The growth of the Green function for random walks and Poincar{\'e} series

TL;DR

This paper studies the asymptotic behavior of the Green function in random walks on groups, constructing examples with diverse behaviors and exploring implications for renewal theory and Poincaré series.

Contribution

It introduces new examples of group behaviors for the Green function's growth, including relatively hyperbolic groups with convergent Poincaré series, expanding understanding of renewal phenomena.

Findings

Constructed examples of groups with various Green function behaviors

Identified classes of groups where Poincaré series converge

Analyzed Green function asymptotics for abelian, nilpotent, lamplighter, and product groups

Abstract

Given a probability measure $\mu$ on a finitely generated group $\Gamma$, the Green function $G(x,y|r)$ encodes many properties of the random walk associated with $\mu$. Finding asymptotics of $G(x,y|r)$ as $y$ goes to infinity is a common thread in probability theory and is usually referred as renewal theory in literature. Endowing $\Gamma$ with a word distance, we denote by $H_r(n)$ the sum of the Green function $G(e,x|r)$ along the sphere of radius $n$. This quantity appears naturally when studying asymptotic properties of branching random walks driven by $\mu$ on $\Gamma$ and the behavior of $H_r(n)$ as $n$ goes to infinity is intimately related to renewal theory. Our motivation in this paper is to construct various examples of particular behaviors for $H_r(n)$. First, our main result exhibits a class of relatively hyperbolic groups with convergent Poincar{\'e} series generated by $H_r(n)$, which answers some questions raised in a previous paper of the authors. Along the way, we investigate the behavior of $H_r(n)$ for several classes of finitely generated groups, including abelian groups, certain nilpotent groups, lamplighter groups, and Cartesian products of free groups.