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Communications in Mathematical Physics

, Volume 288, Issue 2, pp 731–744 | Cite as

Phase Transition in the 1d Random Field Ising Model with Long Range Interaction

  • Marzio Cassandro
  • Enza Orlandi
  • Pierre PiccoEmail author
Article

Abstract

We study one–dimensional Ising spin systems with ferromagnetic, long–range interaction decaying as n −2+α , \({\alpha \in\left(\frac 12,\frac{\ln 3}{\ln2}-1\right)}\), in the presence of external random fields. We assume that the random fields are given by a collection of symmetric, independent, identically distributed real random variables, gaussian or subgaussian. We show, for temperature and strength of the randomness (variance) small enough, with IP = 1 with respect to the random fields, that there are at least two distinct extremal Gibbs measures.

Keywords

Long Range Gibbs Measure Range Interaction Gibbs State Interface Point 
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.
    Aizenman M., Chayes J., Chayes L., Newman C.: Discontinuity of the magnetization in one–dimensional 1/|xy|2 percolation, Ising and Potts models. J. Stat. Phys. 50(1–2), 1–40 (1988)zbMATHCrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Aizenman M., Wehr J.: Rounding of first order phase transitions in systems with quenched disorder. Commun. Math. Phys. 130, 489–528 (1990)zbMATHCrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Bovier, A.: Statistical Mechanics of Disordered Systems. Cambridge Series in Statistical and Probabilistic Mathematics 18, Cambridge: Cambridge Univ. Press, 2006Google Scholar
  4. 4.
    Bricmont J., Kupiainen A.: Phase transition in the three-dimensional random field Ising model. Commun. Math. Phys. 116, 539–572 (1988)CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Cassandro M., Ferrari P.A., Merola I., Presutti E.: Geometry of contours and Peierls estimates in d = 1 Ising models with long range interaction. J. Math. Phys. 46(5), 053305 (2005)CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Dobrushin R.: The description of a random field by means of conditional probabilities and conditions of its regularity. Theory Probability Appl. 13, 197–224 (1968)CrossRefGoogle Scholar
  7. 7.
    Dobrushin R.: The conditions of absence of phase transitions in one-dimensional classical systems. Matem. Sbornik 93(N1), 29–49 (1974)Google Scholar
  8. 8.
    Dobrushin R.: Analyticity of correlation functions in one-dimensional classical systems with slowly decreasing potentials. Commun. Math. Phys. 32(N4), 269–289 (1973)CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Dyson F.J.: Existence of phase transition in a one-dimensional Ising ferromagnetic. Commun. Math. Phys. 12, 91–107 (1969)CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Ellis R.S.: Entropy, Large Deviation and Statistical Mechanics. Springer, New York (1988)Google Scholar
  11. 11.
    Fröhlich J., Spencer J.: The phase transition in the one-dimensional Ising model with \({\frac 1 {r^2}}\) interaction energy. Commun. Math. Phys. 84, 87–101 (1982)zbMATHCrossRefADSGoogle Scholar
  12. 12.
    Gallavotti G., Miracle Sole S.: Statistical mechanics of Lattice Systems. Commun. Math. Phys. 5, 317–323 (1967)zbMATHCrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Imry Y., Ma S.: Random field instability of the ordered state of continuous symmetry. Phys. Rev. Lett. 35, 1399–1401 (1975)CrossRefADSGoogle Scholar
  14. 14.
    Rogers J.B., Thompson C.J.: Absence of long range order in one dimensional spin systems. J. Stat. Phys. 25, 669–678 (1981)CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Ruelle D.: Statistical mechanics of one-dimensional Lattice gas. Commun. Math. Phys. 9, 267–278 (1968)zbMATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Dipartimento di FisicaUniversitá di Roma “La Sapienza”RomaItaly
  2. 2.Dipartimento di MatematicaUniversitá di Roma TreRomaItaly
  3. 3.LATP, CMI, UMR 6632, CNRS, Université de ProvenceMarseille Cedex 13France

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