Skip to main content
SpringerLink
Log in

Decay of Correlations in 1D Lattice Systems of Continuous Spins and Long-Range Interaction

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We consider a one-dimensional lattice system of unbounded and continuous spins. The Hamiltonian consists of a perturbed strictly-convex single-site potential and a product term with longe-range interaction. We show that if the interactions have an algebraic decay of order \(2+\alpha \), \(\alpha >0\), then the correlations also decay algebraically of order \(2+ \tilde{\alpha }\) for some \(\alpha > \tilde{\alpha }> 0\). For the argument we generalize a method due to Zegarlinski from finite-range to infinite-range interaction to get a preliminary decay of correlations, which is improved to the correct order by a recursive scheme based on Lebowitz inequalities. Because the decay of correlations yields the uniqueness of the Gibbs measure, the main result of this article yields that the one-phase region of a continuous spin system is at least as large as for the Ising model. This shows that there is no-phase transition in one-dimensional systems of unbounded and continuous spins as long as the interaction decays algebraically of order \(2+\alpha \), \(\alpha >0\).

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. Georgii, H.O.: Gibbs Measures and Phase Transitions, de Gruyter Studies in Mathematics, vol. 9, 2nd edn. Walter de Gruyter & Co., Berlin (2011). doi:10.1515/9783110250329

  2. Simon, B.: The Statistical Mechanics of Lattice Gases. Princeton Series in Physics, vol. I. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  3. Weitz, D.: Combinatorial criteria for uniqueness of Gibbs measures. Random Struct. Algorithms 27(4), 445–475 (2005). doi:10.1002/rsa.20073

  4. Dobrušin, R.L.: Description of a random field by means of conditional probabilities and conditions for its regularity. Teor. Verojatnost. i Primenen 13, 201–229 (1968)

    MathSciNet  Google Scholar 

  5. Dobrušin, R.L.: Conditions for the absence of phase transitions in one-dimensional classical systems. Mat. Sb. (N.S.) 93(135), 29–49, 151 (1974).

  6. Ruelle, D.: Statistical mechanics of a one-dimensional lattice gas. Commun. Math. Phys. 9, 267–278 (1968)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Cassandro, M., Orlandi, E., Picco, P.: Phase transition in the 1D random field Ising model with long range interaction. Commun. Math. Phys. 288(2), 731–744 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Dyson, F.J.: Existence of a phase-transition in a one-dimensional Ising ferromagnet. Commun. Math. Phys. 12(2), 91–107 (1969)

    Article  ADS  MathSciNet  Google Scholar 

  9. Fröhlich, J., Spencer, T.: The phase transition in the one-dimensional Ising model with \(1/r^{2}\) interaction energy. Commun. Math. Phys. 84(1), 87–101 (1982)

    Article  ADS  MATH  Google Scholar 

  10. Cassandro, M., Ferrari, P.A., Merola, I., Presutti, E.: Geometry of contours and Peierls estimates in \(d=1\) Ising models with long range interactions. J. Math. Phys. 46(5), 053305–053322 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  11. Imbrie, J.Z.: Decay of correlations in the one-dimensional Ising model with \(J_{ij}=\mid i-j\mid ^{-2}\). Commun. Math. Phys. 85(4), 491–515 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  12. Zegarlinski, B.: The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice. Commun. Math. Phys. 175(2), 401–432 (1996)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. Yoshida, N.: Phase transition from the viewpoint of relaxation phenomena. Rev. Math. Phys. 15(7), 765–788 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Yoshida, N.: The equivalence of the log-Sobolev inequality and a mixing condition for unbounded spin systems on the lattice. Ann. Inst. H. Poincaré Probab. Stat. 37(2), 223–243 (2001)

    Article  ADS  MATH  Google Scholar 

  15. Menz, G.: The approach of Otto–Reznikoff revisited (2013). arXiv:1309.0862.

  16. Simon, B.: Correlation inequalities and the decay of correlations in ferromagnets. Commun. Math. Phys. 77(2), 111–126 (1980)

    Article  ADS  Google Scholar 

  17. Lebowitz, J.L.: GHS and other inequalities. Commun. Math. Phys. 35, 87–92 (1974)

    Article  ADS  MathSciNet  Google Scholar 

  18. Royer, G.: An initiation to logarithmic Sobolev inequalities, SMF/AMS Texts and Monographs, vol. 14. American Mathematical Society, Providence (2007). Translated from the 1999 French original by Donald Babbitt.

  19. Zitt, P.A.: Functional inequalities and uniqueness of the Gibbs measure—from log-Sobolev to Poincaré. ESAIM Probab. Stat. 12, 258–272 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Béllissard, J., Høegh-Krohn, R.: Compactness and the maximal Gibbs state for random Gibbs fields on a lattice. Commun. Math. Phys. 84(3), 297–327 (1982)

    Article  ADS  MATH  Google Scholar 

  21. Menz, G., Otto, F.: Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential. Ann. Probab. 41(3B), 2182–2224 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  22. Horiguchi, T., Morita, T.: Upper and lower bounds to a correlation function for an Ising model with random interactions. Phys. Lett. A 74(5), 340–342 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  23. Kelly, D.G., Sherman, S.: General Griffiths’ inequalities on correlations in Ising ferromagnets. J. Math. Phys. 9(3), 466 (1968)

    Article  ADS  Google Scholar 

  24. Sylvester, G.S.: Inequalities for continuous-spin Ising ferromagnets. J. Stat. Phys. 15(4), 327–341 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  25. Otto, F., Reznikoff, M.G.: A new criterion for the logarithmic Sobolev inequality and two applications. J. Funct. Anal. 243(1), 121–157 (2007)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

Both authors want to thank the anonymous referee for the plentiful and valuable comments on the manuscript. The first author wants to thank Maria Westdickenberg (neé Reznikoff), Felix Otto, Nobuo Yoshida, and Chris Henderson for the fruitful and inspiring discussions on this topic. The second author is grateful to Felix Otto for bringing this subject to his attention. Both authors are indebted to the Max-Planck Institute for Mathematics in the Sciences in Leipzig for funding during the years 2010 to 2012, where most of the content of this article originated.

Author information

Authors and Affiliations

  1. Department of Mathematics, Stanford University, Building 380, Stanford, CA , 94305, USA

    Georg Menz

  2. Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 , Leipzig, Germany

    Robin Nittka

Authors
  1. Georg Menz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Robin Nittka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Georg Menz.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Menz, G., Nittka, R. Decay of Correlations in 1D Lattice Systems of Continuous Spins and Long-Range Interaction. J Stat Phys 156, 239–267 (2014). https://doi.org/10.1007/s10955-014-1011-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-014-1011-1

Keywords

Mathematics Subject Classification (2000)

Search

Navigation