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 $\widetilde{\Gamma}$ 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.