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

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.