Communications in Mathematical Physics

, Volume 327, Issue 3, pp 951–991 | Cite as

One-Dimensional Ising Models with Long Range Interactions: Cluster Expansion, Phase-Separating Point

  • Marzio Cassandro
  • Immacolata Merola
  • Pierre PiccoEmail author
  • Utkir Rozikov


We consider the phase separation problem for the one-dimensional ferromagnetic Ising model with long-range two-body interaction, J(n) = n −2+α , where \({n\in {\rm {I\!N}}}\) denotes the distance of the two spins and \({\alpha \in [0,\alpha_+[}\) with α + = (log 3)/(log 2) −1. We prove that when α = 0 the localization of the phase separation fluctuates macroscopically with a non-uniform explicit limiting law, while when 0 < α < α + the macroscopic fluctuations disappear and mesoscopic ones appear with a gaussian behavior when conveniently scaled. The mean magnetization profile is also given.


Ising Model Range Interaction Gibbs State Cluster Expansion External Contour 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abraham, D.B.: Surfaces Structures and Phase Transition-Exact Results. Phase Transitions and Critical Phenomena, vol. 10, pp. 1–74. Academic Press, London (1986)Google Scholar
  2. 2.
    Abraham D.B., Reed P.: Interface profile of the Ising ferromagnet in two dimensions. Commun. Math. Phys. 49, 35–46 (1976)ADSCrossRefMathSciNetGoogle Scholar
  3. 3.
    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)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Ben Arous G., Guionnet A.: Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy. Probab. Theory Relat. Fields 108, 517–542 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bricmont J., Lebowitz J., Pfister C.E.: On the equivalence of boundary conditions. J. Stat. Phys. 21, 573–582 (1979)ADSCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bissacot R., Fernández R., Procacci A.: On the convergence of cluster expansions for polymer gases. J. Stat. Phys. 139, 598–617 (2010)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Burkov S.E., Sinai Ya.G.: Phse diagrams of one-dimensional lattice models with long-range antiferromagnetic interaction. Russ. Math Survey 38(4), 235–257 (1983)ADSCrossRefMathSciNetGoogle Scholar
  8. 8.
    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)Google Scholar
  9. 9.
    Cassandro M., Olivieri E.: Renormalization group and analyticity in one dimension: a proof of Dobrushin’s theorem. Commun. Math. Phys. 80, 255–270 (1981)ADSCrossRefMathSciNetGoogle Scholar
  10. 10.
    Cassandro M., Orlandi E., Picco P.: Phase transition in the 1d random field Ising model with long range interaction. Commun. Math. Phys. 2, 731–744 (2009)ADSCrossRefMathSciNetGoogle Scholar
  11. 11.
    Cassandro, M., Orlandi, E., Picco, P.: Typical Gibbs configurations for the 1d random field Ising model with long range interaction. Commun. Math. Phys. 309, 229–253 (2012)Google Scholar
  12. 12.
    Cellarosi F., Sinai Ya.G.: The Möbius fonction and statistical mechanics. Bull. Math. Sci. 1, 245–275 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Coquille L., Velenik Y.: A finite-volume version of Aizenman Higuchi theorem for the 2d Ising model. Probab. Theory Relat. Fields 153, 25–44 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Dobrushin R.: The description of a random field by means of conditional probabilities and conditions of its regularity. Theory Probab. Appl. 13, 197–224 (1968)CrossRefGoogle Scholar
  15. 15.
    Dobrushin R.: The conditions of absence of phase transitions in one-dimensional classical systems. Matem. Sbornik 93(N1), 29–49 (1974)Google Scholar
  16. 16.
    Dobrushin R.: Analyticity of correlation functions in one-dimensional classical systems with slowly decreasing potentials. Commun. Math. Phys. 32(N4), 269–289 (1973)ADSCrossRefMathSciNetGoogle Scholar
  17. 17.
    Dobrushin R.: Gibbs state describing coexistence of phases for a three-dimensional Ising model. Theory Probab. Appl. 17, 582–600 (1972)CrossRefzbMATHGoogle Scholar
  18. 18.
    Dobrushin R., Hryniv O.: Fluctuations of the phase boundary in the 2D Ising ferromagnet. Commun. Math. Phys. 189, 395–445 (1997)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Dyson F.J.: Existence of phase transition in a one-dimensional Ising ferromagnetic. Commun. Math. Phys. 12, 91–107 (1969)ADSCrossRefMathSciNetGoogle Scholar
  20. 20.
    Dyson F.J.: Non-existence of spontaneous magnetization in a one-dimensional Ising ferromagnet. Commun. Math. Phys. 12, 212–215 (1969)ADSCrossRefMathSciNetGoogle Scholar
  21. 21.
    Dyson F.J.: An Ising ferromagnet with discontinuous long-range order. Commun. Math. Phys. 21, 269–283 (1971)ADSCrossRefMathSciNetGoogle Scholar
  22. 22.
    Fannes M., Vanheuverzwijn P., Verbeure A.: Energy-entropy inequalities for classical lattice systems. J. Stat. Phys. 29(3), 547–560 (1982)ADSCrossRefMathSciNetGoogle Scholar
