PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

Percolation and Disordered Systems 271
At the next stages we shall need to work slightly harder. Fix a positive
integer m, and let Ri = im for 0 ≤ i ≤ K where K = ⌊n/m⌋. Let Li =
{∂B(Ri) ↔ ∂B(Ri+1)} and Hi = FLi. By (13.21),
(13.28) Fn ≥
K−1 �
i=0
Hi.
Now there exists a constant η (< ∞) such that
(13.29) φp,q(Li) ≤ |∂B(Ri)|φp,q(Am) ≤ ηn d−1 φp,q(Am)
for 0 ≤ i ≤ K − 1.
Let r < s < pg, and let c2 = c2(s), ∆2 = ∆2(s) as in (13.27). From
(13.28)–(13.29),
φs,q(Fn) ≥
We now choose m by
K−1 �
i=0
φs,q(Li) ≥ K � 1 − ηn d−1 φp,q(Am) �
�
≥ K 1 − ηn d−1
c2
m d−1+∆2
�
.
m = � (2ηc2)n d−1� 1/(d−1+∆2)
(actually, an integer close to this value) to find that
φs,q(Fn) ≥ 1
2K ≥ Dn∆3
for some D > 0, 0 < ∆3 < 1. Substitute into (13.24) to obtain
(13.30) φr,q(An) ≤ exp{−c3n ∆3 }
for some positive c3 = c3(r), ∆3 = ∆3(r). This improves (13.27) substantially.
We repeat the last step, using (13.30) in place of (13.27), to obtain
�
(13.31) φr,q(An) ≤ exp − c4n
(log n) ∆4
�
if r < pg
for some c4 = c4(r) > 0 and 1 < ∆4 = ∆4(r) < ∞.
At the next stage, we use (13.28)–(13.29) more carefully. This time, set
m = (log n) 2 , and let r < s < t < pg. By (13.29) and (13.31),
φt,q(Li) ≤ ηn d−1 �
exp − c4m
(log m) ∆4
