  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.

Probability