Random walks on co-compact fuchsian groups

[Marches aléatoires sur des groupes fuchsiens co-compacts]

Gouëzel, Sébastien ; Lalley, Steven P.

Annales scientifiques de l'École Normale Supérieure, Tome 46 (2013), p. 131-175 / Harvested from Numdam

Résumé

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, $p^n(x,y)\sim C_{x,y}R^{-n}n^{-3/2}$.

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, $p^n(x,y)\sim C_{x,y}R^{-n}n^{-3/2}$.

Détail

Publié le : 2013-01-01
DOI : https://doi.org/10.24033/asens.2186
Classification:  31C20,  31C25,  60J50,  60B99
Mots clés: 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

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