Modified Macdonald polynomials and the multispecies zero range process: II

Ayyer, A and Mandelshtam, O and Martin, JB (2024) Modified Macdonald polynomials and the multispecies zero range process: II. In: Mathematische Zeitschrift, 308 (2).

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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 x1,�,xn 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. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.

Item Type:	Journal Article
Publication:	Mathematische Zeitschrift
Publisher:	Springer Science and Business Media Deutschland GmbH
Additional Information:	The copyright for this article belongs to Springer Science and Business Media Deutschland GmbH
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	15 Oct 2024 18:02
Last Modified:	15 Oct 2024 18:02
URI:	http://eprints.iisc.ac.in/id/eprint/86326

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