TITLE: Divergence of Green function of random walks at spectral radius
QUESTION [5 upvotes]: Let $P=(p(x,y))_{x, y\in N}$ be the transition matrix over countable states $N$.
Consider the generating Green function $G(x, y|t)=\sum_{0}^{\infty} p^n(x, y) t^n$, where $p^n(x,y)$ is the $(x,y)$-entry of the matrix $P^n$.
The spectral radius is given by $\rho(P)=\limsup_{n\to\infty} p^n(x,y)^{1/n}$. It is clear that $G(x,y|t)$ has convergence radius $1/\rho(P)$.
I'm now interested in when $G(x,y|t)$ diverges at $t=1/\rho(P)$. In particular, I'm considering that $P$ induces a random walk on a non-amenable group $N$. Does there any results related to this divergence in some interesting class of groups?
I'm newbie to random walk on graphs. Any comments would be extremely welcome!

REPLY [5 votes]: Concerning random walk on non-amenable groups, the Green function converges at the spectral radius: this is a result of Guivarc'h, quoted
in Wolgang Woess's book (Random walks on infinite graphs and groups), chapter IIB (in particular Theorem 7.8). 
For a bit more explicit statement, one can laso look at the introduction of this paper of Gouezel and Lalley:
https://arxiv.org/pdf/1107.5591.pdf