Advertisement

Journal of Statistical Physics

, Volume 130, Issue 5, pp 983–1009 | Cite as

Sharpness of the Phase Transition and Exponential Decay of the Subcritical Cluster Size for Percolation on Quasi-Transitive Graphs

  • Tonći Antunović
  • Ivan Veselić
Article

Abstract

We study homogeneous, independent percolation on general quasi-transitive graphs. We prove that in the disorder regime where all clusters are finite almost surely, in fact the expectation of the cluster size is finite. This extends a well-known theorem by Menshikov and Aizenman & Barsky to all quasi-transitive graphs. Moreover we deduce that in this disorder regime the cluster size distribution decays exponentially, extending a result of Aizenman & Newman. Our results apply to both edge and site percolation, as well as long range (edge) percolation. The proof is based on a modification of the Aizenman & Barsky method.

Keywords

Random graphs Edge percolation Site percolation Quasi-transitive graphs Phase transition 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aizenman, M., Barsky, D.J.: Sharpness of the phase transition in percolation models. Commun. Math. Phys. 108(3), 489–526 (1987) MATHCrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Aizenman, M., Newman, C.M.: Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys. 36(1-2), 107–143 (1984) MATHCrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Antunović, T., Veselić, I.: Equality of Lifshitz and van Hove exponents on amenable Cayley graphs. http://www.arxiv.org/abs/0706.2844
  4. 4.
    Antunović, T., Veselić, I.: Spectral asymptotics of percolation Hamiltonians on amenable Cayley graphs. In: Proceedings of OTAMP 2006. Operator Theory: Advances and Applications (2007, in press) Google Scholar
  5. 5.
    Biskup, M., König, W.: Long-time tails in the parabolic Anderson model with bounded potential. Ann. Probab. 29(2), 636–682 (2001) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Grimmett, G.: Percolation, Grundlehren der Mathematischen Wissenschaften, vol. 321. Springer, Berlin (1999) Google Scholar
  7. 7.
    Hof, A.: Percolation on Penrose tilings. Can. Math. Bull. 41(2), 166–177 (1998) MATHMathSciNetGoogle Scholar
  8. 8.
    Kesten, H.: The critical probability of bond percolation on the square lattice equals 1/2. Commun. Math. Phys. 74(1), 41–59 (1980) MATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Kesten, H.: Percolation Theory for Mathematicians. Progress in Probability and Statistics, vol. 2. Birkhäuser, Boston (1982) MATHGoogle Scholar
  10. 10.
    Kirsch, W., Müller, P.: Spectral properties of the Laplacian on bond-percolation graphs. Math. Z. 252(4), 899–916 (2006). http://www.arXiv.org/abs/math-ph/0407047 MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Klopp, F., Nakamura, S.: A note on Anderson localization for the random hopping model. J. Math. Phys. 44(11), 4975–4980 (2003) MATHCrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Men’shikov, M.: Coincidence of critical points in percolation problems. Sov. Math. Dokl. 33, 856–859 (1986) MATHGoogle Scholar
  13. 13.
    Men’shikov, M.V., Molchanov, S.A., Sidorenko, A.F.: Percolation theory and some applications. In: Probability Theory. Mathematical Statistics. Theoretical Cybernetics (Russian), Itogi Nauki i Tekhniki, vol. 24, pp. 53–110. Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow (1986). Translated in J. Sov. Math. 42(4) 1766–1810. http://dx.doi.org/10.1007/BF01095508 Google Scholar
  14. 14.
    Müller, P., Stollmann, P.: Spectral asymptotics of the Laplacian on supercritical bond-percolation graphs. http://www.arxiv.org/math-ph/0506053
  15. 15.
    Müller, P., Richard, C.: Random colourings of aperiodic graphs: ergodic and spectral properties. http://www.arxiv.org/abs/0709.0821
  16. 16.
    Russo, L.: On the critical percolation probabilities. Z. Wahrscheinlichkeitstheor. Verw. Geb. 56(2), 229–237 (1981) MATHCrossRefGoogle Scholar
  17. 17.
    van den Berg, J., Kesten, H.: Inequalities with applications to percolation and reliability. J. Appl. Probab. 22(3), 556–569 (1985) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Veselić, I.: Quantum site percolation on amenable graphs. In: Proceedings of the Conference on Applied Mathematics and Scientific Computing, pp. 317–328. Springer, Dordrecht (2005). http://arXiv.org/math-ph/0308041 CrossRefGoogle Scholar
  19. 19.
    Veselić, I.: Spectral analysis of percolation Hamiltonians. Math. Ann. 331(4), 841–865 (2005). http://arXiv.org/math-ph/0405006 CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Emmy-Noether-Programme of the Deutsche Forschungsgemeinschaft & Fakultät für Mathematik, TU ChemnitzChemnitzGermany

Personalised recommendations