An exactly solvable asymmetric simple inclusion process

We study a generalization of the asymmetric simple inclusion process (ASIP) on a periodic one-dimensional lattice, where the integers in the particles rates are deformed to their $t$-analogues. We call this the $(q, t, \theta)$~ASIP, where $q$ is the asymmetric hopping parameter and $\theta$ is the diffusion parameter. We show that this process is a misanthrope process, and consequently the steady state is independent of $q$. We compute the steady state, the one-point correlation and the current in the steady state. In particular, we show that the single-site occupation probabilities follow a \emph{beta-binomial} distribution at $t=1$. We compute the two-dimensional phase diagram in various regimes of the parameters $(t, \theta)$ and perform simulations to justify the results. We also show that a modified form of the steady state weights at $t \neq 1$ satisfy curious palindromic and antipalindromic symmetries. Lastly, we define an enriched process at $t=1$ and $\theta$ an integer which projects onto the $(q, 1, \theta)$~ASIP and whose steady state is uniform, which may be of independent interest.