Weyl Law in Liouville Quantum Gravity

Can you hear the shape of 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 area of the domain (times a certain deterministic constant depending on γ∈ (0, 2). At the heart of the proof we obtain estimates of independent interest on the small-time behaviour of the on-diagonal heat kernel. Interestingly, we show that the scaled heat kernel displays nontrivial pointwise fluctuations, which however vanish when integrating over a domain.

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