PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

Percolation and Disordered Systems 227
not simply use the FKG inequality at (9.81). Instead, let Aπ be the event
that the path π is open. Then, in the notation of [G],
�
3
P1 LR(
2 2l, l)� ≥ P1
2
�
N + � �
∩
≥ P1
2 (N+ )P1
2
≥ P1
≥ P1
2 (N+ ) �
π
π∈T −
[Aπ ∩ M − π ]
� �
��
[Aπ ∩ M − �
π ]
π
P1
2 (Lπ ∩ M − π )
2 (N+ )(1 − √ 1 − τ) �
= P1
2 (N+ )(1 − √ 1 − τ)P1
2 (L− )
≥ (1 − √ 1 − τ) 3
π
by FKG
P1
2 (Lπ) by (9.84)
as in [G].
There are several applications of the RSW lemma, of which we present
one.
Theorem 9.5. There exist constants A, α satisfying 0 < A, α < ∞ such
that
(9.6)
1
2n−1/2 � � −α
≤ P1 0 ↔ ∂B(n) ≤ An .
2
Similar power-law estimates are known for other macroscopic quantities
at and near the critical point pc = 1
2 . In the absence of a proof that quantities
have ‘power-type’ singularities near the critical point, it is reasonable to look
for upper and lower bounds of the appropriate type. As a general rule, one
such bound is usually canonical, and applies to all percolation models (viz.
the inequality (7.36) that θ(p) − θ(pc) ≥ a(p − pc)). The complementary
bound is harder, and is generally unavailable at the moment when d ≥ 3
(but d is not too large).
Proof. Let R(n) = [0, 2n] × [0, 2n − 1], and let LR(n) be the event that R(n)
is traversed from left to right by an open path. We have by self-duality that
1
P1 (LR(n)) =
2 2 . On the event LR(n), there exists a vertex x with x1 = n such
that x ↔ x + ∂B(n) by two disjoint open paths. See Figure 9.3. Therefore
1
2
2n−1
� � �
= P1 LR(n) ≤
2
k=0
P1
2 (Ak
� �2 ◦ Ak) ≤ 2nP1 0 ↔ ∂B(n)
2
where Ak = {(n, k) ↔ (n, k) + ∂B(n)}, and we have used the BK inequality.
This provides the lower bound in (9.6).
For the upper bound, we have from (9.4) that P1 (O(l)) ≥ ξ for all l,
2
where ξ > 0. Now, on the event {0 ↔ ∂B(n)}, there can be no closed dual