A variational principle for domino tilings

Cohn, Henry; Kenyon, Richard; Propp, James

doi:10.1090/s0894-0347-00-00355-6

Cited by 296 publications

(435 citation statements)

References 16 publications

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“…Note that our result for the partition function for the checkerboard-C model for the case M = N reproduces the result of Cohn, Kenyon and Proop [32]. In our notation, that result reads as…”

Section: The Checkerboard-c Modelsupporting

confidence: 85%

“…Additionally, we present the exact solutions for the dimer model on checkerboard-B and checkerboard-C lattices using the Pfaffian method. The latter solution recovers that of Cohn, Kenyon, and Propp [32] in the case M = N . Another checkerboard lattice is the so-called generalized-K model [33,34], which is characterized by two parameters x and y.…”

Section: Introductionsupporting

confidence: 82%

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Exact solution of the dimer model on the generalized finite checkerboard lattice

Измаилян

Kenna

2015

Phys. Rev. E

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We present the exact closed-form expression for the partition function of a dimer model on a generalized finite checkerboard rectangular lattice under periodic boundary conditions. We investigate three different sets of dimer weights, each with different critical behaviors. We then consider different limits for the model on the three lattices. In one limit, the model for each of the three lattices is reduced to the dimer model on a rectangular lattice, which belongs to the c = −2 universality class. In another limit, two of the lattices reduce to the anisotropic Kasteleyn model on a honeycomb lattice, the universality class of which is given by c = 1. The result that the dimer model on a generalized checkerboard rectangular lattice can manifest different critical behaviors is consistent with early studies in the thermodynamic limit and also provides insight into corrections to scaling arising from the finite-size versions of the model.

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“…Note that our result for the partition function for the checkerboard-C model for the case M = N reproduces the result of Cohn, Kenyon and Proop [32]. In our notation, that result reads as…”

Section: The Checkerboard-c Modelsupporting

confidence: 85%

Section: Introductionsupporting

confidence: 82%

Exact solution of the dimer model on the generalized finite checkerboard lattice

Измаилян

Kenna

2015

Phys. Rev. E

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“…4. Lanes 1, 4, 7, 10, and 13, no cold competitor; lanes 2,5,8,11, and 14, indicated cold competitors at 10-fold molar excess; lanes 3, 6, 9, 12, and 15, indicated cold competitors at 30-fold molar excess. Three closely migrating specific complexes are visualized in the gel shift (lane 1).…”

Section: Figmentioning

confidence: 99%

“…Lanes 1,4,7,10,13,16,19,22,25, and 28, basal binding. Lanes 2,5,8,11,14,17,20,23,26, and 29, binding in the presence of the indicated antibody. Note that the anti-USF1 antibody (lane 26) markedly diminished formation of this protein-DNA complex.…”

Section: A7r5 Aortic Vsmc Dna Binding Activities Containing C-fos C-mentioning

confidence: 99%

Osteopontin Transcription in Aortic Vascular Smooth Muscle Cells Is Controlled by Glucose-regulated Upstream Stimulatory Factor and Activator Protein-1 Activities

Bidder

Shao

Charlton-Kachigian

et al. 2002

Journal of Biological Chemistry

112

126

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“…In this paper we address this question and show that equilibrium and non-equilibrium effects in the growth of a rhombus tiling [10][11][12][13][14][15][16][17] may be distinguished using tiletile correlations of arrays simulated using a lattice gas model [18]. Direct growth to a configuration with equilibrium statistics occurs when entropic terms dominate the free energy, while non-equilibrium effects result in faceted islands and clustering of topological defects.…”

mentioning

confidence: 99%

Entropically stabilized growth of a two-dimensional random tiling

et al. 2010

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The assembly of molecular networks into structures such as random tilings and glasses has recently been demonstrated for a number of two-dimensional systems. These structures are dynamicallyarrested on experimental timescales so the critical regime in their formation is that of initial growth. Here we identify a transition from energetic to entropic stabilisation in the nucleation and growth of a molecular rhombus tiling. Calculations based on a lattice gas model show that clustering of topological defects and the formation of faceted boundaries followed by a slow relaxation to equilibrium occurs under conditions of energetic stabilisation. We also identify an entropicallystabilised regime in which the system grows directly into an equilibrium configuration without the need for further relaxation. Our results provide a methodology for identifying equilibrium and nonequilibrium randomness in the growth of molecular tilings, and we demonstrate that equilibrium spatial statistics are compatible with exponentially slow dynamical behaviour.The properties of two-dimensional supramolecular networks have been the focus of growing interest in recent years with most efforts directed towards the controlled introduction of translational order into such systems [1,2]. However there have been several recent observations of surface-bound supramolecular arrays which assemble into dynamically-arrested structures akin to glasses [3][4][5] which lack translational order. Such arrangements raise many interesting questions related to the growth of random systems [6,7]. In particular it is important to distinguish randomising effects which arise from kinetic effects, such as nucleation [8,9], from equilibrium disorder due to entropic terms in the free energy. Entropically-stabilised disorder may be regarded as intrinsic randomness, whereas kinetically-driven disorder is often determined by sample history and preparative conditions. In one recent study [3] a random molecular rhombus tiling was shown to have equilibrium (maximum entropy) spatial correlations, despite being frozen on an experimental timescale. In such a system the maximum entropy configuration must form, and be frozen in, during the initial growth, since the spatio-temporal fluctuations which normally facilitate the evolution of kinetically-trapped configurations to equilibrium are absent. However, it is not clear, a priori, that there is a set of local rules for molecular attachment which can lead to the direct growth of a 'perfect', i.e. maximum entropy, configuration.In this paper we address this question and show that equilibrium and non-equilibrium effects in the growth of a rhombus tiling [10-17] may be distinguished using tiletile correlations of arrays simulated using a lattice gas model [18]. Direct growth to a configuration with equilibrium statistics occurs when entropic terms dominate the free energy, while non-equilibrium effects result in faceted islands and clustering of topological defects.The parameters which control growth are the tile-tile interaction...

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A variational principle for domino tilings

Cited by 296 publications

References 16 publications

Exact solution of the dimer model on the generalized finite checkerboard lattice

Exact solution of the dimer model on the generalized finite checkerboard lattice

Osteopontin Transcription in Aortic Vascular Smooth Muscle Cells Is Controlled by Glucose-regulated Upstream Stimulatory Factor and Activator Protein-1 Activities

Entropically stabilized growth of a two-dimensional random tiling

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