Skip to main content
SpringerLink
Log in

The Near-Critical Planar FK-Ising Model

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We study the near-critical FK-Ising model. First, a determination of the correlation length defined via crossing probabilities is provided. Second, a phenomenon about the near-critical behavior of the FK-Ising is highlighted, which is completely missing from the case of standard percolation: in any monotone coupling of FK configurations ω p (e.g., in the one introduced in Grimmett (Ann Probab 23(4):1461–1510, 1995)), as one raises p near p c , the new edges arrive in a self-organized way, so that the correlation length is not governed anymore by the number of pivotal edges at criticality.

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. Borgs C., Chayes J.T., Kesten H., Spencer J.: Uniform boundedness of critical crossing probabilities implies hyperscaling. Random Struct. Algorithms 15(3-4), 368–413 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Borgs C., Chayes J.T., Kesten H., Spencer J.: The birth of the infinite cluster: finite-size scaling in percolation. Commun. Math. Phys. 224(1), 153–204 (2001)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  3. Beffara V., Duminil-Copin H.: The self-dual point of the two-dimensional random-cluster model is critical for q ≥ 1. Probability Theory Relat. Fields 153(3–4), 511–542 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  4. Beffara V., Duminil-Copin H.: Smirnov’s fermionic observable away from criticality. Ann. Prob. 40(6), 2667–2689 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Boutillier C., de Tilière B.: The critical Z-invariant Ising model via dimers: the periodic case. Probab. Theory Relat. Fields 147(3-4), 379–413 (2010)

    Article  MATH  Google Scholar 

  6. Boutillier C., de Tilière B.: The critical Z-invariant Ising model via dimers: locality property. Commun. Math. Phys. 301(2), 473–516 (2011)

    Article  ADS  MATH  Google Scholar 

  7. Chelkak, D., Duminil-Copin, H., Hongler, C., Kemppainen, A., Smirnov, S.: Convergence of Ising interfaces to Schramm’s SLE’s. In preparation, 2012

  8. Chelkak, D., Hongler, C., Izyurov, K.: Conformal invariance of spin correlations in the planar Ising model. http://arxiv.org/abs/1202.2838v1 [math-ph], 2012

  9. Duminil-Copin, H., Garban, C.: Critical exponents in FK-I sing percolation. In preparation

  10. Duminil-Copin H., Hongler C., Nolin P.: Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model. Commun. Pure App. Math. 64(9), 1165–1198 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. Duminil-Copin, H., Smirnov, S.: Conformal invariance of lattice models. arXiv:1109.1549, 2011. Probability and Statistical Physics in Two and More Dimensions, Editors David Ellwood, Charles Newman, Vladas Sidoravicius, Wendelin Werner, Clay Mathematics Proceedings, Vol. 15, Amer. Math. Soc., Providence, RI, 2012

  12. Deng Y., Garoni T.M., Sokal A.D.: Critical speeding-up in the local dynamics of the random-cluster model. Phys. Rev. Lett. 98(23), 230602 (2007)

    Article  ADS  Google Scholar 

  13. Ferdinand A., Fisher M.: Bounded and inhomogeneous ising models. I. Specific-heat anomaly of a finite lattice. Phys. Rev. 185(2), 832–846 (1969)

    Article  ADS  Google Scholar 

  14. Garban, C., Hongler, C.: Specific Heat of the Ising model. In preparation

  15. Garban, C., Pete, G.: The scaling limit of dynamical FK-percolation. In preparation

  16. Garban C., Pete G., Schramm O.: The Fourier spectrum of critical percolation. Acta Math. 205(1), 19–104 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. Garban C., Pete G., Schramm O.: Pivotal, cluster and interface measures for critical planar percolation. J. Amer. Math. Soc. 26, 939–1024 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  18. Garban, C., Pete, G., Schramm, O.: The scaling limits of near-critical and dynamical percolation. http://arxiv.org/abs/1305.5526v2 [math.PR], 2013

  19. Grimmett G.: The stochastic random-cluster process and the uniqueness of random-cluster measures. Ann. Probab. 23(4), 1461–1510 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  20. Grimmett, G.: Percolation. Grundlehren der mathematischen Wissenschaften 321, 2nd edn. Berlin: Springer, 1999

  21. Grimmett, G.: The random-cluster model. Grundlehren der Mathematischen Wissenschaften 333. Berlin: Springer-Verlag, 2006

  22. Häggström O., Jonasson J., Lyons R.: Coupling and Bernoullicity in random-cluster and Potts models. Bernoulli 8(3), 275–294 (2002)

