Stochastic Loewner evolution in multiply connected domains

Robert O. Bauer; Roland M. Friedrich

doi:10.1016/j.crma.2004.08.010

Probability Theory

Stochastic Loewner evolution in multiply connected domains

Presented by: Paul Malliavin

Robert O. Bauer ¹; Roland M. Friedrich ²

¹ Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA
² Institute for Advanced Study, Princeton, NJ 08540, USA

Comptes Rendus. Mathématique, Volume 339 (2004) no. 8, pp. 579-584.

Abstract
Résumé

We construct radial stochastic Loewner evolution in multiply connected domains, choosing the unit disk with concentric circular slits as a family of standard domains. The natural driving function or input is a diffusion on the associated moduli space. The diffusion stops when it reaches the boundary of the moduli space. We show that for this driving function the family of random growing compacts has a phase transition for $κ = 4$ and $κ = 8$ , and that it satisfies locality for $κ = 6$ .

Nous construisons l'évolution stochastique radiale de Loewner dans des domaines multiple connexes, en choisissant le disque d'unité avec des segments concentriqueés, comme famille de référence. La fonction naturelle qui fait croître les traces de l'évolution est une diffusion sur l'espace associé des modules. La diffusion s'arrête dès qu'il touche le bord de l'espace des modules. Nous démontrons que pour cette fonction qui engendre la croissance de ces compacts aléatoires, on trouve une transition de phase pour $κ = 4$ et $κ = 8$ , et qu'ils satisfont la propriété de localité pour $κ = 6$ .

Received: 2004-08-12
Accepted: 2004-08-28
Published online: 2004-10-15

DOI: 10.1016/j.crma.2004.08.010

Author's affiliations:

Robert O. Bauer ¹; Roland M. Friedrich ²

¹ Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA
² Institute for Advanced Study, Princeton, NJ 08540, USA

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     author = {Robert O. Bauer and Roland M. Friedrich},
     title = {Stochastic {Loewner} evolution in multiply connected domains},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {579--584},
     publisher = {Elsevier},
     volume = {339},
     number = {8},
     year = {2004},
     doi = {10.1016/j.crma.2004.08.010},
     language = {en},
}

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Robert O. Bauer; Roland M. Friedrich. Stochastic Loewner evolution in multiply connected domains. Comptes Rendus. Mathématique, Volume 339 (2004) no. 8, pp. 579-584. doi : 10.1016/j.crma.2004.08.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.08.010/

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