Toroidal Dimer Model and Temperley's Bijection

Sun, Wangru

doi:10.48550/arxiv.1603.00690

Cited by 2 publications

(3 citation statements)

References 12 publications

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“…REMARK 5. A similar result for a more general construction was proved in [25], however that approach requires embedding spanning forests (cycle-rooted spanning forests) on the torus and taking the toroidal exhaustion. Our approach bypasses this.…”

Section: Convergence To the Full-plane Smooth Phase The Kasteleyn Mat...mentioning

confidence: 55%

Local geometry of the rough-smooth interface in the two-periodic Aztec diamond

2022

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Random tilings of the two-periodic Aztec diamond contain three macroscopic regions: frozen, where the tilings are deterministic; rough, where the correlations between dominoes decay polynomially; smooth, where the correlations between dominoes decay exponentially. In a previous paper, the authors found that a certain averaging of height function differences at the rough-smooth interface converged to the extended Airy kernel point process. In this paper, we augment the local geometrical picture at this interface by introducing well-defined lattice paths which are closely related to the level lines of the height function. We show, after suitable centering and rescaling, that a point process from these paths converge to the extended Airy kernel point process provided that the natural parameter associated to the two-periodic Aztec diamond is small enough.

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Section: Convergence To the Full-plane Smooth Phase The Kasteleyn Mat...mentioning

confidence: 55%

Local geometry of the rough-smooth interface in the two-periodic Aztec diamond

2022

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“…The theory we develop below is similar to [29] but with a few important modifications, related in particular to the fact that the Temperleyan forest is typically not connected. See also [38] for a version on the torus.…”

Section: Winding and Height Functionmentioning

confidence: 99%

“…In [29], this connection was established for trees with straight line embeddings in the simply connected case. In [38], the toroidal case was treated, but with only straight line embeddings. In what follows, we need to define the height function properly not necessarily for just straight line embeddings, but any arbitrary embedding which is smooth except at the vertices.…”

Section: Winding and Height Functionmentioning

confidence: 99%

The dimer model on Riemann surfaces, I

Berestycki¹,

Laslier²,

Ray³

2019

Preprint

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We develop a framework to study the dimer model on Temperleyan graphs embedded on a Riemann surface with finitely many holes and handles. We extend Temperley's bijection to this setting and show that the dimer model can be understood in terms of an object which we call Temperleyan forests. Extending our earlier work [4] to the setup of Riemann surfaces, we show that if the Temperleyan forest has a scaling limit then the fluctuations of the height one-form of the dimer model also converge.Furthermore, if the Riemann surface is either a torus or an annulus, we show that Temperleyan forests reduce to cycle-rooted spanning forests and show convergence of the latter to a conformally invariant, universal scaling limit. This generalises a result of . As a consequence, the dimer height one-form fluctuations also converge on these surfaces, and the limit is conformally invariant. Combining our results with those of Dubédat [17], this implies that the height one-form on the torus converges to the compactified Gaussian free field, thereby settling a question in [18]. This is the first part in a series of works on the scaling limit of the dimer model on general Riemann surfaces. Contents

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Toroidal Dimer Model and Temperley's Bijection

Cited by 2 publications

References 12 publications

Local geometry of the rough-smooth interface in the two-periodic Aztec diamond

Local geometry of the rough-smooth interface in the two-periodic Aztec diamond

The dimer model on Riemann surfaces, I

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