# Random matrices and their applications

Kyoto university – May 21-25 2018

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The former editions of this workshop took place in Hong-Kong (2015) and Paris (2012, 2010).

Participants Affiliation
Chafaï, Djalil Université Paris-Dauphine
Collins, Benoit Kyoto University
Kumari, Sushma Kyoto University
Sakuma, Noriyoshi Aichi University of Education
Sapra, Gunjan Kyoto university
ZHENG, GUANGQU University of Luxembourg
Hasebe, Takahiro Hokkaido University
Fevre, Mathieu Kyoto University
Shirai, Tomoyuki Kyushu University
Dobriban, Edgar University of Pennsylvania
Matsumoto, Sho Kagoshima University
Roberts, Dale Australian National University
Najim, Jamal CNRS and Université Paris Est
Andraus, Sergio Chuo University
Teh, Terrence University of the Philippines
Molag, Leslie KU Leuven
male, camille Université de Bordeaux
Wang, Ke Hong Kong University of Science and Technology
Vidotto, Anna University of Luxembourg
Rossi, Maurizia Université Paris Descartes
Kajino, Naotaka Kobe University
Tanemura, Hideki Keio university
Byun, Sung-Soo Seoul National University
Seo, Seong-Mi Korea Institute for Advanced Study
forrester, peter university of melbourne
Jalowy, Jonas Bielefeld University
Graczyk, Piotr LAREMA Université d'Angers
Katori, Makoto Chuo University
Feng, Renjie Peking University
Sasamoto, Tomohiro Tokyo Institute of Technology
Endo, Taiki Chuo University
Yahagi, Shu Chuo University
Baba, Hiroya Chuo University
Yang, Wooseok Korea Advanced Institute of Science and Technology
Yoshida, Hiroaki Ochanomizu University
Park, Jaewhi Korea Advanced Institute of Science and Technology
Hora, Akihito Hokkaido University
Lee, Jinyeop Korea Advanced Institute of Science and Technology
TIAN, Peng University of Paris East Marne la Vallee
Kumagai, Takashi RIMS, Kyoto University
Dartois, Stephane University of Melbourne
Fukushima, Ryoki RIMS Kyoto University
Jung, Paul KAIST
Tezuka, Masaki Kyoto University
Lionni, Luca YITP Kyoto
Tieplova, Daria University of Paris East Marne la Vallee
Butez, Raphael Université Paris Dauphine
Gaudreau Lamarre, Pierre Yves Princeton University
Hiroyuki, Ochiai Kyushu University
Richards, Donald Institute of Statistical Mathematics, and Penn State University
Kuriki, Satoshi Institute of Statistical Mathematics
Trinh, Khanh Duy Tohoku University
Goel, Akshay Kyushu University
Wang, Zhenggang The University of Hong Kong
Komatsu, Kazunobu Kyushu University
Benigni, Lucas LPSM, Université Paris Diderot
Yabuoku, Satoshi Chiba University
Kawamoto, Yosuke Kyushu University
Yoo, Hyun Jae Hankyong National University
Mucciconi, Matteo Tokyo Institute of Technology
Wang, Jingming Hong Kong University of Science and Technology
Rahman, Anas University of Melbourne
HAYASE, Tomohiro University of Tokyo
Choda, Marie Osaka Kyoiku University
Maosheng, Xiong Hong Kong University of Science and Technology
Parraux, Félix ÉNS Lyon
Imamura, Takashi Chiba University
Suzaki, Kiyotaka Kyushu University
Hatano, Takahiro University of Tokyo
Hirao, Masatake Aichi Prefectural University
Mizuta, Rei University of Tokyo
Yabuoku, Satoshi Chiba University
Moriyama, Sanefumi Osaka City University
Uchiyama, Mitsuru Shimane / Ritsumeikan
Esaki, Syota Fukuoka University
• A workshop dinner is organized on Tuesday May 22 from 6pm to 8pm.
• Schedule proposal (last update):
Mon 21 Tue 22 Wed 23 Thu 24 Fri 25
09:00-09:20 Registration / Coffee Coffee Coffee Coffee Coffee
09:20:09:30 Welcoming
09:30-10:20 Kuijlaars PDF Bourgade Knowles PDF Forrester PDF Tanemura PDF
10:20:10:50 Coffee break Coffee break Coffee break (+photo) Coffee break Coffee break
10:50-11:40 Dumaz PDF Wang PDF Matsumoto PDF Hora Hardy PDF
Graczyk PDF
11:40-12:30 Sasamoto PDF Katori PDF Nakano PDF Bao PDF Bufetov
12:30-13:40 Lunch Lunch Lunch Lunch Lunch
13:40-14:30 Salez PDF Posters presentations Free Nagao Free
14:30-15:00 Coffee break Coffee + posters session Coffee break
15:00-16:00 Kawamoto PDF
Hasebe PDF
Fukuda PDF
Butez PDF
16:00-16:20 Pause Pause
16:20-17:10 Trinh PDF
Jung PDF
Ueda PDF Yao PDF
Evening (18:00) Banquet
• Sergio Andraus, Chuo University, Tokyo
Dunkl jump processes: relaxation and a phase transition. PDF
• Lucas Benigni, NYU and Paris-Diderot
Eigenvector distribution and QUE for deformed wigner matrices PDF
• Pierre Yves Gaudreau Lamarre, Princeton University
The Stochastic Semigroup Approach to the Edge of Beta-Ensembles. PDF
• Tomohiro Hayase, University of Tokyo
Cauchy Noise Loss for Stochastic optimization of random matrix models via free deterministic equivalents. PDF
• Satoshi Kuriki, Institute of Statistical Mathematics
The Euler characteristic method for the largest eigenvalues of random matrices. PDF
• Camille Male, University of Bordeaux and CNRS
Eigenvalues of functions in random matrices - Analytic aspect of traffic-independence. PDF
• Leslie Molag, K.U. Leuven
A Riemann-Hilbert approach to the Muttalib-Borodin ensemble. PDF
• Matteo Mucciconi, Tokyo Institute of Technology
Stationary KPZ fluctuations for the Higher Spin Six Vertex Model. PDF
• Anas Rahman, University of Melbourne
Topological Recursion. PDF
• Donald Richards, Penn State University
Integral transform methods in hypothesis testing for Wishart distributions. PDF
• Ryosuke Sato University of Kyushu
Quantized Vershik-Kerov Theory and q-deformed Gelfand-Tsetlin graph. PDF
• Masaki Tezuka, Kyoto University
Random-matrix behavior in the energy spectrum of the Sachdev-Ye-Kitaev model and in the Lyapunov spectra of classical chaos systems. PDF
• Peng Tian, Université de Marne-la-Vallée
Large Random Matrices of Long Memory Stationary Processes: Asymptotics and fluctuations of the largest eigenvalue. PDF

