Modified Macdonald polynomials and the multispecies zero-range process: I

Source abstract

In this paper we prove a new combinatorial formula for the modified Macdonald polynomials H ˜ λ ( X ; q , t ) , motivated by connections to the theory of interacting particle systems from statistical mechanics. The formula involves a new statistic called queue inversions on fillings of tableaux. This statistic is closely related to the multiline queues which were recently used to give a formula for the Macdonald polynomials P λ ( X ; q , t ) . In the case q = 1 and X = ( 1 , 1 , ⋯ , 1 ) , that formula had also been shown to compute stationary probabilities for a particle system known as the multispecies ASEP on a ring, and it is natural to ask whether a similar connection exists between the modified Macdonald polynomials and a suitable statistical mechanics model. In a sequel to this work, we demonstrate such a connection, showing that the stationary probabilities of the multispecies totally asymmetric zero-range process (mTAZRP) on a ring can be computed using tableaux formulas with the queue inversion statistic. This connection extends to arbitrary X = ( x 1 , ⋯ , x n ) ; the x i play the role of site-dependent jump rates for the mTAZRP.