Abstract
We prove a formula relating the Hausdorff dimension of a deterministic Borel subset of {mathbb {R}} and the Hausdorff dimension of its image under a conformal map from the upper half-plane to a complementary connected component of an hbox {SLE}_kappa curve for kappa not =4. Our proof is based on the relationship between SLE and Liouville quantum gravity together with the one-dimensional KPZ formula of Rhodes and Vargas (ESAIM Probab Stat 15:358–371, 2011) and the KPZ formula of Gwynne et al. (Ann Probab, 2015). As an intermediate step we prove a KPZ formula which relates the Euclidean dimension of a subset of an hbox {SLE}_kappa curve for kappa in (0,4)cup (4,8) and the dimension of the same set with respect to the gamma -quantum natural parameterization of the curve induced by an independent Gaussian free field, gamma = sqrt{kappa }wedge (4/sqrt{kappa }).
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