Asymptotic entropy and Green speed for random walks on countable groups (Q2427060)

Let \(\Gamma\) be an infinite countable group, and let \((Z_n)\) be a transient random walk on \(\Gamma\). The Green metric on \(\Gamma\) is defined as \[ d(x,y)=-\ln \mathbb P^x[\tau_y<\infty ],\quad x,y\in \Gamma , \] where \(\tau_y\) is the hitting time of the element \(y\) by the random walk starting at \(x\). The rate of escape of the random walk computed in the Green metric (or the Green speed) is defined by the almost sure limit \[ l=\lim\limits_{n\to \infty}n^{-1}d(e,Z_n). \] The authors prove that the above rate of escape equals the asymptotic entropy (if the entropy is finite). The proof relies on integral representations of both quantities. In the case of finitely generated groups, where the result was proved earlier by \textit{I. Benjamini} and \textit{Y. Peres}, [Probab. Theory Relat. Fields 98, No. 1, 91--112 (1994; Zbl 0794.60068)], the authors give an alternative proof based on an inequality relating the rate of escape, the entropy, and the logarithmic volume growth.