Volume of metric balls in Liouville quantum gravity

Abstract

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 $p\in (-\mathrm{\infty },4/{\gamma }^2)$. 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. Gwynne-Miller-Sheffield (2020) prove that the metric measure space structure of $\gamma$-LQG a.s. determines its conformal structure when $\gamma =\sqrt{8/3}$; their argument and our estimate yield the result for all $\gamma \in (0,2)$.