Asymptotics of Height Change on Toroidal Temperleyan Dimer Models

Dubédat, Julien; Gheissari, Reza

doi:10.1007/s10955-014-1181-x

Cited by 16 publications

(20 citation statements)

References 34 publications

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“…) is closely related to the homology class of (F, F * ). This fact is already shown by authors of [DG15].…”

Section: On the Torus Temperley's Bijection Maps A Dimer Configurationsupporting

confidence: 69%

Toroidal Dimer Model and Temperley's Bijection

Sun

2016

Preprint

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Temperley's bijection relates the toroidal dimer model to cycle rooted spanning forests (CRSF ) on the torus. The height function of the dimer model and the homology class of CRSF are naturally related. When the size of the torus tends to infinity, we show that the measure on CRSF arising from the dimer model converges to a measure on (disconnected) spanning forests or spanning trees. There is a phase transition, which is determined by the average height change.

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“…) is closely related to the homology class of (F, F * ). This fact is already shown by authors of [DG15].…”

Section: On the Torus Temperley's Bijection Maps A Dimer Configurationsupporting

confidence: 69%

Toroidal Dimer Model and Temperley's Bijection

Sun

2016

Preprint

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“…Recently, Finski [14,15] obtained, by a different method, a slightly weaker version of Theorem 1.1 in the case of the square lattice quadrangulations of Riemann surfaces with Neumann boundary conditions and cone angles restricted to integer multiples of π. For other related recent work, see [17,26,27,29,10] Theorem 1.1 above is both sharper and more general than the previous results, and we propose a new, relatively short and elementary proof. The idea is similar to that used by Chinta-Jorgenson-Karlsson [5,6] and Friedli [16] who studied the square lattice Laplacians on a torus: we use an integral representation for log det ∆ Ω δ ,ϕ in terms of theta function and then break the integral into parts that we analyze separately.…”

Section: Introductionsupporting

confidence: 50%

Asymptotics of the determinant of discrete Laplacians on triangulated and quadrangulated surfaces

Izyurov¹,

Khristoforov²

2020

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Consider a surface Ω with a boundary obtained by gluing together a finite number of equilateral triangles, or squares, along their boundaries, equipped with a flat unitary vector bundle. Let Ω δ be the discretization of this surface by a bi-periodic lattice with enough symmetries, scaled to have mesh size δ. We show that the logarithm of the product of non-zero eigenvalues of the discrete Laplacian acting on the sections of the bundle is asymptotic toHere A and B are lattice-dependent constants; C is an explicit constant depending on the bundle, the angles at conical singularities and at corners of the boundary, and D is a sum of lattice-dependent contributions from singularities and a universal term that can be interpreted as a zeta-regularization of the continuum Laplacian on Ω. We allow for Dirichlet or Neumann boundary conditions, or mixtures thereof. Our proof is based on an integral formula for the determinant in terms of theta function, and the functional Central limit theorem.

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“…To our knowledge, Theorem 1.9 is the first result to establish limiting height fluctuations for an unconditioned dimer model with local edge weights on a domain topologically equivalent to an annulus (as mentioned, for a conditioned dimer model such results were shown previously in [BG18]). The only other non-simply connected domain for which we are aware of such results is the torus, on which height fluctuations for domino tilings were studied previously in [Dub15,DG15,BLR19]. In these works, the limiting height fluctuations are also given by a Gaussian free field on the torus and an additional discrete component.…”

Section: H(τ Y ) :=mentioning

confidence: 99%

Lozenge tilings and the Gaussian free field on a cylinder

Ahn,

Russkikh,

Van Peski

2021

Preprint

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We use the periodic Schur process, introduced in [Bor07], to study the random height function of lozenge tilings (equivalently, dimers) on an infinite cylinder distributed under two variants of the q 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 in [Gor21] for dimer models on planar domains with holes. Contents1. Introduction 1.1. Background 1.2. Lozenge tilings and cylindric partitions 1.3. Main results 1.4. Methods 1.5. Organization. 2. Gaussian free field and Green's function on cylinder 2.1. Gaussian free field 2.2. Green's function 3. The model 3.1. Periodic Schur processes 3.2. Lozenge tilings 3.3. q vol Measure on cylindric partitions 4. Moment formula 5. Asymptotics 6. Limit shape and fluctuations 6.1. Limit shape 6.2. Gaussian free field convergence for the unshifted model 6.3. Gaussian free field convergence for the shift-mixed model 7. Matching with conjectures on height fluctuations 7.1. Kenyon-Okounkov conjecture 7.2. Hole height fluctuations for multiply connected domains References

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Asymptotics of Height Change on Toroidal Temperleyan Dimer Models

Cited by 16 publications

References 34 publications

Toroidal Dimer Model and Temperley's Bijection

Toroidal Dimer Model and Temperley's Bijection

Asymptotics of the determinant of discrete Laplacians on triangulated and quadrangulated surfaces

Lozenge tilings and the Gaussian free field on a cylinder

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