Collisions of several walkers in recurrent random environments

We consider d independent walkers on 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 ≤ 1 or m = d = 2. In the transient case with r ≥ 1, we prove that the walkers meet infinitely often, almost surely, if and only if m = 2 and r ≥ 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.