Browse issues
no. 6
A strange five vertex model and multispecies ASEP on a ring
Kuniba, Atsuo1; Okado, Masato2; Scrimshaw, Travis3
1 Institute of Physics, Graduate School of Arts and Sciences, University of Tokyo, Komaba, Tokyo 153-8902 (Japan)
2 Osaka Central Advanced Mathematical Institute & Department of Mathematics, Osaka Metropolitan University, Osaka 558-8585 (Japan)
3 Department of Mathematics, Hokkaido University, Sapporo 060-0808 (Japan)
Algebraic Combinatorics, Volume 8 (2025) no. 6, pp. 1713-1741

We revisit the problem of constructing the stationary states of the multispecies asymmetric simple exclusion process on a one-dimensional periodic lattice. Central to our approach is a quantum oscillator weighted five vertex model which features a strange weight conservation distinct from the conventional one. Our results clarify the interrelations among several known results and refine their derivations. For instance, the stationary probability derived from the multiline queue construction by Martin (2020) and Corteel–Mandelshtam–Williams (2022) is identified with the partition function of a three-dimensional system. The matrix product operators by Prolhac–Evans–Mallick (2009) acquire a natural diagrammatic interpretation as corner transfer matrices (CTM). The origin of their recursive tensor structure, as questioned by Aggarwal–Nicoletti–Petrov (2023), is revealed through the CTM diagrams. Finally, the derivation of the Zamolodchikov–Faddeev algebra by Cantini–de Gier–Wheeler (2015) is made intrinsic by elucidating its precise connection to a solution to the Yang–Baxter equation originating from quantum group representations.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.457
Classification: 60J27, 82B23, 05E05, 81R12
Keywords: multispecies ASEP, stationary state, five vertex model, Zamolodchikov–Faddeev algebra

Kuniba, Atsuo 1; Okado, Masato 2; Scrimshaw, Travis 3

1 Institute of Physics, Graduate School of Arts and Sciences, University of Tokyo, Komaba, Tokyo 153-8902 (Japan)
2 Osaka Central Advanced Mathematical Institute & Department of Mathematics, Osaka Metropolitan University, Osaka 558-8585 (Japan)
3 Department of Mathematics, Hokkaido University, Sapporo 060-0808 (Japan)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2025__8_6_1713_0,
     author = {Kuniba, Atsuo and Okado, Masato and Scrimshaw, Travis},
     title = {A strange five vertex model and multispecies {ASEP} on a ring},
     journal = {Algebraic Combinatorics},
     pages = {1713--1741},
     year = {2025},
     publisher = {The Combinatorics Consortium},
     volume = {8},
     number = {6},
     doi = {10.5802/alco.457},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.457/}
}
TY  - JOUR
AU  - Kuniba, Atsuo
AU  - Okado, Masato
AU  - Scrimshaw, Travis
TI  - A strange five vertex model and multispecies ASEP on a ring
JO  - Algebraic Combinatorics
PY  - 2025
SP  - 1713
EP  - 1741
VL  - 8
IS  - 6
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.457/
DO  - 10.5802/alco.457
LA  - en
ID  - ALCO_2025__8_6_1713_0
ER  - 
%0 Journal Article
%A Kuniba, Atsuo
%A Okado, Masato
%A Scrimshaw, Travis
%T A strange five vertex model and multispecies ASEP on a ring
%J Algebraic Combinatorics
%D 2025
%P 1713-1741
%V 8
%N 6
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.457/
%R 10.5802/alco.457
%G en
%F ALCO_2025__8_6_1713_0
Kuniba, Atsuo; Okado, Masato; Scrimshaw, Travis. A strange five vertex model and multispecies ASEP on a ring. Algebraic Combinatorics, Volume 8 (2025) no. 6, pp. 1713-1741. doi: 10.5802/alco.457

[1] Aboumrad, W.; Scrimshaw, T. On the structure and representation theory of q-deformed Clifford algebras, Math. Z., Volume 306 (2024) no. 1, Paper no. 10, 28 pages | DOI | MR | Zbl

[2] Aggarwal, A.; Nicoletti, M.; Petrov, L. Colored interacting particle systems on the ring: stationary measures from Yang-Baxter equations, Volume 161, 2025 no. 8, pp. 1855-1922 | DOI | MR | Zbl

[3] Arita, C.; Kuniba, A.; Sakai, K.; Sawabe, T. Spectrum of a multi-species asymmetric simple exclusion process on a ring, J. Phys. A, Volume 42 (2009) no. 34, Paper no. 345002, 41 pages | DOI | MR | Zbl

[4] Baxter, R. J. Exactly solved models in statistical mechanics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1989, xii+486 pages | MR | Zbl

