Random Walk on a Co-Compact Fuchsian Group: Behavior of the Green's Function at the Spectral Radius

Sébastien Gouëzel (1) , Steven P. Lalley (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 University of Chicago, Department of Statistics

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

Mots clés: hyperbolic group; surface group; random walk; Green's function; Gromov boundary; Martin boundary; Ruelle operator theorem; Gibbs state; local limit theorem

Domaines: Probabilités [math.PR]; Théorie des groupes [math.GR]

Sébastien Gouëzel, Steven P. Lalley. Random Walk on a Co-Compact Fuchsian Group: Behavior of the Green's Function at the Spectral Radius. 2011. ⟨hal-00617199v1⟩