Modified Macdonald polynomials and the multispecies zero range process: II

Abstract

In a previous part of this work, we gave a new tableau formula for the modified Macdonald polynomials \(\widetilde{H}_{\lambda }(X;q,t)\), using a weight on tableaux involving the queue inversion (quinv) statistic. In this paper we explicitly describe a connection between these combinatorial objects and a class of multispecies totally asymmetric zero range processes (mTAZRP) on a ring, with site-dependent jump-rates. We construct a Markov chain on the space of tableaux of a given shape, which projects to the mTAZRP, and whose stationary distribution can be expressed in terms of quinv-weighted tableaux. We deduce that the mTAZRP has a partition function given by the modified Macdonald polynomial \(\widetilde{H}_{\lambda }(X;1,t)\). The novelty here in comparison to previous works relating the stationary distribution of integrable systems to symmetric functions is that the variables \(x_1,\ldots ,x_n\) are explicitly present as hopping rates in the mTAZRP. We also obtain interesting symmetry properties of the mTAZRP probabilities under permutation of the jump-rates between the sites. Finally, we explore a number of interesting special cases of the mTAZRP, and give explicit formulas for particle densities and correlations of the process purely in terms of modified Macdonald polynomials.