also obtain the perturbation to any other variables. The perturbation to the distance, for instance, is $$\langle {\overline {\delta_d}} \rangle = - \langle v\rangle / z + 4 \langle v^2\rangle / z^2.
\label{eq:generalisedHMBdeltad}$$
As a check, we can apply (\[eq:generalisedHMB\]) and (\[eq:generalisedHMBdeltad\]) to the log-normal model of Lynden-Bell et al. 1988 that is known to give exactly $\delta \ln d = 3 \Delta^2$ (or $\langle d \mu | z \rangle = (15 / \ln 10) \Delta^2$). In this model the probability distribution for the estimated log-distance $l_e$ given a true log-distance $l = \ln d$ is a Gaussian: $$P(l_e | l) = (2 \pi \Delta^2)^{-1/2} \exp(-(l_e - l)^2 / (2 \Delta^2)).$$ With $d = e^l$ and $z = e^{l_e}$ the moments of the estimated distance distribution are $\langle z^n \rangle = d^n \exp(n^2 \Delta^2 / 2)$. The first moment is $\langle z \rangle = d (1 + \Delta^2 / 2 + \ldots)$, so the mean of the distance error is $\langle v \rangle = d \Delta^2 / 2 + \ldots$, which is non-zero, and the second moment is $\langle z^2 \rangle = d^2 (1 + 2 \Delta^2 + \ldots)$ so the distance error variance is $\langle v^2 \rangle = d^2 \Delta^2 + \ldots$, where the notation $\ldots$ indicates quantities that are of higher order in the assumed small logarithmic variance $\Delta^2$. Using these in (\[eq:generalisedHMB\]) gives $\langle d \mu | z \rangle = (15 / \ln 10) \Delta^2$ in agreement with Lynden-Bell et al. 1988, while (\[eq:generalisedHMBdeltad\]) gives ${\overline {\delta_d}} = (7/2) \Delta^2$, in accord with equation 185 of Strauss & Willick (1995).
Equations (\[eq:generalisedHMB\]) and (\[eq:generalisedHMBdeltad\]) provide a generalisation of the standard results in that they do not assume a perfectly log-normal distribution, though they are limited to the regime where the fractional variance $\Delta^2 \ll 1$. They apply only to the ‘forward’ method where one averages the peculiar velocity of objects as a function of estimated distance. The more popular ‘inverse’ methods do not suffer this bias. Instead they have the much smaller residual bias from random motions causing scatter in distance that is the focus of the main part of this paper.
|
---
abstract: 'Accurate Monte Carlo data from a set of isotherms near the critical point are analyzed using two RG based complementary representations, given respectively in terms of $\bar h$=$h$/$\mid$$t$$\mid^{\beta\delta}$ and $\bar \tau$=$t/h^{1/\beta\delta}$. Scaling plots for data on simple cubic Ising lattices are compared with plots of $ML^{\beta\nu}$ vs. $\mid$$t$$\mid$$L^{1/\nu}$ for increasing $L$ values and with high quality experimental data on $CrBr_{3}$. Finite size effects and the equation of state are discussed.'
author:
- 'J.G. García'
- 'J.A. Gonzalo'
bibliography:
- 'apssamp.bib'
title: Finite size scaling and equation of state for Ising lattices
---
Monte Carlo methods [@Binder] using Wolff algorithms [@Wolff] have been extensively used to describe $M(T)$ for $H$=0 in the whole range of temperatures both below and above the critical temperature (specially at $T$$\cong$$T_c$) but few, if any, Monte Carlo simulations of magnetic isotherms $M(H)$ at $T$$\cong$$T_c$ have been reported in the literature. It is clear, however, that with recent improvements in computing facilities (larger memory, greater speed, better availability) accurate, well thermalized, closely spaced data can provide very substantial contribution to describing the phase transition and better understanding of finite size effects.
Simulations of this type are reported in this work, performed using Metropolis algorithms [@Metropolis], which are specially convenient to describe the system evolution at constant temperature and for small field increments[@Garcia; @Garcia2]. Using relatively large lattices with 70$\times$70$\times$70 spins, an accurate characterization of the scaling behavior, and, therefore, the equation of state in the vicinity of the critical point ($H$=0, $T_c$=4.511523785) [@Blote] for simple cubic Ising lattices can be obtained. Details of the Monte Carlo calculations for isotherms taken at $T$ near $T_c$ are given in reference [@Garcia]. Periodic boundary conditions were used and 140,000 Monte Carlo steps were taken at each field/temperature to ensure equilibrium.
For the scaling representation of the raw $M(H)$ data at each $T$ we did use two complementary ways: (a) the usual way [@Yeomans], involving $\bar m$$\equiv$$M(t,h)$/$\mid$$t$$\mid^\beta$ and $\bar h$$\equiv$$h$/$\mid$$t$$\mid^{\beta\delta}$, where $t\equiv(T-T_c)/T_c$ involves the temperature gap to $T_c$, $h$ (reduced field) is proportional to $H$, $\beta$ is the spontaneous magnetization critical exponent $\beta\equiv\partial$log$M/\partial$log$\mid$$t$$\mid$ at $T$$\rightarrow$$T_c$, and $\delta$ is the critical isotherm exponent $\delta^{-1}$$\equiv\partial$log$M/\partial$log$h$ at $T$$\rightarrow$$T_c$; and (b) a complementary way involving $\bar \mu$$\equiv$$M(h,t)/h^{1/\delta}$ and $\bar \tau$$\equiv$$t/h^{1/\beta\delta}$.
Near the fixed point corresponding to the critical point, the singular part of the reduced free energy per spin may be written [@Yeomans] in scaling form as
$$\label{eq:eq1}
\bar f_s(g_{1}, g_{2}, g_{3}, \dots) \sim \bar b^{d} \bar f_{s}(b^{y_{1}}g_{1}, b^{y_{2}}g_{2}, b^{y_{3}}g_{3}, \dots)$$
where $b$ is an arbitrary scale factor and the $y_{i}$ relate to the usual critical exponents. To get (a) Equation (\[eq:eq1\]) is differentiated with respect to the field in the usual way to obtain
$$\label{eq:eq2}
M(t, h, g_{3}, \dots) \sim \bar b^{-d+y_{2}} M(b^{y_{1}}t, b^{y_{2}}h, b^{y_{3}}g_{3}, \dots)$$
Taking $b^{y_{1}}$$\mid$$t$$\mid$=1 and setting the irrelevant variables equal to zero
$$\label{eq:eq3}
M(t, h) \sim \mid t \mid^{(d-y_{2})/y_{1}} M(\pm 1, h \mid t \mid^{-y_{2}/y_{1}})$$
and using the well known scaling relationships [@Yeomans] giving $y_1$ and $y_2$ in terms of $\beta$ and $\delta$ one gets
$$\label{eq:eq4}
M(t, h) \sim \mid t \mid^{\beta} M(\pm 1, h \mid t \mid^{-\beta\delta})$$
which implies scaling using the ordinary scaling variables
$$\label{eq:eq5}
\bar m \equiv M(t, h) / \mid t \mid^{\beta}, \bar h \equiv h / \mid t \mid^{\beta\delta}$$
To get (b), on the other hand, we can argue in an analogous way. Taking $b^{y_2}h$=1 in Equation (\[eq:eq2\]) and setting the irrelevant variables equal to zero,
$$\label{eq:eq6}
M(t, h) \sim h^{(d-y_2)/y_2} M(t\cdot h^{-y_{1}/y_{2}},1)$$
and using $y_1$ and $y_2$ in terms of $\beta$ and $\delta$, one finally gets
$$\label{eq:eq7}
M(t, h) \sim h^{1/\delta} M(t^{-1/\beta\delta},1)$$
which implies as alternative scaling variables
$$\label{eq:eq8}
\bar \mu \equiv M(t, h) / h^{1/\delta}, \bar \tau \equiv t / h^{1/\beta\delta}$$
![\[fig:epsart1\] Scaling plots of Monte Carlo data for a s.c. Ising lattice of $70^3$ spins. (a) $\bar m$$\equiv$$M$/$\mid$$t$$\mid^{\beta}$ vs $\bar h$$\equiv$$h$/$\mid$$t$$\mid^{\beta\delta}$, and (b) $\bar \mu$$\equiv$$M$/$h^{1/\delta}$ vs $\bar \tau$$\equiv$$t$/$h^{1/\beta\delta}$. The data include 30 isotherms in the intervals 4 $< T <$ 5 and 0 $< h <$ 0.02, the same symbol has been used for all of them. The critical temperature [@Blote] was taken as $T_c$=4.511523785 and the critical exponents [@Garcia] as $\beta$=5/16 and $\delta$=5. ](fig1a "fig:"){width="6.1cm" height="7.9cm"}\
![\[fig:epsart1\] Scaling plots of Monte Carlo data for a s.c. Ising lattice of $70^3$ spins. (a) $\bar m$$\equiv$$M$/$\mid$$t$$\mid^{\beta}$ vs $\bar h$$\equiv$$h$/$\mid$$t$$\mid^{\beta\delta}$, and (b) $\bar \mu$$\equiv$$M$/$h^{1/\delta}$ vs $\bar \tau$$\equiv$$t$/$h^{1/\beta\delta}$. The data include 30 isotherms in the intervals 4 $< T <$ 5 and 0 $< h <$ 0.02, the same symbol has been used for all of them. The critical temperature [@Blote] was taken as $T_c$=4.511523785 and the critical exponents [@Garcia] as $\beta$=5/16 and $\delta$=5. ](fig1b "fig:"){width="6.1cm" height="7.9cm"}
Figure 1(a) gives our Monte Carlo data using $\bar m$ and $\bar h$ as scaling variables as given by Equation (\[eq:eq5\]) with $T_c$=4.511523785 and the exponents $\beta$=5/16=0.3125 and $\delta$=5. The usual behavior is observed. We may note that the data scale extremely well and that only very minor deviations at lower $h$, attributable to finite size effects are perceptible. Figure 1(b) gives the same data using the alternative scaling representation. It may be noted immediately that the plot in Figure 1(b) resembles closely [@Binder] typical scaling plots of $ML^{\beta/\nu}$ vs $\mid$$t$$\mid$$L^{1/\nu}$ for lattices with linear dimension $L$, in our case, for Ising systems of $L^3$ spins, suggesting [@Marques] formal relationships between $H$ and $L$, $M$ and $L$, and $\mid$$t$$\mid$ and $L$, which, through the scaling plot branches defining the critical isotherm, the spontaneous magnetization (coexistence curve) below $T_c$, and the low field susceptibility above, imply respectively
$$\label{eq:eq9}
H \sim L^{-\beta\delta/\nu}, M \sim L^{-\beta/\nu} and \mid t \mid \sim L^{-1/\nu}$$
Hence
$ML^{\beta/\nu}\sim$const ($t$$\gtrless$0) $\longrightarrow$ critical isotherm
$ML^{\beta/\nu}\sim$const$\times$($\mid$$t$$\mid$$L^{1/\nu}$$)^{\beta}$ ($t<0$) $\longrightarrow$ spontaneous magnetization
$ML^{\beta/\nu}\sim$const$\times$($\mid$$t$$\mid$$L^{1/\nu}$$)^{\beta(\delta-1)/2}$ ($t>0$) $\longrightarrow$ susceptibility
![\[fig:epsart2\] Scaling plot of Monte Carlo data $ML^{\beta/\nu}$ vs $\mid$$t$$\mid$$L^{1/\nu}$ for s.c. Ising lattices with linear size $L$=30, 60, 90, 115. Note that finite size effects for $L$=30 show up closer to $\mid$$t$$\mid$$\rightarrow$0.](fig2){width="6.1cm" height="7.9cm"}
Figure \[fig:epsart2\] shows data of $ML^{\beta/\nu}$ vs $\mid$$t$$\mid$$L^{1/\nu}$ for $H$=0 and $L$=30, 60, 90, 115 which mimic the behavior shown in Figure 1(b) implying that simulations using periodic boundary conditions of phase transitions with finite size show the effects of an effective straining contribution to the magnetic field $H_{fs}\sim$$L^{-\beta\delta/\nu}$, i.e. $H_{eff}$=$H$+$H_{fs}$ such that for $L$$\rightarrow$$\infty$, $H_{eff}$$\cong$$H$.
![\[fig:epsart3\] Scaling plots of experimental data for $CrBr_3$ (a) $\bar m$$\equiv$$M$/$\mid$$t$$\mid^{\beta}$ vs $\bar h$$\equiv$$h$/$\mid$$t$$\mid^{\beta\delta}$, and (b) $\bar \mu$$\equiv$$M$/$h^{1/\delta}$ vs $\bar \tau$$\equiv$$t$/$h^{1/\beta\delta}$. The data are made up of 30 isotherms in the interval $T_c$-0.9K $< T <$ $T_c$+6.7K. The critical temperature was $T_c$=32.844 K and the critical exponents used were $\beta$=5/16, $\delta$=5 (Ising 3D) and $\beta$=0.368, $\delta$=4.28, as given in Reference [@Ho]. ](fig3a "fig:"){width="6.1cm" height="7.9cm"}\
![\[fig:epsart3\] Scaling plots of experimental data for $CrBr_3$ (a) $\bar m$$\equiv$$M$/$\mid$$t$$\mid^{\beta}$ vs $\bar h$$\equiv$$h$/$\mid$$t$$\mid^{\beta\delta}$, and (b) $\bar \mu$$\equiv$$M$/$h^{1/\delta}$ vs $\bar \tau$$\equiv$$t$/$h^{1/\beta\delta}$. The data are made up of 30 isotherms in the interval $T_c$-0.9K $< T <$ $T_c$+6.7K. The critical temperature was $T_c$=32.844 K and the critical exponents used were $\beta$=5/16, $\delta$=5 (Ising 3D) and $\beta$=0.368, $\delta$=4.28, as given in Reference [@Ho]. ](fig3b "fig:"){width="6.1cm" height="7.9cm"}
Figure 3 gives scaling plots of the high quality data of 30 isotherms at the vicinity of the Curie temperature pertaining to the insulating ferromagnet $CrBr_{3}$ measured by Ho and Litster [@Ho; @Litster] which are a classical example of experimental scaling data. We show plots of $\bar m$ vs $\bar h$ (Figure 3a) and $\bar \mu$ vs $\bar \tau$ (Figure 3b) with $T=T_{c}$=32.844K for two sets of critical exponents, Ising 3d (fractional values) and Ho & Litster (experimental values) summarized in Table \[tab:table1\]. Both sets of critical exponents produce good scaling plots in (a) as well as in (b), but the critical exponents of Ho and Litster produce somewhat better scaling plots. The accuracy of the magnetization measurements for the set of isotherms was comparable to that of nuclear magnetic resonance data and it was sufficiently precise to establish the form of the scaling function. Our Monte Carlo data, shown in Figures 1(a) and 1(b) are of comparable quality to establish the 3d Ising scaling function.
$\beta$ $\delta^{-1}$ $\gamma$
-------------------------- ------------- --------------- ----------
Ising 3D [@Garcia] 5/16=0.3125 1/5=0.2 5/4=1.25
Heisenberg 3D [@Yeomans] 0.340 0.208 1.39
$CrBr_3$ [@Ho] 0.368 0.233 1.215
: \[tab:table1\] Critical exponents
![\[fig:epsart4\] Optimized fits of the 30 Monte Carlo isotherms to the given Ising 3D equation of state. ](fig4){width="6.1cm" height="7.9cm"}
Finally we address the question of the form equation of state for 3D Ising lattices in the light of the information provided by the set of isotherms in the vicinity of the critical temperature obtained by the Monte Carlo method in our $70^3$ s.c. lattice. Figure 4 gives the plot of $M(h,t)$, Equation (\[eq:eq5\]), rewritten as
$$\label{eq:eq10}
\frac{\bar h}{\bar m} = f(\bar m) = A(1 \pm B \bar m^{z})^{(\delta-1)/z}$$
Here $A$ can be reduced to unity just by choosing properly the units for the field $H$. $B$ is a more meaningful coefficient which, in the particular case of a phase transition describable by means of the mean field approximation (such as the phase transition in a uniaxial ferroelectric) is equal to $1/\delta$=1/3. And $z$, as pointed out in Reference [@Marques] is given by $z$$\cong$$\beta\delta/\nu$=2.5 for $T<T_c$ and $z$$\cong$$\beta\delta$=1.562 for $T>T_c$. Figure 4 shows the excellent fit obtained by means of Equation (\[eq:eq10\]) with ($B/A$)=0.102. The equation of state put in the form given by Equation (\[eq:eq10\]) is specially good to show directly the most relevant information: (a) the critical isotherm for $T<T_c$ and $T>T_c$, (b) the spontaneous magnetization curve ($T<T_c$) as a vertical line, and (c) the zero field susceptibility ($T>T_c$) as a horizontal line. The quality of the fit is comparable or better than those obtained with traditional expressions of the scaling function [@Ho; @Arrot; @Vicentini; @Schofield; @Gaunt; @Gonzalo; @Milosevic; @Milosevic2].
Work is in progress to obtain Monte Carlo data in larger 3D Ising lattices at closer field/temperature intervals, and to extend the investigation of scaling plots in the vicinity of the transition to higher dimensionalities Ising 4D, Ising 5D, etc, in order to monitor closely how the approach to mean field behavior takes place. Of course we will be limited to more reduced sizes (smaller L’s) as the dimensionality increases, but we have excellent experimental data [@Jota] on a complete set of isotherms in uniaxial ferroelectric TGS to produce excellent scaling plots with $T_c$=321.470 and mean field critical exponents $\beta$=1/2 and $\delta$=3.
We specially acknowledge helpful comments and software by M.I. Marqués. Support from the Spanish MECyD through Grant Number BFM2000-0032 is gratefully acknowledged.
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---
abstract: 'The distribution of the return intervals $\tau$ between price volatilities above a threshold height $q$ for financial records has been approximated by a scaling behavior. To explore how accurate is the scaling and therefore understand the underlined non-linear mechanism, we investigate intraday datasets of 500 stocks which consist of the Standard & Poor’s 500 index. We show that the cumulative distribution of return intervals has systematic deviations from scaling. We support this finding by studying the $m$-th moment $\mu_m \equiv \langle(\tau/\langle\tau\rangle)^m\rangle^{1/m}$, which show a certain trend with the mean interval $\langle\tau\rangle$. We generate surrogate records using the Schreiber method, and find that their cumulative distributions almost collapse to a single curve and moments are almost constant for most range of $\langle\tau\rangle$. Those substantial differences suggest that non-linear correlations in the original volatility sequence account for the deviations from a single scaling law. We also find that the original and surrogate records exhibit slight tendencies for short and long $\langle\tau\rangle$, due to the discreteness and finite size effects of the records respectively. To avoid as possible those effects for testing the multiscaling behavior, we investigate the moments in the range $10<\langle\tau\rangle\leq100$, and find the exponent $\alpha$ from the power law fitting $\mu_m\sim\langle\tau\rangle^\alpha$ has a narrow distribution around $\alpha\neq0$ which depend on $m$ for the 500 stocks. The distribution of $\alpha$ for the surrogate records are very narrow and centered around $\alpha=0$. This suggests that the return interval distribution exhibit multiscaling behavior due to the non-linear correlations in the original volatility.'
author:
- 'Fengzhong Wang$^1$, Kazuko Yamasaki$^{1,2}$, Shlomo Havlin$^{1,3}$ and H. Eugene Stanley$^1$'
date: '30 July 2007 version wyhs.tex'
title: Indication of multiscaling in the volatility return intervals of stock markets
---
Introduction
============
The price dynamics of financial markets has long been a focus of economics and econophysics research [@Mandelbrot63; @Mantegna95; @Kondor99; @Mantegna00; @Takayasu97; @Liu99; @Weber07; @Bouchaud03]. Studying the volatility time series is not only crucial for revealing the underlined mechanism of financial markets dynamics, but also useful for traders. For example, it helps traders to estimate the risk and optimize the portfolio [@Bouchaud03; @Johnson03]. The volatility series is known to be long-term power-law correlated [@Ding83; @Wood85; @Harris86; @Admati88; @Schwert89; @Dacorogna93; @Granger96; @Pagan96; @Liu97; @Cizeau97; @Cont98; @Pasquini99]. To better understand these correlations and characterize temporal scaling features in volatilities, recently Yamasaki et al. [@Yamasaki05] and Wang et al. [@Wang06; @Wang07] studied the statistics of return intervals $\tau$ between volatilities that are above a given threshold $q$, which is an alternative way to analyze long-term correlated time series (see Ref [@Altmann05] and references therein). They find that scaling and memory in the return intervals of daily and intraday financial records are similar to that found in the climate and earthquake data [@Bunde04; @Bunde05; @Livina05].
Studies of financial records show that the scaling in the return intervals distribution can be well approximated by a scaling function [@Yamasaki05; @Wang06; @Wang07]. However, financial time series are known to show complex behavior and are not of uniscaling nature [@Matteo07] and non-linear features [@Cao92]. Recent studies [@Ivanov04; @Eisler06A; @Eisler06B] of stock markets show that the distribution of activity measure such as the intertrade time has multiscaling behavior. Thus, a detailed analysis of the scaling properties of the volatility return intervals is of interest. It might improve our understanding of the return intervals statistics and shed light on the underlined complex mechanism of the volatility. Our analysis suggests that for all Standard & Poor’s index constituents, the cumulative distributions of the return intervals depart slightly but systematically from a single scaling law. We also find that the moments $\mu_m\equiv\langle(\tau/\langle\tau\rangle)^m\rangle^{1/m}$ are consistent with the deviations from scaling. However, using the corresponding surrogate records [@Schreiber96; @Makse96; @Schreiber00] which remove the non-linearities, $\mu_m$ almost does not depend on $\langle\tau\rangle$ and no deviation from scaling occur. Therefore, our results suggest that non-linear correlations in the volatility account for the deviations from a scaling law.
The paper is organized as follows: In section II we introduce the database and define the volatility. In section III we discuss the scaling and investigate the deviations from scaling in the cumulative distributions of the return intervals. We also describe the stretched exponential form suggested for the distribution and the generation of the surrogate records. Section IV deals with the moments of the return intervals. We quantify the deviation from the scaling that exhibits multiscaling behavior. We simulate the return intervals with different sizes and show the finite size effect for long $\langle\tau\rangle$. We also study the discreteness effect for short $\langle\tau\rangle$ and explore the relation between the moment and its order. In Section V we present a discussion.
Database
========
We analyze the Trades And Quotes (TAQ) database from New York Stock Exchange (NYSE), which records every trade for all the securities in United States stock markets. The duration is from Jan 1, 2001 to Dec 31, 2002, which has a total of 500 trading days. We study all 500 companies which consist of the Standard & Poor’s 500 index (S&P 500) [@Note1], the benchmark for American stock markets. The volatility is defined the same as in [@Wang06]. First we take the absolute value for the logarithmic price change, then remove the intraday U-shape pattern, and finally normalize it with the standard deviation. Here the price is the closest tick to a minute mark. Thus the sampling time is 1 minute and a trading day usually has 391 points after removing the market closing hours. For each stock, the size of dataset is about 200,000 records.
Scaling in Return Intervals
===========================
The probability density function (PDF) for the return intervals $\tau$ of the financial volatilities is well-approximated by the following form, $$P_q(\tau)=\frac{1}{\langle\tau\rangle}f(\tau/\langle\tau\rangle),
\label{pdf.eq}$$ as analyzed by Yamasaki et al. [@Yamasaki05] and Wang et al. [@Wang06; @Wang07]. Here $\langle\cdot\rangle$ stands for the average over the dataset and $\tau/\langle\tau\rangle$ depends on the threshold $q$. It was suggested that the scaling function can be approximated by a stretched exponential [@Yamasaki05; @Wang06; @Wang07], $$f(x)=c e^{-(ax)^\gamma}
\label{scaling.eq}$$ for financial records, which is consistent with other long-term correlated records [@Altmann05; @Bunde04; @Bunde05; @Livina05]. Here $a$ and $c$ are fitting parameters and $\gamma$ is the exponent characterizing the long-term correlation [@Altmann05; @Bunde04; @Bunde05; @Livina05]. From the normalization of PDF follows [@Note2], $$1 = \int^\infty_0 P_q(\tau) d\tau = \int^\infty_0
\frac{1}{\langle\tau\rangle} c e^{-(a\tau/\langle\tau\rangle)^\gamma}
d\tau.
\label{normal.eq}$$ From the definition of $\langle\tau\rangle$ follows, $$\langle\tau\rangle = \int^\infty_0 \tau\cdot P_q(\tau)
d\tau=\int^\infty_0 \tau\cdot \frac{1}{\langle\tau\rangle} c
e^{-(a\tau/\langle\tau\rangle)^\gamma}d\tau.
\label{mean.eq}$$ Thus, using Eqs. (\[normal.eq\]) and (\[mean.eq\]), the parameters $a$ and $c$ can be expressed by $\gamma$, $$\begin{aligned}
a & = & \Gamma(2/\gamma)/\Gamma(1/\gamma), \nonumber\\
c & = & \gamma a/\Gamma(1/\gamma) = \gamma \Gamma(2/\gamma)/\Gamma(1/\gamma)^2.
\label{parameter.eq}\end{aligned}$$ Here $\Gamma(a)\equiv\int^\infty_0 t^{a-1}e^{-t}dt$ is the Gamma function. Thus, if the stretched exponential distribution is valid for the scaled interval $\tau/\langle\tau\rangle$, it is completely determined by $\gamma$. For $\gamma=1$, the record has no long-term correlations and the return interval distribution indeed follows an exponential distribution, represented by a Poissonian statistics, as expected.
Though the scaling in the return intervals distribution is a good approximation, we find slight deviations that as shown below are attributed to non-linear features. To explicitly explore the quality of the scaling in return interval distributions, we study all S&P 500 constituents and show the results of four representative stocks, Citigroup (C), General Electric (GE), Coca Cola (KO) and Exxon Mobil (XOM). All other stocks studied here usually show similar features. First, we examine the cumulative distribution of the scaled intervals. $$D(\tau/\langle\tau\rangle)\equiv\int^\infty_{\tau}
P_q(\tau)d\tau=\int^\infty_{\tau/\langle\tau\rangle} f(x)dx.$$
If the scaling function $f(\tau/\langle\tau\rangle)$ is valid, the cumulative distributions should also collapse to a single curve. Otherwise, the cumulative distributions, which integrate deviations, may show clearer deviations from scaling. Indeed, in Fig. \[Fig1\] we show cumulative distributions for three thresholds $q=2$, $4$ and $6$. Note that the volatility is normalized by its standard deviation, the threshold $q$ is in units of standard deviations and therefore $q=6$ is a quite large volatility. It is clearly seen that those distributions are close to each other but do not collapse to a single curve. More important, they show apparent deviations from the scaling, which are systematic with the threshold. For small scaled intervals ($\tau/\langle\tau\rangle<1$), the cumulative distribution decreases with $q$, while for large scaled intervals ($\tau/\langle\tau\rangle>1$), it increases with $q$ [@Note3]. In other words, the scaled interval prefers to be larger for higher threshold. This systematic trend suggests multiscaling in the return intervals, which might be related to the non-linear correlations in the volatility.
To better understand the systematic trends and test if it is not due to finite size effect or discreteness of minutes, we also measure the cumulative distribution of return intervals for surrogate records of volatilities using the Schreiber method [@Schreiber96; @Makse96; @Schreiber00] where non-linearities are removed. For a given time series, we store the power spectrum and randomly shuffle the sequence, then we apply the following iterations. Each iteration consists of two consecutive steps:
\(i) We perform the Fourier transform of the shuffled series, replace its power spectrum with the original one, then take the inverse Fourier transform to achieve a series. This step enforces the desired power spectrum to the series, while the distribution of volatilities usually is modified.
\(ii) By ranking, we exchange the values of the resulting series from step (i) with that of the original record. The largest value in the resulting series is replaced by the largest one in the original series, the second largest value is replaced by the second largest one, and so on. This step restores the original distribution but now the power spectrum is changed.
To achieve the convergence to the desired power spectrum and distribution, we repeat these two steps 30 times. By this way, a “surrogate” series is generated. Because of the Wiener-Kinchine theorem [@Kampen92], the surrogate record has the same linear correlations as the original, as well as the distribution. The only difference is that the original record has the non-linear correlations (if they exist) but the surrogate does not have any non-linear features.
In Fig. \[Fig1\] we also plot the cumulative distribution for the surrogate with the same three thresholds as the original. Since the surrogate records lost the non-linear correlations, they are similar to each other, we only show results for GE’s surrogate. It is seen that the collapse of the surrogate for different $q$ values is significantly better than that of the original and the deviation tendency with the threshold in the original records disappears. This indicates that the scaling deviations in the original are due to the non-linear correlations in the volatility. To further test this hypothesis, we analyze the moments $\mu_m\equiv\langle(\tau/\langle\tau\rangle)^m\rangle^{1/m}$ in Sec. (IV) and show similar and consistent deviations from scaling. We also compare our results to the stretched exponential distribution (dashed lines). This curve is very close to the empirical results, in particular for the surrogate records which contain only the linear correlations. This suggests that PDF of return intervals is well approximated by a stretched exponential.
The Moments of Scaled Intervals
===============================
The cumulative distribution shows clear systematic trend with $q$, which is difficult to see from the PDF directly [@Yamasaki05; @Wang06; @Wang07; @Altmann05; @Bunde04; @Bunde05]. To further analyze the systematic tendency in the distribution, we calculate the moments $\mu_m$ averaged over a stock dataset as a function of $\langle\tau\rangle$, where a mean interval $\langle\tau\rangle$ corresponds to a threshold $q$ and therefore characterizes a return interval series. We study moments for a wide range of $\langle\tau\rangle$, from $3$ minutes (to avoid the artificial effects due to discreteness close to $\tau=1$) to thousands minutes (few trading days or even a week). Assuming a single scaling function for the PDF $P_q(\tau)$, Eq. (\[pdf.eq\]), it follows $$\mu_m\equiv\langle(\tau/\langle\tau\rangle)^m\rangle^{1/m}=\{\int^\infty_0
(\tau/\langle\tau\rangle)^m\cdot \frac{1}{\langle\tau\rangle}
f(\tau/\langle\tau\rangle) d\tau\}^{1/m}=\{\int^\infty_0 x^m f(x)
dx\}^{1/m},
\label{moment.eq}$$ which only depends on $m$ and on the form of the scaling function $f(x)$ but independent of $\langle\tau\rangle$. Thus, if $\mu_m$ depends on $\langle\tau\rangle$, it suggests deviation from the assumption of scaling.
Moments vs. Mean Interval $\langle\tau\rangle$
----------------------------------------------
First we examine the relation between the moments $\mu_m$ and the mean interval $\langle\tau\rangle$. Fig. \[Fig2\] shows four representative moments $m=0.25$, $0.5$, $2$ and $4$ for stock C, GE, KO and XOM. Ignoring small fluctuations, which is usually due to limited size data, all moments $\mu_m$ for the original records deviate significantly from a horizontal line, which is expected for a perfect scaling of the PDF. They depend on $\langle\tau\rangle$ and show some systematic tendency. For $m>1$, moments have similar convex structure, first $\mu_m$ increases with $\langle\tau\rangle$ and then decreases, where the crossover starts earlier for larger $m$. For $m<1$, moments also show similar tendency but in the opposite direction compared to $m>1$. These deviations from scaling in $\mu_m$ are consistent with the deviations seen in the cumulative distributions shown in Fig. \[Fig1\]. Moments of large $m$ ($m>1$) represent large $\tau/\langle\tau\rangle$ in the PDF and they initially (for $\langle\tau\rangle\leq100$) increase with $\langle\tau\rangle$, while moments of small $m$ ($m<1$) represent small $\tau/\langle\tau\rangle$ and they initially (for $\langle\tau\rangle\leq100$) decrease with $\langle\tau\rangle$.
To further test if the systematic deviations are not due to finite size effects and discreteness, we also examine moments for the surrogate records which are more flat for most range, as shown in Fig. \[Fig2\]. For the same order $m$, the moment of the surrogate obviously differs from that of the original, especially in the medium range of $\langle\tau\rangle$ ($10<\langle\tau\rangle\leq100$). This discrepancy suggests that the non-linear correlations exist in the original volatility and accounts for the scaling deviations. Nevertheless, all moments of surrogate show small curvature from a perfect straight line at both short and long $\langle\tau\rangle$, which are much weaker compared to the original records. The weak curvature suggests that some additional effects, not related to the non-linear correlations, affect the moments. For small $\langle\tau\rangle$, the resolution discrete limit seems to have some influence on the moments. We will discuss this effect in section C. For large $\langle\tau\rangle$, the moments are gradually approaching the horizontal line and are more fluctuating, the effect seems to be related to limited size of the record. This effect will be discussed in section B.
Multiscaling
------------
For the original volatility records, the systematic tendency in the distribution of $\tau$ and the moments implies that the return intervals may have multiscaling features. To avoid as much as possible the effect of discreteness and finite size, we calculate the moments only for some medium range of $\langle\tau\rangle$ where the effects are small. Since there is no non-linear correlations in the surrogate records, the curvature in their moments is only due to the additional effects, we use the surrogate curve as our reference. For small $\langle\tau\rangle$, the increasing (decreasing) range for $m>1$ ($m<1$) almost ends at $\langle\tau\rangle=10$ minutes. For large $\langle\tau\rangle$, the curves start to decrease (increase) from different positions, but at $\langle\tau\rangle=100$, all curves do not or just start to decrease (increase). Thus we choose to study $\mu_m$ in the region, $10<\langle\tau\rangle\leq100$, represented by the shadow areas in Fig. \[Fig2\]. In this range, we find a clear trend for the original records while the surrogate is almost horizontal. To quantify the tendency, we fit the moments with a power-law, $$\mu_m \sim \langle\tau\rangle ^ \alpha.
\label{alpha.eq}$$ If the distribution of $\tau/\langle\tau\rangle$ follows a scaling law, the exponent $\alpha$ should be some value very close to $0$. If $\alpha$ is significantly different from $0$, it suggests multiscaling.
To examine the multiscaling behavior for the whole market, we calculate $\alpha$ for all 500 stocks of S&P 500 constituents and plot the histogram for $m=0.25$ to $2$. Fig. \[Fig3\] shows that each histogram has a narrow distribution, which suggests that $\alpha$ are similar for the 500 stocks. For the original records, almost all $\alpha$ significantly differ from $0$, thus the moments clearly depend on the mean interval. Moreover, the mean value of $\alpha$ shifts with order $m$ from $\langle\alpha\rangle\simeq-0.2$ for $m=0.25$ to $\langle\alpha\rangle\simeq0.1$ for $m=2$ which means the dependence varies with the order $m$. This behavior suggests multiscaling in the return intervals distribution. Indeed, histograms for the surrogate records are more centered around values close to $\alpha=0$. The uniscaling behavior for the surrogate suggests that the non-linear correlations in the volatility are responsible for the multiscaling behavior in the original.
To remove fluctuations and show the tendency clearer, we plot the dependence of $\langle\alpha\rangle$ on $m$, where $\langle\alpha\rangle$ is the average $\alpha$ over all 500 stocks. In Fig. \[Fig4\] we show this relation for a wide range of $m$, $0.1\leq m\leq 10$, and the plot shows two different behaviors. For small $m$ (roughly $m\leq2$), $\langle\alpha\rangle$ for the original records clearly deviates from 0 and demonstrates the multiscaling behavior, while $\langle\alpha\rangle$ for the surrogate is closer to $0$. For large $m$ ($m>2$), the two curves have similar decreasing trend. Since large $\tau$ dominates high order moments, this similarity may be due to finite size effects. To test the finite size effects we simulate surrogate return intervals by assuming a stretched exponential distribution i.i.d. process with 3 sizes (number of all $\tau$ in the series), $2\times 10^6$, $2\times 10^5$ (the size of the empirical dataset) and $2\times 10^4$. Without loss of generality, we choose $\gamma=0.3$, which is the correlation exponent for GE of $q=2$. To be consistent with the 500 stocks, we perform 500 realizations and plot their average exponent $\langle\alpha\rangle$. As shown in the inset of Fig. \[Fig4\], all $\langle\alpha\rangle$ show a similar decreasing trend as that of the empirical curves. However, it is seen that the trend starts earlier for smaller size, and thus the size limit has a strong influence on high order moments. Fig. \[Fig4\] also shows the error bars for the two records, which is the standard deviation of 500 $\alpha$ values. Note that the error bars for the volatility records do not overlap those of their corresponding surrogate, indicating the significance of our results.
Discreteness Effect
-------------------
For small $\langle\tau\rangle$ ($\langle\tau\rangle\le10$), the behavior of $\mu_m$ as a function of $\langle\tau\rangle$ was attributed to the discreteness. Here we examine this effect. Due to the limits in recording, we can not have a continuous but discrete record. In our study the volatility is recorded in 1 minute. The relative errors in moments will be considerable large for small $\langle\tau\rangle$ close to $1$ minute. By starting from $\langle\tau\rangle=3$, we only partially avoid the discreteness in the moments. To test the discreteness effects, since we can not increase the resolution, we reduce it and compare the moments $\mu_m$ with 3 resolutions, 1 minute, 5 minutes and 10 minutes. Fig. \[Fig5\](a) shows this comparison for GE for $m=0.5$ and $m=2$. The three resolutions have a similar trend, showing that the curves become flatter for the higher resolution. For other stocks, we find similar behavior. This systematic tendency suggests that the recording limit (1 minute) strongly affects the moments at short $\langle\tau\rangle$. To reduce it, we should raise the recording precision or study the moments of larger $\langle\tau\rangle$.
