Some sufficient conditions for infinite collisions of simple random walks on a wedge comb

TL;DR

This paper establishes conditions under which multiple independent simple random walks on a wedge comb graph collide infinitely often, highlighting the impact of the graph's profile growth and randomness on collision behavior.

Contribution

It provides new sufficient conditions for infinite collisions of simple random walks on wedge combs, including growth rate and probabilistic profile assumptions.

Findings

Infinite collisions occur if the profile grows as n log n.

Three walks collide infinitely often for almost all profiles with i.i.d. finite mean.

Profiles with i.i.d. variables lead to infinite collisions in most cases.

Abstract

In this paper, we give some sufficient conditions for the infinite collisions of independent simple random walks on a wedge comb with profile $\{f(n), n\in \ZZ\}$. One interesting result is that if $f(n)$ has a growth order as $n\log n$, then two independent simple random walks on the wedge comb will collide infinitely many times. Another is that if $\{f(n); n\in \ZZ\}$ are given by i.i.d. non-negative random variables with finite mean, then for almost all wedge comb with such profile, three independent simple random walks on it will collide infinitely many times.