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Perfect Matchings on Doubly Free Boundary Rail-Yard Graph with Macdonald Weights

Zhongyang Li, Kaili Shi

TL;DR

This work analyzes Jack-weighted perfect matchings on rail-yard graphs with fully free boundary conditions, mapping dimer configurations to sequences of partitions via a particle–hole framework. Departing from determinantal techniques, the authors develop a Negut-operator–based approach and new Macdonald polynomial identities to obtain a rigorous limit shape and Gaussian fluctuations in the liquid region, with explicit frozen-boundary characterizations. The results establish the first rigorous limit shape and fluctuation theory for Jack-weighted tilings under general free boundaries, and recover the half-space Macdonald process with Jack weights as a special case. The methodology extends the asymptotic theory of symmetric-function–deformed models beyond determinantal structures, providing a robust toolbox for analyzing non-determinantal integrable probability models and their random-surface limits.

Abstract

We investigate the asymptotic behavior of perfect matchings on rail-yard graphs with doubly free boundary conditions and Jack weights. While a special case of this model reduces to the half space Macdonald process with Jack weights introduced by Barraquand, Borodin, and Corwin [3], the asymptotic behavior in the general Jack-weighted free boundary setting considered here has, to our knowledge, remained open in the literature; perhaps due to the absence of determinantal structure and the analytic complexity of boundary interactions that distinguish this setting from previously tractable cases. Our analysis is inspired by the asymptotic framework developed around the Negut operator by Gorin, Zhang, and Ahn, but it is adapted in new directions to address the challenges posed by the fully free boundary Jack-weighted regime. In particular, we establish novel identities for Macdonald polynomials and analyze infinite-product expansions not previously studied in this context. These tools enable us to rigorously establish the existence of a limit shape and to prove that the height fluctuations converge to the Gaussian Free Field (GFF) in the liquid region. These results, to the best of our knowledge, provide the first rigorous limit shape and fluctuation analysis in Jack-weighted tiling models with general free boundary conditions. In doing so, we expand the asymptotic theory of symmetric-function-deformed models beyond previously accessible, determinantal frameworks.

Perfect Matchings on Doubly Free Boundary Rail-Yard Graph with Macdonald Weights

TL;DR

This work analyzes Jack-weighted perfect matchings on rail-yard graphs with fully free boundary conditions, mapping dimer configurations to sequences of partitions via a particle–hole framework. Departing from determinantal techniques, the authors develop a Negut-operator–based approach and new Macdonald polynomial identities to obtain a rigorous limit shape and Gaussian fluctuations in the liquid region, with explicit frozen-boundary characterizations. The results establish the first rigorous limit shape and fluctuation theory for Jack-weighted tilings under general free boundaries, and recover the half-space Macdonald process with Jack weights as a special case. The methodology extends the asymptotic theory of symmetric-function–deformed models beyond determinantal structures, providing a robust toolbox for analyzing non-determinantal integrable probability models and their random-surface limits.

Abstract

We investigate the asymptotic behavior of perfect matchings on rail-yard graphs with doubly free boundary conditions and Jack weights. While a special case of this model reduces to the half space Macdonald process with Jack weights introduced by Barraquand, Borodin, and Corwin [3], the asymptotic behavior in the general Jack-weighted free boundary setting considered here has, to our knowledge, remained open in the literature; perhaps due to the absence of determinantal structure and the analytic complexity of boundary interactions that distinguish this setting from previously tractable cases. Our analysis is inspired by the asymptotic framework developed around the Negut operator by Gorin, Zhang, and Ahn, but it is adapted in new directions to address the challenges posed by the fully free boundary Jack-weighted regime. In particular, we establish novel identities for Macdonald polynomials and analyze infinite-product expansions not previously studied in this context. These tools enable us to rigorously establish the existence of a limit shape and to prove that the height fluctuations converge to the Gaussian Free Field (GFF) in the liquid region. These results, to the best of our knowledge, provide the first rigorous limit shape and fluctuation analysis in Jack-weighted tiling models with general free boundary conditions. In doing so, we expand the asymptotic theory of symmetric-function-deformed models beyond previously accessible, determinantal frameworks.

Paper Structure

This paper contains 15 sections, 50 theorems, 363 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results
  3. Weighted rail yard graphs
  4. Dimer coverings
  5. Partition Function
  6. Partitions
  7. Laplace Transform of Height Function
  8. Scaling Limits
  9. Limit Shape and Frozen Boundary
  10. Height Fluctuations and Gaussian Free Field
  11. Gaussian free field
  12. $\mathbf{w}_{+}$ as a mapping from $\mathcal{L}$ to $\mathbb{H}$
  13. Example: Half-space Macdonald Process
  14. Technical Results
  15. Duality

Key Result

Theorem 2.3

Let $M$ be a random dimer covering on the rail yard graph $RYG(l,r,\underline{a},\underline{b})$ with probability distribution given by (ppt). Let $h_M$ be the height function associated to $M$ as defined in (dhm). Suppose Assumption ap5 holds. Then the rescaled random height function $\epsilon h_M\ where the contour $\mathcal{C}$ satisfies the conditions of Proposition p57, $\mathcal{G}_{\chi}(w)

Figures (1)

  • Figure 2.1: A rail yard graph with LR sequence $\underline{a}=\{L,R,R,L\}$and sign sequence $\underline{b}=\{+,+,-,-\}$. Odd vertices are represented by red points, and even vertices are represented by blue points. Dark lines represent a pure dimer covering. Assume that above the horizontal line $y=4$, only horizontal edges with an odd vertex on the left are present in the dimer configuration; and below the horizontal line $y=-4$, only horizontal edges with an even vertex on the left are present in the dimer configuration. The corresponding sequence of partitions (from the left to the right) is given by $\emptyset\prec(2,0,\ldots)\prec' (3,1,1,\ldots)\succ'(2,0,\ldots)\succ \emptyset$.

Theorems & Definitions (92)

  • Definition 2.1
  • Theorem 2.3
  • Definition 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • ...and 82 more