Can you hear the shape of Liouville quantum gravity (LQG)?

We obtain a Weyl law for the eigenvalues of Liouville Brownian motion: the n-th eigenvalue grows linearly with n, with the proportionality

constant given by the Liouville measure of the domain and a certain deterministic constant which is computed explicitly and is,

surprisingly, strictly greater than its Riemannian counterpart. After explaining this result and its context, as well as some related

estimates pertaining to the small-time behaviour of the heat kernel, I hope to also present a number of conjectures on the spectral geometry

of LQG.

These relate both to the behaviour of eigenfunctions (suggesting intriguing connections with so-called “quantum chaos”) and to that of

eigenvalues, for which we conjecture a connection to random matrix statistics.

This is joint work with Mo-Dick Wong (Durham).

**Date**: 16 October 2023, 14:00 (Monday, 2nd week, Michaelmas 2023)**Venue**: Mathematical Institute

Woodstock Road OX2 6GGSee location on maps.ox**Speaker**: Nathanaël Berestycki (University of Vienna)**Organising department**: Department of Statistics**Organisers**: Matthias Winkel (Department of Statistics, University of Oxford), Julien Berestycki (University of Oxford), Christina Goldschmidt (Department of Statistics, University of Oxford), James Martin (Department of Statistics, University of Oxford)**Part of**: Probability seminar**Booking required?**: Not required**Audience**: Public- Editors: Christina Goldschmidt, Julien Berestycki