Two repelling random walks on $\mathbb Z$

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.Comment: 17 pages. Added references and corrected typos. Revised the argument for the convergence to equilibria of the vector field. Improved the proof for the recurrence when beta belongs to (0,1); leading to the removal of a previous conjectur