Probability and Statistics Seminar—— Liouville quantum gravity from random matrix dynamics
报告人：Hugo Falconet (NYU Courant)
时间：2023-05-15 14:00-15:00
地点：Room 1114, Sciences Building No. 1
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.
Short bio: Hugo Falconet is currently an Assistant Professor/Courant Instructor at the Courant Institute of New York University. Prior to this appointment, he completed his undergraduate studies at Ecole Normale Superieure in Paris, and then obtained a PhD in mathematics from Columbia University supervised by Julien Dubedat. His research interests are mainly focused on probability theory and its applications to problems in mathematical physics whose universality class is encoded by the Gaussian free field, such as the Liouville quantum gravity metric, random matrices, random surfaces, and spin models.