All posters presentations in a single file: PDF

A couple of minutes from the last talk, by A. Bufetov.

Speaker Title Abstract
Bao, Zhigang Local single ring theorem on optimal scale The celebrated single ring theorem asserts that the empirical eigenvalue distribution of a non-Hermitian random matrix with given singular values converges weakly to a deterministic measure which is supported on a single ring centered at the origin in the complex plane. In this talk, I will present a local version of this theorem on the optimal scale. This is a joint work with Laszlo Erdos and Kevin Schnelli.
Wang, Ke Limiting eigenvalue distribution of the non-backtracking matrices of Erdos-Renyi random graphs In this talk, we give a precise description of the eigenvalues of the non-backtracking matrix of an Erdos-Renyi random graph on n vertices, where edges present independently with probability p. We allow p to be constant or decreasing with n, so long as np/log n tends to infinity. The key observation in the proof is that derandomizing part of the non-backtracking random matrix simplifies the spectrum considerably. Then we use Tao and Vu’s replacement principle and the Bauer-Fike theorem to show that the partly derandomized spectrum is, in fact, very close to the original spectrum.
Jung, Paul Levy-Khintchine Random Matrices We study a class of Hermitian random matrices which includes Wigner matrices, heavy-tailed matrices, and sparse random matrices such as adjacency matrices of Erdos-Renyi graphs with p=1/N. The entries are i.i.d. (up to symmetry) and their distribution may depend on N; however, the sums of rows should converge in distribution to an infinitely divisible law. The LSD exists, via local weak convergence of associated graphs, and it corresponds to the spectral measure associated to the root of a graph which is formed by connecting infinitely many Poisson weighted infinite trees using a backbone structure of special edges. One example covered are matrices with i.i.d. entries having infinite second moments, but normalized to be in the Gaussian domain of attraction. In this case, the LSD is a semi-circle law.
Salez, Justin Spectral atoms of unimodular random trees We use the Mass Transport Principle to analyze the local recursion governing the resolvent $(A−z)^{−1}$ of the adjacency operator of unimodular random trees. In the limit where the complex parameter $z$ approaches a given location $\lambda$ on the real axis, we show that this recursion induces a decomposition of the tree into finite blocks whose geometry directly determines the spectral mass at $\lambda$. We then exploit this correspondence to obtain precise information on the pure-point support of the spectrum, in terms of expansion properties of the tree. In the special case of Galton-Watson trees, this allows us to settle a conjecture of Bordenave, Sen and Vir\'ag (2013).
Kuijlaars, Arno The two-periodic Aztec diamond and matrix valued orthogonal polynomials Uniform domino tilings of the Aztec diamond have the arctic circle phenomenon: near the corners the pattern is fixed and only one type of domino appears, while in the middle there is disorder and all types appear. The transition is sharp with fluctuations described by the Tracy-Widom distributions. In the two-periodic Aztec diamond the dominos have a two-periodic weighting and this creates a new phase in the large size limit, where correlations decay at an exponential rate. In recent work with Maurice Duits (KTH Stockholm) we analyze this model with the help of matrix valued orthogonal polynomials. We obtain a remarkably simple double contour integral formula for the correlation kernel that we can analyze in the limit to recover the three phases of the model and the fluctuations near the transition curves.
Forrester, Peter Applications of decomposition of measure in random matrix theory to integral geometry and number theory Fundamental to random matrix theory is various factorisations of Lebesgue product measure implied by matrix change of variables. In number theory, factorisation of Siegel's invariant measure for SL${}_N(\mathbb R)$ is an ingredient in Duke, Rudnik and Sarnak's asymptotic computation of the number of matrices in SL${}_N(\mathbb Z)$, with a bounded norm. It allows for calculations in the space of integral lattices SL${}_N(\mathbb R)/{\rm SL}_N(\mathbb Z)$ and generalisations such as SL${}_N(\mathbb C)/{\rm SL}_N(\mathbb Z[i])$ Factorisation of measure is also fundamental to integral geometry, with one of the most important results due to Blaschke and Petkantschin. Following recent work of Moghadasi, we show how the latter is related to matrix polar decomposition, and can be applied to the calculation of the moments of the volume content of the convex hull of random points in higher dimensional spaces.
