Asymmetric Simple Exclusion Process with Open Boundaries and Koornwinder Polynomials

In this paper, we analyze the steady state of the asymmetric simple exclusion process with open boundaries and second class particles by deforming it through the introduction of spectral parameters. The (unnormalized) probabilities of the particle configurations get promoted to Laurent polynomials in the spectral parameters and are constructed in terms of non-symmetric Koornwinder polynomials. In particular, we show that the partition function coincides with a symmetric Macdonald–Koornwinder polynomial. As an outcome, we compute the steady current and the average density of first class particles.