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 γ-Liouville quantum gravity (LQG) surface, γ∈ (0, 2), decorated with a space-filling form of Schramm’s SLEκ, κ= 16/γ2 ∈ (4,∞), η 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 η, which can be defined as a function of η (modulo time parameterization) to the Hausdorff dimension of the corresponding time set η−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 SLEκ curve for κ ≠ 4; the double points and cut points of SLEκ for κ > 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 SLEκ for κ > 4 and m≥ 3 by computing the Hausdorff dimension of the so-called (m− 2)-tuple π/ 2-cone times of a correlated planar Brownian motion.