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.

• Charles Bordenave, CNRS & Aix-Marseille Université, France
• Benoit Collins, Kyoto University, Japan
• Laure Dumaz, CNRS & Université Paris-Dauphine - PSL, France
• Ioana Dumitriu, UCSD, USA
• Zhou Fan, Yale University, USA
• Yan Fyodorov, King's College, UK
• Brian Hall, University of Notre Dame, USA
• Jonathan Keating, Oxford University, UK
• Antti Knowles, Université de Genève, Switzerland
• Torben Krüger, Bonn University, Germany
• Jun Yin, UCLA, USA
• Van Vu, Yale University, USA
• Ofer Zeitouni, Weizmann Institute of Science, Israel

This is a draft proposal NY local time.

• Monday May 25
  • 09:50-10:00 Opening
  • 10:00-10:45 Zeitouni - A central limit theorem for characteristic polynomial of the GbetaU ensemble
  • 11:00-11:45 Fan - Applications of random matrix theory to graph matching and neural networks
  • 12:00-12:30 Coffee break
  • 12:30-13:15 Bordenave - Detection thresholds in very sparse matrix completion
• Tuesday May 26
  • 10:00-10:45 Knowles - Fluctuations of extreme eigenvalues of sparse Erdös-Rényi graphs
  • 11:00-11:45 Yin - Universality and delocalization of random band matrices
  • 12:00-12:30 Coffee break
  • 12:30-13:15 Dumitriu - Random Regular Hypergraphs
  • 13:30-14:15 Vu - Recent progress in combinatorial random matrix theory
• Wednesday May 27
  • No talk
• Thursday May 28
  • 10:00-10:45 Fyodorov - Statistics of extremes in eigenvalue-counting staircases
  • 11:00-11:45 Dumaz - Localization of the continuous Anderson hamiltonian in 1-d and its transition towards delocalization
  • 12:00-12:30 Coffee break
  • 12:30-13:15 Krüger - Non-selfadjoint random matrices: spectral statistics and applications
• Friday May 29
  • 10:00-10:45 Collins - Norm estimates for polynomials in random matrices: new results
  • 11:00-11:45 Hall - Eigenvalues for sums of self-adjoint and skew-self-adjoint random matrices
  • 12:00-12:30 Coffee break
  • 12:30-13:15 Keating - Joint Moments of Characteristic Polynomials of Random Unitary Matrices

• 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
  • Eigenvalues for sums of self-adjoint and skew-self-adjoint random matrices
    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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• Paul Bourgade, New York University, USA
• Djalil Chafaï, Paris-Dauphine / PSL, France
• Alice Guionnet, CNRS & ÉNS Lyon, France
• Jamal Najim, CNRS & Université Gustave Eiffel, France