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Multispecies TAZRP and modified Macdonald polynomials

Ayyer, A and Mandelshtam, O and Martin, J (2021) Multispecies TAZRP and modified Macdonald polynomials. In: Seminaire Lotharingien de Combinatoire (85).

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Official URL: https://doi.org/10.5802/alco.248


In this paper we prove a new combinatorial formula for the modified Macdonald polynomials Heλ(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 Pλ(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 = (x1, . . ., xn); the xi play the role of site-dependent jump rates for the mTAZRP.

Item Type: Journal Article
Publication: Seminaire Lotharingien de Combinatoire
Publisher: Universitat Wien, Fakultat fur Mathematik
Additional Information: The copyright for this article belongs to the Centre Mersenne.
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 11 Jul 2023 07:03
Last Modified: 11 Jul 2023 07:03
URI: https://eprints.iisc.ac.in/id/eprint/82430

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