Lozenge tilings and the Gaussian free field on a cylinder

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 $q^{\mathrm{vol}}$ 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.

43 pages, 6 figures. Journal version, to appear in Communications in Mathematical Physics

References in corpus (5)

• Loop statistics in the toroidal honeycomb dimer model
• Asymptotics of height change on toroidal Temperleyan dimer models
• Height Fluctuations of Random Lozenge Tilings Through Nonintersecting Random Walks
• Bipartite dimer model: perfect t-embeddings and Lorentz-minimal surfaces
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Cited by in corpus (2)

• Identity between Restricted Cauchy Sums for the $q$-Whittaker and Skew Schur Polynomials
• Peaks of cylindric plane partitions