The inhomogeneous t-PushTASEP and Macdonald polynomials

We study a multispecies t-PushTASEP system on a finite ring of n sites with site-dependent rates x₁,,xₙ. Let λ=(λ₁,,λₙ) be a partition whose parts represent the species of the n particles on the ring. We show that for each composition η obtained by permuting the parts of λ, the stationary probability of being in state η is proportional to the ASEP polynomial Fη(x₁,,xₙ; q,t) at $q=1$; the normalizing constant (or partition function) is the Macdonald polynomial Pλ(x₁,,xₙ;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.