# Online Random Matrices and Their Applications 2020

Online RMTA 2020 - May 25-29 - New York Time

Random matrix theory is at the confluence of pure mathematics, theoretical physics, multivariate statistics, electrical engineering and so forth. The aim of this one week meeting is to gather prominent specialists of the field for exchange and stimulation. The former editions of this workshop took place in Kyoto (2018), Hong Kong (2015), and Paris (2012, 2010).

This online conference is an electronic replacement of the conventional face-to-face onsite conference RMTA 2020, May 25-29, New York, that was canceled due to the Covid-19 outbreak.

This is a draft proposal NY local time.

• The Zoom room opens 30 minutes before the first talk
• A Slack workspace is made available in parallel
• The reference time is NY local time
• New York 10:00 AM
• Los Angeles 07:00 AM (-3)
• Paris 04:00 PM (+6)
• Beijing & HK 10:00 PM (+12)
• Kyoto 11:00 PM (+13)
• Bordenave
• Detection thresholds in very sparse matrix completion
This is a joint work with Simon Coste and Raj Rao Nadakuditi. Let X be a rectangular matrix of size n x m and Y be the random matrix where each entry of X is multiplied by an independent 0-1 Bernoulli random variable with parameter 1/2. In many practical settings, the spectrum of the matrix Y(X-Y)* conveys more relevant information on the structure of X than the spectrum of XX* used in principal component analysis. We illustrate this striking phenomenon on the matrix completion problem where the matrix X is equal to a matrix T on a random subset of entries of size dn and all other entries of X are equal to zero. In the regime where the ratio n/m is of order 1 and provided that a usual incoherence assumption holds for the matrix T, we show that the eigenvalues of Y(X-Y)* with modulus greater than an explicit threshold are asymptotically equal to the eigenvalues of TT* greater than this threshold, and the associated eigenvectors are aligned. It notably holds in a very sparse regime where d is of order 1. This breaks the theoretical-information limit d of order log n for recovery well-known in the literature. We also define an improved version of this asymmetric principal component analysis which allows to remove the Bernoulli random variables and improve by a constant factor the detection threshold at the cost of increasing the dimension of the asymmetric matrix.
• Collins
• Norm estimates for polynomials in random matrices: new results
Estimating the operator norm of a random matrix — and more generally, investigating the problem of the existence of outliers — is an important problem in random matrix theory, and the first breakthrough in the case of i.i.d GUEs was achieved by Haagerup and Thorbjornsen, about 15 years ago. They relied heavily on innovative linearization techniques and matrix valued stieltjes transforms. We revisit these results with substantially different and arguably simpler methods, that allow more precise and quantitative statements. This talk is based on joint works with Alice Guionnet and Felix Parraud. We will also mention recent developments, including an extension to the unitary case by Felix Parraud, and applications to quantum information theory.
• Dumaz
• Localization of the continuous Anderson hamiltonian in 1-d and its transition towards delocalization
We consider the continuous Schrödinger operator - d^2/d^x^2 + B’(x) on the interval [0,L] where the potential B’ is a white noise. We study the spectrum of this operator in the large L limit. We show the convergence of the smallest eigenvalues as well as the eigenvalues in the bulk towards a Poisson point process, and the localization of the associated eigenvectors in a precise sense. We also find that the transition towards delocalization holds for large eigenvalues of order L, where the limiting law of the point process corresponds to Sch_tau, a process introduced by Kritchevski, Valko and Virag for discrete Schrodinger operators. In this case, the eigenvectors behave like the exponential Brownian motion plus a drift, which proves a conjecture of Rifkind and Virag. Joint works with Cyril Labbé.
• Dumitriu
• Random Regular Hypergraphs
In the last couple of decades, random graph theory has been fully adopted into the large family that is random matrix theory. Hypergraphs are the next thing on the horizon. For some of them, the results that allow us to analyze them are already here, all we need is translation. This is joint work with Yizhe Zhu.
• Fan
• Applications of random matrix theory to graph matching and neural networks
I will discuss two applications of random matrix theory to statistics and machine learning. The first is the “graph matching” problem of identifying the vertex correspondence between two correlated random graphs. We study a spectral algorithm, based on the pairwise alignments between eigenvectors of the two graph adjacency matrices. Our main result establishes exact recovery of the underlying matching in a correlated Erdos-Renyi model on n vertices when the edge correlation is 1-1/polylog(n). The design of this algorithm has some connections to the eigenvector moment flow under Dyson Brownian motion, and our analysis relies on local law estimates in the Erdos-Renyi model. This is joint work with Cheng Mao, Yihong Wu, and Jiaming Xu.
