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

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.