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

Let \Gamma = (G; E) be an infinite weighted graph which is Ahlfors ff-regular, so thatthere exists a constant c such that c, where V (x; r) is the volume of theball centre x and radius r. Define the escape time T (x; r) to be the mean exit time of a simplerandom walk on \Gamma starting at x from the ball centre x and radius r. We say \Gamma has escapetime exponent fi ? 0 if there exists a constant c such that c T (x; r) crfor r 1.Well known estimates for random walks on...