Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

An $\mathrm{SLE}_{2}$ loop measure

Stéphane Benoist and Julien Dubédat

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There is an essentially unique way to associate to any Riemann surface a measure on its simple loops, such that the collection of measures satisfy a strong conformal invariance property (see (J. Amer. Math. Soc. 21 (2008) 137–169)). These random loops are constructed as the boundary of Brownian loops, and so correspond in the zoo of statistical mechanics models to central charge $0$, or Schramm–Loewner Evolution (SLE) parameter $\kappa=8/3$. The goal of this paper is to construct a family of measures on simple loops on Riemann surfaces that satisfies a conformal covariance property, and that would correspond to SLE parameter $\kappa=2$ (central charge $-2$). On planar annuli, this loop measure was already built by Adrien Kassel and Rick Kenyon in (Random curves on surfaces induced from the Laplacian determinant (2012) ArXiv e-prints). We will give an alternative construction of this loop measure on planar annuli, investigate its conformal covariance, and finally extend this measure to general Riemann surfaces. This gives an example of a Malliavin–Kontsevich–Suhov loop measure (Tr. Mat. Inst. Steklova 258 (2007) 107–153) in non-zero central charge.


Il y a une manière essentiellement unique d’associer à toute surface de Riemann une mesure sur ses boucles simples de telle sorte que la collection de mesures soit invariante conforme en un sens fort (voir (J. Amer. Math. Soc. 21 (2008) 137–169)). Ces boucles aléatoires peuvent être construites comme frontières de boucles browniennes, et correpondent donc, dans la classification des modèles de mécanique statistique à la valeur de charge centrale $0$, ou alternativement au paramètre d’évolution de Schramm–Loewner (SLE) $\kappa=8/3$. Dans cet article, nous construisons une famille de mesures sur les boucles simples dans les surfaces de Riemann qui est covariante conforme, et qui correpond à la valeur $2$ du paramètre SLE (ou de maniére équivalente, à la charge centrale $-2$). Cette mesure de boucles avait été précédemment construite dans les anneaux planaires par Adrien Kassel et Rick Kenyon (Random curves on surfaces induced from the Laplacian determinant (2012) ArXiv e-prints). Nous donnerons une construction alternative de cette mesure de boucles dans les anneaux planaires et étudierons ses propriétés de covariance conforme, avant d’étendre cette mesure à toute surface de Riemann. En particulier, cela donne un exemple de mesure de Malliavin–Kontsevich–Suhov (Tr. Mat. Inst. Steklova 258 (2007) 107–153) en charge centrale non nulle.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1406-1436.

Received: 22 July 2014
Revised: 17 December 2014
Accepted: 8 April 2015
First available in Project Euclid: 28 July 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J67: Stochastic (Schramm-)Loewner evolution (SLE) 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

SLE UST Loop Restriction


Benoist, Stéphane; Dubédat, Julien. An $\mathrm{SLE}_{2}$ loop measure. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1406--1436. doi:10.1214/15-AIHP681.

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