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

Abstract

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 $\tilde{\Gamma }$ which is Ahlfors ${\alpha }_0$-regular and has escape time exponent ${\beta }_0$. In addition we can make $\tilde{\Gamma }$ sufficiently uniform so that it satisfies an elliptic Harnack inequality.