### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 008, 24 pages      arXiv:2104.06471      https://doi.org/10.3842/SIGMA.2022.008

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

Eunghyun Lee and Temirlan Raimbekov
Department of Mathematics, Nazarbayev University, Nur-sultan, Kazakhstan

Received April 15, 2021, in final form January 24, 2022; Published online January 29, 2022

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.

Key words: multi-species ASEP; transition probability; Bethe ansatz; symmetric group.

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