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!



Divergence of Green function of random walks at spectral radius
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>


