Crossing Probabilities in Asymmetric Exclusion Processes

We consider the one-dimensional asymmetric simple exclusion process in which particles jump to the right at rate p and to the left at rate 1 − p, interacting by exclusion. Suppose that the initial state has first-class particles to the left of the origin, a second class particle at the origin, a third class particle at site 1 and holes to the right of site 1. We show that the probability that the second-class particle overtakes the third-class particle is (1+p)/3p. We obtain various limiting results about the joint behavior of the second-class and third-class particles, and a partial extension to a system with a further (fourth-class) particle.