# Conference: ProbabLY ON Random Matrices

## Abstracts

Wednesday 9:30 – 10:15

Abstract:

Consider the problem of recovering an invertible $n×n$ matrix $A$ and a sparse $n×p$ random matrix $X$ based on the observation of $Y= AX$ (up to a scaling and permutation of columns of $A$ and rows of $X$). Spielman, Wang and Wright have proposed an algorithm which with high probability recovers both $A$ and $X$ (in the above sense) provided that the matrix $X$ is sufficiently sparse and $p$ is sufficiently large. In my talk I will present a simple modification of this algorithm and provide an analysis of its sample complexity, proving that up to universal constants it performs optimally with respect to the dimension $n$.

Monday 16:15 – 17:00

Abstract:

We will revisit Borell's proof of the large deviations principle of Wiener chaoses. We will argue that some heavy-tail phenomena observed in large deviations can be explained under the paradigm of Borell, meaning that the deviations are created, as for the Wiener chaoses, in a sense by translations. More precisely, we prove a general large deviations principle for a certain class of functionals $f_n : \mathbb{R}^n \to \mathcal{X}$, where $\mathcal{X}$ is some metric space, under the probability measure $\nu_{\alpha}^n$, where $\nu_{\alpha} =Z_{\alpha}^{-1}e^{-|x|^{\alpha}}dx$, for which the large deviations are explained by translations. We retrieve, as an application, the large deviations principles known for the spectral measure, the largest eigenvalue and traces of polynomials of Wigner matrices without Gaussian tails .

Thursday 14:00 – 14:45

Abstract:

Voiculescu's bi-free probability is an extension of freeness designed to encode the joint distribution of pairs of random variables (called faces by Voiculescu) that "behave like" pairs of left and right creation/annihilation operators on a full Fock space. Since Voiculescu's founding paper "Free Probability for Pairs of Faces I" (2013), much progress has been achieved in the understanding of bi-free probability: at this moment there is a theory of bi-freeness with amalgamation and bi-free cumulants, bi-free central limits, bi-free convolutions and their analytic transforms, bi-free infinite divisibility, and of random matrix models for bi-freeness. After a shot review of a few of these elements, we will show that the subordination functions from "classical" free probability play a role in bi-freeness, and indicate how one proves certain regularity results for bi-free convolutions using the analytic subordination property. This is part of joint work with H. Bercovici, Y. Gu and P. Skoufranis.

Monday 11:45 – 12:30

Abstract:

In this joint work with Romain Couillet (Central-Supelec), we give an analysis of kernel spectral clustering methods in the regime where the dimension $p$ of the data vectors to be clustered and their number $n$ grow large at the same rate. We demonstrate, under a $k$-class Gaussian mixture model, that the normalized Laplacian matrix associated with the kernel matrix asymptotically behaves similar to a so-called spiked random matrix. Some of the isolated eigenvalue-eigenvector pairs in this model are shown to carry the clustering information upon a separability condition classical in spiked matrix models. We evaluate precisely the position of these eigenvalues and the content of the eigenvectors, which unveil important properties concerning spectral clustering, in particular in simple toy models. Our results are then compared to the practical clustering of images from the MNIST database, thereby revealing an important match between theory and practice.

Tuesday 9:30 – 10:15
Tuesday 14:00 – 14:45

Abstract:

How do determinantal point processes behave under conditioning with respect to fixing the configuration in a subset of the phase space? The talk will first address this question for specific examples such as the sine-process or the process with the Bessel kernel, where one can explicitl y write the analogue of the Gibbs property in our situation. We will then consider processes induced by general self-adjoint kernels, for which, in joint work with Yanqi Qiu and Alexander Shamov, it is shown that conditional measures of such processes are themselves determinantal and gove rned by self-adjoint kernels, that thethe tail sigma-algebra for such proces ses is trivial (a result independently and by a completely different method obtained by Osada--Osada) and proof is given of the Lyons-Peres conjecture o n completeness of the system of kernels sampled at the particles of a random configuration. The talk is based on the preprint arXiv:1605.01400 as well as on the preprint arXiv:1612.06751 joint with Yanqi Qiu and Alexan der Shamov.

