Divergence of Green function of random walks at spectral radius

Question

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!

Answer 1

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

Answer 2

I would advise you to first look on random walk on trees. I suspect non-amenable group has been treated a lot as well but I can't find a good reference.

For example the random walk on the 4 regular tree (which is the free group $F_2$) can be also as a random walk on $\mathbb{N}$ with drift (the probability to go away from 0 is $3/4$ and to go toward $0$ is $1/4$). At $t=1/\rho(P)$, the drift disappears and we should obtain $p^{2n}(0,0)t^{2n}\sim \frac{1}{2^n}\frac{1}{n+1}\begin{pmatrix}2n \\n\end{pmatrix}\sim n^{-3/2}$

My intuition is then the following. We note $S_n= \{ y:d(y,x_0)=n \}$ with $d(x_0,y)$ the graph distance from a root point $x_0$. Because for any non-amenable graph, there should exist $r>1$ such that $|S_n|\approx r^n$. If one focus then only on the distance between the walker $X_n$ and $x_0$, one should recover somehow the previous picture of a random walk on $\mathbb{N}$ with a drift away from zero. (but this is only an educated guess).