Distribution of a second-class particle's position in the two-species ASEP with a special initial configuration

In this paper, we consider the two-species asymmetric simple exclusion process consisting of $N-1$ first-class particles and one second-class particle. We assume that the second-class particle is the rightmost particle at t=0. We provide an alternative method to Tracy and Widom’s method in [J. Phys. A, 42, 425002, (2009)] to find the probability distribution of the second-class particle’s position.