Almost sure multifractal spectrum of Schramm–Loewner evolution

Abstract

Suppose that $\eta$ is a Schramm–Loewner evolution (${SLE}_{\kappa }$) in a smoothly bounded simply connected domain $D\subset \mathbf{C}$ and that $\varphi$ is a conformal map from $\mathbf{D}$ to a connected component of $D\setminus \eta \left(\right[0,t\left]\right)$ 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 $x\in \partial \mathbf{D}$ such that $\left|\varphi \text{'}\right((1-ϵ)x\left)\right|={ϵ}^{-s+o\left(1\right)}$ as $ϵ\to 0$. 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 $\kappa \le 4$. Our results also hold for the ${SLE}_{\kappa }\left(\underline{\rho }\right)$ processes with general vectors of weight $\underline{\rho }$.