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 $\kappa \ne 4$. 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, $\gamma =\sqrt{\kappa }\wedge (4/\sqrt{\kappa })$.

arXiv comment15 pages, 1 figure