Some conditional probabilities in the TASEP with second class particles

TL;DR

This paper analyzes the TASEP with second class particles, deriving probabilities of specific configurations over time, including explicit formulas for step initial conditions, extending previous results for a single particle case.

Contribution

It provides new explicit formulas for the probability of certain configurations in TASEP with second class particles, generalizing earlier single-particle results.

Findings

Derived probability formulas for the position of first class particles over time.

Extended previous single-particle results to multiple particles.

Provided explicit formulas for step initial conditions.

Abstract

In this paper we consider the TASEP with second class particles with the initial order is such that $k$ first class particles are located to the left of $N-k$ second class particles. Under this assumption of the initial state of order, we find the probability that the first class particles are at $x,x+1,\cdots, x+k-1$ and the first class particles are still located to the left of all second class particles at later time $t$. Moreover, we obtain an explicit formula of this probability for the step initial condition. This probability generalizes a result for $k=1$ in Lee (arXiv:1705.10544).