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Colored interacting particle systems on the ring: Stationary measures from Yang–Baxter equations

Part of: Algebraic combinatorics Markov processes Special processes Hopf algebras, quantum groups and related topics Equilibrium statistical mechanics Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  03 October 2025

Amol Aggarwal ,
Matthew Nicoletti  and
Leonid Petrov
Amol Aggarwal
Affiliation:
Columbia University, 2990 Broadway, New York, NY 10027, USA Clay Mathematics Institute, 1624 Market Street, Suite 226 #17261, Denver, CO 80202, USA amolagga@gmail.com
Matthew Nicoletti
Affiliation:
Massachusetts Institute of Technology, Cambridge MA, USA Current address: Department of Statistics, University of California, Berkeley, 367 Evans Hall, Berkeley, CA 94720, USA mnicoletti@berkeley.edu
Leonid Petrov
Affiliation:
Department of Mathematics, University of Virginia, 141 Cabell Drive, Kerchof Hall, P.O. Box 400137, Charlottesville, VA 22904, USA lenia.petrov@gmail.com
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Abstract

Recently, there has been much progress in understanding stationary measures for colored (also called multi-species or multi-type) interacting particle systems, motivated by asymptotic phenomena and rich underlying algebraic and combinatorial structures (such as nonsymmetric Macdonald polynomials). In this paper, we present a unified approach to constructing stationary measures for most of the known colored particle systems on the ring and the line, including (1) the Asymmetric Simple Exclusion Process (multi-species ASEP, or mASEP); (2) the $q$-deformed Totally Asymmetric Zero Range Process (TAZRP) also known as the $q$-Boson particle system; (3) the $q$-deformed Pushing Totally Asymmetric Simple Exclusion Process ($q$-PushTASEP). Our method is based on integrable stochastic vertex models and the Yang–Baxter equation. We express the stationary measures as partition functions of new ‘queue vertex models’ on the cylinder. The stationarity property is a direct consequence of the Yang–Baxter equation. For the mASEP on the ring, a particular case of our vertex model is equivalent to the multiline queues of Martin (Stationary distributions of the multi-type ASEP, Electron. J. Probab. 25 (2020), 1–41). For the colored $q$-Boson process and the $q$-PushTASEP on the ring, we recover and generalize known stationary measures constructed using multiline queues or other methods by Ayyer, Mandelshtam and Martin (Modified Macdonald polynomials and the multispecies zero range process: II, Algebr. Comb. 6 (2022), 243–284; Modified Macdonald polynomials and the multispecies zero-range process: I, Algebr. Comb. 6 (2023), 243–284) and Bukh and Cox (Periodic words, common subsequences and frogs, Ann. Appl. Probab. 32 (2022), 1295–1332). Our proofs of stationarity use the Yang–Baxter equation and bypass the Matrix Product Ansatz (used for the mASEP by Prolhac, Evans and Mallick (The matrix product solution of the multispecies partially asymmetric exclusion process, J. Phys. A. 42 (2009), 165004)). On the line and in a quadrant, we use the Yang–Baxter equation to establish a general colored Burke’s theorem, which implies that suitable specializations of our queue vertex models produce stationary measures for particle systems on the line. We also compute the colored particle currents in stationarity.

Keywords

Yang–Baxter equationstationary measuresstochastic vertex modelsmulti-species ASEP

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82B23: Exactly solvable models; Bethe ansatz 16T25: Yang-Baxter equations 05E05: Symmetric functions and generalizations 60J27: Continuous-time Markov processes on discrete state spaces 82C22: Interacting particle systems

Information

Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 8 , August 2025 , pp. 1855 - 1922
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© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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