Volume of metric balls in Liouville quantum gravity

2020/01/30 by Morris Ang, Ang, Morris, Hugo Falconet +3
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #math-ph #math.MP #math.PR

paper · pdf · doi:10.48550/arxiv.2001.11467

44 pages; 2 figures. To appear in EJP

arxiv created 2020/11/30 · arxiv updated 2020/12/01

Abstract

We study the volume of metric balls in Liouville quantum gravity (LQG). For γ∈ (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 ∈ (-∞, 4/γ²). 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 γ= √(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_γ+o_r(1), where d_γ 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 γ-LQG a.s. determines its conformal structure when γ=√(8/3); their argument and our estimate yield the result for all γ∈ (0,2).

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