Skip to main content
SpringerLink
Log in

Functional relations in lattice statistical mechanics, enumerative combinatorics, and discrete dynamical systems

  • Published:
Annals of Combinatorics Aims and scope Submit manuscript

Abstract

We recall some non-trivial, non-linear functional relations appearing in various domains of mathematics and physics, such as lattice statistical mechanics, quantum mechanics, or enumerative combinatorics. We focus, more particularly, on the analyticity properties of the solutions of these functional relations. We then consider discrete dynamical systems corresponding to birational transformations. The rational expressions for dynamical zeta functions obtained for a particular two-dimensional birational mapping, depending on two parameters, are recalled, as well as some non-trivial functional relations satisfied by these dynamical zeta functions. We finally give some functional equations corresponding to some singled out orbits of this two-dimensional birational mapping for particular values of the two parameters. This example shows that functional equations associated with curves, for real values of the variables, are actually compatible with a chaotic dynamical system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. Abarenkova, J.-C. Anglès d'Auriac, S. Boukraa, S. Hassani, and J.-M. Maillard, Rational dynamical zeta functions for birational transformations, chao-dyn/9807014.

  2. N. Abarenkova, J.-C. Anglès d'Auriac, S. Boukraa, S. Hassani, and J.-M. Maillard, Topological entropy and complexity for discrete dynamical systems, chao-dyn/9806026.

  3. N. Abarenkova, J.-C. Anglès d'Auriac, S. Boukraa and J.-M. Maillard, Complexity spectrum of some discrete dynamical systems, chao-dyn/9807031.

  4. N. Abarenkova, J.-C. Anglès d'Auriac, S. Boukraa, S. Hassani, and J.-M. Maillard, From Yang-Baxter equations to dynamical zeta functions for birational transformations, to appear.

  5. R. Abraham and J.E. Mardsen, Foundations of Mathematics, Addison-Wesley, 1978.

  6. R.C. Adler, A.C. Konheim, and M.H. McAndrew, Topological entropy, Trans. Amer. Math. Soc.114 (1965) 309.

    Google Scholar 

  7. K.T. Alligood, T.D. Sauer, and J.A. Yorke, Chaos: An Introduction to Dynamical Systems, Springer-Verlag, New York, 1997.

    Google Scholar 

  8. G.E. Andrews, On the proofs of the Rogers-Ramanujan identities, In:q-Series and Partitions, D.W. Stanton, Ed., The IMA Volumes in Mathematics and its Applications, Vol. 18, Springer-Verlag, 1989, pp. 1–14.

  9. G.E. Andrews, R.J. Baxter, and P.J. Forrester, Eight vertex SOS model and generalized Rogers-Ramanujan-Type identities, J. Stat. Phys.35 (1984) 193.

    Google Scholar 

  10. anonymous@crtbt.polycnrs-gre.fr

  11. V. Arnold, Problems on singularities and dynamical systems, In: Developments in Mathematics, The Moscow School, V. Arnold and M. Monastyrsky, Eds., Chapman & Hall, 1989, pp. 261–274.

  12. V.I. Arnold and A. Avez, Problèmes Ergodiques de la Mécanique Classique, Gauthier-Villars, Paris, 1967.

    Google Scholar 

  13. M. Artin and B. Mazur, On periodic points, Ann. Math.81 (1965) 82.

    Google Scholar 

  14. V. Baladi, Periodic orbits and dynamical spectra, Ergod. Theor. and Dynam. Sys.18 (1998) 255–292.

    Google Scholar 

  15. R.J. Baxter, The inversion relation method for some two-dimensional exactly solved models in lattice statistics, J. Stat. Phys.28 (1982) 1–41.

    Google Scholar 

  16. R.J. Baxter, Proc. of the 1980 Enschede Summer School: Fundamental Problems in Statistical Mechanics V, E.G.D. Cohen, Ed., North-Holland, Amsterdam, 1981.

    Google Scholar 

  17. R.J. Baxter, Partition function of the eight-vertex lattice model, Ann. Phys.70 (1972) 193.

    Google Scholar 

  18. R.J. Baxter, On Zamolodchikov's solution of the Tetrahedron equations, Commun. Math. Phys.88 (1983) 185–205.

    Google Scholar 

  19. R.J. Baxter and Enting, 399th solution of the Ising model, J. Phys. A11 (1978) 2463–2473.

    Google Scholar 

  20. R.J. Baxter and P.A. Pearce, Hard hexagon model, J. Phys. A15 (1982) 897–910.

    Google Scholar 

  21. V.V. Bazhanov, Inversion and symmetry relations for a three-dimensional model, Int. J. Mod. Phys B7 (1993) 3501–3515.

