On the range of the Campanino and P\'etritis random walk

TL;DR

This paper investigates the range behavior of a specialized random walk with random horizontal orientations on a 2D lattice and explores related random scenery models to understand their asymptotic properties.

Contribution

It introduces analysis of the range of the Campanino and Pétritis random walk and connects it to the behavior of random walks in random scenery, providing new insights into their asymptotics.

Findings

Characterization of the range of the Campanino and Pétritis random walk.

Asymptotic behavior of the first coordinate's range in the model.

Connections established between the walk's range and random scenery models.

Abstract

We are interested in the behaviour of the range of the Campanino and P\'etritis random walk, namely a simple random walk on the lattice $\mathbb Z^2$ with random orientations of the horizontal layers. We also study the range of random walks in random scenery, from which the asymptotic behaviour of the range of the first coordinate of the Campanino and P\'etritis random walk can be deduced.