Probabilités, Physique Mathématique et Analyse (2PMA)

Limit shapes are known to occur in various models in statistical mechanics, for example dimer models or vertex models. In this talk, I will discuss how those appear in quantum spin chains or quantum fermionic models initialized in a domain wall state. My main example will be the XXZ spin chain, for which some exact results may be obtained using Integrability techniques. This is done by taking a non trivial limit in the six vertex model with domain wall boundary conditions, and working out this limit in terms of orthogonal polynomials. I will also discuss simpler quantum models where a probabilistic meaning may be lost.

Crossed study of zeros of random polynomials and electrons of a Coulomb gas Abstract: In this talk I will present two models of systems of particles: on one hand the zeros of random polynomials with i.i.d. gaussian coefficients and on the other hand a finite system of electrons in dimension 2 interacting with some positive background. Those two models are popular in the statistical physics litterature as examples of disordered systems. In this talk, we will see that these two particle systems have very similar behaviors and that studying them toguether gives some new results and many more conjectures. I will focus on 2 aspects of these particle systems: first the macroscopic limit of the system (convergence of empirical measures) and second the existence of outliers.

Dans cet exposé, j'introduirai un modèle de particles évoluant sur un cercle discret sans s'intersecter. Ce modèle présente de nombreuses propriétés d'intégrabilité qui permettent de prouver des résultats d'universalité. Après avoir brièvement expliqué ces propriétés d'un point de vue algébrique, je montrerai comment on peut en déduire des théorèmes de type Berry-Esseen ou limite locale pour l'évolution du système en temps long. Je montrerai également une conséquence géométrique de ces résultats sur un problème de comptage d'applications holomorphes à valeurs dans une Grassmannienne (tous ces termes seront expliqués au préalable). Cet exposé relate un travail réalisé en collaboration avec Jérémie Guilhot et Cédric Lecouvey.

The Harish-Chandra-Itzykson-Zuber integral, also called spherical integral is defined as the expectation of exp(Tr(AUBU*)) for A and B two self adjoint matrices and U Haar-distributed on the unitary/orthogonal/symplectic group. It was initially introduced by Harish-Chandra to study Lie groups and it also has an interpretation in terms of Schur functions. Since then, it has had many kinds of applications, from physics to statistical learning. In this talk we will look at the asymptotics of these integrals when one of the matrices remains of small rank. We will also see how to use these asymptotics to prove large deviation principles for the largest eigenvalues for random matrix models that satisfy a sub-Gaussian bound. This talk is mainly based on a collaboration with Justin Ko.

We will consider two approaches for constructing non-abelian analogs of the Painleve equations, regarding the Painleve-4 equation as an example. Namely, we are going to present “a fully non-commutative” analog and three non-equivalent matrix analogs, that were obtained in the papers arXiv:2205.05107 (joint with Vladimir Retakh, Vladimir Rubtsov and Georgy Sharygin) and arXiv:2107.11680, 2110.12159 (joint with Vladimir Sokolov), respectively.

The aim of the talk is to show that the Janossy densities of a suitably thinned Airy process are governed by the Schrodinger and (cylindrical) KdV equations. In order to achieve this, first we will review known results for the gap probability of the same thinned Airy process and then we will use them (by means of Darboux transformations) to characterize also the Janossy densities. The talk is based on ongoing work with T. Claeys, G. Glesner and G. Ruzza (UCLouvain).