Weyl's law in Liouville quantum gravity

Abstract

Can you hear the shape of Liouville quantum gravity? 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 and a certain deterministic constant $c_\gamma$ depending on $\gamma \in (0, 2)$. The constant $c_\gamma$, initially a complicated function of Sheffield's quantum cone, can be evaluated explicitly and is strictly greater than the equivalent Riemannian constant. At the heart of the proof we obtain sharp asymptotics of independent interest for the small-time behaviour of the on-diagonal heat kernel. Interestingly, we show that the scaled heat kernel displays nontrivial pointwise fluctuations. Fortunately, at the level of the heat trace these pointwise fluctuations cancel each other, which leads to the result. We complement these results with a number of conjectures on the spectral geometry of Liouville quantum gravity, notably suggesting a connection with quantum chaos.