Sasamoto, Tomohiro Fluctuations of stationary KPZ models and multiple integral The purpose of this talk is two fold. The first is to present analysis of KPZ models, in particular the q-TASEP, for the stationary situation. After recalling a difficulty of using a standard method of using q-deformed moments, we explain our method to represent the q-Laplace transform of the particle position as a multiple integral, using Ramanujan’s summation formula and Cauchy determinant for theta functions. The second is to explain that the above type of multiple integral is ubiquitous and provides a unified approach to study various models including the standard GUE. Reference: [1] T. Imamura, T. Sasamoto, Fluctuations for stationary $q$-TASEP, arxiv:1701: 1701.05991
Hora, Akihito Dynamical scaling limit of the restriction-induction chain on Young diagrams in terms of free probability The restriction-induction chain on Young diagrams is a Markov chain caused by the branching rule for irreducible representations of symmetric groups, in which one step transition admits a non-local movement of a corner box. This model was previously treated in works of Fulman and Borodin-Olshanski. We consider diffusive scaling limit of the continuous time restriction-induction chain in the regime of an appearance of the well-known limit shape. Our time evolution is described by using machinery of free probability theory.
Nagao, Taro Spectral analysis of scale free networks In the Goh-Kahng-Kim (GKK) model of a scale free network, the degree (the number of edges attached to a vertex) exhibits a power-law behavior (the degree density has a power-law tail). Because of this asymptotic behavior, the network is said to be scale free. Each adjacency matrix element of the GKK model is independently distributed but the probability density function (p.d.f.) is not identical to the p.d.f. of the other elements. This inhomogeneity is an outstanding difference between scale free networks and ordinary random networks. We will discuss the eigenvalue distributions of the adjacency matrices of scale free networks. The emphasis will be put on the similarity and difference between inhomogeneous random matrices and ordinary homogeneous ones.
Katori, Makoto Macdonald denominators for affine root systems, orthogonal theta functions, and elliptic determinantal processes Rosengren and Schlosser (2006) gave determinantal expressions to the Macdonald denominators for the reduced affine root systems. We will explain the following two facts, (i) the entries of their determinants construct the orthogonal-function systems expressed by the Jacobi theta functions, and (ii) up to trivial factors, their determinants are identified with the Karlin-McGregor determinants of noncolliding Brownian motions in intervals pinned at specified configurations at the final time. Based on these observations, we will discuss the time evolution of interacting particle systems which we call the elliptic determinantal processes and the elliptic Dyson models. This is an extension of the work by Forrester found in his textbook of random matrix theory (2010) (Section 5.6. Log-gas systems with periodic boundary conditions).
Matsumoto, Sho Weingarten calculus and counting paths on Weingarten graphs We consider matrix elements in a random matrix from compact Lie groups and symmetric spaces. Weingarten calculus, triggered by Weingarten's paper in 1978, is a general method for computations of mixed moments of these matrix elements. It has been widely applied to classical/free probability, representation theory, quantum information theory, and others. In the first half of this talk, I will present an introduction to Weingarten calculus. It is based on harmonic analysis for symmetric groups. In the second half, I will give a connection between Weingarten calculus and some related infinite graphs, called Weingarten graphs. Various Weingarten functions are expressed as generating functions counting paths on the graphs. The latter one is a joint work with Benoit Collins.
Tanemura, Hideki Systems of infinitely many hard balls with long range interaction A particle system with hard core interaction can be regarded as a system of hard balls. An infinite Brownian particle system has been studied in the case where interaction between balls is of finite or short range. The main purpose of this talk is to show the existence and uniqueness of infinite dimensional stochastic differential equation (ISDE) describing a system of infinitely many hard balls with long range interaction.
Nakano, Fumihiko Density of states and level statistics for 1d Schroedinger operators We consider the 1d Schroedinger operator with random potential decaying of order ¥alpha. The results include : (1) the fluctuation of density of states with different behavior depending on ¥alpha, (2) the level statistics asymptotically obeys clock, Sine_{¥beta}, and Poisson processes for super-critical, critical, and sub-critical cases, respectively. (3) if the time permits, we discuss some recent results on eigenfunction statistics. Joint work with S. Kotani and T. K. Duy
Ueda, Yoshimichi Matrix liberation process and a free probability question Matrix liberation process is a natural random matrix counterpart of the so-called liberation process in free probability. We emphasize that any information about it can never be captured by the associated eigenvalue process (which indeed becomes a constant process). In the talk, I’ll start with some backgrounds to the matrix liberation process and then go to some details about a large deviation phenomenon for the matrix liberation process. I’ll also touch on my on-going project toward the unification between Voiculescu’s and our approaches to mutual information in free probability as far as time will permit.
Bourgade, Paul The overlaps between Ginibre eigenvectors Eigenvectors of non-hermitian matrices are non-orthogonal, and their distance to a unitary basis can be quantified through the matrix of overlaps. These variables quantify the stability of the spectrum, and characterize the joint eigenvalues increments under Dyson-type dynamics. They first appeared in the physics literature; well known work by Chalker and Mehlig calculated the expectation of these overlaps for complex Ginibre matrices. For the same model, we extend their results by deriving the distribution of the overlaps and their correlations. As a corollary, at equilibrium, eigenvalues move with diffusive scaling. (Joint work with G. Dubach).
Yao, Jianfeng Random matrices and some recent progresses in high-dimensional statistics This talk is dedicated to show how random matrix theory has been recently used to solve a few problems in high-dimensional statistics. The focus would be on the type of matrices and type of results from RMT which are "useful" in such statistical problems. Some unsolved and interesting problems (to author's opinion) would be also discussed.
Trinh, Khanh Duy Classical beta ensembles at high temperature Three classical beta ensembles in the real line (beta Hermite, beta Laguerre and beta Jacobi ensembles) are now realized as eigenvalues of certain random tridiagonal matrices. This talk introduces an approach to identify the limiting distribution in the global regime via spectral measures. Besides recovering all classical laws for the three beta ensembles for fixed beta, this approach can work even when the parameter beta, which is regarded as the inverse temperature, varies as the size of the system tends to infinity. In particular, when the temperature is proportional to the system size, the limiting distributions are shown to be the probability measures of associated Hermite, Laguerre and Jacobi orthogonal polynomials, respectively.
Kawamoto, Yosuke Dynamical universality for the Airy random point field Abstrct: The universality of random matrices (or log-gases), which is a central issue in random matrix theory, has been developed rapidly in the several decades. From these results, we obtain universal random point fields as a limit. Let us show a example for the universality results. Consider a log-gas with finite particle, here free potential is given by suitable one. For free potential of wide class, the log-gas converges to the Airy random point field as the soft-edge scaling limit . Here the Airy random point field does not depend on free potential, and in this sense the Airy random point field is universal. Our purpose is to establish a dynamical counterpart for this universality result. In other words, we would like to show the distorted Brownian motion associated with the log-gas converges to a solution of the Airy interacting infinite-dimentional stochastic differential equation as the particle number to infinity. Here the limit dynamics is associated with the Airy random point field. Therefore whilst the dynamics with finite particle depends on free potential, the limit dynamics does not depend on free potential. We are going to introduce this kind of dynamical universality and discuss about what is necessary for our framework in this talk.
Hardy, Adrien Energy of the Coulomb gas on the sphere at low temperature The 7th Smale's problem is about constructing point configurations on the sphere whose discrete logarithmic energy approximates the minimal energy up to a logarithmic error. After briefly reviewing what is known on the topic, I will explain that the Coulomb gas on the sphere does the job provided the inverse temperature beta is of the same order than the configuration size. This is a joint work with Carlos Beltran.
