On the TASEP with Second Class Particles

Abstract

In this paper, we study some conditional probabilities for the totally asymmetric simple exclusion processes (TASEP) with second-class particles. To be more specific, we consider a finite system with one first-class particle and N−1 second-class particles, and we assume that the first-class particle is initially at the leftmost position. In this case, we find the probability that the first class particle is at x and it is still the leftmost particle at time t. In particular, we show that this probability is expressed by the determinant of an N×N matrix of contour integrals if the initial positions of particles satisfy the step initial condition. The resulting formula is very similar to a known formula in the (usual) TASEP with the step initial condition, which was used for asymptotics by Nagao and Sasamoto [Nuclear Phys. B 699 (2004), 487-502].