Almost sure multifractal spectrum of SLE

Abstract

Suppose that $\eta$ is a Schramm-Loewner evolution (SLE${}_{\kappa }$) in a smoothly bounded simply connected domain $D\subset \mathbb{C}$ and that $\varphi$ is a conformal map from $\mathbb{D}$ 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 $x\in \partial \mathbb{D}$ such that $|{\varphi }^{\prime }((1-\epsilon )x)|={\epsilon }^{-s+o(1)}$ 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 and we obtain a new derivation of the a.s. Hausdorff dimension of the SLE curve for $\kappa \le 4$. Our results also hold for the SLE${}_{\kappa }(\underline{\rho })$ processes with general vectors of weight $\underline{\rho }$.

92 pages and 18 figures

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• Regularity of the SLE${}_4$ uniformizing map and the SLE${}_8$ trace