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Slab percolation for the Ising model
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  • Published: 09 October 2004

Slab percolation for the Ising model

  • T. Bodineau1 

Probability Theory and Related Fields volume 132, pages 83–118 (2005)Cite this article

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Abstract.

For the FK representation of the Ising model, we prove that the slab percolation threshold coincides with the critical temperature in any dimension d≥3.

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  1. Aizenman, M.: Rigorous studies of critical behavior. II. Statistical physics and dynamical systems. Progr. Phys. 10, Birkhäuser Boston, Boston, MA, 1985, pp. 453–481

  2. Aizenman, M.: Absence of an intermediate phase for a general class of one-component ferromagnetic models. Phys. Rev. Lett. 54, 839–842 (1985)

    Article  Google Scholar 

  3. Aizenman, M., Barsky, D.: Sharpness of the phase transition in percolation models. Commun. Math. Phys. 108, 489–526 (1987)

    Google Scholar 

  4. Aizenman, M., Barsky, D., Fernandez, R.: The phase transition in a general class of Ising-type models is sharp. J. Stat. Phys. 47, 343–374 (1987)

    Google Scholar 

  5. Aizenman, M., Chayes, J.T., Chayes, L. Fröhlich, J., Russo, L.: On a sharp transition from area law to perimeter law in a system of random surfaces. Commun. Math. Phys. 92, 19–69 (1983)

    MATH  Google Scholar 

  6. Barsky, D., Grimmett, G., Newman, C.: Percolation in half-spaces: equality of critical densities and continuity of the percolation probability. Probab. Theory Related Fields 90, 111–148 (1991)

    MATH  Google Scholar 

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

    Google Scholar 

  8. Bodineau, T., Ioffe, D., Velenik, Y.: Rigorous probabilistic analysis of equilibrium crystal shapes. J. Math. Phys. 41, 1033–1098 (2000)

    Google Scholar 

  9. Borgs, C., Kotecký, R., Medved, I.: Finite-size effects for the Potts model with weak boundary conditions. J. Stat. Phys. 109, 67–131 (2002)

    Article  MATH  Google Scholar 

  10. Cerf, R.: Large deviations for three dimensional supercritical percolation. Astérisque No. 267, 2000

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

    Google Scholar 

  12. Dobrushin, R.L., Shlosman, S.: Completely analytical Gibbs fields. Progr. Phys. 10, Birkhäuser Boston, Boston, MA, 1985, pp. 371–403

  13. Edwards, R., Sokal, A.: Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. Phys. Rev. D (3) 38, 2009–2012 (1988)

    Google Scholar 

  14. Fortuin, C., Kasteleyn, P.: On the random-cluster model. I. Introduction and relation to other models. Physica 57, 536–564 (1972)

    Article  Google Scholar 

  15. Fröhlich, J., Pfister, C.: Semi-infinite Ising model. II. The wetting and layering transitions. Commun. Math. Phys. 112, 51–74 (1987)

    Google Scholar 

  16. Grimmett, G.: Percolation. Second edition, 321, Springer-Verlag, Berlin, 1999

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

    MATH  Google Scholar 

  18. Grimmett, G., Hiemer, P.: Directed percolation and random walk. Progr. Probab. 51, Birkhäuser Boston, Boston, MA, 2002, pp. 273–297

  19. Grimmett, G., Marstrand, J.: The supercritical phase of percolation is wellbehaved. Proc. Roy. Soc. London Ser. A 430, 439–457 (1990)

    MATH  Google Scholar 

  20. Kotecký, R., Laanait, L., Messager, A., Ruiz, J.: The q-state Potts model in the standard Pirogov-Sinai theory: surface tensions and Wilson loops. J. Stat. Phys. 58, 199–248 (1990)

    Google Scholar 

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

    Article  Google Scholar 

  22. Meester, R., Steif, J.: On the continuity of the critical value for long range percolation in the exponential case. Commun. Math. Phys. 180, 483–504 (1996)

    MATH  Google Scholar 

  23. Messager, A., Miracle-Solé, S., Pfister, C.: On classical ferromagnets with a complex external field. J. Stat. Phys. 34, 279–286 (1984)

    Google Scholar 

  24. Messager, A., Miracle-Solé, S., Ruiz, J.: Surface tension, step free energy and facets in the equilibrium crystal. J. Stat. Phys. 79, 1995

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

    Article  MATH  Google Scholar 

  26. Schonmann, R., Shlosman, S.: Complete analyticity for 2D Ising completed. Commun. Math. Phys. 170, 453–482 (1995)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Probabilités et modèles aléatoires, CNRS-UMR 7599, Universités Paris VI & VII, 4 place Jussieu, Case 188, 75252, Paris, Cedex 05, France

    T. Bodineau

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  1. T. Bodineau
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Corresponding author

Correspondence to T. Bodineau.

Additional information

Mathematics Subject Classification (2000): 82B20

I wish to thank G. Grimmett, D. Ioffe and R. Kotecky for very stimulating discussions and useful comments.

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Bodineau, T. Slab percolation for the Ising model. Probab. Theory Relat. Fields 132, 83–118 (2005). https://doi.org/10.1007/s00440-004-0391-6

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  • Received: 18 September 2003

  • Revised: 30 July 2004

  • Published: 09 October 2004

  • Issue Date: May 2005

  • DOI: https://doi.org/10.1007/s00440-004-0391-6

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Keywords or pharses

  • Ising
  • Percolation
  • FK representation
  • Coarse graining
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