To further test this result, we simulate artificial return intervals with an i.i.d. process from stretched exponential distribution with $\gamma=0.3$, same as the empirical $\gamma$ for GE of $q=2$. We examine moments of $m=0.5$ and $m=2$ with the same three resolutions (1, 5 and 10 time units) as in the empirical test done above. The simulated size is 200 thousands points for each trial and we use the average over 100 trials for each resolution. Fig. \[Fig5\](b) shows curves similar to that of empirical (Fig. \[Fig5\](a)). For the higher resolution, the curve is closer to the horizontal line and finally may reach the line when we raise the resolution high enough. To show this, we also simulate continuous return intervals and find constant moments, as expected (Fig. \[Fig5\](b)). Therefore, the discreteness effects can be overcome if the resolution is improved enough. Note that for the empirical data, we expect the moment not to be constant for small $\langle\tau\rangle$ even if we have a much better resolution, since the return intervals has the multiscaling behavior, as shown for larger $\langle\tau\rangle$ in the range $10<\langle\tau\rangle\leq100$ which is not affected by discreteness.
Moments vs. Order $m$
---------------------
The moments $\mu_m$ have systematic dependence on $m$, as seen in Figs. \[Fig2\] and \[Fig3\] where the moments are plotted as the function of $\langle\tau\rangle$. It is of interest to explore the relation between the moments and $m$ directly. For a fixed $\langle\tau\rangle$, representing a given threshold $q$ one can study, the return intervals and their moments of various orders which exhibit information on different scales of $\tau$. Moments of large $m$ represent large $\tau$ and vice versa. If $\tau/\langle\tau\rangle$ follows a single distribution without corrections due to effects such as discreteness and finite size, curves of $\mu_m$ vs. $m$ for different $\langle\tau\rangle$ should collapse to a single one, which only depends on the scaling function $f(x)$ from Eq. (\[moment.eq\]). In Fig. \[Fig6\] we plot $\mu_m$ vs. $m$ for both the original and surrogate records. We plot $\mu_m$ for $m$ between $0.1$ and $10$ for three $\langle\tau\rangle$ values: 10, 80 and 400 minutes. For the original (Fig. \[Fig6\](a)), there is substantial deviations from a single curve. This supports our suggestion that the return intervals has multiscaling behavior. Moments for the surrogate (Fig. \[Fig6\](b)) converge to a single curve for $m\le2$ but become diverse for high orders, which agrees with the strong influence of the finite size effects. As a reference, we also plot the analytical moments (Fig. \[Fig6\](c)) from the stretched exponential distribution. Substituting Eq. (\[scaling.eq\]) into Eq. (\[moment.eq\]), we obtain $$\mu_m=\frac{1}{a}\{\frac{\Gamma((m+1)/\gamma)}{\Gamma(1/\gamma)}\}^{1/m}.
\label{moment1.eq}$$ Fig. \[Fig6\](c) shows analytical curves for various correlation exponent $\gamma$.
Discussion
==========
We study the scaling properties of the distribution of the volatility return intervals for all S&P 500 constituents. We find small but systematic deviations from scaling assumption with the threshold $q$ in the cumulative distribution. Compared to the good collapse for the surrogate records where non-linearities are removed, this suggests that the origin of this trend is due to non-linear correlations in the original volatility. Moreover, we find similar systematic deviations for the moments $\mu_m$, which are also attributed to the non-linear correlations in the volatility. We distinguish these deviations from the deviations due to the discreteness for small $\langle\tau\rangle$ and finite size effect for large $\langle\tau\rangle$. Further, we explore the dependence of the moment $\mu_m$ on its order $m$. When compare to surrogate records and to analytical curves, the results support the multiscaling hypothesis of the return intervals. Thus, the scaling assumption in the return interval distributions although it is a good approximation can not be exact. Also, the stretched exponential form of the scaling function can only be an approximation. Recently Eisler et al. [@Eisler06A; @Eisler06B] exhibits that the distribution of intertrade times has similar multiscaling behavior and the market activity depends on the company capitalization. It would be interesting to connect the intertrade times with the return intervals and test size dependence in the return intervals.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Y. Ashkenazy for his kind help in the simulations, R. Mantegna, J. Kertész and Z. Eisler for fruitful discussions, and the NSF and Merck Foundation for financial support.
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|
---
abstract: 'External pilot trials of complex interventions are used to help determine if and how a confirmatory trial should be undertaken, providing estimates of parameters such as recruitment, retention and adherence rates. The decision to progress to the confirmatory trial is typically made by comparing these estimates to pre-specified thresholds known as progression criteria, although the statistical properties of such decision rules are rarely assessed. Such assessment is complicated by several methodological challenges, including the simultaneous evaluation of multiple endpoints, complex multi-level models, small sample sizes, and uncertainty in nuisance parameters. In response to these challenges, we describe a Bayesian approach to the design and analysis of external pilot trials. We show how progression decisions can be made by minimising the expected value of a loss function, defined over the whole parameter space to allow for preferences and trade-offs between multiple parameters to be articulated and used in the decision making process. The assessment of preferences is kept feasible by using a piecewise constant parametrisation of the loss function, the parameters of which are chosen at the design stage to lead to desirable operating characteristics. We describe a flexible, yet computationally intensive, nested Monte Carlo algorithm for estimating operating characteristics. The method is used to revisit the design of an external pilot trial of a complex intervention designed to increase the physical activity of care home residents.'
author:
- 'Duncan T. Wilson^1^'
- 'James M. S. Wason^2,3^'
- Julia Brown^1^
- 'Amanda J. Farrin^1^'
- 'Rebecca E. A. Walwyn^1^'
bibliography:
- 'C:/Users/meddwilb/Documents/Literature/Databases/DTWrefs.bib'
date: |
1 - Leeds Institute of Clinical Trials Research, University of Leeds, Leeds, UK\
2 - Institute of Health and Society, Newcastle University, Newcastle, UK\
3 - MRC Biostatistics Unit, University of Cambridge, Cambridge, UK
title: Bayesian design and analysis of external pilot trials for complex interventions
---
Introduction {#sec:introduction}
============
Complex interventions, defined as those comprised of several interacting components [@Craig2008], can be challenging to evaluate in randomised controlled trials (RCTs) due to factors such as slow patient recruitment, poor levels of adherence to the intervention, and low completeness of follow-up data. To identify these problems prior to the main RCT we often conduct small trials [@Craig2008] known as pilots. These typically take the same form as the planned RCT but with a considerably lower sample size [@Eldridge2016]. If there is a seamless transition between the pilot and the main RCT, with all data being pooled and used in the final analysis, they are known as internal pilots. External pilots, in contrast, are carried out separately to the main RCT with a clear gap between the two trials. Pilot trials, which aim to inform the feasibility and optimal design of a subsequent definitive trial [@Lancaster2004], are distinct from phase II trials, which focus instead on assessing potential efficacy and safety.
The data generated by an external pilot trial is used to help decide if the main RCT should go ahead, and if so, whether the intervention or the trial design should be adjusted to ensure success. In the UK, the National Institute for Health Research ask that these *progression criteria* are pre-specified and included in the research plan [@NIHR2017], and the recent CONSORT extension to randomised pilot trials requires their reporting [@Eldridge2016a]. A single pilot trial can collect data on several progression criteria, often focused on the aforementioned areas of recruitment, protocol adherence, and data collection [@Avery2017]. Although they may take the form of single threshold values leading to binary stop/go decision rules, investigators are increasingly using two thresholds to accommodate an intermediate decision between stopping altogether and progressing straight to the main trial, which would allow progression but only after some adjustments have been made [@Eldridge2016a]. The need for appropriate progression criteria is clear when we consider the consequences of poor post-pilot progression decisions. If the criteria are too lax, there is a greater risk that the main trial will go ahead but found to be infeasible and thus a waste of resources; if the criteria are too strict, a promising intervention may be discarded under the mistaken belief that the main trial would be infeasible. Despite this, there is little published guidance about how they should be determined[@Avery2017; @Hampson2017].
In addition to pre-specifying progression criteria, another key design decision is the choice of pilot sample size. Conventional methods of sample size determination, which focus on ensuring the trial will have sufficient power to detect a target difference in the primary outcome, are rarely used since they would lead to a pilot sample size comparable with the main trial sample size. Several methods for pilot sample size determination instead aim to provide a sufficiently precise estimate of the variance in the primary outcome measure to inform the sample size of the main trial [@Browne1995; @Julious2005; @Sim2012; @Teare2014; @Eldridge2015; @Whitehead2015]. Others have suggested a simple rule of thumb for when the goal is to identify unforeseen problems [@Viechtbauer2015]. While some have noted that the low sample size in pilots may lead to a considerable probability that a certain progression criterion will be met (or missed) due to random sampling variation [@Eldridge2015; @Cooper2018], and despite the consequences of making the wrong progression decision, the statistical properties of pilot decision rules are rarely used to inform the choice of sample size. This may be due to the methodological challenges commonly found in pilot trials of complex interventions, including the simultaneous evaluation of multiple endpoints, complex multi-level models, small sample sizes, and prior uncertainty in nuisance parameters [@Wilson2015].
In this paper we will describe a method for designing and analysing external pilot trials which addresses these challenges. We take a Bayesian view, where progression decisions are made to minimise the expected value of a loss function. We propose a loss function with three parameters whose values can be determined either through direct elicitation of preferences or by considering the pilot trial operating characteristics they lead to. The operating characteristics we propose are all unconditional probabilities (with respect to a prior distribution) of making incorrect decisions, also known as assurances [@OHagan2005]. Using assurances rather than the analogous frequentist error rates brings several benefits, including the ability to make use of existing knowledge whilst allowing for any uncertainty, and a more natural interpretation [@Crisp2018]. As we will show, assurances are also useful when our preferences for different end-of-trial decisions are based on several attributes in a complex way that involves trading off some against others.
The remainder of this paper is organised as follows. In Section \[sec:methods\] we describe the general framework for pilot design and analysis, some operating characteristics used for evaluation, and a routine for optimising the design. Two illustrative examples are then described in Sections \[sec:TIGA\] and \[sec:REACH\]. Finally, we discuss implications and limitations in Section \[sec:discussion\].
Methods {#sec:methods}
=======
Prior specification
-------------------
Consider a pilot trial which will produce data $x$ according to model $p(x | \theta)$. We decompose the parameters into $\theta = (\phi, \psi)$, where $\phi$ denotes the parameters of substantive interest and $\psi$ the nuisance parameters. We follow [@Wang2002] and assume that two joint prior distributions of $\theta$ have been specified. First, the *analysis* prior $p_A(\theta)$ is that which will be used when fitting the model once the pilot data is obtained. It has been argued that regulators are unlikely to accept the prior beliefs of the trial sponsor for analysis of the data [@OHagan2005; @Walley2015], and as such a weakly or non-informative prior should be used for $p_A(\theta)$ in order to “let the data drive the inference” [@Wang2002]. The choice of such a prior will be depend on the specific model being used, although methodological guidance for various specific cases such as logistic regression [@Gelman2008] and hierarchical models [@Spiegelhalter2001] is available. It should be emphasised, however, that the typically small sample size of a pilot trial can mean the effect of the analysis prior is non-negligible. As such, the analysis prior should provide a credible and justifiable representation of prior ignorance, avoiding extreme default choices which may place too much prior weight on infeasible regions of the parameter space.
The *design* prior $p_D(\theta)$ will be used when evaluating the statistical performance of a proposed pilot trial design. It may be considered as purely hypothetical in the spirit of a ‘what-if’ analysis [@Wang2002], in which case several candidate design priors may be suggested and performance evaluated under each of these. Alternatively, and as we will assume in the remainder of this paper, $p_D(\theta)$ can be a completely subjective prior which fully expresses our knowledge and uncertainty in the parameters at the design stage. Although eliciting such a prior is potentially challenging, many examples describing successful practical applications of expert elicitation for clinical trial design are available [@Walley2015; @Crisp2018; @Dallow2018], as are tools for its conduct such as the Sheffield Elicitation Framework (SHELF) [@OHagan2006a].
Analysis and progression decisions {#sec:analysis}
----------------------------------
After observing the pilot data $x$, we must decide whether or not to progress to the main RCT. We consider three possible actions following the aforementioned ‘traffic light’ system commonly used in pilot trials:
- red - discard the intervention and stop all future development or evaluation;
- amber - proceed to the main RCT, but only after some modifications to the intervention, the planned trial design, or both; or
- green - proceed immediately to the main RCT.
In what follows we will denote these decisions by $r, a$ and $g$ respectively. We assume that our preferences between the three possible decisions are influenced by $\phi$ but independent of $\psi$, formalising the separation of $\theta$ into substantive and nuisance components. We partition the substantive parameter space $\Phi$ into three subspaces $\Phi_I$, for $I=R,A,G$. Each subspace label corresponds to the decision we would make if we knew the true value of $\phi$. For example, if $\phi \in \Phi_R$ then the optimal decision is $r$(ed) - halt development and do not proceed to a definitive trial. We will henceforth refer to these three subsets as *hypotheses*. Throughout, we will distinguish hypothesis $I$ from the corresponding optimal decision $i$ by using upper and lower case letters respectively.
When $\phi \in \Phi_{I}$ and we choose a decision $j \neq i$, there will be negative consequences. In particular, we may make three kinds of mistakes: proceed to an infeasible main RCT; discard a promising intervention; or make unnecessary adjustments to the intervention or trial design. We denote these errors as $E_1$, $E_2$, $E_3$ respectively. The occurrence of error $j$ will be denoted by $E_j = 1$, otherwise $E_j = 0$. An error’s occurrence will be a function of the decision made $d$ and the true parameter value $\phi$, i.e. $E_j(d, \phi): \{r, a, g\} \times \Phi \rightarrow \{0,1\}$ for $j = 1,2,3$. We then use a loss function to express the preferences of the decision maker(s) on the space of possible events $E_1 \times E_2 \times E_3$ under uncertainty, defined as $$L(d, \phi) = c_1 E_1(d, \phi) + c_2 E_2(d, \phi) + c_3 E_3(d, \phi).$$ Note that the additive form of the loss function implies that the our preferences for any one of the attributes $E_1, E_2, E_3$ are independent of the values taken by the others [@French2000].
To determine appropriate values of the parameters $c_1, c_2, c_3$ we first scale the loss function by setting $c_1 + c_2 + c_3 = 1$. Thus, a loss of 0 is obtained if no errors occur, and a loss of 1 is obtained if all errors occur (although note that this is not possible in this setting). We then follow the procedure described by French and Rios Insua (page 99) [@French2000], eliciting some judgements from the decision maker(s) and using these to determine the values of $c_1, c_2, c_3$. One such judgement involves a simple gamble of obtaining the event $(E_1 = 0, E_2 = 0, E_3 = 0)$ with probability $1 - p_1$ and the event $(E_1 = 1, E_2 = 0, E_3 = 1)$ with probability $p_1$. The decision maker is asked to compare this gamble against an alternative of obtaining the event $(E_1 = 1, E_2 = 0, E_3 = 0)$ for certain, and to adjust the value of $p_1$ until they feel indifferent between the two options. Since this indifference implies the expected losses of the two options are equal, we will then have $$p_1 (c_1 + c_3) = c_1.$$ Similarly, we can ask the decision maker(s) to consider a gamble between the event $(E_1 = 0, E_2 = 0, E_3 = 0)$ with probability $1 - p_2$ and the event $(E_1 = 1, E_2 = 1, E_3 = 0)$ with probability $p_2$, and compare this against the option of obtaining $(E_1 = 1, E_2 = 0, E_3 = 0)$ for certain. Again, by determining the value of $p_2$ which corresponds to indifference and thus equal expected loss, we deduce that $$p_2(c_1 + c_2) = c_1.$$ This gives three equations which can be solved to obtain $$c_1 = \frac{-p_1 p_2}{p_1 p_2 - p_1 - p_2}, ~ c_2 = \frac{p_1 p_2 - p_1}{p_1 p_2 - p_1 - p_2} ~ c_3 = \frac{p_1 p_2 - p_2}{p_1 p_2 - p_1 - p_2}.$$ Note that the two specific judgements suggested here are only two of many possible similar questions which could be posed to the decision maker(s). It is recommended that more indifferences are elicited in order to seek out any inconsistencies and further clarify their true preferences.
The loss function will then take values as given in Table \[tab:loss\]. For example, suppose we make a ‘green’ decision under the ‘amber’ hypothesis. The subsequent trial will be infeasible because the necessary adjustments will not have been made; but we have also discarded a promising intervention, since it would have been redeemed had the adjustments been made. The overall loss is therefore $c_{1} + c_{2}$.
-- ----- --------------------- --------------------- ---------------------
$\phi \in \Phi_{R}$ $\phi \in \Phi_{A}$ $\phi \in \Phi_{G}$
$r$ 0 $c_{2}$ $c_{2}$
$a$ $c_{1} + c_{3}$ 0 $c_{3}$
$g$ $c_{1}$ $c_{1} + c_{2}$ 0
-- ----- --------------------- --------------------- ---------------------
: Losses associated with each decision under each hypothesis.
\[tab:loss\]
Given a loss function with parameters $\mathbf{c} = (c_1, c_2, c_3)$ we follow the principle of maximising expected utility (or in our case, minimising the expected loss) when making a progression decision. We first use the pilot data in conjugation with the analysis prior $p_{A}(\theta)$ to obtain a posterior $p(\phi ~|~ x)$, and then choose the decision $i^{*}$ such that $$\begin{aligned}
i^{*} & = \operatorname*{arg\,min}_{i \in \{r,a,g\}} \mathbb{E}_{\phi | x} [ L(i, \phi) ] \\
& = \operatorname*{arg\,min}_{i \in \{r,a,g\}} \int L(i, \phi) p(\phi | x) d\phi.\end{aligned}$$ We can simplify this expression by noting that, given the piecewise constant nature of the loss function, the expected loss of each decision depends only on the posterior probabilities $p_{I} = Pr[\phi \in \Phi_{I} ~|~ x]$ for $I = R, A, G$. We then have $$\begin{aligned}
\label{eqn:exp_loss}
\mathbb{E}_{\phi | x} [ L(r, \phi) ] & = p_{A}c_{3} + p_{G}c_{3}, \\
\mathbb{E}_{\phi | x} [ L(a, \phi) ] & = p_{R}c_{1} + p_{R}c_{2} + p_{G}c_{2}, \\
\mathbb{E}_{\phi | x} [ L(g, \phi) ] & = p_{R}c_{1} + p_{A}c_{1} + p_{A}c_{3}.\end{aligned}$$ For some simple models which admit a conjugate analysis, the posterior probabilities $p_I$ can be obtained exactly. Otherwise, Monte Carlo estimates can be computed based on the samples from the joint posterior distribution generated by an MCMC analysis of the pilot data. Specifically, given $M$ samples $\phi^{(1)}, \phi^{(2)}, \ldots , \phi^{(M)} \sim p(\phi ~|~ x)$, $$p_I \approx \frac{1}{M} \sum_{k = 1}^{M} \mathbb{I}(\phi^{(k)} \in \Phi_I),$$ where $\mathbb{I}(.)$ is the indicator function.
Operating characteristics {#sec:evaluation}
-------------------------
Defining a loss function and following the steps of the preceding section effectively prescribes a decision rule mapping the pilot data sample space $\mathcal{X}$ to the decision space $\{r, a, g\}$. To gain some insight at the design stage into the properties of this rule, we propose to calculate some trial operating characteristics. These take the form of unconditional probabilities of making an error when following the rule, calculated with respect to the design prior $p_D(\theta)$. We consider the following:
- $OC_1 = Pr[a ~\&~ \phi \in \Phi_R] + Pr[g ~\&~ \phi \in \Phi_R \cup \Phi_A]$ - probability of proceeding to an infeasible main RCT;
- $OC_2 = Pr[r ~\&~ \phi \in \Phi_A \cup \Phi_G] + Pr[g ~\&~ \phi \in \Phi_A]$ - probability of discarding a promising intervention;
- $OC_3 = Pr[a ~\&~ \phi \in \Phi_R \cup \Phi_G]$ - probability of making unnecessary adjustments to the intervention or the trial design.
These operating characteristics can be estimated using simulation. First, we draw $N$ samples $(\theta^{(1)}, x^{(1)}), (\theta^{(2)}, x^{(2)}), \ldots , (\theta^{(N)}, x^{(N)})$ from the joint distribution $p(\theta, x) = p(x | \theta)p_D(\theta)$. For each data set we then apply the analysis and decision making procedure described in Section \[sec:analysis\], using some vector $\mathbf{c}$ to parametrise the loss function. This results in $N$ decisions $i^{(k)}$ which can be contrasted with the corresponding true parameter value $\theta^{(k)}$ and in which hypothesis it resides, noting if any of the three types of errors have been made. MC estimates of the operating characteristics can then be calculated as the proportion of occurrences of each type of error in the $N$ simulated cases. Assuming that $N$ is large, the unbiased MC estimate of an operating characteristic with true probability $p$ will be approximately normally distributed with variance $p(1-p)/N$.[^1]
Optimisation {#sec:optimisation}
------------
Elicitation of the loss function parameters $\mathbf{c} = (c_1, c_2, c_3)$ in the manner described in Section \[sec:analysis\] may be challenging, particularly when multiple decision-makers are involved [@Keeney1976]. An alternative way to determine $\mathbf{c}$ is through examining the operating characteristics it leads to (for some fixed pilot design). As $\mathbf{c}$ is adjusted, the balance between the conflicting objectives of minimising each OC will change, and the task is then to find the $\mathbf{c}$ which returns the best balance from the perspective of the decision-maker. Formally, and thinking of operating characteristics as functions of $\mathbf{c}$, we wish to solve the multi-objective optimisation problem $$\label{eqn:opt}
\min_{\mathbf{c} \in \mathcal{C}} ~ \left( OC_{1}(\mathbf{c}),~ OC_{2}(\mathbf{c}),~ OC_{3}(\mathbf{c}) \right)$$ where $\mathcal{C} = \{c_{1}, c_{2} \in [0,1] ~|~ c_{1} + c_{2} \leq 1\}$.
Since the three objectives are in conflict there will be no single solution which simultaneously minimises each one. We would instead like to find a set $\mathcal{C}^* = \{ \mathbf{c}^{(1)}, \mathbf{c}^{(2)}, \ldots, \mathbf{c}^{(K)} \}$ such that each member provides a different balance between minimising the three operating characteristics. If there exist $\mathbf{c}, \mathbf{c}' \in \mathcal{C}^*$ such that $OC_i(\mathbf{c}') \leq OC_i(\mathbf{c})$ for all $i \in \{1, 2, 3\}$ and $OC_i(\mathbf{c}') < OC_i(\mathbf{c})$ for some $i \in \{1, 2, 3\}$, we say that $\mathbf{c}'$ dominates $\mathbf{c}$. In this case, because $\mathbf{c}$ leads to worse (or at least no better) values of all three operating characteristics when compared to $\mathbf{c}'$, we have no reason to include it in our set $\mathcal{C}^*$. Because the search space $\mathcal{C}$ has only two dimensions, problem (\[eqn:opt\]) can be approximately solved by generating a uniform random sample of $\mathbf{c}$’s and estimating the operating characteristics for each. Any parameters which are dominated in this set can then be discarded, and the operating characteristics of those which remain can be illustrated graphically. The decision maker(s) can then view the range of available options, all providing different trade-offs amongst the three operating characteristics, and choose from amongst them.
To solve the problem in a timely manner we must be able to estimate operating characteristics quickly. Noting from equation (\[eqn:exp\_loss\]) that the expected loss of each decision depends only on $\mathbf{c}$ and the posterior probabilities $p_R, p_A$ and $p_G$, we first generate $N$ samples of these posterior probabilities and then use this same set of samples for every evaluation. This approach not only ensures that optimisation is computationally feasible, but also means that differences in operating characteristics are entirely due to differences in costs, as opposed to differences in the random posterior probability samples.
Illustrative example - Child psychotherapy (TIGA-CUB) {#sec:TIGA}
=====================================================
TIGA-CUB (Trial on Improving Inter-Generational Attachment for Children Undergoing Behaviour problems) was a two-arm, individually-randomised, controlled pilot trial informing the feasibility and design of a confirmatory RCT comparing Child Psychotherapy (CP) to Treatment as Usual (TaU), for children with treatment resistant conduct disorders. The trial aimed to recruit $60$ primary carer-child dyads, to be randomised equally to each arm. This sample size was chosen to give desired levels of precision in the estimates of the common standard deviation of the primary outcome, the follow-up rate, and the adherence rate. Here, we focus on the latter two parameters and consider how our proposed method could have informed the design of TIGA-CUB.
We model the number of participants successfully followed-up (denoted $f$) using a binomial distribution with parameter $p_f$, and similarly the number successfully adhering to the intervention (denoted $a$) with a binomial distribution with parameter $p_a$. For a fixed pilot trial per-arm sample size $n$, the parameters of the model are $\phi = (p_f, p_a)$, with no nuisance parameters. Further assuming that the numbers followed-up and adhering are independent, the likelihood is then $$p(f, a | p_f, p_a) = \left[{2n \choose f}p_f^{f}(1-p_f)^{2n-f}\right] \times \left[{n \choose a}p_a^{a}(1-p_a)^{n-a}\right].$$
At the design stage, the follow-up rate $p_f$ was thought to be somewhere in the range 62% to 92%, while the adherence rate $p_a$ was thought to lie between 40% and 95%. We reflect these ranges of uncertainty in our design priors by using beta distributions $p_f \sim Beta(40, 10)$ (thus giving a prior mean of 0.8), and $p_a \sim Beta(11.2, 4.8)$ (giving a prior mean of 0.7). We assume that a uniform ‘non-informative’ prior $Beta(1,1)$ will be used for each parameter in the analysis.
TIGA-CUB’s progression criteria included only simple stop/go thresholds, with no intermediate ‘amber’ decisions. As such, in this example we partition the parameter space into two hypotheses, $\Phi_G$ and $\Phi_R$. For the purposes of illustration we define the hypothesis $\Phi_G$ as the subset of the parameter space where $p_f >= 0.8$ and $p_a >= 0.7$, hypothesis $\Phi_R$ being its complement. Thus, in this example we do not consider there to be a trade-off between the two parameters of interest. For the main trial to be feasible, both must be above their respective thresholds. The prior distributions on parameters $p_f$ and $p_a$ imply an *a priori* probability of 0.28 that $\phi \in \Phi_G$, i.e. that both follow-up and adherence are sufficiently high.
In this special case, the loss function is $$L(d, \phi) = c_1 E_1(d, \phi) + c_2 E_2(d, \phi)$$ and the expected losses of decisions $g$ and $r$ will be $\mathbb{E}_{\phi | x}[L(g, \phi)] = c_1 p_R$ and $\mathbb{E}_{\phi | x}[L(r, \phi)] = c_2 p_G $, where $p_R + p_G = 1$ and $c_1 + c_2 = 1$. Decision $g$ is therefore optimal whenever $p_G > c_1$. The posterior probability $p_G$ can be easily calculated given the pilot data due to the beta prior distributions being conjugate. Specifically, given a total sample size $n$ and observing $x_f$ participants with follow-up and $x_a$ participants with adherence, the posterior probability $Pr[\phi \in \Phi_G ~|~ x]$ is given by $$p_G = [1 - F(0.8; 1+x_f, 1+n-x_f)] \times [1 - F(0.7; 1+x_a, 1 + n/2 - x_a)],$$ where $F(y; \alpha, \beta)$ denotes the cumulative probability function of the beta distribution with parameters $\alpha, \beta$.
At the design stage we can calculate the probability of an infeasible trial ($OC_1$), $$\begin{aligned}
Pr[g, \phi \in \Phi_R] &= \int_{\Phi_R} Pr[g ~|~ \phi] p(\phi) d\phi \\
&= \int_{\Phi_R} \left( \sum_{x_f = 0}^{n} \left[ \sum_{x_a = 0}^{n/2} \mathbb{I}(p_G < c_1 ~|~ x_f, x_a, n) p(x_a ~|~ \phi) \right]p(x_f ~|~ \phi) \right)p(\phi) d\phi,\end{aligned}$$ and similarly for the probability of discarding a promising intervention. As these calculations can be computationally expensive for moderate $n$ due to the nested summation term, we use Monte Carlo approximations as described in Section \[sec:methods\].
Keeping the sample size fixed at $n = 30$ per arm, we estimated the operating characteristics using a range of cost parameters values $c_1 = 0, 0.02, 0.04, \ldots , 1$ using $M = 10^6$ Monte Carlo samples. The results are plotted in Figure \[fig:tiga\_n60\], with some specific values of $c_1$ highlighted. The decision-maker can decide which point on the operating characteristic curve best reflects their own priorities in terms of the two types of error. For example, if the consequences of running an infeasible main RCT are considered less important than those of needlessly discarding a potentially effective intervention, the decision-maker may choose to set $c_1 = 0.2$ and would obtain $OC_1 = 0.19, OC_2 = 0.05$.
![Probabilities of an infeasible main trial ($OC_1$) and of discarding a promising intervention ($OC_2$) for a range of loss parameters $c_1$ when sample size is fixed at $n=30$.[]{data-label="fig:tiga_n60"}](tiga_n30)
To examine the effect of adjusting the sample size, we evaluated the operating characteristics obtained for $n = 10, 12, 14, \ldots , 50$ per arm whilst setting $c_1 = 0.2, 0.36, 0.5$. The results are shown in Figure \[fig:tiga\_ocs\]. Each line includes a shaded area denoting the 95% Monte Carlo error intervals, although these are so small as to be illegible given the high number ($M = 10^6$) of MC samples used for each calculation. Although operating characteristics generally improve as the sample size is increased, we see that for $c_1 = 0.36$ and 0.5 the probability of an infeasible main trial, $OC_1$, remains flat whilst $OC_2$ has a downward trend. As we would expect, the the expected loss reduces smoothly as $n$ increases in all cases. In contrast, there is some variability beyond that explained by MC error in the OCs. This can be explained by the discrete nature of simulated adherence and follow-up data. Our results show that, for the design priors and hypotheses used in this example, the chosen sample size in TIGA-CUB of $n=30$ can provide error rates broadly in line with conventional type I and II error rates under the usual hypothesis testing framework.
Illustrative example - Physical activity in care homes (REACH) {#sec:REACH}
==============================================================
The REACH (Research Exploring Physical Activity in Care Homes) trial aimed to inform the feasibility and design of a future definitive RCT assessing a complex intervention designed to increase the physical activity of care home residents [@Forster2017]. The trial was cluster randomised at the care home level, with twelve care homes in total randomised equally between treatment as usual (TaU) and the intervention plus TaU.
Data on several feasibility outcomes were collected. Here, we focus on four: recruitment (measured in terms of the average number of residents in each care home who participate in the trial, or average cluster size); adherence (a binary indicator at the care home level indicating if the intervention was fully implemented); data completion (a binary indicator for each resident of successful follow-up at the planned primary outcome time of 12 months); and potential efficacy (a continuous measure of physical activity at the resident level). Progression criteria using the traffic light system were pre-specified for all of these outcomes except potential efficacy, as detailed in Table \[tab:pcs\]
Outcome Red Amber Green
---------------------------------- --------------- -------------------- --------------
Recruitment (avg. per care home) Less than 8 Between 8 and 10 At least 10
Adherence Less than 50% Between 50 and 75% At least 75%
Follow-up Less than 65% Between 65 and 75% At least 75%
: Pre-specified progression criteria used in the original REACH design.
\[tab:pcs\]
Denoting the size of the $j$th cluster by $m_j$, we assume that cluster sizes are normally distributed, $m_j \sim N(\mu_c, \sigma^2), j = 1, \ldots , 2k$. We further assume that the probability of a participant being followed-up is constant across clusters and arms, and that the total number follows a binomial distribution $f \sim Bin(\sum_{j=1}^{2k} m_j, p_f)$. The number of care homes which successfully adhere to the intervention is assumed to binomially distributed, $a \sim Bin(k, p_a)$.
The continuous measure of physical activity is expected to be correlated within care homes. We model this using a random intercept, where the outcome $y_{ij}$ of resident $i$ in care home $j$ is $$y_{ij} = X_{j} \times Y_{j} \times \mu + u_{j} + \varepsilon_{i}.$$ Here, $X_{j}$ is a binary indicator of care home $j$ being randomised to the intervention arm, $Y_{j}$ is a binary indicator of care home $j$ successfully adhering to the intervention, $\mu$ is the mean treatment effect, $u_{j} \sim \mathcal{N}(0, \sigma_{B}^{2})$ is the random effect for care home $j$, and $\varepsilon_{i} \sim \mathcal{N}(0, \sigma_{W}^{2})$ is the residual for resident $i$. We parametrise the model using the intracluster correlation coefficient, $\rho = \sigma_{B}^{2} / (\sigma_{B}^{2} + \sigma_{W}^{2})$.
The parameters describing average cluster size, follow-up and adherence rates, and mean treatment effect are of substantive interest when making progression decisions, giving $\phi = (\mu_c, p_f, p_a, \mu)$. The remainder are nuisance parameters, $\psi = (\sigma^2, \rho, \sigma_W^2)$.
Model specification
-------------------
To begin specifying a model for the REACH trial, we first note that the four substantive parameters can be divided into two pairs. Firstly, mean cluster size and follow-up rate relate to the amount of information which a confirmatory trial will gather. Secondly, potential efficacy and adherence relate to the effectiveness of the intervention, where effectiveness is thought of as the effect which will be obtained in practice when the effect of non-adherence is accounted for. We expect that a degree of trade-off between adherence and potential efficacy will be acceptable, with a decrease in one being compensated by an increase in the other. Likewise, low mean cluster size could be compensated to some extent by higher follow-up rate, and vice versa.
While there may be trade-offs within these pairs of parameters, we do not expect trade-offs between them. A trial with no effectiveness will be futile regardless of the amount of information collected, and so should not be conducted. Similarly, a confirmatory trial should not be conducted if it is highly unlikely to produce enough information for the research question to be adequately answered. We therefore consider the sub-spaces of $\Phi$ formed by these parameter pairs, partition these into hypotheses, and combine these together. Constructing hypotheses in these two-dimensional spaces is cognitively simpler than working in the original four dimensional space, not least because they can be easily illustrated graphically.
Formally, let $\Phi^i$ be the sub-space of mean cluster size and follow-up rate, and $\Phi^e$ be that of adherence and potential efficacy. Having specified hypotheses $\Phi^i_I, \Phi^e_I$ for $I = R,A,G$, we then have $$\begin{aligned}
\label{eqn:comb_hyp}
\phi \in \begin{cases}
\Phi_R \text{ if } \phi^i \in \Phi^i_R \text{ or } \phi^e \in \Phi^e_R \\
\Phi_G \text{ if } \phi^i \in \Phi^i_G \text{ and } \phi^e \in \Phi^e_G \\
\Phi_A \text{ otherwise}. \\
\end{cases}\end{aligned}$$
### Follow-up and cluster size
Recall that cluster sizes are assumed to be normally distributed with mean $\mu_c$ and variance $\sigma^2)$. A normal-inverse-gamma prior $$\sigma^{2} \sim \Gamma^{-1} (\alpha_{0}, \beta_{0}), ~ \mu_{c} \sim N(\mu_{0}, \sigma^{2}/\nu_{0})$$ is placed on the mean and variance to allow for prior uncertainty in both parameters. It was anticipated that an average of 8 - 12 residents would be recruited in each care home. To reflect this prior belief we set the hyper-parameters to $\mu_{0} = 10, \nu_{0} = 6, \alpha_{0} = 20, \beta_{0} = 39$, giving a prior cluster size of 10 with mean variance 2.05.
For the probability of successful follow-up, $p_f$, we take a Beta distribution with hyper-parameters $\alpha_{0} = 22.4, \beta_{0} = 9.6$ as the prior. This gives a prior with a mean of 0.7 and a standard deviation of 0.08.
To partition the parameter space into hypotheses, we first consider the case where follow-up is perfect, i.e. $p_{f} = 1$. Conditional on this, we reason that a mean cluster size of below 5 should lead to a red decision (stop development), whereas a size of above 7 should lead to a green decision (proceed to the main trial). As the probability of successful follow-up decreases, we suppose that this can be compensated by an increase in mean cluster size. We assume the nature of this trade-off is linear and decide that if $p_{f}$ were reduced to 0.8, we would want to have a mean cluster size of at least 8 to consider decisions $a$ or $g$. We further decide that a follow-up rate of less than $p_{f} = 0.6$ would be critically low, regardless of the mean cluster size, and should always lead to decision $r$. Similarly, a follow-up rate of $0.6 \leq p_{f} < 0.66$ should lead to modification of the intervention or trial design. Together, these conditions lead to the following partitioning of the parameter space: $$(p_{f}, \mu_{c}) \in \begin{cases}
\Phi^i_R \text{ if } p_{f} < 0.6 \text{ or } 20-15p_{f} > \mu_{c} \\
\Phi^i_G \text{ if } p_{f} > 0.66 \text{ and } 22-15p_{f} < \mu_{c} \\
\Phi^i_A \text{ otherwise.}
\end{cases}$$
The hypotheses are illustrated in Figure \[fig:hyps\] (a). Having specified both the hypotheses and the prior distribution for these two parameters, we can obtain prior probabilities of each hypothesis by sampling from the prior and calculating the proportion of these samples falling into the regions $\Phi^i_R, \Phi^i_A$ and $\Phi^i_G$. We have plotted 1000 samples from the prior in Figure \[fig:hyps\] (a), falling into hypotheses $\Phi^i_R, \Phi^i_A$ and $\Phi^i_G$ in proportions 0.354, 0.517, 0.129 respectively. This demonstrates that there is significant prior uncertainty regarding the optimal decision, indicating the potential value of the pilot trial.
