Random Walk on a Co-Compact Fuchsian Group

TL;DR

This paper proves exponential decay of Green's function, extends Ancona's inequalities, and establishes a local limit theorem for symmetric finite range random walks on co-compact Fuchsian groups, linking probabilistic and geometric boundaries.

Contribution

It demonstrates the extension of Ancona's inequalities to the radius of convergence and characterizes the Martin boundary for these groups.

Findings

Green's function decays exponentially at the radius R

Martin boundary coincides with the geometric boundary S^1

Local limit theorem for transition probabilities

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