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

Abstract

We study a multispecies -PushTASEP system on a finite ring of n sites with site-dependent rates x₁,…,x_n. Let λ = (λ₁,…,λ_n) 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 F_η(x₁,…,x_n; q,t) at q=1; the normalising constant (or partition function) is the Macdonald polynomial P_λ( x₁,…,x _n; q,t) at q=1 . Our approach involves new relations between the families of ASEP polynomials and of nonsymmetric Macdonald polynomials at . 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.