\
### Adherence and potential efficacy
Having defined priors and hypotheses with respect to cluster size and follow-up, we now consider adherence and potential efficacy. Recall that the number of care homes which successfully adhere to the intervention delivery plan is assumed to be binomially distributed with probability $p_{a}$. We assume that adherence is absolute in the sense that all residents in a care home which does not successfully deliver the intervention will not receive any of the treatment effect. We place a Beta prior on $p_{a}$, with hyper-parameters $\alpha = 28.8$ and $\beta = 3.2$ giving a prior mean of 0.9 and a standard deviation of 0.05.
For the continuous measure of physical activity, we place priors on the mean effect $\mu$, the intracluster correlation coefficient $\rho$, and the within-cluster variance $\sigma_{W}^{2}$ in the manner suggested in [@Spiegelhalter2001]. Specifically, we choose $$\begin{aligned}
\mu & \sim N(0.2, 0.25^{2}) \\
\sigma_{W}^{2} & \sim \Gamma^{-1}(50, 45) \\
\rho & \sim Beta(1.6, 30.4).\end{aligned}$$ To reflect prior expectation of an ICC around 0.05 but possibly as large as 0.1, the hyperparameters give a prior mean of 0.05 for the ICC with a prior probability of 0.104 that it will exceed 0.1.
While there is potential for adherence to be improved after the pilot, we assume there will be little opportunity to improve the potential efficacy of the intervention. Moreover, we suppose an absolute improvement in adherence of up to around 0.1 is feasible. To define the hypotheses in this subspace we first set a minimal level of potential efficacy to be 0.1, and decide that we would be happy to make decision $g$ at this point if and only if adherence is perfect. As $p_{a}$ reduces from 1, a corresponding linear increase in potential efficacy is considered to maintain the overall effectiveness of the intervention. The rate of substitution for this trade-off is determined to be approximately 0.57 units of potential efficacy per unit of adherence probability. We consider an absolute lower limit in adherence of $p_{a} = 0.5$, below which we will always consider decision $r$ to be optimal. Taking these considerations together, the marginal hypotheses are defined as
$$(p_{a}, \mu) \in \begin{cases}
\Phi^e_R \text{ if } p_{a} < 0.5 \text{ or } 0.96-0.57\mu > p_{a} \\
\Phi^e_G \text{ if } p_{a} > 0.6 \text{ and } 1.06-0.57\mu < p_{a} \\
\Phi^e_A \text{ otherwise.}
\end{cases}$$
The hypotheses are illustrated in Figure \[fig:hyps\] (b). Again, a sample of size 1000 from the joint marginal prior distribution $p(p_{a}, \mu)$ is also plotted, falling into hypotheses $\Phi^e_R, \Phi^e_A$ and $\Phi^e_G$ in proportions 0.234, 0.470, 0.296 respectively. As before, this indicates substantial prior uncertainty regarding the optimal decision and thus supports the use of a pilot study.
The marginal hypotheses are combined together using equation (\[eqn:comb\_hyp\]). Considering the same 1000 samples from the design prior plotted in Figure \[fig:hyps\], these now fall into the regions $\Phi_R, \Phi_A$ and $\Phi_G$ in proportions 0.507, 0.458, and 0.035 respectively. Note that the prior probabilities of these overall hypotheses are quite different to those of the marginal hypotheses. In particular, there is a considerable increase in the probability that decision $r$ will be optimal, and a considerable decrease that decision $g$ will be.
Evaluation
----------
### Weakly informative analysis
We applied the proposed method assuming that a weakly informative joint prior distribution will be used at the analysis stage[^2]. We took the sample size of the trial to be $k = 6$ clusters per arm. For calculating operating characteristics we generated $N = 10^4$ samples from the joint distribution $p(\theta, x) = p(x | \theta)p_D(\theta)$. We analysed each simulated data set using Stan via the R package rstan [@rstan], in each case generating 5000 samples in four chains and discarding the first 2500 samples in each to allow for burn-in, leading to $M = 10^4$ posterior samples in total. This gave a maximum Monte Carlo error of approximately 0.005 when estimating a posterior probability $Pr[\phi \in \Phi_I ~|~ x]$, which we considered sufficient. These posterior samples were then used to find the posterior probabilities of each hypothesis, for each simulated data set.
We evaluated the operating characteristics for a sample of parameters $(c_1, c_2, c_3)$ as described in Section \[sec:optimisation\]. A total of 254 parameter vectors were evaluated, of which 62 led to operating characteristics which were worse in every respect than some other vector (i.e. dominated) and were discarded. The operating characteristics of the non-dominated parameters are shown in Figure \[fig:p\_front\]. The three operating characteristics are found to be highly correlated. In particular, changing the parameters to give a lower probability of discarding a promising intervention ($OC_2$) tends to lead to a reduction in the probability of making an unnecessary adjustment ($OC_3$). When selecting $(c_1, c_2)$, the key decision appears to be trading off the probability of an infeasible trial, ($OC_{1}$), against $OC_{2}$. There is a very limited opportunity to minimise $OC_{3}$ at the expense of these. For example, compare points $b$ and $c$ in Figure \[fig:p\_front\], details of which are given in Table \[tab:costs\]. We see that point $c$ reduces $OC_3$ by 0.078 in comparison to point $b$, but only at the expense of increase in $OC_1$ and $OC_2$ of 0.13 and 0.145 respectively.
![Operating characteristics of the example pilot trial for a range of loss parameter vectors, when a weakly informative analysis prior is used.[]{data-label="fig:p_front"}](p_front)
\[tab:costs\]
We would expect to see a clear relationship between the value of parameters $c_1, c_2, c_3$ and the operating characteristics they relate to. We explore this in Figure \[fig:cost\_OCs\] with scatter plots of each parameter against each operating characteristic. The results show that there is indeed a strong relationship between the loss assigned to discarding a promising intervention, $c_2$, and the probability that this event will occur, $OC_2$ (see centre plot). Moreover, $c_2$ also seems to be the main determinant of operating characteristics $OC_1$ and $OC_3$. The implication is that once the $c_2 \in [0,1]$ has been chosen, the operating characteristics of the trial depend only weakly on the way in which the remaining $1-c_3$ is allocated to $c_1$ and $c_3$. This appears to be due to the fact that, regardless of how errors are weighted, the way we have defined our prior distributions and hypotheses means we are much more likely to make the error of discarding a promising intervention than the other types of error. The cost we assign to this error is therefore more influential on the overall operating characteristics than the other costs.
![Relationships between the three loss parameters ($x$ axes) and resulting operating characteristics ($y$ axes).[]{data-label="fig:cost_OCs"}](cost_OCs)
To illustrate the effect of varying sample size in the REACH trial, we set the loss function parameters to that of point $a$ in Figure \[fig:p\_front\] and Table \[tab:costs\], $(c_1, c_2, c_3) = (0.07, 0.9, 0.03)$. We then estimated the operating characteristics obtained for $k = 6, 12, 18$ clusters per arm. Note that we considered only three choices of sample size due to the significant computational burden of each evaluation. The results are plotted in Figure \[fig:k\_comp\]. Increasing the sample size appears to have little effect on $OC_1$ and $OC_3$, while leading to a decrease in $OC_2$, the probability of discarding a promising intervention. This behaviour reflects the priorities encoded by the costs parameter, where $c_2 = 0.9$.
![Operating characteristics of the REACH trial for per-arm sample sizes $k = 6, 12, 18$ and setting $(c_1, c_2, c_3) = (0.069, 0.116, 0.815)$. Error bars denote 95% confidence intervals. All points have been adjusted horizontally to avoid overlap.[]{data-label="fig:k_comp"}](k_comp)
### Incorporating subjective priors
Rather than use weakly or non-informative priors when analysing the pilot data, we may instead want to make use of the (subjective) elicited knowledge of parameter values described in the design prior $p_D(\theta)$. Anticipating criticisms of a fully subjective analysis, we can envisage two particular cases where this might be appropriate. Firstly, using the components of the design prior which describe the nuisance parameters $\psi$ while maintaining weakly informative priors on substantive parameters $\phi$. Secondly, when very little data on a specific substantive parameter is going to be collected in the pilot, using the informative design prior for that parameter could substantially improve operating characteristics.
We replicated the above analysis for these two scenarios. For the second, we used informative priors for all nuisance parameters and for the probability of adherence, $p_a$. Recall that this is informed by a binary indicator at the care home level and only in the intervention arm, and will therefore have very little pilot data bearing on it. For each case we used the same $N$ samples of parameters and pilot data which were used in the weakly informative case, repeating the Bayesian analysis using the appropriate analysis prior and obtaining estimated posterior probabilities $p_R, p_A$ and $p_G$ as before. These were used in conjunction with the same set of loss parameter vectors $\mathcal{C}$ to obtain corresponding operating characteristics.
For brevity we will refer to the three cases as Weakly Informative (WI), Informative Nuisance (IN), and Informative Nuisance and Adherence (INA). Comparing the operating characteristics of cases WI and IN, we found very little difference (further details are provided in the appendix). When we contrast cases WI and INA, however, there is a clear distinction. Using the INA analysis prior will lead to larger probabilities of an infeasible trial ($OC_1$) and of unnecessary adjustment ($OC_2$), while reducing the probability of discarding a promising intervention ($OC_3$), for almost all loss parameters. The expected loss is always lower for the INA analysis than for WI, as we would expect.
![Operating characteristics and expected utilities for weakly (WI) and partially informative (INA)[]{data-label="fig:an_prior_comp"}](an_prior_comp)
Discussion {#sec:discussion}
==========
When deciding if and how a definitive RCT of a complex intervention should be conducted, and basing this decision on an analysis of data from a small pilot trial, there is a risk we will inadvertently make the wrong choice. A Bayesian analysis of pilot data followed by decision making based on a loss function can help ensure this risk is minimised. The expected results of such a pilot can be evaluated through simulation at the design stage, producing operating characteristics which help us understand the potential for the pilot to lead to better decision making. These evaluations can in turn be used to find the loss function which leads to the most desirable operating characteristics, and to inform the choice of sample size.
Our proposal has been motivated by some salient characteristics of complex intervention pilot trials, and offers several potential benefits over standard pilot trial design and analysis techniques. The Bayesian approach to analysis means that complex multi-level models can be used to describe the data, even when the sample size is small. In contrast to the usual application of independent progression criteria for several parameters of interest, we provide a way for preferential relationships between parameters to be articulated and used when making decisions. Using a subjective prior distribution on unknown parameters at the design stage allows both our knowledge and our uncertainty to be fully expressed, meaning we can leverage external information whilst also avoiding decisions which are highly sensitive to imprecise point estimates.
Our proposed design is related to the literature on assurance calculations for clinical trials [@OHagan2005], applying the idea of using unconditional event probabilities as operating characteristics to the pilot trial setting. In doing so we have shown how assurances can be defined for multiple substantive parameters with trade-offs between them, and with respect to the ‘traffic light’ red/amber/green decision structure commonly found in pilot trials. The multi-objective optimisation framework we have used to inform trial design allows the decision-maker to explicitly consider the different trade-offs between operating characteristics which are available, and select that which best reflects their own preferences. A similar approach has been taken in the context of phase II trials using the statistical concept of admissible designs [@Jung2004; @Mander2012]. This can be contrasted with the conventional and much criticised approach common in the frequentist context, where arbitrary constraints are placed on type I and II error rates in order to define a single optimal design [@Bacchetti2010].
The benefits brought by the Bayesian approach must be set against the challenges it brings, particularly in terms of computation time and implementation. In terms of the latter, we are required to specify a joint prior distribution over the parameters $\theta$ and a partitioning of the parameter space into the three hypotheses. The specification of the prior distribution may be a challenging and time-consuming task. Although some relevant data relating to similar contexts may be available, for example in systematic reviews or observational studies, expert opinion may still be required to articulate the relevance of such data to the problem at hand. When no data are available, which is not unlikely given the early phase nature of pilot studies, expert opinion will be the only source of information. Although potentially challenging, many examples describing successful practical applications of elicitation for clinical trial design are available [@Walley2015; @Crisp2018; @Dallow2018], as are tools for its conduct such as the Sheffield Elicitation Framework (SHELF) [@OHagan2006a]. Dividing the parameter space into three hypotheses may also prove challenging in practice, particularly when trade-offs between more than two parameters are to be elicited. There is a need for methodological research investigating how methods for multi-attribute preference elicitation, such as those set out in [@Keeney1976], can be applied in this context.
The computational burden of the proposed method is significant, particularly when the model is too complex to allow a conjugate analysis to be used when sampling from the posterior distribution. We have used a nested Monte Carlo sampling scheme to estimate operating characteristics, as seen elsewhere [@Wang2002; @OHagan2005; @Sutton2007]. One potential approach to improve efficiency is to use non-parametric regression to predict the expected losses of Equation (\[eqn:exp\_loss\]) based on some simulated data, thus bypassing the need to undertake a full MCMC analysis for each of the $N$ samples in the outer loop. This approach has been shown to be successful in the context of expected value of information calculations [@Strong2014; @Strong2015]. The computational difficulties will be particularly pertinent when using our approach to determine sample size, as several evaluations of different sample size choices will be required. If the choice of sample size can be framed as an optimisation problem, methods for efficient global optimisation of computationally expensive functions such as those described in [@Jones2001; @Roustant2012] may be useful [@Wilson2015]. Alternatively, one of several rules-of-thumb for choosing pilot sample size [@Lancaster2004; @Julious2005; @Teare2014; @Whitehead2015] could be used, with the resulting operating characteristics evaluated using the proposed method.
We have defined our procedure in terms of a loss function, where the decision making following the pilot will minimise the expected loss. However, the piecewise constant loss function we have proposed may not adequately represent the preferences of the decision maker. For example, we may object to the loss associated with discarding a promising intervention being independent of exactly how effective the intervention is. An alternative is to try to define a richer representation of the loss function through direct elicitation of the decision makers preferences under uncertainty [@French2000], leading to a fully decision-theoretic approach to design and analysis [@Lindley1997]. However, as previously noted by others [@Joseph1997a; @Bacchetti2008; @Whitehead2008], implementation of these approaches has been limited in practice and this may be indicative of their feasibility.
The proposed method could be extended in several ways. More operating characteristics could be defined and used in design optimisation, more complicated trade-off relationships between multiple parameters could be addressed, or the hypotheses could be expanded to include nuisance parameters which would be used as part of the sample size calculation in the main RCT. A particularly interesting avenue for future research is to consider how to model post-pilot trial actions in more detail. For example, while we allow for the possibility of making an ‘amber’ decision, indicating that modifications to the intervention or trial design should be made, we do not model what that decision will actually look like and how it should relate to the observed pilot data. Methodology for jointly modelling a pilot and subsequent main RCT in this manner could be informed by developments for designing phase II/III programs in the drug setting [@Stallard2012; @Wason2013; @Goette2015; @Kirchner2015].
Acknowledgements {#acknowledgements .unnumbered}
----------------
We would like to thank Alex Wright-Hughes, Robert Cicero, and the TIGA-CUB and REACH trial teams for discussions which helped shape the scope of this paper.
Data availability statement {#data-availability-statement .unnumbered}
---------------------------
All simulated data used in this manuscript, together with the code used to generate it, is available at <https://github.com/DTWilson/Bayesian_pilot>.
Funding {#funding .unnumbered}
-------
This work was supported by the Medical Research Council under Grant MR/N015444/1 to D.T.W. and Grant MC\_UU\_00002/6 to J.M.S.W.
[^1]: Note that in the case of complex models which do no admit a conjugate analysis, the posterior probabilities obtained using an MCMC analysis will themselves be approximate and as such the optimal decision will be subject to error, which may increase the variance of the operating characteristic estimates. However, this issue can be sidestepped by assuming that, for each data set, the analysis that is simulated corresponds exactly to the analysis that would be carried out in practice. In particular, we assume that exactly $M$ posterior samples will be generated by the same MCMC algorithm, using the same seed in the random number generator.
[^2]: Full details of the weakly informative prior are given in the appendix.
|
---
author:
- 'Ward Melis, Thomas Rey and Giovanni Samaey'
bibliography:
- 'refs.bib'
title: Projective integration for nonlinear BGK kinetic equations
---
Introduction {#sec:introduction}
============
The Boltzmann equation constitutes the cornerstone of the kinetic theory of rarefied gases. In a dimensionless, scalar setting, it describes the evolution of the one-particle mass distribution function ${f^{\varepsilon}}({\mathbf{x}},{\mathbf{v}},t) \in {\mathbb{R}}^{+}$ as: $$\label{eq:Boltzmann_equation}
\partial_t {f^{\varepsilon}}+ {\mathbf{v}}\cdot \nabla_{{\mathbf{x}}} {f^{\varepsilon}}= \frac{1}{{\varepsilon}}{{\mathcal{Q}}{\mathopen{}\mathclose\bgroup\originalleft}({f^{\varepsilon}}{\aftergroup\egroup\originalright})({\mathbf{v}})},$$ where $t \ge 0$ represents time, and $({\mathbf{x}},{\mathbf{v}}) \subset {{\mathbb{R}}^{D_x \times D_v}}$ are the $D_x$-dimensional particle positions and $D_v$-dimensional particle velocities. In equation , the dimensionless constant ${\varepsilon}> 0$ determines the regime of the gas flow, for which we roughly identify the hydrodynamic regime $({\varepsilon}\le 10^{-4})$, the transitional regime $({\varepsilon}\in [10^{-4},10^{-1}])$, and the kinetic regime $({\varepsilon}\ge 10^{-1})$. Furthermore, the left hand side of corresponds to a linear transport operator that comprises the convection of particles in space, whereas the right hand side contains the Boltzmann collision operator that entails velocity changes due to particle collisions. However, due to its high-dimensional and complicated structure, the Boltzmann collision operator is often replaced by simpler collision models that capture most essential features of the former. The most well-known such model is the BGK model [@Bhatnagar1954], which models collisions as a linear relaxation towards thermodynamic equilibrium, and is given by: $$\label{eq:bgk_equation}
\partial_t {f^{\varepsilon}}+ {\mathbf{v}}\cdot \nabla_{{\mathbf{x}}} {f^{\varepsilon}}= \frac{1}{{\varepsilon}}({{{\mathcal{M}}_{{\mathbf{v}}}{\mathopen{}\mathclose\bgroup\originalleft}({f^{\varepsilon}}{\aftergroup\egroup\originalright})}}- {f^{\varepsilon}}),$$ in which ${{{\mathcal{M}}_{{\mathbf{v}}}{\mathopen{}\mathclose\bgroup\originalleft}({f^{\varepsilon}}{\aftergroup\egroup\originalright})}}$ denotes the local Maxwellian distribution, which, for a $D_v$-dimensional velocity space, is given by: $$\label{eq:maxwellian}
{{{\mathcal{M}}_{{\mathbf{v}}}{\mathopen{}\mathclose\bgroup\originalleft}({f^{\varepsilon}}{\aftergroup\egroup\originalright})}}= \frac{\rho}{(2\pi T)^{D_v/2}} \exp{{\mathopen{}\mathclose\bgroup\originalleft}(-\frac{|{\mathbf{v}}-{\mathbf{{\bar{v}}}}|^2}{2T}{\aftergroup\egroup\originalright})} := {\mathcal{M}}_{\mathbf{v}}^{\rho,{\mathbf{{\bar{v}}}},T}.$$ The Maxwellian distribution contains the velocity moments of the distribution function ${f^{\varepsilon}}$, which are calculated as: $$\label{eq:f_moments}
\rho = \int_{{\mathbb{R}}^{D_v}} {f^{\varepsilon}}d{\mathbf{v}}, \qquad
{\bar{v}}^d = \frac{1}{\rho}\int_{{\mathbb{R}}^{D_v}} v^d {f^{\varepsilon}}d{\mathbf{v}}, \qquad
T = \frac{1}{D_v\rho}\int_{{\mathbb{R}}^{D_v}} {{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}-{\mathbf{{\bar{v}}}}{\aftergroup\egroup\originalright}\vert}^2 {f^{\varepsilon}}d{\mathbf{v}},$$ where $\rho \in {\mathbb{R}}^+$, ${\mathbf{{\bar{v}}}}= {{\mathopen{}\mathclose\bgroup\originalleft}({\bar{v}}^d{\aftergroup\egroup\originalright})_{d=1}^{D_v}} \in {\mathbb{R}}^{D_v}$ and $T \in {\mathbb{R}}^+$ are the density, macroscopic velocity and temperature, respectively, which all depend on space ${\mathbf{x}}$ and time $t$. Then, in the limit ${{\varepsilon}\to 0}$, the solution to equation converges towards ${\mathcal{M}}_{\mathbf{v}}^{\rho,{\mathbf{{\bar{v}}}},T}$, whose moments in are solution to the compressible Euler system: $$\label{eq:fluid_euler}
{\mathopen{}\mathclose\bgroup\originalleft}\{ \begin{aligned}
& \partial_t \rho + \operatorname{div}_{\mathbf{x}}(\rho \, {\mathbf{{\bar{v}}}}) \,=\, 0, \\
& \partial_t(\rho \, {\mathbf{{\bar{v}}}}) + \operatorname{div}_{\mathbf{x}}{\mathopen{}\mathclose\bgroup\originalleft}(\rho \, {\mathbf{{\bar{v}}}}\otimes {\mathbf{{\bar{v}}}}\,+\, \rho \, T \,{\rm\bf I}{\aftergroup\egroup\originalright}) \, =\, \bm{0},
\\
& \partial_t E + \operatorname{div}_{\mathbf{x}}{\mathopen{}\mathclose\bgroup\originalleft}( {\mathbf{{\bar{v}}}}{\mathopen{}\mathclose\bgroup\originalleft}( E +\rho \, T{\aftergroup\egroup\originalright}) {\aftergroup\egroup\originalright}) \,=\, 0,
\end{aligned} {\aftergroup\egroup\originalright}.$$ in which $E$ is the second moment of ${f^{\varepsilon}}$, namely its total energy.
In this paper, we construct a fully explicit, asymptotic-preserving, arbitrary order time integration method for the stiff equation . For a comprehensive review of numerical schemes for collisional kinetic equations such as equation , we refer to [@DimarcoPareschi15]. The asymptotic-preserving property [@Jin1999] implies that, in the limit when ${\varepsilon}$ tends to zero, an ${\varepsilon}$-independent time step constraint, of the form ${\Delta t}= O({\Delta x})$, can be used, in agreement with the classical hyperbolic CFL constraint for the limiting fluid equations . To achieve this, we will use a projective integration method, which was introduced in [@Gear2003projective] and first applied to kinetic equations in [@lafitte2012].
The remainder of this paper is structured as follows. We describe the projective integration method in more detail in section \[sec:projective\_integration\], after which we discuss (in section \[sec:spectral\_properties\]) the spectral properties of the linearized BGK operator, which are needed to ensure stability of the method. Some numerical experiments are done in section \[sec:results\].
Projective integration {#sec:projective_integration}
======================
Projective integration [@Gear2003projective; @lafitte2012] combines a few small time steps with a naive (*inner*) timestepping method (here, a direct forward Euler discretization) with a much larger (*projective, outer*) time step. The idea is sketched in figure \[fig:proj\_int\].
**Inner integrators.** We discretize equation on a uniform, constant in time, periodic spatial mesh with spacing ${\Delta x}$, consisting of $I$ mesh points $x_i=i{\Delta x}$, ${1 \le i \le I}$, with $I{\Delta x}=1$, and a uniform time mesh with time step ${\delta t}$ and discrete time instants $t^k=k{\delta t}$. Furthermore, we discretize velocity space by choosing $J$ discrete components denoted by ${\mathbf{v}}_j$. The numerical solution on this mesh is denoted by $f_{i,j}^k$, where we have dropped the superscript ${\varepsilon}$ on discretized quantities. We then obtain a semidiscrete system of ODEs of the form: $$\label{eq:semidiscrete}
\dot{{\mathbf{f}}} = {\mathrm{D}_t}({\mathbf{f}}), \qquad
{\mathrm{D}_t}({\mathbf{f}}) = -{\mathrm{D}_{{\boldsymbol{x}},{\boldsymbol{v}}}}({\mathbf{f}}) + \frac{1}{{\varepsilon}}{\mathopen{}\mathclose\bgroup\originalleft}({{{\mathcal{M}}_{{\boldsymbol{v}}}{\mathopen{}\mathclose\bgroup\originalleft}({\mathbf{f}}{\aftergroup\egroup\originalright})}} - {\mathbf{f}}{\aftergroup\egroup\originalright}),$$ where ${\mathrm{D}_{{\boldsymbol{x}},{\boldsymbol{v}}}}(\cdot)$ represents a suitable discretization of the convective derivative ${\mathbf{v}}\cdot \nabla_{\mathbf{x}}$ (for instance, using upwind differences), and ${\mathbf{f}}$ is a vector of size $I \cdot J$.
As inner integrator, we choose the (explicit) forward Euler method with time step ${\delta t}$, for which we will, later on, use the shorthand notation: $$\label{eq:fe_scheme}
{\mathbf{f}}^{k+1} = {S_{{\delta t}}}({\mathbf{f}}^{k}) = {\mathbf{f}}^k + {\delta t}{\mathrm{D}_t}({\mathbf{f}}^k), \qquad k = 0, 1, \ldots.$$
**Outer integrators.** In system , the small parameter ${\varepsilon}$ leads to the classical time step restriction of the form ${\delta t}= O({\varepsilon})$ for the inner integrator. However, as ${\varepsilon}$ goes to $0$, we obtain the limiting system for which a standard finite volume/forward Euler method only needs to satisfy a stability restriction of the form ${\Delta t}\le C{\Delta x}$, with $C$ a constant that depends on the specific choice of the scheme.
In [@lafitte2012], it was proposed to use a projective integration method to accelerate such a brute-force integration; the idea, originating from [@Gear2003projective], is the following. Starting from a computed numerical solution ${\mathbf{f}}^n$ at time $t^n=n{\Delta t}$, one first takes $K+1$ *inner* steps of size ${\delta t}$ using , denoted as ${\mathbf{f}}^{n,k+1}$, in which the superscripts $(n,k)$ denote the numerical solution at time ${t^{n,k}=n{\Delta t}+k{\delta t}}$. The aim is to obtain a discrete derivative to be used in the *outer* step to compute ${\mathbf{f}}^{n+1} = {\mathbf{f}}^{n+1,0}$ via extrapolation in time: $$\label{eq:pfe_scheme}
{\mathbf{f}}^{n+1} = {\mathbf{f}}^{n,K+1} + ({\Delta t}- (K + 1){\delta t})\frac{{\mathbf{f}}^{n,K+1} - {\mathbf{f}}^{n,K}}{{\delta t}}.$$
Higher-order projective Runge-Kutta (PRK) methods can be constructed by replacing each time derivative evaluation $\mathbf{k}_s$ in a classical Runge-Kutta method by $K+1$ steps of an inner integrator as follows: $$\begin{aligned}
s = 1 :\;\; &
\begin{dcases}
{\mathbf{f}}^{n,k+1} &= {\mathbf{f}}^{n,k} + {\delta t}{\mathrm{D}_t}({\mathbf{f}}^{n,k}), \qquad 0 \le k \le K \\
\mathbf{k}_1 &= \dfrac{{\mathbf{f}}^{n,K+1} - {\mathbf{f}}^{n,K}}{{\delta t}}
\end{dcases} \label{eq:PRK_stage_1} \\
2 \le s \le S :\;\; &
\begin{dcases}
{\mathbf{f}}^{n+c_s,0}_s &= {\mathbf{f}}^{n,K+1} + (c_s{\Delta t}-(K+1){\delta t}) \sum_{l=1}^{s-1}\dfrac{a_{s,l}}{c_s} \mathbf{k}_l, \\
{\mathbf{f}}^{n+c_s,k+1}_s &= {\mathbf{f}}^{n+c_s,k}_s + {\delta t}{\mathrm{D}_t}({\mathbf{f}}^{n+c_s,k}_s), \qquad 0 \le k \le K \\
\mathbf{k}_s &= \dfrac{{\mathbf{f}}^{n+c_s,K+1}_s - {\mathbf{f}}^{n+c_s,K}_s}{{\delta t}}
\end{dcases} \label{eq:PRK_stage_s} \\
& {\mathbf{f}}^{n+1} = {\mathbf{f}}^{n,K+1} + ({\Delta t}-(K+1){\delta t})\sum_{s=1}^{S}b_s \mathbf{k}_s.\end{aligned}$$ To ensure consistency, the Runge-Kutta matrix $\mathbf{a}=(a_{s,i})_{s,i=1}^S$, weights ${\mathbf{b}=(b_s)_{s=1}^S}$, and nodes $\mathbf{c}=(c_s)_{s=1}^S$ satisfy the conditions $0\le b_s \le 1$ and $0 \le c_s \le 1,$ as well as: $$\label{eq:RK_conditions}
\sum_{s=1}^Sb_s=1, \qquad \sum_{i=1}^{S-1} a_{s,i} =c_s, \quad 1 \le s \le S.$$
Spectral properties {#sec:spectral_properties}
===================
To choose the method parameters (the size of the small and large time steps $\delta t$ and $\Delta t$, as well as the number $K$ of small steps), one needs to analyze the spectrum of the collision operator. In [@Melis2016], this was done in the hyperbolic scaling for a system with a linear Maxwellian that serves as a relaxation of a nonlinear hyperbolic conservation law.
By linearizing the Maxwellian around the global Maxwellian distribution ${{\mathcal{M}}^{{\rho^{\infty}},{{\mathbf{{\bar{v}}}}^{\infty}},{T^{\infty}}}_{{\mathbf{v}}}}= {\mathcal{M}}_{\mathbf{v}}^{1,0,1}$, it is shown in [@Cercignani1988 p.206] that the resulting linearized equilibrium can be written as: $$\label{eq:linearized_bgk_maxwellian}
{\mathcal{M}}_\text{lin}({f^{\varepsilon}})({\mathbf{x}},{\mathbf{v}},t) = \sum_{k=0}^{D_v+1} \Psi_k({\mathbf{v}}) (\Psi_k, {f^{\varepsilon}})({\mathbf{x}},t),$$ in which the scalar product is defined by: $$\label{eq:scalar_product_Hilbert}
(g,h) = \int_{{\mathbb{R}}^{D_v}} g({\mathbf{v}})\overline{h({\mathbf{v}})} \frac{1}{(2\pi)^{D_v/2}}\exp{\mathopen{}\mathclose\bgroup\originalleft}(\frac{-{{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}{\aftergroup\egroup\originalright}\vert}^2}{2}{\aftergroup\egroup\originalright}) d{\mathbf{v}}.$$ Furthermore, the orthonormal set of basis functions $\Psi_k({\mathbf{v}})$ in are obtained from a straightforward application of the Gram-Schmidt process to the $D_v+1$ collision invariants $(1,{\mathbf{v}},{{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}{\aftergroup\egroup\originalright}\vert}^2)$, yielding: $$\label{eq:Psi_normalized}
\big(\Psi_0({\mathbf{v}}), \ldots, \Psi_{D_v+1}({\mathbf{v}})\big) = {\mathopen{}\mathclose\bgroup\originalleft}(1, v^1, ..., v^{D_v}, \frac{{{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}{\aftergroup\egroup\originalright}\vert}^2 - D_v}{2^{D_v/2}}{\aftergroup\egroup\originalright}).$$ Using the linearized Maxwellian , the linearized version of the full BGK equation reads: $$\label{eq:linearized_bgk_equation_framework}
\partial_t {f^{\varepsilon}}+ {\mathbf{v}}\cdot \nabla_{{\mathbf{x}}} {f^{\varepsilon}}= -\frac{1}{{\varepsilon}}({\mathcal{I}}- \Pi_\text{BGK}){f^{\varepsilon}},$$ where ${\mathcal{I}}$ denotes the identity operator and $\Pi_\text{BGK}$ is the following rank-$(D_v+2)$ projection operator: $$\label{eq:projection_operator_bgk}
\Pi_\text{BGK} {f^{\varepsilon}}= \sum_{k=0}^{D_v+1} \Psi_k({\mathbf{v}})(\Psi_k, {f^{\varepsilon}}).$$
This shows that the structure of the linearized Maxwellian and the linearized BGK projection operator are almost identical to those in [@Melis2016]. We can actually view these linear kinetic models as a special simplified case of the linearized BGK equation. Therefore, it is expected that the construction of stable, asymptotic-preserving projective integration methods for the full BGK equation is practically identical to that in [@Melis2016]. In particular, the conclusion is that, when choosing $\delta t={\varepsilon}$, one is able to choose $\Delta t=O(\Delta x)$ and $K$ independent of ${\varepsilon}$, resulting in a scheme with computational cost independent of ${\varepsilon}$.
Numerical experiments {#sec:results}
=====================
**BGK in 1D.** As a first experiment, we focus on the nonlinear BGK equation in 1D. We consider a Sod-like test case for $x \in [0,1]$ consisting of an initial centered Riemann problem with the following left and right state values: $$\label{eq:bgk_1d_sod}
\big(\rho_L, {\bar{v}}_L, T_L\big) = (1, 0, 1), \qquad\quad
\big(\rho_R, {\bar{v}}_R, T_R\big) = (0.125, 0, 0.25).$$ The initial distribution ${f^{\varepsilon}}(x,v,0)$ is then chosen as the Maxwellian corresponding to the above initial macroscopic variables. We impose outflow boundary conditions and perform simulations for $t \in [0,0.15]$. As velocity space, we take the interval $[-8,8]$, which we discretize on a uniform grid using $J=80$ velocity nodes. In all simulations, space is discretized using the WENO3 spatial discretization with ${\Delta x}= 0.01$. Below, we compare solutions for three gas flow regimes: ${\varepsilon}= 10^{-1}$ (kinetic regime), ${\varepsilon}= 10^{-2}$ (transitional regime) and ${\varepsilon}= 10^{-5}$ (fluid regime).
In the kinetic $({\varepsilon}= 10^{-1})$ and transitional $({\varepsilon}= 10^{-2})$ regimes, we compute the numerical solution using the fourth order Runge-Kutta (RK4) time discretization with time step ${\delta t}= 0.1{\Delta x}$. In the fluid regime $({\varepsilon}= 10^{-5})$, direct integration schemes such as RK4 become too expensive due to a severe time step restriction, which is required to ensure stability of the method. Exploiting that the spectrum of the linearized BGK equation is close to that of the linear kinetic models used in [@Melis2016], see section \[sec:spectral\_properties\], we construct a projective integration method to accelerate time integration in the fluid regime. As inner integrator, we select the forward Euler time discretization with ${\delta t}= {\varepsilon}$. As outer integrator, we choose the fourth-order projective Runge-Kutta (PRK4) method, using $K=2$ inner steps and an outer step of size ${\Delta t}= 0.4{\Delta x}$.
The results are shown in figure \[fig:bgk\_1d\], where we display the density $\rho$, macroscopic velocity ${\bar{v}}$ and temperature $T$ as given in at $t = 0.15$. In addition, we plot the heat flux $q$, which, in a general $D_v$-dimensional setting, is a vector ${\mathbf{q}}= {{\mathopen{}\mathclose\bgroup\originalleft}(q^d{\aftergroup\egroup\originalright})_{d=1}^{D_v}}$ with components given by: $$\label{eq:heat_flux}
q^d = \frac{1}{2}\int_{{\mathbb{R}}^{D_v}} {{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{c}}{\aftergroup\egroup\originalright}\vert}^2c^d{f^{\varepsilon}}d{\mathbf{v}},$$ in which ${\mathbf{c}}= {{\mathopen{}\mathclose\bgroup\originalleft}(c^d{\aftergroup\egroup\originalright})_{d=1}^{D_v}} = {\mathbf{v}}- {\mathbf{{\bar{v}}}}$ is the peculiar velocity. The different regimes are shown by blue (kinetic), purple (transitional) and green (fluid) dots. The red line in each plot denotes the limiting $({{\varepsilon}\to 0})$ solution of each macroscopic variable, which all converge to the solution of the compressible Euler equations with ideal gas law $P = \rho T$ and heat flux $q = 0$. From this, we observe that the BGK solution is increasingly dissipative for increasing values of ${\varepsilon}$ since the rate with which ${f^{\varepsilon}}$ converges to its equilibrium ${{{\mathcal{M}}_{v}{\mathopen{}\mathclose\bgroup\originalleft}({f^{\varepsilon}}{\aftergroup\egroup\originalright})}}$ becomes slower. In contrast, for sufficiently small ${\varepsilon}$, relaxation to thermodynamic equilibrium occurs practically instantaneous and the Euler equations yield a valid description. Since this is a hyperbolic system, it allows for the development of sharp discontinuous and shock waves which are clearly seen in the numerical solution.
