  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
H\"older continuous. Finally, it is proved that this implies a local limit
theorem for the transition probabilities.
