PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

178 Geoffrey Grimmett
Now φ(p) = 0 when p > pc (since Pp(An) ≥ θ(p) > 0). Therefore the
above discussion needs more thought in this case. In defining the supercritical
correlation length, it is normal to work with the ‘truncated’ probabilities
Pp(An, |C| < ∞). It may be shown ([94, 164]) that the limit
�
(6.8) φ(p) = lim −
n→∞
1 � �
log Pp 0 ↔ ∂B(n), |C| < ∞
n �
exists for all p, and satisfies φ(p) > 0 if and only if p �= pc. We now define
the correlation length ξ(p) by
(6.9) ξ(p) = 1/φ(p) for 0 < p < 1.
Proof of Theorem 6.2. Define the (two-point) connectivity function τp(x, y) =
Pp(x ↔ y). Using the FKG inequality,
�
�
τp(x, y) ≥ Pp {x ↔ z} ∩ {z ↔ y} ≥ τp(x, z)τp(z, y)
for any z ∈ Zd . Set x = 0, z = me1, y = (m + n)e1, to obtain that
τp(r) = Pp(0 ↔ re1) satisfies τp(m + n) ≥ τp(m)τp(n). Therefore the limit
φ2(p) exists by the subadditive inequality.
The existence of φ1(p) may be shown similarly, using the BK inequality
as follows. Note that
{0 ↔ ∂B(m + n)} ⊆ � � �
{0 ↔ x} ◦ {x ↔ x + ∂B(n)}
x∈∂B(m)
� �
(this is geometry). Therefore βp(r) = Pp 0 ↔ ∂B(r) satisfies
βp(m + n) ≤ �
τp(0, x)βp(n).
x∈∂B(m)
Now τp(0, x) ≤ βp(m) for x ∈ ∂B(m), so that
βp(m + n) ≤ |∂B(m)|βp(m)βp(n).
With a little ingenuity, and the subadditive inequality, we deduce the existence
of φ1(p) in (6.3). That φ2(p) ≥ φ1(p) follows from the fact that
τp(0, ne1) ≤ βp(n). For the converse inequality, pick x ∈ ∂B(n) such that
τp(0, x) ≥
1
|∂B(n)| βp(n),
and assume that x1 = +n. Now
�
�
τp(0, 2ne1) ≥ Pp {0 ↔ x} ∩ {x ↔ 2ne1} ≥ τp(0, x) 2
by the FKG inequality. �