Infinitely many collisions between a recurrent simple random walk and arbitrary many transient random walks in a subballistic random environment

Abstract

We consider $d$ random walks $(S_n^{(j)}{)}_{n\in \mathbb{N}}$, $1\le j\le d$, in the same random environment $\omega$ in $\mathbb{Z}$, and a recurrent simple random walk $(Z_n{)}_{n\in \mathbb{N}}$ on $\mathbb{Z}$. We assume that, conditionally on the environment $\omega$, all the random walks are independent and start from even initial locations. Our assumption on the law of the environment is such that a single random walk in the environment $\omega$ is transient to the right but subballistic, with parameter $0<\kappa <1/2$. We show that - for every value of $d$ - there are almost surely infinitely many times for which all these random walks, $(Z_n{)}_{n\in \mathbb{N}}$ and $(S_n^{(j)}{)}_{n\in \mathbb{N}}$, $1\le j\le d$, are simultaneously at the same location, even though one of them is recurrent and the $d$ others ones are transient.

11 pages ; applying and extending some results of arXiv:1811.12763 ; to appear in Electronic Communications in Probability