Communications in Mathematical Physics

, Volume 122, Issue 4, pp 597–607 | Cite as

A note on the Ising model in high dimensions

  • J. Bricmont
  • H. Kesten
  • J. L. Lebowitz
  • R. H. Schonmann


We consider thed-dimensional Ising model with a nearest neighbor ferromagnetic interactionJ(d)=1/4d. We show that asd→∞ the+phase (and the — phase) approaches a product measure with density given by the mean field approximation. In particular the spontaneous magnetization converges to its mean field value. A similar result holds for the unique Gibbs measure of the system subject to an external fieldh≠0.


Neural Network Statistical Physic Complex System Nonlinear Dynamics High Dimension 
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  1. [AF]
    Aizenman, M., Fernandez, R.: On the critical behavior of the magnetization in high-dimensional Ising models. J. Stat. Phys.44, 395–454 (1986)Google Scholar
  2. [Bi]
    Billingsley, P.: Probability and measure, 2nd ed. New York: John Wiley 1986Google Scholar
  3. [Br]
    Brout, R.: Statistical mechanical theory of ferromagnetism high density behavior. Phys. Rev.118, 1009–1019 (1960)Google Scholar
  4. [BF]
    Bricmont, J., Fontaine, J. R.: Perturbation about the mean field critical point. Commun. Math. Phys.86, 337–362 (1982)Google Scholar
  5. [E]
    Ellis, R.: Entropy, large deviations and statistical mechanics. Berlin, Heidelberg, New York: Springer 1985Google Scholar
  6. [F]
    Fisher, M.: Critical temperatures of anisotropic Ising lattices. II. General upper bounds. Phys. Rev.162, 480–485 (1967)Google Scholar
  7. [FILS]
    Fröhlich, J., Israel, R., Lieb, E. H., Simon, B.: Phase transition and reflexion positivity. I. General theory and long range lattice models. Commun. Math. Phys.62, 1–34 (1978)Google Scholar
  8. [FSS]
    Fröhlich, J., Simon, B., Spencer, T.: Infrared bounds, phase transitions and continuous symmetry breaking. Commun. Math. Phys.50, 79–85 (1976)Google Scholar
  9. [G]
    Griffiths, R. B.: Correlations in Ising ferromagnets. III. Commun. Math. Phys.6, 121–127 (1967)Google Scholar
  10. [K1]
    Kesten, H.: Asymptotics in high dimension for percolation. To appear in Festschrift in honor of J. M. HammersleyGoogle Scholar
  11. [K2]
    Kesten, H.: Asymptotics in high dimension for the Fortuin-Kasteleyn random cluster model. To appear in Festschrift in honor of T. E. HarrisGoogle Scholar
  12. [KS]
    Kesten, H., Schonmann, R. H.: Behavior in large dimensions of the Potts and Heisenberg models. To appear in Rev. Math. Phys.Google Scholar
  13. [LM]
    Lebowitz, J. L., Martin-Löf, A.: On the uniqueness of the equilibrium state for Ising spin systems. Commun. Math. Phys.25, 276–282 (1972)Google Scholar
  14. [LP]
    Lebowitz, J. L., Penrose, O.: Rigorous treatment of the van der Waals-Maxwell theory of liquid-vapor transition. J. Math. Phys.7, 98–113 (1966)Google Scholar
  15. [N]
    Newman, C. M.: Shock waves and mean field bounds. Concavity and analyticity of the magnetization at low temperatures. Appendix to Percolation theory: A selective survey of rigorous results, pp. 147–167. In Advances in multiphase flow and related problems. Papanicolaou, G., (ed.): SIAM 1987Google Scholar
  16. [Pe]
    Pearce, P. A.: Mean field bounds on the magnetization for ferromagnetic spin models. J. Stat. Phys.25, 309–320 (1981)Google Scholar
  17. [Pr]
    Preston, C. J.: An application of the GHS inequalities to show the absence of phase transition for Ising spin systems. Commun. Math. Phys.35, 253–255 (1974)Google Scholar
  18. [PT]
    Pearce, P. A., Thompson, C. J.: The high density limit for lattice spin models. Commun. Math. Phys.58, 131–138 (1978)Google Scholar
  19. [R]
    Ruelle, D.: Thermodynamic formalism. Reading, MA: Addison Wesley 1978Google Scholar
  20. [SV]
    Schonmann, R. H., Vares, M. E.: The survival of the large dimensional basic contact process. Probab. Th. Rel. Fields72, 387–393 (1986)Google Scholar
  21. [Si]
    Simon, B.: Mean field upper bound on the transition temperature in multicomponent ferromagnets. J. Stat. Phys.22, 491–493 (1980)Google Scholar
  22. [S1]
    Slawny, J.: On the mean field theory bound on the magnetization. J. Stat. Phys.32, 375–388 (1983)Google Scholar
  23. [T1]
    Thompson, C. J.: Upper bounds for Ising model correlation functions. Commun. Math. Phys.24, 61–66 (1971)Google Scholar
  24. [T2]
    Thompson, C. J.: Mathematical statistical mechanics. Princeton, NJ: Princeton University Press (1972)Google Scholar
  25. [T3]
    Thompson, C. J.: Ising model in the high density limit. Commun. Math. Phys.36, 255–262 (1974)Google Scholar
  26. [TH]
    Tasaki, H., Hara, T.: Mean field bound and GHS inequality. J. Stat. Phys.35, 99–107 (1984)Google Scholar
  27. [V]
    Vigfusson, J. O.: New upper bounds for the magnetization in ferromagnetic one-component systems. Lett. Math. Phys.10, 71–77 (1985)Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • J. Bricmont
    • 1
  • H. Kesten
    • 2
  • J. L. Lebowitz
    • 3
  • R. H. Schonmann
    • 2
  1. 1.Institut de Physique TheoriqueUniversite Catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.Department of Mathematics, White HallCornell UniversityIthacaUSA
  3. 3.Department of MathematicsRutgers UniversityNew BrunswickUSA

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