Séminaire Lotharingien de Combinatoire, 89B.70 (2023), 12 pp.
Arvind Ayyer, Olya Mandelshtam and James Martin
The Multispecies Zero Range Process and Modified Macdonald Polynomials
Abstract.
In a previous part of this work (FPSAC 2021), 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 the 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 which projects to the mTAZRP, and whose stationary distribution can be expressed in terms of quinv-weighted tab\-leaux. We also obtain interesting symmetry properties of the mTAZRP probabilities under permutation of the jump rates between the sites. Finally, we give explicit formulas for particle densities and correlations of the process purely in terms of modified Macdonald polynomials.
Received: November 15, 2022.
Accepted: February 20, 2023.
Final version: April 1, 2023.
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