Volume of metric balls in Liouville quantum gravity

Morris Ang, Hugo Falconet, Xin Sun

Published 2020-01-30Version 1

We study the volume of metric balls in Liouville quantum gravity (LQG). For $\gamma \in (0,2)$, it has been known since the early work of Kahane (1985) and Molchan (1996) that the LQG volume of Euclidean balls has finite moments exactly for $p \in (-\infty, 4/\gamma^2)$. Here, we prove that the LQG volume of LQG metric balls admits all finite moments. This answers a question of Gwynne and Miller and generalizes a result obtained by Le Gall for the Brownian map, namely, the $\gamma = \sqrt{8/3}$ case. We use this moment bound to show that on a compact set the volume of metric balls of size $r$ is given by $r^{d_{\gamma}+o_r(1)}$, where $d_{\gamma}$ is the dimension of the LQG metric space. Using similar techniques, we prove analogous results for the first exit time of Liouville Brownian motion from a metric ball. Our result implies that the metric measure space structure of $\gamma$-LQG determines its conformal structure.