Almost sure multifractal spectrum of Schramm–Loewner evolution

Abstract

Suppose that η is a Schramm–Loewner evolution (SLEκ) in a smoothly bounded simply connected domain D⊂C and that ϕ is a conformal map from D to a connected component of D∖η([0,t]) for some t>0. The multifractal spectrum of η is the function (−1,1)→[0,∞) which, for each s∈(−1,1), gives the Hausdorff dimension of the set of points x∈∂D such that |ϕ'((1−ϵ)x)|=ϵ−s+o(1) as ϵ→0. We rigorously compute the almost sure multifractal spectrum of SLE, confirming a prediction due to Duplantier. As corollaries, we confirm a conjecture made by Beliaev and Smirnov for the almost sure bulk integral means spectrum of SLE, we obtain the optimal Hölder exponent for a conformal map which uniformizes the complement of an SLE curve, and we obtain a new derivation of the almost sure Hausdorff dimension of the SLE curve for κ≤4. Our results also hold for the SLEκ(ρ̲) processes with general vectors of weight ρ̲.