Dimension transformation formula for conformal maps into the complement of an SLE curve
Authors
Gwynne, Ewain
Holden, Nina
Miller, Jason
Publication Date
2019-11-02Journal Title
Probability Theory and Related Fields
ISSN
0178-8051
Publisher
Springer Berlin Heidelberg
Volume
176
Issue
1-2
Pages
649-667
Language
en
Type
Article
This Version
VoR
Metadata
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Gwynne, E., Holden, N., & Miller, J. (2019). Dimension transformation formula for conformal maps into the complement of an SLE curve. Probability Theory and Related Fields, 176 (1-2), 649-667. https://doi.org/10.1007/s00440-019-00952-y
Description
Funder: University of Cambridge
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).
Keywords
Article, Schramm-Loewner evolution, Liouville quantum gravity, KPZ formula, Hausdorff dimension, Conformal map, Peanosphere, 60J67
Identifiers
s00440-019-00952-y, 952
External DOI: https://doi.org/10.1007/s00440-019-00952-y
This record's URL: https://www.repository.cam.ac.uk/handle/1810/312312
Rights
Licence:
https://creativecommons.org/licenses/by/4.0/