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Fixed-Boundary Octagonal Random Tilings: A Combinatorial Approach

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Abstract

Some combinatorial properties of fixed boundary rhombus random tilings with octagonal symmetry are studied. A geometrical analysis of their configuration space is given as well as a description in terms of discrete dynamical systems, thus generalizing previous results on the more restricted class of codimension-one tilings. In particular this method gives access to counting formulas, which are directly related to questions of entropy in these statistical systems. Methods and tools from the field of enumerative combinatorics are used.

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REFERENCES

  1. D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53:1951 (1984).

    Google Scholar 

  2. R. Penrose, The role of aesthetics in pure and applied mathematical research, Bull. Inst. Math. Appl. 10:226 (1974).

    Google Scholar 

  3. N. Destainville, R. Mosseri, and F. Bailly, Configurational entropy of codimension-one tilings and directed membranes, J. Stat. Phys. 87(3/4):697 (1997).

    Google Scholar 

  4. N. Wang, H. Chen, and K. H. Kuo, Two-dimensional quasicrystal with eightfold rotational symmetry, Phys. Rev. Lett. 59:1010 (1987).

    Google Scholar 

  5. W. Li, H. Park, and M. Widom, Phase diagram of a random tiling quasicrystal, J. Stat. Phys. 66(1/2):1 (1992).

    Google Scholar 

  6. M. Widom, Bethe ansatz solution of the square-triangle random tiling model, Phys. Rev. Lett. 70:2094 (1993).

    Google Scholar 

  7. P. A. Kalugin, The square-triangle random-tiling model in the thermodynamic limit, J. Phys. A: Math. Gen. 27:3599 (1994).

    Google Scholar 

  8. J. de Gier and B. Nienhuis, Exact solution of an octagonal random tiling model, Phys. Rev. Lett. 76:2918 (1996).

    Google Scholar 

  9. J. de Gier and B. Nienhuis, Bethe ansatz solution of a decagonal rectangle-triangle random tiling, J. Phys. A: Math. Gen. 31:2141 (1998).

    Google Scholar 

  10. M. Widom, N. Destainville, R. Mosseri, and F. Bailly, Two-dimensional random tilings of large codimension, in Proceedings of the 6th International Conference on Quasicrystals (World Scientific, 1998).

  11. R. Mosseri and F. Bailly, Configurational entropy in octagonal tiling models, Int. J. Mod. Phys. B 7(6/7):1427 (1993).

    Google Scholar 

  12. V. Elser, Comment on“Quasicrystals: a new class of ordered structures,” Phys. Rev. Lett. 54:1730 (1985).

    Google Scholar 

  13. M. Duneau and A. Katz, Quasiperiodic patterns, Phys. Rev. Lett. 54:2688 (1985).

    Google Scholar 

  14. A. P. Kalugin, A. Y. Kitaev, and L. S. Levitov, Al0.86Mn0.14: A six-dimensional crystal, JETP Lett. 41:145 (1985); A. P. Kalugin, A. Y. Kitaev, and L. S. Levitov, 6-dimensional properties of Al0.86Mn0.14, J. Phys. Lett. France 46:L601 (1985).

    Google Scholar 

  15. N. G. de Bruijn, Algebraic theory of Penrose's non-periodic tilings of the plane, Kon. Nederl. Akad. Wetensch. Proc. Ser. A 43:84 (1981).

    Google Scholar 

  16. N. G. de Bruijn, Dualization of multigrids, J. Phys. France 47:C3-9 (1986).

    Google Scholar 

  17. J. E. S. Socolar, P. J. Steinhardt, and D. Levine, Quasicrystals with arbitrary orientational symmetry, Phys. Rev. B 32(8):5547 (1985).

    Google Scholar 

  18. F. Gähler and J. Rhyner, Equivalence of the generalized grid and projection methods for the construction of quasiperiodic tilings, J. Phys. A: Math. Gen. 19:267 (1986).

    Google Scholar 

  19. V. Elser, Solution of the dimer problem on an hexagonal lattice with boundary, J. Phys. A: Math. Gen. 17:1509 (1984).

    Google Scholar 

  20. R. Mosseri, F. Bailly, and C. Sire, Configurational entropy in random tiling models, J. Non-Cryst. Solids 153-54:201 (1993).

    Google Scholar 

  21. N. Destainville, Ph.D. Thesis: “Entropie configurationnelle des pavages aléatoires et des membranes dirigées,” Thése de l'Université Paris 6 (1997).

  22. N. Destainville, Entropy and boundary conditions in random rhombus tilings, J. Phys. A: Math. Gen. 31:6123 (1998).

    Google Scholar 

  23. G. D. Bailey, Tilings of Zonotopes: Discriminental Arrangements, Oriented Matroids and Enumeration (Minnesota University Thesis, 1997).

