Speaker: **Felix Parraud** (ENS Lyon)

Title: Interpolation between random matrices and their free limit with the help of free stochastic processes

Time/Date: 4:45-6:15pm, April 30 (Thu.), 2020

Room: This will be an online seminar on Zoom. Ask Kawahigashi for the link.

Abstract:
Given several independent random matrices X^{N}, we consider a polynomial P evaluated in those matrices.
It is well-known since the nineties, thanks to the work of Voiculescu, that under some assumptions on the
law of X^{N}, the renormalized trace of P(X^{N}) converges towards the trace of P evaluated
in free operators. This kind of results gives us a rough understanding of the asymptotic behavior of the
spectrum of P(X^{N}). However it does not say anything about the local properties of the spectrum.
In this talk we focus in particular on the operator norm of P(X^{N}). The earliest result on this
matter dates back to 2005 where Haagerup and Thorbjornsen proved the convergence of the operator norm of
any polynomials in GUE random matrices. Typically to get this kind of result we need to study the
non-renormalized trace of a smooth function in our matrices. Our strategy to do so is to interpolate
our random matrices and the free limit with the help of well-chosen free stochastic processes. We apply
this strategy to two models of random matrices, GUE matrices which we interpolate with free semicircular
with the help of usual free Ornstein Uhlenbeck processes, and unitary Haar matrices which we interpolate with free Haar unitaries with the help of free unitary Brownian motions.