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 $2  \frac{\gamma}{d_\gamma}\left(\frac{2}{\gamma} + \frac{\gamma}{2} \right) + \frac{\gamma^2}{d_\gamma^2}$ (resp.\ $d_\gamma1$), 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2019
 arXiv:
 arXiv:1909.08588
 Bibcode:
 2019arXiv190908588G
 Keywords:

 Mathematics  Probability;
 Mathematical Physics
 EPrint:
 61 pages, 5 figures