    MATH  MathSciNet  Google Scholar 

  23. Henkel, M.: Conformal Invariance and Critical Phenomena. Berlin-Heidelberg-New York: Springer, 1999

  24. Hongler, C.: Conformal invariance of Ising model correlations. PhD thesis, 2010

  25. Kadanoff L.P.: Correlations along a line in the two-dimensional Ising model. Phys. Rev. 188, 859–863 (1969)

    Article  ADS  Google Scholar 

  26. Kesten H.: Scaling relations for 2D-percolation. Commun. Math. Phys. 109(1), 109–156 (1987)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  27. Laanait L., Messager A., Miracle-Solé S., Ruiz J., Shlosman S.: Interfaces in the Potts model. I. Pirogov–Sinai theory of the Fortuin-Kasteleyn representation. Commun. Math. Phys. 140(1), 81–91 (1991)

    Article  ADS  MATH  Google Scholar 

  28. Lawler, G.F., Schramm, O., Werner, W.: One-arm exponent for critical 2D percolation. Electron. J. Probab. 7(2) (electronic) (2002)

  29. McCoy B.M., Tracy C.A., Wu T.T.: Painlevé functions of the third kind. J. Math. Phys. 18, 1058–1092 (1977)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  30. McCoy, B.M., Wu, T.-T.: The two-dimensional Ising model. Cambridge, MA: Harvard University Press, 1973

  31. Messikh, R.: The surface tension near criticality of the 2d-Ising model. http://arxiv.org/abs/math/0610.636v1 [math.PR], 2006

  32. Nolin P.: Near-critical percolation in two dimensions. Electron. J. Probab. 13(55), 1562–1623 (2008)

    MATH  MathSciNet  Google Scholar 

  33. Nolin P., Werner W.: Asymmetry of near-critical percolation interfaces. J. Amer. Math. Soc. 22(3), 797–819 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  34. Onsager L.: Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. (2) 65, 117–149 (1944)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  35. Palmer, J.: Planar Ising correlations. Basel-Boston: Birkhäuser, 2007

  36. Schramm O.: Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, 221–288 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  37. Smirnov, S.: Conformal invariance in random cluster models. II. Scaling limit of the interface. In preparation

  38. Smirnov S.: Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333(3), 239–244 (2001)

    Article  ADS  MATH  Google Scholar 

  39. Smirnov S.: Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. Math. (2) 172(2), 1435–1467 (2010)

    Article  MATH  Google Scholar 

  40. Schramm O., Steif J.: Quantitative noise sensitivity and exceptional times for percolation. Ann. Math. 171(2), 619–672 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  41. Smirnov S., Werner W.: Critical exponents for two-dimensional percolation. Math. Res. Lett. 8(5-6), 729–744 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  42. Tracy C.A.: Asymptotics of a τ-function arising in the two-dimensional Ising model. Commun. Math. Phys. 142, 297–311 (1991)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  43. Werner, W.: Lectures on two-dimensional critical percolation. IAS Park City Graduate Summer School, 2007. http://arxiv.org/abs/0710.0856v3 [math.PR], 2008

  44. Werner, W.: Private communication, 2009

  45. Werner, W.: Percolation et modèle d’Ising. Volume 16 of Cours Spécialisés [Specialized Courses]. Paris: Soc. Math. de France, 2009

  46. Wu F.Y.: The Potts model. Rev. Mod. Phys. 54(1), 235–268 (1982)

    Article  ADS  Google Scholar 

  47. Wu T.T., Mc Coy B.M., Tracy C.A., Barouch E.: Spin–spin correlation functions for the two-dimensional Ising model: exact theory in the scaling region. Phys. Rev. B 13, 316–375 (1976)

    Article  ADS  Google Scholar 

  48. Yang C.N.: The spontaneous magnetization of a two-dimensional Ising model. Phys. Rev. (2) 85, 808–816 (1952)

    Article  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, Université de Genève, 2-4 rue du lièvre, Case postate 64, 1211, Geneva 4, Switzerland

    Hugo Duminil-Copin

  2. CNRS & Ecole Normale Supérieure de Lyon, UMPA 46, allée d’Italie, 69364, Lyon Cedex 07, France

    Christophe Garban

  3. Institute of Mathematics, Technical University of Budapest, Egry József u. 1, 1111, Budapest, Hungary

    Gábor Pete

  4. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary

    Gábor Pete

Authors
  1. Hugo Duminil-Copin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christophe Garban
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gábor Pete
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christophe Garban.

Additional information

Communicated by M. Aizenman

Rights and permissions

Reprints and permissions

About this article

Cite this article

Duminil-Copin, H., Garban, C. & Pete, G. The Near-Critical Planar FK-Ising Model. Commun. Math. Phys. 326, 1–35 (2014). https://doi.org/10.1007/s00220-013-1857-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-013-1857-0

Keywords

Search

Navigation