# 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}
}
• 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… 8 Citations ## Figures from this paper Existence and uniqueness of the Liouville quantum gravity metric for $$\gamma \in (0,2)$$ γ ∈ ( 0 , • Mathematics Inventiones mathematicae • 2020 We show that for each $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , there is a unique metric (i.e., distance function) associated with $$\gamma$$ γ -Liouville quantum gravity (LQG). More precisely, we show Geodesic networks in Liouville quantum gravity surfaces • Ewain Gwynne • Mathematics Probability and Mathematical Physics • 2021 Recent work has shown that for$\gamma \in (0,2)$, a Liouville quantum gravity (LQG) surface can be endowed with a canonical metric. We prove several results concerning geodesics for this metric. In What is a random surface? Given 2n unit equilateral triangles, there are finitely many ways to glue each edge to a partner. We obtain a random sphere-homeomorphic surface by sampling uniformly from the gluings that produce a An invariance principle for ergodic scale-free random environments • Mathematics Acta Mathematica • 2022 There are many classical random walk in random environment results that apply to ergodic random planar environments. We extend some of these results to random environments in which the length scale Metric growth dynamics in Liouville quantum gravity • Mathematics • 2021 We consider the metric growth in Liouville quantum gravity (LQG) for γ ∈ (0, 2). We show that a process associated with the trace of the free field on the boundary of a filled LQG ball is stationary, Introduction to the Liouville quantum gravity metric • Mathematics, Physics • 2021 Liouville quantum gravity (LQG) is a one-parameter family of models of random fractal surfaces which first appeared in the physics literature in the 1980s. Recent works have constructed a metric The volume measure of the Brownian sphere is a Hausdorff measure We prove that the volume measure of the Brownian sphere is equal to a constant multiple of the Hausdorff measure associated with the gauge function h(r) = r log log(1/r). This shows in particular Spine representations for non-compact models of random geometry • Mathematics Probability Theory and Related Fields • 2021 We provide a unified approach to the three main non-compact models of random geometry, namely the Brownian plane, the infinite-volume Brownian disk, and the Brownian half-plane. This approach allows ## References SHOWING 1-10 OF 56 REFERENCES KPZ formulas for the Liouville quantum gravity metric • Mathematics Transactions of the American Mathematical Society • 2020 Let$\gamma\in (0,2)$, let$h$be the planar Gaussian free field, and let$D_h$be the associated$\gamma$-Liouville quantum gravity (LQG) metric. We prove that for any random Borel set$X \subset
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