PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

202 Geoffrey Grimmett
Lemma 7.24. Suppose there exists a constant γ such that γ > pc(F, site)
and
(7.25) P � Z(xt+1) = 1 � �
�S1, S2, . . . , St ≥ γ for all t.
Then P(|A∞| = ∞) > 0.
We omit a formal proof of this lemma (but see [164]). Informally, (7.25)
implies that, uniformly in the past history, the chance of extending At exceeds
the critical value of a supercritical site percolation process on F. Therefore
A∞ stochastically dominates the open cluster at x1 of a supercritical site percolation
cluster. The latter cluster is infinite with strictly positive probability,
whence P(|A∞| = ∞) > 0.
Having established the three basic lemmas, we turn to the construction
itself. Recall the notation and hypotheses of Theorem 7.8. Let 0 < η < pc,
and choose
(7.26) p = pc + 1
�
1 1
2η, δ = 12η, ǫ = 24 1 − pc(F, site) � .
Note that pc(F, site) < 1 since by assumption pc(F) = pc(F, bond) < 1
(cf. Theorem 5.13). Since p > pc, we have that θ(p) > 0, and we apply
Lemma 7.17 with the above ǫ, δ to find corresponding integers m, n. We
define N = m + n + 1, and we shall define a process on the blocks of Z 3
having side-length 2N.
Consider the set {4Nx : x ∈ Z d } of vertices, and the associated boxes
Bx(N) = {4Nx + B(N) : x ∈ Z d }; these boxes we call site-boxes. A pair
Bx(N), By(N) of site-boxes is deemed adjacent if x and y are adjacent in
L d . Adjacent site-boxes are linked by bond-boxes, i.e., boxes Nz + B(N)
for z ∈ Z d exactly one component of which is not divisible by 4. If this
exceptional component of z is even, the box Nz + B(N) is called a half-way
box. See Figure 7.5.
We shall examine site-boxes one by one, declaring each to be ‘occupied’ or
‘unoccupied’ according to the existence (or not) of certain open paths. Two
properties of this construction will emerge.
(a) For each new site-box, the probability that it is occupied exceeds the
critical probability of a certain site percolation process. This will imply
that, with strictly positive probability, there is an infinite occupied
path of site-boxes.
(b) The existence of this infinite occupied path necessarily entails an infinite
open path of L d lying within some restricted region.
The site-boxes will be examined in sequence, the order of this sequence being
random, and depending on the past history of the process. Thus, the
renormalisation is ‘dynamic’ rather than ‘static’.
As above, let F be an infinite connected subset of Z d ; we shall assume for
neatness that F contains the origin 0 (otherwise, translate F accordingly).
As above, let e(1), e(2), . . . be a fixed ordering of the edges joining vertices in