Volume of metric balls in Liouville quantum gravity
Volume of metric balls in Liouville quantum gravity
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DOI：
10.1214/20-ejp564
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发表时间：
2020-01
影响因子：
1.4
通讯作者：
M. Ang;Hugo Falconet;Xin Sun
中科院分区：
文献类型：
--
作者：
M. Ang;Hugo Falconet;Xin Sun
研究了Liouville量子引力中度规球的体积。对于$\gamma \in（0，2）$，自从Kahane（1985）和Molchan（1996）的早期工作以来，人们已经知道欧几里得球的LQG体积对$p \in（-\infty，4/\gamma^2）$具有有限矩。在这里，我们证明了LQG度量球的LQG体积允许所有有限矩。这回答了Gwynne和米勒的一个问题，推广了Le Gall关于布朗映射的一个结果，即$\gamma = \sqrt{8/3}$的情形。我们利用这个矩界证明了在紧集上，尺寸为r的度量球的体积由r^{d_{\gamma}+o_r（1）}$给出，其中d_{\gamma}$是LQG度量空间的维数。利用类似的技巧，我们证明了类似的结果，从度量球的刘维尔布朗运动的第一次出口时间。我们的结果表明，$\gamma$-LQG的度量测度空间结构决定了它的共形结构.
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.