An exactly solvable asymmetric simple inclusion process

Abstract

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 $(q,t,\hat{I}¸)$~ASIP, where $q$ is the asymmetric hopping parameter and $\hat{I}¸$ 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 $(t,\hat{I}¸)$ and perform simulations to justify the results. We also show that a modified form of the steady state weights at $t\ne 1$ satisfy curious palindromic and antipalindromic symmetries. Lastly, we define an enriched process at $t=1$ and $\hat{I}¸$ an integer which projects onto the $(q,1,\hat{I}¸)$~ASIP and whose steady state is uniform, which may be of independent interest.

30 pages, 10 figures, many changes in response to referee comments, final version

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