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 $\mathrm{SLE}_{\kappa }$, $\kappa =16/\gamma^{2}\in (4,\infty)$, $\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 $\mathrm{SLE}$, $\mathrm{CLE}$ 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 $\mathrm{SLE}_{\kappa}$ curve for $\kappa \neq4$; the double points and cut points of $\mathrm{SLE}_{\kappa }$ for $\kappa >4$; and the intersection of two flow lines of a Gaussian free field. We obtain the Hausdorff dimension of the set of $m$-tuple points of space-filling $\mathrm{SLE}_{\kappa }$ 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.