PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

270 Geoffrey Grimmett
That is to say, conditional on the first component of a pair (π, ω) sampled
according to µ, the measure of the second component dominates a non-trivial
product measure.
Now suppose that ξ (∈ ΩE) is such that FA(ξ) ≤ k, and find a set B =
B(ξ) of edges, such that |B| ≤ k, and with the property that ξ B ∈ A, where
ξ B is the configuration obtained from ξ by declaring all edges in B to be
open. By (13.23),
ψs(A) ≥ �
ξ:FA(ξ)≤k
≥ β k ψr(FA ≤ k)
µ � {(π, ω) : ω(e) = 1 for e ∈ B, π = ξ} �
as required. �
Proof of Theorem 13.17. (a) Write An = {0 ↔ ∂B(n)} and ψp = φ 0 B(m),p,q
where p < pc = pc(q). We apply Lemma 13.20 (in an integrated form), and
pass to the limit as m → ∞, to obtain that the measures φp,q = φ 0 p,q satisfy
(13.24) φr,q(An) ≤ φs,q(An)exp � −4(s − r)φs,q(Fn) � , if r ≤ s,
where Fn = FAn (we have used Theorem 13.2(a) here, together with the fact
that Fn is a decreasing random variable).
Similarly, by summing the corresponding inequality of Lemma 13.22 over
k, and letting m → ∞, we find that
(13.25) φr,q(Fn) ≥
− logφs,q(An)
log c
− c
c − 1
if r < s.
We shall use (13.24) and (13.25) in an iterative scheme. At the first stage,
assume r < s < t < pg = pg(q). Find c1(t) such that
(13.26) φp,q(An) ≤ c1(t)
n d−1
By (13.25),
φs,q(Fn) ≥
which we insert into (13.24) to obtain
(13.27) φr,q(An) ≤
(d − 1)log n
log c
c2(r)
n d−1+∆2(r)
for all n.
+ O(1),
for all n,
and for some constants c2(r), ∆2(r) (> 0). This is an improvement over
(13.26).