Liouville quantum gravity from random matrix dynamics

Liouville quantum gravity from random matrix dynamics

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Hugo Falconet, New York University
Fine Hall 224

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 $|\det(U_t - e^{i \theta})|^\gamma dt \, d\theta$ 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.