On the disconnection of a discrete cylinder by a biased random walk

TL;DR

This paper studies how a biased random walk on a high-dimensional discrete cylinder affects the time it takes to disconnect the structure, revealing different asymptotic behaviors depending on the bias strength.

Contribution

It establishes the asymptotic behavior of the disconnection time for biased random walks on discrete cylinders, extending previous results to include the effect of drift.

Findings

For drift exponent α > 1, disconnection time behaves like N^{2d+o(1)}.

For α < 1, disconnection time grows exponentially with N.

The behavior transitions at the critical drift exponent α=1.

Abstract

We consider a random walk on the discrete cylinder $({\mathbb{Z}}/N{\mathbb{Z}})^d\times{\mathbb{Z}}$, $d\geq3$ with drift $N^{-d\alpha}$ in the $\mathbb{Z}$-direction and investigate the large $N$-behavior of the disconnection time $T^{\mathrm{disc}}_N$, defined as the first time when the trajectory of the random walk disconnects the cylinder into two infinite components. We prove that, as long as the drift exponent $\alpha$ is strictly greater than 1, the asymptotic behavior of $T^{\mathrm{disc}}_N$ remains $N^{2d+o(1)}$, as in the unbiased case considered by Dembo and Sznitman, whereas for $\alpha<1$, the asymptotic behavior of $T^{\mathrm{disc}}_N$ becomes exponential in $N$.