Percolation pp 169-185 | Cite as

Near the Critical Point: Rigorous Results

  • Geoffrey Grimmett


It is the presence of circuits in L d which causes difficulties in exact calculations: there are many different paths joining two specified vertices, and pairs of such paths generally have edges in common In attempting to understand critical phenomena, it is usual to tackle first the corresponding problem on a rather special lattice, being a lattice which is devoid of circuits; we do this in the hope that the ensuing calculation will be simple but will help in the development of insight into more general situations. Such a lattice is called a tree, and it is to percolation on trees that this section is devoted.


Critical Exponent Binary Tree Open Cluster Tauberian Theorem Critical Probability 
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  1. Harris, T. E. 1963 The Theory of Branching Processes, Springer, Berlin.zbMATHCrossRefGoogle Scholar
  2. Athreya, K. B. and Ney, P. E. 1972 Branching Processes, Springer, Berlin.zbMATHCrossRefGoogle Scholar
  3. Appel, M. J. and Wierman, J. C. 1987 On the absence of infinite AB percolation clusters in bipartite graphs, Journal of Physics A: Mathematical and General 20, 2527–2531.MathSciNetCrossRefGoogle Scholar
  4. Asmussen, S. and Hering, H. 1983 Branching Processes, Birkhäuser, Boston.zbMATHGoogle Scholar
  5. Durrett, R. 1985a Some general results concerning the critical exponents of percolation processes, Zeitschrift für Wahrscheinlichkeitstheorie and Verwandte Gebiete 69, 421–437.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Kesten, H. 1987b Scaling relations for 2D-percolation, Communications in Mathematical Physics 109, 109–156.MathSciNetzbMATHCrossRefGoogle Scholar
  7. Aizenman, M. and Newman, C. M. 1984 Tree graph inequalities and critical behavior in percolation models, Journal of Statistical Physics 36, 107–143.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 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
  9. Menshikov, M. V. 1987b Critical points in the mathematical theory of percolation, Doctoral Dissertation, University of Moscow.Google Scholar
  10. Aizenman, M., Barsky, D. J., and Fernandez, R. 1987 The phase transition in a general class of Ising-type models is sharp, Journal of Statistical Physics 47, 343–374.MathSciNetCrossRefGoogle Scholar
  11. Newman, C. M. 1987c Another critical exponent inequality for percolation: ß > 2/6, Journal of Statistical Physics 47, 695–699.MathSciNetCrossRefGoogle Scholar
  12. Newman, C. M. 1987b Inequalities for y and related exponents in short-and long-range percolation, in Percolation Theory and Ergodic Theory of Infinite Particle Systems, ed. H. Kesten, 229–244, Springer, Berlin.CrossRefGoogle Scholar
  13. Newman, C. M. 1986 Some critical exponent inequalities for percolation Journal of Statistical Physics 45, 359–368. Google Scholar
  14. Chayes, J. T., Chayes, L., and Newman, C. M. 1987 Bernoulli percolation above threshold: an invasion percolation analysis, Annals of Probability 15, 1272–1287.MathSciNetzbMATHCrossRefGoogle Scholar
  15. Nguyen, B. G. 1987b Gap exponents for percolation processes, Journal of Statistical Physics 49, 235–243.MathSciNetzbMATHCrossRefGoogle Scholar
  16. Hara, T. and Slade, G. 1988 Mean-field critical phenomena for percolation in high dimensions, preprint.Google Scholar

Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

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

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