## Electronic Journal of Probability

### On the domination of a random walk on a discrete cylinder by random interlacements

Alain-Sol Sznitman

#### Abstract

We consider simple random walk on a discrete cylinder with base a large $d$-dimensional torus of side-length $N$, when $d$ is two or more. We develop a stochastic domination control on the local picture left by the random walk in boxes of side-length almost of order $N$, at certain random times comparable to the square of the number of sites in the base. We show a domination control in terms of the trace left in similar boxes by random interlacements in the infinite $(d+1)$-dimensional cubic lattice at a suitably adjusted level. As an application we derive a lower bound on the disconnection time of the discrete cylinder, which as a by-product shows the tightness of the laws of the ratio of the square of the number of sites in the base to the disconnection time. This fact had previously only been established when $d$ is at least 17.

#### Article information

Source
Electron. J. Probab., Volume 14 (2009), paper no. 56, 1670-1704.

Dates
Accepted: 25 July 2009
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464819517

Digital Object Identifier
doi:10.1214/EJP.v14-679

Mathematical Reviews number (MathSciNet)
MR2525107

Zentralblatt MATH identifier
1196.60170

Rights

#### Citation

Sznitman, Alain-Sol. On the domination of a random walk on a discrete cylinder by random interlacements. Electron. J. Probab. 14 (2009), paper no. 56, 1670--1704. doi:10.1214/EJP.v14-679. https://projecteuclid.org/euclid.ejp/1464819517

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