We study the asymptotic speed of a second class particle in the two-species asymmetric simple exclusion process (ASEP) on $\mathbb{ Z}$ with each particle belonging either to the first class or the second class. For any fixed non-negative integer $L$, we consider the two-species ASEP started from the initial data with all the sites of $\mathbb{ Z}_{ <-L}$ occupied by first class particles, all the sites of $\mathbb{ Z}_{ [-L,0]}$ occupied by second class particles, and the rest of the sites of $\mathbb{ Z}$ unoccupied. With these initial conditions, we show that the speed of the leftmost second class particle converges weakly to a distribution supported on a symmetric compact interval $\Gamma\subset \mathbb{ R}$. Furthermore, the limiting distribution is shown to have the same law as the minimum of $L+1$ independent random samples drawn uniformly from the interval $\Gamma$.
Categories: math.PR math-ph math.MP