Almost sure multifractal spectrum of SLE

Understand

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.