2015/12/03 by Gwynne, Ewain, Holden, Nina, Miller, Jason · 3 citations
#Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.1512.01223
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\not=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 also 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.