Percolation pp 44-71 | Cite as

The Uniqueness of the Critical Point

  • Geoffrey Grimmett
Chapter

Abstract

Let us consider bond percolation on L d with edge-density p, where d ≥ 2. As we have remarked, the following two functions are two of the principal characters in the action:
$$\theta (p) = p_p (|C| = \infty ),\:\chi (p) = E_p |C|,$$
where C is the open cluster containing the origin and |C| is the number of vertices in C. We have seen that there exists p c = p c (d) in (0, 1) such that
$$\theta (p)\left\{ {\begin{array}{*{20}c} { = 0\:if\:p < p_c ,} \\ { > 0\:if\:p > p_c .} \\ \end{array} } \right.$$
(3.1)
.

Keywords

Open Cluster Differential Inequality Open Path Open Edge Radius Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

  1. Harris, T. E. 1963 The Theory of Branching Processes, Springer, Berlin.MATHCrossRefGoogle Scholar
  2. Grimmett, G. R. and Kesten, H. 1988 An Introduction to the Theory of Coverage Processes, Wiley, New York.Google Scholar
  3. Menshikov, M. V. 1985 Estimates of percolation thresholds for lattices in F8“ Soviet Mathematics Doklady 32, 368–370. Google Scholar
  4. 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
  5. Lieb, E. H. 1980 A refinement of Simon’s correlation inequality Communications in Mathematical Physics 77127–135. Google Scholar
  6. Menshikov, M. V. 1985 Estimates of percolation thresholds for lattices in F8“ Soviet Mathematics Doklady 32, 368–370. Google Scholar
  7. Menshikov, M. V. 1986 Coincidence of critical points in percolation problems Soviet Mathematics Doklady 33, 856–859. Google Scholar
  8. Grigorchuk, R. I. 1983 On Milnor’s problem of group growth, Soviet Mathematics Doklady 28, 23–26.MATHGoogle Scholar
  9. Chayes, J. T., Chayes, L., and Newman, C. M. 1987 Bernoulli percolation above threshold: an invasion percolation analysis, Annals of Probability 15, 1272–1287.MathSciNetMATHCrossRefGoogle Scholar
  10. Frisch, H. L., Gordon, S. B., Vyssotsky, V. A., and Hammersley, J. M. 1962 Monte Carlo solution of bond percolation processes on various crystal lattices, The Bell System Technical Journal 41, 909–920.Google Scholar
  11. Griffiths, R. B. 1967 Correlations in Ising ferromagnets. II. External magnetic fields Journal of Mathematical Physics 8, 484–489. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

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

Personalised recommendations