Special Analysis/Probability Seminar

The large-time behavior of the return probabilities of a random walk is controlled by the behavior of the Green's function $G_r (x,y)$ at the radius $r=R$ of convergence. For nearest neighbor random walks on virtually free groups it is known that the the Green's function is algebraic, and that the singularity at the radius of convergence is of square-root type. For other Fuchsian groups, however, the Green's function is likely not algebraic. Nevertheless, we show that for simple random walk on a surface group of large genus the singularity is still of square-root type. A number of interesting related results concerning the behavior of $G_R (x,y)$ as $y$ approaches the geometric boundary are obtained: (1) $G_R (x,y)$ decays exponentially in distance$(x,y)$. (2) Ancona's inequalities persist at $r=R$. (3) The Martin boundary for $R-$potentials coincides with the geometric boundary.