Liouville quantum gravity from random matrix dynamics
Liouville quantum gravity from random matrix dynamics
The Liouville quantum gravity measure is a properly renormalized exponentialÌýof the 2d GFF. In this talk, I will explain how it appears as a limit ofÌýnatural random matrix dynamics: if $(U_t)$ is a Brownian motion on the unitaryÌýgroup 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 limitÌýof large dimension. This extends results from Webb, Nikula and Saksman forÌýfixed time. The proof relies on a newÌýmethod for Fisher-Hartwig asymptoticsÌýof Toeplitz determinants with real symbols, which extends to multi-timeÌýsettings. I will explain this method and how to obtain multi-time loopÌýequations by stochastic analysis on Lie groups.Ìý
Based on a joint work with Paul Bourgade.