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Almost sure multifractal spectrum of SLE

Wednesday 28th January 2015 - 10:00 to 11:00
INI Seminar Room 1
Co-authors: Jason Miller (Massachusetts Institute of Technology), Xin Sun (Massachusetts Institute of Technology)

Suppose that $\eta$ is an SLE$_\kappa$ in a smoothly bounded simply connected domain $D \subset \mathbb C$ and that $\phi$ is a conformal map from the unit disk $\mathbb D$ to a connected component of $D \setminus \eta([0,t])$ for some $t>0$. The multifractal spectrum of $\eta$ is the function $(-1,1) \rightarrow [0,\infty)$ 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 $\epsilon \rightarrow 0$. I will present a rigorous computation of the a.s. multifractal spectrum of SLE (joint with J. Miller and X. Sun), which confirms a prediction due to Duplantier. The proof makes use of various couplings of SLE with the Gaussian free field. As a corollary, we also confirm a conjecture of Beliaev and Smirnov for the a.s. bulk integral means spectrum of SLE.

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons