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

Abstract

Abstract We prove a formula relating the Hausdorff dimension of a deterministic Borel subset of $${\mathbb {R}}$$R and the Hausdorff dimension of its image under a conformal map from the upper half-plane to a complementary connected component of an $$\hbox {SLE}_\kappa $$SLEκ curve for $$\kappa \not =4$$κ≠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 $$\hbox {SLE}_\kappa $$SLEκ curve for $$\kappa \in (0,4)\cup (4,8)$$κ∈(0,4)∪(4,8) and the dimension of the same set with respect to the $$\gamma $$γ-quantum natural parameterization of the curve induced by an independent Gaussian free field, $$\gamma = \sqrt{\kappa }\wedge (4/\sqrt{\kappa })$$γ=κ∧(4/κ).