Lozenge tilings and the Gaussian free field on a cylinder

TL;DR

This paper investigates lozenge tilings on an infinite cylinder using the periodic Schur process, demonstrating convergence to a Gaussian free field with specific fluctuation behaviors under different measures, confirming conjectures in dimer models.

Contribution

It applies the periodic Schur process to analyze the height function of lozenge tilings on a cylinder, revealing new fluctuation results and confirming the Gaussian free field behavior in this setting.

Findings

Height function converges to a deterministic shape.

Fluctuations are described by the Gaussian free field.

Additional discrete Gaussian shift appears in one variant.

Abstract

We use the periodic Schur process, introduced in arXiv:math/0601019v1, to study the random height function of lozenge tilings (equivalently, dimers) on an infinite cylinder distributed under two variants of the $q^{\operatorname{vol}}$ measure. Under the first variant, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under the second variant, corresponding to an unrestricted dimer model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured for dimer models on planar domains with holes.