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Annales scientifiques de l'ENS - Titles - série 4, 46 (2013)

Titles < série 4, 46

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 46, fascicule 1 (2013)

Sébastien Gouëzel, Steven P. Lalley
Random Walks on Co-Compact Fuchsian Groups
Annales scientifiques de l'ENS 46, fascicule 1 (2013), 129-173

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Résumé :
Marches aléatoires sur des groupes fuchsiens co-compacts
Considérons une marche aléatoire symétrique à support fini sur un groupe fuchsien co-compact. Nous montrons que la fonction de Green à son rayon de convergence R décroît exponentiellement vite en fonction de la distance à l'origine. Nous montrons également que les inégalités d'Ancona s'étendent jusqu'au paramètre R, et par conséquent que la frontière de Martin pour les R-potentiels s'identifie avec la frontière géométrique S^1. De plus, le noyau de Martin correspondant est höldérien. Ces résultats sont utilisés pour démontrer un théorème limite local pour les probabilités de transition : dans le cas apériodique, p^n(x,y)C_x,yR^-nn^-3/2.

Mots-clefs : Groupe hyperbolique, groupe de surface, marche aléatoire, fonction de Green, frontière de Gromov, frontière de Martin, opérateur de Ruelle, états de Gibbs, théorème limite local.

Abstract:
It is proved that the Green's function of a symmetric finite range random walk on a co-compact Fuchsian group decays exponentially in distance at the radius of convergence R. It is also shown that Ancona's inequalities extend to R, and therefore that the Martin boundary for R-potentials coincides with the natural geometric boundary S^1, and that the Martin kernel is uniformly Hölder continuous. Finally, this implies a local limit theorem for the transition probabilities: in the aperiodic case, p^n(x,y)C_x,yR^-nn^-3/2.

Keywords: Hyperbolic group, surface group, random walk, Green's function, Gromov boundary, Martin boundary, Ruelle operator theorem, Gibbs state, local limit theorem.

Class. math. : 31C20; 31C25 60J50 60B99


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

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