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 $λ$ if it jumps at rate $λ$ according to the matrix $P$. Fix $λ_X,λ_Y,λ_Z\geq 0$, then let $X,Y$ and $Z$ be independent Markov chains with transition matrix $P$ and speeds $λ_X,λ_Y$ and $λ_Z$ 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 $λ_X,λ_Y$ and $λ_Z$ with $\max(λ_X,λ_Y)>0$, we prove a lower bound for this probability which is uniform over all transitive, irreducible and reversible chains. In the case that $λ_X=λ_Y=1$ and $λ_Z=0$ 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 $λ_X,λ_Y$ and $λ_Z$.