An exactly solvable asymmetric K-exclusion process

Abstract

We study an interacting particle process on a finite ring with L sites with at most K particles per site, in which particles hop to nearest neighbors with rates given in terms of t-deformed integers and asymmetry parameter q, where t > 0 and q⩾0 are parameters. This model, which we call the (q, t) asymmetric simple K-exclusion process (ASEP), reduces to the usual ASEP on the ring when K = 1 and to a model studied by Schütz and Sandow (Phys. Rev. E, 1994) when t=q=1 . This is a special case of the misanthrope process and as a consequence, the steady state does not depend on q and is of product form, generalizing the same phenomena for the ASEP. What is interesting here is the steady state weights are given by explicit formulas involving t-binomial coefficients, and are palindromic polynomials in t. Interestingly, although the (q, t) K-ASEP does not satisfy particle-hole symmetry, its steady state does. We analyze the density and calculate the most probable number of particles at a site in the steady state in various regimes of t. Lastly, we construct a two-dimensional exclusion process on a discrete cylinder with height K and circumference L which projects to the (q, t) K-ASEP and whose steady state distribution is also of product form. We believe this model will serve as an illustrative example in constructing two-dimensional analogues of misanthrope processes.Simulations are attached as ancillary files.