Some Sufficient Conditions for Infinite Collisions of Simple Random Walks on a Wedge Comb

Dayue Chen,Xinxing Chen

doi:10.1214/ejp.v16-907

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.

Highlights

We study the number of collisions of two independent simple random walks on an infinite connected graph with finite degrees
Let X = {Xn} and X = {Xn} be independent simple random walks starting from the same vertex
We say that the graph has the infinite collision property if X and X collide infinitely often, i.e., |{n : Xn = Xn}| = ∞, almost surely

Summary

Introduction

We study the number of collisions of two independent simple random walks on an infinite connected graph with finite degrees. Let X = {Xn} and X = {Xn} be independent simple random walks starting from the same vertex. If β > 2, the total number of collisions by two independent simple random walks on Comb(Z, f ) is a.s. finite.

Results

Conclusion