  24. H. Cohn, M. Larsen, and J. Propp, The shape of a typical boxed plane partition, New York J. of Math. 4:137 (1998).

    Google Scholar 

  25. H. Cohn, R. Kenyon, and J. Propp, A variational principle for domino tilings, J. of AMS. (to appear).

  26. M. Latapy, Generalized integer partitions, tilings of zonotopes and lattices, preprint.

  27. R. Kenyon, Tilings of polygons with parallelograms, Algorithmica 9, 382 (1993).

    Google Scholar 

  28. S. Elnitsky, Rhombic tilings of polygons and classes of reduced words in Coxeter groups, J. Combinatorial Theory A 77:193 (1997).

    Google Scholar 

  29. R. P. Stanley, Ordered structures and partitions, Memoirs of the AMS 119 (1972).

  30. V. Strehl, Combinatorics of special functions: facets of Brock's identity, in Séries Formelles et Combinatoire Algébrique, P. Leroux and C. Reutanauer, eds. (University of Québec, Montreal, 1992).

    Google Scholar 

  31. C. L. Henley, Relaxation time for a dimer covering with height representation, J. Stat. Phys. 89:483 (1997).

    Google Scholar 

  32. D. Randall and P. Tetali, Analyzing Glauber dynamics by comparison of Markov chains, in Proceedings of the 3rd Latin American Theoretical Informatics Symposium, Springer Lecture Notes in Computer Science, Vol. 1380, p. 292 (1998).

  33. M. Luby, D. Randall, and A. Sinclair, Markov chain algorithms for planar lattice structures, preprint.

  34. D. B. Wilson, Mixing times of lozenge tiling and card shuffling Markov chains, preprint.

  35. R. Mosseri and J.-F. Sadoc, Glass-like properties in quasicrystals, in Proceedings of the 5th International Conference on Quasicrystals, C. Janot and R. Mosseri, eds. (World Scientific, 1995), p. 747.

  36. L. Leuzzi and G. Parisi, A Tiling Model for Glassy Systems, preprint (cond-mat/9911020).

  37. K. J. Strandburg and P. R. Dressel, Thermodynamic behavior of a Penrose-tiling quasicrystal, Phys. Rev. B 41:2469 (1990).

    Google Scholar 

  38. Y. Ishii, Dynamics of phason relaxation on Penrose lattices, in Proceedings of the 5th International Conference on Quasicrystals, C. Janot and R. Mosseri, eds. (World Scientific, 1995), p. 359.

  39. R. P. Stanley, On the number of reduced decompositions of elements of Coxeter groups, European J. of Comb. 5:359 (1984).

    Google Scholar 

  40. P. Edelman and C. Greene, Balanced tableaux, Adv. in Math. 63:42 (1987).

    Google Scholar 

  41. B. E. Sagan, The Symmetric Group (Wadsworth and Brooks, California, 1991).

    Google Scholar 

  42. D. M. Knuth, Axioms and hulls, Lect. Notes in Computer Sci. 606:35 (1992).

    Google Scholar 

  43. A. Young, On quantitative substitutional analysis, Proc. London Math. Soc. 2 28:255 (1927).

    Google Scholar 

  44. C. Greene, A. Nijenhuis, and H. S. Wilf, A probabilistic proof of a formula for the number of Young tableaux of a given shape, Adv. in Math. 31:104 (1979).

    Google Scholar 

  45. I. Gessel and G. Viennot, Binomial determinants, paths and hook lenght formulae, Adv. in Math. 58:300 (1985).

    Google Scholar 

  46. A. Björner, M. Las Vergnas, B. Sturmfels, N. White, and G. M. Ziegler, Oriented Matroids (Cambridge University Press, 1993).

  47. B. Sturmfels and G. M. Ziegler, Extention spaces of oriented matroids, Discrete & Computational Geom. 10:23 (1993).

    Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Physique Quantique—IRSAMC—UMR 5626, Université Paul Sabatier, 118, route de Narbonne, 31062, Toulouse Cedex 04, France

    N. Destainville

  2. Groupe de Physique du Solide, Tour 23, 5ième étage, Universités Paris 7 et 6, 2, place Jussieu, 75251, Paris Cedex 05, France

    R. Mosseri

  3. Laboratoire de Physique du Solide—CNRS, 1, place Aristide Briand, 92195, Meudon Cedex, France

    F. Bailly

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  1. N. Destainville
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  2. R. Mosseri
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  3. F. Bailly
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Destainville, N., Mosseri, R. & Bailly, F. Fixed-Boundary Octagonal Random Tilings: A Combinatorial Approach. Journal of Statistical Physics 102, 147–190 (2001). https://doi.org/10.1023/A:1026564710037

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