A probabilistic interpretation for interpolation Macdonald polynomials

TL;DR

This paper extends probabilistic interpretations of Macdonald polynomials to the interpolation setting by introducing the interpolation $t$-Push TASEP and its signed multiline queue representation. It proves that, at $q=1$, the stationary distribution of the interpolation $t$-Push TASEP on content $oldsymbol{ ho}$ is proportional to the interpolation ASEP polynomial $F^*_ u(oldsymbol{x};1,t)$ and that the partition function equals the interpolation Macdonald polynomial $P^*_oldsymbol{ ho}(oldsymbol{x};1,t)$. The authors first establish the result for partitions with distinct parts using signed multiline queues, then extend to all partitions via recoloring (lumping) and a weak reordering property, thereby linking the dynamics to $F^*_ u$ and $P^*_oldsymbol{ ho}$. They also develop density formulas and connect to $t$-interpolation Schur polynomials, providing explicit expressions for particle densities in the stationary state. The work broadens the algebraic-probabilistic bridge between inhomogeneous interpolation polynomials and interacting particle systems, with potential implications for representation theory and combinatorial probabilistic models.

Abstract

Previous work of Ayyer, Martin, and Williams gave a probabilistic interpretation of the Macdonald polynomials $P_λ(x_1,\dots,x_n;1,t)$ at $q=1$ in terms of a Markov chain called the multispecies $t$-Push TASEP, a Markov chain involving particles of types $λ_1,\dots,λ_n$ hopping around a ring. In particular, they showed that for each composition $η$ obtained by permuting the parts of $λ$, the stationary probability of being in state $η$ is proportional to the ASEP polynomial $F_η(x_1,\dots,x_n; 1,t)$, and the normalizing constant (or partition function) is $P_λ(x_1,\dots,x_n; 1,t)$. There is an inhomogeneous generalization of Macdonald polynomials due to Knop and Sahi called interpolation Macdonald polynomials $P^*_λ(x_1,\dots,x_n;q,t)$, as well as an inhomogeneous generalization of ASEP polynomials called interpolation ASEP polynomials $F^*_η(x_1,\dots,x_n;q,t)$ 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.

Figures (8)

• Figure 1: An example of the interpolation $t$-Push TASEP dynamics with $n=8$ and $\eta=(1,0,3,4,3,1,2,0)$. The figure on the left illustrates \ref{['Step0']} and \ref{['Step1']}: the bell rings at position $j=5$, then balls in positions 5, 7 and 1 and 2 are activated successively (the quantities next to the arrows give the transition rates). The figure in the middle illustrates \ref{['Step2']}: the ball at position $j=5$ is activated and travels clockwise starting from position 1. Then balls in positions 1, 2, 4<$j$ are activated successively. The figure on the right gives the final configuration.
• Figure 2: The transition graph of \ref{['Step1']} and \ref{['Step2']} of the $t$-Push$^*$ TASEP for $\lambda=(2,1,0)$ when $j=3$. The transition edges corresponding to \ref{['Step1']} (respectively \ref{['Step2']}) are represented in plain edges (respectively dashed edges).
• Figure 3: An example of a two-line queue in $\mathcal{Q}^{(4,0,2,0,0,3,0,0,4)}_{(0,4,4,0,2,3,0,1,0)}.$
• Figure 4: An example of a signed two-line queue in $\mathcal{G}_{(0,0,4,4,0,0,3,2,1)}^{(0,4,-4,0,-2,-3,0,1,0)}.$
• Figure 5: The unique paired ball system in $\widebar{\mathcal{G}}^{(2,7,1,5,0,6)}_{(0,7,2,1,5,6)}.$

Theorems & Definitions (72)

• Theorem 1.1
• Theorem 1.2: BenDaliWilliams2025
• Proposition 1.3: BenDaliWilliams2025
• Definition 1.5: The interpolation $t$-Push TASEP
• Remark 1.6
• Remark 1.7
• Example 1.8
• Remark 1.9
• Theorem 1.10
• Corollary 1.11