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

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.

63 pages, 5 figures; final version, to appear in CMP