Random Walk on a Co-Compact Fuchsian Group

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, it is proved that this implies a local limit theorem for the transition probabilities.

arXiv admin note: text overlap with arXiv:0710.5745

References in corpus (2)

• Asymptotic entropy and green speed for random walks on countable groups
• The ratio set of the harmonic measure of a random walk on a hyperbolic group