Translator Disclaimer
March 2020 An almost sure KPZ relation for SLE and Brownian motion
Ewain Gwynne, Nina Holden, Jason Miller
Ann. Probab. 48(2): 527-573 (March 2020). DOI: 10.1214/19-AOP1385

Abstract

The peanosphere construction of Duplantier, Miller and Sheffield provides a means of representing a $\gamma $-Liouville quantum gravity (LQG) surface, $\gamma \in (0,2)$, decorated with a space-filling form of Schramm’s $\mathrm{SLE}_{\kappa }$, $\kappa =16/\gamma^{2}\in (4,\infty)$, $\eta $ 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 $\eta $, which can be defined as a function of $\eta $ (modulo time parameterization) to the Hausdorff dimension of the corresponding time set $\eta^{-1}(A)$. This result serves to reduce the problem of computing the Hausdorff dimension of any set associated with an $\mathrm{SLE}$, $\mathrm{CLE}$ 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 $\mathrm{SLE}_{\kappa}$ curve for $\kappa \neq4$; the double points and cut points of $\mathrm{SLE}_{\kappa }$ for $\kappa >4$; 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 $\mathrm{SLE}_{\kappa }$ for $\kappa >4$ and $m\geq 3$ by computing the Hausdorff dimension of the so-called $(m-2)$-tuple $\pi /2$-cone times of a correlated planar Brownian motion.

Citation

Download Citation

Ewain Gwynne. Nina Holden. Jason Miller. "An almost sure KPZ relation for SLE and Brownian motion." Ann. Probab. 48 (2) 527 - 573, March 2020. https://doi.org/10.1214/19-AOP1385

Information

Received: 1 February 2016; Revised: 1 November 2017; Published: March 2020
First available in Project Euclid: 22 April 2020

zbMATH: 07199854
MathSciNet: MR4089487
Digital Object Identifier: 10.1214/19-AOP1385

Subjects:
Primary: 60G60, 60J67

Rights: Copyright © 2020 Institute of Mathematical Statistics

JOURNAL ARTICLE
47 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.48 • No. 2 • March 2020
Back to Top