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

TL;DR

This paper studies the asymptotic behavior of perfect matchings on rail-yard graphs with doubly free boundaries and Jack weights, establishing a limit shape and Gaussian Free Field fluctuations, expanding the understanding of symmetric-function-deformed models.

Contribution

It introduces new identities for Macdonald polynomials and analyzes infinite-product expansions to rigorously determine limit shapes and fluctuations in Jack-weighted free boundary models.

Findings

Established the existence of a limit shape.

Proved height fluctuations converge to the Gaussian Free Field.

First rigorous analysis of Jack-weighted tiling models with free boundaries.

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.