PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

Percolation and Disordered Systems 269
Lemma 13.20. Let 0 < p < 1 and q ≥ 1. For any non-empty increasing
event A,
d �
log ψp(A)
dp
� ≥ ψp(FA)
p(1 − p)
where
�
�
FA(ω) = inf
e
� ω ′ (e) − ω(e) � : ω ′ ≥ ω, ω ′ ∈ A
Proof. It may be checked that FA1A = 0, and that N + FA is increasing.
Therefore, by the FKG inequality,
�
.
� �
ψp(N1A) − ψp(N)ψp(A) = ψp (N + FA)1A − ψp(N)ψp(A)
≥ ψp(FA)ψp(A).
Now use Theorem 13.19. �
The quantity FA is central to the proof of Theorem 13.17. In the proof,
we shall make use of the following fact. If A is increasing and A ⊆ B1 ∩B2 ∩
· · · ∩ Bm, where the Bi are cylinder events defined on disjoint sets of edges,
then
(13.21) FA ≥
m�
FBi.
i=1
Lemma 13.22. Let q ≥ 1 and 0 < r < s < 1. There exists a function
c = c(r, s, q), satisfying 1 < c < ∞, such that
and for all increasing events A.
ψr(FA ≤ k) ≤ c k ψs(A) for all k ≥ 0
Proof. We sketch this, which is similar to the so called ‘sprinkling lemma’ of
[14]; see also [G, 167].
Let r < s. The measures ψr and ψs may be coupled together in a natural
way. That is, there exists a probability measure µ on Ω2 E = {0, 1}E × {0, 1} E
such that:
(a) the first marginal of µ is ψr,
(b) the second marginal of µ is ψs,
(c) µ puts measure 1 on the set of configurations (π, ω) ∈ Ω 2 E
such that
π ≤ ω.
Furthermore µ may be found such that the following holds. There exists a
positive number β = β(r, s, q) such that, for any fixed ξ ∈ ΩE and subset B
of edges (possibly depending on ξ), we have that
(13.23)
µ � {(π, ω) : ω(e) = 1 for e ∈ B, π = ξ} �
µ � {(π, ω) : π = ξ} � ≥ β|B| .