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Probability Theory and Related Fields

, Volume 130, Issue 3, pp 289–318 | Cite as

Critical resonance in the non-intersecting lattice path model

  • Richard W. Kenyon
  • David B. Wilson
Article

Abstract.

We study the phase transition in the honeycomb dimer model (equivalently, monotone non-intersecting lattice path model). At the critical point the system has a strong long-range dependence; in particular, periodic boundary conditions give rise to a “resonance” phenomenon, where the partition function and other properties of the system depend sensitively on the shape of the domain.

Keywords

Boundary Condition Phase Transition Partition Function Periodic Boundary Periodic Boundary Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Bateman, H., Erdélyi, A., et al.: Higher Transcendental Functions. Volume 1, McGraw-Hill, 1953. Based in part on notes left by Bateman, edited by Erdélyi et alGoogle Scholar
  2. 2.
    Blöte, H.W.J., Hilhorst, H.J.: Roughening transitions and the zero-temperature triangular Ising antiferromagnet. J. Phys. A 15, L631–L637 (1982)Google Scholar
  3. 3.
    Borgs, C., Chayes, J.T., King, C.: Meissner phase for a model of oriented flux lines. J. Phys. A 28, 6483–6499 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Cerf, R., Kenyon, R.: The low-temperature expansion of the Wulff crystal in the 3D Ising model. Communications in Mathematical Physics 222, 147–179 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Chang, J.T., Peres, Y.: Ladder heights, Gaussian random walks and the Riemann zeta function. Annals of Probability 25, 787–802 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Cohn, H., Kenyon, R., Propp, J.: A variational principle for domino tilings. Journal of the American Mathematical Society 14, 297–346 (2001), arXiv:math.CO/0008220CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Comtet, L.: Advanced Combinatorics. D. Reidel Publishing Co., 1974Google Scholar
  8. 8.
    den Nijs, M.: The domain wall theory of two-dimensional commensurate-incommensurate phase transitions. In: C.Domb and J.L. Lebowitz, (eds.), Phase Transitions and Critical Phenomena, Volume 12, Academic Press, 1988, pp. 219–333Google Scholar
  9. 9.
    Dingle, R.B.: On the Bose-Einstein integrals Open image in new window Applied Scientific Research 6, 240–244 (1957)Google Scholar
  10. 10.
    Dingle, R.B.: On the Fermi-Dirac integrals Open image in new window Applied Scientific Research 6, 225–239 (1957)Google Scholar
  11. 11.
    Fisher., M.E.: Walks, walls, wetting and melting. Journal of Statistical Physics 34, 667–729 (1984)Google Scholar
  12. 12.
    Galluccio, A., Loebl, M.: On the theory of Pfaffian orientations. I. Perfect matchings and permanents. Electronic Journal of Combinatorics 6, #R6, (1999)Google Scholar
  13. 13.
    Huang, H.Y., Wu, F.Y., Kunz, H., Kim, D.: Interacting dimers on the honeycomb lattice: An exact solution of the five-vertex model. Physica A 228, 1–32 (1996), arXiv:cond-mat/9510161MathSciNetzbMATHGoogle Scholar
  14. 14.
    Kasteleyn, P.W.: Graph theory and crystal physics. In: Frank Harary, (ed.), Graph Theory and Theoretical Physics. Academic Press, 1967Google Scholar
  15. 15.
    Kenyon, R.: Local statistics of lattice dimers. Annales de l’Institut Henri Poincaré – Probabilités et Statistiques 33, 591–618 (1997), arXiv:math.CO/0105054CrossRefzbMATHGoogle Scholar
  16. 16.
    Lewin, L.: Polylogarithms and Associated Functions. North Holland, 1981Google Scholar
  17. 17.
    Lindelöf, E.: Le Calcul des Résidus et ses Applications a la Théorie des Fonctions. Gauthier-Villars, 1905Google Scholar
  18. 18.
    Lu, W.T., Wu, F.Y.: Dimer statistics on the Möbius strip and the Klein bottle. Physics Letters A 259, 108–114 (1999), arXiv:cond-mat/9906154CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Lu, W.T., Wu, F.Y.: Close-packed dimers on nonorientable surfaces (2001). arXiv:cond-mat/0110035Google Scholar
  20. 20.
    Nagle, J.F.: Theory of biomembrane phase transitions. Journal of Chemical Physics 58, 252–264, (1973)CrossRefGoogle Scholar
  21. 21.
    Pickard, W.F.: On polylogarithms. Publicationes Mathematicae 15, 33–43 (1968)zbMATHGoogle Scholar
  22. 22.
    Popkov, V., Kim, D., Huang, H.Y., Wu, F.Y.: Lattice statistics in three dimensions: solution of layered dimer and layered domain wall models. Physical Review E 56, 3999–4008 (1997). arXiv:cond-mat/9703065CrossRefMathSciNetGoogle Scholar
  23. 23.
    Prähofer, M., Spohn, H.: An exactly solved model of three-dimensional surface growth in the anisotropic KPZ regime. Journal of Statistical Physics 88, 999–1012 (1997). arXiv:cond-mat/9612209MathSciNetGoogle Scholar
  24. 24.
    Regge, T., Zecchina, R.: Combinatorial and topological approach to the 3D Ising model. Journal of Physics A 33, 741–761 (2000). arXiv:cond-mat/9909168CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Sheffield, S.: Gibbs measure uniqueness results for Lipschitz random surfaces, (2001). In preparationGoogle Scholar
  26. 26.
    Tabachnikov, S.: Billiards. Panoramas et Synthèses. Société Mathématique de France, 1995Google Scholar
  27. 27.
    Tesler, G.: Matchings in graphs on non-orientable surfaces. Journal of Combinatorial Theory, Series B 78, 198–231 (2000)Google Scholar
  28. 28.
    Truesdell, C.: On a function which occurs in the theory of the structure of polymers. The Annals of Mathematics 46, 144–157 (1945)zbMATHGoogle Scholar
  29. 29.
    Wu, F.Y.: Exactly soluable model of the ferroelectric phase transition in two dimensions. Physical Review Letters 18, 605–607 (1967)CrossRefGoogle Scholar
  30. 30.
    Wu, F.Y.: Remarks on the modified potassium dihydrogen phosphate model of a ferroelectric. Physical Review 168, 539–543 (1968)CrossRefGoogle Scholar
  31. 31.
    Wu, F.Y., Huang, H.Y.: Exact solution of a vertex model in d dimensions. Letters in Mathematical Physics 29, 205–213 (1993)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Wu, F.Y., Huang, H.Y.: Exact solution of a lattice model of flux lines in superconductors. Physika A 205, 31–40 (1994)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.CNRS UMR 8628, Laboratoire de MathématiquesUniversité Paris-SudFrance
  2. 2.Microsoft ResearchOne Microsoft WayRedmondUSA

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