Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
A finite-volume version of Aizenman–Higuchi theorem for the 2d Ising model
Download PDF
Download PDF
  • Published: 02 February 2011

A finite-volume version of Aizenman–Higuchi theorem for the 2d Ising model

  • Loren Coquille1 &
  • Yvan Velenik1 

Probability Theory and Related Fields volume 153, pages 25–44 (2012)Cite this article

  • 258 Accesses

  • 11 Citations

  • Metrics details

Abstract

In the late 1970s, in two celebrated papers, Aizenman and Higuchi independently established that all infinite-volume Gibbs measures of the two-dimensional ferromagnetic nearest-neighbor Ising model at inverse temperature \({\beta\geq 0}\) are of the form \({\alpha\mu^{+}_\beta + (1-\alpha)\mu^{-}_\beta}\) , where \({\mu^{+}_\beta}\) and \({\mu^{-}_\beta}\) are the two pure phases and \({0\leq\alpha\leq 1}\) . We present here a new approach to this result, with a number of advantages: (a) We obtain an optimal finite-volume, quantitative analogue (implying the classical claim); (b) the scheme of our proof seems more natural and provides a better picture of the underlying phenomenon; (c) this new approach might be applicable to systems for which the classical method fails.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Aizenman M.: Translation invariance and instability of phase coexistence in the two-dimensional Ising system. Comm. Math. Phys. 73, 83–94 (1980)

    Article  MathSciNet  Google Scholar 

  2. Bodineau T.: Translation invariant Gibbs states for the Ising model. Probab. Theory Relat. Fields 135, 153–168 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bricmont J., Lebowitz J.L.: On the continuity of the magnetization and energy in Ising ferromagnets. J. Stat. Phys. 42, 861–869 (1986)

    Article  MathSciNet  Google Scholar 

  4. Bricmont J., Lebowitz J.L., Pfister C.E.: On the local structure of the phase separation line in the two-dimensional Ising system. J. Stat. Phys. 26, 313–332 (1981)

    Article  MathSciNet  Google Scholar 

  5. Campanino M., Ioffe D.: Ornstein–Zernike theory for the Bernoulli bond percolation on \({\mathbb{Z}^d}\) . Ann. Probab. 30, 652–682 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Campanino M., Ioffe D., Velenik Y.: Ornstein–Zernike theory for finite range Ising models above T c . Probab. Theory Relat. Fields 125, 305–349 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Campanino M., Ioffe D., Velenik Y.: Fluctuation theory of connectivities for subcritical random cluster models. Ann. Probab. 36, 1287–1321 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chayes J.T., Chayes L., Schonmann R.H.: Exponential decay of connectivities in the two-dimensional Ising model. J. Stat. Phys. 49, 433–445 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  9. Coquille, L., Duminil-Copin, H., Ioffe, D. and Velenik, Y.: Work in progress (2011)

  10. Dobrushin, R.L. and Shlosman, S.B.: The problem of translation invariance of Gibbs states at low temperatures, In: Mathematical physics reviews, Vol. 5, Soviet Sci. Rev. Sect. C Math. Phys. Rev., vol. 5, pp. 53–195. Harwood Academic Publ., Chur (1985)

  11. Dobrušin R.L.: The Gibbs state that describes the coexistence of phases for a three-dimensional Ising model. Teor. Verojatnost. i Primenen 17, 619–639 (1972)

    MathSciNet  Google Scholar 

  12. Dunlop F., Topolski K.: Cassie’s law and concavity of wall tension with respect to disorder. J. Stat. Phys. 98, 1115–1134 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gallavotti G.: The phase separation line in the two-dimensional Ising model. Comm. Math. Phys. 27, 103–136 (1972)

    Article  MathSciNet  Google Scholar 

  14. Gallavotti G., Miracle-Sole S.: Equilibrium states of the ising model in the two-phase region. Phys. Rev. B 5, 2555–2559 (1972)

    Article  Google Scholar 

  15. Georgii, H.-O., Higuchi, Y.: Gibbs measures and phase transitions. In: de Gruyter Studies in Mathematics, vol. 9. Walter de Gruyter & Co, Berlin

  16. Georgii, H.-O., Higuchi, Y.: Percolation and number of phases in the two-dimensional Ising model. J. Math. Phys. 41, 1153–1169 (2000) (Probabilistic techniques in equilibrium and nonequilibrium statistical physics)

    Google Scholar 

  17. Greenberg L., Ioffe D.: On an invariance principle for phase separation lines. Ann. Inst. H. Poincaré Probab. Stat. 41, 871–885 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Higuchi Y.: On some limit theorems related to the phase separation line in the two-dimensional Ising model. Z. Wahrsch. Verw. Gebiete 50, 287–315 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  19. Higuchi, Y.: On the absence of non-translation invariant Gibbs states for the two-dimensional Ising model, In: Random fields, Vol. I, II (Esztergom, 1979), Colloq. Math. Soc., vol. 27, pp. 517–534. János Bolyai, North-Holland, Amsterdam (1981)

  20. Lebowitz J.L., Pfister C.E.: Surface tension and phase coexistence. Phys. Rev. Lett. 46, 1031–1033 (1981)

    Article  MathSciNet  Google Scholar 

  21. Messager A., Miracle-Sole S.: Equilibrium states of the two-dimensional Ising model in the two-phase region. Comm. Math. Phys. 40, 187–196 (1975)

    Article  MathSciNet  Google Scholar 

  22. Pfister, C.-E. and Velenik, Y.: Mathematical theory of the wetting phenomenon in the 2D Ising model. Helv. Phys. Acta, 69, 949–973 (1996) (Papers honouring the 60th birthday of Klaus Hepp and of Walter Hunziker, Part I (Zürich, 1995))

  23. Pfister C.-E., Velenik Y.: Interface, surface tension and reentrant pinning transition in the 2D Ising model. Comm. Math. Phys. 204, 269–312 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Russo L.: The infinite cluster method in the two-dimensional Ising model. Comm. Math. Phys. 67, 251–266 (1979)

    Article  MathSciNet  Google Scholar 

  25. Enter A.C.D., Netočný K., Schaap H.G.: On the Ising model with random boundary condition. J. Stat. Phys. 118, 997–1056 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Geneva, 2-4 rue du Lièvre, Case Postale 64, 1211, Geneva 4, Switzerland

    Loren Coquille & Yvan Velenik

Authors
  1. Loren Coquille
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yvan Velenik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yvan Velenik.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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). https://doi.org/10.1007/s00440-011-0339-6

Download citation

  • Received: 18 June 2010

  • Revised: 20 December 2010

  • Published: 02 February 2011

  • Issue Date: June 2012

  • DOI: https://doi.org/10.1007/s00440-011-0339-6

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Ising model
  • Gibbs states
  • Translation invariance

Mathematics Subject Classification (2000)

  • 60K35
  • 82B20
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature