zbMATH — the first resource for mathematics

Sharpness of the phase transition and exponential decay of the subcritical cluster size for percolation on quasi-transitive graphs. (English) Zbl 1214.82028
Summary: 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 and 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 and 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 Aizenman and Barsky’s method.

82B26 Phase transitions (general) in equilibrium statistical mechanics
82B43 Percolation
82B27 Critical phenomena in equilibrium statistical mechanics
Full Text: DOI arXiv
[1] Aizenman, M., Barsky, D.J.: Sharpness of the phase transition in percolation models. Commun. Math. Phys. 108(3), 489–526 (1987) · Zbl 0618.60098 · doi:10.1007/BF01212322
[2] Aizenman, M., Newman, C.M.: Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys. 36(1-2), 107–143 (1984) · Zbl 0586.60096 · doi:10.1007/BF01015729
[3] Antunović, T., Veselić, I.: Equality of Lifshitz and van Hove exponents on amenable Cayley graphs. http://www.arxiv.org/abs/0706.2844 · Zbl 1171.05027
[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) · Zbl 1170.05314
[5] Biskup, M., König, W.: Long-time tails in the parabolic Anderson model with bounded potential. Ann. Probab. 29(2), 636–682 (2001) · Zbl 1018.60093 · doi:10.1214/aop/1008956688
[6] Grimmett, G.: Percolation, Grundlehren der Mathematischen Wissenschaften, vol. 321. Springer, Berlin (1999)
[7] Hof, A.: Percolation on Penrose tilings. Can. Math. Bull. 41(2), 166–177 (1998) · Zbl 0915.60091 · doi:10.4153/CMB-1998-026-0
[8] Kesten, H.: The critical probability of bond percolation on the square lattice equals 1/2. Commun. Math. Phys. 74(1), 41–59 (1980) · Zbl 0441.60010 · doi:10.1007/BF01197577
[9] Kesten, H.: Percolation Theory for Mathematicians. Progress in Probability and Statistics, vol. 2. Birkhäuser, Boston (1982) · Zbl 0522.60097
[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 · Zbl 1087.60073 · doi:10.1007/s00209-005-0895-5
[11] Klopp, F., Nakamura, S.: A note on Anderson localization for the random hopping model. J. Math. Phys. 44(11), 4975–4980 (2003) · Zbl 1062.82049 · doi:10.1063/1.1616998
[12] Men’shikov, M.: Coincidence of critical points in percolation problems. Sov. Math. Dokl. 33, 856–859 (1986) · Zbl 0615.60096
[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 · Zbl 0647.60103
[14] Müller, P., Stollmann, P.: Spectral asymptotics of the Laplacian on supercritical bond-percolation graphs. http://www.arxiv.org/math-ph/0506053 · Zbl 1127.60090
[15] Müller, P., Richard, C.: Random colourings of aperiodic graphs: ergodic and spectral properties. http://www.arxiv.org/abs/0709.0821
[16] Russo, L.: On the critical percolation probabilities. Z. Wahrscheinlichkeitstheor. Verw. Geb. 56(2), 229–237 (1981) · Zbl 0457.60084 · doi:10.1007/BF00535742
[17] van den Berg, J., Kesten, H.: Inequalities with applications to percolation and reliability. J. Appl. Probab. 22(3), 556–569 (1985) · Zbl 0571.60019 · doi:10.2307/3213860
[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 · Zbl 1072.82532
[19] Veselić, I.: Spectral analysis of percolation Hamiltonians. Math. Ann. 331(4), 841–865 (2005). http://arXiv.org/math-ph/0405006 · Zbl 1117.81064 · doi:10.1007/s00208-004-0610-6
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.