Random matrices and Free probability

Liste des contributions

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

Alice Guionnet

19/06/2024 10:00

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....

2. Multi-spiked tensor models

Gérard Ben Arous

19/06/2024 11:30

5. Tensors of free variables and random quantum channels

Pierre Youssef

19/06/2024 14:00

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...

4. Secular coefficients and the holomorphic multiplicative chaos.

Joseph Najnudel

19/06/2024 15:00

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...

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

Mylène Maïda

19/06/2024 16:30

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...

6. Freeness to all orders

Gaëtan Borot

20/06/2024 10:00

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...

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

Denis Bernard

20/06/2024 11:30

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...

8. Free cumulants everywhere

Philippe Biane

20/06/2024 14:00

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.

9. Large deviations for macroscopic observables of heavy-tailed matrices

Charles Bordenave

20/06/2024 15:00

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...

10. Notions of Non-Commutative Independence

Marwa Banna

20/06/2024 16:30

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...

13. Diner de conférence

20/06/2024 19:30

11. Universality of the Large Deviations of the Longest Increasing Subsequence of Random Permutations

Slim Kammoun

21/06/2024 09:00

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...

12. Rate of convergence of empirical measures of hyperuniform point processes

Sandrine Dallaporta

21/06/2024 10:00

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...

14. Outliers of perturbations of banded Toeplitz matrices

François Chapon

21/06/2024 11:30

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...