Determinant representation for some transition probabilities in the TASEP with second class particles

TL;DR

This paper derives explicit formulas for transition probabilities in the multi-class TASEP with first and second class particles using Bethe ansatz and determinantal representations, advancing understanding of multi-species exclusion processes.

Contribution

It provides a novel explicit determinantal formula for transition probabilities in multi-class TASEP, extending previous results to more complex particle interactions.

Findings

Explicit Bethe ansatz-based formulas for transition probabilities.

Determinantal form for cases with fixed particle order.

Generalization to multi-class TASEP through geometric insights.

Abstract

We study the transition probabilities for the totally asymmetric simple exclusion process (TASEP) on the infinite integer lattice with a finite, but arbitrary number of first and second class particles. Using the Bethe ansatz we present an explicit expression of these quantities in terms of the Bethe wave function. In a next step it is proved rigorously that this expression can be written in a compact determinantal form for the case where the order of the first and second class particles does not change in time. An independent geometrical approach provides insight into these results and enables us to generalize the determinantal solution to the multi-class TASEP.