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Let $G$ be a $d$-regular graph of exponential growth. By exponential growth I mean that $$ \liminf_{r \to \infty} | B(o, r)|^{1/r} >1. $$ Here $B(o,r)$ is the ball of radius $r$ centered at a given vertex $v$ of $G$.

My question is the following: is it true that the SRW on $G$ is transient? One can see that this holds in the special case that $G$ the Cayley graph of a group.

I am mostly interested in a positive answer, so even if the statement is false at this level of generality, I would be interested in special cases (more general than that of groups) in which such a statement holds. Any counter-example to the general case would also be appreciated.

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No. A counterexample can be obtained in the following way. Let us begin with an integer line $\mathbb Z$. Then take a sequence of finite graphs $\Gamma_n$, and for each $n\in\mathbb Z$ attach the graph $\Gamma_n$ to the point $n$. It can be done in such a way that the resulting graph $\Gamma$ be regular and have exponential growth. On the other hand, the simple random walk on $\Gamma$ is recurrent as it is just the simple random walk on $\mathbb Z$ with added excursions to the graphs $\Gamma_n$.

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  • $\begingroup$ Thank you veru much for the counter-example. Are there any positive results going in this direction? $\endgroup$
    – Keivan Karai
    Commented Apr 25 at 9:40

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