# 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…

## 9 Citations

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## 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…