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

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

Ewain Gwynne, Nina Holden, Jason Miller

Mathematics

Research output: Contribution to journal › Article › peer-review

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 SLE _κ 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 SLE _κ 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).

Original language	English (US)
Pages (from-to)	649-667
Number of pages	19
Journal	Probability Theory and Related Fields
Volume	176
Issue number	1-2
DOIs	https://doi.org/10.1007/s00440-019-00952-y
State	Published - Feb 1 2020

Keywords

Conformal map
Hausdorff dimension
KPZ formula
Liouville quantum gravity
Peanosphere
Schramm-Loewner evolution

ASJC Scopus subject areas

Analysis
Statistics and Probability
Statistics, Probability and Uncertainty

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10.1007/s00440-019-00952-y

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title = "Dimension transformation formula for conformal maps into the complement of an SLE curve",

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 SLE κ 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 SLE κ 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 = "Conformal map, Hausdorff dimension, KPZ formula, Liouville quantum gravity, Peanosphere, Schramm-Loewner evolution",

author = "Ewain Gwynne and Nina Holden and Jason Miller",

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doi = "10.1007/s00440-019-00952-y",

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T1 - Dimension transformation formula for conformal maps into the complement of an SLE curve

AU - Gwynne, Ewain

AU - Holden, Nina

AU - Miller, Jason

PY - 2020/2/1

Y1 - 2020/2/1

N2 - 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 SLE κ 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 SLE κ 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).

AB - 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 SLE κ 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 SLE κ 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).

KW - Conformal map

KW - Hausdorff dimension

KW - KPZ formula

KW - Liouville quantum gravity

KW - Peanosphere

KW - Schramm-Loewner evolution

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U2 - 10.1007/s00440-019-00952-y

DO - 10.1007/s00440-019-00952-y

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JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 1-2

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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