**Shock-bubble interaction in 2D.** Here, we consider the BGK equation in 2D and we investigate the interaction between a moving shock wave and a stationary smooth bubble, which was proposed in [@Torrilhon2006], see also [@Cai2010]. This problem consists of a shock wave positioned at $x = -1$ in a spatial domain ${\mathbf{x}}= (x,y) \in [-2,3] \times [-1,1]$ traveling with Mach number ${\mathit{Ma}}= 2$ into an equilibrium flow region. Over the shock wave, the following left $(x \le -1)$ and right $(x > -1)$ state values are imposed [@Cai2010]: $$\label{eq:bgk_2d_shock}
\big(\rho_L, {\bar{v}}^x_L, {\bar{v}}^y_L, T_L\big) = {\mathopen{}\mathclose\bgroup\originalleft}(\frac{16}{7}, \sqrt{\frac{5}{3}}\frac{7}{16}, 0, \frac{133}{64}{\aftergroup\egroup\originalright}), \qquad\quad
\big(\rho_R, {\mathbf{{\bar{v}}}}_R, T_R\big) = {\mathopen{}\mathclose\bgroup\originalleft}(1, \bm{0}, 1{\aftergroup\egroup\originalright}).$$ Due to this initial profile, the shock wave will propagate rightwards into the flow region at rest $(x > -1)$. Moreover, in this equilibrium region, a smooth Gaussian density bubble centered at ${\mathbf{x}}_0 = (0.5,0)$ is placed, given by: $$\label{eq:bgk_2d_bubble}
\rho({\mathbf{x}},0) = 1 + 1.5\exp{\mathopen{}\mathclose\bgroup\originalleft}(-16{{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{x}}- {\mathbf{x}}_0{\aftergroup\egroup\originalright}\vert}^2{\aftergroup\egroup\originalright}).$$ Then, the initial distribution ${f^{\varepsilon}}({\mathbf{x}},{\mathbf{v}},0)$ is chosen as the Maxwellian corresponding to the initial macroscopic variables in -. We impose outflow and periodic boundary conditions along the $x$- and $y$-directions, respectively, and we perform simulations for $t \in [0,0.8]$. As velocity space, we take the domain $[-10,10]^2$, which we discretize on a uniform grid using $J_x = J_y = 30$. We discretize space using the WENO2 spatial discretization with $I_x = 200$ and $I_y = 25$. Furthermore, we consider a fluid regime by taking ${\varepsilon}= 10^{-5}$.
We construct a PRK4 method with FE as inner integrator to speed up simulation in time. The inner time step is fixed as ${\delta t}= {\varepsilon}$ and we use $K = 2$ inner steps in each outer integrator iteration. The outer time step is chosen as ${\Delta t}= 0.4{\Delta x}$. To compare our results with those in [@Torrilhon2006], where the smallest value of ${\varepsilon}$ is chosen as ${\varepsilon}= 10^{-2}$, we regard the one-dimensional evolution of density and temperature along the axis $y = 0$. For ${t \in \{0, 0.2, 0.4, 0.6, 0.8\}}$, we plot these intersections in figure \[fig:bgk\_2d\_shock\_bubble\_y0\]. We conclude that we obtain the same solution structure at $t = 0.8$ as in [@Torrilhon2006]. However, our results are sharper and less dissipative supposedly due to the particular small value of ${\varepsilon}$ ($10^{-5}$ versus $10^{-2}$). In contrast to [@Cai2010], we nicely capture the swift changes in the temperature profile for $x \in [0.5,1]$ at $t = 0.8$.
|
---
author:
- |
Joan-C. Lario and Anna Somoza\
(with an appendix by Francesc Fité)
bibliography:
- 'ref.bib'
nocite: '[@*]'
title: |
The Sato-Tate conjecture for a Picard curve\
with Complex Multiplication
---
Introduction
============
Serre [@Ser12] provides a vast generalization of the Sato-Tate conjecture, which is known to be true for varieties with complex multiplication [@Joh13]. As a down-to-earth example, in this paper we consider the Picard curve defined over ${\mathbb Q}$ given by the affine model $$C \colon y^3 = x^4 -x \,.$$ One easily checks that $[0:1:0]$ is the unique point of $C$ at infinite, and that $C$ has good reduction at all primes different from $3$. The Jacobian variety of $C$ is absolutely simple and it has complex multiplication by the cyclotomic field $K={\mathbb Q}(\zeta)$ where $\zeta$ is a primitive $9$th root of unity.
With the help of Sage, we compile information on the number of points that the reduction of $C$ has over finite fields of small characteristic:
$$\begin{array}{ccccc}
\hline
\;\;p\;\; & |C({\mathbb F}_p)| & |C({\mathbb F}_{p^2})| & |C({\mathbb F}_{p^3})| \\
\hline
2 & 3 & 5 & 9 \\
5 & 6 & 26 & 126 \\
7 & 8 & 50 & 365 \\
11 & 12 & 122 & 1332 \\
13 & 14 & 170 & 2003 \\
17 & 18 & 392 & 4914 \\
19 & 14 & 302 & 6935 \\
\hline
\end{array}$$
For every primer $p$ of good reduction, we consider the local zeta function $$\zeta (C/{\mathbb F}_p ; s) = \exp\left(\sum_{k\geq 1} |C({\mathbb F}_{p^k})| \, \frac{p^{-ks}}{k}\right)\, .$$
It follows from Weil’s conjectures [@We49] that the zeta function is a rational function of $T = p^{-s}$. That is, $$\zeta (C/{\mathbb F}_p ; T) = \exp\left(\sum_{k\geq 1} |C({\mathbb F}_{p^k})| \,
\frac{T^k}{k}\right) = \frac{L_p(C,T)}{(1-T)(1-pT)}$$ where the so-called *local factor* of $C$ at $p$ $$L_p(C,T) = \sum_{i=0}^6 b_i T^i = \prod_{i=1}^6 (1-\alpha_i T)$$ is a polynomial of degree $6$ with integral coefficients and the complex numbers $\alpha_i$ satisfy $\vert\alpha_i\vert = \sqrt{p}$. In particular, it is determined by the three numbers $|C({\mathbb F}_p)|$, $|C({\mathbb F}_{p^2})|$, $|C({\mathbb F}_{p^3})|$ according to:
$$\begin{array}{c@{\,=\,}l}
b_0 & 1 \\[1pt]
b_1 & |C({\mathbb F}_p)| - (p+1) \\[1pt]
b_2 & (|C({\mathbb F}_{p^2})| - (p^2+1)+b_1^2)/2\\[1pt]
b_3 & (|C({\mathbb F}_{p^3})| - (p^3+1)-b_1^3+3b_2b_1)/3 \\[1pt]
b_4 & p b_2 \\[1pt]
b_5 & p^2 b_1\\[1pt]
b_6 & p^3\,.
\end{array}$$ For all $m\geq 1$ it holds $$| C({\mathbb F}_{p^m}) | = 1+p^m - \sum_{i=1}^6 \alpha_i^m \,.$$ The local factors for small good primes are:
$$\begin{array}{cl}
\hline
\;\;p\;\; & L_p(C,T) \\
\hline
2 & (1+2T^2)(1-2T^2+4T^4) \\
5 & (1+5T^2)(1-5T^2+25T^4) \\
7 & 1+7T^3+343T^6 \\
11 & (1+11T^2)(1-11T^2+121T^4) \\
13 & 1-65T^3+2197T^6 \\
17 & (1+17T^2)^3 \\
19 & 1-6T-12T^2+169T^3-228T^4-2166T^5+6859T^6 \\
\hline
\end{array}$$
0.2truecm
Even if for every such prime $p$ all terms of the sequence $$|C({\mathbb F}_p)|, |C({\mathbb F}_{p^2})|, |C({\mathbb F}_{p^3})|,
\dots, |C({\mathbb F}_{p^m})|, \, \dots
\quad (m\geq 1)$$ are determined by the first three, the obtention of these first three can be a hard computational task as soon as the prime $p$ gets large. However, the presence of complex multiplication enables the fast computation of the local factors $L_p(C,T)$ (see Section \[numeric\]).
For future use, we introduce some notation. The ring of integers of $K$ will be denoted by ${\mathcal O}={\mathbb Z}[\zeta]$, and the unit group ${\mathcal O}^* \simeq {\mathbb Z}/18{\mathbb Z}\times {\mathbb Z}\times {\mathbb Z}$ has generators $\epsilon_0=-\zeta^2$, $\epsilon_1= \zeta^4 - \zeta^3 + \zeta$, $\epsilon_2= \zeta^5 + \zeta^2 - \zeta$. Let $\sigma_i$ denote the automorphism of ${\operatorname{Gal}}(K/{\mathbb Q})$ determined by $\sigma_i(\zeta)= \zeta^i$; one has that $\sigma_2$ generates the Galois group ${\operatorname{Gal}}(K/{\mathbb Q})\simeq ({\mathbb Z}/9{\mathbb Z})^*$. The unique ramified prime in $K/{\mathbb Q}$ is$3{\mathcal {O}}= (1+\zeta+\zeta^4)^6$.
Since the Jacobian variety $\operatorname{Jac}(C)$ has complex multiplication, the work of Shimura and Taniyama [@ST] ensures the existence of an ideal ${\mbox{\gotip m}}$ of the ring of integers ${\mathcal {O}}$ and a Grössencharakter $\psi \colon I_K({\mbox{\gotip m}}) \to {\mathbb C}^\ast$, where $I_K({\mbox{\gotip m}})$ stands for the group of fractional ideals coprime with ${\mbox{\gotip m}}$, $$\psi(\alpha {\mathcal {O}}) = \prod_{\sigma \in \Phi^*} {\,}^\sigma \alpha
\quad \text{if $\alpha \equiv 1 \,(\!\bmod^{\ast} \,{\mbox{\gotip m}})$,
}$$ such that $L(\psi,s) = L(C,s) $. The infinite type $\Phi^*$ is the reflex of the CM-type $\Phi$ of $\operatorname{Jac}(C)$. Up to a finite number of Euler factors, one has $$L(\psi,s) =
\prod_{{{\mathfrak{p}}}} \left( 1 -
\psi({{\mathfrak{p}}})
\operatorname{N}({{\mathfrak{p}}})^{-s} \right)^{-1}
\text{ and \ \ }
L(C,s) =
\prod_{p} L_p(C,p^{-s})^{-1} \,.$$ Hence, the local factor $L_p(C,T)$ can be obtained from the (monic) irreducible polynomial of $\psi({{\mathfrak{p}}})$ over ${\mathbb Q}$ according to $$L_p(C,T)=T^6 \operatorname{Irr}(\psi({{\mathfrak{p}}}), 1/T^f;{\mathbb Q})^{6/(fd)}\,,$$ where $f$ is the residual class degree of $p$ in $K$, and $d=[{\mathbb Q}(\psi({{\mathfrak{p}}}))\colon {\mathbb Q}]$.
There exists a Grössencharakter $\psi\colon I_K({\mbox{\gotip m}})\to {\mathbb C}^*$ of conductor ${\mbox{\gotip m}}= (1+\zeta+\zeta^4)^4$ and infinite type $\Phi^*=
\{ \sigma_1,\sigma_5,\sigma_7 \}=
\{ \sigma_2^0,\sigma_2^4,\sigma_2^5 \}$.
The following holds $$\epsilon_0^{18} \equiv 1 \pmod {\mbox{\gotip m}}\,,\qquad\epsilon_1^9 \equiv 1 \pmod {\mbox{\gotip m}}\,,\qquad \epsilon_2^3
\equiv 1 \pmod {\mbox{\gotip m}}\,.$$ Moreover, one readily checks that $
\epsilon_0^a \epsilon_1^b \epsilon_2^c \equiv 1 \,(\!\bmod {\mbox{\gotip m}})$ if and only if $$\begin{array}{l@{\,\equiv\,}l}
(a,b,c) \quad \, & \quad (0,0,0), \quad (2,1,2), \quad (4,2,1), \\[3pt]
& \quad (6,3,0), \quad (8,4,2), \quad (10,5,1), \\[3pt]
& \quad (12,6,0), \quad (14,7,2), \quad (16,8,1),
\end{array}$$ mod $(18,9,3)$, respectively. Now an easy computation case-by-case shows that if $\epsilon_0^a \epsilon_1^b \epsilon_2^c \equiv 1 \pmod {\mbox{\gotip m}}$, then $$\prod_{\sigma \in \Phi ^*} {\,}^\sigma (\epsilon_0^a \epsilon_1^b \epsilon_2^c) = 1
\,.$$
By using that $K$ has class number one, we define $\psi({{\mathfrak{p}}})$ over prime ideals ${{\mathfrak{p}}}$ of ${\mathcal {O}}$ coprime with ${\mbox{\gotip m}}$ as follows. First we find a generator of ${{\mathfrak{p}}}=(\alpha)$, and then search for $$\epsilon_0^a \epsilon_1^b \epsilon_2^c \alpha \equiv 1 \pmod {\mbox{\gotip m}}$$ with $0\leq a < 18$, $0\leq b < 9$, and $0\leq c < 3$. The existence of such triple $(a,b,c)$ is guaranteed by the fact that $(\alpha,{\mbox{\gotip m}})=1$ and the classes of the 486 possible products $\epsilon_0^a \epsilon_1^b \epsilon_2^c$ exhaust the all the elements in $({\mathcal {O}}/{\mbox{\gotip m}})^*$. It follows that $$\psi({{\mathfrak{p}}})= \prod_{\sigma \in \Phi^*}
{\,}^\sigma (\epsilon_0^a \epsilon_1^b \epsilon_2^c\alpha)$$ is well-defined. Finally, one extends $\psi$ over all ideals prime to ${\mbox{\gotip m}}$ multiplicatively. An argument along the same lines shows the non existence of a Grössencharakter of $K$ of modulus $(1+\zeta+\zeta^4)^i$ for $i<4$. Thus, $\psi$ has conductor ${\mbox{\gotip m}}$.
[max width = .9]{} $
\begin{array}{rrr}
\hline
p& \psi({{\mathfrak{p}}})
& L_p(C,T)=T^6\operatorname{Irr}(\psi({{\mathfrak{p}}}), 1/T^f;{\mathbb Q})^{6/(fd)} \\
\hline
5 & -5 & (1+5T^2)(1-5T^2+25T^4) \\
7 & -3 \zeta^{3} - 2 & 1+7T^3+343T^6 \\
11 & -1331 & (1+11T^2)(1-11T^2+121T^4) \\
13 & 3 \zeta^{3} + 4 & 1-65T^3+2197T^6 \\
17 & 2 \zeta^{5} + \zeta^{4} + \zeta^{2} - \zeta + 1 &
{\left(1 +17 T^{2} \right)}^{3}\\
19 & -\zeta^{4} - 2 \zeta^{3} - 2 \zeta & 1-6T-12T^2+169T^3-228T^4-2166T^5+6859T^6 \\
23 & -23 & {\left(1+23 T^{2} \right)}
{\left(1 - 23 \, T^{2} + 529T^{4} \right)} \\
29 & -29 & {\left(1+29T^{2}\right)}
{\left( 1 - 29 \, T^{2} + 841 T^{4}\right)} \\
31 & 6 \zeta^{3} + 1 & 1 + 124 \, T^{3} + 29791 T^{6} \\
37 & \zeta^{5} - \zeta^{3} + 2 \zeta^{2} - 1 &
1 - 6 \, T + 42 \, T^{2}
- 47 \, T^{3} + 1554 \, T^{4} - 8214 \, T^{5} + 50653 T^{6} \\
\hline
\end{array}
$
\[prop1.1\] Let $\psi$ be the above Grössencharakter. Then, one has $L(C,s)=L(\psi,s)$.
For every prime ${{\mathfrak{p}}}$ in $I_K({\mbox{\gotip m}})$, let ${\mathbb F}_{{{\mathfrak{p}}}} = {\mathcal {O}}/{{\mathfrak{p}}}$ be the residue field of ${{\mathfrak{p}}}$ and consider the character $\chi_{{{\mathfrak{p}}}} \colon {\mathbb F}_{{{\mathfrak{p}}}}^* \to K^*$ such that $$\chi_{{{\mathfrak{p}}}}(x) \equiv x^{({\operatorname{N}}({{\mathfrak{p}}})-1)/9}
\,(\!\bmod \, {{\mathfrak{p}}})\,,$$ that we extend by $\chi_{{{\mathfrak{p}}}}(0)=0$. By Hasse [@Has], the Jacobi sum $${\operatorname{J}}({{\mathfrak{p}}}) :=
- \sum_{x\in {\mathbb F}_{{{\mathfrak{p}}}}} \chi_{{{\mathfrak{p}}}}^3(x) \chi_{{{\mathfrak{p}}}}(1-x)$$ is uniquely determined by the three properties:
- $|J({{\mathfrak{p}}})| = \sqrt{{\operatorname{N}}({{\mathfrak{p}}})}\,$;
- $J({{\mathfrak{p}}}) \equiv 1 \,(\!\bmod \,{\mbox{\gotip m}})$;
- $J({{\mathfrak{p}}}) \,{\mathcal {O}}= ({{\mathfrak{p}}}\cdot {{\mathfrak{p}}}^{\sigma_2^4} \cdot {{\mathfrak{p}}}^{\sigma_2^5})$.
One the one hand, it is easy to check that $\psi({{\mathfrak{p}}})$ satisfies (i), (ii), and (iii). On the other hand, Holzapfel and Nicolae [@Hol-Nic03] show that for a primer power $q$ such that $q\not\equiv 1 \,(\!\bmod \,9)$ one has $|C({\mathbb F}_q)| = q +1$, while for $q\equiv 1 \,(\!\bmod \, 9)$ it follows $$|C({\mathbb F}_{{{\mathfrak{p}}}})| = {\operatorname{N}}({{\mathfrak{p}}}) +1 - {\operatorname{Tr}}_{K/{\mathbb Q}}(J({{\mathfrak{p}}})) \,,$$ where ${{\mathfrak{p}}}$ is any prime ideal of the factorization of $q\mathcal{O}$. The claim follows.
The proof of the last equalities takes 4 pages in the referenced article [@Hol-Nic03]. We are grateful to Francesc Fité for a more concise proof included in the appendix of the present paper.
The Grössencharakter $\psi$ satisfies ${}^\sigma \psi({{\mathfrak{p}}}) = \psi({}^\sigma {{\mathfrak{p}}}) $ for every prime ideal ${{\mathfrak{p}}}$ and $\sigma \in {\operatorname{Gal}}(K/{\mathbb Q})$. The $L$-function of the curve $C$ over $K$ satisfies $$L(C_K,s) = \prod_{\sigma\in {\operatorname{Gal}}(K/{\mathbb Q})} L({\,}^\sigma\psi, s) = L(C,s)^6\,.$$ The CM-type of ${\operatorname{Jac}}(C)$ is $\Phi=\{ \sigma_2, \sigma_4, \sigma_8 \}$, i.e. the reflex of $\Phi^*$.
The Sato-Tate group $\operatorname{ST}(C)$
==========================================
For every prime $p\neq 3$, let us normalize the polynomials $$L_p^{\operatorname{ST}}(C,T) = L_p\left(C,\frac{T}{\sqrt{p}}\right)$$ and call them *normalized local factors* of $C$. Since they are monic, palindromic with real coefficients, roots lying in the unit circle and Galois stable, one can think of them as the characteristic polynomials of (conjugacy classes of) matrices in the unitary symplectic group $$\operatorname{USp}(6,{\mathbb C})
=\{ M \in {\mathrm{GL}}(6,{\mathbb C})\colon M^{-1}= J^{-1} M^{t} J = M^*\}\,,$$ where $M^*$ denotes the complex conjugate transpose of $M$, and $J$ denotes the skew-symmetric matrix $$J = \begin{pmatrix}
0 & 1 & 0 & 0 & 0 & 0 \\
-1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & -1 & 0 \\
\end{pmatrix} \,.$$ Roughly, the Sato-Tate group attached to $C$ is defined to be a compact subgroup $\operatorname{ST}(C) \subseteq \operatorname{USp}(6,{\mathbb C})$ such that the characteristic polynomials of the matrices in $\operatorname{ST}(C)$ [*fit well*]{} with the normalized local factors $L_p^{\operatorname{ST}}(C,T)$, in the sense that the normalized local factors $L_p^{\operatorname{ST}}(C,T)$, as $p$ varies, are equidistributed with respect to the Haar measure of $\operatorname{ST}(C)$ projected on the set of its conjugacy classes.
In analogy with Galois theory, the presence of some extra structure on $C$ gives rise to proper subgroups of the symplectic group; moreover, the distribution of $L_p^{\operatorname{ST}}(C,T)$ can be viewed as a generalization of the classical Chebotarev distribution. Serre [@Ser12] proposes a vast generalization of the Sato-Tate conjecture (born for elliptic curves) giving a precise recipe for $\operatorname{ST}(C)$. In this section, we calculate the Sato-Tate group $\operatorname{ST}(C)$ for our Picard curve $C$.
\[STgroup\] Up to conjugation in $\operatorname{USp}(6,{\mathbb C})$, the Sato-Tate group of $C$ is $$\operatorname{ST}(C) = \left\langle \begin{pmatrix}
u_1&&&&&\\
&\bar{u}_1&&&&\\
&&u_2&&&\\
&&&\bar{u}_2&&\\
&&&&u_3&\\
&&&&&\bar{u}_3\\
\end{pmatrix},
\begin{pmatrix}
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 \\
0 & -1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}, |u_i|=1 \right\rangle$$ In particular, there is an isomorphism $\operatorname{ST}(C) \simeq {U}(1)^3 \rtimes
({\mathbb Z}/9{\mathbb Z})^*$.
The recipe of Serre in [@Ser12] is as follows. Fix an auxiliary prime $\ell$ of good reduction (say $\ell>3$), and fix an embedding $\iota \colon {\mathbb Q}_\ell \hookrightarrow {\mathbb C}$. Let $$\rho_\ell \colon {{\operatorname{Gal}}({{\overline{\mathbb Q}}}/{\mathbb Q})}\to {\mathrm{GL}}( V_\ell({\operatorname{Jac}}(C)))\simeq {\mathrm{GL}}(6,{\mathbb Q}_\ell)$$ be the $\ell$-adic Galois representation attached to the $\ell$-adic Tate module of the Jacobian variety of $C$. Denote by $G$ the Zariski closure of the image $\rho_\ell({{\operatorname{Gal}}({{\overline{\mathbb Q}}}/{\mathbb Q})})$, and let $G_1$ be the Zariski closure of $G\cap \operatorname{Sp}_6({\mathbb Q}_\ell)$, where $\operatorname{Sp}_6$ denotes the symplectic group. By definition, the Sato-Tate group $\operatorname{ST}(C)$ is a maximal compact subgroup of $G_1\otimes_\iota {\mathbb C}$. In general, one hopes that this construction does not depend on $\ell$ and $\iota$, and this is the case for our Picard curve $C$. Indeed, since the CM-type of ${\operatorname{Jac}}(C)$ is non-degenerate then the twisted Lefschetz group $\operatorname{TL}(C)$ satisfies $G_1 = \operatorname{TL}(C)\otimes {\mathbb Q}_\ell$ for all primes $\ell$ (see [@FGL14 Lemma 3.5]). Recall that the twisted Lefschetz group is defined as $$\operatorname{TL}(C) = \bigcup_{\tau\in {{\operatorname{Gal}}({{\overline{\mathbb Q}}}/{\mathbb Q})}} \operatorname{L}(C)(\tau)\,,$$ where $\operatorname{L}(C)(\tau) = \{ \gamma \in \operatorname{Sp}_6({\mathbb Q}) \colon
\gamma \alpha \gamma^{-1} = \tau(\alpha) \text{ for all } \alpha \in {\operatorname{End}}({\operatorname{Jac}}(C)_{{{\overline{\mathbb Q}}}})\otimes {\mathbb Q}\}$, where ${\operatorname{Jac}}(C)_{{\overline{\mathbb Q}}}$ denotes the base change to ${{\overline{\mathbb Q}}}$. Here, $\alpha$ is seen as an endomorphism of $H_1({\operatorname{Jac}}(C)_{{\mathbb C}} ,{\mathbb Q}) $. The reason why the CM-type of ${\operatorname{Jac}}(C)$ is non-degenerate is due to the fact that $\Phi^*$ is simple and $\dim {\operatorname{Jac}}(C)=3$ (see [@K; @R]); alternatively, one checks that the ${\mathbb Z}$-linear map: $${\mathbb Z}[{\operatorname{Gal}}(K/{\mathbb Q})] \to {\mathbb Z}[{\operatorname{Gal}}(K/{\mathbb Q}) ], \qquad \sigma_a
\mapsto \sum_{\sigma_b\in \Phi} \sigma_{b}^{-1}\sigma_a$$ has maximal rank $1+\dim({\operatorname{Jac}}(C))=4$. Then, by combining [@BGK03] and [@FKRS12 Thm.2.16(a)], it follows that the connected component of the identity $\operatorname{TL}(C)^0$ satisfies $$G_1^0 = \operatorname{TL}(C)^0 \otimes {\mathbb Q}_\ell
= \left\{ \operatorname{diag}(x_1,y_1,x_2,y_2,x_3,y_3) \mid x_i,y_i\in {\mathbb Q}_\ell^*\,,
x_i y_i = 1 \right\}\,.$$ Thus, the connected component of the Sato-Tate group for $C$ is equal to $$\operatorname{ST}(C)^0 =
\left\{
\operatorname{diag}(u_1,\overline{u}_1,u_2,
\overline{u}_2,u_3,\overline{u}_3)
\colon u_i \in U(1)
\right\} \simeq U(1)^3\,.$$ According to [@FKRS12 Prop. 2.17], it also follows that the group of components of $\operatorname{ST}(C)$ is isomorphic to ${\operatorname{Gal}}(K/{\mathbb Q})$. We claim that $
\operatorname{ST}(C) =
\operatorname{ST}(C)^0 \rtimes \langle \gamma \rangle
$, where $$\gamma =
\begin{pmatrix}
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 \\
0 & -1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}
\,.$$ To this end, we consider the automorphism of the Picard curve $C$ determined by $\alpha(x,y)=(\zeta^6x,\zeta^2\,y)$. We still denote by $\alpha$ the induced endomorphism of ${\operatorname{Jac}}(C)$. Under the basis of regular differentials of $\Omega^1(C)$:
$$\omega_1 = \frac{dx}{y^2} \,,\quad
\omega_2 = \frac{dx}{y} \,,\quad
\omega_3 = \frac{xdx}{y^2} \,,$$ the action induced is given by $\alpha^*(\omega_1)= \zeta^2 \omega_1$, $\alpha^*(\omega_2)= \zeta^4 \omega_2$, $\alpha^*(\omega_3)= \zeta^8 \omega_3$. By taking the symplectic basis of $H_1({\operatorname{Jac}}(C)_{\mathbb C},{\mathbb C})$ corresponding to the above basis (with respect to the skew-symmetric matrix $J$), we get the matrix $$\alpha =
\begin{pmatrix}
\zeta^2 & 0 & 0 & 0 & 0 & 0 \\
0 & \overline{\zeta}^2 & 0 & 0 & 0 & 0 \\
0 & 0 & \zeta^4 & 0 & 0 & 0 \\
0 & 0 & 0 & \overline{\zeta}^4 & 0 & 0 \\
0 & 0 & 0 & 0 & \zeta^8 & 0 \\
0 & 0 & 0 & 0 & 0 & \overline{\zeta}^8 \\
\end{pmatrix} \,.$$ One checks that the matrix $\gamma$ satisfies $$\gamma \alpha \gamma^{-1} = {\,}^{\sigma_2} \alpha \,,$$ which implies that $\gamma\in \operatorname{TL}(\sigma_2)$. Hence, $\gamma$ belongs to $\operatorname{ST}(C)$; finally, a short computations shows that $\gamma^6=-\operatorname{Id} \in \operatorname{ST}(C)^0$, but $\gamma^i$ is not in $\operatorname{ST}(C)^0$ for $1\leq i < 6$.
0.3truecm
\[Poly-shape\] For future use, we compute the shape of the characteristic polynomials in each component of the Sato-Tate group. To this end, we take a random matrix $\operatorname{diag}(u_1,\overline{u}_1,u_2,\overline{u}_2,u_3,\overline{u}_3)$ in the connected component $\operatorname{ST}^0(C)$, and we get:
$$\begin{array}{l@{\, :\quad} l}
\operatorname{ST}^0(C)\cdot \operatorname{Id} & \prod_{i=1}^3 (T-u_i)(T-\overline{u}_i) \\[3pt]
{\operatorname{ST}}^0(C)\cdot \gamma & T^6+1 \\[3pt]
{\operatorname{ST}}^0(C)\cdot \gamma^2 & T^{6} + \left(u_{1} \overline{u}_2 u_{3} + \overline{u}_1 u_{2}
\overline{u}_3 \right) T^{3} + 1 \\[3pt]
{\operatorname{ST}}^0(C)\cdot \gamma^3 & {\left(T^{2} + 1\right)}^{3} \\[3pt]
{\operatorname{ST}}^0(C)\cdot \gamma^4 & T^{6} - \left(u_{1} \overline{u}_2 u_{3} + \overline{u}_1 u_{2}
\overline{u}_3 \right) T^{3} + 1 \\[3pt]
{\operatorname{ST}}^0(C)\cdot \gamma^5 & T^6+1 \,.\\[3pt]
\end{array}$$
\[Remark\] As a consequence of [@FKRS12 Prop. 2.17], we also obtain that, for every subextension $K/K'/{\mathbb Q}$, one has ${\operatorname{ST}}(C_{K'}) = {\operatorname{ST}}(C)^0 \rtimes \langle \gamma^{[K':{\mathbb Q}]} \rangle$, where $C_{K'}$ denotes the base change $C\times_{\mathbb Q}K'$.
Sato-Tate distribution
======================
A general strategy to prove the expected distribution is due to Serre [@Ser68]. For every non-trivial irreducible representation $\phi\colon \operatorname{ST}(C) \to {\mathrm{GL}}_m({\mathbb C})$, one needs to consider the $L$-function $$L(\phi, s) = \prod_{p\neq 3} {\det(1-\phi(x_p) p^{-s} ) }^{-1}\,,$$ where $x_p = \frac{1}{\sqrt{p}}
\rho_{\ell}({\operatorname{Frob}}_p) \in
\operatorname{ST}(C)$, and then show that $L(\phi, s)$ is invertible, in the sense that it has meromorphic continuation to $\operatorname{Re}(s)\geq 1$ and it holds $$L(\phi, 1) \neq 0\,.$$
\[prop3.1\] The Picard curve $C\colon y^3 = x^4 -x$ satisfies the generalized Sato-Tate conjecture. More explicitly, the sequence $$\left\{
\left( \frac{{\,}^{\sigma_2}\psi({{\mathfrak{p}}})}{\sqrt{N({{\mathfrak{p}}})} }
, \frac{{\,}^{\sigma_4}\psi({{\mathfrak{p}}})}{\sqrt{N({{\mathfrak{p}}})} } ,
\frac{{\,}^{\sigma_8}\psi({{\mathfrak{p}}})}{\sqrt{N({{\mathfrak{p}}})} },\, p
\right) \right\}_{p\neq 3} \subseteq
U(1)^3 \rtimes ({\mathbb Z}/9{\mathbb Z})^* \simeq {\operatorname{ST}}(C)\,,$$ where ${{\mathfrak{p}}}$ is any prime ideal of the factorization of $p\mathcal{O}$, is equidistributed over $U(1)^3 \rtimes ({\mathbb Z}/9{\mathbb Z})^*$ with respect to the Haar measure.
The irreducible representations of $\operatorname{ST}(C)\simeq U(1)^3 \rtimes ({\mathbb Z}/9{\mathbb Z})^*$ can be described as follows (see [@Ser77 §8.2]). For every triple $\underline{b}=(b_1,b_2,b_3)$ in ${\mathbb Z}^3$, we consider the irreducible character of $U(1)^3$ given by $$\phi_{\underline{b}} \colon U(1)^3 \to {\mathbb C}^* \,,\quad
\phi_{\underline{b}}(u_1,u_2,u_3) = \prod_{i=1}^3 u_i^{b_i}\,,$$ and let $$H_{\underline{b}} =
\{ h \in ({\mathbb Z}/9{\mathbb Z})^* \colon
\phi_{\underline{b}}(u_1,u_2,u_3) =
\phi_{\underline{b}}({\,}^h (u_1,u_2,u_3)) \}\,.$$ The action of $({\mathbb Z}/9{\mathbb Z})^*$ on $U(1)^3$ is given by conjugation through powers of the matrix $\gamma$; more precisely, for the generator $g=2$ of $({\mathbb Z}/9{\mathbb Z})^*$ we have ${\,}^g (u_1,u_2,u_3) = (u_2,u_3,\overline{u}_1)$ since $$\gamma
\begin{pmatrix}
u_1&&&&&\\
&\bar{u}_1&&&&\\
&&u_2&&&\\
&&&\bar{u}_2&&\\
&&&&u_3&\\
&&&&&\bar{u}_3\\
\end{pmatrix} \gamma^{-1}
=
\begin{pmatrix}
u_2&&&&&\\
&\bar{u}_2&&&&\\
&&u_3&&&\\
&&&\bar{u}_3&&\\
&&&& \bar{u}_1 &\\
&&&&& u_1\\
\end{pmatrix}
\,.$$ An easy computation shows that $H_{\underline{b}}=
\langle 2 \rangle$ or $\langle 2^3 \rangle$ if and only if $\underline{b}=(0,0,0)$, while $H_{\underline{b}} = \langle 2^2 \rangle$ for $\underline{b}=(b_1,-b_1,b_1)$ with $b_1\neq 0$, and $H_{\underline{b}}$ is trivial otherwise. Then, one has that $$\phi_{\underline{b}}(u_1,u_2,u_3,h) = \prod_{i=1}^3 u_i^{b_i}$$ is a character of $H:=U(1)^3 \rtimes H_{\underline{b}}$. By [@Ser77 Prop. 25] every irreducible representation of $G:=U(1)^3 \rtimes ({\mathbb Z}/9{\mathbb Z})^*$ is of the form $\theta := {\operatorname{Ind}}_H^G ( \phi_{\underline{b}}\otimes \chi )$, where $\chi$ is a character of $H_{\underline{b}}$ that may be viewed as a character of $H$ by composing with the projection $H \to H_{\underline{b}}$.
Let $\theta={\operatorname{Ind}}_H^G ( \phi_{\underline{b}}\otimes \chi )$ be an irreducible representation of $U(1)^3 \rtimes ({\mathbb Z}/9{\mathbb Z})^*$ as above. If we denote the sequence by $$x_{p} =
\left( \frac{{\,}^{\sigma_2}\psi({{\mathfrak{p}}})}{\sqrt{N({{\mathfrak{p}}})} }
, \frac{{\,}^{\sigma_4}\psi({{\mathfrak{p}}})}{\sqrt{N({{\mathfrak{p}}})} } ,
\frac{{\,}^{\sigma_8}\psi({{\mathfrak{p}}})}{\sqrt{N({{\mathfrak{p}}})} },\, p
\right) \in U(1)^3 \rtimes ({\mathbb Z}/9{\mathbb Z})^*$$ where ${{\mathfrak{p}}}$ is any prime ideal of the factorization of $p\mathcal{O}$, our claim is equivalent to show that the corresponding $L$-function $$L(\theta, s) =
\prod_{p \neq 3}
(1-\det (\theta(x_{p})) p^{-s})^{-1}$$ is invertible provided that $(b_1,b_2,b_3)
\neq (0,0,0) $. Assume first that $H_{\underline{b}}$ is trivial. Then, also $\chi$ is trivial and one has $$L(\theta,s) = L(\phi_{\underline{b}},s) =
\prod_{p\neq 3} \left(
1- \frac{{\,}^{\sigma_2}\psi({{\mathfrak{p}}})^{b_1}
{\,}^{\sigma_4}\psi({{\mathfrak{p}}})^{b_2}
{\,}^{\sigma_8}\psi({{\mathfrak{p}}})^{b_3}}
{\sqrt{{\operatorname{N}}({{\mathfrak{p}}})}^{b_1+b_2+b_3}} \right)\,.$$ This can be seen as the $L$-function of the unitarized Grössencharakter $$\Psi := \frac{
{\,}^{\sigma_2}\psi(\cdot)^{b_1}
{\,}^{\sigma_4}\psi(\cdot)^{b_2}
{\,}^{\sigma_8}\psi(\cdot)^{b_3} }
{{{\operatorname{N}}(\cdot)}^{(b_1+b_2+b_3)/2}}$$ Under our assumption $(b_1,b_2,b_3)
\neq (0,0,0) $ and by using the factorization of $\psi({{\mathfrak{p}}}){\mathcal {O}}$ into prime ideals (see property (iii) in the proof of Proposition \[prop1.1\]), an easy computation shows that $\Psi$ is non-trivial. Hecke showed [@He20] that the $L$-function of a non-trivial unitarized Grössencharkter is holomorphic and non-vanishing for ${\operatorname{Re}}(s)\geq 1$. In the remaining case, that is for $H_{\underline{b}}$ of order $3$, one gets $L(\Psi,s)=L(\theta,s)^3$ and the claim also follows by the same argument.
0.3truecm
The moment sequences
====================
In this section we will compute the moment sequences in two independent ways, one (exact) from the Sato-Tate group and the other one (numerically) by computing the local factors of our curve up to some bound.
Let $\mu$ be a positive measure on $I = [-d,d]$. Then, on the one hand, for every integer $n\geq 0$, the $n$th moment $M_n[\mu]$ is by definition $\mu(\varphi_n)$, where $\varphi_n$ is the function $z \mapsto
z^n$. That is, we have $$M_n[\mu] = \int_{I} \, z^n\mu(z)$$ The measure $\mu$ is *uniquely* determined by its moment sequence $M_n[\mu]$.
On the other hand, if a sequence $\{a(p)\}_p$ is $\mu$-equidistributed, then the following equality holds: $$M_n[\mu] = \lim_{x\to\infty} \frac{1}{\pi(x)}\sum_{p\leq x} a_p^n \,.$$
From now on, we shall denote by $a_1(p)$, $a_2(p)$, $a_3(p)$ the [*higher*]{} traces according to $$L_p^{\operatorname{ST}}(C,T)=
1+ a_1(p)T+ a_2(p) T^2+ a_3(p) T^3 +a_2(p)T^4
+a_1(p)T^5+T^6 \,.$$
Recall that due to the Weil’s conjectures, we know that $$a_1(p) \in [-6,6]\,,\quad
a_2(p) \in [-15,15]\,,\quad
a_3(p) \in [-20,20]\,.$$
The distribution of $ST(C)$
---------------------------
For each $i$ in $\{ 1,2,3 \}$, let $\mu_i$ denote the projection on the interval $I_i=[-\binom{6}{i},+\binom{6}{i} ]$ obtained from the Haar measure of the Sato-Tate group $\operatorname{ST(C)}\simeq {U}(1)^3 \rtimes
({\mathbb Z}/9{\mathbb Z})^*$.
