The Uniqueness of the Critical Point

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)

.