# 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.

## Key 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 $d\ge2$.

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Source: https://tomesphere.com/paper/0909.4422