Simplified Forms of the Transition Probabilities of the Two-Species ASEP with Some Initial Orders of Particles

TL;DR

This paper derives simplified formulas for the transition probabilities of multi-species ASEP under specific initial orders, reducing the complexity from potentially many terms to at most N! contour integrals.

Contribution

It provides explicit formulas for transition probabilities of multi-species ASEP with certain initial orders, simplifying the known complex sums.

Findings

Transition probabilities expressed as at most N! contour integrals for specific initial orders.

Explicit formulas for these simplified transition probabilities.

Reduction of complexity in multi-species ASEP calculations.

Abstract

It has been known that the transition probability of the single species ASEP with $N$ particles is expressed as a sum of $N!$ $N$-fold contour integrals which are related to permutations in the symmetric group $S_N$. On other hand, the transition probabilities of the multi-species ASEP, in general, may be expressed as a sum of much more terms than $N!$. In this paper, we show that if the initial order of species is given by $2\cdots 21$, $12\cdots 2$, $1\cdots 12$ or $21\cdots 1$, then the transition probabilities can be expressed as a sum of at most $N!$ contour integrals, and provide their formulas explicitly.