In the second application, I will discuss a spectral analysis of the Neural Tangent Kernel (NTK) in multi-layer feedforward neural networks. Recent theory in machine learning connects the spectral decomposition of the NTK to the training and generalization properties of the underlying network. We analyze the empirical eigenvalue distributions of the NTK and of the related Conjugate Kernel at random initialization, under a condition for the input data that encompasses independent sub-Gaussian samples, and in an asymptotic regime where the hidden layer widths are proportional to the sample size and input dimension. Our main result establishes the weak convergence of the eigenvalue distribution of the NTK to a deterministic limit, and characterizes this limit by fixed-point equations in its Stieltjes transform. This is joint work with Zhichao Wang.
• Fyodorov
• Statistics of extremes in eigenvalue-counting staircases
We consider the counting function (“spectral staircase”) for eigenvalues of a random unitary matrix, drawn from the corresponding beta-ensemble. Our goal is to characterize the statistics of maximum deviation of this staircase from its mean slope in a fixed interval, when size of the matrix $N»1$. We will show that one-sided extremes can be addressed by exploiting a mapping onto the statistical mechanics of log-correlated random processes and using an extended Fisher-Hartwig conjecture. The resulting statistics exhibits combined features of counting statistics of Fermions with Sutherland-type interaction and extremal statistics of the fractional Brownian motion with Hurst index $H=0$. Some of the features are expected to be universal. The talk is based on the paper Fyodorov-Le Doussal arXiv:2001.04135.
• Hall
I will discuss the problem of finding the eigenvalues of a non-normal random matrix of the form $X + i S$, where X is an arbitrary self-adjoint random matrix and S is an independent GUE matrix. Specifically, I will describe what we expect to be the large-N eigenvalue distribution, namely the Brown measure of $x_0 +i s$ where $x_0$ is self-adjoint and $s$ is a freely independent semicircular element. As a point of comparison, Biane has computed the distribution of the self-adjoint operator $x_0 + s$ (without the factor of $i$) using the subordination method. For the case of $x_0 +i s$, I will describe the support of the Brown measure and the density of the Brown measure in its support. One striking result is that the density of the Brown measure inside its support is constant in the imaginary direction. The proof is based on the PDE method developed by the speaker with B. Driver and T. Kemp. The results are joint work with Ching Wei Ho. The talk will be self-contained and have lots of pictures.
• Keating
• Joint Moments of Characteristic Polynomials of Random Unitary Matrices
I will review what is known and not known about the joint moments of the characteristic polynomials of random unitary matrices and their derivatives. I will then explain some recent results, obtained with Theo Assiotis and Jon Warren, which relate the joint moments to an interesting class of measures, known as Hua-Pickrell measures. This allows us to prove a conjecture, due to Chris Hughes in 2000, concerning the asymptotics of the joint moments, as well as to establish a connection between the measures in question and one of the Painlevé equations.
• Knowles
• Fluctuations of extreme eigenvalues of sparse Erdös-Rényi graphs
I discuss the fluctuations of individual eigenvalues of the adjacency matrix of the Erdös-Rényi graph $G(N,p)$. I show that if $N^{\epsilon} \leq Np \leq N^{1/3-\epsilon}$ then all nontrivial eigenvalues away from 0 have asymptotically Gaussian fluctuations. These fluctuations are governed by a single random variable, which has the interpretation of the total degree of the graph. The main technical tool of the proof is a rigidity bound of accuracy $N^{-1/2-\epsilon} (Np)^{-1/2}$ for the extreme eigenvalues, which avoids the $(Np)^{-1}$-expansions from previous works. Joint work with Yukun He.
• Krüger
• Non-selfadjoint random matrices: spectral statistics and applications
The empirical spectral distribution of a non-selfadjoint random matrix concentrates around a deterministic probability measure on the complex plane as its dimension increases. Despite the inherent spectral instability of such models, this approximation is valid all the way down to local scales just above the typical eigenvalue spacing distance. We will present recent results on eigenvalue spectra for non-selfadjoint random matrices with correlated entries and their application to systems of randomly coupled differential equations that are used to model a wide range of disordered dynamical systems ranging from neural networks to food webs. Joint work with Johannes Alt, László Erdős and David Renfrew.
