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 $p \in (\infty, 4/\gamma^2)$. 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. GwynneMillerSheffield (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 $\gamma \in (0,2)$.
 Publication:

arXiv eprints
 Pub Date:
 January 2020
 DOI:
 10.48550/arXiv.2001.11467
 arXiv:
 arXiv:2001.11467
 Bibcode:
 2020arXiv200111467A
 Keywords:

 Mathematics  Probability;
 Mathematical Physics
 EPrint:
 44 pages