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Lozenge tilings and the Gaussian free field on a cylinder
Probability · Submitted 2 May 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.69 | 0.87 | 0.07 | asserted 0.96 |
| s1 | 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. | result 1.00 | 0.87 | 0.42 | asserted 1.00 |
| s2 | 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. | result 1.00 | 0.88 | 0.27 | asserted 1.00 |
| s3 | Fluctuations of the latter type have been previously conjectured for dimer models on planar domains with holes. | background 0.90 | 0.03 | 0.03 | asserted 0.51 |
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Original 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.
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How the literature uses it
2 arXiv papers cite this in 3 sentences
Papers citing it
Compares with it 1 paper
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A tiling model on a cylinder was studied in [ref] , and a result exactly analogous to our Theorems [ref] and [ref] below is obtained; this work provided the first computation of fluctuations in a non simply connected setup.
Temperleyan Domino Tilings with Holes · Overview · 1.00
Cites it as background 1 paper
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In recent years the same model has received some attention and its properties and generalizations have been considered in [ref] .
Identity between Restricted Cauchy Sums for the $q$-Whittaker and Skew Schur Polynomials · Background from integrable probability · 0.98
What it cites · 8 sentences without a confident label
Uses the method or tool of 1 paper
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The formulas for the joint moments were inspired by [ref] , which obtained formulas for observables for periodic Macdonald processes.
Free field theory and observables of periodic Macdonald processes · Methods · 0.83
Compares with 1 paper
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While the results of [ref] do not imply our formulas, there is agreement between the two in certain special cases.
Free field theory and observables of periodic Macdonald processes · Methods · 1.00
Cites as background 1 paper
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Let us first give a characterization which can be seen as an equivalent definition of the Green's function, see e.g. [ref] .
Lecture notes on the Gaussian Free Field · Computing the Green's function. · 0.89
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