# Revista Matemática Iberoamericana

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**Volume 20, Issue 1, 2004, pp. 1–31**

**DOI: 10.4171/RMI/378**

Published online: 2004-04-30

Which values of the volume growth and escape time exponent are possible for a graph?

Martin T. Barlow^{[1]}(1) The University of British Columbia, Vancouver, Canada

Let $\Gamma=(G,E)$ be an infinite weighted graph which is Ahlfors $\alpha$-regular, so that there exists a constant $c$ such that $c^{-1} r^\alpha\le V(x,r)\le c r^\alpha$, where $V(x,r)$ is the volume of the ball centre $x$ and radius $r$. Define the escape time $T(x,r)$ to be the mean exit time of a simple random walk on $\Gamma$ starting at $x$ from the ball centre $x$ and radius $r$. We say $\Gamma$ has escape time exponent $\beta>0$ if there exists a constant $c$ such that $c^{-1} r^\beta \le T(x,r) \le c r^\beta$ for $r\ge 1$. Well known estimates for random walks on graphs imply that $\alpha\ge 1$ and $2 \le \beta \le 1+\alpha$. We show that these are the only constraints, by constructing for each $\alpha_0$, $\beta_0$ satisfying the inequalities above a graph $\widetilde{\Gamma}$ which is Ahlfors $\alpha_0$-regular and has escape time exponent $\beta_0$. In addition we can make $\widetilde{\Gamma}$ sufficiently uniform so that it satisfies an elliptic Harnack inequality.

*Keywords: *Graph, random walk, volume growth, anomalous diffusion

Barlow Martin: Which values of the volume growth and escape time exponent are possible for a graph?. *Rev. Mat. Iberoam.* 20 (2004), 1-31. doi: 10.4171/RMI/378