KPZ formulas for the Liouville quantum gravity metric.

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. [ABSTRACT FROM AUTHOR]