15 April 2018 Almost sure multifractal spectrum of Schramm–Loewner evolution
Ewain Gwynne, Jason Miller, Xin Sun
Duke Math. J. 167(6): 1099-1237 (15 April 2018). DOI: 10.1215/00127094-2017-0049


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'((1ϵ)x)|=ϵs+o(1) as ϵ0. 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 κ4. Our results also hold for the SLEκ(ρ̲) processes with general vectors of weight ρ̲.


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Ewain Gwynne. Jason Miller. Xin Sun. "Almost sure multifractal spectrum of Schramm–Loewner evolution." Duke Math. J. 167 (6) 1099 - 1237, 15 April 2018. https://doi.org/10.1215/00127094-2017-0049


Received: 8 March 2016; Revised: 6 October 2017; Published: 15 April 2018
First available in Project Euclid: 28 March 2018

zbMATH: 06870402
MathSciNet: MR3786302
Digital Object Identifier: 10.1215/00127094-2017-0049

Primary: 60J67
Secondary: 60G17

Keywords: Gaussian free field , multifractal spectrum , Schramm–Loewner evolution

Rights: Copyright © 2018 Duke University Press


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Vol.167 • No. 6 • 15 April 2018
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