Almost sure multifractal spectrum of SLE

TL;DR

This paper rigorously computes the almost sure multifractal spectrum of SLE curves, confirming predictions and conjectures, and provides new insights into the Hausdorff dimension and integral means spectrum of SLE for various parameters.

Contribution

It offers the first rigorous computation of the almost sure multifractal spectrum of SLE, validating prior predictions and extending results to SLE with general weight vectors.

Findings

Confirmed the multifractal spectrum of SLE matches Duplantier's prediction.

Validated Beliaev and Smirnov's conjecture on the bulk integral means spectrum.

Derived the Hausdorff dimension of SLE curves for   4.

Abstract

Suppose that $\eta$ is a Schramm-Loewner evolution (SLE$_\kappa$) in a smoothly bounded simply connected domain $D \subset {\mathbb C}$ and that $\phi$ 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 $|\phi'( (1-\epsilon) x)| = \epsilon^{-s+o(1)}$ 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 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$_\kappa(\underline \rho)$ processes with general vectors of weight $\underline\rho$.