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The 2D-Ising model near criticality: a FK-percolation analysis
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  • Published: 11 March 2010

The 2D-Ising model near criticality: a FK-percolation analysis

  • R. Cerf1 &
  • R. J. Messikh2 nAff3 

Probability Theory and Related Fields volume 150, pages 193–217 (2011)Cite this article

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Abstract

We study the 2D-Ising model defined on finite boxes at temperatures that are below but very close from the critical point. When the temperature approaches the critical point and the size of the box grows fast enough, we establish large deviations estimates on FK-percolation events that concern the phenomenon of phase coexistence.

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References

  1. Alexander K.S.: On weak mixing in lattice models. Probab. Theory Relat. Fields 110, 441–471 (1998)

    Article  MATH  Google Scholar 

  2. Alexander K.S.: Mixing properties and exponential decay for lattice systems in finite volumes. Ann. Probab. 32, 441–487 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alexander K.S.: Stability of the Wulff minimum and fluctuations in shape for large finite clusters in two-dimensional percolation. Probab. Theory Relat. Fields 91, 507–532 (1992)

    Article  MATH  Google Scholar 

  4. Alexander K.S.: Cube-root boundary fluctuations for droplets in random cluster models. Comm. Math. Phys. 224, 733–781 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alexander K.S., Chayes J.T., Chayes L.: The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation. Comm. Math. Phys. 131, 1–50 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  6. Barbato, D.: Tesi di Laurea. Universita di Pisa (2001)

  7. Bodineau T.: The Wulff construction in three and more dimensions. Comm. Math. Phys. 207, 197–229 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bodineau T.: Slab percolation for the Ising model. Probab. Theory Relat. Fields 132, 83–118 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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 

  10. Camia, F., Newman, C.M.: The Full Scaling Limit of Two-Dimensional Critical Percolation. Preprint (2005)

  11. Cerf, R.: Large deviations for three-dimensional supercritical percolation. Astérisque 267 (2000)

  12. Cerf, R.: The Wulff crystal in Ising and percolation models. Ecole d’été de probabilités, Saint Flour (2004)

  13. Cerf, R., Messikh, R.J.: On the Wulf crystal of the 2D-Ising model near criticality (2006)

  14. Cerf R., Pisztora Á.: On the Wulff crystal in the Ising model. Ann. Probab. 28, 947–1017 (2000)

    MathSciNet  MATH  Google Scholar 

  15. Cerf R., Pisztora Á.: Phase coexistence in Ising, Potts and percolation models. Ann. I. H. P. PR 37, 643–724 (2001)

    MathSciNet  MATH  Google Scholar 

  16. 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 

  17. Couronné O., Messikh R.J.: Surface order large deviations for 2D FK-percolation and Potts models. Stoch. Proc. Appl. 113, 81–99 (2004)

    Article  MATH  Google Scholar 

  18. Dobrushin, R.L., Kotecký, R., Shlosman, S.B.: Wulff construction: a global shape from local interaction. Am. Math. Soc. Transl. Ser. (1992)

  19. Edwards R.G., Sokal A.D.: Generalization of the Fortuin–Kasteleyn–Swenden–Wang representation and Monte Carlo algorithm. Phys. Rev. D 38, 2009–2012 (1988)

    Article  MathSciNet  Google Scholar 

  20. Ellis R.S.: Entropy, Large Deviations, and Statistical Mechanics. Springer, New York (1985)

    MATH  Google Scholar 

  21. Fortuin C.M., Kasteleyn R.W.: On the random-cluster model I. Introduction and relation to other models. Physica 57, 125–145 (1972)

    Article  MathSciNet  Google Scholar 

  22. Grimmett, G.R.: Percolation and disordered systems. Lectures on Probability Theory and Statistics. In: Bertrand, P. (ed.) Lectures from the 26th Summer School on Probability Theory held in Saint Flour, August 19–September 4, 1996. Lecture Notes in Mathematics 1665 (1997)

  23. Grimmett, G.R.: The random cluster model. In: Kelly, F.P. (ed.) Probability, Statistics and Optimization: A Tribute to Peter Whittle, pp. 49–63 (1994)

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Grimmett G.R., Marstrand J.M.: The supercritical phase of percolation is well behaved. Proc. R. Soc. Lond. Ser. A 430, 439–457 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hoeffding W.: Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc. 58, 13–30 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ioffe D.: Large deviation for the 2D Ising model: a lower bound without cluster expansions. J. Stat. Phys. 74, 411–432 (1993)

    Article  MathSciNet  Google Scholar 

  28. Ioffe D.: Exact large deviation bounds up to T c for the Ising model in two dimensions. Probab. Theory Relat. Fields 102, 313–330 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ioffe D., Schonmann R.: Dobrushin–Kotecký–Shlosman theorem up to the critical temperature. Comm. Math. Phys. 199, 117–167 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. Laanait L., Messager A., Ruiz J.: Phase coexistence and surface tensions for the Potts model. Comm. Math. Phys. 105, 527–545 (1986)

    Article  MathSciNet  Google Scholar 

  31. Mc Coy B.M., Wu T.T.: The Two Dimensional Ising Model. Harvard University Press, Cambridge (1973)

    Google Scholar 

  32. Messikh, R.J.: The surface tension of the 2D Ising model near criticality (2006, submitted)

  33. Messikh, R.J.: From the Ising model towards the Black and White Mumford–Shah functional. Ph.D. thesis (2004)

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Pfister C.E.: Large deviations and phase separation in the two-dimensional Ising model. Helv. Phys. Acta 64, 953–1054 (1991)

    MathSciNet  Google Scholar 

  36. Pfister C.E., Velenik Y.: Large deviations and continuum limit in the 2D Ising model. Probab. Theory Relat. Fields 109, 435–506 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  37. Pisztora Á.: Surface order large deviations for Ising, Potts and percolation models. Probab. Theory Relat. Fields 104, 427–466 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  39. Slade, G.: The Lace Expansion and its Applications. Ecole d’été de probabilités, Saint Flour (2004)

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

    MathSciNet  MATH  Google Scholar 

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Author information

Author notes
  1. R. J. Messikh

    Present address: , 64, Rue de rive, 1260, Nyon, Switzerland

Authors and Affiliations

  1. Mathématiques, Université Paris-Sud, 91405, Orsay, France

    R. Cerf

  2. Ecole Polytechnique Federale de Lausanne, CMOS, 1015, Lausanne, Switzerland

    R. J. Messikh

Authors
  1. R. Cerf
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  2. R. J. Messikh
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Correspondence to R. Cerf.

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Cerf, R., Messikh, R.J. The 2D-Ising model near criticality: a FK-percolation analysis. Probab. Theory Relat. Fields 150, 193–217 (2011). https://doi.org/10.1007/s00440-010-0272-0

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  • Received: 27 November 2008

  • Revised: 14 December 2009

  • Published: 11 March 2010

  • Issue Date: June 2011

  • DOI: https://doi.org/10.1007/s00440-010-0272-0

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Keywords

  • Large deviations
  • Criticality
  • Phase coexistence

Mathematics Subject Classification (2000)

  • 60F10
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