Weyl law in Liouville quantum gravity
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 6GG
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