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 XN, 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 XN, the renormalized trace of P(XN) 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(XN). 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(XN). 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.