An exactly solvable asymmetric $K$-exclusion process

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 $q \geq 0$ are parameters. This model, which we call the $(q, t)$~$K$-ASEP, reduces to the usual ASEP on the ring when $K = 1$ and to a model studied by Sch\"utz and Sandow (\emph{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.