## Almost sure multifractal spectrum of SLE

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##### Authors

Gwynne, E

Miller, Jason

Sun, X

##### Journal Title

Duke Mathematical Journal

##### Publisher

Duke University Press

##### Type

Article

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AM

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Gwynne, E., Miller, J., & Sun, X. (2018). Almost sure multifractal spectrum of SLE. Duke Mathematical Journal https://doi.org/10.17863/CAM.17783

##### 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 $\underline\rho$.

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This record's DOI: https://doi.org/10.17863/CAM.17783

This record's URL: https://www.repository.cam.ac.uk/handle/1810/270852

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