Deuxième rencontre de probabilités intégrables

La deuxième rencontre de probabilités intégrables aura lieu le vendredi 21 mars à l'Institut Henri Poincaré.

Un mini-cours sera donné par Béatrice de Tilière (Université Paris-Dauphine PSL) et un exposé invité par Giulio Ruzza (Université de Lisbonne). Nous aurons également quelques exposés courts de 7 minutes, comme à la première rencontre.

Titres et résumés

Mini-cours (2x1h) de Béatrice de Tilière (Université Paris-Dauphine PSL) :

Lecture 1. The dimer model: introduction

Lecture 2. Dimer model on minimal graphs: the elliptic case and beyond

The dimer model represents the adsorption of diatomic molecules on the surface of a crystal. This is described through perfect matchings of a planar graph, chosen according to the Boltzmann measure. When the graph is periodic and bipartite, Kenyon, Okounkov and Sheffield prove that the phase diagram is given by the spectral curve of the model, which has the remarkable property of being Harnack. Another important result is the local formula obtained by Kenyon for the maximal entropy Gibbs measure, when the underlying graph is isoradial and the model is critical. In a series of papers with Cédric Boutillier (Sorbonne université) and David Cimasoni (Université de Genève), we extend these results in a unified framework. We consider the model on minimal graphs and prove an explicit correspondence with the set of Harnack curves; we also prove local formulas for a two-parameter family of Gibbs measures. The first lecture will be an introduction to the dimer model; the second will be devoted to our results with Cédric Boutillier and David Cimasoni. 

Exposé invité (1h) de Giulio Ruzza (Université de Lisbonne) :

Integrable Probability and Integrable Equations 

I will survey a variety of results connecting multiplicative statistics (and, more generally, Jánossy densities) of determinantal point processes (such as the Airy or Bessel processes) to integrable equations (such as the KdV and Toda equations). This connection generalizes the well-known relationship between the Tracy-Widom distribution and the Painlevé II equation. I will explain how this connection manifests explicitly through Riemann-Hilbert problems which underlie both the determinantal point processes ("IIKS theory of integrable operators") and the integrable equations ("inverse scattering"). These Riemann-Hilbert problems provide highly effective tools, e.g., for analyzing asymptotic questions that are meaningful in both probabilistic and equation-theoretic contexts. This talk is based on joint work with M. Cafasso, C. Charlier, T. Claeys, G. Glesner, and S. Tarricone.