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

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 𝑆𝑁. On the other hand, the transition probabilities of the multi-species ASEP, in general, may be expressed as a sum of many more terms than 𝑁!. In this paper, we show that if the initial order of species is given by 2⋯21, 12⋯2, 1⋯12, or 21⋯1, then the transition probabilities can be expressed as a sum of at most 𝑁! contour integrals, and provide their formulas explicitly.