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Almost sure multifractal spectrum of Schramm-Loewner evolution

Accepted version
Peer-reviewed

Type

Article

Change log

Authors

Gwynne, E 
Miller, J 
Sun, X 

Abstract

Suppose that η is a Schramm-Loewner evolution (SLEκ) in a smoothly bounded simply connected domain DC and that ϕ is a conformal map from D to a connected component of Dη([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 xD such that |ϕ′((1−ϵ)x)|=ϵs+o(1) as ϵ→0. We rigorously compute the a.s. multifractal spectrum of SLE, confirming a prediction due to Duplantier. As corollaries, we confirm a conjecture made by Beliaev and Smirnov for the a.s. bulk integral means spectrum of SLE and we obtain a new derivation of the a.s. Hausdorff dimension of the SLE curve for κ≤4. Our results also hold for the SLEκ(ρ―) processes with general vectors of weight ρ.

Description

Keywords

4901 Applied Mathematics, 4902 Mathematical Physics, 49 Mathematical Sciences

Journal Title

Duke Mathematical Journal

Conference Name

Journal ISSN

0012-7094
1547-7398

Volume Title

167

Publisher

Duke University Press