    Google Scholar 

  22. V.V. Bazhanov and R.J. Baxter, Star-Triangle relation for a three-dimensional model, J. Stat. Phys. 71 (1993) 839–963.

    Google Scholar 

  23. M.P. Bellon, J.-M. Maillard, and C.M. Viallet, Quasi integrability of the sixteen-vertex model, Phys. Lett. B281 (1992) 315–319.

    Google Scholar 

  24. M.P. Bellon, J.-M. Maillard, and C.M. Viallet, Integrable Coxeter groups, Phys. Lett. A159 (1991) 221–232.

    Google Scholar 

  25. M.P. Bellon, J.-M. Maillard, and C.M. Viallet, Higher dimensional mappings, Phys. Lett. A159 (1991) 233–244.

    Google Scholar 

  26. M.P. Bellon, J.-M. Maillard, and C.M. Viallet, Infinite discrete symmetry group for the Yang-Baxter equations: Spin models, Phys. Lett. A157 (1991) 343–353.

    Google Scholar 

  27. M.P. Bellon, J.-M. Maillard, and C.M. Viallet, Infinite discrete symmetry group for the Yang-Baxter equations: Vertex models, Phys. Lett. B260 (1991) 87–100.

    Google Scholar 

  28. M.P. Bellon, J.-M. Maillard, and C.M. Viallet, Rational mappings, arborescent iterations, and the symmetries of integrability, Phys. Rev. Lett.67 (1991) 1373–1376.

    Google Scholar 

  29. A. Berkovich, B.M. Mc Coy, and A. Schilling, Rogers-Schur-Ramanujan type identities for theM (p, p′) minimal models of conformal field theory, Commun. Math. Phys.191 (1998) 325–395.

    Google Scholar 

  30. S. Boukraa, S. Hassani, and J.-M. Maillard, New integrable cases of a Cremona transformation: A finite order orbit analysis, Physica A240 (1997) 586.

    Google Scholar 

  31. S. Boukraa and J.-M. Maillard, Factorization properties of birational mappings, Physica A220 (1995) 403.

    Google Scholar 

  32. S. Boukraa, J.-M. Maillard, and G. Rollet, Discrete symmetry groups of vertex models in statistical mechanics, J. Stat. Phys.78 (1995) 1195.

    Google Scholar 

  33. S. Boukraa, J.-M. Maillard, and G. Rollet, Integrable mappings and polynomial growth, Physica A208 (1994) 115–175.

    Google Scholar 

  34. S. Boukraa, J.-M. Maillard, and G. Rollet, Determinantal identities on integrable mappings, Int. J. Mod. Phys. B8 (1994) 2157–2201.

    Google Scholar 

  35. S. Boukraa, J.-M. Maillard, and G. Rollet, Almost integrable mappings, Int. J. Mod. Phys. B8 (1994) 137.

    Google Scholar 

  36. R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math.95 (1973) 429.

    Google Scholar 

  37. R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin, 1975.

    Google Scholar 

  38. B.L.J. Braaksma, G.K. Immimk, and E.M. van der Put, The Stokes Phenomenon and Hilbert's 16th Problem, World Scientific, 1996.

  39. A. Cayley, A theorem on trees, Quar. J. Pure Appl. Math.23 (1889) 376–378.

    Google Scholar 

  40. J. Chazy, Sur les équations différentielles dont l'intégrale est uniforme et admet des singularités essentielles mobiles, Comptes Rendus de l'Académie des Sciences149 (1909) 563–565.

    Google Scholar 

  41. J. Chazy, Sur les équations différentielles don l'intégrale générale possède une coupure essentielle mobile, Comptes Rendus de l'Académie des Sciences150 (1910) 456–458.

    Google Scholar 

  42. D.J. Collins, R.I. Grigorchuk, P.F. Kurchanov, and H. Zieschang, Combinatorial Group Theory and Applications to Geometry, Springer-Verlag, 1998.

  43. C. Conley and R. Easton, Isolated invariant sets and isolated blocks, Trans. Amer. Math. Soc.158 (1971) 35.

    Google Scholar 

  44. P. Cvitanović, Universality in Chaos, Adam Hilger Publishing, Bristol, New York, 1989.

    Google Scholar 

  45. P. Cvitanović, et al., Classical and Quantum Chaos, Niels Bohr Institute, Copenhagen, 1998; http://www.nbi.dk/ChaosBook/.

