Liouville quantum gravity from random matrix dynamics

11/09/2022 3:30 pm - 4:30 pm
CMSA Room G10
Address: CMSA, 20 Garden Street, Cambridge, MA 02138 USA

Probability Seminar

Speaker: Hugo Falconet (Courant Institute, NYU)

Title: Liouville quantum gravity from random matrix dynamics

Abstract: 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. I will explain this method and how to obtain multi-time loop equations by stochastic analysis on Lie groups.

Based on a joint work with Paul Bourgade.