An almost sure KPZ relation for SLE and Brownian motion

Abstract

The peanosphere construction of Duplantier, Miller, and Sheffield provides a means of representing a $\gamma$-Liouville quantum gravity (LQG) surface, $\gamma \in (0,2)$, decorated with a space-filling form of Schramm's SLE$_\kappa$, $\kappa = 16/\gamma^2 \in (4,\infty)$, $\eta$ as a gluing of a pair of trees which are encoded by a correlated two-dimensional Brownian motion $Z$. We prove a KPZ-type formula which relates the Hausdorff dimension of any Borel subset $A$ of the range of $\eta$ which can be defined as a function of $\eta$ (modulo time parameterization) to the Hausdorff dimension of the corresponding time set $\eta^{-1}(A)$. This result serves to reduce the problem of computing the Hausdorff dimension of any set associated with an SLE, CLE, or related processes in the interior of a domain to the problem of computing the Hausdorff dimension of a certain set associated with a Brownian motion. For many natural examples, the associated Brownian motion set is well-known. As corollaries, we obtain new proofs of the Hausdorff dimensions of the SLE$_\kappa$ curve for $\kappa \not=4$; the double points and cut points of SLE$_\kappa$ for $\kappa >4$; and the intersection of two flow lines of a Gaussian free field. We also obtain the Hausdorff dimension of the set of $m$-tuple points of space-filling SLE$_\kappa$ for $\kappa>4$ and $m \geq 3$ by computing the Hausdorff dimension of the so-called $(m-2)$-tuple $\pi/2$-cone times of a correlated planar Brownian motion.