Random walks colliding before getting trapped

TL;DR

This paper investigates the probability that two independent Markov chains collide before either meets a third, providing bounds and conditions under which these probabilities are uniform across certain classes of chains.

Contribution

It establishes a uniform lower bound for collision probabilities in transitive, irreducible, reversible Markov chains, and strengthens results using martingale techniques for specific speeds.

Findings

Provides a lower bound for collision probability before trapping.

Shows the importance of transitivity in the bounds.

Uses martingale methods to strengthen main results.

Abstract

Let $P$ be the transition matrix of a finite, irreducible and reversible Markov chain. We say the continuous time Markov chain $X$ has transition matrix $P$ and speed $\lambda$ if it jumps at rate $\lambda$ according to the matrix $P$. Fix $\lambda_X,\lambda_Y,\lambda_Z\geq 0$, then let $X,Y$ and $Z$ be independent Markov chains with transition matrix $P$ and speeds $\lambda_X,\lambda_Y$ and $\lambda_Z$ respectively, all started from the stationary distribution. What is the chance that $X$ and $Y$ meet before either of them collides with $Z$? For each choice of $\lambda_X,\lambda_Y$ and $\lambda_Z$ with $\max(\lambda_X,\lambda_Y)>0$, we prove a lower bound for this probability which is uniform over all transitive, irreducible and reversible chains. In the case that $\lambda_X=\lambda_Y=1$ and $\lambda_Z=0$ we prove a strengthening of our main theorem using a martingale argument. We provide an example showing the transitivity assumption cannot be removed for general $\lambda_X,\lambda_Y$ and $\lambda_Z$.