[5] Borodin, A.; Wheeler, M. Colored stochastic vertex models and their spectral theory, Astérisque (2022) no. 437, p. ix+225 | DOI | MR | Zbl

[6] Cantini, L.; de Gier, J.; Wheeler, M. Matrix product formula for Macdonald polynomials, J. Phys. A, Volume 48 (2015) no. 38, Paper no. 384001, 25 pages | DOI | MR | Zbl

[7] Corteel, S.; Mandelshtam, O.; Williams, L. From multiline queues to Macdonald polynomials via the exclusion process, Amer. J. Math., Volume 144 (2022) no. 2, pp. 395-436 | DOI | MR | Zbl

[8] Crampe, N.; Ragoucy, E.; Vanicat, M. Integrable approach to simple exclusion processes with boundaries. Review and progress, J. Stat. Mech. Theory Exp. (2014) no. 11, Paper no. P11032, 42 pages | DOI | MR | Zbl

[9] de Gier, J.; Wheeler, M. A summation formula for Macdonald polynomials, Lett. Math. Phys., Volume 106 (2016) no. 3, pp. 381-394 | DOI | MR | Zbl

[10] Drinfel’d, V. G. Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI (1987), pp. 798-820 | MR | Zbl

[11] Ferrari, P. A.; Martin, J. B. Stationary distributions of multi-type totally asymmetric exclusion processes, Ann. Probab., Volume 35 (2007) no. 3, pp. 807-832 | DOI | MR | Zbl

[12] Hayashi, T. q-analogues of Clifford and Weyl algebras—spinor and oscillator representations of quantum enveloping algebras, Comm. Math. Phys., Volume 127 (1990) no. 1, pp. 129-144 | MR | DOI | Zbl

[13] Holstein, T.; Primakoff, H. Field dependence of the intrinsic domain magnetization of a ferromagnet, Phys. Rev., Volume 58 (1940) no. 12, pp. 1098-1113 | DOI | Zbl

[14] Inoue, R.; Kuniba, A.; Okado, M. A quantization of box-ball systems, Rev. Math. Phys., Volume 16 (2004) no. 10, pp. 1227-1258 | DOI | MR | Zbl

[15] Iwao, S.; Motegi, K.; Ohkawa, R. Tetrahedron equation and Schur functions, Volume 58, 2025 no. 1, Paper no. 015201, 31 pages | MR | Zbl

[16] Jimbo, M. A q-difference analogue of U(𝔤) and the Yang–Baxter equation, Lett. Math. Phys., Volume 10 (1985) no. 1, pp. 63-69 | DOI | MR | Zbl

[17] Kashiwara, M. Crystalizing the q-analogue of universal enveloping algebras, Comm. Math. Phys., Volume 133 (1990) no. 2, pp. 249-260 | MR | Zbl | DOI

[18] Kuniba, A. Quantum groups in three-dimensional integrability, Theoretical and Mathematical Physics, Springer, Singapore, 2022, xi+331 pages | DOI | MR | Zbl

[19] Kuniba, A.; Mangazeev, V. V.; Maruyama, S.; Okado, M. Stochastic R matrix for U q (A n (1) ), Nuclear Phys. B, Volume 913 (2016), pp. 248-277 | DOI | MR | Zbl

[20] Kuniba, A.; Maruyama, S.; Okado, M. Multispecies TASEP and combinatorial R, J. Phys. A, Volume 48 (2015) no. 34, Paper no. 34FT02, 19 pages | DOI | MR | Zbl

[21] Kuniba, A.; Maruyama, S.; Okado, M. Multispecies TASEP and the tetrahedron equation, J. Phys. A, Volume 49 (2016) no. 11, Paper no. 114001, 22 pages | DOI | MR | Zbl

[22] MacDonald, J. T.; Gibbs, J. H.; Pipkin, A. C. Kinetics of biopolymerization on nucleic acid templates, Biopolymers, Volume 6 (1968) no. 1, pp. 1-25 | DOI | Zbl

[23] Martin, J. B. Stationary distributions of the multi-type ASEP, Electron. J. Probab., Volume 25 (2020), Paper no. 43, 41 pages | DOI | MR | Zbl

[24] Prolhac, S.; Evans, M. R.; Mallick, K. The matrix product solution of the multispecies partially asymmetric exclusion process, J. Phys. A, Volume 42 (2009) no. 16, Paper no. 165004, 25 pages | DOI | MR | Zbl

[25] Spitzer, F. Interaction of Markov processes, Adv. Math., Volume 5 (1970), pp. 246-290 | DOI | MR | Zbl

[26] Sage Mathematics Software (Version 10.7) (2025) (https://www.sagemath.org)

Cited by Sources: