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.
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