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A KPZ formula for SLE and Brownian motion

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We prove a KPZ -type formula which relates the a.s. Hausdorff dimension of any set associated with the Schramm-Loewner evolution (SLE) or the conformal loop emsemble (CLE) to the a.s. Hausdorff dimension of a time set for a two-dimensional correlated Brownian motion. As applications of our formula we get new short proofs of the Hausdorff dimension of a number of special sets for SLE . Based on a joint work with Ewain Gwynne and Jason Miller.

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