    Google Scholar 

  46. R.W. Easton, Geometrical methods for discrete dynamical systems, Oxford Engineering Science Series50 (1998).

  47. J. Ecalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Hermann, 1992; Collection Actualités Mathématiques.

  48. D. Fried, Rationality for isolated expansive sets, Adv. Math.65 (1987) 35.

    Google Scholar 

  49. A. Gaaf and J. Hijmans, Symmetry relations in the sixteen-vertex model, Physica A80 (1975) 149–171.

    Google Scholar 

  50. R.W. Ghrist, P.J. Holmes, and M.C. Sullivan, Knots and Links in Three-Dimensional Flows, Lecture Notes in Mathematics, Vol. 1654, Springer-Verlag, 1998.

  51. B. Grammaticos, F.W. Nijhoff, and A. Ramani, Discrete Painlevé Equations, Lectures presented at the Cargèse School: The Painlevé Property, One Century Later, R. Conte and M. Musette, Eds., Springer-Verlag, to be published.

  52. J. Guckenheimer, Axiom A+no cycles implies ζ j (t) is rational, Bull. Amer. Math. Soc.76 (1970) 592.

    Google Scholar 

  53. A.J. Guttmann and I.G. Enting, Solvability of some statistical mechanical systems, Phys. Rev. Lett.76 (1996) 344.

    Google Scholar 

  54. D. Hansel, J.-M. Maillard, J. Oitmaa, and M. Velgakis, Analytical properties of the anisotropic cubic Ising model, J. Stat. Phys.48 (1987) 69.

    Google Scholar 

  55. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, 1977.

  56. M.T. Jaekel and J.-M. Maillard, Inverse functional relations on the Potts model, J. Phys. A15 (1982) 2241.

    Google Scholar 

  57. M.T. Jaekel and J.-M. Maillard, Largeq expansion on the anisotropic Potts model, J. Phys. A16 (1983) 1975.

    Google Scholar 

  58. M.T. Jaekel and J.-M. Maillard, Inversion relations and disorder solutions on Potts models, J. Phys. A17 (1984) 2079.

    Google Scholar 

  59. M.T. Jaekel and J.-M. Maillard, Symmetry relations in exactly soluble models, J. Phys. A15 (1982) 1309–1325.

    Google Scholar 

  60. P. Kaliappan and M. Lakshmanan, J. Phys. A12 (1979) L. 249.

  61. E.H. Lieb and F.Y. Wu, Two-Dimensional Ferroelectric Models, In: Exact Results, Phase Transitions and Critical Phenomena, Vol. 1, C. Domb and M.S. Green, Eds., Academic Press, New York, 1972, pp. 331–490.

    Google Scholar 

  62. J.-M. Maillard, The inversion relation: Some simple examples (I) and (II), J. de Physique46 (1984) 329.

    Google Scholar 

  63. A. Manning, Axiom A diffeomorphisms have rational zeta functions, Bull. London. Math. Soc.3 (1971) 215.

    Google Scholar 

  64. B.M. McCoy, C.A. Tracy, and T.T. Wu, J. Math. Phys.18 (1977) 1068–1092.

    Google Scholar 

  65. J. Moser and A.P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Commun. Math. Phys.139 (1991) 217.

    Google Scholar 

  66. D. Mumford, Tata Lectures on Theta II.; Tata Lectures on Theta III., Progress in Mathematics, Birkhäuser, 1991.

  67. F.W. Nijhoff, Similarity reduction on the lattice and discrete Painlevé equations, In: Proceedings of the 1996 Workshop on the Theory of Nonlinear Special Functions: The Painlevé Transcendents, CRM, Montréal, P. Winternitz, Ed., Springer-Verlag, to be published.

  68. F.W. Nijhoff, Discrete Painlevé equations and symmetry reduction on the lattice, In: Discrete Integrable Geometry and Physics, A.I. Bobenko and R. Seiler, Eds., Oxford University Press, to be published.

  69. F.W. Nijhoff and V.G. Papageorgiou, Similarity reductions of integrable lattices and discrete analogues of the Painlevé II equation, Phys. Lett. A153 (1991) 337–344.

    Google Scholar 

  70. P. Painlevé, Comptes Rendus de l'Académie des Sciences143 (1926) 1111.

    Google Scholar 

  71. H. Poincaré, Mémoires sur les fonctions Fuchsiennes, Acta Math.1 (1882) 193–294; Oeuvres, Gauthier-Villars Vol. 2, 1916, pp. 167–257.

