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2020 Volume of metric balls in Liouville quantum gravity
Morris Ang, Hugo Falconet, Xin Sun
Electron. J. Probab. 25: 1-50 (2020). DOI: 10.1214/20-EJP564


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. 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 $\gamma \in (0,2)$.


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Morris Ang. Hugo Falconet. Xin Sun. "Volume of metric balls in Liouville quantum gravity." Electron. J. Probab. 25 1 - 50, 2020.


Received: 7 April 2020; Accepted: 29 November 2020; Published: 2020
First available in Project Euclid: 30 December 2020

Digital Object Identifier: 10.1214/20-EJP564

Primary: 60D05


Vol.25 • 2020
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