  23. 23.
    Fröhlich J., Spencer T.: The phase transition in the one-dimensional Ising model with \({\frac{1}{r^2}}\) interaction energy. Commun. Math. Phys. 84, 87–101 (1982)ADSCrossRefzbMATHGoogle Scholar
  24. 24.
    Gallavotti G.: The phase separation line in the two-dimensional Ising model. Commun. Math. Phys. 27, 103–136 (1972)ADSCrossRefMathSciNetGoogle Scholar
  25. 25.
    Gallavotti G., Miracle-Solé S.: Statistical mechanics of lattice systems. Commun. Math. Phys. 5, 317–323 (1967)ADSCrossRefzbMATHGoogle Scholar
  26. 26.
    Gallavotti G., Martin-Löf A., Miracle-Solé S.: Some pro blems connected with the description of coexisting phases at low temperatures in Ising models. In: Lenard, A. (eds) Mathematical Methods in Statistical Mechanics, pp. 162–202. Springer, Berlin (1973)Google Scholar
  27. 27.
    Greenberg L., Ioffe D.: On an invariance principle for phase separation lines. Ann. Inst. H. Poincaré Probab. Stat. 45, 871–885 (2005)ADSCrossRefMathSciNetGoogle Scholar
  28. 28.
    Higuchi Y.: On some limit theorems related to the phase separation line in the two-dimensional Ising model. Z. Wahrscheinlichkeitstheorie verw. Gebiete. 50, 287–315 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Hryniv O.: On local behavior of the phase separation line in the 2D Ising model. Probab. Theory Relat. Fields 110, 91–107 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Imbrie J.Z.: Decay of correlations in the one-dimensional Ising model with J ij = | ij|−2. Commun. Math. Phys. 85, 491–515 (1982)ADSCrossRefMathSciNetGoogle Scholar
  31. 31.
    Imbrie J.Z., Newman C.M.: An intermediate phase with slow decay of correlations in one-dimensional 1/| xy| 2 percolation, Ising and Potts models. Commun. Math. Phys. 118, 303–336 (1988)ADSCrossRefMathSciNetGoogle Scholar
  32. 32.
    Johanson K.: Condensation of a one-dimensional lattice gas. Commun. Math. Phys. 141, 41–61 (1991)ADSCrossRefGoogle Scholar
  33. 33.
    Johanson K.: Separation of phases at low temperatures in a one-dimensional continuous gas. Commun. Math. Phys. 141, 259–278 (1991)ADSCrossRefGoogle Scholar
  34. 34.
    Johanson K.: On the separation of phases in one-dimensional gases. Commun. Math. Phys. 169, 521–561 (1995)ADSCrossRefGoogle Scholar
  35. 35.
    Minlos, R.A., Sinai, Ya. G.: The phenomenon of phase separation at low temperatures in certain lattice models of a gas. I Math. USSR Sbornik 2, 339–395 (1967) and II Trans. Moscow Math. Soc. 19, 121–196 (1968)Google Scholar
  36. 36.
    Pfister Ch.-E.: Large deviations and phase separation in the two-dimensional Ising model. Helv. Phys. Acta 64(7), 953–1054 (1991)MathSciNetGoogle Scholar
  37. 37.
    Pfister C.-E., Velenik Y.: Large deviations and continuum limit in the 2D Ising model. Probab. Theory Relat. Fields 109, 435–506 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Pfister C.-E., Velenik Y.: Interface, surface tension and reentrant pinning transition in the 2D Ising model. Commun. Math. Phys. 204(2), 269–312 (1999)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Procacci A., Scoppola B.: Polymer gas approach to N-body lattice systems. J. Stat. Phys. 96, 49–68 (1999)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Rota G.-C.: On the foundation of combinatorial theory: theory of Möbius function. Z. Wahrsch. Verw. Gebiete 2, 340–368 (1964)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Rogers J.B., Thompson C.J.: Absence of long range order in one dimensional spin systems. J. Stat. Phys. 25, 669–678 (1981)ADSCrossRefMathSciNetGoogle Scholar
  42. 42.
    Ruelle D.: Statistical mechanics of one-dimensional lattice gas. Commun. Math. Phys. 9, 267–278 (1968)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    Thouless D.J.: Long-range order in one-dimensional Ising systems. Phys. Rev. 187, 732–733 (1969)ADSCrossRefGoogle Scholar
  44. 44.
    van Beijeren H.: Interface sharpness in the Ising system. Commun. Math. Phys. 40, 1–6 (1975)ADSCrossRefMathSciNetGoogle Scholar
  45. 45.
    Wigner E.P.: On the distribution of the roots of certain symmetric matrices. Ann. Math. 67(2), 325–327 (1958)CrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    Wigner E.P.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math. 62, 548–564 (1955)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Marzio Cassandro
    • 1
  • Immacolata Merola
    • 2
  • Pierre Picco
    • 3
    Email author
  • Utkir Rozikov
    • 4
  1. 1.Gran Sasso Science Institute, INFN Center for Advanced StudiesL’AquilaItaly
  2. 2.Dipartimento di MatematicaUniversità di L’AquilaCoppito (AQ)Italy
  3. 3.LATP, CMI, UMR 6632, CNRS, Université de ProvenceMarseille Cedex 13France
  4. 4.Institute of MathematicsTashkentUzbekistan

Personalised recommendations