Global Universality of Macdonald Plane Partitions

TL;DR

This paper investigates the scaling limits of Macdonald plane partitions with general boundary conditions, demonstrating convergence to a limit shape and Gaussian free field fluctuations, and verifies a conjecture in a special case.

Contribution

It establishes the universality of the Gaussian free field fluctuations for Macdonald plane partitions and verifies the Kenyon-Okounkov conjecture in a specific noninteracting case.

Findings

Height functions converge to a deterministic limit shape.

Global fluctuations are given by the 2D Gaussian free field.

Verification of the Kenyon-Okounkov conjecture for the volume measure case.

Abstract

We study scaling limits of periodically weighted skew plane partitions with semilocal interactions and general boundary conditions. The semilocal interactions correspond to the Macdonald symmetric functions which are $(q,t)$-deformations of the Schur symmetric functions. We show that the height functions converge to a deterministic limit shape and that the global fluctuations are given by the $2$-dimensional Gaussian free field as $q,t\to 1$ and the mesh size goes to $0$. Specializing to the noninteracting case, this verifies the Kenyon-Okounkov conjecture for the case of the $r^{\mathrm{volume}}$ measure under general boundary conditions. Our approach uses difference operators on Macdonald processes.