How universal are asymptotics of disconnection times in discrete cylinders?

Abstract

We investigate the disconnection time of a simple random walk in a discrete cylinder with a large finite connected base. In a recent article of A. Dembo and the author it was found that for large N the disconnection time of G_N×ℤ has rough order |G_N|², when G_N=(ℤ/Nℤ)^d. In agreement with a conjecture by I. Benjamini, we show here that this behavior has broad generality when the bases of the discrete cylinders are large connected graphs of uniformly bounded degree.