Random Walk on a Co-Compact Fuchsian Group
  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. 
S\'ebastien Gou\"ezel and Steven P. Lalley
math.PR math.GR