• Yin
• Universality and delocalization of random band matrices
From numerical stimulation and some heuristic proof, we know there is famous random band matrix conjecture. This conjecture predicts a phase transition on the eigenvalue and eigenvector behaviors based on the band width of the matrices (and the dimension). Especially, the eigenvector - delocalization - localization conjecture was believed to be related to the Anderson’s conductor-insulator transition problem. In this talk, we will introduce some new work on this topic. They are joint work with H.T. Yau and Yang Fan.
• Vu
• Recent progress in combinatorial random matrix theory
I introduce some key problems in the combinatorial side of random matrix theory, and discuss many exciting recent progress. Part of the talk is based on my recent survey https://arxiv.org/abs/2005.02797.
• Zeitouni
• A central limit theorem for characteristic polynomial of the GbetaU ensemble
We consider the characteristic polynomial $W_n(z)=det(zI-X_n)$ where $X_n$ is distributed according to the GbetaE ensemble. We prove a CLT for $(\log W_n(z)-F_n(z))/\sqrt{\log n}$, for $z$ inside the support of the equilibrium measure, and appropriate deterministic \$F_n(z). The proof proceeds through the Dumitriu-Edelman three diagonal representation, and an analysis of the recursions satisfied by the determinant. In contrast to the CbetaE case, the recursions involve products of two dimensional matrices, and the main part of the proof is a control of the oscillatory regime of the recursions. Joint work with Fanny Augeri and Raphael Butez.

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Participants Affiliation
Chafaï, Djalil Paris-Dauphine - PSL
D, Yogeshwaran Indian Statistical Institute, Bangalore
Alam, Kazi University of Florida
Najim, Jamal Université Gustave Eiffel (Paris-Est) - CNRS
Bailey, Emma University of Bristol
Forkel , Johannes Mathematical Institute, University of Oxford
Assiotis, Theo University of Oxford
Wong, Mo Dick University of Oxford
Loeffler, Matthias ETH Zürich
Abdalla Teixeira, Pedro PUC-RIO
Arpino, Gabriel ETH Zürich
Grometto, Nicolò ETH Zurich
Ding, Yunzi New York University
Sakuma, Noriyoshi Aichi University of Education
SINGH, SUDHIR National Institute of Technology, Trichy, India
Ueda, Yuki National Institute of Technology, Ichinoseki
Lee, Jinyeop Korea Institute for Advanced Study
Kovaleva, Valeriya University of Oxford
Coste, Simon INRIA (Paris)
Cook, Nicholas Stanford
Benigni, Lucas University of Chicago
Pain, Michel NYU Courant Institute
Renfrew, David SUNY Binghamton
Macera, Davide Università degli Studi Roma Tre and Queen Mary University of London
Swiech, Artur Optiver
Coghi, Francesco Queen Mary, University of London
Yizhe, Zhu University of California, San Diego
Mody, Krishnan Courant, NYU
Farid, Asma Queen Mary University of London
Borodin, Alexei MIT
Ho, Ching Wei Indiana University Bloomington
Yang, Wei SUNY Binghamton
Yang, Fan University of Pennsylvania
WEI, Lu University of Michigan - Dearborn
Maida, Mylene Université de Lille
Driver, Bruce UCSD
Carugno, Giorgio King's College London
Malysheva, Svetlana Queen Mary University of London
Li, Zeng Southern University of Science and Technology, China
Le, Trieu University of Toledo
Nemish, Yuriy University of California, San Diego
Zhang, Jiyuan University of Melbourne
Shi-Hao, Li The University of Melbourne
Jiaxin, QIU Southern University of Science and Technology
Augeri, Fanny Weizmann Institute of Science
Basak, Anirban ICTS-TIFR
Alt, Johannes Université de Genève
Johansson, Kurt KTH Royal Institute of Technology
Renaud, Rivier University of Geneva
Dartois, Stephane LaBRI, University of Bordeaux
Lancien, Cécilia Institut de Mathématiques de Toulouse & CNRS
Butez, Raphael Weizmann Institute of science
Reda, Chhaibi Université Paul Sabatier - Toulouse
