A probabilistic interpretation for interpolation Macdonald polynomials

TL;DR

This paper introduces a new Markov chain called the interpolation t-Push TASEP, providing a probabilistic interpretation of interpolation Macdonald polynomials at q=1, extending previous work on Macdonald and ASEP polynomials.

Contribution

It develops the interpolation t-Push TASEP Markov chain and links its steady state to interpolation Macdonald polynomials, generalizing prior probabilistic interpretations.

Findings

Steady state probabilities are given by interpolation ASEP polynomials.

Partition function equals the interpolation Macdonald polynomial at q=1.

Generalizes previous probabilistic interpretations of Macdonald polynomials.

Abstract

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 $P^*_{\lambda}(x_1,\dots,x_n;q,t)$, as well as an inhomogeneous generalization of ASEP polynomials called interpolation ASEP polynomials $F^*_{\eta}(x_1,\dots,x_n;q,t)$ 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.