A probabilistic interpretation for interpolation Macdonald polynomials

Abstract

Previous work of Ayyer, Martin, and Williams gave a probabilistic interpretation of the Macdonald polynomials $P_{\hat{I}}»(x_1,\dots ,x_n;1,t)$ at $q=1$ in terms of a Markov chain called the multispecies $t$-Push TASEP, a Markov chain involving particles of types $\hat{I}{»}_1,\dots ,\hat{I}{»}_n$ hopping around a ring. In particular, they showed that for each composition $\hat{I}\cdot$ obtained by permuting the parts of $\hat{I}»$, the stationary probability of being in state $\hat{I}\cdot$ is proportional to the ASEP polynomial $F_{\hat{I}}\cdot (x_1,\dots ,x_n;1,t)$, and the normalizing constant (or partition function) is $P_{\hat{I}}»(x_1,\dots ,x_n;1,t)$. There is an inhomogeneous generalization of Macdonald polynomials due to Knop and Sahi called interpolation Macdonald polynomials $P_{\hat{I}}^{\ast }»(x_1,\dots ,x_n;q,t)$, as well as an inhomogeneous generalization of ASEP polynomials called interpolation ASEP polynomials $F_{\hat{I}}^{\ast }\cdot (x_1,\dots ,x_n;q,t)$ 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.

29 pages, 8 Figures. This paper provided one of the problems to the "First Proof" paper (arXiv:2602.05192)