The second class particle process at shocks

TL;DR

This paper analyzes the behavior of a second class particle in TASEP with a shock, providing exact distribution identities and joint distributions without relying on last passage percolation models.

Contribution

It derives exact identities and joint distributions for the second class particle in TASEP with shocks, extending previous results through a direct approach.

Findings

Exact distribution identity for the second class particle

Limiting joint distributions of the second class particle

Extension of one-point distribution results

Abstract

We consider the totally asymmetric simple exclusion process (TASEP) starting with a shock discontinuity at the origin, with asymptotic densities $\lambda$ to the left of the origin and $\rho$ to the right of it and $\lambda<\rho$. We find an exact identity for the distribution of a second class particle starting at the origin. Then we determine the limiting joint distributions of the second class particle. Bypassing the last passage percolation model, we work directly in TASEP, allowing us to extend previous one-point distribution results via a more direct and shorter ansatz.