PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT
PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT
PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT
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216 <strong>Geoffrey</strong> Grimmett<br />
Lemma 8.11. Let τp(u, v) = Pp(u ↔ v), and<br />
Q(a, b) = �<br />
v,w∈Z d<br />
τp(a, v)τp(v, w)τp(w, b) for a, b ∈ Z d .<br />
Then Q is a positive-definite form, in that<br />
�<br />
f(a)Q(a, b)f(b) ≥ 0<br />
a,b<br />
for all suitable functions f : Z d → C.<br />
Proof. We have that<br />
�<br />
f(a)Q(a, b)f(b) = �<br />
g(v)τp(v, w)g(w)<br />
a,b<br />
v,w<br />
�<br />
�<br />
�<br />
= Ep g(v)1 {v↔w}g(w)<br />
v,w<br />
� � �<br />
���<br />
�2�<br />
= Ep �<br />
� g(x) �<br />
�<br />
where g(v) = �<br />
a f(a)τp(a, v), and the penultimate summation is over all<br />
open clusters C. �<br />
C<br />
x∈C<br />
We note the consequence of Lemma 8.11, that<br />
(8.12) Q(a, b) 2 ≤ Q(a, a)Q(b, b) = T(p) 2<br />
by Schwarz’s inequality.<br />
Theorem 8.13. If d ≥ 2 and T(pc) < ∞ then<br />
χ(p) ≃ (pc − p) −1<br />
as p ↑ pc.<br />
Proof. This is taken from [26]; see also [G] and [180]. The following sketch<br />
is incomplete in one important regard, namely that, in the use of Russo’s<br />
formula, one should first restrict oneself to a finite region Λ, and later pass<br />
to the limit as Λ ↑ Zd ; we omit the details of this.<br />
Write τp(u, v) = Pp(u ↔ v) as before, so that<br />
χ(p) = �<br />
τp(0, x).<br />
By (ab)use of Russo’s formula,<br />
(8.14)<br />
dχ<br />
dp<br />
= d<br />
dp<br />
x∈Z d<br />
x∈Z d<br />
�<br />
τp(0, x) = � �<br />
Pp(e is pivotal for {0 ↔ x}).<br />
x∈Zd e∈Ed
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