An almost sure KPZ relation for SLE and Brownian motion

The peanosphere construction of Duplantier, Miller and Sheffield provides a means of representing a .-Liouville quantum gravity (LQG) surface, ..(0,2), decorated with a space-filling form of Schramm.s SLE., .=16/.2.(4,.), . 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 ., which can be defined as a function of . (modulo time parameterization) to the Hausdorff dimension of the corresponding time set ..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. curve for ..4; the double points and cut points of SLE. for .>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 SLE. for .>4 and m.3 by computing the Hausdorff dimension of the so-called (m.2)-tuple ./2-cone times of a correlated planar Brownian motion.