KPZ formulas for the Liouville quantum gravity metric

Abstract

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 $X \subset \mathbb{C}$ which is independent from $h$, the Hausdorff dimensions of $X$ with respect to the Euclidean metric and with respect to the $\gamma$-LQG metric $D_h$ are a.s. related by the (geometric) KPZ formula. As a corollary, we deduce that the Hausdorff dimension of the continuum $\gamma$-LQG metric is equal to the exponent $d_\gamma > 2$ studied by Ding and Gwynne (2018), which describes distances in discrete approximations of $\gamma$-LQG such as random planar maps. We also derive "worst-case" bounds relating the Euclidean and $\gamma$-LQG dimensions of $X$ when $X$ and $h$ are not necessarily independent, which answers a question posed by Aru (2015). Using these bounds, we obtain an upper bound for the Euclidean Hausdorff dimension of a $\gamma$-LQG geodesic which equals $1.312\dots$ when $\gamma = \sqrt{8/3}$; and an upper bound of $1.9428\dots$ for the Euclidean Hausdorff dimension of a connected component of the boundary of a $\sqrt{8/3}$-LQG metric ball. We use the axiomatic definition of the $\gamma$-LQG metric, so the paper can be understood by readers with minimal background knowledge beyond a basic level of familiarity with the Gaussian free field.