Solution of the Dimer Problem by the Transfer Matrix Method

Lieb, Élliott H.

doi:10.1007/978-3-662-06390-3_34

Cited by 12 publications

(20 citation statements)

References 4 publications

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“…The 1-dimensional case is easy, namely e h is equal to the spectral radius ρ(A) of a certain matrix A called the "transfer matrix". There are very few 2-dimensional models where the value of h is known in closed form [2,10,23,26,27]. In all other cases there are estimates based on: (a) asymptotic expansions, e.g., [1,17,29]; (b) Monte Carlo methods, e.g., [3,20]; (c) bounds, e.g., [6,7,11,19,28,30].…”

Section: Introductionmentioning

confidence: 99%

Theory of computation of multidimensional entropy with an application to the monomer–dimer problem

Friedland

Peled

2005

Advances in Applied Mathematics

106

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Consider all colorings of a finite box in a multidimensional grid with a given number of colors subject to given local constraints. We outline the most recent theory for the computation of the exponential growth rate of the number of such colorings as a function of the dimensions of the box. As an application we compute the monomer-dimer constant for the 2-dimensional grid to 9 decimal digits, agreeing with the heuristic computations of Baxter, and for the 3-dimensional grid with an error smaller than 1.35%.  2004 Elsevier Inc. All rights reserved.

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Section: Introductionmentioning

confidence: 99%

Theory of computation of multidimensional entropy with an application to the monomer–dimer problem

Friedland

Peled

2005

Advances in Applied Mathematics

106

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“…See for example [1,2,4,5,6,12,14,15,16,17,19,20,21,23,24,25,26,28,30]. Let G = (V, E) be an undirected graph with vertices V and edges E. G can be a finite or infinite graph.…”

Section: Introductionmentioning

confidence: 99%

On the Validations of the Asymptotic Matching Conjectures

et al. 2008

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In this paper we review the asymptotic matching conjectures for rregular bipartite graphs, and their connections in estimating the monomerdimer entropies in d-dimensional integer lattice and Bethe lattices. We prove new rigorous upper and lower bounds for the monomer-dimer entropies, which support these conjectures. We describe a general construction of infinite families of r-regular tori graphs and give algorithms for computing the monomer-dimer entropy of density p, for any p ∈ [0, 1], for these graphs. Finally we use tori graphs to test the asymptotic matching conjectures for certain infinite r-regular bipartite graphs.2000 Mathematics Subject Classification: 05A15, 05A16, 05C70, 05C80, 82B20 Keywords and phrases: Matching and asymptotic growth of average matchings for r-regular bipartite graphs, monomer-dimer partitions and entropies.

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“…However, in our computations m can be any natural number up to 30, so M can have 2 30 × 2 30 = 2 60 ≈ 10 18 entries. Luckily, the matrix M is highly structured (see [24]), so there exists a faster algorithm for computing P (m,n+1) from P (m,n) .…”

Section: 2mentioning

confidence: 99%

Power series expansions for the planar monomer-dimer problem

Pogudin¹

2017

Phys. Rev. E

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Abstract. We compute the free energy of the planar monomer-dimer model. Unlike the classical planar dimer model, an exact solution is not known in this case. Even the computation of the lowdensity power series expansion requires heavy and nontrivial computations. Despite of the exponential computational complexity, we compute almost three times more terms than were previously known. Such an expansion provides both lower and upper bound for the free energy, and allows to obtain more accurate numerical values than previously possible. We expect that our methods can be applied to other similar problems.

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Solution of the Dimer Problem by the Transfer Matrix Method

Cited by 12 publications

References 4 publications

Theory of computation of multidimensional entropy with an application to the monomer–dimer problem

Theory of computation of multidimensional entropy with an application to the monomer–dimer problem

On the Validations of the Asymptotic Matching Conjectures

Power series expansions for the planar monomer-dimer problem

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