PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

Percolation and Disordered Systems 217
If e = 〈a, b〉 is pivotal for {0 ↔ x}, then one of the events {0 ↔ a} ◦ {b ↔ x}
and {0 ↔ b} ◦ {a ↔ x} occurs. Therefore, by the BK inequality,
(8.15)
dχ
dp
≤ �
x
= �
e=〈a,b〉
�
e=〈a,b〉
� τp(0, a)τp(b, x) + τp(0, b)τp(a, x) �
� τp(0, a) + τp(0, b) � χ(p) = 2dχ(p) 2 .
This inequality may be integrated, to obtain that
1 1
−
χ(p1) χ(p2) ≤ 2d(p2 − p1) for p1 ≤ p2.
Take p1 = p < pc and p2 > pc, and allow the limit p2 ↓ pc, thereby obtaining
that
1
(8.16) χ(p) ≥
2d(pc − p)
for p < pc.
In order to obtain a corresponding lower bound for χ(p), we need to obtain
a lower bound for (8.14). Let e = 〈a, b〉 in (8.14), and change variables
(x ↦→ x − a) in the summation to obtain that
(8.17)
dχ
dp
� �
=
x,y |u|=1
�
Pp 0 ↔ x, u ↔ y off Cu(x) �
where the second summation is over all unit vectors u of Z d . The (random
set) Cu(x) is defined as the set of all points joined to x by open paths not
using 〈0, u〉.
In the next lemma, we have a strictly positive integer R, and we let
B = B(R). The set CB(x) is the set of all points reachable from x along
open paths using no vertex of B.
Lemma 8.18. Let u be a unit vector. We have that
�
Pp 0 ↔ x, u ↔ y off Cu(x) � �
≥ α(p)Pp 0 ↔ x, u ↔ y off CB(x) � ,
where α(p) = {min(p, 1 − p)} 2d(2R+1)d.
Proof of Lemma 8.18. Define the following events,
(8.19)
E = � 0 ↔ x, u ↔ y off Cu(x) � ,
F = � 0 ↔ x, u ↔ y off CB(x) � ,
G = � B ∩ C(x) �= ∅, B ∩ C(y) �= ∅, CB(x) ∩ CB(y) = ∅ � ,