Random walks colliding before getting trapped

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 $\lambda _Z$.