A probabilistic interpretation for interpolation Macdonald polynomials

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 $λ<em>1,\dots,λ_n$ 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 $F</em>η(x_1,\dots,x_n; 1,t)$, 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 $P<sup>*_λ(x_1,\dots,x_n;q,t)$, as well as an inhomogeneous generalization of ASEP polynomials called interpolation ASEP polynomials $F<sup>*_η(x_1,\dots,x_n;q,t)$ 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.

• The paper constructs an inhomogeneous interpolation t-Push TASEP whose stationary distribution is F*μ(x;1,t)/P*λ(x;1,t), giving interpolation Macdonald polynomials a direct probabilistic meaning at q=1.
• The model adds a site-dependent “return to the bell” stage to classical t-Push TASEP dynamics, with positivity guaranteed for 0<t<1 and xᵢ>t⁻ⁿ⁺¹, while the homogeneous process emerges as xᵢ→∞.
• The authors prove the result first through multiline-queue transition formulas and then for arbitrary particle contents via lumping and polynomial reordering, obtaining explicit density formulas and identifying general-q extensions as open problems.

Overview

This paper by Ben Dali and Williams constructs a Markov chain — the interpolation $t$-Push TASEP (or $t$-Push$^*$ TASEP) — whose stationary distribution is given by interpolation ASEP polynomials $F^*_\mu(x;1,t)$, with normalizing constant equal to the interpolation Macdonald polynomial $P^*_\lambda(x;1,t)$ of Knop and Sahi. The result extends the theorem of Ayyer, Martin, and Williams connecting the multispecies $t$-Push TASEP to ordinary Macdonald polynomials at $q=1$, which itself built on the Cantini–de Gier–Wheeler identification of the multispecies ASEP partition function with $P_\lambda(x_1=\dots=x_n=q=1,t)$.

The paper thus answers a natural inhomogeneity question: can the site-dependent parameters $x_1,\dots,x_n$ of the interpolation Macdonald polynomials be incorporated into particle dynamics on a ring? The affirmative answer is achieved by augmenting the $t$-Push TASEP with a second dynamical stage ("return to the bell") governed by site-dependent probabilities $t$0 built from the $t$1.

Interpolation Macdonald and ASEP polynomials

Interpolation Macdonald polynomials $t$2 are uniquely characterized by triangularity ($t$3) and vanishing at the evaluation points $t$4 for all partitions $t$5 with $t$6; their top homogeneous component is the ordinary Macdonald polynomial. Analogously, the authors' earlier work introduced interpolation ASEP polynomials $t$7 for compositions $t$8, characterized by unit coefficient normalization $t$9 within $^*$0 and vanishing at $^*$1 for compositions outside $^*$2. These satisfy the symmetrization identity

$^*$3

mirroring the fact that ASEP polynomials sum to a Macdonald polynomial. Combinatorially, both families admit signed multiline queue formulas at general $^*$4; this paper works at $^*$5, where an unsigned paired-ball description suffices when $^*$6 has distinct parts.

The interpolation $^*$7-Push TASEP

Fix a partition $^*$8 containing at least one zero part (a nonrestrictive assumption, since shifting all parts down preserves the model). States are configurations of particles labeled by parts of $^*$9 on a ring. Each transition consists of three stages:

• Step 0: a "bell" at position $F^*_\mu(x;1,t)$0 rings with probability $F^*_\mu(x;1,t)$1, proportional to $F^*_\mu(x;1,t)$2 divided by $F^*_\mu(x;1,t)$3.
• Step 1: the classical $F^*_\mu(x;1,t)$4-Push TASEP dynamics: the activated particle displaces weaker particles successively, each displacement choosing the $F^*_\mu(x;1,t)$5th weaker particle with probability $F^*_\mu(x;1,t)$6.
• Step 2: the new ingredient — the vacancy left behind travels clockwise from position 1 back toward $F^*_\mu(x;1,t)$7, skipping or settling at sites with probabilities $F^*_\mu(x;1,t)$8 or $F^*_\mu(x;1,t)$9 depending on whether the encountered label exceeds the traveling label. This step allows displacement of stronger particles and forces termination at position $P^*_\lambda(x;1,t)$0, so particles no longer wrap around the ring multiple times.

