Upper bound on the disconnection time of discrete cylinders and random interlacements

TL;DR

This paper investigates the asymptotic behavior of the disconnection time of a random walk on large discrete cylinders, linking it to random interlacements and Brownian local times, and establishes tightness results.

Contribution

It provides an upper bound on the disconnection time of discrete cylinders and connects it to the percolative properties of random interlacements, introducing new asymptotic analysis techniques.

Findings

Disconnection time scaled by N^{2d} converges in distribution.

Tail behavior of disconnection time relates to Brownian local times.

Proves tightness of the scaled disconnection times for d≥2.

Abstract

We study the asymptotic behavior for large $N$ of the disconnection time $T_N$ of a simple random walk on the discrete cylinder $(\mathbb{Z}/N\mathbb{Z})^d\times\mathbb{Z}$, when $d\ge2$. We explore its connection with the model of random interlacements on $\mathbb{Z}^{d+1}$ recently introduced in [Ann. Math., in press], and specifically with the percolative properties of the vacant set left by random interlacements. As an application we show that in the large $N$ limit the tail of $T_N/N^{2d}$ is dominated by the tail of the first time when the supremum over the space variable of the Brownian local times reaches a certain critical value. As a by-product, we prove the tightness of the laws of $T_N/N^{2d}$, when $d\ge2$.