Collisions of several walkers in recurrent random environments

TL;DR

This paper analyzes the recurrence and meeting properties of multiple walkers in random environments, revealing precise conditions under which they meet infinitely often or are recurrent, with new localization results for single walkers in recurrent environments.

Contribution

It provides new criteria for recurrence and meeting of multiple walkers in recurrent random environments, including localization results for single walkers.

Findings

Product of walkers is recurrent if and only if m ≤ 1 or m = d = 2.

Walkers meet infinitely often in transient environments if and only if m=2 and r≥1.

The environment parameter I influences meeting probability but not recurrence/transience.

Abstract

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 $\le$ 1 or m = d = 2. In the transient case with r $\ge$ 1, we prove that the walkers meet infinitely often, almost surely, if and only if m = 2 and r $\ge$ 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.