Percolation pp 148-168 | Cite as

Near the Critical Point: Scaling Theory

  • Geoffrey Grimmett


The behaviour of the percolation process on L d depends dramatically on whether p < p c or p > p c . In the former subcritical case, all open clusters are almost surely finite and their size-distribution has a tail which decays exponentially. In the latter supercritical case, there exists almost surely an infinite open cluster and the size-distribution of the remaining finite open clusters has a tail which decays slower than exponentially. Some of the major differences between these two phases are highlighted in the following table.


Correlation Length Critical Exponent Open Cluster Percolation Process Asymptotic Relation 
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  1. Essam, J. W. 1980 Percolation theory, Reports on Progress in Physics 43, 833–949.MathSciNetCrossRefGoogle Scholar
  2. Stauffer, D. 1979 Scaling theory of percolation clusters, Physics Reports 54, 1–74.CrossRefGoogle Scholar
  3. Wierman, J. C.1987 Duality for k-degree percolation on the square lattice, in Percolation Theory and Ergodic Theory of Infinite Particle Systems,ed. H. Kesten, 311–323, Springer, Berlin.Google Scholar
  4. Aizenman, M. 1987 General results in percolation theory, in Proceedings of Taniguchi Workshop and Symposium on Probabilistic Methods in Mathematical Physics, ed. N. Ikeda, 1–22, Kataka and Kyoto.Google Scholar
  5. Aizenman, M., Kesten, H., and Newman, C. M. 1987 Uniqueness of the infinite cluster and continuity of connectivity functions for short-and long-range percolation, Communications in Mathematical Physics 111, 505–532.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Stauffer, D. 1981 Scaling properties of percolation clusters, in Disordered Systems and Localization, eds. C. Castellani, C. DiCastro and L. Pettiti, 9–25, Springer, Berlin.CrossRefGoogle Scholar
  7. Stauffer, D. 1985 Introduction to Percolation TheoryTaylor and Francis, London. Google Scholar
  8. Toulouse, G. 1974 Perspectives from the theory of phase transitions, Il Nuovo Cimento B 23, 234–240.CrossRefGoogle Scholar
  9. Tasaki, H. 1987a Geometric critical exponent inequalities for general random cluster models, Journal of Statistical Physics 49, 841–847.MathSciNetCrossRefGoogle Scholar
  10. Tasaki, H. 1988 On the upper critical dimension of random spin systems, preprint. Tempel’man, A. A.Google Scholar
  11. Aizenman, M., Chayes, J. T., Chayes, L., and Newman, C. M. 1988 Discontinuity of the magnetization in one-dimensional 1/jx — y1 2 Ising and Potts models, Journal of Statistical Physics 50, 1–40.MathSciNetzbMATHCrossRefGoogle Scholar
  12. Stanley, H. E. 1971 Introduction to Phase Transitions and Critical Phenomena, Clarendon Press, Oxford.Google Scholar
  13. Fisher, M. E. 1983 Scaling, universality and renormalization group, in Critical Phenomena, ed. F. J. W. Hahne, 1–139, Lecture Notes in Physics no. 186, Springer, Berlin.Google Scholar
  14. Kesten, H. 1987a Surfaces with minimal random weights and maximal flows: A higher dimensional version of first-passage percolation, Illinois Journal of Mathematics 31, 99–166.MathSciNetzbMATHGoogle Scholar
  15. Chayes, J. T. and Chayes, L. 1986a Percolation and random media, in Critical Phenomena, Random Systems and Gauge Theories, Les Houches, Session XLIII, 1984, eds. K. Osterwalder and R. Stora, 1001–1142, Elsevier, Amsterdam.Google Scholar
  16. Chayes, J. T. and Chayes, L. 1987 The mean field bound for the order parameter of Bernoulli percolation, in Percolation Theory and Ergodic Theory of Infinite Particle Systems, ed. H. Kesten, 49–71, Springer, Berlin.CrossRefGoogle Scholar
  17. Coniglio, A. 1985 Shapes, surfaces and interfaces in percolation clusters, in Physics of Finely Divided Matter, Les Houches, Session XLIV, 1985, eds. N. Boccara and M. Daoud, 84–101, Springer, Berlin.Google Scholar
  18. Ben-Avraham, D. and Havlin, S. 1982 Diffusion on percolation clusters at criticality, Journal of Physics A: Mathematical and General 15, L691 — L697.CrossRefGoogle Scholar
  19. Kapitulnik, A., Aharony, A., Deutscher, G., and Stauffer, D. 1983 Self similarity and correlations in percolation, Journal of Physics A: Mathematical and General 16, L269 — L274.CrossRefGoogle Scholar
  20. Kesten, H. 1986a Aspects of first-passage percolation, in Ecole d’Eté de Probabilités de Saint Flour XIV-1984, ed. P. L. Hennequin, 125–264, Lecture Notes in Mathematics no. 1180, Springer, Berlin.Google Scholar
  21. Kesten, H. 1986b The incipient infinite cluster in two-dimensional percolation, Probability Theory and Related Fields 73, 369–394.MathSciNetzbMATHCrossRefGoogle Scholar
  22. De Masi, A., Ferrari, P. A., Goldstein, S., and Wick, W. D. 1985 Invariance principle for reversible Markov processes with application to diffusion in the percolation regime, in Particle Systems, Random Media and Large Deviations, ed. R. Durrett, 71–85, Contemporary Mathematics 41, American Mathematical Society, Providence, Rhode Island.Google Scholar
  23. Chayes, J. T. and Chayes, L. 1987 The mean field bound for the order parameter of Bernoulli percolation, in Percolation Theory and Ergodic Theory of Infinite Particle Systems, ed. H. Kesten, 49–71, Springer, Berlin.CrossRefGoogle Scholar
  24. Menshikov, M. V. 1987a Numerical bounds and strict inequalities for critical points of graphs and their subgraphs, Theory of Probability and its Applications (in Russian) 32, 599–602 (544–547 in translation).Google Scholar

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© Springer Science+Business Media New York 1989

Authors and Affiliations

  • Geoffrey Grimmett
    • 1
  1. 1.School of MathematicsUniversity of BristolBristolEngland

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