Collisions of several walkers in recurrent random environments

2018/07/11 by Devulder, Alexis, Gantert, Nina, Pene, Françoise · 1 citation
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1807.04019

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 ≤ 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.

Cited by

• Collisions of random walks on comb graphs with a planar base

Related

• Infinitely many collisions between a recurrent simple random walk and arbitrary many transient random walks in a subballistic random environment
• On the recurrence of some random walks in random environment
• Arbitrary many walkers meet infinitely often in a subballistic random environment
• Collisions of Random Walks in Dynamic Random Environments
• Excited random walks with non-nearest neighbor steps