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The inhomogeneous two-species TASEP on a ring is an exclusion process thatdescribes particles of different species hopping clockwise on a ring withparameters giving the hopping rates for different species. We introduce acombinatorial object that we call \emph{toric rhombic alternative tableaux},which are certain fillings of tableaux on a triangular lattice tiled withrhombi, and are in bijection with the well-studied \emph{multiline queues} ofFerrari and Martin. Using the tableaux, we obtain a formula for the stationaryprobabilities of this TASEP, which specializes to results of Ayyer andLinusson. We obtain, in addition, an explicit determinantal formula for theseprobabilities, and define a Markov chain on the tableaux that projects to thetwo-species TASEP on a ring.

Toric tableaux and the inhomogeneous two-species TASEP on a ring