Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Brownian motion correlation in the peanosphere for $\kappa>8$

Ewain Gwynne, Nina Holden, Jason Miller, and Xin Sun

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Abstract

The peanosphere (or “mating of trees”) construction of Duplantier, Miller, and Sheffield encodes certain types of $\gamma$-Liouville quantum gravity (LQG) surfaces ($\gamma\in(0,2)$) decorated with an independent $\operatorname{SLE}_{\kappa}$ ($\kappa=16/\gamma^{2}>4$) in terms of a correlated two-dimensional Brownian motion and provides a framework for showing that random planar maps decorated with statistical physics models converge to LQG decorated with an $\operatorname{SLE}$. Previously, the correlation for the Brownian motion was only explicitly identified as $-\cos(4\pi/\kappa)$ for $\kappa\in(4,8]$ and unknown for $\kappa>8$. The main result of this work is that this formula holds for all $\kappa>4$. This supplies the missing ingredient for proving convergence results of the aforementioned type for $\kappa>8$. Our proof is based on the calculation of a certain tail exponent for SLE$_{\kappa}$ on a quantum wedge and then matching it with an exponent which is well-known for Brownian motion.

Résumé

La sphère de Peano (ou <<Accouplement d’arbres>>) construite par Duplantier, Miller, et Sheffield encode certains types de surfaces de $\gamma$-gravité quantique de Liouville (LQG) décorées par un $\operatorname{SLE}_{\kappa}$ (pour $\gamma\in(0,2)$ et $\kappa=16/\gamma^{2}>4$), en termes d’un mouvement Brownien $2$-dimensionnel corrélé et fournit un cadre pour montrer que les cartes planaires décorées par un modèle de physique statistique convergent vers un LQG décoré par un $\operatorname{SLE}$. Précédemment, la corrélation du mouvement Brownien était seulement explicitement identifiée à $-\cos(4\pi/\kappa)$ pour $\kappa\in(4,8]$, mais inconnue pour $\kappa>8$. Le résultat principal de ce travail est que cette formule reste vraie pour $\kappa>8$. Cela donne l’ingrédient manquant pour prouver les résultats de convergence mentionnés précédemment pour $\kappa>8$. Notre preuve est basée sur le calcul d’un exposant de queue pour le $\operatorname{SLE}_{\kappa}$ sur un coin quantique et sur son identification avec un exposant bien connu pour le mouvement Brownien.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 4 (2017), 1866-1889.

Dates
Received: 19 January 2016
Revised: 17 May 2016
Accepted: 9 June 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1511773729

Digital Object Identifier
doi:10.1214/16-AIHP774

Mathematical Reviews number (MathSciNet)
MR3729638

Zentralblatt MATH identifier
06847065

Subjects
Primary: 60J67: Stochastic (Schramm-)Loewner evolution (SLE) 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Schramm–Loewner evolution Liouville quantum gravity Peanosphere

Citation

Gwynne, Ewain; Holden, Nina; Miller, Jason; Sun, Xin. Brownian motion correlation in the peanosphere for $\kappa&gt;8$. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 1866--1889. doi:10.1214/16-AIHP774. https://projecteuclid.org/euclid.aihp/1511773729


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