Capitaine, Mireille CNRS, IMT, Université Toulouse 3
Berezin, Sergey CNRS & Aix-Marseille Université, France
GRACZYK, PIOTR University Angers
Rouault, Alain UVSQ Paris-Saclay
Donati-Martin, Catherine UVSQ - Paris-Saclay
Cipolloni, Giorgio IST Austria
Nonnenmacher, Stéphane Université Paris-Saclay
Yanqi, QIU Institute of Mathematics, AMSS, Chinese Academy of Sciences;CNRS
Lytova, Anna Opole University
Liu, Dang-Zheng University of Science and Technology of China
Wang, Yanhui Henan University, China
Yandong, Gu University of science and technology of china
Junshan, Xie Henan University
Dembczak-Kołodziejczyk, Alicja Opole University
Barraquand, Guillaume Ecole Normale Supérieure
TIAN, Peng HongKong University
O'Rourke, Sean University of Colorado Boulder
Wang, Zhichao UCSD
Zhang, Lu University of Science and Technology of China
Dai, Dan City University of Hong Kong
Wu, Qiang UIUC
Zhang, Lun Fudan University
Xu, Shuai-Xia Sun Yat-sen University
Couillet, Romain GIPSA-lab, Université Grenoble-Alpes
Guillaud, Maxime Huawei Technologies
Gérard, Besson CNRS-Université Grenoble Alpes
Day, Huw University of Bristol
SÉJOURNÉ, Charles Gipsa-Lab, UGA Grenoble
Chatelain, Florent Univ. Grenoble-Alpes
Seddik , Mohamed El Amine CEA
Zuber, Jean-Bernard Sorbonne Université
TIOMOKO, Malik Université Paris Sud
Gaudreau Lamarre, Pierre Yves Princeton University
McSwiggen, Colin Brown University
Yao, Jeff The University of Hong Kong
Wang, Zhenggang The University of Hong Kong
Kuijlaars, Arno Katholieke Universiteit Leuven
Betea, Dan KU Leuven
Świderski, Grzegorz Katholieke Universiteit Leuven
Yang, Meng UCLouvain
Glesner, Gabriel Université Catholique de Louvain
Liao, Zhenyu University of California, Berkeley
Zarrouk, Tayeb GIPSA-lab, Université Grenoble-Alpes
Minakov, Alexander UCLouvain
Yuan, Wangjun The University of Hong Kong
Wang, Jingming Hong Kong University of Science and Technology
Byun, Sung-Soo Seoul National University
Kahn, Ezechiel ENPC
Seo, Seong-Mi Korea Institute for Advanced Study
Hulbert, Christopher George Mason University
Ruzza, Giulio Université Catholique de Louvain
Kafetzopoulos, Anastasis Queen Mary University of London
He, Yukun University of Zurich
Kammoun, Slim Université de Lille
Semerjian, Guilhem ENS - PSL
Deaño, Alfredo Universidad Carlos III de Madrid (Spain) / University of Kent (UK)
Kammoun, Abla KAUST
FitzGerald, Will University of Sussex
Yifei, Li peking university
Wang, Haoyu Courant Institute
Akjouj, Imane Université de Lille
Rahman, Anas University of Melbourne
Yu, Tingzhou University of Victoria
Chavali, Vaibhav George Mason University
Wage, Kathleen George Mason University
Mergny, Pierre LPTMS
Collins, Benoit Kyoto University
Lysychkin, Serhii Queen Mary University of London
Groot, Alan KU Leuven
Maltsev, Anna Queen Mary University of London
Yabuoku, Satoshi Chiba University
Guo, Jiongji HKUST
Hughes, Christopher University of York
Thompson, Matthew University of York
Gordina, Masha University of Connecticut
Gross, Leonard Cornell Univeersity
Park, Jaewhi KAIST
Ji, Hong Chang Korea Advanced Institute of Science and Technology
Jung, Ji Hyung KAIST
Yang, Wooseok KAIST
Nechita, Ion CNRS, University of Toulouse, France
Cammarota, Valentina Sapienza University of Rome
Barucca, Paolo UCL
Di Nardo, Elvira University of Turin, Italy
Diaz-Santos, Jose George Mason University
Guenzani, Roberto Insubria University
Marinucci, Domenico Department of Mathematics, University of Rome Tor Vergata
Brini, Alessio Scuola Normale Superiore
Breuer, Jonathan The Einstein Institute of Mathematics, The Hebrew University of Jerusalem
Aydogan, Burcu Middle East Technical University
Parraud, Felix ENS Lyon
Gordin, Mira Brown University
De Canditiis, Daniela CNR- Rome
Andreis, Luisa Wias Berlin
Greco, Giacomo Sapienza University of Rome
Keleş, Ahmet Abdullah Bilkent University
Campbell, Andrew University of Colorado at Boulder
