Collisions of several walkers in recurrent random environments

Abstract

We consider $d$ independent walkers on $\mathbb{Z} $, $m$ of them performing simple symmetric random walk and $r= d-m$ of them performing recurrent RWRE (Sinai walk), in $I$ independent random environments. We show that the product is recurrent, almost surely, if and only if $m\leq 1$ or $m=d=2$. In the transient case with $r\geq 1$, we prove that the walkers meet infinitely often, almost surely, if and only if $m=2$ and $r \geq I= 1$. In particular, while $I$ does not have an influence for the recurrence or transience, it does play a role for the probability to have infinitely many meetings. To obtain these statements, we prove two subtle localization results for a single walker in a recurrent random environment, which are of independent interest.