How universal are asymptotics of disconnection times in discrete cylinders?

TL;DR

This paper explores the disconnection times of random walks in large discrete cylinders, demonstrating that the rough order of these times is broadly applicable across various large connected base graphs with bounded degree.

Contribution

It extends previous results on disconnection times from specific base graphs to a wide class of large connected graphs with bounded degree, confirming conjectures about their asymptotic behavior.

Findings

Disconnection time scales as the square of the base graph size.

Broad applicability of asymptotic behavior across different graph structures.

Supports conjecture by I. Benjamini on universality.

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 $G_N\times\mathbb{Z}$ has rough order $|G_N|^2$, when $G_N=(\mathbb{Z}/N\mathbb{Z})^d$. 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.