    Google Scholar 

  72. G. Polya and R.C. Read, Combinatorial Enumeration of Groups, Graphs and Chemical Compounds, Springer-Verlag, 1987.

  73. G. Polya, R.E. Tarjan, and D.R. Woods, Notes on Introductory Combinatorics, Birkhäuser, 1983.

  74. R. Rammal and J.-M. Maillard,q-state Potts model on the checkerboard lattice, J. Phys. A16 (1983) 1073–1081.

    Google Scholar 

  75. A. Rechnitzer, W. Orrick, and A.J. Guttmann, in preparation.

  76. K.V. Rerikh, Non-algebraic integrability of the Chew-Low reversible dynamical system of the Cremona type and the relation with the 7th Hilbert problem (non-resonant case), Physica D82 (1995) 60–78.

    Google Scholar 

  77. K.V. Rerikh, Algebraic-geometry approach to integrability of birational plane mappings, Integrable birational quadratic reversible mappings. I, J. Geometry and Phys.24 (1998) 265–290.

    Google Scholar 

  78. K.V. Rerikh, Integrability of functional equations defined by birational mappings. II, Mathematical Physics, Analysis and Geometry, submitted.

  79. J. Riordan, Combinatorial Identities, R.E. Krieger Publishing Company Inc., 1979.

  80. D. Ruelle, Dynamical zeta fonctions: Where do they come from and what are they good for?, IHES/P/91/56, preprint.

  81. D. Ruelle, Thermodynamic Formalism, Addison-Wesley, Reading, 1978.

    Google Scholar 

  82. L. Schlesinger, Kadomstev-Petviashvile and two-dimensional Sino-Gordon equations: Reduction to Painlevé equations, Journal für die Reine Math.123 (1901) 138.

    Google Scholar 

  83. B. Shawyer and B. Watson, Borel's Methods of Summability (Theory and Applications), Clarendon Press, Oxford, 1994.

    Google Scholar 

  84. Y. Sibuya, Global theory of a second order linear differential equation with a polynomial coefficient, Math. Studies, Vol. 18, North-Holland, 1975.

  85. Y. Sibuya and R.H. Cameron, Symposium in Ordinary Differential Equations, Lecture Notes in Mathematics, Vol. 312, Springer-Verlag, 1973, p. 194.

  86. C. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc.73 (1967) 747.

    Google Scholar 

  87. Y.G. Stroganov, A new calculation method for partition functions in some lattice models, Phys. Lett. A74 (1979) 116.

    Google Scholar 

  88. H. Tsukahara and T. Inami, Test of Guttmann and Enting's conjecture in the eight-vertex model, J. Phys. Soc. Japan67 (1998) 1067–1070.

    Google Scholar 

  89. A.P. Veselov, Growth and integrability in the dynamics of mappings, Commun. Math. Phys.145 (1992) 181.

    Google Scholar 

  90. A.P. Veselov, Cremona group and dynamical systems, Math. Z.45 (1989) 118.

    Google Scholar 

  91. A. Voros, Compte Rendus Acad. Sciences,293, 1 (1981).

    Google Scholar 

  92. H.S. Wilf, Generatingfunctionology, Academic Press, 1994.

  93. Y.K. Zhou, Fusion hierarchies with open boundaries and exactly solvable models, In: Statistical Models Yang-Baxter Equation and Related Topics, M.L. Ge and F.Y. Wu, Eds., World Scientific, 1995, p. 351

Download references

Author information

Authors and Affiliations

  1. Centre de Recherches sur les Très Basses Températures, B.P. 166, F-38042, Grenoble, France

    J. -Ch. Anglès d'Auriac

  2. Institut d'Aéronautique, Université de Blida, BP 270, Blida, Algeria

    S. Boukraa

  3. LPTHE, Tour 16, ler étage, 4 Place Jussieu, 75252, Paris Cedex, France

    J. -M. Maillard

Authors
  1. J. -Ch. Anglès d'Auriac
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Boukraa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. -M. Maillard
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Anglès d'Auriac, J.C., Boukraa, S. & Maillard, J.M. Functional relations in lattice statistical mechanics, enumerative combinatorics, and discrete dynamical systems. Annals of Combinatorics 3, 131–158 (1999). https://doi.org/10.1007/BF01608780

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01608780

AMS Subject Classification

Keywords

Search

Navigation