Thursday 15:00 – 15:45

Abstract:

We will consider Hermitian polynomials in independent Wigner and deterministic matrices and will present some ideas of the proof that, a.s., for large dimension, there is no eigenvalue in any interval lying outside the support of some deterministic probability measure which is computed with the tools of free probability. We will also present some subsequent results concerning Information-Plus-Noise type matrices.

Monday 17:15 – 18:00

Abstract:

Free stochastic analysis can be seen as the large N limit of stochastic analysis on NxN matrices. A tricky aspect of free stochastic analysis, proved by Biane in 1998, is that the transition kernel of a free increment does not only depend on the law of the jump but also on the law of the random variable we started from. I will discuss how to get around this problem, and I will present applications, including the large-N limit of processes on Lie groups, and the q-deformation of the Segal-Bargmann transform (joint work with Ching Wei Ho).

Tuesday 10:45 – 11:30

Abstract:

We study the dynamics of a finite system of interacting stochastic particles, in dimension two, confined by an external field and subject to a singular pair repulsion. The invariant law is a Coulomb gas known as the Ginibre ensemble. Despite the apparent simplicity, the interaction is not convex and the invariant law is not product. We study the long-time behavior of the system as well as the behavior when the number of particles tends to infinity, seeing the system as a singular McKean-Vlasov model. We identify in particular two natural regimes depending on the asymptotic behavior of the noise. This is a joint work in progress with Francois Bolley and Joaquin Fontbona.

Friday 10:45 – 11:30

Abstract:

In this talk, I want to report on the work https://arxiv.org/abs/1607.00243 [arxiv.org] in collaboration with J. Najnudel and T. Madaule. There, we investigate the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according the Circular Beta Ensemble (CβE)

Monday 10:45 – 11:30

Abstract: We study the relation in finite dimension, and in the large dimension limit, between pairs of unitarily invariant random matrices, and quantum matrices with values on enveloping algebras. We also consider applications to asymptotics of $GL_N(\mathbb{C})$ tensor products. This talk is based on the preprint arXiv:1611.01892 in collaboration with Novak and Śniady and previous results in collaboration with Śniady.

Tuesday 16:15 – 17:00

Abstract:

A Coulomb gas in $\mathbb{R}^d$ is the canonical Gibbs measure associated with a system of particles in electrostatic interaction. As the number of particles $N$ grows to infinity, the empirical measure of a Coulomb gas converges weakly towards an equilibrium measure, characterized by a variational principle. We obtain sub-gaussian concentration inequalities around this equilibrium measure, in the weak and $W_1$ topologies, at rate $N^2$. This yields for instance a concentration inequality at the correct rate for the Ginibre ensemble. The proof mainly relies on new functional inequalities, which are counterparts of Talagrand's transport inequality $T_1$ in the Coulomb interaction setting. Joint work with Djalil Chafaï and Mylène Maïda.

Friday 11:45 – 12:30

Abstract:

In this talk, I will discuss the recent work with R. Bauerschmidt and H.-T. Yau on the local Kesten-Mckay law for random regular graphs with large but fixed degrees. We proved that the Kesten–McKay law holds for the bulk spectral density down to small scales and the delocalization of bulk eigenvectors. Our method is based on estimating the Green’s function of the adjacency matrices and a resampling of the boundary edges of large balls in the graphs, which combines the almost deterministic tree-like structure of random regular graphs at small distances with methods from random matrix theory for large distances.

