PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

7.3 Limit of Slab Critical Points
Percolation and Disordered Systems 195
Material in this section is taken from [164]. We assume that d ≥ 3 and that
p is such that θ(p) > 0; under this hypothesis, we wish to gain some control
of the (a.s.) unique open cluster. In particular we shall prove the following
theorem, in which pc(A) denotes the critical value of bond percolation on the
subgraph of Z d induced by the vertex set A. In this notation, pc = pc(Z d ).
Theorem 7.8. If F is an infinite connected subset of Z d with pc(F) < 1,
then for each η > 0 there exists an integer k such that
� �
pc 2kF + B(k) ≤ pc + η.
Choosing F = Z 2 × {0} d−2 , we have that 2kF + B(k) = {x ∈ Z d : −k ≤
xj ≤ k for 3 ≤ j ≤ d}. The theorem implies that pc(2kF + B(k)) → pc as
k → ∞, which is a stronger statement than the statement that pc = pc(S).
In the remainder of this section, we sketch the salient features of the
block construction necessary to prove the above theorem. This construction
may be used directly to obtain further information concerning supercritical
percolation.
The main idea involves working with a ‘block lattice’ each point of which
represents a large box of L d , these boxes being disjoint and adjacent. In this
block lattice, we declare a vertex to be ‘open’ if there exist certain open paths
in and near the corresponding box of L d . We shall show that, with positive
probability, there exists an infinite path of open vertices in the block lattice.
Furthermore, this infinite path of open blocks corresponds to an infinite open
path of L d . By choosing sufficiently large boxes, we aim to find such a path
within a sufficiently wide slab. Thus there is a probabilistic part of the proof,
and a geometric part.
There are two main steps in the proof. In the first, we show the existence
of long finite paths. In the second, we show how to take such finite paths
and build an infinite cluster in a slab.
The principal parts of the first step are as follows. Pick p such that
θ(p) > 0.
1. Let ǫ > 0. Since θ(p) > 0, there exists m such that
� �
Pp B(m) ↔ ∞ > 1 − ǫ.
This is elementary probability theory.
2. Let n ≥ 2m, say, and let k ≥ 1. We may choose n sufficiently large to
ensure that, with probability at least 1 −2ǫ, B(m) is joined to at least
k points in ∂B(n).
3. By choosing k sufficiently large, we may ensure that, with probability
at least 1 − 3ǫ, B(m) is joined to some point of ∂B(n), which is itself
connected to a copy of B(m), lying ‘on’ the surface ∂B(n) and every
edge of which is open.