Infinitely many collisions between a recurrent simple random walk and arbitrary many transient random walks in a subballistic random environment
Alexis Devulder
TL;DR
This work addresses collisions between a recurrent simple random walk and multiple independent transient RWREs on ℤ within a shared random environment, focusing on the subballistic regime κ∈(0,1/2). It leverages potential theory, ladder epochs, and quenched techniques, building on prior transient-collision results to prove that for any number d≥1 of transient RWREs (with same-parity starts), there are almost surely infinitely many times when all d RWREs and the recurrent walk meet at the same site. Conversely, it shows that when κ>1/2, the collision set is almost surely finite, with the transition explained by the relative scale n^{κ} versus the SRW’s √n fluctuations. The paper also outlines extensions, including the κ=1/2 border case, ballistic scenarios, and multi-walk generalizations, and raises open questions about the precise behavior in borderline regimes.
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 $ω$ 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 $ω$, 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 $ω$ is transient to the right but subballistic, with parameter $0<κ<1/2$. 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.