## 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 | |

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