PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

190 Geoffrey Grimmett
where δ 2 = n0g0.
We are essentially finished. Let n > n0, and find an integer k such that
nk−1 ≤ n < nk; this is always possible since gk → 0 as k → ∞, and therefore
nk−1 < nk for all large k. Then
gp(n) ≤ gpk−1 (nk−1) since p ≤ pk−1
= gk−1
≤ δn −1/2
k
≤ δn −1/2
by (3.35)
since n < nk
as required. This is valid for n > n0, but we may adjust the constant δ so
that a similar inequality is valid for all n ≥ 1. �
6.3 Ornstein–Zernike Decay
The connectivity function τp(x, y) = Pp(x ↔ y) decays exponentially when
p < pc, which is to say that the limits
�
(6.12) φ(p, x) = lim −
n→∞
1
n log τp(0,
�
nx)
exist and satisfy φ(p, x) > 0 for 0 < p < pc and x ∈ Z d \ {0} (cf. Theorem
6.2).
In one direction, this observation may lead to a study of the function
φ(p, ·). In another, one may ask for finer asymptotics in (6.12). We concentrate
on the case x = e1, and write φ(p) = φ(p, e1).
Theorem 6.13 (Ornstein–Zernike Decay). Suppose that 0 < p < pc.
There exists a positive function A(p) such that
τp(0, ne1) = � 1 + O(n −1 ) � A(p)
n 1
2 (d−1)e−nφ(p)
as n → ∞.
The correction factor n − 1
2 (d−1) occurs similarly in many other disordered
systems, as was proposed by Ornstein and Zernike [300]. Theorem 6.13, and
certain extensions, was obtained for percolation by Campanino, Chayes, and
Chayes [85].