## Abstract

The peanosphere construction of Duplantier, Miller and Sheffield provides a means of representing a γ-Liouville quantum gravity (LQG) surface, γ∈ (0, 2), decorated with a space-filling form of Schramm’s SLEκ, κ= 16/γ^{2} ∈ (4,∞), η as a gluing of a pair of trees which are encoded by a correlated two-dimensional Brownian motion Z. We prove a KPZ-type formula which relates the Hausdorff dimension of any Borel subset A of the range of η, which can be defined as a function of η (modulo time parameterization) to the Hausdorff dimension of the corresponding time set η^{−1}(A). This result serves to reduce the problem of computing the Hausdorff dimension of any set associated with an SLE, CLE or related processes in the interior of a domain to the problem of computing the Hausdorff dimension of a certain set associated with a Brownian motion. For many natural examples, the associated Brownian motion set is well known. As corollaries, we obtain new proofs of the Hausdorff dimensions of the SLEκ curve for κ ≠ 4; the double points and cut points of SLEκ for κ > 4; and the intersection of two flow lines of a Gaussian free field. We obtain the Hausdorff dimension of the set of m-tuple points of space-filling SLEκ for κ > 4 and m≥ 3 by computing the Hausdorff dimension of the so-called (m− 2)-tuple π/ 2-cone times of a correlated planar Brownian motion.

Original language | English (US) |
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Pages (from-to) | 527-573 |

Number of pages | 47 |

Journal | Annals of Probability |

Volume | 48 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2020 |

## Keywords

- Hausdorff dimension
- Kpz formula
- Liouville quantum gravity
- mating of trees
- peanosphere
- planar Brownian motion
- Schramm–Loewner evolution

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty