PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

172 Geoffrey Grimmett
Proof of Theorem 5.1. Assume that µ satisfies (5.2), and let f and g be
increasing functions. By adding a constant to the function g, we see that it
suffices to prove (5.3) under the extra hypothesis that g is strictly positive.
We assume this holds. Define positive probability measures µ1 and µ2 on
(ΩE, FE) by µ2 = µ and
g(ω)µ(ω)
µ1(ω) = �
ω ′ g(ω′ )µ(ω ′ )
for ω ∈ ΩE.
Since g is increasing, (5.6) follows from (5.2). By Holley’s inequality,
which is to say that
µ1(f) ≥ µ2(f),
�
ω f(ω)g(ω)µ(ω)
�
ω ′ g(ω′ )µ(ω ′ )
�
≥ f(ω)µ(ω)
as required. �
5.2 Disjoint Occurrence
Van den Berg has suggested a converse to the FKG inequality, namely that,
for some interpretation of the binary operation ◦,
Pp(A ◦ B) ≤ Pp(A)Pp(B) for all increasing events A, B.
The correct interpretation of A◦B turns out to be ‘A and B occur disjointly’.
We explain this statement next.
As usual, E is a finite set, ΩE = {0, 1} E , and so on. For ω ∈ ΩE, let
K(ω) = {e ∈ E : ω(e) = 1},
so that there is a one–one correspondence between configurations ω and sets
K(ω). For increasing events A, B, let
�
A ◦ B =
ω : for some H ⊆ K(ω), we have that ω ′ ∈ A and ω ′′ ∈ B,
where K(ω ′ ) = H and K(ω ′′ ) = K(ω) \ H
and we call A ◦ B the event that A and B occur disjointly.
The canonical example of disjoint occurrence in percolation theory concerns
the existence of disjoint open paths. If A = {u ↔ v} and B = {x ↔ y},
then A ◦ B is the event that are two edge-disjoint paths, one joining u to v,
and the other joining x to y.
ω
�
,