Advertisement

Journal of Statistical Physics

, Volume 122, Issue 1, pp 59–72 | Cite as

Slab Percolation and Phase Transitions for the Ising Model

  • Emilio De Santis
  • Rossella Micieli
Article
  • 53 Downloads

Abstract

We prove, using the random-cluster model, a strict inequality between site percolation and magnetization in the region of phase transition for the d-dimensional Ising model, thus improving a result of [5]. We extend this result also at the case of two plane lattices \({\mathbb Z}\) (slabs) and give a characterization of phase transition in this case. The general case of N slabs, with N an arbitrary positive integer, is partially solved and it is used to show that this characterization holds in the case of three slabs with periodic boundary conditions.

Key Words

Percolation infinite clusters magnetization Gibbs measure random-cluster measure 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Aizenman, Translation invariance and instability of phase coexistence in the two-dimensional Ising system, Comm. Math. Phys. 73(1), 83–94 (1980).CrossRefMathSciNetGoogle Scholar
  2. 2.
    M. Aizenman, J. Bricmont and J. Lebowitz, Percolation of the minority spins in high dimensional Ising models. J. Stat. Phys. 49(1), 859–865 (1987).Google Scholar
  3. 3.
    B. Bollob´s and G. Brightwell, Random walks and electrical resistances in products of graphs. Discrete Appl. Math. 73(1), 69–79 (1997).MathSciNetGoogle Scholar
  4. 4.
    T. Bodineau, Slab percolation for the ising model. arXiv:math.PR/0309300 pp. 1–33, (2003).Google Scholar
  5. 5.
    A. Coniglio, C.R. Nappi, F. Peruggi and L. Russo, Percolation and phase transitions in the Ising model. Comm. Math. Phys. 51(3), 315–323 (1976).CrossRefMathSciNetGoogle Scholar
  6. 6.
    A. Coniglio, C.R. Nappi, F. Peruggi and L. Russo, Percolation points and critical point in the Ising model. J. Phys. A 10(2), 205–218 (1977).ADSMathSciNetGoogle Scholar
  7. 7.
    M. Campanino and L. Russo, An upper bound on the critical percolation probability for the three-dimensional cubic lattice. Ann. Probab. 13(2), 478–491 (1985).MathSciNetGoogle Scholar
  8. 8.
    R. Edwards and A. Sokal, Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. Phys. Rev. D 38(6), 2009–2012 (1988).CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    C. M. Fortuin, P. W. Kasteleyn and J. Ginibre, Correlation inequalities on some partially ordered sets. Comm. Math. Phys. 22, 89–103 (1971).CrossRefMathSciNetGoogle Scholar
  10. 10.
    C. M. Fortuin, On the random-cluster model. III. The simple random-cluster model. Physica 59, 545–570 (1972).CrossRefMathSciNetGoogle Scholar
  11. 11.
    H.-O. Georgii, Gibbs measures and phase transitions, Vol. 9 of Gruyter Studies in Mathematics. Walter de Gruyter & Co., (Berlin 1988).Google Scholar
  12. 12.
    H.-O. Georgii, O. Häggströmm and C. Maes, The random geometry of equilibrium phases, Vol. 18 of Domb Lebowitz: Critical Phenomena (2001). Google Scholar
  13. 13.
    H.-O. Georgii and Y. Higuchi, Percolation and number of phases in the two-dimensional Ising model. J. Math. Phys. 41(3), 1153–1169 (2000). Probabilistic techniques in equilibrium and nonequilibrium statistical physics.CrossRefADSMathSciNetGoogle Scholar
  14. 14.
    A. Gandolfi, M. Keane and L. Russo, On the uniqueness of the infinite occupied cluster in dependent two-dimensional site percolation. Ann. Probab. 16(3), 1147–1157 (1988).MathSciNetGoogle Scholar
  15. 15.
    G. Grimmett, The random-cluster model. In Probability on discrete structures, Vol. 110 of Encyclopaedia Math. Sci. (Springer, Berlin 2004) pp. 73–123.Google Scholar
  16. 16.
    O. H´ggström, Probability on bunkbed graphs. Formal Power Series and Algebraic Combinatorics, (2003) pp. 19–27,Google Scholar
  17. 17.
    Y. Higuchi, On the absence of non-translation invariant Gibbs states for the two-dimensional Ising model. In Random fields, Vol. I, II (Esztergom, 1979), Vol. 27 of Colloq. Math. Soc. J´nos Bolyai, (North-Holland, Amsterdam 1981) pp. 517–534.Google Scholar
  18. 18.
    Y. Higuchi, A weak version of RSW theorem for the two-dimensional Ising model.In Particle systems, random media and large deviations (Brunswick, Maine, 1984), Vol. 41 of Contemp. Math. Amer. Math. Soc. Providence RI, (1985) pp. 207–214.Google Scholar
  19. 19.
    Y. Higuchi Percolation of the two-dimensional Ising model. In Stochastic processes–-mathematics and physics, II (Bielefeld, 1985), Vol. 1250 of Lecture Notes in Math., (Springer, Berlin 1987) pp. 120–127.Google Scholar
  20. 20.
    Y. Higuchi, Coexistence of infinite (*)-clusters. II. Ising percolation in two dimensions. Probab. Theory Related Fields 97(1–2), 1–33 (1993).MATHMathSciNetGoogle Scholar
  21. 21.
    Y. Higuchi, A sharp transition for the two-dimensional Ising percolation. Probab. Theory Related Fields 97(4), 489–514 (1993).CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    T.M. Liggett, Interacting particle systems, Vol. 276 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].(Springer-Verlag, New York 1985).Google Scholar
  23. 23.
    C.M. Newman, Topics in disordered systems. Lectures in Mathematics ETH Zürich. (Birkhäuser Verlag, Basel 1997).Google Scholar
  24. 24.
    L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65(2), 117–149 (1944).ADSMATHMathSciNetGoogle Scholar
  25. 25.
    L. Russo, The infinite cluster method in the two-dimensional Ising model. Comm. Math. Phys. 67(3), 251–266 (1979).CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Dipartimento di Matematica “Guido Castelnuovo”Università di Roma “La Sapienza”.RomaItalia

Personalised recommendations