Probability Theory/Geometry
SLEs as boundaries of clusters of Brownian loops
Comptes Rendus. Mathématique, Volume 337 (2003) no. 7, pp. 481-486.

In this research announcement, we show that SLE curves can in fact be viewed as boundaries of certain clusters of Brownian loops (of the clusters in a Brownian loop soup). For small densities c of loops, we show that the outer boundaries of the clusters created by the Brownian loop soup are SLEκ-type curves where κ∈(8/3,4] and c related by the usual relation c=(3κ−8)(6−κ)/2κ (i.e., c corresponds to the central charge of the model). This gives (for any Riemann surface) a simple construction of a natural countable family of random disjoint SLEκ loops, that behaves “nicely” under perturbation of the surface and is related to various aspects of conformal field theory and representation theory.

Nous étudions certaines propriétés de connectivité de la « soupe » de lacets browniens dans un domaine. On montre l'exitence d'une transition de phase : lorsque l'intensité c est petite, il y a un ensemble dénombrable d'amas disjoints alors que lorsque c est grand, il n'y a presque sûrement qu'un seul amas. Nous montrons que pour les petites valeurs de c, les frontières de ces amas sont des courbes simples de type SLE (Evolution de Loewner–Schramm) de paramètre κ∈(8/3,4] avec c=(6−κ)(3κ−8)/2κ. Ceci permet de construire une famille aléatoire de boucles de SLE disjointes sur toute surface de Riemann et est étroitement relié à la théorie conforme des champs.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2003.08.003
Werner, Wendelin 1

1 Université Paris-Sud and IUF, laboratoire de mathématiques, 91405 Orsay cedex, France
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Werner, Wendelin. SLEs as boundaries of clusters of Brownian loops. Comptes Rendus. Mathématique, Volume 337 (2003) no. 7, pp. 481-486. doi : 10.1016/j.crma.2003.08.003. http://www.numdam.org/articles/10.1016/j.crma.2003.08.003/

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