PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

268 Geoffrey Grimmett
Evidently pg(q) ≤ pc(q), and it is believed that equality is valid here. Next,
we define the kth iterate of (natural) logarithm by
λ1(n) = log n, λk(n) = log + {λk−1(n)} for k ≥ 2
where log + x = max{1, logx}.
We present the next theorem in two parts, and shall give a full proof of
part (a) only; for part (b), see [167].
Theorem 13.17. Let 0 < p < 1 and q ≥ 1, and assume that p < pg(q).
(a) If k ≥ 1, there exists α = α(p, q, k) satisfying α > 0 such that
(13.18) φ 0 � � �
p,q 0 ↔ ∂B(n) ≤ exp −αn/λk(n) � for all large n.
(b) If (13.18) holds, then there exists β = β(p, q) satisfying β > 0 such
that
φ 0 � � −βn
p,q 0 ↔ ∂B(n) ≤ e for all large n.
The spirit of the theorem is close to that of Hammersley [175] and Simon–
Lieb [236, 331], who derived exponential estimates when q = 1, 2, subject to
a hypothesis of finite susceptibility (i.e., that �
x φ0p,q(0 ↔ x) < ∞). The
latter hypothesis is slightly stronger than the assumption of Theorem 13.17
when d = 2.
Underlying any theorem of this type is an inequality. In this case we use
two, of which the first is a consequence of the following version of Russo’s
formula, taken from [74].
Theorem 13.19. Let 0 < p < 1, q > 0, and let ψp be the corresponding
random-cluster measure on a finite graph G = (V, E). Then
d
dp ψp(A)
1 �
= ψp(N1A) − ψp(N)ψp(A)
p(1 − p)
�
for any event A, where N = N(ω) is the number of open edges of a configuration
ω.
Here, ψp is used both as probability measure and expectation operator.
Proof. We express ψp(A) as
ψp(A) =
�
ω 1A(ω)πp(ω)
�
ω πp(ω)
where πp(ω) = p N(ω) (1 −p) |E|−N(ω) q k(ω) . Now differentiate throughout with
respect to p, and gather the terms to obtain the required formula. �