Two repelling random walks on $\mathbb Z$

TL;DR

This paper studies two interacting random walks on the integer line, showing how their recurrence or transience depends on a parameter modulating their mutual repulsion, with results obtained via stochastic approximation methods.

Contribution

It introduces a model of two mutually repelling random walks with a parameter controlling the strength of repulsion and analyzes their long-term behavior.

Findings

Both walks are recurrent for  < eta .

Walks diverge in opposite directions for \u03b2 > 2.

Behavior for  <   remains open.

Abstract

We consider two interacting random walks on $\mathbb{Z}$ such that the transition probability of one walk in one direction decreases exponentially with the number of transitions of the other walk in that direction. The joint process may thus be seen as two random walks reinforced to repel each other. The strength of the repulsion is further modulated in our model by a parameter $\beta \geq 0$. When $\beta = 0$ both processes are independent symmetric random walks on $\mathbb{Z}$, and hence recurrent. We show that both random walks are further recurrent if $\beta \in (0,1]$. We also show that these processes are transient and diverge in opposite directions if $\beta > 2$. The case $\beta \in (1,2]$ remains widely open. Our results are obtained by considering the dynamical system approach to stochastic approximations.