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

Ayyer, A and Mandelshtam, O and Martin, JB (2023) Modified Macdonald polynomials and the multispecies zero-range process: I. In: Algebraic Combinatorics, 6 (1). pp. 243-284.

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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: Algebraic Combinatorics
Publisher: Centre Mersenne
Additional Information: The copyright for this article belongs to the Authors.
Keywords: modified Macdonald polynomials; tableaux; TAZRP; zero range process
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 29 May 2023 03:11
Last Modified: 29 May 2023 03:11
URI: https://eprints.iisc.ac.in/id/eprint/81510

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