In general it is difficult to obtain the explicit distribution function, but because of the isomorphism stated in Proposition \[STgroup\], we can easily compute the moment sequence of the Sato-Tate measure.
Similarly as in [@FGL14] we shall split each measure as a sum of its restrictions to each component of ${\operatorname{ST}}(C)^0 \cdot \gamma^k$, where $0\leq k \leq 5$.
Therefore one has $$\mu_i = \frac{1}{6} \sum_{0\leq k \leq 5} {\,}^k\mu_{i} \,, \quad M_n[\mu_i] = \frac{1}{6}\sum_{0\leq k \leq 5} M_n[{\,}^k\mu_{i}]$$ so we can compute the moments $M_n[{\,}^k\mu_{i}]$ separately for every $k$ and then get the *total* moments $M_n[\mu_{i}]$. To ease notation, we shall denote the moment sequences by $$M[\mu_i]:= ( M_0[\mu_i], M_1[\mu_i], M_1[\mu_i], \dots , M_n[\mu_i], \dots ) \,,$$ and similarly for every $M[{\,}^k\mu_{i}]$.
In what follows, the characteristic polynomial of a matrix in ${\operatorname{USp}}(6)$ we will be denoted by $$P(T)=
1+ a_1T+ a_2 T^2+ a_3 T^3 +a_2T^4+a_1T^5+T^6 \,.$$
Case $k=1,5$: In these components, according to Remark \[Poly-shape\] one has that $P(T) = T^6+1$, so that $$a_1 = a_2 = a_3=~0\,.$$ Hence, $$M_n[{\,}^k\mu_{1}] = M_n[{\,}^k\mu_{2}] = M_n[{\,}^k\mu_{3}] = 0 \,
\text{ for all }
n\geq 1\,.$$
Case $k=2,4$: In these components, we have $$P(T) = T^6 \pm (u_1\overline{u}_2u_3 + \overline{u}_1u_2\overline{u}_3)T^3 + 1 \,.$$ So that $a_1 = a_2 = 0$. Hence, it follows that $$M_n[{\,}^k\mu_{1}] = M_n[{\,}^k\mu_{2}] = 0 \text{ for all }
n\geq 1\,.$$ To get the distribution of the third trace, since $u_1$, $u_2$, and $u_3$ are independent elements of ${\operatorname{U}}(1)$, the distribution of $a_3(p)$ will correspond to the distribution of $\alpha := u + \overline{u}$ for $u\in{\operatorname{U}}(1)$, and hence its associated moment sequence is $$M[{\,}^k\mu_{3}] = (1,0,2,0,6,0,20,0,\dots)\,.$$
Case $k=3$: In this case, one has $P(T)= (1 + T^2)^3$, so that we have $a_1 = a_3 = 0$, while $a_2 = 3$. Hence, we obtain $$M_n[{\,}^3\mu_{1}] = M_n[{\,}^3\mu_{3}] = 0\,,\
M_n[{\,}^3\mu_{2}] = 3^n \ \text{ for all }
n\geq 1\,.$$
Case $k=0$: In this case one has that $P(T) = \prod_{i=1}^3 (T-u_i)(T-\overline{u}_i)$. If we develop this expression we get the following coefficients, where as above $\alpha_i$ stands for the sum of $u_i$ and its complex conjugate: $$\label{aidealfa}
\begin{array}{l@{\,=\,}l}
a_1 & \alpha_1+\alpha_2+\alpha_3\,,\\[3pt]
a_2 & 3 + \alpha_1\alpha_2 + \alpha_2\alpha_3 + \alpha_1\alpha_3\,,\\[3pt]
a_3 & 2\,\alpha_1 + 2\,\alpha_2 + 2\,\alpha_3 + \alpha_1\alpha_2\alpha_3.
\end{array}$$
To get the sequences we proceed as follows. Recall that if $X$ and $Y$ denote independent random variables, then $M_n[X] = E(X^n)$, $E(X+Y) = E(X)+E(Y)$, and $E(XY) = E(X)E(Y)$. Hence, one has $$\begin{aligned}
M_n[X+Y] = E((X+Y)^n) =& E\left(\sum_{k=0}^n\binom{n}{k}X^kY^{n-k}\right)\\
=& \sum_{k=0}^n\binom{n}{k}E(X^k)E(Y^{n-k})\\ =& \sum_{k=0}^n\binom{n}{k}M_k[X]M_{n-k}[Y]\,.\end{aligned}$$
Since we know that $M[\alpha]:=M[\alpha_i] = (1,0,2,0,6,0,20,0,\dots)$ for $i=1,2,3$, one gets: $$\begin{aligned}
M_n[{\,}^0 \mu_{1}] =& \sum_{a+b+c=n}\binom{n}{a,b,c}M_a[\alpha]M_b[\alpha]M_c[\alpha],\\
M_n[{\,}^0\mu_{2}] =& \sum_{a+b+c+d=n}\binom{n}{a,b,c,d}3^aM_{b+d}[\alpha]M_{b+c}[\alpha]M_{c+d}[\alpha]\,,\\
M_n[{\,}^0\mu_{3}] =& \sum_{a+b+c+d=n}\binom{n}{a,b,c,d}2^{a+b+c}M_{a+d}[\alpha]M_{b+d}[\alpha]M_{c+d}[\alpha].\end{aligned}$$
Therefore we obtain the sequences: $$\begin{aligned}
M[{\,}^0\mu_{1}] =& (1,0,6,0,90,0,1860,\dots),\\
M[{\,}^0\mu_{2}] =& (1,3,21,183,1845,\dots),\\
M[{\,}^0\mu_{3}] =& (1,0,32,0,4920,0,1109120,\dots).\end{aligned}$$
We can summarize the above results in the following proposition.
With the above notations, the first moments of the measures of ${\,}^k\mu_i$ and $\mu_i$ are as follows:
- [The moments of the first trace are:]{} $$M[{\,}^k\mu_1] = \begin{cases}
(1,0,0,\dots) &\text{if } k = 1,\dots,5\,;\\
(1,0,6,0,90,0,1860,\dots) & \text{if } k = 0\,.
\end{cases}$$
Hence, $M[\mu_1] = (1,0,1,0,15,0,310,\dots)$.
- [The moments of the second trace are:]{} $$M[{\,}^k\mu_2] = \begin{cases}
(1,0,0,\dots) &\text{if } k = 1,2,4,5\,;\\
(1,3,9,27,\dots) &\text{if } k = 3\,;\\
(1,3,21,183,1845,\dots) &\text{if } k = 0\,.
\end{cases}$$
Hence, $M[\mu_2] = (1,1,5,35,321,\dots)$.
- [The moments of the third trace are:]{} $$M[{\,}^k\mu_3] = \begin{cases}
(1,0,0,\dots) &\text{if } k = 1,3,5\,;\\
(1,0,2,0,6,0,20,0,\dots) &\text{if } k = 2,4\,;\\
(1,0,32,0,4920,0,1109120,\dots) &\text{if } k = 0\,.
\end{cases}$$
Hence, $M[\mu_3] = (1,0,6,0,822,0,184860,0\dots)$.
The numerical sequences for $C$\[numeric\]
------------------------------------------
Once we have computed the theoretical moment sequences from the Sato-Tate group ${\operatorname{ST}}(C)$, we wish to compute for every prime (up to some bound) its associated normalized local factor $L_p^{{\operatorname{ST}}}(A,T)$ to get the corresponding traces $a_1(p), \, a_2(p)$ and $a_3(p)$ and do the experimental equidistribution matching.
The Grössencharakter $\psi$ attached to the Picard curve $C$ permits us to perform this numerical experimentation within a reasonable time, in this case $p\leq 2^{26}$ (about two hours of a standard laptop). We display the data obtained:
$$\begin{array}{rcrrcrrcrr}
\hline
&&\multicolumn{2}{c}{a_1}&&\multicolumn{2}{c}{a_2}&&\multicolumn{2}{c}{a_3}\\
n&&M_n[\mu_1]&M_n[\mu_1]_{\leq 2^{26}}&&M_n[\mu_2]&M_n[\mu_2]_{\leq 2^{26}}&&M_n[\mu_3]&M_n[\mu_3]_{\leq 2^{26}}\\
\hline
0&&1&1&&1&1&&1&1\\
1&&0&-0.000&&1&0.999&&0&-0.000\\
2&&1&0.998&&5&4.991&&6&5.984\\
3&&0&-0.005&&35&34.868&&0&-0.147\\
4&&15&14.946&&321&319.058&&822&815.937\\
5&&0&-0.151&&&&&&\\
6&&310&308.160&&&&&&\\
\hline
\end{array}$$
We include graphics to display the histograms (for primes up to $p\leq 2^{26}$) showing the nondiscret components of the three distributions $\mu_i$.
![Histogram of the first trace for primes $p\equiv 1 \, (\!\bmod 9)$.[]{data-label="fig:digraph1"}](tr1_p1.eps)
![Histogram of the second trace for primes $p\equiv 1 \, (\!\bmod 9)$.[]{data-label="fig:digraph2"}](tr2_p1.eps)
![Histogram of the third trace for primes $p\equiv 1 \, (\!\bmod 9)$.[]{data-label="fig:digraph3"}](tr3_p1.eps)
![Histogram of the third trace for primes $p\equiv 4,7 \, (\!\bmod 9)$.[]{data-label="fig:digraph4"}](tr3_p47.eps)
Appendix (by F. Fité) {#appendix-by-f.-fité .unnumbered}
=====================
We keep the notation of the article. Let $C\colon y^ 3=x^ 4-x$. Let $K$ denote the cyclotomic field ${\mathbb Q}(\zeta)$, where $\zeta$ is a 9th root of unity. For every prime ${\mathfrak{p}}$ of $K$ coprime to 3, consider the character $\chi_{\mathfrak{p}}\colon {\mathbb F}_{\mathfrak{p}}^*\rightarrow K^*$ such that $\chi_{\mathfrak{p}}(x)$ is the only 9th root of unity satisfying $$\chi_{\mathfrak{p}}(x)\equiv x^{(N({\mathfrak{p}})-1)/9}\pmod {\mathfrak{p}}\,.$$ For $a,b\in {\mathbb Z}/9{\mathbb Z}$, define $$J_{(a,b)}({\mathfrak{p}}):=\sum_{x\in{\mathbb F}_{\mathfrak{p}}}\chi_{\mathfrak{p}}^a(x)\chi^b_{\mathfrak{p}}(1-x)\,.$$
The number of points of $ C$ defined over the finite field ${\mathbb F}_{\mathfrak{p}}$ is $$| C({\mathbb F}_{\mathfrak{p}})|=
\begin{cases}
1 + N({\mathfrak{p}}) & \text{if $N({\mathfrak{p}}) \not\equiv 1 \pmod 9$,}\qquad \text{(i)}\\
1+N({\mathfrak{p}})+ {\operatorname{Tr}}_{K/{\mathbb Q}}(J_{(6,1)}({\mathfrak{p}})) & \text{if $N({\mathfrak{p}}) \equiv 1 \pmod 9$.}\qquad \text{(ii)}\\
\end{cases}$$
Case $(i)$ is considered in Proposition 1 and Proposition 2 of [@HN02]. We now show case (ii), by giving an alternative and shorter proof of Proposition 3 of [@HN02]. Let $
C'\colon v^ 9=u(u+1)^ 6$. There is an isomorphism between $ C$ and $ C'$ given by $$\phi\colon C \rightarrow C'\,, \qquad \phi(x,y)=\left(-\frac{1}{x^3},-\frac{y^ 2}{x^3} \right)\,.$$ One easily sees that the inverse of $\phi$ is given by $$\phi^{-1}\colon C' \rightarrow C\,, \qquad \phi^{-1}(u,v)=\left(-\frac{(u+1)^2}{v^3},-\frac{(u+1)^3}{v^4} \right)\,.$$ Note that if $N({\mathfrak{p}}) \not\equiv 1 \pmod 3$, then exponentiation by 9 is an isomorphism of ${\mathbb F}_{\mathfrak{p}}$. Thus $ C'$ has $N({\mathfrak{p}})$ affine points plus one point at infinity. Assume now that $N({\mathfrak{p}}) \equiv 1 \pmod 9$. By [@IR90 Prop. 8.1.5], we have that $$\begin{aligned}
| C'({\mathbb F}_{\mathfrak{p}})| &=1+\sum_{u\in{\mathbb F}_{\mathfrak{p}}}\sum_{a\in{\mathbb Z}/9{\mathbb Z}}\chi_{\mathfrak{p}}^a(u)\chi_{\mathfrak{p}}^{6a}(u+1)\\
&=1+N({\mathfrak{p}})+\sum_{a\in({\mathbb Z}/9{\mathbb Z})^*}\sum_{u\in{\mathbb F}_{\mathfrak{p}}}\chi_{\mathfrak{p}}^a(u)\chi_{\mathfrak{p}}^{6a}(u+1)\,,
\end{aligned}$$ where for the second equality we have used [@IR90 Thm. 1 (b), p. 93]. But writing $x=u+1$, we obtain $$\sum_{u\in{\mathbb F}_{\mathfrak{p}}}\chi_{\mathfrak{p}}(u)\chi_{\mathfrak{p}}^{6}(u+1)=\chi_{\mathfrak{p}}(-1)\sum_{x\in{\mathbb F}_{\mathfrak{p}}}\chi_{\mathfrak{p}}^{6}(x)\chi_{\mathfrak{p}}(1-x)\,.$$ Case $(ii)$ of the proposition is a consequence of the equality $\chi_{\mathfrak{p}}(-1)=1$ (this follows form the fact that the order of $\chi_{\mathfrak{p}}$ is odd).
To show that our result agrees with Proposition 3 of [@HN02] it remains to show that $
{\operatorname{Tr}}_{K/{\mathbb Q}}(J_{(6,1)}({\mathfrak{p}}))={\operatorname{Tr}}_{K/{\mathbb Q}}(J_{(3,1)}({\mathfrak{p}}))\,.
$ Indeed, by [@BEW98 Thm. 2.1.5], one has $$J_{(6,1)}({\mathfrak{p}})=J_{(2,1)}({\mathfrak{p}})\,,\qquad J_{(3,1)}({\mathfrak{p}})=J_{(5,1)}({\mathfrak{p}})\,.$$ Since $5\cdot 2\equiv 1 \pmod{9}$, we deduce that $${\operatorname{Tr}}_{K/{\mathbb Q}}(J_{(2,1)}({\mathfrak{p}}))={\operatorname{Tr}}_{K/{\mathbb Q}}(J_{(5,1)}({\mathfrak{p}}))\,.$$
Finally, note that $J({\mathfrak{p}})=-J_{(3,1)}({\mathfrak{p}})$ in the notation of the article.
[McK-Sta]{} B.C. Berndt, R.J. Evans, K.S. Williams, *Gauss and Jacobi sums*, John Wiley, Canada, 1998.
R-P. Holzapfel, F. Nicolae *Arithmetic on a family of Picard curves*, Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, Springer Verlag, Berlin Heidelberg, 2002.
K. Ireland, M. Rosen, *A Classical Introduction to Modern Number Theory*, Springer-Verlag, New York, 1990.
|
---
abstract: 'In this paper, we study the emergence of a Landauer transport regime from the quantum-mechanical dynamics of free electrons in a disordered tight-binding chain, which is coupled to finite leads with open boundaries. Both *partitioned* and *partition-free* initial conditions are analyzed and seen to give rise, for large enough leads, to the same spatially uniform quasi-steady-state current, which agrees with the Landauer value. The quasi-steady-state regime is preceded by a transient regime, which last for a time proportional to the length of the disordered sample, and followed by recursions, after a time that is proportional to the lead size. We also observe finite-size current oscillations, superimposed on the quasi-steady-state, whose behavior depends crucially on the conditions initially imposed on the system. Finally, we show how a time-resolved Kubo formula is able to reproduce this Landauer transport regime, as the leads grow bigger.'
author:
- 'J. P. Santos Pires'
- 'B. Amorim'
- 'J. M. Viana Parente Lopes'
bibliography:
- 'References\_NEW.bib'
title: 'Landauer transport as a quasi-steady-state on finite chains under unitary quantum dynamics'
---
\#1 \#1[|\#1|]{} \#1[\#1]{} \#1[|\#1]{} \#1[\#1|]{} \#1\#2[\#1\#2]{}
\[sec:Introduction\]Introduction
================================
The study of electronic transport is amongst the main goals of condensed matter physics. In the regime of small length scales and low temperatures, the mesoscopic transport regime, quantum coherence effects play a dominant role in the propagation of electron states. In such a case, transport can no longer be seen as a bulk phenomenon, but instead depends on device-specific details such as the geometry of and nature of the electrodes, as well as the specific distribution of disorder in the sample.
A theoretical description of mesoscopic transport was first developed by Landauer[@landauer_electrical_1970] and later generalized by Büttiker[@buttiker_four-terminal_1986]. In the now called, Landauer-Büttiker formalism, the problem of stationary mesoscopic transport is recast as a scattering problem, where single-electron states incoming from the leads are transmitted across a central device. The current may then be expressed as a sum over the transmission probabilities of the occupied incoming lead states. In parallel to this work, Caroli[@caroli_direct_1971] applied the non-equilibrium Green’s function formalism of Kadanoff-Baym[@Kadanoff_Baym_book] and Keldysh[@Keldysh_1964] to the calculation of mesoscopic transport. The obtained expression has a structure similar to the Landauer-Büttiker one, but with the transmission coefficient now expressed in terms of Green’s functions of the central device and spectral functions of the leads. While apparently distinct, the two approaches lead to the same result, as implied by the Fisher-Lee relation[@fisher_relation_1981; @stone_what_1988; @baranger_electrical_1989] between transmission coefficients and Green’s functions (for a detailed proof see Ref. [@wimmer_quantum_2009]). Central to both approaches are the assumptions that the leads attached to the central device are semi-infinite and that the occupation of the incoming single-electron states is determined by independent Fermi energies on each lead. Moreover, both methods are only able to describe steady-state transport.
If one is interested in the transient dynamics and how this steady-state is reached, the matter of what the initial condition of the system are, becomes relevant. At the theoretical level, two initial conditions have been historically considered: *(i)* In the *partitioned approach*[@caroli_direct_1971; @meir_landauer_1992; @jauho_time-dependent_1994], the leads and the central device are assumed to be initially disconnected, each being in equilibrium with independent Fermi-levels. This Fermi-level imbalance takes into account the applied bias. Then the leads and the device are suddenly brought into contact allowing a charge current to flow. *(ii)* In the *partition-free approach*[@Cini1980; @stefanucci_time-dependent_2004; @Odashima], the leads are assumed to be connected from the beginning, and in global equilibrium with a common Fermi energy. Then a potential bias between the leads is suddenly applied to the connected system. It has been shown that a steady-state transport regime is reached in both approaches and the current coincides in both cases[@stefanucci_time-dependent_2004]. Crucial to this result is the fact that the leads have a continuum spectrum (as occurs for semi-infinite leads), which leads to a loss of memory about the initial conditions.
In more recent years, a significant effort was devoted to the study of time-dependent transport and transient dynamics in mesoscopic systems attached to infinite leads[@jauho_time-dependent_1994; @Stefanucci_2004; @Tuovinen_2013; @tuovinen_time-dependent_2014; @Latini_2014; @popescu_efficient_2016; @popescu_emergence_2017]. In the work of Pal *et al* [@pal_emergence_2018], this assumption was relaxed and time-dependent transport through a quantum-dot connected to two systems with a quasi-continuum spectrum (discrete, but dense), which take the role of finite leads, was considered in the partitioned approach. It was found that even in this case, after transients died-out, a steady-state transport regime across the dot emerges. Previously, Di Ventra *et al*[@ventra_transport_2004; @bushong_approach_2005] also explored this possibility and developed a micro-canonical method to deal with quasi-steady-state transport in finite systems. However, many questions remained to be answered about which finite-size effects arise, the dependence of the current’s dynamics on the initial preparation of the system and also the precise conditions in which one expects the emergence of a transport quasi-steady-state in the absence of decoherence mechanisms.
The purpose of the present work is to further explore how a steady-state transport regime emerges from quantum time-evolution in systems with finite, but large leads with open boundaries. By combining numerical and analytical work, we study the time-dependent current dynamics in a prototypical one-dimensional non-interacting tight-binding model with disorder, analyzing in detail how the current dynamics depend on the initial conditions (*partitioned* vs *partition-free*) and on the size of the finite leads. We employ both a full quantum time-evolution, starting from both initial conditions, and a time-dependent Kubo formula, for the partition-free approach, to study the time-dependent current upon the sudden connection of the appropriate perturbation. The latter allows us to see rigorously how a linear Landauer-Büttiker formula, involving only quantum transmittances, emerges from an unitary time-evolution in the limit of very large leads.
Finally we remark that, besides its theoretical interest, the present paper may also offer predictions for future experiments. It is well established that the use of fermionic ultra-cold atoms trapped in optical lattices allows a very precise tuning of both the interactions and hopping parameters governing the particles’ motion [@ott_collisionally_2004; @rom_free_2006]. As discussed by Chien *et al*[@chien_bosonic_2012], such atomic gases can be prepared in controlled initial states, far away from thermodynamic equilibrium, but their subsequent dynamics must be studied by taking into account the finite number of particles in the gas. Therefore, the finite and closed nature of the leads we are considering, may actually be a relevant feature for the physics of such systems, thus rendering our “finite-size effects” potentially observable.
![\[Scheme\_of\_the\_system\]Scheme of the setup used to simulate the time-dependent LB transport using an one-dimensional sample coupled to finite leads. The red dots stand for the places where there is a disordered potential and the blue curve represents the profile of the externally applied potential. The chain has open boundary conditions. (color online)](fig1)
The text is organized as follows. In Sec. \[sec:Hamiltonian\], we introduce the one-dimensional tight-binding model Hamiltonian that will be used throughout the rest of the paper and detail both the *partitioned* and *partition-free* approaches. In Sec. \[subsec:Method\_TimeEvolution\], we describe the numerical methods used for calculating the time-dependent local current from the unitary dynamics of the finite system and also the steady-state Landauer current for infinite leads. The main numerical results are then presented in Sec. \[sec:NumericalResults\], where the time-evolution of the non-equilibrium current is systematically analyzed as a function of the bias, the size of the finite leads and the central sample’s disorder and size. Finally, in Sec. \[sec:Emergence-of-Landauer\], we provide analytical insight into the numerical results of Sec. \[sec:NumericalResults\], by developing a time-dependent Kubo formula for the *partition-free* approach and expressing it in terms of complex reflection and transmission coefficients of the central sample. In Sec. \[sec:Conclusions\], we discuss the obtained results and conclude the paper.
\[sec:Hamiltonian\]Model Hamiltonian and initial conditions
===========================================================
We will study the current dynamics of non-interacting electrons in a finite one-dimensional tight-binding model, with nearest neighbor hoppings. The tight-binding chain is composed by a total of $L$ sites, with the central $L_{s}$ sites, the sample, having an on-site Anderson disorder and being subject to a constant electric field. The sites outside the sample region form the left and right leads (each with $L_{l}=\left(L-L_{s}\right)/2$ sites), are not disordered and hold a constant electrostatic potential. The will refer to the different regions in the chain as Left Lead (LL), Sample (S) and Right Lead (RL). An illustrative scheme of this setup is shown in Fig. \[Scheme\_of\_the\_system\]. For times $t>0$, the dynamics of the system is governed by the time-independent Hamiltonian $$\begin{gathered}
\mathcal{H}\left(t>0\right)=\sum_{n=0}^{L-1}\left(\epsilon_{n}^{\text{d}}-ev_{n}^{\text{e}}\right)c_{n}^{\dagger}c_{n}\\
-w\sum_{n=0}^{L-2}\left(c_{n+1}^{\dagger}c_{n}+c_{n}^{\dagger}c_{n+1}\right),\label{eq:time-evolution_Hamiltonian}\end{gathered}$$ where $c_{n}^{\dagger}$$\left(c_{n}\right)$ are creation (annihilation) operators for an electron at the chain site $n$, $w$ is the nearest-neighbor hopping amplitude, $e>0$ is the fundamental charge and $v_{n}^{\text{e}}$ is the electrostatic potential. According to the previous discussion $v_{n}^{\text{e}}$ has the form $$v_{n}^{\text{e}}=\begin{cases}
\frac{\Delta V}{2} & ,\,n\in0,...,L_{l}-1\\
\left(\frac{1}{2}-\frac{n-L_{l}+1}{L_{s}+1}\right)\Delta V & ,\,L_{l}\leq n<L_{l}+L_{s}\\
-\frac{\Delta V}{2} & ,\,n\in L_{l}+L_{s},...,L
\end{cases},\label{PotentialProfile}$$ where $\Delta V$ is the applied potential bias, and $\epsilon_{n}^{\text{d}}$ is the Anderson on-site potential disorder, which is only present in the sample sites, and are taken as random numbers uniformly distributed inside $\left[-\frac{W}{2},\frac{W}{2}\right]$.
We will study the current dynamics in this system both in the *partitioned* and *partition-free* approaches. In both cases, the dynamics for $t>0$ are governed by the Hamiltonian of Eq. (\[eq:time-evolution\_Hamiltonian\]), with only the initial state being different.
In the *partitioned approach* the initial state if formed by occupied states for the partitioned system, with the bias already applied. The *partitioned system* is described by the Hamiltonian, $$\mathcal{H}^{\text{P}}(t=0)=\mathcal{H}_{\text{LL}}^{\text{P}}+\mathcal{H}_{\text{S}}^{\text{P}}+\mathcal{H}_{\text{RL}}^{\text{P}},\label{eq:partitioned_Hamiltonian}$$ with $\mathcal{H}_{\text{LL}}$, $\mathcal{H}_{\text{S}}$ and $\mathcal{H}_{\text{RL}}$ the Hamiltonians, respectively, for the decoupled left lead, sample and right lead, to wit $$\begin{aligned}
\mathcal{H}_{\text{LL}}^{\text{P}} & =\sum_{n=0}^{L_{l}-1}\left(-ev_{n}^{\text{e}}\right)c_{n}^{\dagger}c_{n}-w\sum_{n=0}^{L_{l}-2}\left(c_{n+1}^{\dagger}c_{n}+\text{h.c.}\right),\label{eq:partitioned_Hamiltonian_LL}\\
\mathcal{H}_{\text{S}}^{\text{P}} & =\;\qquad\mathllap{\sum_{n=L_{l}}^{L_{l}+L_{s}-1}}\qquad\quad\;\;\mathllap{\left(\epsilon_{n}^{\text{d}}-ev_{n}^{\text{e}}\right)}c_{n}^{\dagger}c_{n}-w\qquad\;\;\mathllap{\sum_{n=L_{l}}^{L_{l}+L_{s}-2}}\qquad\qquad\;\qquad\mathllap{\left(c_{n+1}^{\dagger}c_{n}+\text{h.c.}\right)},\label{eq:partitioned_Hamiltonian_S}\\
\mathcal{H}_{\text{RL}}^{\text{P}} & =\;\;\;\qquad\mathllap{\sum_{n=L_{l}+L_{s}}^{L-1}}\qquad\mathllap{\left(-ev_{n}^{\text{e}}\right)}c_{n}^{\dagger}c_{n}-w\qquad\;\;\mathllap{\sum_{n=L_{l}+L_{s}}^{L-2}}\qquad\qquad\;\qquad\mathllap{\left(c_{n+1}^{\dagger}c_{n}+\text{h.c.}\right)}.\label{eq:partitioned_Hamiltonian_RL}\end{aligned}$$ The occupation of the single-electron states is determined by the independent Fermi levels for each region. Hence, we write $\varepsilon_{\text{F},\text{LL}}=\varepsilon_{\text{F}}+\nicefrac{\Delta V}{2}$, $\varepsilon_{\text{F},\text{S}}=\varepsilon_{\text{F}}$ and $\varepsilon_{\text{F},\text{RL}}=\varepsilon_{\text{F}}-\nicefrac{\Delta V}{2}$, as the chemical potential for the left lead, central sample and right lead, respectively. $\varepsilon_{\text{F}}$ is a reference chemical potential. The initial state is thus described by the reduced density matrix $$\rho^{\text{P}}(t=0)=\sum_{r=\text{LL},\text{S},\text{RL}}\sum_{\alpha_{r}}f_{r,\alpha_{r}}^{\text{P}}\left|\Psi_{r,\alpha_{r}}^{\text{P}}\right\rangle \left\langle \Psi_{r,\alpha_{r}}^{\text{P}}\right|.\label{eq:initial_condition_partitioned}$$ where $\left|\Psi_{r,\alpha_{r}}^{\text{P}}\right\rangle $ are the independent single-electron eigenstates of the initial partitioned Hamiltonian belonging to region $r$, $\mathcal{H}_{r}^{\text{P}}$, with an energy $\varepsilon_{r,\alpha_{r}}^{\text{P}}$. At any temperature, the initial occupation of the states is given by the factor $f_{r,\alpha_{r}}^{\text{P}}=f\left(\varepsilon_{r,\alpha_{r}}^{\text{P}}-\varepsilon_{\text{F},r}\right)$, with $r=\text{LL},\text{S},\text{RL}$ and $f(\varepsilon)=\left(e^{\beta\varepsilon}+1\right)^{-1}$ being the Fermi-Dirac distribution function. Throughout this work, we will restrict ourselves to the $T=0$ case, where $f(\varepsilon)=\Theta\left(-\varepsilon\right)$ and $\Theta\left(x\right)$ being the usual Heaviside step-function. The hoppings between the leads and the sample are then suddenly switched on and the time-evolution of these states is generated by the Hamiltonian in Eq.(\[eq:time-evolution\_Hamiltonian\]).
In the *partition-free approach*, the contact between the sample and the leads is already established in the initial state, but the bias is not yet applied. Therefore, the initial condition is determined by populating the eigenstates of the *partition-free* Hamiltonian $$\mathcal{H}^{\text{PF}}(t=0)=\sum_{n=0}^{L-1}\epsilon_{n}^{\text{d}}c_{n}^{\dagger}c_{n}-w\sum_{n=0}^{L-2}\left(c_{n+1}^{\dagger}c_{n}+\text{h.c.}\right),\label{eq:partition-free_Hamiltonian}$$ up to a commonly defined Fermi energy, $\epsilon_{\text{F}}$. This initial state is thus described by the reduced density matrix $$\rho^{\text{PF}}(t=0)=\sum_{\alpha}f_{\alpha}^{\text{PF}}\left|\Psi_{\alpha}^{\text{PF}}\right\rangle \left\langle \Psi_{\alpha}^{\text{PF}}\right|,\label{eq:initial_condition_partition-free}$$ with $\left|\Psi_{\alpha}^{\text{PF}}\right\rangle $ being eigenstates of Eq. (\[eq:partition-free\_Hamiltonian\]) with an eigenenergy $\varepsilon_{\alpha}^{\text{PF}}$. The occupation factor of the states is similarly given by $f_{\alpha}^{\text{PF}}=f\left(\varepsilon_{\alpha}^{\text{PF}}-\varepsilon_{\text{F}}\right)$. In this case, the sudden perturbation driving the current is the connection of the bias potential, $v_{n}^{\text{e}}$, at $t=0$, after which the time-evolution is governed by the Hamiltonian of Eq.(\[eq:time-evolution\_Hamiltonian\]).
We end this section, by noting that the charge current flowing from site $n$ to site $n+1$, for the Hamiltonian of Eq. (\[eq:time-evolution\_Hamiltonian\]), is given by $$\mathcal{I}^{n}=\frac{ew}{i\hbar}\left(c_{n+1}^{\dagger}c_{n}-c_{n}^{\dagger}c_{n+1}\right).\label{eq:local_current}$$
\[sec:Numerical-methods\]Numerical methods for current evaluation
=================================================================
\[subsec:Method\_TimeEvolution\]Time-resolved current from quantum evolution of eigenstates
-------------------------------------------------------------------------------------------
The dynamics of the system, in the partitioned approach after suddenly switching on the lead-sample hoppings, or in the partition-free approach after suddenly switching on the external bias, is governed by the Hamiltonian Eq. (\[eq:time-evolution\_Hamiltonian\]). Therefore, in both approaches and for $t>0$, the reduced density matrix of the system evolves according to
$$i\hbar\frac{d\rho(t)}{dt}=\left[\mathcal{H}(t>0),\rho(t)\right].\label{eq:RDM_eom}$$
The solution for this equation, with initial condition given by either Eq. (\[eq:initial\_condition\_partitioned\]) or (\[eq:initial\_condition\_partition-free\]), is given by
$$\rho(t)=\sum_{\alpha}f_{\alpha}\left|\Psi_{\alpha}(t)\right\rangle \left\langle \Psi_{\alpha}(t)\right|,$$
with the single-electron states evolving according to Eq. (\[eq:time-evolution\_Hamiltonian\]): $\left|\Psi_{\alpha}(t)\right\rangle =e^{-\frac{i}{\hbar}\mathcal{H}(t>0)t}\left|\Psi_{\alpha}\right\rangle $, with $\left|\Psi_{\alpha}\right\rangle $ being the single-electron eigenstates of either $\mathcal{H}^{\text{P}}(t=0)$ or $\mathcal{H}^{\text{PF}}(t=0)$ (with occupation $f_{\alpha}$), for the *partitioned* and *partition-free* approaches, respectively.
The expected value of the current as a function of time, is given by $$\begin{gathered}
I^{n}(t)=\frac{ew}{i\hbar}\sum_{\alpha\in\text{occupied}}\left(\left\langle \left.\Psi_{\alpha}(t)\right|n+1\right\rangle \left\langle n\left|\Psi_{\alpha}(t)\right.\right\rangle \right.\\
\left.-\left\langle \left.\Psi_{\alpha}(t)\right|n\right\rangle \left\langle n+1\left|\Psi_{\alpha}(t)\right.\right\rangle \right),\label{eq:LocalCurrent_withSPES}\end{gathered}$$ where $\left|n\right\rangle $ represents the state localized at site $n$. We also used the fact that at $t=0$, $f_{\alpha}=1$ for initially occupied states and $f_{\alpha}=0$ for empty states. The above expression, allows us to evaluate the current flowing from site $n$ to site $n+1$ provided we know the time-evolution of the initial single-electron states. Although correct, Eq.(\[eq:LocalCurrent\_withSPES\]) is not very convenient from a numerical point of view, since for each initially occupied state, we would have to perform one time-evolution. A more convenient expression is obtained by writing $\left\langle \left.\Psi_{\alpha}(t)\right|n\right\rangle =\left\langle \Psi_{\alpha}\right|e^{\frac{i}{\hbar}\mathcal{H}\left(t>0\right)t}\left|n\right\rangle =\left\langle \left.\Psi_{\alpha}\right|n(-t)\right\rangle $, such that instead of evolving the initial eigenstate forwards in time, we evolve the localize states backwards in time[^1] The current can therefore be expressed as $$I^{n}(t)=\frac{2ew}{\hbar}\text{Im}\sum_{\alpha}\left\langle n(-t)\left|\Psi_{\alpha}\right.\right\rangle \left\langle \left.\Psi_{\alpha}\right|n+1(-t)\right\rangle .$$ Despite being equivalent to Eq. (\[eq:LocalCurrent\_withSPES\]), this last expression allows for a great gain in computational efficiency, as the the number of required time-evolutions is reduced from $\mathcal{O}\left(L\right)$ to only two, for each single-time calculation of the current between sites $n$ and $n+1$.
