• DOI:10.1214/20-ejp564
• Corpus ID: 210966305

Volume of metric balls in Liouville quantum gravity

@article{Ang2020VolumeOM,
  title={Volume of metric balls in Liouville quantum gravity},
  author={Morris Ang and Hugo Falconet and Xin Sun},
  journal={arXiv: Probability},
  year={2020}
}

• M. Ang, Hugo Falconet, Xin Sun
• Published 30 January 2020
• Mathematics
• arXiv: Probability

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.