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 R and the Hausdorff dimension of its image under a conformal map from the upper half-plane to a complementary connected component of an SLEκ curve for 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 SLEκ curve for κ∈(0, 4)∪(4, 8) and the dimension of the same set with respect to the γ-quantum natural parameterization of the curve induced by an independent Gaussian free field, γ=κ∧(4/κ).