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.