Hasebe, Takahiro Free Levy processes in large and small time limits In classical probability, if the law of a Levy process with an affine transformation converges as time goes to 0 or infinity, then the limit distribution is stable. A similar theorem holds for free Levy processes with respect to addition. However, for free Levy processes with respect to multiplication, the situation becomes different in large time as proved by Tucci and Haagerup-Moeller. I will explain what happens in the small time limit. This is a joint work with Octavio Arizmendi.
Fukuda, Motohisa Random matrices in quantum information In this talk, we learn how random matrices and free probability are used in the research of quantum information, especially quantum channel, where one could investigate typical properties of randomly generated quantum channels (random quantum channels) instead of individual ones. This viewpoint, for example, led to proofs which show existence of counterexamples for a famous conjecture on additivity. In addition, we discuss PPT (positive partial transpose) property of random quantum channels, and connection between random quantum states and meandric systems.
Graczyk, Piotr New results on Squared Bessel particle systems and Wishart processes (joint work with J. Malecki and E. Mayerhofer) We study the existence, uniqueness, collisions and other properties of Squared Bessel particle systems in full generality, i.e. admitting that they take also negative values. We extend the results obtained by Going-Jaeschke and Yor in 1-dimensional case. Squared Bessel particle systems may be interpreted as eigenvalues of Wishart processes. We determine the parameter set of Wishart processes, i.e. the stochastic Gindikin set and deduce the first characterization of analytical non-central Gindikin set. Our techniques use elementary symmetric polynomials and, for Wishart processes, the theory of affine stochastic processes.
Dumaz, Laure Localization of the continuous Anderson Hamiltonian We consider the continuous Schrodinger operator (also named Hill operator or Anderson Hamiltonian) $-\Delta + B'$ on the interval $[0,L]$ where the potential $B'$ is a white noise. In the large $L$ limit, we show that the smallest eigenvalues converge towards a Poisson point process and the localization of the associated eigenvectors in a precise sense: when we zoom around their maximum, their shape becomes deterministic and does not depend on the eigenvalue. Joint work with Cyril Labbé.
Butez, Raphael The largest root of Kac random polynomials is heavy tailed In this short talk we will study the behavior of the root of largest modulus of random Kac polynomials. Random polynomials and random matrices share a lot of results and techniques in common but we will see that the largest root of random Kac polynomials has heavy tails. The largest root of random polynomials converges towards a non-universal random variable whose number of finite moments is controlled.
Knowles, Antti Mesoscopic eigenvalue correlations of random matrices. Ever since the pioneering works of Wigner, Gaudin, Dyson, and Mehta, the correlations of eigenvalues of large random matrices on short scales have been a central topic in random matrix theory. On the microscopic spectral scale, comparable with the typical eigenvalue spacing, these correlations are now well understood for Wigner matrices thanks to the recent solution of the Wigner-Gaudin-Dyson-Mehta universality conjecture. In this talk I focus on eigenvalue density-density correlations between eigenvalues whose separation is much larger than the microscopic spectral scale; here the correlations are much weaker than on the microscopic scale. I discuss to what extent the Wigner-Gaudin-Dyson-Mehta universality remains valid on such larger scales, for Wigner matrices and, if time allows, random band matrices.
Bufetov, Alexander Determinantal point processes and the reconstruction of holomorphic functions In joint work with Yanqi Qiu and Alexander Shamov we prove that the sequence of reproducing kernels sampled along a random trajectory of a determinant point process is complete in the ambient Hilbert space. From this result and the Peres-Virag Theorem it follows, in particular, that the zero set of a Gaussian Analytic Function is almost surely a uniqueness set in the Bergman space on the unit disc — equivalently, that any square-integrable holomorphic function is uniquely determined by its restriction to our set. In joint work with Yanqi Qiu, we show that the Patterson-Sullivan construction recovers the value of any Hardy function at any point of the disc from its restriction to a random configuration of the determinant point process with the Bergman kernel. This extrapolation result is then extended to real and complex hyperbolic spaces of higher dimension. Recovering continuous functions by the Patterson-Sullivan construction is also shown to be possible in more general Gromov hyperbolic spaces.