Numerically, the time-evolution of the localized states $\left|n\right\rangle $, for very large chains is computed efficiently using a polynomial Chebyshev expansion [@KPM_Timeevolution_Talezer1984; @fehske_numerical_2009] of the time-evolution operator $\mathcal{U}(t)=e^{-\frac{i}{\hbar}\mathcal{H}t}$ (for details, see Appendix\[appx:TimeEvolutionMethod\]). Finally, the single-electron eigenstates of $\mathcal{H}\left(t=0\right)$ were calculated “on-the-fly” using a memory-efficient algorithm developed by Fernando [@fernando_computing_1997].
\[subsec:Steady-state-Landauer\]Landauer-Büttiker formula for the steady-state current
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In sec.\[sec:NumericalResults\], the time-resolved current across the sample with finite leads, calculated using the quantum evolution of the occupied states, will be compared with the steady-state value for the current as given by the Landauer-Büttiker formula for the same sample attached to infinite leads. We evaluate this current using the Caroli-Meir-Wingreen form of the Landauer-Büttiker formula[@caroli_direct_1971; @meir_landauer_1992], which for a two-terminal device, at zero temperature, reads
$$I_{\text{LB}}=\frac{e}{2\pi\hbar}\int_{\varepsilon_{\text{F}}-e\frac{\Delta V}{2}}^{\varepsilon_{\text{F}}+e\frac{\Delta V}{2}}d\varepsilon T\left(\varepsilon\right),\label{eq:Mier-WingreenFormula-1}$$
where the energy dependent transmission function is expressed in terms of Green’s functions as
$$T(\varepsilon)=\text{Tr}\left[\bm{G}^{A}(\varepsilon)\cdot\bm{\Gamma}_{\text{RL}}(\varepsilon)\cdot\bm{G}^{R}(\varepsilon)\cdot\bm{\Gamma}_{\text{LL}}(\varepsilon)\right],$$
where $\bm{\Gamma}_{\text{LL}/\text{RL}}(\varepsilon)$ are real-space spectral functions of the unattached leads and $\bm{G}^{R/A}(\varepsilon)$ are the real-space retarded/advanced Green’s function of the central sample, in the presence of the leads. For our particular one-dimensional model, the leads’ spectral functions are matrices with the only non-zero elements between boundary sites, i.e. $\Gamma_{\text{LL}}^{L_{l}-1,L_{l}-1}(\varepsilon)=\Gamma_{\text{LL}}(\varepsilon)$ and $\Gamma_{\text{RL}}^{L_{l}+L_{s},L_{l}+L_{s}}(\varepsilon)=\Gamma_{\text{LL}}(\varepsilon)$ , where $\Gamma_{\text{LL}/\text{RL}}(\varepsilon)=w^{2}\rho_{\text{LL}/\text{RL}}(\varepsilon)$. The functions $\rho_{\text{LL}/\text{RL}}(\varepsilon)$ are surface density of states of the leads which may be computed analytically yielding:
![\[fig:PartitionedVsNonPartitioned\]Comparison of the time-dependent current traversing the center of a sample with $L_{s}=256$ sites, obtained in both the *partition-free* (*(a)* and *(b)*) and *partitioned* approaches (*(c)* and *(d)*), with the steady-state current obtained from the Landauer formula with semi-infinite leads (dashed magenta lines). In both approaches, the current is shown for different lead sizes, $\varepsilon_{\text{F}}=0$ and a bias of $\Delta V=0.1w$, without [\[]{}*(a)* and *(c)*[\]]{} and with disorder [\[]{}*(b)* and *(d)*[\]]{}. The insets represent the zooms of the current in the quasi-steady-state regime. As can be seen, the superposed finite-size oscillations have an amplitude which is less than $1\%$ of the Landauer current in both approaches, although their nature is different. In the *partition-free* approach there is a decrease of their amplitude with $L_{l}$, while the *partitioned* approach, they die-out only as $t\to+\infty$. In all four panels, the quasi-steady-state regime is limited by the recurrence time $t_{\text{r}}=2L_{l}/v_{\text{F}}$ (vertical dashed lines). (color online)](fig2)
$$\begin{gathered}
\rho_{\text{LL}/\text{RL}}(\varepsilon)=\Theta\left(4w^{2}-\left(\varepsilon\mp\frac{e\Delta V}{2}\right)^{2}\right)\\
\times\frac{1}{w^{2}}\sqrt{4w^{2}-\left(\varepsilon\mp\frac{e\Delta V}{2}\right)^{2}}.\label{eq:Gamma-1}\end{gathered}$$
Therefore, the final form for the transmission function reads
$$T(\varepsilon)=w^{4}\rho_{\text{LL}}(\varepsilon)\rho_{\text{RL}}(\varepsilon)\left|G_{L_{l}-1,L_{l}+L_{s}}^{R}(\varepsilon)\right|^{2}.\label{eq:LandauerFormula1D}$$
For each central sample, the retarded Green’s function in Eq. (\[eq:LandauerFormula1D\]) was calculated by using the well-known *recursive Green’s function method*[@MacKinnon1985; @wimmer_quantum_2009; @lewenkopf_recursive_2013], using the surface Green function of the semi-infinite one-dimensional leads as boundary conditions, as detailed in Appendix \[appx:RecursiveTransferMatrixAppendix\].
\[sec:NumericalResults\]Numerical results and comparison with the Landauer formula
==================================================================================
We evaluated the time-dependent current in finite open chains, using the method described in Sec. \[subsec:Method\_TimeEvolution\], for both clean and disordered samples and considering both the *partitioned* and *partition-free* initial conditions. This current was then compared with the Landauer expression for the steady-state current flowing through the same sample attached to infinite leads, as described in Sec. \[subsec:Steady-state-Landauer\].
As we can see in Fig. (\[fig:PartitionedVsNonPartitioned\]), three regimes are clearly distinguished for large enough leads: *(i)* initially, we have a transient regime up to a time $t_{\text{stab}}$, after which *(ii)* the current eventually approaches an approximately constant quasi-steady-state value, which last up to *(iii)* a recurrence time, $t_{\text{r}}$, after which an inversion of the current occurs.
Transient Behavior and Stabilization Times
------------------------------------------
As one could expect, the transient behavior depends on the initial preparation of the system, being different for the *partitioned* and *partition-free* approaches, as evident in Fig.\[fig:PartitionedVsNonPartitioned\]. On the one hand, after an initial time, during which the current is approximately zero, the transient fluctuations of the current are much more violent in the partitioned case. These fluctuations kick in after a build-up time, $t_{\text{b}}$, determined by the propagation of a Fermi-level plane-wave from the lead-sample boundaries to the center of the sample, where the current is being probed. The upper panel of Fig.\[fig:Transient\], where the current is now probed at a fixed distance $L_{\text{meas}}$ from the left lead, confirms this interpretation, as $t_{\text{b}}=v_{\text{F}}^{-1}L_{\text{meas}}$.
On the other hand, for the *partition-free* setup one observes a monotonous build-up of the current from the beginning, which is due to the local dynamics induced inside the central sample, by the sudden connection of the potential ramp. This build-up phase lasts up to a time $t_{\text{b}}\simeq v_{\text{F}}^{-1}L_{\text{meas}}$, at which the effects of the amplitudes initially on the leads start reaching the point of measurement (represented as vertical lines in Fig.\[fig:Transient\]) At this time, we also observe small inflections in the current, as indicated by the arrows in the bottom panel Fig. \[fig:Transient\]. After this initial build-up, the current enters a sample-specific damped oscillatory phase which relaxes towards the appropriate Landauer quasi-steady-state.
![\[fig:Transient\]Plots of the normalized time-dependent current at the $64^{\text{th}}$ hopping of a disordered central sample, calculated using the unitary quantum dynamics method in the *partitioned* (upper panel) and *partition-free* approach (lower panel) for a bias of $\Delta V=0.1w$ and different sample sizes. The time coordinate is rescaled by the stabilization time-scale, i.e. $t_{\text{stab}}=2v_{\text{F}}^{-1}L_{\text{s}}$, which turns the onset time of the quasi-steady-state roughly independent of the sample’s size in both approaches. The vertical pointed lines mark the time taken for a Fermi energy state to propagate from the left lead to the point where the current is being measured, i.e. $T=t_{\text{b}}/t_{\text{stab}}$, where the colored arrows highlight the inflection which occurs at this point for all the curves in *partition-free* case. In the lower panel, the dashed curves correspond to different seed used for generating the central disorder. (color online)](fig3){width="8.5cm"}
In both approaches, after a time which grows with the central sample size, $t_{\text{stab}}\simeq2v_{\text{F}}^{-1}L_{s}$, the value of the current stabilizes to an approximately constant value, as shown by the collapse of the curves in Fig.\[fig:Transient\], where the time axis is shown in units of $t_{\text{stab}}$. Physically, this time can be interpreted as the one needed for the quantum single-particle states near $\varepsilon_{\text{F}}$ to travel back and forth inside the central sample and thus probing the existing disorder landscape. Curiously, this time-scale is roughly independent of the particular disorder configuration in question.
Landauer Quasi-Steady-State Transport, Finite-Size Effects and Recurrence Times
-------------------------------------------------------------------------------
In both approaches, if the leads are large enough, the initial build-up and stabilization of the current is followed, for $t>t_{\text{stab}}$, by an intermediate time period where the current through the sample stabilizes. As the size of the leads increases, the value of this quasi-steady-state current tends to the sample-specific Landauer value, independently of the initial preparation of the system (*partitioned* or *partition-free*). Hence, the present result are numerical checks of the memory-loss theorem of Stefanucci *et al* [@stefanucci_time-dependent_2004]. This theorem states that, provided the leads have a continuum spectrum, a steady-state value of the current is achieved for long times and that this value is independent of the initial state of the system.
![\[fig:DifferentEFs\]Plots of the time-dependent current from the unitary quantum dynamics in the *partition-free* approach and for a single disordered central sample at different values of the common Fermi energy. The dashed magenta lines correspond, once again, to the respective steady-state current obtained from the Landauer formula with semi-infinite leads. The bias used was $\Delta V=0.1w$. (color online)](fig4){width="8.6cm"}
In our case, a quasi-steady-state only occurs for intermediate times, due to the finite nature of the leads, which makes their spectrum discrete. One expects that for sufficiently large leads, the value of the current in the quasi-steady-state approaches the Landauer value. However, the way in which this occurs depends crucially on the initial condition of the system. In the *partition-free* approach, we observe that the current in the quasi-steady-state regime does not approach the Landauer value monotonically in time. Instead, there is a small oscillatory component, with approximately constant amplitude in time, superposed on its steady-state value, which persists up to the first recursion time. The amplitude of these oscillations is seen to decrease as $L_{l}\to\infty$, which identifies them with a finite-size effect that disappears in the limit of semi-infinite leads. We also observed, that the period of this oscillations is roughly inversely proportional to the applied bias. For the *partitioned* approach, a rather different behavior is observed. In this setup, the amplitude of the oscillations does not decay as $L_{l}$ is increased, being roughly independent of the size of the leads for a fixed observation time. Instead, it is the amplitude of the oscillations that decays over time. As the size of the leads is increased, the quasi-steady-state can be observed for longer times, for which the amplitude of the oscillations is smaller. Despite being roughly independent of $L_{l}$ for a fixed observation time, these are also finite-size effects since they only disappear as $t\to\infty$, hence being limited by the recursion time $t_{\text{r}}$. All the previous effects are depicted in the four insets of Fig. \[fig:PartitionedVsNonPartitioned\].
Pal *et al* [@pal_emergence_2018] pointed out that the recurrence time is inversely proportional to the level spacing of the leads’ spectra, which measures how close the finite leads are to a true continuous spectrum. Our present results allow for an alternative interpretation. As demonstrated in Figs. \[fig:PartitionedVsNonPartitioned\] — where we show the current for fixed Fermi energy and different sizes of the leads — and in Fig. \[fig:DifferentEFs\] — where we show the current for fixed $L_{l}$ but different Fermi energies — the recurrence time is roughly given by $t_{\text{r}}\sim2v_{\text{F}}^{-1}L_{l}$, where $v_{\text{F}}$ is the Fermi velocity. Notice that $2v_{\text{F}}^{-1}L_{l}$ is just the time a Fermi-level electron takes in a round trip inside of a lead, in agreement with what was previously reported in Ref. @bushong_approach_2005. Furthermore, one also sees that the recursion time is roughly independent of the disorder on the sample, which is consistent with its previous physical interpretation.
![\[fig:Different sites\]Plots of the electric current traversing the central bond (blue curves) and the boundaries (red and green curves) of a disordered sample of length $L_{s}=256$ and disorder strength of $W=0.3w$. The leads had a length of $L_{l}=2^{13}$ and a potential difference of $\Delta V=0.01w$. The top panel depicts the calculation done with the *partitioned* setup, while in the bottom panel the *partition-free* configuration was used. The dashed magenta lines stand for the value of the steady-state current obtained using the Landauer formula for this sample. (color online)](fig5){width="8.6cm"}
In a true steady-state, the value of the current is not only time-independent but must also position-independent, as no charge accumulation can occur. Hence, we also investigated whether or not this emergent quasi-steady-state current in finite chains is homogeneous over the sample. Indeed, we found out that in the quasi-steady-state the current is approximately homogeneous in space, independently of the initial preparation of the system, for large enough leads and provided we are far away from the chain’s open extremities.This observation is exemplified in Fig. \[fig:Different sites\], where we show the time-dependent current for a disordered central sample, measured at three different bonds: center, left and right boundaries of a randomly picked disordered sample. As can be seen, after the disappearance of the initial transients, the same quasi-steady-state current is reached at the three positions, apart from the finite-size oscillations which are different.
![\[fig:LinearResponseExample\]Plots of the time-dependent current across a disordered sample coupled to finite leads with $L_{l}=16384$ sites and for different values of $\Delta V\ll w$. The full lines stand for the results of a fully non-linear calculation using the quantum dynamics method of last section in the *partitioned* (upper panel) and *partition-free* approach (lower panel), while the points stand for the raw evaluation of the linear response Eq.(\[eq:LRTCurrent-1\]). The last are only present in the *partition-free* case, where the time-dependent Kubo formula is valid. The value of the current is normalized to the corresponding Landauer steady-state value. (color online)](fig6)
Interestingly, the establishment of a well defined quasi–steady-state might not occur for very small biases, where we would expect linear response theory to hold, depending on the initial conditions. This is illustrated in Fig.\[fig:LinearResponseExample\]. There, we can see that for the *partition-free* setup, no clear quasi-steady-state is observed for very small biases. This occurs because, for fixed leads size, the period and amplitudes (relative to the infinite leads’ Landauer value) of the finite-size oscillations increases with the applied bias. Therefore, for small enough bias, the period of the oscillations might become larger than the recursion time, and no quasi-steady-state is observed. In the *partitioned* setup, the situation is a bit different and for large times: the current always tends to the Landauer value with the amplitude of the finite-size oscillations decreasing over time. These observations seem to be in agreement with the interpretation of Ref. @bushong_approach_2005, where it is put forward that the observation of a quasi-steady-state requires the change in the initial spread of the electrons’ momenta. In Ref. @bushong_approach_2005, this occurs either due to a geometrical constriction at the lead-sample contact, or due to an initial applied energy barrier. In our case, it seems that the applied bias is the mechanism by which electrons change their initial momenta. As a general “rule-of-thumb”, we can tell that, in order to observe a quasi-steady-state current regime with minor finite-size effects, one must always consider biases that are much larger than the level spacing of the whole system’s spectrum.
Sample-Specific $I-V$ curves at large biases
--------------------------------------------
We finally point out, that the coincidence between the Landauer value for the current and the value of the current in the quasi-steady-state occurs for any value of the bias potential, as long as it is smaller than the bandwidth and provided a quasi-steady-state is established.
![\[fig:IVCurves\]Plots of the $I\left(\Delta V\right)$ curves of two independent disordered samples. The black curves in the main plots were obtained using the Landauer formula of Eq.(\[eq:LandauerFormula1D\]). The red dots were obtained from the quasi-steady-state current of a quantum dynamics calculation, using the *partition-free* approach with $L_{l}=2^{14}$ sites. The use of a partitioned approach could also be done, but would be redundant given that we proved the numerical equivalence of both approaches in the previous discussion. In the insets, we highlight with a red arrow the time of measurement in a plot of $I(t)$. (color online)](fig7){width="8cm"}
This is illustrated in Fig. \[fig:IVCurves\], where we show values for the time-dependent current in the quasi-steady-state regime as a function of the applied bias, for two random disordered samples, and compare the results with the value of the Landauer current. The results clearly confirm that the quasi-steady-state current seen in the quantum dynamics calculations with finite leads indeed corresponds to the Landauer transport predicted for samples coupled to semi-infinite leads. The agreement between the two approaches was seen to be perfect for all the range of bias tested and well beyond linear response.
\[sec:Emergence-of-Landauer\]Emergence of Landauer transport within linear response in the partition-free approach
==================================================================================================================
The numerical studies of the previous section show that a quasi-steady-state transport regime, with an approximately uniform and time-independent current, emerges across finite systems subjected to a potential bias and coupled to finite but large leads. Moreover, the results also show that for large enough leads, the value of this quasi-steady-state current coincides with the Landauer result for the transport’s steady-state with semi-infinite leads. In this section, we will try to shed further light on these numerical results using a semi-analytical procedure. In order to make as much analytical progress as possible, we shall restrict ourselves to the *partition-free* case and small biases, such that we can study the current using Kubo linear response theory in the applied bias, $\Delta V$.
Time-dependent Kubo formula for a sudden connection
---------------------------------------------------
We will always consider the *partition-free* Hamiltonian at $t=0$ as the unperturbed Hamiltonian for this case, i.e.
$$\begin{aligned}
\mathcal{H}_{0} & =\mathcal{H}^{\text{PF}}(t=0)\nonumber \\
& =\sum_{n=0}^{L-1}\epsilon_{n}^{\text{d}}c_{n}^{\dagger}c_{n}-w\sum_{n=0}^{L-2}\left(c_{n+1}^{\dagger}c_{n}+c_{n}^{\dagger}c_{n+1}\right)\label{eq:unperturbed_Hamiltonian}\end{aligned}$$ and treat the applied potential bias as the current-driving perturbation,
$$\mathcal{V}(t)=-e\Theta(t)\sum_{n=0}^{L-1}v_{n}^{\text{e}}c_{n}^{\dagger}c_{n}.\label{eq:PerturbationHamiltonian}$$
with the electrostatic potential profile, $v_{n}^{\text{e}}$, given by Eq. (\[PotentialProfile\]).
In order to derive a time-dependent Kubo formula for the current, we will start by writing the equation of motion for the reduced density matrix, Eq. (\[eq:RDM\_eom\]), in the eigenbasis of the unperturbed Hamiltonian. Thus, we obtain $$\frac{d}{dt}\rho_{\alpha\beta}(t)=-\frac{i}{\hbar}\left(\varepsilon_{\alpha}-\varepsilon_{\beta}\right)\rho_{\alpha\beta}(t)-\frac{i}{\hbar}\left[\mathcal{V}(t),\rho(t)\right]_{\alpha\beta}\label{eq:EOM_RDM_eigenbasis}$$ where $O_{\alpha\beta}(t)=\left\langle \Psi_{\alpha}\left|O\right|\Psi_{\beta}\right\rangle $ and $\left|\Psi_{\alpha}\right\rangle $ is an eigenstate of $\mathcal{H}_{0}$ with energy $\varepsilon_{\alpha}$. Within linear response theory, we write the reduced density matrix as $$\rho_{\alpha\beta}\left(t\right)=\delta_{\alpha\beta}f\left(\varepsilon_{\alpha}\right)+\delta\rho_{\alpha\beta}\left(t\right),$$ where $\rho_{\alpha\beta}\left(0\right)=\delta_{\alpha\beta}f\left(\epsilon_{\alpha}\right)$ is the initial equilibrium reduced density matrix and $\delta\rho_{\alpha\beta}\left(t\right)$ is a small correction, which in linear response is assumed to be $\propto\mathcal{V}(t)$. Disregarding any contributions of $\mathcal{O}\left(\mathcal{V}^{2}\right)$ in the equation of motion, we obtain
$$\begin{gathered}
\frac{d}{dt}\delta\rho_{\alpha\beta}(t)=-\frac{i}{\hbar}\left(\varepsilon_{\alpha}-\varepsilon_{\beta}\right)\delta\rho_{\alpha\beta}(t)\\
-\frac{ie}{\hbar}\Theta(t)\Gamma_{\alpha\beta}\left(f(\varepsilon_{\alpha})-f(\varepsilon_{\beta})\right).\label{eq:EMRDM-1-2}\end{gathered}$$
where $\Gamma_{\alpha\beta}$ are the matrix elements of the applied potential bias,
$$\Gamma_{\alpha\beta}=\sum_{n}\psi_{\alpha}^{*}(n)\psi_{\beta}(n)v_{n}^{\text{e}},$$ and $\psi_{\alpha}(n)$ is the amplitude of the eigenstate $\ket{\Psi_{\alpha}}$ on site $n$, i.e. $\psi_{\alpha}(n)=\left\langle n\left|\Psi_{\alpha}\right.\right\rangle $. Now, using the fact that $\delta\rho_{\alpha\beta}(t<0)=0$, it is possible to integrate Eq. (\[eq:EMRDM-1-2\]), obtaining $$\delta\rho_{\alpha\beta}(t)=-e\Gamma_{\alpha\beta}\frac{\Delta f_{\alpha\beta}}{\Delta\varepsilon_{\alpha\beta}}\left(1-e^{-\frac{i}{\hbar}\Delta\varepsilon_{\alpha\beta}t}\right),\label{eq:EMRDM-1-2-1-1-1-1}$$ where $\Delta f_{\alpha\beta}=f(\varepsilon_{\alpha})-f(\varepsilon_{\beta})$ and $\Delta\varepsilon_{\alpha\beta}=\varepsilon_{\alpha}-\varepsilon_{\beta}$. The expected value of the current that flows from site $n$ to $n+1$, is thus given by $$I^{n}\left(t\right)=\frac{ie^{2}w}{\hbar}\sum_{\alpha,\beta}\Pi_{\alpha\beta}^{n}\Gamma_{\beta\alpha}\frac{\Delta f_{\alpha\beta}}{\Delta\varepsilon_{\alpha\beta}}\left(1-e^{-\frac{i}{\hbar}\Delta\varepsilon_{\alpha\beta}t}\right),\label{eq:LRTCurrent}$$ where we introduced $$\Pi_{\alpha\beta}^{n}=\psi_{\alpha}^{*}(n+1)\psi_{\beta}(n)-\psi_{\alpha}^{*}(n)\psi_{\beta}(n+1),\label{eq:Gamma_definition}$$ which are the matrix elements of the local current operator between sites $n$ and $n+1$, up to a dimension-full multiplicative factor.
By further noticing that the amplitudes $\psi_{\alpha}(n)$ may be chosen as all real and $\Pi^{s}$ is an anti-symmetric matrix, one can rewrite Eq. (\[eq:LRTCurrent\]) in the following way:
$$I^{n}\left(t\right)=\frac{2e^{2}w}{\hbar}\qquad\qquad\quad\qquad\qquad\qquad\qquad\qquad\mathllap{\sum_{\overset{\alpha}{\left(\varepsilon_{\alpha}\leq\varepsilon_{F}\right)}}\sum_{\overset{\beta}{\left(\varepsilon_{\beta}>\varepsilon_{F}\right)}}\Pi_{\alpha\beta}^{n}\Gamma_{\alpha\beta}\frac{\sin\left(\frac{\Delta\varepsilon_{\alpha\beta}t}{\hbar}\right)}{\Delta\varepsilon_{\alpha\beta}}.}\label{eq:LRTCurrent-1}$$
which is our final time-dependent Kubo formula for the current.
Obviously, one cannot give a general rule for establishing the validity regime of Eq.(\[eq:LRTCurrent-1\]), since that will depend crucially on the properties of the central disordered sample. However, for each sample, there is always a value of $\Delta V$ sufficiently small, such that a linear response theory for the current is valid. We depict such an example in the upper panel of Fig.\[fig:LinearResponseExample\], where the current traversing the central bond of a disordered sample, as obtained from Eq.(\[eq:LRTCurrent-1\]), is compared with the one obtained from the fully nonlinear quantum dynamics of sec.\[sec:Numerical-methods\] in the *partition-free* approach. As a further short comment on the plots of Fig.\[fig:LinearResponseExample\], it is interesting to note that, for the parameters used, it seems that no quasi-steady-state plateau emerges from the quantum dynamics close to the linear response regime. As referred before, this is simply a consequence of a greater relevance of the finite-size oscillations which, now, have a period larger than the recurrence time and a much larger relative amplitude.
Representation of the eigenstates in terms of the sample’s quantum reflection/transmission coefficients
-------------------------------------------------------------------------------------------------------
In order to make an effective use of Eq. (\[eq:LRTCurrent-1\]) and make analytic progress we must be able to find a semi-analytical expression for the matrix elements $\Pi_{\alpha\beta}^{n}$ and $\Gamma_{\alpha\beta}$, which, in principle, requires the knowledge of the eigenfunctions in the whole chain. These wavefunctions usually present a very complicated structure inside the disordered central sample, but for large enough leads, we actually only need to know their form in the leads. On the one hand, the $\Pi_{\alpha\beta}^{n}$ matrix elements only require the knowledge of local amplitudes in the two adjacent sites across which the current is being measured. Hence, we can simply choose to measure it outside the sample. On the other hand, we expect the current to be dominated by states that are not localized in the disordered sample, but instead are delocalized in the leads. Hence, we only need to calculate the $\Gamma_{\alpha\beta}$ matrix elements between delocalized states. For such states, and provided the leads are much larger than the disordered sample region, we can approximate
$$\Gamma_{\alpha\beta}=\sum_{n}\psi_{\alpha}^{*}(n)\psi_{\beta}(n)v_{n}^{\text{e}}\simeq\qquad\mathllap{\sum_{n\in\text{Leads}}}\psi_{\alpha}^{*}(n)\psi_{\beta}(n)v_{n}^{\text{e}}.\label{eq:large_lead:approx}$$
This approximation, allows us to evaluate the current $I^{n}\left(t\right)$ in the leads, without knowing the shape of the eigenwavefunctions inside the central sample.
Next, we notice that the form of the scattering eigenstates in the leads can be expressed in terms of the complex reflection and transmission coefficients of the central sample. For perfect leads, the wavefunctions of the eigenstates will have the form of a coherent superposition of left and right propagating plane-waves. With a change of notation from the previous section, we will relabel sites of the left lead with indices $n=-L_{l},...,-1$ and the ones of the right lead with $n=1,...,L_{l}$. Using this notation, the form of the eigenstate wavefunction $\left|\Psi_{k}\right\rangle $ in the leads have the form
$$\begin{gathered}
\psi_{k}(n)=\left\langle n\left|\Psi_{k}\right.\right\rangle \\
=\begin{cases}
\Psi_{+}^{L}e^{ik\left(n-1\right)}+\Psi_{-}^{L}e^{-ik\left(n-1\right)}, & \quad\qquad\qquad\mathllap{-L_{l}\leq n\leq-1}\\
\Psi_{+}^{R}e^{ik\left(n-L_{s}\right)}+\Psi_{-}^{R}e^{-ik\left(n-L_{s}\right)}, & 1\leq n\leq L_{l}
\end{cases},\label{eq:wavefunctions_leads}\end{gathered}$$
being labeled by a crystal momentum $k$, and with $\Psi_{+/-}^{L(R)}$ being the amplitude of a right/left propagating state in the left (right) lead. Notice that the time-independent Schrödinger equation inside the leads, still allows us to relate the crystal momentum $k$ to the energy of the state as $E=-2t\cos(k)$, i.e. the same as for an infinite periodic chain. As usual in one-dimensional scattering problems, the amplitudes of propagating states on the left and right leads can be related by a transfer matrix, $\mathcal{M}(k)$: $$\left(\begin{array}{c}
\Psi_{+}^{R}\\
\Psi_{-}^{R}
\end{array}\right)=\mathcal{M}\left(k\right)\cdot\left(\begin{array}{c}
\Psi_{+}^{L}\\
\Psi_{-}^{L}
\end{array}\right),\label{eq:S_Matrixef-4-1}$$ In the presence of time-reversal symmetry, the transfer matrix has the general form[ $$\mathcal{M}\left(k\right)=\left(\begin{array}{cc}
\frac{1}{\abs{t\left(k\right)}}e^{i\phi\left(k\right)} & -\frac{\abs{r\left(k\right)}}{\abs{t\left(k\right)}}e^{-i\theta\left(k\right)+i\phi\left(k\right)}\\
-\frac{\abs{r\left(k\right)}}{\abs{t\left(k\right)}}e^{i\theta\left(k\right)-i\phi\left(k\right)} & \frac{1}{\abs{t\left(k\right)}}e^{-i\phi\left(k\right)}
\end{array}\right),$$ ]{}where $\abs{t\left(k\right)}/\abs{r\left(k\right)}$ and $\phi\left(k\right)/\theta\left(k\right)$ are the moduli and phases of the transmission/reflection coefficients, respectively. Moreover, for any sample one has $\det\mathcal{M}=1$, which implies the conservation of current, i.e. $\abs t^{2}+\abs r^{2}=1$. These coefficients are physical characteristics of the central sample only and, thus, may be rightfully calculated by assuming the leads as semi-infinite. The determination of the reflection and transmission coefficients of a specific sample, in general, can only be done numerically, using the method detailed in Appendix\[appx:RecursiveTransferMatrixAppendix\]. The great advantage of this method is that, once this calculation is done, the wavefunctions in the leads can be expressed in terms of only a few parameters. Additionally, to obtain the eigenstates, we must further impose open boundary conditions at the ends of the leads, i.e.
$$\psi_{k}\left(-L_{l}-1\right)=\psi_{k}\left(L_{l}+1\right)=0.\label{eq:boundary_condition}$$ Combining Eqs. (\[eq:wavefunctions\_leads\])-(\[eq:boundary\_condition\]) one arrives at the following general expression for the wavefunctions:
$$\psi_{k}(n)=\frac{1}{\sqrt{N_{k}}}\begin{cases}
\left|t\left(k\right)\right|\sin\left[k\left(n+L_{l}+1\right)\right] & n<0\\
f_{2}\left(k\right)\sin\left[k\left(n-L_{l}-1\right)\right] & n>0
\end{cases},\label{eq:RealSpaceAmplitudes}$$
where $N_{k}$ is a normalization factor, which can be determined in the limit of large leads by approximating, in the same spirit of Eq. (\[eq:large\_lead:approx\]), $$\sum_{n}\left|\psi_{k}(n)\right|^{2}\simeq\sum_{n\in\text{Leads}}\left|\psi_{k}(n)\right|^{2}.$$ This finally leads to
$$N_{k}\simeq L_{l}f_{1}\left(k\right).\label{eq:normalization}$$ The functions $f_{1}(k)$ and $f_{2}\left(k\right)$ are defined as
$$\begin{aligned}
f_{1}\left(k\right) & =1+\abs{r\left(k\right)}\cos\left[2k\left(L_{l}+1\right)+\theta\left(k\right)\right]\\
f_{2}\left(k\right) & =\cos\left[2k\left(L_{l}+1\right)+\phi\left(k\right)\right]\nonumber \\
& +\abs{r\left(k\right)}\cos\left[\theta\left(k\right)-\phi\left(k\right)\right],\end{aligned}$$
and where $k$ is constrained to verify the following quantization condition:
[ $$\sin\left[2k\left(L_{l}+1\right)+\phi\left(k\right)\right]=\abs{r\left(k\right)}\sin\left[\theta\left(k\right)-\phi\left(k\right)\right].\label{eq:QuantizationCondition}$$ ]{}
Notice that the solution of this last condition, together with the relation $k=\arccos\left(-E/\left(2w\right)\right)$, allows us to determine the eigenenergies corresponding to delocalized states. In the lower panels of Fig. \[fig:ComparisonwithNumericals\] (b), we exemplify the validity of this statement by comparing the wavefunctions obtained from the numerical diagonalization of $\mathcal{H}_{0}$, with $L_{l}=8192$ sites, to the semi-analytical expressions of Eq. (\[eq:RealSpaceAmplitudes\]). The wavenumbers obtained from the numerical diagonalization, i.e. $k=\arccos\left(-E/2\right)$, are also seen to coincide perfectly with the roots of Eq. (\[eq:QuantizationCondition\]) (see the upper panels of Fig. \[fig:ComparisonwithNumericals\](b)).
Note also that Eqs. (\[eq:RealSpaceAmplitudes\]) and (\[eq:QuantizationCondition\]) reduce to the usual result for the eigenstates of a finite open chain, when $\abs{r\left(k\right)}=0$ and $\phi\left(k\right)=k\left(L_{l}-1\right)$ is the phase accumulated by a plane-wave crossing the internal bonds of an ordered sample, i.e.
$$\psi_{k}(n)=\frac{1}{\sqrt{L_{l}}}\begin{cases}
\sin\left[k\left(n+L_{l}+1\right)\right],\qquad\;\;n<0\\
\left(-1\right)^{p}\sin\left[k\left(n-L_{l}-1\right)\right],n>0
\end{cases}\label{RealSpaceAmplitudes-1}$$
with $k=\pi p/\left(L+1\right)$ and $p=1,...,L$. Moreover, these states are non-degenerate and also alternately symmetrical and antissymetrical under parity ($n\to-n$), which just reflects that same symmetry of the clean Hamiltonian.
With the knowledge of the eigenstates wavefunctions of the leads, Eq. (\[eq:RealSpaceAmplitudes\]), we can write the matrix elements $\Gamma_{k,q}$ and $\Pi_{k,q}^{n}$. With the approximation of Eq. (\[eq:large\_lead:approx\]), we can evaluate $\Gamma_{k,q}$ analytically, obtaining
As for the matrix elements $\Pi_{k,q}^{n}$, from the definition Eq. (\[eq:Gamma\_definition\]), for bonds in the left lead ($n<-1$), and after some simple manipulations, we obtain[ $$\begin{gathered}
\Pi_{k,q}^{n<-1}=\frac{\left|t\left(k\right)\right|\left|t\left(q\right)\right|}{L_{l}\sqrt{f_{1}\left(k\right)f_{1}\left(q\right)}}\times\\
\times\left\{ \sin\left(\frac{k-q}{2}\right)\sin\left[\left(k+q\right)\left(n+L_{l}+\frac{3}{2}\right)\right]\right.\\
\left.-\sin\left(\frac{k+q}{2}\right)\sin\left[\left(k-q\right)\left(n+L_{l}+\frac{3}{2}\right)\right]\right\} ,\label{eq:PiTransferMatrix}\end{gathered}$$ ]{}while for the current in bonds of the right lead ($n>1$), we obtain a similar result after replacing $\left|t\left(k\right)\right|\left|t\left(q\right)\right|\rightarrow f_{2}\left(k\right)f_{2}\left(q\right)$ and $L_{l}\rightarrow-L_{l}-2$ in Eq.(\[eq:PiTransferMatrix\]).
![\[fig:ComparisonwithNumericals\]*(a)* Scheme of the procedure of replacing the central sample by an effective momentum-dependent transfer matrix, $\mathcal{M}\left(k\right)$. *(b)* Comparison between the exact eigenvalues and eigenstates obtained from the numerical diagonalization of a system with finite leads of size $L_{l}=8192$ and a sample with $L_{s}=512$ sites, and the ones obtained using the transfer matrix method. The left panels correspond to a case without disorder, while the right ones to disordered central sample. The upper panels compare the wavenumbers obtained from the eigenvalues of the numerical diagonalization with the zeros of the analytical quantization condition (Eq.(\[eq:QuantizationCondition\])), while the lower panels compare the corresponding wavefunctions of one of eigenstates (signaled by the red arrow). (color online)](fig8a){width="7.8cm"}
![\[fig:ComparisonwithNumericals\]*(a)* Scheme of the procedure of replacing the central sample by an effective momentum-dependent transfer matrix, $\mathcal{M}\left(k\right)$. *(b)* Comparison between the exact eigenvalues and eigenstates obtained from the numerical diagonalization of a system with finite leads of size $L_{l}=8192$ and a sample with $L_{s}=512$ sites, and the ones obtained using the transfer matrix method. The left panels correspond to a case without disorder, while the right ones to disordered central sample. The upper panels compare the wavenumbers obtained from the eigenvalues of the numerical diagonalization with the zeros of the analytical quantization condition (Eq.(\[eq:QuantizationCondition\])), while the lower panels compare the corresponding wavefunctions of one of eigenstates (signaled by the red arrow). (color online)](fig8b){width="8.6cm"}
The continuum regime of the Kubo formula
----------------------------------------
When analyzing the time-dependent Kubo formula of Eq. (\[eq:LRTCurrent-1\]), we must take into account that there are actually two distinct time scales: *1)* the observation time, $t$, and *2)* the scale associated with the spacing between the discrete energy levels of the finite chain. The latter is proportional to the length of the leads and, as discussed in the Sec. \[sec:NumericalResults\], is associated to the recurrence time $t_{r}\sim2L_{l}/v_{\text{F}}$.
As expected and confirmed in Sec. \[sec:NumericalResults\], the quasi-steady-state regime which approximates the Landauer transport regime of semi-infinite leads, emerges when we take $T,L_{l}\rightarrow\infty$ (with $T=tw/\hbar$ being the time in dimensionless units), but while keeping $T\ll L_{l}$. In such case, all the transients have died out, but the system is still far away from getting into the regime where current inversions occur. Furthermore, Eq. (\[eq:LRTCurrent-1\]) includes a factor of $\sin\left(\Delta\varepsilon_{\alpha\beta}t/\hbar\right)/\Delta\varepsilon_{\alpha\beta}$, which is an emergent $\delta$-function in the limit $t\to+\infty$, with a broadening of $\hbar t^{-1}$ in energy. This factor actually acts as a spectral filter which kills-off the contributions coming from pairs of eigenstates having an energy separation larger than $\hbar t^{-1}$. Hence, we will show in this section how the approximately time-independent quasi-steady-state current emerges, when we are in the limit $T,L_{l}\rightarrow\infty$, with $T\ll L_{l}$, such that there are many eigenvalues inside the interval $\left[\varepsilon_{\text{F}}-\hbar t^{-1},\varepsilon_{\text{F}}+\hbar t^{-1}\right]$. We will refer to this limit as the *continuum regime*.
