SECOND AND THIRD CLASS PARTICLES IN TASEP

Abstract. We consider the nearest neighbors one-dimensional totally asymmetric simple ex-clusion process starting with ones to the left of the origin, a second class particle at the origin, a third class particle at site 1 and no particles to the right of site 1. We show that the probability that the third class particle is to the right of the second class particle at time t converges to 2/3 as t → ∞. We also consider the asymmetric exclusion process with transition rates having a positive mean and show that if the system starts with a product measure with densities λ> ρ to the left and right of the origin, respectively, then the position of the second class particle at time t divided by t converges in distribution to a uniform random variable in the interval [−(ρ − λ), ρ − λ], extending a result by the first author and Kipnis. 1