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- Title
Modified Macdonald polynomials and the multispecies zero range process: II.
- Authors
Ayyer, Arvind; Mandelshtam, Olya; Martin, James B.
- Abstract
In a previous part of this work, we gave a new tableau formula for the modified Macdonald polynomials H ~ λ (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 H ~ λ (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 , … , 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.
- Publication
Mathematische Zeitschrift, 2024, Vol 308, Issue 2, p1
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/s00209-024-03548-y