Random Walk on a Surface Group: Boundary Behavior of the Green's Function at the Spectral Radius

It is proved that the Green’s function of the simple random walk on a surface group of large genus decays exponentially at the spectral radius. It is also shown that Ancona’s inequalities extend to the spectral radius R, and therefore that the Martin boundary for R-potentials coincides with the natural geometric boundary S^1. This implies that the Green’s function obeys a power law with exponent 1/2 at the spectral radius.