A universal interacting particle system in discrete random matrix theory

• Roger Van Peski (KTH Stockholm)

Random matrices over the integers, finite fields, and p-adic integers have been studied since the late 1980s as natural models for random groups appearing in number theory and combinatorics. I will discuss the appearance of a new universal process, the <em>reflecting Poisson sea</em>, governing local limits in several discrete random matrix settings. This object is a discrete-space local interacting particle system playing a role analogous to the extended sine and Airy processes in classical random matrix theory. If time permits I will discuss the tools from integrable probability which make these results possible.

The 2D Toda lattice hierarchy for multiplicative statistics of Schur measures

• Pierre Lazag (Université d'Angers)

Schur measures are probability measures on Young diagrams depending on two countable sets of parameters. Introduced by Okounkov in 2003 and containing the poissonized Plancherel measure as the most known example, they lead to determinantal point processes on the one dimensional lattice. Okounkov also proved that the gap probabilities of these measures are tau functions for the 2D Toda lattice hierarchy, that is to say, they satisfy a hierarchy of particular bilinear PDEs. In this talk, I will show how to extend this result to expectations of more general multiplicative functionals. I will try to give a comprehensive exposition of the techniques of the proof which use the fermionic Fock space formalism. As an application, we recover a recent result of Cafasso-Ruzza on the finite temperature discrete Bessel process, which corresponds to a deformation of the poissonized Plancherel measure, and also obtain a hierarchy for more general finite temperature Schur measures.