Collisions of several walkers in recurrent random environments

Alexis Devulder,Françoise Pène,Nina Gantert

doi:10.1214/18-ejp192

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.

Highlights

We consider d independent walkers on Z, m of them performing simple symmetric random walk and r = d − m of them performing recurrent random walks in random environment (RWRE) (Sinai walk), in I independent random environments
Recurrence and transience of products of simple symmetric random walks on Zd is well-known since the works of Pólya [30]
Pólya’s original interest in recurrence/transience of simple random walk came from a question about collisions of two independent walkers on the same grid, see [31], “Two incidents”

Summary

Introduction and statement of the main results

Recurrence and transience of products of simple symmetric random walks on Zd is well-known since the works of Pólya [30]. (iii) If m = 2 and r ≥ I ≥ 2, for P-almost every environment ω, Pω0 Sn(1) = Sn(2) = Zn(1) = Zn(2) infinitely often = 0, i.e. almost surely, the walks S(1), S(2), Z(1), Z(2), and a fortiori the walks S(1), S(2), Z(1), ..., Z(r), meet simultaneously only a finite number of times. This last result can be summarized in the following manner. The proofs concerning the simultaneous meetings of random walks are based on the above-mentioned two key localization results for recurrent RWRE, proved in Sections 4 and 5

A return probability estimate for the RWRE

Direct product of walks