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

Ewain Gwynne,Nina Holden,Jason Miller

doi:10.1007/s00440-019-00952-y

Ewain Gwynne, Nina Holden + Show 1 more

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https://doi.org/10.1007/s00440-019-00952-y

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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 hbox {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 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 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 }).

Highlights

The Schramm–Loewner evolution (SLEκ ) is a family of conformally invariant random fractal curves in two dimensions, originally introduced in [46]
The first version of the KPZ formula relates the Euclidean dimension of a subset X of an SLEκ curve η for κ ∈ (0, 4) to the dimension of η−1(X ), when η is parameterized by γ -quantum length with respect to an independent GFF
In this subsection we briefly review some facts about SLE and Liouville quantum gravity (LQG) which will be needed for the proofs of our main results

Summary

Overview

The Schramm–Loewner evolution (SLEκ ) is a family of conformally invariant random fractal curves in two dimensions, originally introduced in [46]. The first version of the KPZ formula (stated as Theorem 2.1 below) relates the Euclidean dimension of a subset X of an SLEκ curve η for κ ∈ (0, 4) to the dimension of η−1(X ), when η is parameterized by γ -quantum length with respect to an independent GFF. This formula will be deduced from another.

Main results

Proofs

Open questions