Dimension transformation formula for conformal maps into the complement of an SLE curve

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 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-Vargas (2011) and the KPZ formula of Gwynne-Holden-Miller (2015). As an intermediate step we prove a KPZ formula which relates the Euclidean dimension of a subset of an 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, $\gamma = \sqrt \kappa \wedge (4/\sqrt\kappa)$.