Random Walks on Co-Compact Fuchsian Groups
Résumé
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, $p^n(x,y)\sim C_{x,y}R^{-n}n^{-3/2}$.
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