Percolation pp 44-71 | Cite as

The Uniqueness of the Critical Point

  • Geoffrey Grimmett


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
$$\theta (p)\left\{ {\begin{array}{*{20}c} { = 0\:if\:p < p_c ,} \\ { > 0\:if\:p > p_c .} \\ \end{array} } \right.$$


Open Cluster Differential Inequality Open Path Open Edge Radius Distribution 
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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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