### Approximate form of $\Gamma_{k,q}$ and $\Pi_{k,q}^{n}$ matrices in the continuum regime
We start by noting that, in the continuum limit, since only states close to the Fermi energy contribute, it suffices to obtain the matrix elements $\Gamma_{k,q}$ and $\Pi_{k,q}^{n}$ between states where $k-q$ is small and $k,q\simeq k_{\text{F}}$. In the limit of $k-q\rightarrow0$, the first term of Eq. (\[eq:GammaTransferMatrix\]) dominates over the second. Therefore, we can approximate it as
$$\Gamma_{k,q}\simeq\Delta V\frac{\abs{t\left(k\right)}\abs{t\left(q\right)}-f_{2}\left(k\right)f_{2}\left(q\right)}{8L_{l}\sqrt{f_{1}\left(k\right)f_{1}\left(q\right)}}\frac{\sin\left[\left(k-q\right)L_{l}\right]}{\sin\left(\frac{k-q}{2}\right)},\label{eq:Gamma_approx}$$ where, in the of limit $L_{l}\rightarrow\infty$ , we approximated $\sin\left[\left(q-k\right)\left(L_{l}+\frac{1}{2}\right)\right]\simeq\sin\left[\left(q-k\right)L_{l}\right]$. Doing the same for $\Pi_{k,q}^{n}$, we obtain
$$\Pi_{k,q}^{n<-1}\simeq-\frac{\abs{t\left(k\right)}\abs{t\left(q\right)}}{L_{l}\sqrt{f_{1}\left(k\right)f_{1}\left(q\right)}}\sin\left(k_{\text{F}}\right)\sin\left[\left(k-q\right)L_{l}\right],\label{eq:Pi_left_approx}$$ where we assumed that $\left|n\right|\ll L_{l}$, when approximating $\sin\left[\left(k-q\right)\left(n+L_{l}+\frac{3}{2}\right)\right]\simeq\sin\left[\left(k-q\right)L_{l}\right]$. This justifies why in the quasi-steady-state regime, the current is approximately uniform, if we are away from the chain’s extremities. For the current on the right lead, we obtain a similar result, namely,
$$\Pi_{k,q}^{n>1}\simeq\frac{f_{2}\left(k\right)f_{2}\left(q\right)}{L_{l}\sqrt{f_{1}\left(k\right)f_{1}\left(q\right)}}\sin\left(k_{\text{F}}\right)\sin\left[\left(k-q\right)L_{l}\right].\label{eq:Pi_right_approx}$$
Now, we note that for a chain without any disorder, the matrix elements of $\Gamma_{k,q}$ will only be non-zero if the states labeled to $k$ and $q$ have opposite parities. This selection rule stems from the fact that the fully ordered chain is symmetric under inversion and therefore its eigenstates will have a well-defined parity. Since the applied potential $v_{n}^{\text{e}}$ is an odd perturbation, it only couples states of opposite parities. In the presence of a general disorder in the central sample, we no longer have inversion symmetry. Nevertheless, one may still expect that in the limit $L_{l}\gg L_{s}$, the breaking of the symmetry is small and an approximate selection rule should emerge. Indeed, this is the case. In order to obtain this approximate selection rule for a sample with disorder, we notice that although we can no longer classify the states as even and odd, given the quantization condition Eq. (\[eq:QuantizationCondition\]), which involves $\sin\left[2k\left(L_{l}+1\right)+\phi\left(k\right)\right]$[, ]{}we can classify the states as $+$ and $-$ according to the sign of $\cos\left[2k\left(L_{l}+1\right)+\phi\left(k\right)\right]$:[ $$\begin{gathered}
\cos\left[2k^{\pm}\left(L_{l}+1\right)+\phi\left(k^{\pm}\right)\right]=\\
=\pm\sqrt{1-\abs{r\left(k^{\pm}\right)}^{2}\sin^{2}\left[\theta\left(k^{\pm}\right)-\phi\left(k^{\pm}\right)\right]}.\label{eq:CosClassification}\end{gathered}$$ ]{}For an ordered or symmetrically disordered sample, this reduces to a labelling of eigenstates as even or odd, respectively, under a parity transformation, $n\to-n$. With such a classification, it can be shown (see Appendix \[appx:SelectionRule\]) that in the limits of $k-q\rightarrow0$ and $L_{l}\rightarrow\infty$, one obtains the following effective selection rule: $$\begin{gathered}
\lim_{L_{l}\rightarrow\infty}\left|\sin\left[\left(k^{\sigma}-q^{\sigma^{\prime}}\right)L_{l}\right]\right|=\left(1-\delta_{\sigma,\sigma^{\prime}}\right)\times\\
\times\sqrt{1-\abs{r\left(k_{\text{F}}\right)}^{2}\sin^{2}\left(\theta\left(k_{\text{F}}\right)-\phi\left(k_{\text{F}}\right)\right)}.\label{eq:Effective_SelectionRule}\end{gathered}$$ with $\sigma,\sigma'=\pm$ and which immediately implies that $\Gamma_{k,q}\simeq0$, if $k,q$ are in the same class as $L_{l}\rightarrow\infty$. This is an approximate selection rule, analogous to the one which exists in the clean case, but which only emerges when $L_{l}\to\infty$. In Fig. \[fig:SinScaling\], we represent the values of $\left|\sin\left[\left(k-q\right)L_{l}\right]\right|$ as a function of $\left(k-q\right)L_{l}$, for allowed values of $k$ and $q$. We can clearly see that for some data points $\left|\sin\left[\left(k-q\right)L_{l}\right]\right|\rightarrow0$ as $L_{l}$ increases, while other data points tend to a finite value, which is given by the sample-specific value, $\sqrt{1-\abs{r\left(k_{\text{F}}\right)}^{2}\sin^{2}\left(\theta\left(k_{\text{F}}\right)-\phi\left(k_{\text{F}}\right)\right)}$.[^2]
We point out that in the case of symmetric disorder profile, one can derive from the properties of the transfer matrix that $\phi\left(k\right)-\theta\left(k\right)=\pm\pi/2$. In this case, one immediately sees that the scattering wavefunctions of Eq.(\[eq:RealSpaceAmplitudes\]) reduce to the same form as in Eq. (\[RealSpaceAmplitudes-1\]), with the parity determined by the class to which it belongs. In such a case, the $\abs{t\left(k\right)}$ factor of the $\Gamma$ and $\Pi$ matrices comes only from this effect, since the functions $f_{1}\left(k\right)$ and $f_{2}\left(k\right)$ are exactly the same as in the non-disordered case. Just the allowed $k$’s are different.
Having established this effective selection rule, we can expand the prefactors of Eqs. (\[eq:Gamma\_approx\])-(\[eq:Pi\_right\_approx\]) around $k_{\text{F}}$. Taking into account that the only significant contributions come from pairs of states belonging to different classes, we may use Eqs. (\[eq:QuantizationCondition\]) and (\[eq:CosClassification\]) to write, for $k,q\simeq k_{\text{F}}$,
$$\begin{gathered}
f_{1}\left(k\right)f_{1}\left(q\right)\simeq\\
\simeq\left(1-\abs{r\left(k_{\text{F}}\right)}^{2}\sin^{2}\left[\theta\left(k_{\text{F}}\right)-\phi\left(k_{\text{F}}\right)\right]\right)\left|t\left(k_{\text{F}}\right)\right|^{2},\end{gathered}$$
$$f_{2}\left(k\right)f_{2}\left(q\right)\simeq-\left|t\left(k_{\text{F}}\right)\right|^{2}.$$
Using these approximations in Eq. (\[eq:Gamma\_approx\]), we obtain $$\begin{aligned}
\Gamma_{k,q} & \simeq\Delta V\frac{\left|t\left(k_{\text{F}}\right)\right|}{4L_{l}}\frac{1}{\sin\left(\frac{k-q}{2}\right)}\nonumber \\
& \simeq\Delta V\frac{\left|t\left(k_{\text{F}}\right)\right|}{2L_{l}}\frac{1}{k-q},\label{eq:Gamma_approx_final}\end{aligned}$$ and from Eqs. (\[eq:Pi\_left\_approx\])-(\[eq:Pi\_right\_approx\]) we obtain $$\Pi_{k,q}^{n<-1}\simeq\Pi_{k,q}^{n>1}\simeq-\frac{\left|t\left(k_{\text{F}}\right)\right|}{L_{l}}\sin\left(k_{\text{F}}\right).\label{eq:Pi_approx_final}$$ In the following, we will use Eqs.(\[eq:Gamma\_approx\_final\])-(\[eq:Pi\_approx\_final\]) to obtain the Landauer current from the Kubo formula of Eq.(\[eq:LRTCurrent-1\])
![\[fig:SinScaling\]Scatter plot of $\protect\abs{\sin\left[\left(k-q\right)L_{l}\right]}$ versus $\left(q-k\right)L_{l}$ for the allowed values of $k,q$ and different lead sizes ($L_{l}=2^{16}$—$2^{19}$ sites). The different data sets correspond to four different samples, one ordered ($\times$) and two disordered ones ([$\bigcirc$]{} and $\triangle$), which were randomly chosen. The dashed magenta curves correspond to the asymptotic limits of $k-q\rightarrow0$ and $L_{l}\rightarrow\infty$, as given by Eq. (\[eq:Effective\_SelectionRule\]). (color online)](fig9){width="8.6cm"}
### Continuum limit expression for the stationary current: the emergence of Landauer transport
Using Eqs. (\[eq:Gamma\_approx\_final\]) and (\[eq:Pi\_approx\_final\]), we can write the time-dependent Kubo formula Eq. (\[eq:LRTCurrent-1\]) as $$I^{n}\left(t\right)=\frac{e}{2}^{2}\sideset{}{^{\prime}}\sum_{k,q}\mathcal{P}_{k,q}\frac{\sin\left(\Delta\varepsilon_{k,q}t/\hbar\right)}{\Delta\varepsilon_{k,q}},\label{eq:emergent_Landauer}$$ where the primed sum in Eq. (\[eq:emergent\_Landauer\]) means that only pairs of states $\left(k,q\right)$ of opposite classes are included in the sum, due to the emergent selection rule. We also introduced the quantity $\mathcal{P}_{k,q}$, which is defined as $$\begin{aligned}
\mathcal{P}_{k,q} & =\frac{2w}{\hbar}\Pi_{k,q}^{n}\Gamma_{k,q}\Delta f_{k,q}\nonumber \\
& \simeq-\frac{1}{2L_{l}^{2}}\frac{\left|t\left(k_{\text{F}}\right)\right|^{2}v_{\text{F}}^{2}\hbar}{\abs{\Delta\varepsilon_{k,q}}}\Delta V,\label{eq:FullPrefactorEq}\end{aligned}$$ where we approximated $v_{\text{F}}\hbar\left(k-q\right)\simeq\Delta\varepsilon_{k,q}$, with $v_{\text{F}}=2w\sin\left(k_{\text{F}}\right)/\hbar$. To make Eqs.(\[eq:emergent\_Landauer\]) and (\[eq:FullPrefactorEq\]) more clear, we remark that this definition of the current is no longer dependent on the condition $k<k_{\text{F}}<q$, and $\mathcal{P}_{k,q}$ is actually symmetrical upon exchange of the indexes. The above equation also shows that, for $k,q\simeq k_{\text{F}}$, the latter approximately only a function of the difference in eigenenergies. This result was checked numerically as seen in Fig. \[FullPrefactors\], where we can see that for a wide variety of disordered samples and different values of the Fermi energy, all the values of $\mathcal{P}_{k,q}$ (calculated directly from the wavefunctions in Eq. (\[RealSpaceAmplitudes-1\])) fall into the curve Eq.(\[eq:FullPrefactorEq\]).
The time-dependent current in the continuum regime can thus be written as
$$\begin{gathered}
I^{n}\left(t\right)=\frac{e^{2}}{\hbar}\left|t\left(k_{\text{F}}\right)\right|^{2}\left(v_{\text{F}}\hbar\right)^{2}\Delta V\\
\times\int_{-\infty}^{+\infty}d\left(\Delta\varepsilon\right)\frac{\sin\left(\Delta\varepsilon t/\hbar\right)}{\Delta\varepsilon\abs{\Delta\varepsilon}}\varrho\left(\Delta\varepsilon\right),\label{eq:current_JDoCS}\end{gathered}$$
where we introduced the *joint density of contributing states* (JDoCS), $\varrho$, as
$$\varrho\left(\epsilon_{\text{F}},\Delta\varepsilon\right)=\frac{1}{4L_{l}^{2}}\sideset{}{^{'}}\sum_{\overset{k,q}{}}\delta\left(\Delta\varepsilon-\Delta\varepsilon_{k,q}\right).\label{eq:JDoCS}$$
The restricted summation in Eq.(\[eq:JDoCS\]) already takes into account the emergent selection rule of Eq. (\[eq:Effective\_SelectionRule\]). In Appendix \[appx:CalculationJDoCS\], we show that this quantity, in the limit $L_{l}\to\infty$, can be written in terms of the density of states of each class in a fully clean system and its expression for small enough $\abs{\Delta\varepsilon}$ is simply
$$\lim_{L_{l}\to\infty}\left[\varrho\left(\varepsilon_{\text{F}},\Delta\varepsilon\right)\right]=\frac{\abs{\Delta\varepsilon}}{2\pi^{2}\left(4w^{2}-\varepsilon_{\text{F}}^{2}\right)}+\mathcal{O}\left[\Delta\varepsilon^{2}\right].\label{LinearizedDPEP}$$
Hence, when Eq. (\[LinearizedDPEP\]) is plugged into Eq. (\[eq:current\_JDoCS\]), we get
$$\begin{gathered}
I^{n}\left(t\right)=\frac{e^{2}}{2\pi^{2}\hbar}\frac{\left|t\left(k_{\text{F}}\right)\right|^{2}\left(v_{\text{F}}\hbar\right)^{2}\Delta V}{4w^{2}-\varepsilon_{\text{F}}^{2}}\\
\times\int_{-\infty}^{\infty}d\left(\Delta\varepsilon\right)\frac{\sin\left(\Delta\varepsilon t/\hbar\right)}{\Delta\varepsilon}.\label{eq:current_JDoCS-1}\end{gathered}$$
Finally, Eq.(\[eq:current\_JDoCS-1\]) together with the facts that
$$\lim_{T\to\infty}\left[\frac{\sin\left[xT\right]}{x}\right]=\pi\delta\left(x\right),$$
and $v_{F}\hbar=\sqrt{4w^{2}-\varepsilon_{F}^{2}}$, yields a steady-state current $$I^{n}\left(t\right)=\frac{e^{2}}{h}\left|t\left(k_{F}\right)\right|^{2}\Delta V,\label{eq:Landauer_Current}$$ which is precisely the linear Landauer steady-state current for a two-terminal one-dimensional device.
Notice, that in the derivation of this result from the time-dependent Kubo formula, it is essential that $t,L_{l}\rightarrow\infty$ with $wt/\hbar\ll L_{l}$, such that the $\varrho\left(\Delta\varepsilon\right)$ can be evaluated in the limit of $L_{l}\rightarrow\infty$, while the factor $\sin\left(\Delta\varepsilon t/\hbar\right)/\Delta\varepsilon$ is treated as as emergent $\delta$-function. When $wt/\hbar\apprge L_{l}$, then there will be few pairs of states with $\Delta\varepsilon_{k,q}\in\left[\varepsilon_{F}-\hbar t^{-1},\varepsilon_{F}+\hbar t^{-1}\right]$, and we can no longer threat $\sin\left(\Delta\varepsilon t/\hbar\right)/\Delta\varepsilon$ as a $\delta$-function. When this happens, we start observing recurrences in the current as reported in Sec. (\[sec:NumericalResults\]).
![\[FullPrefactors\]Plots of the full prefactors $\mathcal{P}\left(\varepsilon_{\text{F}},k-q\right)$ for different central samples at half-filling (upper panels) and different Fermi energies, $\varepsilon_{\text{F}}$ (lower panels). The collapse of all the data into the red dashed curves justifies the validity of expression of Eq.(\[eq:FullPrefactorEq\]) for states close to the Fermi level. In both panels $\mathcal{P}_{k,q}$ is measured in units of $w^{2}/\hbar$. (color online)](fig10a){width="8.6cm"}
![\[FullPrefactors\]Plots of the full prefactors $\mathcal{P}\left(\varepsilon_{\text{F}},k-q\right)$ for different central samples at half-filling (upper panels) and different Fermi energies, $\varepsilon_{\text{F}}$ (lower panels). The collapse of all the data into the red dashed curves justifies the validity of expression of Eq.(\[eq:FullPrefactorEq\]) for states close to the Fermi level. In both panels $\mathcal{P}_{k,q}$ is measured in units of $w^{2}/\hbar$. (color online)](fig10b){width="8.6cm"}
\[sec:Conclusions\]Conclusions
==============================
In this work, we investigated how a quasi-steady-state particle transport regime emerges across disordered samples coupled to large, but finite leads which are subjected to a potential bias. In order to do so, we have studied time-dependent transport, both numerically and semi-analytically, in a non-interacting and one-dimensional tight-binding chain, with open boundary conditions, where the central region is an extended disordered sample, and the rest of the chain acts as a pair of finite, but otherwise perfect leads.
For large lead size, and sufficiently large bias, a quasi-steady-state regime emerges at intermediate times, after the transient behavior has died out and before inversions in the current are observed. The current in the quasi-steady-state is approximately constant in time and homogeneous in space (if measure at points far away from the chain’s extremities). Furthermore, the value of the current in the quasi-steady-state coincides with the one predicted by the Landauer formula for semi-infinite leads, independently of the initial condition of the system (*partitioned* or *partition-free*). These results constitute an exemplification and extension to finite systems of the results of Stefanucci *et al*[@stefanucci_time-dependent_2004] on the establishment of a steady-state regime of transport in samples which are attached to infinite leads.
We have found that the quasi-steady-steady is established, for both initial conditions, after a stabilization time $t_{\text{stab}}\approx2v_{\text{F}}^{-1}L_{\text{s}}$. Physically, this can be interpreted as the time taken by a Fermi-level state to probe the disordered landscape inside the central sample. The quasi-steady-state lasts until a *recurrence time* $t_{\text{r}}\approx2v_{\text{F}}^{-1}L_{\text{l}}$, where current inversions start happening. Besides being related to the inverse spacing of the energy levels in the system[@pal_emergence_2018], this recurrence time may also be interpreted as the time taken by a Fermi-level electron to leave the sample and return to it, by traveling back and forth inside a lead. This conclusion was seen to be independent of the central sample’s features, as long as the leads are much larger than it.
During the quasi-steady-state, persistent finite-size effects are observed in the *partition-free* approach as superposed oscillations, with a period that is inversely proportional to the bias $\Delta V$, and an amplitude which scales to zero as $L_{\text{l}}\to\infty$ but becomes more relevant (relative to $I_{\text{Landauer}}$) for very small values of $\Delta V$. This effect prevents the onset of a quasi-steady-state regime for a system prepared in the *partition-free* setup, if the leads are too small. In the *partitioned* case, the amplitude of the oscillations superposed on the quasi-steady-state plateaux is not influenced by the size of leads, but instead are damped as the observation time is increased (while keeping $t<t_{\text{r}}$). Similarly to the *partition-free* case, the amplitude of the fluctuations increases for smaller biases. These observations seem to indicate that the observation of a clear quasi-steady-state requires some kind of mechanism which scatters the electron’s momenta@bushong_approach_2005. Here it is provided by the applied potential ramp in the sample, which becomes less effective mechanism as $\Delta V\to0$. In both cases, these finite-size oscillations can be made arbitrarily small if $L_{l}$ is large enough.
In order to shine light on the numerical results, a time-dependent Kubo formula for the current in the *partition-free* approach, which is suitable for semi-analytical treatment, was developed in order to describe the local time-dependent current due to a small applied bias. From this formula, it was possible to see that an approximately time-independent and spatially uniform current emerges in the limit of large system’s size and observation times, $L_{l},t\rightarrow\infty$, provided $t\ll v_{\text{F}}^{-1}L_{l}$, in agreement with the recurrence time observed numerically. These conditions are necessary to treat the leads as being effectively infinite, in what respects DC-transport. After expressing the eigenfunctions of the disordered central sample in terms of complex reflection and transmission coefficients, all the matrix elements appearing in the Kubo formula were evaluated semi-analytically. The quasi-steady-state current thus obtained was shown to reproduce the Landauer formula for the current in a two-terminal device.
We hope that these theoretical predictions of the time scales over which the quasi-steady-state occurs and the nature of the finite-size oscillations can be experimentally tested and guide future research on mesoscopic transport in fermionic ultra-cold atomic gases in optical lattices.
Acknowledgments
===============
J.M.V.P.L. and J.P.S.P. acknowledge financing of Fundação da Ciência e Tecnologia, of COMPETE 2020 program in FEDER component (European Union), through projects POCI-01-0145-FEDER-028887 and UID/FIS/04650/2013. J.P.S.P. is supported by the MAP-fis PhD grant PD/BD/142774/2018 of Fundação da Ciência e Tecnologia. B.A. acknowledges financial support from Fundação para a Ciência e a Tecnologia, Portugal, through Project Nº CEECIND/02936/2017. Additionally, J.P.S.P. also acknowledges the hospitality of the University of Central Florida, where part of this work was done, as well as Dr. Eduardo Mucciolo and Dr. Caio H. Lewenkopf, for the enlightening discussions and careful reading of the manuscript. J.P.S.P. also acknowledges the hospitality of Katherine Vasquez during his stay in Orlando, Florida.
\[appx:TimeEvolutionMethod\]Review of the recursive Chebyshev method for quantum time-evolution
===============================================================================================
In this appendix, we wish to describe shortly the algorithm used to time-evolve an arbitrary single-particle state with the full Hamiltonian. As referred in the main text, the Hamiltonian generating the time-evolution for positive times, $\mathcal{H}\left(t>0\right)$, is time-independent and, consequently, the time-evolution operator $\mathcal{U}_{t}$ reads
$$\mathcal{U}_{t}=e^{-i\mathcal{H}\left(t>0\right)t/\hbar}.$$
The method used to calculate $\mathcal{U}_{t}$ for our systems is based on its exact expansion as a series of Chebyshev polynomials in $\mathcal{H}\left(t>0\right)$, due to Tal-Ezer *et al*[@KPM_Timeevolution_Talezer1984]. Namely, one has
$$\mathcal{U}_{t}=\sum_{n=0}^{\infty}\frac{2}{1+\delta_{n,0}}\left(-i\right)^{n}J_{n}(\lambda t)T_{n}(\tilde{\mathcal{H}}),\label{ChebyshevExpansion_of_Ut}$$
where $\mathcal{\tilde{H}}=\left(\nicefrac{1}{\lambda w}\right)\mathcal{H}$ is a dimensionless Hamiltonian, rescaled by a real parameter $\lambda$ which guarantees that its spectrum is contained inside the interval $\left]-1,1\right[$, $T_{n}$ is the $n^{\text{th}}$-order Chebyshev polynomial of the first-kind, $J_{n}(y)$ is a Bessel function of the first kind and $t$ is a time measured in units of $\frac{\hbar}{w}$. The key to the method is to avoid the numerical diagonalization of $\mathcal{H}\left(t>0\right)$, and instead use the recursion relation for the Chebyshev polynomials,
$$T_{n+1}(x)=xT_{n}(x)-T_{n-1}(x),\label{ChebyshevRecursion}$$
in order to evaluate all the needed $T_{n}\left(\tilde{\mathcal{H}}\right)$, recursively. For a generic review on the application of Chebyshev spectral method to physical problems see Ref.[@weise_kernel_2006] and references within.
![\[ChebyshevConvergence\]Comparison between the exact graph for $f(x)=\text{Re}\left[e^{iyx}\right]$ and successive truncated Chebyshev series with the first $M=20,40$ and $60$ polynomials. The colored arrows stand on the values for which the corresponding approximations starts to fail. The imaginary part has an analogous behavior. (color online)](fig11)
Furthermore, the Chebyshev series of Eq.(\[ChebyshevExpansion\_of\_Ut\]) is known to converge rather quickly, meaning that a truncated summation with $M$ terms is usually enough to describe correctly $\mathcal{U}_{t}$, provided $M>t\lambda$. This convergence is illustrated in Fig.\[ChebyshevConvergence\] and in all our calculations, we used $M=8t\lambda$.
Notice that, in order to evaluate the current, we only require to time-evolve a given single-particle state $\ket{\Psi}$. Therefore, we do need the full matrix form of $\mathcal{U}_{t}$, but instead how it acts on an arbitrary state $\ket{\Psi}$. From the expansion of Eq.(\[ChebyshevExpansion\_of\_Ut\]), we know that action to be
$$\ket{\Psi^{M}(t)}=\sum_{n=0}^{M}\frac{2}{1+\delta_{n,0}}\left(-i\right)^{n}J_{n}(\lambda t)\ket{\Psi_{n}},\label{ChebyshevExpansion_of_state-1}$$
where $\ket{\Psi_{n}}=T_{n}\left(\tilde{\mathcal{H}}\right)\ket{\Psi}$ and $M$ is the truncation order of the Chebyshev expansion.[^3] Finally, the first two $\ket{\Psi_{n}}$ can be directly calculated by the simple forms of $T_{0}\left(x\right)$ and $T_{1}\left(x\right)$, i.e.
$$\begin{aligned}
\ket{\Psi_{0}} & =T_{0}(\tilde{\mathcal{H}})\ket{\Psi}=\ket{\Psi}\\
\ket{\Psi_{1}} & =T_{1}(\tilde{\mathcal{H}})\ket{\Psi}=\tilde{\mathcal{H}}\ket{\Psi},\end{aligned}$$
and then the remaining are efficiently calculated by using the operator generalization of the Chebyshev recursion [\[]{}Eq.(\[ChebyshevRecursion\])[\]]{}, i.e.
$$\ket{\Psi_{n+1}}=\tilde{\mathcal{H}}\ket{\Psi_{n}}-\ket{\Psi_{n-1}}.$$
\[appx:RecursiveTransferMatrixAppendix\]Review of the recursive transfer matrix method
======================================================================================
In this Appendix, we explore a very simple algorithm which allows us to calculate the transfer matrix $\mathcal{M}\left(k\right)$ of any given disordered sample, when it is connected to semi-infinite leads. This method is the same used in the early papers of Andereck *et al*[@andereck_numerical_1980] and Pichard[@pichard_one-dimensional_1986] and allows for the calculation of $\mathcal{M}\left(k\right)$ with an $\sim\mathcal{O}\left(L_{S}\right)$ number of operations.
For these purposes, it is more useful to re-express the Hamiltonian of the central sample in a first-quantization language, i.e.
$$\begin{aligned}
\mathcal{H}_{s} & =\sum_{n=1}^{L_{s}}\varepsilon_{n}\left|n\right\rangle \left\langle n\right|\\
& \qquad\qquad-\sum_{n=1}^{L_{s}-1}\left(\left|n\right\rangle \left\langle n+1\right|+\left|n+1\right\rangle \left\langle n\right|\right),\nonumber \end{aligned}$$ where $\left|n\right\rangle $ are the Wannier states of the chain and $\varepsilon_{n}$ is an on-site energy (in units of the hopping $w$). To model the connection between the finite sample to the semi-infinite leads, one has also the following boundary hopping Hamiltonian:
[ $$\mathcal{H}_{s}=-\ket 0\bra 1-\ket 1\bra 0-\ket{L_{s}}\bra{L_{s}+1}-\ket{L_{s}+1}\bra{L_{s}}.$$ ]{}
The main purpose of this method is to find the scattering states associated to a particular disorder realization. For that, one must fix the leads’ propagating states, $\ket{\Psi_{\pm}^{L}}$ and $\ket{\Psi_{\pm}^{R}}$, as the left and right boundary conditions for the problem. This setup is represented in Fig.\[fig:TransferMatrixSetup\], with the counter-propagating plane-waves in the leads being represented as arrows.
![\[fig:TransferMatrixSetup\]Schematic representation of the setup used in the implementation of the transfer matrix method. Red dots represent the disordered scattering region. The leads are represented as the lighter red “ghost” sites on both sides. (color online)](fig12)
Hamiltonian in Real-Space and Boundary Conditions
-------------------------------------------------
The first step towards the definition of the present method is expanding a scattering state (with wavenumber $k$) in the basis of Wannier wavefunctions, i.e.
$$\ket{\Psi_{k}}=\sum_{n}\psi_{n}\ket n,$$
and finally rewriting the time-independent Schrödinger equation — $\mathcal{H}\ket{\Psi_{k}}=E_{k}\ket{\Psi_{k}}$ — in terms of the real-space amplitudes $\psi_{n}$,
$$E_{k}\psi_{n}=\varepsilon_{n}\psi_{n}-\psi_{n-1}-\psi_{n+1},\label{SchroedingerEq_RealSpace}$$
where, by definition, $\varepsilon_{n}=0$ outside of the sample.
As shown in Fig.\[fig:TransferMatrixSetup\], the boundary conditions are to be set as the plane-waves defined in the Eq.(\[eq:wavefunctions\_leads\]) of the main text. Reminding, one has
[ $$\ket{\Psi_{k}^{L}}=\sum_{n=-L_{l}}^{-1}\left[\Psi_{+}^{L}e^{ik\left(n-1\right)}\ket n+\Psi_{-}^{L}e^{-ik\left(n-1\right)}\ket n\right],$$ ]{}
[ $$\ket{\Psi_{k}^{R}}=\sum_{n=1}^{L_{l}}\left[\Psi_{+}^{R}e^{ik\left(n-L_{s}\right)}\ket n+\Psi_{-}^{R}e^{-ik\left(n-L_{s}\right)}\ket n\right].$$ ]{}These states immediately set the amplitudes on the “ghost” sites of Fig.\[fig:TransferMatrixSetup\] to the following values:
$$\begin{aligned}\psi_{-1} & =\Psi_{+}^{L}e^{-2ik}+\Psi_{-}^{L}e^{2ik}\\
\psi_{0} & =\Psi_{+}^{L}e^{-ik}+\Psi_{-}^{L}e^{ik}\\
\psi_{L_{S}+1} & =\Psi_{+}^{R}e^{ik}+\Psi_{-}^{R}e^{-ik}\\
\psi_{L_{S}+2} & =\Psi_{+}^{R}e^{2ik}+\Psi_{-}^{R}e^{-2ik}
\end{aligned}
.\label{BoundaryConditions}$$
Review of the Transfer Matrix Recursive Method
----------------------------------------------
Despite not having the look of a linear algebra problem, Eq.(\[SchroedingerEq\_RealSpace\]) may be turned into a matrix recursion equation, when supplemented by the trivial condition
$$\psi_{n}=\psi_{n}.$$ Hence, we have
$$\left(\begin{array}{c}
\psi_{n+1}\\
\psi_{n}
\end{array}\right)=\underset{\mathbb{T}_{n}\left(k\right)}{\underbrace{\left(\begin{array}{cc}
\varepsilon_{n}-E_{k} & -1\\
1 & 0
\end{array}\right)}}\cdot\left(\begin{array}{c}
\psi_{n}\\
\psi_{n-1}
\end{array}\right).\label{TransferMatrixOneStep}$$
If we now iterate Eq.(\[TransferMatrixOneStep\]), we get the following relation
$$\begin{gathered}
\left(\begin{array}{c}
\psi_{L_{s}+2}\\
\psi_{L_{s}+1}
\end{array}\right)=\mathbb{T}_{L_{s}+1}\left(k\right)\cdot\mathbb{T}_{L_{s}}\left(k\right)\cdot\\
\cdots\cdot\mathbb{T}_{1}\left(k\right)\cdot\mathbb{T}_{0}\left(k\right)\cdot\left(\begin{array}{c}
\psi_{0}\\
\psi_{-1}
\end{array}\right).\label{TransferMatrixOneStep-1}\end{gathered}$$
In the same way, we may write the boundary conditions of Eqs.(\[BoundaryConditions\]), as the following matrix relations:
$$\left(\begin{array}{c}
\psi_{0}\\
\psi_{-1}
\end{array}\right)=\underset{\mathbb{B}_{L}\left(k\right)}{\underbrace{\left(\begin{array}{cc}
e^{-ik} & e^{ik}\\
e^{-2ik} & e^{2ik}
\end{array}\right)}}\cdot\left(\begin{array}{c}
\Psi_{+}^{L}\\
\Psi_{-}^{L}
\end{array}\right),\label{BoundaryLeft}$$
and
$$\left(\begin{array}{c}
\psi_{L_{s}+2}\\
\psi_{L_{s}+1}
\end{array}\right)=\left(\begin{array}{cc}
e^{2ik} & e^{-2ik}\\
e^{ik} & e^{-ik}
\end{array}\right)\cdot\left(\begin{array}{c}
\Psi_{+}^{R}\\
\Psi_{-}^{R}
\end{array}\right),$$
which can be inverted as
$$\left(\begin{array}{c}
\Psi_{+}^{R}\\
\Psi_{-}^{R}
\end{array}\right)=\mathbb{B}_{R}\left(k\right)\cdot\left(\begin{array}{c}
\psi_{L_{s}+2}\\
\psi_{L_{s}+1}
\end{array}\right)\label{BoundaryRight}$$
Using Eqs.(\[BoundaryLeft\]) and (\[BoundaryRight\]) into Eq.(\[TransferMatrixOneStep-1\]), we get to the following final result:
$$\begin{gathered}
\left(\begin{array}{c}
\Psi_{+}^{R}\\
\Psi_{-}^{R}
\end{array}\right)=\mathbb{B}_{R}\left(k\right)\cdot\mathbb{T}_{L_{s}+1}\left(k\right)\cdot\mathbb{T}_{L_{s}}\left(k\right)\cdot\\
\cdots\cdot\mathbb{T}_{1}\left(k\right)\cdot\mathbb{T}_{0}\left(k\right)\cdot\mathbb{B}_{L}\left(k\right)\cdot\left(\begin{array}{c}
\Psi_{+}^{L}\\
\Psi_{-}^{L}
\end{array}\right),\end{gathered}$$
and, by definition, the transfer matrix of the whole sample is written as:
$$\begin{gathered}
\mathcal{M}\left(k\right)=\mathbb{B}_{R}\left(k\right)\cdot\mathbb{T}_{L_{s}+1}\left(k\right)\cdot\mathbb{T}_{L_{s}}\left(k\right)\cdot\\
\cdots\cdot\mathbb{T}_{1}\left(k\right)\cdot\mathbb{T}_{0}\left(k\right)\cdot\mathbb{B}_{L}\left(k\right).\end{gathered}$$
This last equation was the one we implemented to calculate $\mathcal{M}\left(k\right)$ for any given disordered sample.
\[appx:SelectionRule\]Emergence of selection rule
=================================================
In this appendix, we prove the effective selection rule of Eq. (\[eq:Effective\_SelectionRule\]). In order to do so, we will analyze the factor $\sin\left[\left(k-q\right)L_{l}\right]$, when $q,k$ belong to the same or different classes. More precisely, will calculate its absolute value, which can be written as
$$\begin{gathered}
\abs{\sin\left[\left(k-q\right)L_{l}\right]}=\sqrt{\cfrac{1-\cos\left[2L_{l}\left(q-k\right)\right]}{2}}\\
=\frac{1}{\sqrt{2}}\left\{ 1-\cos\left[2\left(L_{l}+1\right)\left(q-k\right)+\phi\left(q\right)-\phi\left(k\right)\right]\cos\left[\phi\left(q\right)-\phi\left(k\right)-2\left(q-k\right)\right]\right.\\
-\left.\sin\left[2\left(L_{l}+1\right)\left(q-k\right)+\phi\left(q\right)-\phi\left(k\right)\right]\sin\left[\phi\left(q\right)-\phi\left(k\right)-2\left(q-k\right)\right]\right\} ^{\frac{1}{2}},\end{gathered}$$
where we summed and subtracted $\phi\left(q\right)-\phi\left(k\right)$ in the argument of the cosine and, then, decomposed it using the rule for the cosine of a sum of angles. The main advantage of this form is that the continuous function $\phi\left(k\right)$ depends solely in the properties of the central sample and the effect of increasing the leads is to populate more densely their domains with allowed values of $k$. This, together with the fact that we are only interested in what happens near $k_{\text{F}}$, allows us to expand it as Taylor series on $\delta q=q-k_{\text{F}}$ and $\delta k=k_{\text{F}}-k$:
$$\phi\left(q\right)-\phi\left(k\right)=\left.\frac{d}{dk}\phi\right|_{k_{\text{F}}}\left(\delta q+\delta k\right)+\cdots\simeq\left.\frac{d}{dk}\phi\right|_{k_{\text{F}}}\left(q-k\right),$$
and, consequently,
$$\begin{aligned}
\abs{\sin\left[\left(q-k\right)L_{l}\right]}\simeq & \sqrt{\cfrac{1-\cos\left[2L_{l}\left(q-k\right)+\phi\left(q\right)-\phi\left(k\right)\right]}{2}},\label{abssin2}\end{aligned}$$
where the corrections are of order $q-k$ and disappear in the limits $L_{l}\to\infty$ and $\hbar t^{-1}\to0$. At this point, all we must do is to decompose the cosine term in Eq.(\[abssin2\]) using the usual rules for the sum of angles and then resort to the quantization condition of Eq.(\[eq:QuantizationCondition\]) to realize that
$$\begin{gathered}
\cos\left[2\left(L_{l}+1\right)\left(q-k\right)+\phi\left(q\right)-\phi\left(k\right)\right]=\\
\mp\sqrt{\left[1-\abs{r\left(q\right)}^{2}\sin^{2}\left(\theta\left(q\right)-\phi\left(q\right)\right)\right]\left[1-\abs{r\left(k\right)}^{2}\sin^{2}\left(\theta\left(k\right)-\phi\left(k\right)\right)\right]}\\
+\abs{r\left(q\right)}\abs{r\left(k\right)}\sin\left[\theta\left(q\right)-\phi\left(q\right)\right]\sin\left[\theta\left(k\right)-\phi\left(k\right)\right],\label{CosDecomposition}\end{gathered}$$
where the $+\left(-\right)$ sign stands for the case when $q$ and $k$ are in the same class (different classes) of states.
Finally, one can evoke the same argument as before to Taylor expand all the sample-specific functions appear in Eq.(\[CosDecomposition\]) (to be clear, $r\left(x\right)$,$\theta\left(x\right)$,$\phi\left(x\right)$) around $k_{\text{F}}$, but noting that $k<k_{\text{F}}<q$ by definition. Up to corrections irrelevant correction in the same limits, this gives rise to Eq.(\[eq:Effective\_SelectionRule\]) of the main text after expanding the $\sin$ functions in powers of $q-k$.
