Let be an infinite weighted graph which is Ahlfors -regular, so that there exists a constant such that , where is the volume of the ball centre and radius . Define the escape time to be the mean exit time of a simple random walk on starting at from the ball centre and radius . We say has escape time exponent if there exists a constant such that for . Well known estimates for random walks on graphs imply that and . We show that these are the only constraints, by constructing for each , satisfying the inequalities above a graph which is Ahlfors -regular and has escape time exponent . In addition we can make sufficiently uniform so that it satisfies an elliptic Harnack inequality.
Cite this article
Martin T. Barlow, Which values of the volume growth and escape time exponent are possible for a graph?. Rev. Mat. Iberoam. 20 (2004), no. 1, pp. 1–31DOI 10.4171/RMI/378