The Annals of Probability

Critical probabilities for site and bond percolation models

G. R. Grimmett and A. M. Stacey

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Abstract

.Any infinite graph $G = (V, E)$ has a site percolation critical probability $p_c^{\rm site}$ and a bond percolation critical probability $p_c^{\rm bond}$. The well-known weak inequality $p_c^{\rm site} \geq p_c^{\rm bond}$ is strengthened to strict inequality for a c c broad category of graphs $G$, including all the usual finite-dimensional lattices in two and more dimensions. The complementary inequality $p_c^{\rm site} \leq 1 - (1 - p_c^{\rm bond})^{\Delta - 1}$ is proved also, where $\Delta$ denotes the supremum of the vertex degrees of $G$.

Article information

Source
Ann. Probab., Volume 26, Number 4 (1998), 1788-1812.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855883

Digital Object Identifier
doi:10.1214/aop/1022855883

Mathematical Reviews number (MathSciNet)
MR1675079

Zentralblatt MATH identifier
0935.60097

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]

Keywords
Percolation enhancement critical probability

Citation

Grimmett, G. R.; Stacey, A. M. Critical probabilities for site and bond percolation models. Ann. Probab. 26 (1998), no. 4, 1788--1812. doi:10.1214/aop/1022855883. https://projecteuclid.org/euclid.aop/1022855883


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References

  • AIZENMAN, M. and GRIMMETT, G. R. 1991. Strict monotonicity for critical points in percolation and ferromagnetic models. J. Statist. Phys. 63 817 835.
  • BENJAMINI, I., LYONS, R., PERES, Y. and SCHRAMM, O. 1997. Group-invariant percolation on graphs. Geom. Funct. Anal. To appear.d
  • BENJAMINI, I. and SCHRAMM, O. 1996. Percolation beyond, many questions and a few answers. Electron. Comm. Probab. 1 71 82.
  • BEZUIDENHOUT, C. E., GRIMMETT, G. R. and KESTEN, H. 1993. Strict inequality for critical values of Potts models and random-cluster processes. Comm. Math. Phys. 158 1 16.
  • BOLLOBAS, B. 1979. Graph Theory. Springer, Berlin. ´
  • DURRETT, R. T. 1988. Lecture Notes on Particle Systems and Percolation. Wadsworth and Brooks Cole, Pacific Grove, CA.
  • GRIMMETT, G. R. 1989. Percolation. Springer, Berlin.
  • GRIMMETT, G. R. 1994. Potts models and random-cluster processes with many-body interactions. J. Statist. Phys. 75 67 121.
  • GRIMMETT, G. R. 1997. Percolation and disordered systems. Ecole d'Ete de Probabilites de Saint ´ ´ Flour XXVI. Lecture Notes in Math. 1665 153 300. Springer, Berlin.
  • HAMMERSLEY, J. M. 1961. Comparison of atom and bond percolation. J. Math. Phys. 2 728 733.Z.
  • HIGUCHI, Y. 1982. Coexistence of the infinite * clusters a remark on the square lattice site percolation.Wahrsch. Verw. Gebiete 61 75 81. Z.
  • HOLROYD, A. E. 1998. Existence and uniqueness of infinite components in generic rigidity percolation. Ann. Appl. Probab. 8 944 973.
  • HUGHES, B. D. 1996. Random Walks and Random Environments. Vol. 2. Random Environments. Oxford Univ. Press.
  • KESTEN, H. 1982. Percolation Theory for Mathematicians. Birkhauser, Boston. ¨
  • MENSHIKOV, M. V. 1987. Quantitative estimates and rigorous inequalities for critical points of a graph and its subgraphs. Theory Probab. Appl. 32 544 547.
  • OXLEY, J. G. and WELSH, D. J. A. 1979. On some percolation results of J. M. Hammersley. J. Appl. Probab. 16 526 540.
  • TOTH, B. 1985. A lower bound for the critical probability of the square lattice site percolation. ´Wahrsch. Verw. Gebiete 69 19 22.