Dimer-monomer model on the Sierpinski gasket

Chang, Shu Chiuan; Chen, Lung Chi

doi:10.1016/j.physa.2007.10.057

Cited by 23 publications

(8 citation statements)

References 40 publications

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“…Fractals are geometric structures of non-integer Hausdorff dimension realized by repeated construction of an elementary shape on progressively smaller length scales [11,12]. A well-known example of fractal is the Sierpinski gasket which has been extensively studied in several contexts [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]. We shall derive the recursion relations for the numbers of ice model and eight-vertex model configurations with Boltzmann factors equal to one on the two-dimensional Sierpinski gasket, and determine the entropies.…”

Section: Introductionmentioning

confidence: 99%

The ice model and the eight-vertex model on the two-dimensional Sierpinski gasket

Chang

Chen

Lee

2013

Physica A: Statistical Mechanics and its Applications

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We present the numbers of ice model and eight-vertex model configurations (with Boltzmann factors equal to one), I(n) and E(n) respectively, on the two-dimensional Sierpinski gasket SG(n) at stage n. For the eight-vertex model, the number of configurations is E(n) = 2 3(3 n +1)/2 and the entropy per site, defined as lim v→∞ ln E(n)/v where v is the number of vertices on SG(n), is exactly equal to ln 2. For the ice model, the upper and lower bounds for the entropy per site lim v→∞ ln I(n)/v are derived in terms of the results at a certain stage. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of the entropy can be evaluated with more than a hundred significant figures accurate. The corresponding result of ice model on the generalized two-dimensional Sierpinski gasket SG b (n) with b = 3 is also obtained. For the generalized vertex model on SG 3 (n), the number of configurations is 2 (8×6 n +7)/5 and the entropy per site is equal to 8 7 ln 2. The general upper and lower bounds for the entropy per site for arbitrary b are conjectured.

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Section: Introductionmentioning

confidence: 99%

The ice model and the eight-vertex model on the two-dimensional Sierpinski gasket

Chang

Chen

Lee

2013

Physica A: Statistical Mechanics and its Applications

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“…Fractals are geometric structures of (generally noninteger) Hausdorff dimension realized by repeated construction of an elementary shape on progressively smaller length scales [18,19]. A well-known example of a fractal is the Sierpinski gasket which has been extensively studied in several contexts [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic enumeration of independent sets on the Sierpinski gasket

Chang¹,

Chen²,

Yan³

2013

Filomat

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The number of independent sets is equivalent to the partition function of the hard-core lattice gas model with nearest-neighbor exclusion and unit activity. We study the number of independent sets m d,b (n) on the generalized Sierpinski gasket SG d,b (n) at stage n with dimension d equal to two, three and four for b = 2, and layer b equal to three for d = 2. The upper and lower bounds for the asymptotic growth constant, defined as z SG d,b = lim v→∞ ln m d,b (n)/v where v is the number of vertices, on these Sierpinski gaskets are derived in terms of the results at a certain stage. The numerical values of these z SG d,b are evaluated with more than a hundred significant figures accurate.We also conjecture the upper and lower bounds for the asymptotic growth constant z SG d,2 with

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“…It appears that the convergence of the lower and upper bounds of the entropy per site for dimer-monomers on T H d (n) becomes a bit faster as d increases.The present results can be compared with the entropy per site for dimer-monomers on the Sierpinski gasket (cf [27]…”

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Study of dimer–monomer on the generalized Hanoi graph

Chang

2020

Comp. Appl. Math.

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We study the number of dimer-monomers M d (n) on the Tower of Hanoi graphs T H d (n) at stage n with dimension d equal to 3 and 4. The entropy per site is defined aswhere v is the number of vertices on T H d (n). We obtain the lower and upper bounds of the entropy per site, and the convergence of these bounds approaches to zero rapidly when the calculated stage increases. The numerical value of z T H d is evaluated to more than a hundred digits correct. Using the results with d less than or equal to 4, we predict the general form of the lower and upper bounds for z T H d with arbitrary d.The dimer-monomer model is an interesting but elusive model in statistical mechanics [1][2][3]. In this model, a dimer is realized by a diatomic molecule with two neighboring sites attaching to a surface or lattice. For the sites that are not occupied by any dimers, they could be regarded as covered by monomers. Let us define N DM (G) to be the number of dimer-monomers on a graph G.The computation of the general dimer-monomer model remains to be a difficult problem [4], in contrast to the closed-packed dimer problem on planar lattices that had been discussed and solved more than fifty years ago [5][6][7]. Recent computation of close-packed dimers, dimers with a single monomer, and general dimer-monomer models on regular lattices are given in Refs. [8][9][10][11][12][13][14][15][16][17][18]. It is also interesting to discuss the dimer-monomer problem on fractals with scaling invariance but not translational invariance. The fractals with noninteger Hausdorff dimension can be constructed from certain basic shape [19,20]. A famous fractal is the Tower of Hanoi graph, and it has been discussed in different contexts [21][22][23].The dimer-monomer problem on the Tower of Hanoi graph with dimension d = 2 was discussed in [24]. In this article, we shall first recall some basic definitions in section II. In section III, we present the recursion relations for the number of dimer-monomers on T H d (n) with dimension d = 3, then enumerate the entropy per site using lower and upper bounds in details. The calculation for T H d (n) with dimension d = 4 will be given in section IV. In the last section, we shall predict the general form of the lower and upper bounds of the entropy per site for dimer-monomers on the Tower of Hanoi graph with arbitrary dimension. II. PRELIMINARIESIn this section, let us review some basic terminology. A graph G = (V, E) that is connected and has no loops is defined by the vertex (site) set V and edge (bond) set E [25,26].Denote v(G) = |V | as the number of vertices in G and e(G) = |E| as the number of edges.Two vertices a and b are neighboring if the edge ab is included in E. A matching of a graph G is an independent edge subset where the edges have no common vertices. The number of matching in G is denoted as N DM (G), which corresponds to the number of dimer-monomers in statistical mechanics. Although monomer and dimer weights can be associated to each

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Dimer-monomer model on the Sierpinski gasket

Cited by 23 publications

References 40 publications

The ice model and the eight-vertex model on the two-dimensional Sierpinski gasket

The ice model and the eight-vertex model on the two-dimensional Sierpinski gasket

Asymptotic enumeration of independent sets on the Sierpinski gasket

Study of dimer–monomer on the generalized Hanoi graph

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