CMSA Probability: Liouville quantum gravity from random matrix dynamics


View Calendar
November 9, 2022 3:30 pm - 4:30 pm
CMSA, 20 Garden St, G10
Address: 20 Garden Street, Cambridge, MA 02138

Hugo Falconet - Courant Institute, NYU

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.