Suppose that is a Schramm–Loewner evolution () in a smoothly bounded simply connected domain and that is a conformal map from to a connected component of for some . The multifractal spectrum of is the function which, for each , gives the Hausdorff dimension of the set of points such that as . We rigorously compute the almost sure multifractal spectrum of , 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 , we obtain the optimal Hölder exponent for a conformal map which uniformizes the complement of an curve, and we obtain a new derivation of the almost sure Hausdorff dimension of the curve for . Our results also hold for the processes with general vectors of weight .
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