Monday 15:00 – 15:45

Abstract:

We establish bounds on the spectral radii for a large class of sparse random matrices, which includes the adjacency matrices of inhomogeneous Erdös-Rényi graphs. For the Erdös-Rényi graph $G(n,d/n)$, our results imply that the smallest and second-largest eigenvalues of the adjacency matrix converge to the edges of the support of the asymptotic eigenvalue distribution provided that $d \gg \log n$. This establishes a crossover in the behaviour of the extremal eigenvalues around $d \sim \log n$. Our results also apply to non-Hermitian sparse random matrices, corresponding to adjacency matrices of directed graphs. Joint work with Florent Benaych-Georges and Charles Bordenave.

Monday 14:00 – 14:45

Abstract:

Gaussian Multiplicative Chaos (GMC) is a theory developed by Kahane in the eighties whose goal is to properly define the exponential of a log-correlated Gaussian process. These exponentials are interpreted as random measures and have important applications in the theories of Liouville quantum gravity or turbulence. It is well-known that for a large class of random matrix ensembles, the asymptotic fluctuations of their eigenvalues are governed by log-correlated Gaussian fields. After reviewing the theory of GMC and some background on unitary invariant matrix ensembles, we will show how such measures can be constructed from the sine process and circular unitary ensemble. This is joint work with Dmity Ostrovsky and Nick Simm.

Thursday 11:45 – 12:30

Abstract:

The convergence in traffic distribution of matrices generalizes the convergence in *-distribution, and makes tractable the description of the limiting distribution of matrices invariant in law by permutation of the elements of the basis. In this presentation I will present three particular situations where the traffic independence reduces respectively to the free, the tensor or the Boolean independence.

Thursday 16:15 – 17:00

Abstract:

For each n, let $A_n = (\sigma_{ij})$ be an $n×n$ deterministic matrix and let $X_n = (X_{ij})$ be an $n×n$ random matrix with i.i.d. centered entries of unit variance. We are interested in the asymptotic behavior of the empirical spectral distribution $\mu^Y_n$ of the rescaled entry-wise product $$Y_n = \frac 1{\sqrt{n}}σ_{ij}X_{ij} .$$ For our main result, we provide a deterministic sequence of probability measures $\mu_n$, each described by a family of Master Equations, such that the difference $\mu_n^Y-\mu_n$ converges weakly in probability to the zero measure. A key feature of our results is to allow some of the entries σij to vanish, provided that the standard deviation profiles An satisfy a certain quantitative irreducibility property. This is a joint work with Nick Cook, Walid Hachem and David Renfrew.

Wednesday 10:45 – 11:30

Abstract:

Let $X_1, ... ,X_n$ be i.i.d. sample in $\mathbb{R}^p$ with zero mean and the covariance matrix S. The problem of recovering the projector onto the eigenspace of S from these observations naturally arises in many applications. Recent technique from [Koltchinskii and Lounici, 2015] helps to study the asymptotic distribution of the distance in the Frobenius norm between the true projector $P_r$ on the subspace of the r-th eigenvalue and its empirical counterpart $\hat{P}_r$ in terms of the effective trace of S. This paper offers a bootstrap procedure for building sharp confidence sets for the true projector $P_r$ from the given data. This procedure does not rely on the asymptotic distribution of $|| P_r - \hat{P}_r ||_2$ and its moments, it applies for small or moderate sample size n and large dimension p. The main result states the validity of the proposed procedure for finite samples with an explicit error bound on the error of bootstrap approximation. This bound involves some new sharp results on Gaussian comparison and Gaussian anti-concentration in high dimension. These are the joint results with V.Spokoiny and V. Ulyanov.

