Random walks colliding before getting trapped
نویسندگان
چکیده
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 λX , λY , λZ ≥ 0, then let X,Y and Z be independent Markov chains with transition matrix P and speeds λX , λY and λZ 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 λX , λY and λZ with max(λX , λY ) > 0, we prove a lower bound for this probability which is uniform over all transitive, irreducible and reversible chains. In the case that λX = λY = 1 and λZ = 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 λX , λY and λZ .
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