$ $
$ $
\[appx:CalculationJDoCS\]Calculation of the joint density of contributing states
================================================================================
In this appendix, we will proceed to calculate the joint density of contributing states (JDoCS), for both positive and negative $\Delta\varepsilon$. For positive energy differences, $\Delta\varepsilon>0$, the JDoCS is defined, from Eq.(\[eq:JDoCS\]), as
$$\varrho\left(\epsilon_{\text{F}},\Delta\varepsilon\right)=\frac{1}{4L_{l}^{2}}\sideset{}{^{'}}\sum_{\overset{k,q}{(\varepsilon_{q}<\varepsilon_{\text{F}}\leq\varepsilon_{k})}}\delta\left(\Delta\varepsilon-\Delta\varepsilon_{k,q}\right).\label{eq:JDoCS-1}$$
Which may be written in terms of the usual density of states for each class, $\sigma=\pm$, i.e.
$$\begin{aligned}
\rho^{\sigma}\left(\varepsilon\right) & =\frac{1}{L_{l}}\sum_{k^{\sigma}}\delta\left(\varepsilon-\varepsilon_{k^{\sigma}}\right),\label{DoSDefinition}\end{aligned}$$ yielding the expression,
$$\begin{gathered}
\varrho\left(\varepsilon_{\text{F}},\Delta\varepsilon\right)=\frac{1}{4}\int_{\varepsilon_{F}}^{2}d\epsilon_{2}\int_{-2}^{\varepsilon_{F}}d\epsilon_{1}\lim_{L_{l}\to\infty}\left\{ \rho_{L_{l}}^{+}\left(\epsilon_{1}\right)\rho_{L_{l}}^{-}\left(\epsilon_{2}\right)\right.\\
\left.+\rho_{L_{l}}^{-}\left(\epsilon_{1}\right)\rho_{L_{l}}^{+}\left(\epsilon_{2}\right)\right\} \delta\left(\Delta\varepsilon-\epsilon_{2}+\epsilon_{1}\right),\label{eq:DPED}\end{gathered}$$ in the limit of semi-infinite leads.
![\[DOSCovergence\]Plots of the DoS calculated using the KPM for a system with leads of different sizes and a central sample without (black curve) and with disorder (colored curves). The number of Chebyshev moments used is $M=4096$ for all the cases. The insets are zooms made to the regions indicated by the black boxes in the main graph, where one can clearly see the spectral weight of the states in the sample being out-weighted by the states coming from the finite clean leads. (color online)](fig13)
To progress beyond Eq.(\[eq:DPED\]) in a general fashion, one starts by recognizing that, since $\rho^{\pm}\left(\varepsilon\right)$ is an intensive quantity. So these must be dominated by the states on the (clean) leads, as $L_{l}\to\infty$. Since we know that, for a clean system, the states of different parities are alternated in $k$-space, with a regular separation given by $\pi/L_{l}$, one concludes that
$$\lim_{L_{l}\to\infty}\rho_{L_{l}}^{\pm}\left(\varepsilon\right)=\rho\left(\varepsilon\right)=\begin{cases}
\frac{1}{\pi\sqrt{4w^{2}-\varepsilon^{2}}} & \text{if}\abs{\varepsilon}\leq2w\\
0 & \text{if}\abs{\varepsilon}>2w
\end{cases},\label{eq:DOS1D}$$
where $\rho\left(\varepsilon\right)$ is the full DoS of a clean infinite chain. In what follows, we will always assume that the expression of Eq.(\[eq:DOS1D\]) may be used to calculate de JDoCS in the limit of very large $L_{l}$. This intuition is confirmed by the plots of the DoS in Fig.\[DOSCovergence\], which were obtained numerically, for a randomly selected disordered sample, using the well-known *kernel polynomial method* with a Jackson kernel and a fixed number of polynomials, $M=4096$, enough to resolve the individual energy levels in the smaller case considered (see Weiße *et al*[@weise_kernel_2006] for more details on the method). Consequently, one has the following expression for the JDoCS
where $\Theta\left(x\right)$ is the Heaviside function and $\Delta\varepsilon\geq0$. The integral in Eq.(\[eq:DPED-1-1\]) can be done numerically and the curves are shown in Fig.\[DPED\] for different values of the Fermi energy $\epsilon_{\text{F}}$. Nevertheless, we are only interested in the shape of $\varrho\left(\varepsilon,\Delta\varepsilon\right)$ when $\Delta\varepsilon\approx0$. For that, we may expand Eq.(\[eq:DPED-1-1\]) in powers of this quantity, yielding
$$\varrho\left(\varepsilon_{\text{F}},\Delta\varepsilon>0\right)=\frac{\Delta\varepsilon}{2\pi^{2}\left(4w^{2}-\varepsilon_{\text{F}}^{2}\right)}+\mathcal{O}\left[\Delta\varepsilon^{2}\right].\label{LinearizedDPEP-1}$$
Finally, we can generalize Eq.(\[LinearizedDPEP-1\]) to $\Delta\varepsilon<0$, which is trivial since, by definition [\[]{}Eq.(\[eq:JDoCS\])[\]]{}, we have $\varrho\left(\Delta\varepsilon\right)=\varrho\left(-\Delta\varepsilon\right)$. Hence, our final expression is simply,
$$\varrho\left(\epsilon_{\text{F}},\Delta\varepsilon\right)=\frac{\abs{\Delta\varepsilon}}{2\pi^{2}\left(4w^{2}-\varepsilon_{\text{F}}^{2}\right)}+\mathcal{O}\left[\Delta\epsilon^{2}\right],\label{LinearizedDPEP-1-1}$$
which is the one we use in the main text [\[]{}Eq. (\[LinearizedDPEP\])[\]]{}.
![\[DPED\]Plots of the JDoCS from the numerical integration of Eqs.(\[eq:DPED-1-1\]) for different values of the Fermi energy and positive values of $\Delta\varepsilon$. The dashed straight lines are plots of the linear approximations near $\varepsilon_{\text{F}}$, as calculated in Eq.(\[LinearizedDPEP-1\]). (color online)](fig14)
[^1]: Note that, despite being an evolution for negative times, the time-evolution operator used to do it is the one which includes the external perturbation: lead-sample hopping in the partitioned approach, and bias in the partition-free approach.
[^2]: As visible in Fig.\[fig:SinScaling\], there is small deviation of the data points obtained for the ordered central sample from the theoretical value $\abs{\sin\left(L_{l}\left(q^{\sigma}-k^{\sigma'}\right)\right)}=1-\delta_{\sigma,\sigma'}$. These results seem incompatible with an exact parity selection rule for $\Gamma_{k,q}$, for this case, however, they are not. This artifact is due to the fact that the data shown was calculated from the numerical diagonalization of $\mathcal{H}_{0}$ and then $k=\arccos\left(-E/2\right)$ was used to obtain the respective wavenumbers. This procedure takes into account the finite dimension of the central sample and hence the allowed wavenumbers are of the form $k^{\sigma}=\pi n/\left(2L_{l}+L_{s}+1\right)$, with $\sigma=\left(-1\right)^{n}$. Since the expression of $\Gamma_{k,q}$ which is proportional to $\sin\left(L_{l}\left(q-k\right)\right)$ is only valid under an approximation which ignores $L_{s}$ [\[]{}i.e. Eq.(\[eq:large\_lead:approx\])[\]]{}, the parity selection rule appears to be approximate as well. At any rate, it may be proven, by symmetry, that this rule is actually true for any value of $L_{l}$, if one considers the full expression for $\Gamma_{k,q}$.
[^3]: As this is needed in the main text, we remark that for backward time-evolutions, one may simply use the fact that $J_{n}\left(-x\right)=\left(-1\right)^{n}J_{n}\left(x\right)$.
|
---
abstract: 'We [give an alternative ]{}definition of quantum fidelity for two density operators on qudits in terms of the Hilbert-Schmidt inner product between them and their purity. It can be regarded as the well-defined operator fidelity for the two operators and satisfies all Jozsa’s four axioms up to a normalization factor. One desire property is that it is not computationally demanding.'
author:
- Xiaoguang Wang
- 'Chang-Shui Yu'
- 'X. X. Yi'
title: An alternative quantum fidelity for mixed states of qudits
---
Fidelity is an important concept in quantum information theory [@Nielsen] and quantum chaos [@fqchaos1]. The well-known quantum fidelity for two general mixed states $\rho_0$ and $\rho_1$ is given by the Uhlmann’s fidelity [Uhlmann,Hubner,Jozsa, Schumacher]{} $$\begin{aligned}
\mathcal{F}\left(\rho_0,\rho_1\right) =\text{tr}\sqrt{\rho_0^{1/2}{\rho_1}%
\rho_0^{1/2}}.\end{aligned}$$This fidelity has many nice properties such as concavity and multiplicativity under tensor product and it satisfies all Josza’s four axioms [@Jozsa]. [However, it is not an easy task to make analytical evaluation of the fidelity and even numerical calculations due to the square roots of Hermitian matrix in the above equation]{}. Quite recently, people tried to define new fidelities to avoid this difficulty. Miszczak et al. and Mendonça et al. defined the following fidelity [@Miszczak; @Mendonca] $$\begin{aligned}
\mathcal{F}_1\left(\rho_0,\rho_1\right) =\text{tr}(\rho_0\rho_1)+\sqrt{1-%
\text{Tr}(\rho_0^2)}\sqrt{1-\text{Tr}(\rho_1^2)},\end{aligned}$$Another fidelity which is essentially is the same as $\mathcal{F}_1$ is defined by Chen et al. as [@Chen] $$\begin{aligned}
\mathcal{F}_2\left(\rho_0,\rho_1\right)=\frac{1-r}2+\frac{1+r}2 \mathcal{F}%
_1\left(\rho_0,\rho_1\right),\end{aligned}$$ where $r=1/(d-1)$ with $d$ being the dimension of the Hilbert space. This fidelity displays a nice property that it has a clear hyperbolic geometric interpretation. Another property is that these two fidelities reduce to Uhlmann’s fidelity in the special case of dimension $d=2$.
One fundamental requirement for a definition of fidelity is that it must obey $F(\rho,\rho)=1$. All the above three definitions satisfies this condition. However, it is not sufficiently emphasized in earlier studies that when two density matrix are orthogonal, the fidelity should be zero. This could be another fundamental requirement for the fidelity. Consider the following density matrices $$\begin{aligned}
\rho_0=\frac{1}2(|0\rangle\langle 0|+|1\rangle\langle 1|), \notag \\
\rho_1=\frac{1}2(|2\rangle\langle 2|+|3\rangle\langle 3|),\end{aligned}$$ acting on four-dimensional Hilbert space spanned by $\{|n\rangle,
n=0,1,2,3\} $. Obviously, these two density matrix is orthogonal and the fidelity should be zero. After some simple calculations, we find that $\mathcal{F}=0, \mathcal{F}_1=1/2$, and $\mathcal{F}_2=2/3$. Thus, in this strict sense it is appropriate to call $\mathcal{F}_1$ super-fidelity which acts as a useful upper bound for the Uhlmann’s fidelity [@Miszczak].
In this paper, we introduce an alternative fidelity defined, which satisfies Josza’s axioms up to a normalization factor. And the fidelity is zero when two density matrices are orthogonal and is 1 when they are identical. We also discuss its other properties such as convexity and multiplicativity under tensor products.
The fidelity can be regarded as the operator fidelity [@wang07] and thus we begin by introducing the definition of operator fidelity between two operators. Let $\mathcal{H}$ be a $d$-dimensional Hilbert space. All linear operators on $\mathcal{H}$ on its own is a $d^2$-dimensional Hilbert space $%
\mathcal{H}_{\text{HS}}$. The inner product in this space is defined as the Hilbert-Schmidt product, i,e., for operators $A$ and $B$, $\langle
A|B\rangle=\text{Tr}(A^\dagger B)$. Thus, any linear operators on $\mathcal{H%
}$ can be considered as a state on $\mathcal{H}_{\text{HS}}$. Thus, the fidelity of two states can be naturally be lifted to the operator level.
To define the operator fidelity between two operators $A$ and $B$, we need to first normalize them as $A/\sqrt{\text{Tr}(AA^\dagger)}$ and $B/\sqrt{\text{Tr}%
(BB^\dagger)}$, respectively. Then, the operator fidelity is defined as $$F(A,B)=\frac{\left\vert \text{Tr}(A^{\dagger }B)\right\vert }{\sqrt{\text{Tr}%
(AA^{\dagger })\text{Tr}(BB^{\dag })}}.$$If we consider two unitary operators $U_{0}$ and $U_{1}$, the above fidelity reduces to $$F(U_{0},U_{1})=\frac{1}{d}|\text{Tr}(U_{0}^{\dagger }U_{1})|, \label{fff}$$which is studied in Ref. [@wang07] and can be applied to measure the sensitivity of quantum systems to perturbations. If two density operator $%
\rho $ and $\sigma $ are considered, the operator fidelity reduces to $$F(\rho_0,\rho_1 )=\frac{\left\vert \text{Tr}(\rho_0 \rho_1 )\right\vert }{\sqrt{%
\text{Tr}(\rho_0^{2})\text{Tr}(\rho_1 ^{2})}}. \label{newfi}$$This is a function of the Hilbert-Schmidt inner product and two purity (equivalent to linear entropy). The fidelity for two density operators can be considered as operator fidelity. On the other hand, it can also be regarded as fidelity between two states $\rho _{0}$ and $\rho _{1}$. This is the alternative definition of the fidelity. [One cannot simply define the fidelity as ]{}$|\text{Tr}(\rho _{0}\rho _{1})|$[ as it becomes less than one when the two density matrices are identical, i.e., ]{}$\text{Tr}(\rho _{0}^{2})<1$.
It is easy to show that the fidelity $F$ has the following desirable properties:
\(1) $F$ is normalized. The maximum 1 is attained if and only if $\rho_0 =\rho_1.$
\(2) $F$ is symmetric under swapping $\rho_0 \ $and $\rho_1$, i.e., $F(\rho_0 ,\rho_1 )=F(\rho_0 ,\rho_1).$
\(3) The fidelity is invariant under unitary transformation $U$ on the state space, i.e., $F(\rho_0 ,\rho_1 )=F(U\rho_0 U^{\dagger
},U\rho_1 U^{\dagger })$.
\(4) When one of the state is pure, say, $\rho_1=|\psi\rangle\langle\psi|$, the fidelity reduces to $F(\rho_0,|\psi\rangle\langle\psi|)=\langle\psi|\rho_0|%
\psi\rangle/\text{Tr}(\rho_0^2)$.
To compare with Jozsa’s four axioms, only the fourth property differs by a normalization factor $1/\text{Tr}(\rho_0^2)$. Then, we see that this fidelity satisfies all Jozsa’s axioms up to a normalization factor [@Jozsa]. Another obvious fact is that if two density matrices are orthogonal, the fidelity is zero. It is easy to check another nice property that the fidelity $F$ is multiplicative under tensor products, i.e., $F(\rho _{1}\otimes \rho
_{2},\sigma _{1}\otimes \sigma _{2})=F(\rho _{1},\sigma _{1})F(\rho
_{2},\sigma _{2})$. The Uhlmann’s fidelity also satisfies this property.
Next, we check that if $F$ satisfies the property of concavity or convexity. By numerical calculations, we find that the following inequality $$p_{1}F(\rho _{1},\sigma )+p_{2}F(\rho _{2},\sigma )\leq F(p_{1}\rho
_{1}+p_{2}\rho _{2},\sigma ),p_{1},p_{2}\geq 0$$is satisfied for most matrices $\rho _{1},\rho _{2}$ and $\sigma $ for $%
p_{1}+p_{2}=1$. [However, the violation can also be found. The most simplest demonstration can be given if ]{}$\rho _{1}=I/2 $, $\rho
_{2}=|0\rangle\langle 0|$ and $\sigma =|1\rangle\langle 1| $ for $%
p_{1},p_{2}\in (0,1)$. Here, states $|0\rangle$ and $|1\rangle$ are orthogonal and $I$ is the $2\times 2$ identity matrix.
To show the violation of concavity, let $$\tilde{\rho}=p\rho _{1}+(1-p)\rho _{2}=(1-p/2)|0\rangle \langle
0|+p/2|1\rangle \langle 1|,$$then $$F(\tilde{\rho},\sigma )=\frac{\langle 1|\tilde{\rho}|1\rangle }{\sqrt{\text{%
Tr}(\tilde{\rho}^{2})}}=\frac{p}{\sqrt{2}[1+(1-p)^{2}]^{1/2}},$$and $$pF(\rho _{1},\sigma )+(1-p)F(\rho _{2},\sigma )=pF(\rho _{1},\sigma )=\frac{p%
}{\sqrt{2}}.$$It is obvious that $$F(\tilde{\rho},\sigma )<pF(\rho _{1},\sigma )+(1-p)F(\rho
_{2},\sigma )$$for $p\in (0,1)$.
That is to say, $F$ satisfies neither concavity nor convexity. Since any measure is monotonically increasing (decreasing) if it is (i) unitarily invariant, (ii) jointly concave (convex) and (iii) invariant under the addition of an ancillary system [@Lieb], $F$ is not monotonically increasing or decreasing under quantum operations.
As an application, we consider thermal equilibrium density matrix $%
\rho_{k}=\exp(-\beta H_k)/Z(\beta) (k=0,1)$ acting on $d$-dimensional Hilbert space \[$Z_k(\beta)=\text{Tr}[\exp(-\beta H_k)]$ is the partition function for $k$-th system, $T=\beta^{-1}$ is the temperature, and the Boltzmann constant is assumed to be one\]. From Eq. (\[newfi\]), the fidelity for the two thermal states is given by $$\begin{aligned}
\label{newthermal}
&F(\rho_0,\rho_1) \notag \\
&=\frac{\text{Tr}(e^{-\beta H_0}e^{-\beta H_1})}{\sqrt{\text{Tr}(e^{-2\beta
H_0})\text{Tr}(e^{-2\beta H_0})}} \notag \\
&=\frac{\text{Tr}[(e^{-\beta H_0})^\dagger e^{-\beta H_1}]}{\sqrt{\text{Tr}%
[(e^{-\beta H_0})^\dagger e^{-\beta H_0}]\text{Tr}[(e^{-\beta H_1})^\dagger
e^{-\beta H_1}]}}.\end{aligned}$$ It is well-known that imaginary time (or imaginary temperature) is essential in connecting quantum mechanics and statistical mechanics. If we make the Wick rotation, i.e., let $\beta=it$, the above equation reduces to $$\label{newthermal1}
F(U_0,U_1)=1/d {\left|\text{Tr}(e^{itH_0}e^{-it H_1})\right|},$$ which is just the operator fidelity for two unitary operators $U_k$ generated by Hamiltonian $H_k$. Thus, we see that the fidelity for two thermal states is connected to the operator fidelity for two unitary evolution operators by the Wick rotation by $\pi/2$. The fidelity introduced here is expected to be applicable to studies of phase transitions and quantum chaos.
In conclusion, we have introduced an alternative fidelity which satisfiies Jozsa’s four axioms up to an normalization factor. It has a desire property that is multiplicative under tensor products and undesire one that it is neither convex nor concave. The relations between this fidelity and the operator fidelity was clarified. Another merit is that it is not computationally demanding. From an measurement point of view, this fidelity is relatively easy to measure as it contains only the Hilbert-Schmidt inner product and two purity.
Acknowledgements
================
We are indebted to S. J. Gu and Z. W. Zhou for fruitful and valuable discussions. X. Wang acknowledges the support from C. N. Yang foundation of CUHK, the Program for New Century Excellent Talents in University (NCET), the NSFC with grant nos. 90503003, the State Key Program for Basic Research of China with grant nos. 2006CB921206, and the Specialized Research Fund for the Doctoral Program of Higher Education with grant No.20050335087.
[99]{} M.A. Nielsen, I.L. Chuang, *Quantum Computation and Quantum Information*, Cambridge Univ. Press, Cambridge, 2000.
T. Gorina, T. Prosen, T. H. Seligman, M. Žnidaric , Phys. Reports **435**, 33 (2006).
D. Bures, Tran. Am. Math. Soc. **135**, 199 (1969); A. Uhlmann, Rep. Math. Phys. **9**, 273 (1976).
M. Hübner, Phys. Lett. A **163**, 229 (1992).
R. Jozsa, J. Mod. Opt. **41**, 2315 (1994).
B. Schumacher, Phys. Rev. A **51**, 2738 (1995).
J. A. Miszczak, Z. Puchala, P. Horodecki, A. Uhlmann, K. Życzkowski, arXiv:0805.2037.
P. E. M. F. Mendonça et al., arXiv:0806.1150.
J. L. Chen, L. B. Fu, A. A. Ungar, and X. G. Zhao, , 054304 (2002).
X. Wang, Z. Sun and Z. D. Wang, arXiv: 0803.2940.
E. A. Carlen and E. H. Lieb, Lett. Math. Phys. **83**, 107 (2008).
|
---
author:
- '[**Denis I. Saveliev**]{}'
date: '2007 August 11, Beijing '
title: |
[**A Note on Singular Cardinals\
in Set Theory without Choice**]{}
---
In this talk, I discuss [*how singular*]{} can cardinals be in absence of AC, the axiom of choice. I shall show that, contrasting with known negative [*consistency*]{} results (of Gitik and others), certain positive results are [*provable*]{}. At the end, I pose some problems.
Given a set $X$, its [*cardinal*]{} number $$|X|$$ is the class of all sets of [*the same size*]{} that $X$, i.e., admitting a one-to-one map onto $X$.
Thus $$|X|=|Y|$$ means “There is a bijection of $X$ onto $Y$”.
Cardinals of nonempty sets are proper classes; so, we have a little technical obstacle: $$\text{How quantify cardinals?}$$ In some happy cases we can represent them by sets:
If $|X|$ is a [*well-ordered cardinal*]{}, i.e., meets the class of (von Neumann’s) ordinals, take the least such ordinal (an [*initial*]{} ordinal).
If $|X|$ is a [*well-founded cardinal*]{}, i.e., meets the class of well-founded sets, take the lower level of the intersection (so-called Scott’s trick).
What is in general? The answer is $$\text{No matter}$$ because instead of cardinals, we can say about sets and bijections.
Thus $$\varphi(|X|,|Y|,\ldots)$$ means $
\varphi(X',Y',\ldots)
$ whenever $|X|=|X'|$, $|Y|=|Y'|$, $\ldots$
Notations:
The German letters $${\mathfrak}l,{\mathfrak}m,{\mathfrak}n,\ldots$$ denote arbitrary cardinals. The Greek letters $$\lambda,\mu,\nu,\ldots$$ denote well-ordered ones (i.e., initial ordinals), while the Greek letters $$\alpha,\beta,\gamma,\ldots$$ denote arbitrary ordinals.
Two basic relations on cardinals (dual in a sense): $$|X|\le|Y|$$ means “$X$ is empty or there is an injection of $X$ into $Y$”, and $$|X|{ {\;\le^{\!*}\,} }|Y|$$ means “$X$ is empty or there is a surjection of $Y$ onto $X$”.
Equivalently,
$|X|\le|Y|$ means “There is a subset of $Y$ of size $|X|$”,
$|X|{ {\;\le^{\!*}\,} }|Y|$ means “$X$ is empty or there is a partition of $Y$ into $|X|$ pieces”.
Clearly:
\(i) Both $\le$ and $\!{ {\;\le^{\!*}\,} }\!$ are reflexive and transitive.
\(ii) $\le$ is antisymmetric (Dedekind; Bernstein), ${ {\;\le^{\!*}\,} }\!$ is not necessarily.
\(iii) $\le$ is stronger than $\!{ {\;\le^{\!*}\,} }\!$. Both relations coincide on well-ordered cardinals.
Two important functions on cardinals (Hartogs and Lindenbaum resp.): $$\aleph({\mathfrak}n)=
\{\alpha:|\alpha|\le{\mathfrak}n\},$$ $$\aleph^*({\mathfrak}n)=
\{\alpha:|\alpha|{ {\;\le^{\!*}\,} }{\mathfrak}n\}.$$
Equivalently,
$\aleph({\mathfrak}n)$ is the least $\alpha$ such that on a set of size ${\mathfrak}n$ there is no well-ordering of length $\alpha$,
$\aleph^*({\mathfrak}n)$ is the least $\alpha$ such that on a set of size ${\mathfrak}n$ there is no pre-well-ordering of length $\alpha$.
Customarily, $\nu^+$ denotes $\aleph(\nu)$ for $\nu$ well-ordered.
Clearly:
\(i) $\aleph({\mathfrak}n)$ and $\aleph^*({\mathfrak}n)$ are well-ordered cardinals.
\(ii) $\aleph({\mathfrak}n)\not\leq{\mathfrak}n$ and $\aleph^*({\mathfrak}n)\not\!\!\!\!{ {\;\le^{\!*}\,} }{\mathfrak}n.$
It follows $\nu<\nu^+$ and so $$\aleph_0<\aleph_1<\ldots<
\aleph_\omega<\ldots<\aleph_{\omega_1}<\ldots$$ (where $\aleph_\alpha$ is $\alpha$th iteration of $\aleph$ starting from $\aleph_0$).
\(iii) $
\aleph({\mathfrak}n)\le
\aleph^*({\mathfrak}n),
$ and both operations coincide on well-ordered cardinals. On other cardinals, the gap can be very large:
Assume ${ {\mathrm {AD}} }$. Then $\aleph(2^{\aleph_0})=\aleph_1$ while $\aleph^*(2^{\aleph_0})$ is a very large cardinal (customarily denoted $\Theta$).
Notations: $${ {\mathrm{Cov}} }({\mathfrak}l,{\mathfrak}m,{\mathfrak}n)$$ means “A set of size ${\mathfrak}n$ can be covered by ${\mathfrak}m$ sets of size ${\mathfrak}l$”.
${ {\mathrm{Cov}} }(\!<\!{\mathfrak}l,{\mathfrak}m,{\mathfrak}n)$ and ${ {\mathrm{Cov}} }({\mathfrak}L,{\mathfrak}m,{\mathfrak}n)$ (where ${\mathfrak}L$ is a class of cardinals) have the appropriate meanings.
$\,$ A cardinal ${\mathfrak}n$ is [*singular*]{} iff\
${ {\mathrm{Cov}} }(\!<\!{\mathfrak}n,<\!{\mathfrak}n,{\mathfrak}n)$, and [*regular*]{} otherwise.
What is under AC?
$\,$ Assume ${ {\mathrm {AC}} }$. Then ${ {\mathrm{Cov}} }({\mathfrak}l,{\mathfrak}m,{\mathfrak}n)$ implies ${\mathfrak}n\le{\mathfrak}l\cdot{\mathfrak}m$.
$\,$ Assume ${ {\mathrm {AC}} }$. Then all the successor alephs are regular.
Thus $\neg{ {\mathrm{Cov}} }(\lambda,\lambda,\lambda^+)$ for all $\lambda\ge\aleph_0$.
What happens without AC?
$\,$ $\aleph_1$ can be singular.
Thus ${ {\mathrm{Cov}} }(\aleph_0,\aleph_0,\aleph_1)$ is consistent.
Moreover, under a large cardinal hypothesis, so can be all uncountable alephs:
$\,$ All uncountable alephs can be singular.
Clearly, then ${ {\mathrm{Cov}} }(\!<\!\lambda,\aleph_0,\lambda)$ for all $\lambda\ge\aleph_0$.
What is the consistency strength?
Without successive singular alephs:
------------------------------------------------------------------------
\
The same as of ZFC.
With $\lambda,\lambda^+$ both singular:
------------------------------------------------------------------------
\
Between 1 Woodin cardinal (Schindler improving Mitchell) and $\omega$ Woodin cardinals (Martin Steel Woodin).
So, in general case:
------------------------------------------------------------------------
\
A proper class of Woodins.
Specker’s problem:
Is ${ {\mathrm{Cov}} }(\aleph_\alpha,\aleph_0,2^{\aleph_\alpha})$ consistent for all $\alpha$ simultaneously?
Partial answer:
$\,$ Let $A\subseteq Ord$ consist either
\(i) of all successor ordinals; or
\(ii) of all limit ordinals and all successor ordinals of form $\alpha=3n, 3n+1, \gamma+3n$, or $\gamma+3n+2$, where $\gamma$ is a limit ordinal.
Then $$(\forall\alpha\in A)\,
{ {\mathrm{Cov}} }(\aleph_\alpha,\aleph_0,2^{\aleph_\alpha})$$ is consistent (modulo large cardinals).
(Really, their technique gives slightly more.)\
In general, the problem remains open.
Question: [*How singular*]{} can cardinals be without AC? in the following sense: How small are ${\mathfrak}l\le{\mathfrak}n$ and ${\mathfrak}m\le{\mathfrak}n$ satisfying
\(i) ${ {\mathrm{Cov}} }(\!<\!{\mathfrak}l,<\!{\mathfrak}n,{\mathfrak}n)$?
\(ii) ${ {\mathrm{Cov}} }(\!<\!{\mathfrak}n,<\!{\mathfrak}m,{\mathfrak}n)$?
\(iii) ${ {\mathrm{Cov}} }(\!<\!{\mathfrak}l,<\!{\mathfrak}m,{\mathfrak}n)$?
On (iii):
------------------------------------------------------------------------
\
Specker’s problem is a partial case.
On (ii):
------------------------------------------------------------------------
\
The answer is $$\text{As small as possible}$$ since Gitik’s model satisfies ${ {\mathrm{Cov}} }(\!<\!{\mathfrak}n,\aleph_0,{\mathfrak}n)$ for all (not only well-ordered) ${\mathfrak}n$.
On (i):
------------------------------------------------------------------------
\
For well-ordered ${\mathfrak}n$, the answer is $${\mathfrak}l<{\mathfrak}n\text{ is impossible.}$$
$\,$ ${ {\mathrm{Cov}} }(\!<\!\lambda,{\mathfrak}m,\nu)$ implies $
\nu{ {\;\le^{\!*}\,} }\lambda\cdot{\mathfrak}m,
$ and so $$\nu^+\le\aleph^*(\lambda\cdot{\mathfrak}m).$$
$\,$ $\neg{ {\mathrm{Cov}} }(\!<\!\lambda,\lambda,\lambda^+)$ for all $\lambda\ge\aleph_0$.
Since ${ {\mathrm{Cov}} }(\lambda,\lambda,\lambda^+)$ is consistent, the result is exact.
$\neg{ {\mathrm{Cov}} }(\aleph_0,\aleph_0,\aleph_2)$ is an old result of Jech. (I am indebted to Prof. Blass who informed me.) By Corollary, really $\neg{ {\mathrm{Cov}} }(\aleph_0,\aleph_1,\aleph_2)$.
Next question: Let ${ {\mathrm{Cov}} }({\mathfrak}l,{\mathfrak}m,{\mathfrak}n)$, is ${\mathfrak}n$ estimated via ${\mathfrak}l$ and ${\mathfrak}m$? (when ${\mathfrak}n$ is not well-ordered). Without Foundation, the answer is $$\text{No}$$ Even in the simplest case ${\mathfrak}l=2$ and ${\mathfrak}m=\aleph_0$ such an estimation of ${\mathfrak}n$ is not provable:
$\,$ It is consistent that for any ${\mathfrak}p$ there exists ${\mathfrak}n\nleq{\mathfrak}p$ such that ${ {\mathrm{Cov}} }(2,\aleph_0,{\mathfrak}n)$.
The proof uses a generalization of permutation model technique to the case of a proper class of atoms. We use non-well-founded sets instead of atoms.
On the other hand, $\aleph({\mathfrak}n)$ and $\aleph^*({\mathfrak}n)$ are estimated via $\aleph({\mathfrak}l)$, $\aleph^*({\mathfrak}l)$, and $\aleph^*({\mathfrak}m)$:
------------------------------------------------------------------------
\
${ {\mathrm{Cov}} }({\mathfrak}L,{\mathfrak}m,{\mathfrak}n)$ implies $$\aleph({\mathfrak}n)\le
\aleph^*(\sup_{{\mathfrak}l\in{\mathfrak}L}\,\aleph({\mathfrak}l)\cdot{\mathfrak}m)$$ and $$\aleph^*({\mathfrak}n)\le
\aleph^*(\sup_{{\mathfrak}l\in{\mathfrak}L}\,\aleph^*({\mathfrak}l)\cdot{\mathfrak}m).$$
------------------------------------------------------------------------
\
$
\neg{ {\mathrm{Cov}} }(\!<\!\lambda,\lambda,2^\lambda)
$ and $
\neg{ {\mathrm{Cov}} }({\mathfrak}n,2^{{\mathfrak}n^2},2^{2^{{\mathfrak}n^2\cdot 2}}).
$
In particular:
$\neg{ {\mathrm{Cov}} }(\lambda,2^\lambda,2^{2^\lambda})$ and $\neg{ {\mathrm{Cov}} }(\beth_\alpha,\beth_{\alpha+1},\beth_{\alpha+2}).$
Since ${ {\mathrm{Cov}} }({\mathfrak}n,{\mathfrak}n,2^{\mathfrak}n)$ is consistent, the result is near optimal.
Another corollary is that Specker’s request, even in a weaker form, gives the least possible evaluation of $\aleph^*(2^\lambda)$ (which is $\lambda^{++}$):
$\,$ ${ {\mathrm{Cov}} }(\lambda,\lambda^+,2^\lambda)$ implies $$\aleph^*(2^\lambda)=
\aleph(2^\lambda)=
\lambda^{++}.$$
So, if there exists a model which gives the positive answer to Specker’s problem, then in it, all the cardinals $\aleph^*(2^\lambda)$ have the least possible values.
As the last corollary, we provide a “pathology” when a set admits [*neither*]{} well-ordered covering (of arbitrary size) by sets of smaller size, [*nor*]{} covering of smaller size by well-orderable sets (of arbitrary size). Moreover, it can be the [*real line*]{}:
$\,$ Assume ${ {\mathrm {CH}} }$ holds and $\Theta$ is limit. (E.g., assume ${ {\mathrm {AD}} }$.) Then for any well-ordered $\lambda$ $$\neg{ {\mathrm{Cov}} }(\!<\!2^{\aleph_0},\lambda,2^{\aleph_0})
\text{\, and \,}
\neg{ {\mathrm{Cov}} }(\lambda,\!<\!2^{\aleph_0},2^{\aleph_0}).$$
(Here CH means “There is no ${\mathfrak}m$ such that\
$\aleph_0<{\mathfrak}m<2^{\aleph_0}$”.)
$\,$ Is $\neg{ {\mathrm{Cov}} }({\mathfrak}n,2^{\mathfrak}n,2^{2^{\mathfrak}n})$ true for all ${\mathfrak}n$?
That holds if ${\mathfrak}n={\mathfrak}n^2$ (by Corollary 1 of Theorem 3).
$\,$ Is $\neg{ {\mathrm{Cov}} }(\!<\!\beth_\alpha,\beth_\alpha,\beth_{\alpha+1})$ true for all $\alpha$?
That near holds if $\alpha$ is successor (again by Corollary 1 of Theorem 3).
$\,$ Is ${ {\mathrm{Cov}} }({\mathfrak}n,\aleph_0,2^{{\mathfrak}n^2})$ consistent for all ${\mathfrak}n$ simultaneously?
This sharps Specker’s problem of course.
$\,$ Can Theorem 2 be proved assuming Foundation? More generally, expand the Transfer Theorem (Jech Sohor) to the case of a proper class of atoms.
$\,$ Is it true that on successor alephs the cofinality can behave anyhow, in the following sense: Let $F$ be any function such that $$F:SuccOrd\to SuccOrd\cup\{0\}$$ and $F$ satisfies $$\begin{array}{rll}
\text{{\rm (i)\/}}&
F(\alpha)\le\alpha\;\text{ and}
\\
\text{{\rm (ii)\/}}&
F(F(\alpha))=F(\alpha)
\end{array}$$ for all successor $\alpha$. Is it consistent $${ {\mathop{\mathrm {cf\,}}\nolimits} }\aleph_\alpha=\aleph_{F(\alpha)}$$ for all successor $\alpha$?
Perhaps if $F$ makes no successive cardinals singular, it is rather easy; otherwise very hard.
\[1\] Arthur W. Apter and Moti Gitik. [*Some results on Specker’s problem*]{}. Pacific Journal of Mathematics, 134, 2 (1988), 227–249.
\[2\] Solomon Feferman and Azriel Lévy. [*Independences results in set theory by Cohen’s method, II.*]{} Notices of the American Mathematical Society, 10 (1963), 593. Abstract.
\[3\] Moti Gitik. [*All uncountable cardinals can be singular*]{}. Israel Journal of Mathematics, 35, 1–2 (1980), 61–88.
\[4\] Moti Gitik. [*Regular cardinals in models of ${ {\mathrm {ZF}} }$*]{}. Transactions of the American Mathematical Society, 290, 1 (1985), 41–68.
\[5\] Donald A. Martin and John R. Steel. [*Projective determinacy*]{}. Proceedings of the National Academy of Sciences of U.S.A., 85, 18 (1988), 6582–6586.
\[6\] Donald A. Martin and John R. Steel. [*A proof of projective determinacy*]{}. Journal of the American Mathematical Society, 2, 1 (1989), 71–125.
\[7\] Ralf Dieter Schindler. [*Successive weakly compact or singular cardinals*]{}. Journal of Symbolic Logic, 64 (1999), 139–146.
\[8\] Ernst P. Specker. [*Zur Axiomatik der Mengenlehre (Fundierungs- und Auswahlaxiom)*]{}.\
Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 3, 3 (1957), 173–210.
\[9\] W. Hugh Woodin. [*Supercompact cardinals, sets of reals, and weakly homogeneous trees*]{}. Proceedings of the National Academy of Sciences of U.S.A., 85, 18 (1988), 6587–6591.
|
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