Combinatorics of the two-species ASEP and Koornwinder moments

In previous work [12], [13], [14], the first and third authors introduced staircase tableaux, which they used to give combinatorial formulas for the stationary distribution of the asymmetric simple exclusion process (ASEP) and for the moments of the Askey-Wilson weight function. The fact that the ASEP and Askey-Wilson moments are related at all is unexpected, and is due to [45]. The ASEP is a model of particles hopping on a one-dimensional lattice of N sites with open boundaries; particles can enter and exit at both left and right borders. It was introduced around 1970 [34], [43] and is cited as a model for both traffic flow and translation in protein synthesis. Meanwhile, the Askey-Wilson polynomials are a family of orthogonal polynomials in one variable which sit at the top of the hierarchy of classical orthogonal polynomials. So from this previous work, we have the relationship ASEP −− staircase tableaux −− Askey-Wilson moments. The Askey-Wilson polynomials can be viewed as the one-variable case of the multivariate Koornwinder polynomials, which are also known as the Macdonald polynomials attached to the non-reduced affine root system (Cn∨,Cn). It is natural then to ask whether one can generalize the relationships among the ASEP, Askey-Wilson moments, and staircase tableaux, in such a way that Koornwinder moments replace Askey-Wilson moments. In [15], we made a precise link between Koornwinder moments and the two-species ASEP, a generalization of the ASEP which has two species of particles with different “weights.” In this article we introduce rhombic staircase tableaux, and show that we have the relationship 2-species ASEP −− rhombic staircase tableaux −− Koornwinder moments. In particular, we give formulas for the stationary distribution of the two-species ASEP and for Koornwinder moments, in terms of rhombic staircase tableaux.75