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

TL;DR

This paper establishes a formula connecting the Hausdorff dimensions of sets under conformal maps related to SLE curves, utilizing KPZ relations and quantum gravity concepts.

Contribution

It introduces a dimension transformation formula for conformal maps into SLE curve complements, linking Hausdorff dimensions via KPZ and quantum gravity techniques.

Findings

Derived a formula relating Hausdorff dimensions under conformal maps and SLE curves.

Proved a KPZ relation connecting Euclidean and quantum dimensions of SLE subsets.

Linked SLE, Liouville quantum gravity, and KPZ formulas in a unified framework.

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