Random walks on co-compact fuchsian groups
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 46 (2013) no. 1, p. 131-175

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,y R -n n -3/2 .

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,y R -n n -3/2 .

DOI : https://doi.org/10.24033/asens.2186
Classification:  31C20,  31C25,  60J50,  60B99
Keywords: hyperbolic group, surface group, random walk, Green's function, Gromov boundary, Martin boundary, Ruelle operator theorem, Gibbs state, local limit theorem
@article{ASENS_2013_4_46_1_131_0,
     author = {Gou\"ezel, S\'ebastien and Lalley, Steven P.},
     title = {Random walks on co-compact fuchsian groups},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 46},
     number = {1},
     year = {2013},
     pages = {131-175},
     doi = {10.24033/asens.2186},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2013_4_46_1_131_0}
}
Gouëzel, Sébastien; Lalley, Steven P. Random walks on co-compact fuchsian groups. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 46 (2013) no. 1, pp. 131-175. doi : 10.24033/asens.2186. http://www.numdam.org/item/ASENS_2013_4_46_1_131_0/

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