The dimension of the boundary of a Liouville quantum gravity metric ball

TL;DR

This paper determines the Hausdorff dimension of the boundary of Liouville quantum gravity (LQG) metric balls, revealing how it varies with the parameter b3 and linking it to the geometry of the underlying Gaussian free field.

Contribution

It provides explicit formulas for the Hausdorff dimensions of LQG metric ball boundaries and their intersections with thick points, advancing understanding of LQG geometry.

Findings

Hausdorff dimension of boundary with respect to Euclidean metric is 2 - (γ/d_γ)(2/γ + γ/2) + γ^2/(2d_γ)^2

Hausdorff dimension of boundary with respect to b3-LQG metric is d_γ - 1

For b3= a8/3, boundary Euclidean dimension is 5/4, LQG dimension is 3

Abstract

Let $\gamma \in (0,2)$, let $h$ be the planar Gaussian free field, and consider the $\gamma$-Liouville quantum gravity (LQG) metric associated with $h$. We show that the essential supremum of the Hausdorff dimension of the boundary of a $\gamma$-LQG metric ball with respect to the Euclidean (resp. $\gamma$-LQG) metric is $2 - \frac{\gamma}{d_\gamma}\left(\frac{2}{\gamma} + \frac{\gamma}{2} \right) + \frac{\gamma^2}{2d_\gamma^2}$ (resp. $d_\gamma-1$), where $d_\gamma$ is the Hausdorff dimension of the whole plane with respect to the $\gamma$-LQG metric. For $\gamma = \sqrt{8/3}$, in which case $d_{\sqrt{8/3}}=4$, we get that the essential supremum of Euclidean (resp. $\sqrt{8/3}$-LQG) dimension of a $\sqrt{8/3}$-LQG ball boundary is $5/4$ (resp. $3$). We also compute the essential suprema of the Euclidean and $\gamma$-LQG Hausdorff dimensions of the intersection of a $\gamma$-LQG ball boundary with the set of metric $\alpha$-thick points of the field $h$ for each $\alpha\in \mathbb R$. Our results show that the set of $\gamma/d_\gamma$-thick points on the ball boundary has full Euclidean dimension and the set of $\gamma$-thick points on the ball boundary has full $\gamma$-LQG dimension.