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 $\big(S_n^{(j)}\big)_{n\in\mathbb{N}}$, $1\leq j \leq d$, 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 $\big(S_n^{(j)}\big)_{n\in\mathbb{N}}$, $1\leq j \leq d$, are simultaneously at the same location, even though one of them is recurrent and the $d$ others ones are transient.