arXiv:0804.1770v1 [math.PR] 10 Apr 2008 CROSSING PROBABILITIES IN ASYMMETRIC EXCLUSION PROCESSES PABLO A. FERRARI, PATRICIA GONC ¸ALVES AND JAMES B. MARTIN Abstract. 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. 1. Introduction In the one-dimensional asymmetric simple exclusion process (asep), particles perform continu- ous time random walks on Z with rate p ∈ (1/2, 1] of jumping to the right and q =1 − p of jumping to the left. Particles interact by exclusion; attempted jumps to occupied sites are suppressed. The resulting process η t is Markov in the state space {0, 1} Z , where for a site x ∈ Z, η t (x) = 1 indicates that the site x is occupied at time t; otherwise it is empty. If p = 1 then all jumps are to the right and the process is known as the totally asymmetric simple exclusion process (tasep). For λ ∈ [0, 1] denote by ν λ the Bernoulli product measure with particle density λ. All translation-invariant stationary distributions for the asep are convex combinations of ν λ , λ ∈ [0, 1]; see Liggett’s book [6]. A second-class particle is a particle that interacts with particles like a hole, and interacts with holes like a particle. That is, a second-class particle jumps right at rate p if there is a hole on its right, and jumps left at rate q if there is a hole on its left; if the site on its left contains a particle, they exchange positions at rate p, and if the site on its right contains a particle, they exchange positions at rate q. Consider an initial state in which every negative site is occupied by a particle, every positive site has a hole, and there is a single second-class particle at the origin. Let X (t) be the position of the second-class particle at time t. Ferrari and Kipnis [1] proved that X (t)/t converges in distribution as t →∞ to a uniform random variable on the interval [−1, 1]. The extension of the method of [1] to the asep is straightforward, but for completeness we give the result and its proof (indeed it can be extended much further, to more general asymmetric exclusion processes and to a larger class of initial distributions). For the particular case of the tasep, almost sure convergence has been proved by Mountford and Guiol [7], Ferrari and Pimentel [5], and Ferrari, Martin and Pimentel [4]. We then consider processes with several different classes of particles. Holes can be considered as particles of class ∞. A class-m particle at x and a class-k particle at x + 1 exchange positions at rate p if m<k and at rate q if m>k. That is, a pair of class-m and class-k particles behaves as particle-hole if m<k and as hole-particle if m>k; particles of the same class interact by exclusion. For example if a second-class particle attempts to jump to a site occupied by a first class particle, the jump is suppressed but if instead it attempts to jump to a site occupied by a third class particle, then they exchange positions. As a consequence, the higher the degree of the class of a particle, the lower its priority. A hole (still denoted by 0) can be considered as a particle with class ∞. Date : April 10, 2008. 2000 Mathematics Subject Classification. 60K35. Key words and phrases. Asymmetric simple exclusion process, rarefaction fan, second-class particle. 1