
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 TwoSpecies ASEP with Some Initial Orders of Particles
Eunghyun Lee and Temirlan Raimbekov
Department of Mathematics, Nazarbayev University, Nursultan, 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 multispecies 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: multispecies ASEP; transition probability; Bethe ansatz; symmetric group.
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