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The dimension of the boundary of a Liouville quantum gravity metric ball
Ewain Gwynne · 2019-09-19 · via math.PR updates on arXiv.org

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

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