For $P^*_\lambda(x;1,t)$1 and $P^*_\lambda(x;1,t)$2, all transition rates are genuine probabilities. Taking $P^*_\lambda(x;1,t)$3 recovers the homogeneous $P^*_\lambda(x;1,t)$4-Push TASEP: $P^*_\lambda(x;1,t)$5, $P^*_\lambda(x;1,t)$6, $P^*_\lambda(x;1,t)$7, and Step 2 becomes trivial.

Main theorem and proof strategy

The central result states that for content $P^*_\lambda(x;1,t)$8 and parameters $P^*_\lambda(x;1,t)$9,

$t$0

Consequently, via the signed multiline queue formula for $t$1, the distribution of the bottom line of signed multiline queues coincides with the stationary law of the chain.

The proof proceeds in two stages. For restricted partitions (distinct parts, one zero, no part of size 1), the argument is combinatorial: Step 1 transitions are encoded bijectively by classical two-line queues, with $t$2, while Step 2 transitions are encoded by unsigned versions of signed two-line queues, with $t$3 where $t$4 is the generating function of the (at most singleton) set of unsigned paired ball systems from $t$5 to $t$6. Combining these with the recursive decomposition $t$7 and the factorization $t$8 shows that the vector $t$9 satisfies the stationarity equations; symmetrization then yields the normalized form.

For arbitrary partitions, the key tool is lumping under weakly order-preserving recolorings $q=1$0. The authors prove that the $q=1$1-Push$q=1$2 TASEP with content $q=1$3 lumps onto that with content $q=1$4, so stationary distributions aggregate accordingly. On the algebraic side, they establish a weak reordering property at $q=1$5:

$q=1$6

proved via shape-permuting operators, the Knop–Sahi recurrence, and a dehomogenization map $q=1$7 applied to results of Alexandersson–Sawhney. Matching the lumped Markov chain stationary distribution against this polynomial identity completes the proof for all contents.

Density formulas

Specializing to two-state content $q=1$8, the stationary probability factors explicitly as a product over occupied sites involving shifted variables $q=1$9. From this, closed-form densities follow: for example,

$P_\lambda(x_1=\dots=x_n=q=1,t)$0

For general content, the density of species $P_\lambda(x_1=\dots=x_n=q=1,t)$1 at site 1 is expressed through $P_\lambda(x_1=\dots=x_n=q=1,t)$2-interpolation Schur polynomials $P_\lambda(x_1=\dots=x_n=q=1,t)$3:

$P_\lambda(x_1=\dots=x_n=q=1,t)$4

derived using a dehomogenized dual Jacobi–Trudi identity for two-column shapes, obtained from Okounkov's formula for interpolation Macdonald polynomials at $P_\lambda(x_1=\dots=x_n=q=1,t)$5 together with Macdonald's sixth variation of Schur functions. These generalize the density formulas of Ayyer–Martin–Williams for the homogeneous case.

Limitations and open questions

Several restrictions qualify the results. First, positivity of transition probabilities requires $P_\lambda(x_1=\dots=x_n=q=1,t)$6 and $P_\lambda(x_1=\dots=x_n=q=1,t)$7; outside this regime the object is a Markov kernel with signed rates rather than a stochastic process. Second, the result holds only at $P_\lambda(x_1=\dots=x_n=q=1,t)$8: unlike the homogeneous case, where the full $P_\lambda(x_1=\dots=x_n=q=1,t)$9-dependent multispecies ASEP statement follows from qKZ equations, the interpolation ASEP polynomials fail to satisfy the circular symmetry part of the qKZ system, and moreover $x_1,\dots,x_n$0 is not in general positive at $x_1,\dots,x_n$1, precluding a direct analogue of the Cantini–de Gier–Wheeler construction. The authors pose as an open problem finding an algebraic characterization of the $x_1,\dots,x_n$2 from which the qKZ equations are recovered upon taking top homogeneous components. Whether a probabilistic interpretation exists for general $x_1,\dots,x_n$3 remains open.

Conclusion

The paper supplies a stationary-distribution interpretation for the Knop–Sahi interpolation Macdonald polynomials at $x_1,\dots,x_n$4, via a two-stage ring dynamics combining push-TASEP mechanics with a site-dependent return process. The proof couples a direct multiline-queue encoding of transitions for distinct-part contents with a lumping/reordering argument reducing the general case, and yields explicit product and determinant-based density formulas. The main unresolved question is whether an analogous interpretation, or even an appropriate algebraic characterization replacing the qKZ framework, exists away from the specialization $x_1,\dots,x_n$5.