Modified Macdonald polynomials and the multispecies zero range process: I

Abstract

In this paper we prove a new combinatorial formula for the modified Macdonald polynomials ${\tilde{H}}_{\lambda }(X;q,t)$, 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 $P_{\lambda }(X;q,t)$. In the case $q=1$ and $X=(1,1,\dots ,1)$, 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 $X=(x_1,\dots ,x_n)$; the $x_i$ play the role of site-dependent jump rates for the mTAZRP.

arXiv comment46 pages, 6 figures

References in corpus (8)

• Asymmetric Simple Exclusion Process with Open Boundaries and Askey-Wilson Polynomials
• Stationary distributions of multi-type totally asymmetric exclusion processes
• Matrix product formula for Macdonald polynomials
• Combinatorics of the two-species ASEP and Koornwinder moments
• Koornwinder polynomials and the stationary multi-species asymmetric exclusion process with open boundaries
• Algebraic construction of multi-species q-Boson system
• Integrable Structure of Multispecies Zero Range Process
• Bijective proofs of some coinversion identities related to Macdonald polynomials

Cited by in corpus (3)

• New formulas for Macdonald polynomials via the multispecies exclusion and zero range processes
• Probabilistic operators for non-attacking tableaux and a compact formula for the symmetric Macdonald polynomials
• Twisted multiline queues for the steady states of TASEP and TAZRP