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

### Martin T. Barlow

The University of British Columbia, Vancouver, Canada

## Abstract

Let $Γ=(G,E)$ be an infinite weighted graph which is Ahlfors $α$-regular, so that there exists a constant $c$ such that $c_{−1}r_{α}≤V(x,r)≤cr_{α}$, 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 $Γ$ starting at $x$ from the ball centre $x$ and radius $r$. We say $Γ$ has escape time exponent $β>0$ if there exists a constant $c$ such that $c_{−1}r_{β}≤T(x,r)≤cr_{β}$ for $r≥1$. Well known estimates for random walks on graphs imply that $α≥1$ and $2≤β≤1+α$. We show that these are the only constraints, by constructing for each $α_{0}$, $β_{0}$ satisfying the inequalities above a graph $Γ$ which is Ahlfors $α_{0}$-regular and has escape time exponent $β_{0}$. In addition we can make $Γ$ 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. 1–31

DOI 10.4171/RMI/378