Axis-Driven Random Walks on $\mathbb{Z}^2$ (transient cases)

TL;DR

This paper studies axis-driven random walks on 2 with a repulsive force, showing that even minimal force causes the walk to be transient and superdiffusive, with detailed tail distribution analysis.

Contribution

It introduces a model with a repulsive force along axes and proves that small forces induce transience and superdiffusivity, extending previous work on localized force fields.

Findings

Walks are transient even with minimal repulsive force.

The walk exhibits superdiffusive behavior.

Derived tail distributions for the walk's position.

Abstract

Axis-driven random walks were introduced by P. Andreoletti and P. Debs [AD23] to provide a rough description of the behaviour of a particle trapped in a localized force field. In contrast to their work, we examine the scenario where a repulsive force (controlled by a parameter $\alpha$) is applied along the axes, with the hypothesis that the walk remains diffusive within the cones. This force gradually pushes the particle away from the origin whenever it encounters an axis. We prove that even with a minimal force (i.e., a small $\alpha$), the walk exhibits transient, superdiffusive behaviour, and we derive the left and right tails of its distribution.