q−Orthogonal dualities for asymmetric particle systems

Abstract

We study a class of interacting particle systems with asymmetric interaction showing a self-duality property. The class includes the ASEP($q,\mathit{\theta }$), asymmetric exclusion process, with a repulsive interaction, allowing up to $\mathit{\theta }\in \mathbb{N}$ particles in each site, and the ASIP$(q,\mathit{\theta })$, $\mathit{\theta }\in {\mathbb{R}}^{+}$, asymmetric inclusion process, that is its attractive counterpart. We extend to the asymmetric setting the investigation of orthogonal duality properties done in [8] for symmetric processes. The analysis leads to multivariate $q-$analogues of Krawtchouk polynomials and Meixner polynomials as orthogonal duality functions for the generalized asymmetric exclusion process and its asymmetric inclusion version, respectively. We also show how the q-Krawtchouk orthogonality relations can be used to compute exponential moments and correlations of ASEP($q,\mathit{\theta }$).