The inhomogeneous $t$-PushTASEP and Macdonald polynomials at $q=1

• Published in: Annales de l'Institut Henri Poincaré. D. Combinatorics, physics and their interactions
• Main Authors: Ayyer, Arvind, Martin, James, Williams, Lauren
• Format: Journal Article
• Language: English
• Published: 11.07.2025
• ISSN: 2308-5827; 2308-5835
• DOI: 10.4171/aihpd/210

We study a multispecies t -PushTASEP system on a finite ring of n sites with site-dependent rates x_{1},\ldots,x_{n} . Let \lambda=(\lambda_{1},\ldots,\lambda_{n}) be a partition whose parts represent the species of the n particles on the ring. We show that, for each composition \eta obtained by permuting the parts of \lambda , the stationary probability of being in state \eta is proportional to the ASEP polynomial F_{\eta}(x_{1},\ldots,x_{n}; q,t) at q=1 ; the normalising constant (or partition function) is the Macdonald polynomial P_{\lambda}(x_{1},\ldots,x_{n};q,t) at q=1 . Our approach involves new relations between the families of ASEP polynomials and of nonsymmetric Macdonald polynomials at q=1 . We also use multiline diagrams , showing that a single jump of the PushTASEP system is closely related to the operation of moving from one line to the next in a multiline diagram. We derive symmetry properties for the system under permutation of its jump rates, as well as a formula for the current of a single-species system.