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} }
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
Existence and uniqueness of the Liouville quantum gravity metric for
$$\gamma \in (0,2)$$
γ
∈
(
0
,
- MathematicsInventiones 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
- MathematicsProbability 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?
- Mathematics
- 2022
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
- MathematicsActa 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
- Mathematics
- 2021
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
- MathematicsProbability 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
- MathematicsTransactions 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…
Confluence of geodesics in Liouville quantum gravity for $\gamma \in (0,2)$
- MathematicsThe Annals of Probability
- 2020
We prove that for any metric which one can associate with a Liouville quantum gravity (LQG) surface for $\gamma \in (0,2)$ satisfying certain natural axioms, its geodesics exhibit the following…
Liouville Brownian motion
- Mathematics
- 2016
We construct a stochastic process, called the Liouville Brownian motion which we conjecture to be the scaling limit of random walks on large planar maps which are embedded in the euclidean plane or…
Weak LQG metrics and Liouville first passage percolation
- MathematicsProbability Theory and Related Fields
- 2020
For $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , we define a weak $$\gamma $$ γ - Liouville quantum gravity ( LQG ) metric to be a function $$h\mapsto D_h$$ h ↦ D h which takes in an instance of the planar…
Existence and uniqueness of the Liouville quantum gravity metric for
$$\gamma \in (0,2)$$
γ
∈
(
0
,
- MathematicsInventiones 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…
Liouville quantum gravity and the Brownian map I: The QLE(8/3,0) metric
- Mathematics
- 2015
Liouville quantum gravity (LQG) and the Brownian map (TBM) are two distinct models of measure-endowed random surfaces. LQG is defined in terms of a real parameter $\gamma$, and it has long been…
The Fractal Dimension of Liouville Quantum Gravity: Universality, Monotonicity, and Bounds
- MathematicsCommunications in Mathematical Physics
- 2019
We prove that for each $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , there is an exponent $$d_\gamma > 2$$ d γ > 2 , the “fractal dimension of $$\gamma $$ γ -Liouville quantum gravity (LQG)”, which describes…
Liouville quantum gravity and the Brownian map III: the conformal structure is determined
- Physics, Mathematics
- 2016
Previous works in this series have shown that an instance of a $$\sqrt{8/3}$$ 8 / 3 -Liouville quantum gravity (LQG) sphere has a well-defined distance function, and that the resulting metric measure…
Liouville first-passage percolation: Subsequential scaling limits at high temperature
- MathematicsThe Annals of Probability
- 2019
Let $\{Y_{\mathfrak{B}}(v):v\in\mathfrak{B}\}$ be a discrete Gaussian free field in a two-dimensional box $\mathfrak{B}$ of side length $S$ with Dirichlet boundary conditions. We study the Liouville…
Liouville quantum gravity spheres as matings of finite-diameter trees
- PhysicsAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques
- 2019
We show that the unit area Liouville quantum gravity sphere can be constructed in two equivalent ways. The first, which was introduced by the authors and Duplantier, uses a Bessel excursion measure…