PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

Percolation and Disordered Systems 201
We may apply this with K = K(m, n), since K(m, n) is defined on the set of
edges exterior to B(n). Therefore, by (7.21), (7.22), and (7.19),
(7.23)
Pp
�
|U � K(m, n) �� �
� > t
≥ 1 − (1 − p) −t � �
Pp ∂R � K(m, n) in B(n)
≥ 1 − (1 − p) −t η > 1 − 1
2ǫ. We now couple together the percolation processes with different values
of p on the same probability space, as described in Section 2.3 and just
prior to the statement of Lemma 7.17. We borrow the notation and results
derived above by specialising to the p-open edges. Conditional on the set
U = U(K(m, n)), the values of X(e), for e ∈ U, are independent and uniform
on [0, 1]. Therefore
�
� �
�
P every e in U is (β(e) + δ)-closed, |U| > t�
H ≤ (1 − δ) t ,
whence, using (7.18) and (7.23),
�
� �
�
P some e in U is (β(e) + δ)-open � H ≥ P(|U| > t | H) − (1 − δ) t
= Pp(|U| > t) − (1 − δ) t
≥ (1 − 1 1
2ǫ) − 2ǫ, and the lemma is proved. �
This completes the two key geometrical lemmas. In moving to the second
part of the proof, we shall require a method for comparison of a ‘dependent’
process and a site percolation process. The argument required at this stage
is as follows.
Let F be an infinite connected subset of Ld for which the associated
(site) critical probability satisfies pc(F, site) < 1, and let {Z(x) : x ∈ F } be
random variables taking values in [0, 1]. We construct a connected subset of
F in the following recursive manner. Let e(1), e(2), . . . be a fixed ordering
of the edges of the graph induced by F. Let x1 ∈ F, and define the ordered
pair S1 = (A1, B1) of subsets of F by
�
({x1}, ∅) if Z(x1) = 1
S1 =
(∅, {x1}) if Z(x1) = 0.
Having defined S1, S2, . . . , St = (At, Bt), for t ≥ 1, we define St+1 as follows.
Let f be the earliest edge in the fixed ordering of the e(i) with the property
that one endvertex, xt+1 say, lies in At and the other endvertex lies outside
At ∪ Bt. Then we declare
�
(At ∪ {xt+1}, Bt) if Z(xt+1) = 1,
St+1 =
(At, Bt ∪ {xt+1}) if Z(xt+1) = 0.
If no such edge f exists, we declare St+1 = St. The sets At, Bt are nondecreasing,
and we set A∞ = limt→∞ At, B∞ = limt→∞ Bt. Think about
A∞ as the ‘occupied cluster’ at x1, and B∞ as its external boundary.