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Recently, there has been much progress in understanding stationary measuresfor colored (also called multi-species or multi-type) interacting particlesystems, motivated by asymptotic phenomena and rich underlying algebraic andcombinatorial structures (such as nonsymmetric Macdonald polynomials).  In this paper, we present a unified approach to constructing stationarymeasures for most of the known colored particle systems on the ring and theline, including (1) the Asymmetric Simple Exclusion Process (multispecies 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 TotallyAsymmetric Simple Exclusion Process (q-PushTASEP). Our method is based onintegrable stochastic vertex models and the Yang-Baxter equation. We expressthe stationary measures as partition functions of new "queue vertex models" onthe cylinder. The stationarity property is a direct consequence of theYang-Baxter equation.  For the mASEP on the ring, a particular case of our vertex model isequivalent to the multiline queues of Martin (arXiv:1810.10650). For thecolored q-Boson process and the q-PushTASEP on the ring, we recover andgeneralize known stationary measures constructed using multiline queues orother methods by Ayyer-Mandelshtam-Martin (arXiv:2011.06117, arXiv:2209.09859),and Bukh-Cox (arXiv:1912.03510). Our proofs of stationarity use the Yang-Baxterequation and bypass the Matrix Product Ansatz used for the mASEP byProlhac-Evans-Mallick (arXiv:0812.3293).  On the line and in a quadrant, we use the Yang-Baxter equation to establish ageneral colored Burke's theorem, which implies that suitable specializations ofour queue vertex models produce stationary measures for particle systems on theline. We also compute the colored particle currents in stationarity.

Colored Interacting Particle Systems on the Ring: Stationary Measures from Yang-Baxter Equation