Mini-Workshop of the ANR STARS

26 juin 2023, 09:00 → 30 juin 2023, 22:30 Europe/Paris

salle Huron (1R1 - 106) (IMT)

Guillaume CEBRON

Random matrices and free probability (see the website https://www.math.univ-toulouse.fr/~gcebron/STARS.php)

 

Participants :
Josué Avajon
Guillaume Cébron
Reda Chhaibi
François Chapon
Clément Chouard
Issa-Mbenard Dabo
Antoine Dahlqvist
Stéphane Dartois
Alexandre Fourment
Slim Kammoun
Pan Kewei
Paul Lacuve
Camille Male
Benjamin McKenna
Ion Nechita
Evangelos Nikitopoulos
Jadi Nisrine

lundi 26 juin

12:30 → 14:00 Lunch

14:00 → 15:00 Talk by Camille Male

Orateur: Camille Male

15:30 → 16:30 Talk by Issa Dabo

Orateur: Issa Dabo

mardi 27 juin

09:45 → 10:45 Large deviations for the top eigenvalue of deformed random matrices

In recent years, the few classical results in large deviations for random matrices have been complemented by a variety of new ones, in both the math and physics literatures, whose proofs leverage connections with Harish-Chandra/Itzykson/Zuber integrals. We present one such result, focusing on extreme eigenvalues of deformed sample-covariance and Wigner random matrices. This confirms recent formulas of Maillard (2020) in the physics literature, precisely locating a transition point whose analogue in non-deformed models is not yet fully understood. Joint work with Jonathan Husson.

Orateur: Benjamin McKenna

11:15 → 12:15 Talk by Stéphane Dartois

Orateur: Stéphane Dartois

12:30 → 14:00 Lunch

mercredi 28 juin

09:30 → 10:30 Martingale Theoretic Approach to Noncommutative Stochastic Calculus

Free -- or more generally noncommutative -- stochastic analysis is often useful for describing the large $N$-limit of an ensemble $X^{(N)} = \big(X_t^{(N)}\big)_{t \geq 0}$ of $N \times N$ matrix stochastic processes. We describe a flexible general theory of noncommutative stochastic calculus that is useful for describing the large-$N$ limits of solutions to $N \times N$ matrix stochastic differential equations. Our theory generalizes the theories of Biane-Speicher for free Brownian motion and Donati-Martin for $q$-Brownian motion. Moreover, it unifies these theories with some aspects of the classical theory of stochastic calculus. This is joint work with D. Jekel and T. Kemp.

Orateur: Vaki Nikitopoulos

11:00 → 12:00 Injective norm of random tensors

I will present some new results about the injective norm of random tensors. The distributions we shall consider range from the simplest Gaussian distribution to that of symmetric Gaussian tensors or random Matrix Product States, which are of interest in quantum information theory. This is joint work in progress with Cécilia Lancien

Orateur: Ion Nechita

12:30 → 14:00 Lunch

20:00 → 22:00 Dinner

jeudi 29 juin

09:30 → 10:30 Pitman's theorem and the quantum group SL2 in infinite curvature

Pitman's theorem states that a Brownian motion minus twice its current minimum is a Markov process. We will consider two a priori distinct approaches to this theorem: Biane's approach, using a non-commutative walk on the quantum group SL2 in the crystal regime "q=0", and Bougerol-Jeulin's approach, using Brownian motion on the hyperbolic space with infinite curvature. A unified version of these two approaches will be given via a presentation of the quantum group isolating a curvature parameter and the Planck's constant. Work in collaboration with R. Chhaibi.

Orateur: François Chapon

11:00 → 12:00 Small cycle structure for words in conjugacy invariant random permutations

We are interested in the cycle structure of words in several random permutations.
The first part of the talk will be dedicated to recall classic results (Nica 1994) when the permutations are i.i.d uniform of size n.

In the second part, we assume that the permutations are independent and that their distribution is conjugacy invariant, with a good control on their short cycles. If, after successive cyclic simplifications, the word w still contains at least two different letters, then we get a universal limiting joint law for small cycles for the word in these permutations.

The third part will be dedicated to the discussion of some open problems.

This talk is based on a joint work with Mylène Maïda (ArXiv 2204.04759).

Orateur: Slim Kammoun

12:30 → 14:00 Lunch

vendredi 30 juin

09:30 → 10:30 How to estimate a covariance matrix? Hopefully in large dimensions.

Consider the basic operation of estimating the spectrum of large covariance matrices.
This estimation has an inherent "large dimensional bias", when one observes a multivariate sample whose size is comparable to the dimension.
Solving this issue amounts to understanding free multiplicative deconvolution.
Our work follows the footsteps of El Karoui, Arizmendi-Tarrago-Vargas and Ledoit-Péché.

After presenting their work, we will discuss the pros and cons of the methods.
Then
1) we will exhibit our own method for computable and statistically consistent estimation.
2) present a cramer-Rao lower bound

This is work in progress. Feedback from the audience will be required.

Orateur: Reda Chhaibi

11:00 → 12:00 Between free groups and surface groups.

We shall consider free groups of arbitrary finite, even rank, and their quotient by one relator leading to surface groups. The aim of the talk is to present different families of traces on the free group, that interpolate between the regular trace on the quotient and the regular trace on the free group, while being motivated by matrix approximations. When the rank is 2, the non-commutative distribution of the generators of the free group interpolates between freely independent and classically independent Haar unitaries.

12:30 → 14:00 Lunch

14:00 → 15:00 Colloquium : Some aspects of Horn's problem

Horn's problem deals with the following question: what can be said about the spectrum
of eigenvalues of the sum 𝐶=𝐴+𝐵 of two Hermitian matrices of given spectrum ? The support of the spectrum of 𝐶 is now well understood, after a long series of works from Weyl (1912) to Horn (1952) to Klyachko (1998) and Knutson and Tao (1999). The problem has also amazing connections with group theory and the decomposition of tensor product of representations. Comparison with the same problem for real symmetric matrices and the action of the orthogonal group reveals similarities but also unexpected differences… In this talk, after a short introduction to the problem, I'll sketch the computation of the probability distribution function of the eigenvalues of 𝐶, when 𝐴 and 𝐵 are independently and uniformly distributed on their orbit under the action of the group. I'll also review some aspects of the connection with representation theory and combinatorics.

Orateur: Jean-Bernard Zuber