A probabilistic interpretation for interpolation Macdonald polynomials

Previous work of Ayyer, Martin, and Williams gave a probabilistic interpretation of the Macdonald polynomials $P_{\lambda }(x_1,\dots ,x_n;1,t)$ at $q=1$ in terms of a Markov chain called the multispecies $t$-Push TASEP, a Markov chain involving particles of types ${\lambda }_1,\dots ,{\lambda }_n$ hopping around a ring. In particular, they showed that for each composition $\eta$ obtained by permuting the parts of $\lambda$, the stationary probability of being in state $\eta$ is proportional to the ASEP polynomial $F_{\eta }(x_1,\dots ,x_n;1,t)$, and the normalizing constant (or partition function) is $P_{\lambda }(x_1,\dots ,x_n;1,t)$. There is an inhomogeneous generalization of Macdonald polynomials due to Knop and Sahi called interpolation Macdonald polynomials $P_{\lambda }^{\ast }(x_1,\dots ,x_n;q,t)$, as well as an inhomogeneous generalization of ASEP polynomials called interpolation ASEP polynomials $F_{\eta }^{\ast }(x_1,\dots ,x_n;q,t)$ that we introduced in previous work. In this article we introduce a new Markov chain called the interpolation $t$-Push TASEP, and show that its steady state probabilities and partition function are given by the interpolation ASEP polynomials and the interpolation Macdonald polynomial, evaluated at $q=1$. This generalizes the previous result of Ayyer, Martin, and Williams.