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

TL;DR

This paper proves exponential decay of the Green's function at the spectral radius for surface groups, extending Ancona's inequalities and characterizing the Martin boundary, revealing a power law decay with exponent 1/2.

Contribution

It establishes the boundary behavior of the Green's function at the spectral radius for large genus surface groups, extending key inequalities and identifying the Martin boundary.

Findings

Green's function decays exponentially at the spectral radius

Ancona's inequalities extend to the spectral radius

Martin boundary coincides with the geometric boundary S^1

Abstract

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.