PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

7.4 Percolation in Half-Spaces
Percolation and Disordered Systems 209
In the last section, we almost succeeded in proving that θ(pc) = 0 when
d ≥ 3. The reason for this statement is as follows. Suppose θ(p) > 0 and
η > 0. There is effectively defined in Section 7.3 an event A living in a finite
box B such that
(a) Pp(A) > 1 − ǫ, for some prescribed ǫ > 0,
(b) the fact (a) implies that θ(p + η) > 0.
Suppose that we could prove this with η = 0, and that θ(pc) > 0. Then
Ppc(A) > 1 − ǫ, which implies by continuity that Pp ′(A) > 1 − ǫ for some
p ′ < pc, and therefore θ(p ′ ) > 0 by (b). This would contradict the definition
of pc, whence we deduce by contradiction that θ(pc) = 0.
The fact that η is strictly positive is vital for the construction, since
we need to ‘spend some extra money’ in order to compensate for negative
information acquired earlier in the construction. In a slightly different setting,
no extra money is required.
Let H = {0, 1, . . . } × Z d−1 be a half-space when d ≥ 3, and write pc(H)
for its critical probability. It follows from Theorem 7.8 that pc(H) = pc, since
H contains slabs of all thicknesses. Let
θH(p) = Pp(0 ↔ ∞ in H).
Theorem 7.35. We have that θH(pc) = 0.
The proof is not presented here, but may be found in [48, 49]. It is closely
related to that presented in Section 7.3, but with some crucial differences.
The construction of blocks is slightly more complicated, owing to the lack
of symmetry of H, but there are compensating advantages of working in a
half-space. For amusement, we present in Figure 7.10 two diagrams (relevant
to the argument of [49]) depicting the necessary constructions.
As observed in Section 3.3, such a conclusion for half-spaces has a striking
implication for the conjecture that θ(pc) = 0. If θ(pc) > 0, then there exists
a.s. a unique infinite open cluster in Z d , which is a.s. partitioned into (only)
finite clusters by any division of Z d into two half-spaces.
7.5 Percolation Probability
Although the methods of Chapter 6 were derived primarily in order to study
subcritical percolation, they involve a general inequality of wider use, namely
g ′ �
�
n
π (n) ≥ gπ(n)
− 1
gπ(i)
� n
i=0
where gπ(n) = Pπ(0 ↔ ∂Sn); see equations (3.10) and (3.18) in Section 6.
We argue loosely as follows. Clearly gπ(n) → θ(π) as n → ∞, whence (cross
your fingers here)
θ ′ � �
1
(π) ≥ θ(π) − 1
θ(π)