Electronic Journal of Probability

Collisions of several walkers in recurrent random environments

Alexis Devulder, Nina Gantert, and Françoise Pène

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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.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 90, 34 pp.

Received: 5 December 2017
Accepted: 3 July 2018
First available in Project Euclid: 12 September 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments 60G50: Sums of independent random variables; random walks

random walk random environment collisions recurrence transience

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Devulder, Alexis; Gantert, Nina; Pène, Françoise. Collisions of several walkers in recurrent random environments. Electron. J. Probab. 23 (2018), paper no. 90, 34 pp. doi:10.1214/18-EJP192. https://projecteuclid.org/euclid.ejp/1536717749

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