Probability and Mathematical Physics Seminar

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Liouville quantum gravity from random matrix dynamics

Speaker: Hugo Falconet, Courant Institute

Location: Warren Weaver Hall 1302

Date: Friday, September 9, 2022, 11:10 a.m.

Synopsis:

The Liouville quantum gravity measure is a properly renormalized exponential of the 2d GFF. In this talk, I will explain how it appears as a limit of natural random matrix dynamics: if (U_t) is a Brownian motion on the unitary group at equilibrium, then the measures $|det(U_t - e^{i theta}|^gamma dt dtheta$ converge to the 2d LQG measure with parameter $gamma$, in the limit of large dimension. This extends results from Webb, Nikula and Saksman for fixed time. The proof relies on a new method for Fisher-Hartwig asymptotics of Toeplitz determinants with real symbols, which extends to multi-time settings. In particular, I will explain how to obtain multi-time loop equations by stochastic analysis on Lie groups.
This is based on a joint work with Paul Bourgade.