Masukawa hall 北部総合教育研究棟 (Monday to Thursday):

Kyoto math departement (Friday):

The closest airports are KIX (Osaka Kansai international airport) and ITM (Osaka Itami airport). If you come from abroad, flying to KIX and then taking a train (e.g. JR Haruka to Kyoto station followed by taxi or common transportation), or a door to door shuttle (e.g. MK taxi) are standard solutions.

Some people prefer to fly to/from Tokyo (HND / NRT) as there are more choices of flights. Although not obvious at first sight, it does not take much longer to reach Kyoto if you land in a Tokyo airport at daytime, thanks to the Japanese bullet train (Shinkansen).

In general train tickets do not need to be reserved in advance in Japan. On the other hand, shuttles must be reserved ahead of time and this can be done online.

Either way, if you land in the evening, plan the domestic part of your trip ahead of time (e.g. if you are not in a standard hotel, confirm the latest check-in time, and if you do not travel with a shuttle, check the schedule of the last train of the day — feel free to talk with the local organizers about your situation if needs be).

• Japanese Grant kakenhi wakate A number 17H04823 (PI: Benoit Collins)
• Japanese Grant kakenhi kiban S number 16H6338 (PI: Hirofumi Osada)
• Japanese Grant kakenhi kiban B number 18H01124 (PI: Tomoyuki Shirai)

• kyoto2018/start.txt