Volume of metric balls in Liouville quantum gravity

Abstract

We study the volume of metric balls in Liouville quantum gravity (LQG). For ${\hat{I}}^3\in (0,2)$, 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 $p \in (-\infty, 4/γ^2)$. 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 ${\hat{I}}^3=\sqrt{8/3}$ case. We use this moment bound to show that on a compact set the volume of metric balls of size $r$ is given by $r^{d_{\hat{I}}^3+o_r(1)}$, where $d_{\hat{I}}^3$ 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. Gwynne-Miller-Sheffield (2020) prove that the metric measure space structure of ${\hat{I}}^3$-LQG a.s. determines its conformal structure when ${\hat{I}}^3=\sqrt{8/3}$; their argument and our estimate yield the result for all ${\hat{I}}^3\in (0,2)$.

44 pages; 2 figures. To appear in EJP