Tuesday 15:00 – 15:45

Abstract:

Two important examples of the determinantal point processes associated with the Hilbert spaces of holomorphic functions are the Ginibre point process and the set of zeros of the Gaussian Analytic Functions on the unit disk. In this talk, I will talk such class of determinantal point processes in greater generality. The main topics concerned are the equivalence of the reduced Palm measures and the quasi-invariance of these point processes under certain natural group action of the group of compactly supported diffeomorphisms of the phase space. This talk is based partly on the joint works with Alexander I. Bufetov and partly on a more recent joint work with Alexander I. Bufetov and Shilei Fan

Friday 9:30 – 10:15

Abstract:

We shall discuss the behavior of statistical mechanics models when the temperature is allowed to take complex values. We shall focus on the phase diagram in the simplest example, the Curie -- Weiss model, and present some results. Based on joint work with Ofer Zeitouni

Thursday 10:45 – 11:30

Abstract:

Free entropy dimension and the first $L^2$ Betti number are both numeric invariants of discrete groups; one is defined by using finite matrices to approximate’’ the group, while the other is of cohomological nature. Somewhat surprisingly, the two numbers are related. I will describe this connection and talk about some applications to von Neumann algebras.

Friday 13:30 – 14:15

Abstract:

A real Ginibre random matrix is defined as an N x N matrix of i.i.d. standard real Gaussian variables. The lack of symmetry leads to complex eigenvalues, but the real matrix elements constrain some of the eigenvalues to the real line. Their expected number is of order $\sqrt{N}$, as shown by Edelman, Kostlan and Shub in 1994, which can be viewed as a law of large numbers for the real eigenvalues. I will show how it is possible to prove central limit theorems (CLTs) for the number of real eigenvalues, and more generally for a certain class of polynomial linear statistics of the eigenvalues. Unlike the usual CLT in random matrix theory, this one requires a normalization. I will also discuss a recent result generalizing Edelman et al's "$\sqrt{N}$ law" to products of random matrices.

Thursday 9:30 – 10:15

Abstract:

The free probability perspective on random matrices is that the large size limit of random matrices is given by some (usually interesting) operators on Hilbert spaces and corresponding operator algebras. The prototypical example for this is that independent GUE random matrices converge to free semicircular operators, which generate the free group von Neumann algebra. The usual convergence in distribution has been strengthened in recent years to a strong convergence, also taking operator norms into account. All this is on the level of polynomials. In my talk I will recall this and then go over from polynomials to rational functions (in non-commuting variables). Unbounded operators will also play a role.

Friday 14:30 – 15:15

Abstract:

We prove a local version of the circular law for products of independent non-Hermitian matrices under weak moment conditions. Our results generalise recent results of Y. Nemish. We apply Stein’s method and some new ideas which help to simplify the proof of the local laws. The talk will be based on joint results with F. Goetze and A. Naumov.
Wednesday 11:45 – 12:30

Abstract:

Let X be an n-dimensional random vector, and assume we are given a sample of its distribution of size N, that is, N independent copies of X. We show that with a large probability the sample covariance matrix approximates the actual covariance matrix of X in the operator norm with any given precision as long as N is a large (constant) multiple of n. The novelty of our result consists in imposing very mild assumptions on the distribution of X, only slightly stronger than the assumption of boundedness of second moments of its one-dimensional projections.

Tuesday 11:45 – 12:30

Abstract:

It was conjectured by Alon, and proved by Friedman, that for a fixed $d$ independent of $n,$ the second largest (in absolute value) eigenvalue $\lambda$ of the random undirected $d$-regular graph on $n$ vertices distributed according to the uniform model satisfies $\lambda\leq 2\sqrt{d-1}+o(1)$ with probability going to $1$ with $n$. This shows that the uniform model is almost Ramanujan for fixed $d$. Vu conjectured that the same phenomenon holds for any $d\leq n/2$. This question was studied by Broder, Frieze, Suen et Upfal who showed that $\lambda\leq O(\sqrt{d})$ for any $d\leq \sqrt{n}$. The range of $d$ was recently extended to $n^{\frac23}$ by Cook, Goldstein and Johnson. We complement these results by showing that for any $\delta\in (0,1)$ and any $n^\delta\leq d\leq n/2$, we have $\lambda\le O(\sqrt{d})$. This answers, up to a multiplicative constant, the conjecture of Vu. This is a joint work with Konstantin Tikhomirov.