The inhomogeneous $t$-PushTASEP and Macdonald polynomials

TL;DR

This paper connects multispecies inhomogeneous $t$-PushTASEP on a ring with Macdonald and ASEP polynomials at $q=1$, revealing new algebraic relations, symmetries, and explicit formulas for stationary probabilities and currents.

Contribution

It introduces novel relations between ASEP and non-symmetric Macdonald polynomials at $q=1$, and links the $t$-PushTASEP stationary distribution to these polynomials.

Findings

Stationary probabilities proportional to ASEP polynomials at $q=1

Partition function given by Macdonald polynomial at $q=1

Derived symmetry properties and current formulas for the system

Abstract

We study a multispecies $t$-PushTASEP system on a finite ring of $n$ sites with site-dependent rates $x_1,\dots,x_n$. Let $\lambda=(\lambda_1,\dots,\lambda_n)$ 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 $F_{\eta}(x_1,\dots,x_n; q,t)$ at $q=1$; the normalizing constant (or partition function) is the Macdonald polynomial $P_{\lambda}(x_1,\dots,x_n;q,t)$ at $q=1$. Our approach involves new relations between the families of ASEP polynomials and of non-symmetric 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.