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Almost sure multifractal spectrum of SLE

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Random Geometry

Co-authors: Jason Miller (Massachusetts Institute of Technology), Xin Sun (Massachusetts Institute of Technology)

Suppose that $ta$ is an SLE $_kappa$ in a smoothly bounded simply connected domain $D ubset mathbb C$ and that $phi$ is a conformal map from the unit disk $mathbb D$ to a connected component of $D etminus ta([0,t])$ for some $t>0$. The multifractal spectrum of $ta$ is the function $(-1,1) ightarrow [0,infty)$ which, for each $s in (1,1)$, gives the Hausdorff dimension of the set of points $x in partial mathbb D$ such that $|phi’( (1psilon) x)| = psilon^{-s+o(1)}$ as $psilon ightarrow 0$. 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 .

This talk is part of the Isaac Newton Institute Seminar Series series.

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