Random matrices and Free probability

19 juin 2024, 09:00 → 21 juin 2024, 18:00 Europe/Paris

Amphi Schwarz (IMT)

Guillaume CEBRON, Mireille Capitaine (IMT, CNRS)

This workshop will highlight not only the interaction of random matrices and free probability with many other branches of mathematics (combinatorics, operator algebra, statistics, etc.), but also the interaction of mathematics with physics and theoretical ecology in particular, which borrow from the theories of random matrices and free probability among other mathematical tools.

The workshop will start on Wednesday 19th of June (morning) and end on Friday 21th of June (12:30pm), and will be preceded by a summer school, which can be accessed at the following link: https://indico.math.cnrs.fr/event/10596/

Speakers:
Marwa Banna (NYU Abu Dhabi)

Gérard Ben Arous (chaire d’excellence CIMI, Courant Institute (New York)) 

Denis Bernard (Laboratoire de Physique de l'Ecole Normale Supérieure, Paris) 

Philippe Biane (Laboratoire d’Informatique Gaspard-Monge, Marne-la-Vallée) 

Charles Bordenave (Institut de Mathématiques de Marseille)

Gaëtan Borot (Institute for Mathematics and Institute for Physics, Berlin, Allemagne)

François Chapon (Institut de Mathématiques de Toulouse)

Sandrine Dallaporta (Université de Poitiers)

Alice Guionnet (Unité de Mathématiques pures et appliquées de l’ENS Lyon) 

Slim Kammoun (Unité de Mathématiques pures et appliquées de l’ENS Lyon) 

Mylène Maïda (Equipe Probabilités et Statistique, Laboratoire Paul Painlevé, Lille)

Joseph Najnudel (University of Bristol)

Pierre Youssef (NYU Abu Dhabi)

mercredi 19 juin

1 Large random matrices, random tilings and large deviations for non-intersecting random walks

Discrete Theta-ensembles generalize the distribution of random
tiling in the same way that beta-ensembles generalize that of Gaussian
matrices. Non-intersecting Bernoulli random walks generalize Dyson
Brownian motion to the discrete ensemble. In this talk, I will discuss
large deviations for these objects, with applications to study the
asymptotics of Jack and Mac Donald polynomials. The techniques are based
on the analysis of the so-called Nekrasov's equations which are discrete
analogues of the loop or Dyson-Schwinger's equations. It generalizes to
the discrete setting the large deviations for the Dyson Brownian motion
derived by G-Zeitouni, see also G-Huang. This talk is based on a recent
joint Jiaoyang Huang.

Orateur: Alice Guionnet

11:00 Coffee break

2 Multi-spiked tensor models

Orateur: Gérard Ben Arous

3 Tensors of free variables and random quantum channels

We study tensor products of free random variables and establish a corresponding central limit theorem. This framework appears naturally as the limiting behavior of some random matrix models associated with quantum channels having random Kraus operators. We study the limiting spectral distribution of those random quantum channels and provide an estimate on the extreme eigenvalues. In particular, those estimates imply that many generic random constructions of quantum channels produce quantum expanders. Based on joint works with Cécilia Lancien and Patrick Oliveira Santos.

Orateur: Pierre Youssef

4 Secular coefficients and the holomorphic multiplicative chaos.

We study the coefficients of the characteristic polynomial of
unitary matrices drawn from the Circular Beta Ensemble. When the inverse
temperature parameter beta is strictly larger than 4, we obtain a new
class of limiting distributions that arise when both the order of the
coefficient and the dimension of the matrix goes to infinity. For beta
equal to 2, we solve an open problem of Diaconis and Gamburd by showing
that the middle coefficient tends to zero in probability when the
dimension goes to infinity. We introduce a new stochastic object
associated to the coefficients of the characteristic polynomial, which
we call Holomorphic Multiplicative Chaos (HMC). Viewing the HMC as a
random distribution, we prove a sharp result about its regularity in an
appropriate Sobolev space. Our proofs expose and exploit several novel
connections with other areas, including random permutations, Tauberian
theorems and combinatorics.

Orateur: Joseph Najnudel

16:00 Coffee break

5 Random partitions and topological expansion of 2D Yang-Mills partition function

In the fifties, Chen Ning Yang and Robert Mills made a major breakthrough in quantum field theory by extending the concept of gauge theory to non-abelian groups. Since then, the study of Yang-Mills theory has been a very active field of research both in mathematics and physics. In particular, in the last two decades, significant progress has been made on the rigorous mathematical understanding of the theory on two-dimensional manifolds with gauge group U(N) or SU(N), and of their limit as N grows to infinity. If this talk, I will how the probabilistic study of well chosen random partitions allows us to give rigorous proofs of some topological expansions of the partition function predicted by physicists Gross and Taylor in the nineties. This is joint work with Thibaut Lemoine (Université de Lille).

Orateur: Mylène Maïda

jeudi 20 juin

6 Freeness to all orders

I will describe a theory of free cumulants to all orders (g,n), both
at the level of combinatorics (surfaced permutations) and generating
series (higher R-transform machinery). Freeness up to order (g,n) is
then defined by the additivity of free cumulants up to order (g,n).
(0,1) is the usual freeness, (1/2,1) is infinitesimal freeness, (0,n)
is the n-order freeness of Collins-Mingo-Speicher-Sniady. This theory
is adapted to address all-order (in particular, beyond leading order)
asymptotic expansions in unitarily invariant ensembles of random
hermitian matrices. I will discuss its application to GUE +
deterministic.

Orateur: Gaëtan Borot

11:00 Coffee break

7 Structured random matrices and cyclic cumulants : a free probability approach (inspired by noisy many-body quantum systems)

We shall discuss a new class of large structured random matrices characterised by the properties of their cyclic cumulants. This class is remarkably stable under non-linear operations. We shall present a simple algorithm, based on an extremization problem, to compute the spectrum of sub-blocks of such matrices, and explain the connection between such algorithm and operator valued free probability. This class of random matrices emerged via the study of the quantum symmetric simple exclusion process or, more generally, noisy many-body quantum systems.
[Work done in collaboration with Ludwig Hruza].

Orateur: Denis Bernard

8 Free cumulants everywhere

I will give examples of free cumulants appearing in various
questions such as matrix integrals, map enumeration,
characters of symmetric
groups, braid enumeration and a quantum version of the
simple exclusion process.

Orateur: Philippe Biane

9 Large deviations for macroscopic observables of heavy-tailed matrices

This is an ongoing joint work with Alice Guionnet and Camille Male. We
consider a finite collection of Hermitian heavy-tailed random matrices
of dimension N. Our model include the Lévy matrices introduced by
Bouchaud and Cizeau or sparse random matrices with O(1) non-zeroes
entries per row. When represented as weighted graphs on N vertices,
these matrices have local weak limits in the Benjamini-Schramm topology.
Thanks to this representation, we establish large deviations principle
for macroscopic observables of such collection of matrices. These
observable include the empirical distribution of the eigenvalues and
empirical distribution of the neighborhood distribution.

Orateur: Charles Bordenave

16:00 Coffee break

10 Notions of Non-Commutative Independence

In this talk, I will start by illustrating matrix models
relative to free, infinitesimal free, monotone, and c-free
independencies. Notions of independence play a key role in studying
joint distributions of non-commutative random variables and hence in
studying limiting distributions of the associated random matrix models.
I will illustrate in particular recent results relative to the monotone
case.

Just as in the classical setting, to each notion of independence
corresponds a central limit theorem. The second part of the talk will
focus on the operator-valued setting, where I show quantitative results
for the relative operator-valued central limit theorems.

Based on joint works with Arizmendi, Gilliers, Mai, & Tseng.

Orateur: Marwa Banna

11 Diner de conférence

vendredi 21 juin

12 Universality of the Large Deviations of the Longest Increasing Subsequence of Random Permutations

The length of the longest increasing subsequence of a uniform permutation displays phenomena similar to those observed in the largest eigenvalue of the Gaussian Orthogonal Ensemble, such as Tracy-Widom fluctuations and large deviations with two speeds.
In this presentation, we establish the universality of the large deviations for both speeds for a class of random permutations with a conjugacy invariant distribution and a low number of cycles.
This work is based on a joint work with A. Guionnet.

Orateur: Slim Kammoun

13 Rate of convergence of empirical measures of hyperuniform point processes

This talk is concerned with the empirical measure of a random point process in R^d, such as the eigenvalues of a random matrix or a Coulomb gas. In several cases, this empirical measure converges towards a deterministic measure. In order to quantify the rate of convergence, we are interested in the p-Wasserstein distance between this random measure and its mean, particularly in dimension 2. We obtain a bound for this distance under some assumption on the p-th centered moment of the number of points in squares, which amounts to hyperuniformity when p=2. In addition, hyperuniform determinantal point processes will satisfy the required assumptions for any p>=1.
This is a joint work with Raphaël Butez (Université de Lille) and David García-Zelada (Sorbonne Université).

Orateur: Sandrine Dallaporta

11:00 Coffee break

14 Outliers of perturbations of banded Toeplitz matrices

This is an ongoing work in collaboration with Charles Bordenave and Mireille Capitaine. Toeplitz matrices are non-normal matrices whose spectral analysis in high dimensions is well understood. The spectrum of these matrices is in particular very sensitive to small perturbations. In this talk, we will focus on banded Toeplitz matrices, whose symbol is given by a Laurent polynomial, and which are perturbed by a random matrix. The goal is to describe "outliers", which are eigenvalues that lie outside the support of the limiting distribution of the perturbed matrix as the dimension tends to infinity. The presence of outliers in some region of the complex plane is specifically related to the winding number of the curve determined by the symbol in that region.

Orateur: François Chapon