Hui, David The Hong Kong University of Science and Technology
Ebina, Masahisa Kyoto University
Lee, SeungYeop University of South Florida
Wang, Dong National University of Singapore
Banna, Marwa Saarland University
Arista, Jonas University College Dublin
Cammarota, Valentina Sapienza University of Rome
Cammarota, Valentina Sapienza University of Rome
Hao, Yifan Fudan University
Xu, Yuanyuan KTH Royal Institute of Technology
Huang, Jiaoyang IAS
Alvarez, Emilia University of Bristol
Zhong, Ping University of Wyoming
Bercovici, Hari Indiana University
Krishnan, Kesav UIUC
Kammoun, Abla KAUST
Memin, Ronan ENS Lyon
Husson, Jonathan ENS Lyon
Speicher, Roland Saarland University
Fang, Kunyu Fudan University
Doz, Cyprien Centrale-Supélec
Snaith, Nina University of Bristol
Youssef, Pierre New York University Abu Dhabi
Cafasso, Mattia Université d'Angers
Ko, Justin University of Toronto
Doan, Luan University of Notre Dame
Walsh, Harriet ENS Lyon
Akemann, Gernot Bielefeld University
Grava, Tamara University of Bristol & SISSA
Dettmann, Carl University of Bristol
Müller, Sebastian University of Bristol
Smolyarenko, Igor Brunel University London
Marklof, Jens University of Bristol
Kawamoto, Naoki Kyoto University
Nakashima, Hideto Nagoya University
Renzi, Bruno Univeristà degli Studi Roma Tre
Jalowy, Jonas Bielefeld University
Khoruzhenko, Boris QMUL
HUSSON, Jonathan ENS Lyon
Chavez Heredia, Victor Eduardo University of Bristol
Savin, Dmitry Brunel University London, UK
Venker, Martin Bielefeld University
Torrisi , Giovanni Luca CNR
Little, Alex University of Bristol
Eaknipitsari, Sorawit University of Notre Dame
Au, Benson University of California, San Diego
le doussal, pierre lpens paris
Buijsman, Wouter University of Amsterdam
Vleeshouwers, Ward Universiteit van Amsterdam
Vogel, Martin CNRS, Universite de Strasbourg
Liao, Yuchen University of Michigan
Ducatez, Raphael Université de Genève
Ryan, Kieran Queen Mary University of London
Peche, Sandrine Univ. Paris Diderot
Wilsher, Michael University of Bristol
Li, Weiming Shanghai University of Finance and Economics
Langhammer, Heide WIAS, Berlin
Vinet, Stephane Université de Montréal
Piccioni, Mauro Sapienza Università di Roma
Magaldi, Hugo Paris-Dauphine - PSL
Matsuda, Junichiro Kyoto University
Li, Zhaoyuan The Chinese University of Hong Kong, Shenzhen
Hou, Zhiqiang Northeast Normal University
Hu, Jiang Northeast Normal University
nie, adam RSFAS, CBE, Australian national university
Katori, Makoto Chuo University
Biane, Philippe CNRS
Shirai, Tomoyuki Kyushu University
Noda, Kohei Kyushu University
Zhang, Zhixiang Nanyang Technological University
Liu, Baisen Dongbei University of Finance and Economics
Erdos, Laszlo IST Austria
Xue, Boqing ShanghaiTech University
Pham, Tuan University of Minnesota
Bhattacharya, Riddhiman University of Minnesota
Xue, Boqing ShanghaiTech University
Trinh, Khanh Duy Waseda University
Caputo, Pietro Roma Tre University
Tsiares, Ioannis McGill Unviersity
Baillard, Valentin Université Paris Sud
Leblé, Thomas Université de Paris
Tang, Chen The Australian National University
Maciej, Nowak Jagiellonian University
Yang, Fan Shanghai Jiaotong University
Xing, HE Shanghai Jiao Tong University
Xin, Shi Shanghai Jiao Tong University
Wątorek, Marcin Institute of Nuclear Physics Polish Academy of Sciences
Simm, Nick University of Sussex
Grela, Jacek Queen Mary University of London
Wojtek, Tarnowski Jagiellonian University
Tsuji, Yuta Kyoto University
Serebryakov, Alexander University of Sussex
Dahlqvist, Antoine University of Sussex
Koenig, Jake OSU
Schlag, Wilhelm Yale
Kim, Yujin Courant Institute of Mathematical Sciences
Nakano, Fumihiko Mathematical Institute, Tohoku Univeristy
Kemp, Todd UC San Diego
Desiraju, Harini SISSA
Vleeshouwers, Ward Universiteit van Amsterdam
Mazzuca, Guido SISSA
Reeves, Thomas Cornell University