Almost sure multifractal spectrum of Schramm–Loewner evolution

Ewain Gwynne University of Chicago Jason Miller University of Cambridge Xin Sun University of Pennsylvania

Probability mathscidoc:2104.28002

Duke Math. J., 167, (6), 2018.4
Suppose that $\eta$ is a Schramm-Loewner evolution ($\SLE_\kappa$) in a smoothly bounded simply connected domain $D \subset \C$ and that $\phi$ is a conformal map from $\D$ to a connected component of $D \setminus \eta([0,t])$ for some $t>0$. The multifractal spectrum of $\eta$ is the function $(-1,1) \to [0,\infty)$ which, for each $s \in (-1,1)$, gives the Hausdorff dimension of the set of points $x \in \partial \D$ such that $|\phi'( (1-\epsilon) x)| = \epsilon^{-s+o(1)}$ as $\epsilon \to 0$. 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 $\SLE$, we obtain the optimal H\"older exponent for a conformal map which uniformizes the complement of an $\SLE$ curve, and we obtain a new derivation of the a.s.\ Hausdorff dimension of the $\SLE$ curve for $\kappa \leq 4$. Our results also hold for the $\SLE_\kappa(\ul\rho)$ processes with general vectors of weight $\ul\rho$.
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  title={Almost sure multifractal spectrum of Schramm–Loewner evolution},
  author={Ewain Gwynne, Jason Miller, and Xin Sun},
  booktitle={Duke Math. J.},
Ewain Gwynne, Jason Miller, and Xin Sun. Almost sure multifractal spectrum of Schramm–Loewner evolution. 2018. Vol. 167. In Duke Math. J..
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