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Random infinite-volume Gibbs states for the Curie-Weiss random field Ising model

Abstract

An approach to the definition of infinite-volume Gibbs states for the (quenched) random-field Ising model is considered in the case of a Curie-Weiss ferromagnet. It turns out that these states are random quasi-free measures. They are random convex linear combinations of the free product-measures “shifted” by the corresponding effective mean fields. The conditional self-averaging property of the magnetization related to this randomness is also discussed.

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References

  1. 1.

    R. L. Dobrushin,Theory Prob. Appl. 13:197 (1968);15:458 (1970).

  2. 2.

    O. E. Lanford and D. Ruelle,Commun. Math. Phys. 13:194 (1969).

  3. 3.

    H.-O. Georgii,Gibbs Measures and Phase Transitions (Walter de Gruyter, Berlin, 1988).

  4. 4.

    Ya. G. Sinai,Theory of Phase Transitions: Rigorous Results (Pergamon, Oxford, 1982).

  5. 5.

    V. A. Malyshev and R. A. Minlos,Gibbs Random Fields. The Method of Cluster Expansions (Dordrecht, Reidel, 1991).

  6. 6.

    R. A. Minlos,Funct. Anal. Appl. 2:60; 3:40 (1967).

  7. 7.

    D. Ruelle,Ann. Phys. (N.Y.)25:109 (1963).

  8. 8.

    J. G. Brankov, V. A. Zagrebnov, and N. S. Tonchev,Theor. Math. Phys. 66:72 (1986).

  9. 9.

    N. Angelescu and V. A. Zagrebnov,J. Stat. Phys. 41:323 (1985).

  10. 10.

    N. Angelescu and V. A. Zagrebnov, inProceedings of the IV Vilnius Conference on Probability Theory and Mathematical Statistics, Yu. V. Prohorovet al., eds. (VNU Science Press, Utrecht, The Netherlands), Vol. 1, p. 69.

  11. 11.

    M. Fannes, H. Spohn, and A. Verbeure,J. Math. Phys. 21:355 (1980).

  12. 12.

    D. Petz, G. A. Raggio, and A. Verbeure,Commun. Math. Phys. 121:271 (1989).

  13. 13.

    G. A. Raggio and R. F. Werner,Helv. Phys. Acta 62:980 (1989).

  14. 14.

    E. Størmer,J. Funct. Anal. 3:48 (1969).

  15. 15.

    R. Griffiths and J. Lebowitz,J. Math. Phys. 9:1284 (1968).

  16. 16.

    L. A. Pastur and A. L. Figotin,Theor. Math. Phys. 35:403 (1978).

  17. 17.

    D. Fisher, J. Fröhlich, and T. Spencer,J. Stat. Phys. 34:863 (1984).

  18. 18.

    G. A. Raggio and R. F. Werner,Europhys. Lett. 9:633 (1989).

  19. 19.

    M. Aizenman and J. Wehr,Commun. Math. Phys. 130:489 (1990).

  20. 20.

    S. R. Salinas and W. F. Wreszinski,J. Stat. Phys. 41:299 (1985).

  21. 21.

    N. Duffield and R. Kühn,J. Phys. A 22:4643 (1989).

  22. 22.

    J. Amaro de Matos and J. F. Perez,J. Stat. Phys. 62:587 (1991).

  23. 23.

    V. A. Zagrebnov, J. Amaro de Matos, and A. E. Patrick, inProceedings of the V Vilnius Conference on Probability Theory and Mathematical Statistics, B. Grigelioniset al., eds. (VSP BV, Utrecht/Mokslas, Vilnius, 1990), Vol. 2, p. 590.

  24. 24.

    A. E. Patrick,Acta Phys. Polonica A 77:527 (1990).

  25. 25.

    A. N. Shiryayev,Probability (Springer-Verlag, New York, 1984).

  26. 26.

    M. Kac,Phys. Fluids 2:8 (1959).

  27. 27.

    M. Reed and B. Simon,Methods of Modern Mathematical Physics, Vol. 1:Functional Analysis (Academic Press, New York, 1972).

  28. 28.

    R. S. Ellis,Entropy, Large Deviations and Statistical Mechanics (Springer-Verlag, New York, 1985).

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This paper is dedicated to Robert A. Minlos on the occasion of his 60th birthday.

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Amaro de Matos, J.M.G., Patrick, A.E. & Zagrebnov, V.A. Random infinite-volume Gibbs states for the Curie-Weiss random field Ising model. J Stat Phys 66, 139–164 (1992). https://doi.org/10.1007/BF01060064

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Key words

  • Random-field
  • Ising model
  • Gibbs states
  • self-averaging