Second and third class particles in TASEP

Abstract

We consider the nearest neighbors one-dimensional totally asymmetric simple exclusion 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 $t\to\infty$. 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 $\lambda>\rho$ 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 $[-(\rho-\lambda),\rho-\lambda]$, extending a result by the first author and Kipnis.