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Dimer problem on the cylinder and torus

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  • Yan, Weigen
  • Yeh, Yeong-Nan
  • Zhang, Fuji

Abstract

We obtain explicit expressions of the number of close-packed dimers and entropy for three types of lattices (the so-called 8.8.6, 8.8.4, and hexagonal lattices) with cylindrical boundary condition and the entropy of the 8.8.6 lattice with toroidal boundary condition. Our results and the one on 8.8.4 and hexagonal lattices with toroidal boundary condition by Salinas and Nagle [S.R. Salinas, J.F. Nagle, Theory of the phase transition in the layered hydrogen-bonded SnCl2⋅2H2O crystal, Phys. Rev. B 9 (1974) 4920–4931] and Wu [F.Y. Wu, Dimers on two-dimensional lattices, Inter. J. Modern Phys. B 20 (2006) 5357–5371] imply that the 8.8.6 (or 8.8.4) lattices with cylindrical and toroidal boundary conditions have the same entropy whereas the hexagonal lattices have not. Based on these facts we propose the following problem: under which conditions do the lattices with cylindrical and toroidal boundary conditions have the same entropy?

Suggested Citation

  • Yan, Weigen & Yeh, Yeong-Nan & Zhang, Fuji, 2008. "Dimer problem on the cylinder and torus," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(24), pages 6069-6078.
    • Handle: RePEc:eee:phsmap:v:387:y:2008:i:24:p:6069-6078
    DOI: 10.1016/j.physa.2008.06.042
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    References listed on IDEAS

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    1. Wu, F.Y. & Wang, Fa, 2008. "Dimers on the kagome lattice I: Finite lattices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(16), pages 4148-4156.
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    Cited by:

    1. Yan, Weigen & Zhang, Zuhe, 2009. "Asymptotic energy of lattices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(8), pages 1463-1471.
    2. Marčetić, Dušanka & Elezović-Hadžić, Sunčica & Živić, Ivan, 2020. "Statistics of close-packed dimers on fractal lattices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 554(C).
    3. Feng, Xing & Zhang, Lianzhu & Zhang, Mingzu, 2018. "Enumeration of perfect matchings of lattice graphs by Pfaffians," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 412-420.
    4. Lu, Fuliang & Zhang, Lianzhu & Lin, Fenggen, 2011. "Dimer statistics on the Klein bottle," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(12), pages 2315-2324.
    5. Lin, Fenggen & Chen, Ailian & Lai, Jiangzhou, 2016. "Dimer problem for some three dimensional lattice graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 443(C), pages 347-354.
    6. Li, Wei & Zhang, Heping, 2012. "Dimer statistics of honeycomb lattices on Klein bottle, Möbius strip and cylinder," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(15), pages 3833-3848.
    7. Li, Shuli & Yan, Weigen, 2016. "Dimers on the 33.42 lattice," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 452(C), pages 251-257.

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