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

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 normalising 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 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.