PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

Percolation and Disordered Systems 255
It follows that π(p0) ≥ x0, yielding θ(p0) > 0. Therefore pc ≤ p0, as required
in (11.5).
The proof that θ(pc) > 0 is more delicate; see [113]. �
Consider now the more general setting in which the random step involves
replacing a typical square of side-length M −k by a M ×M grid of subsquares
of common side-length M −(k+1) (the above concerns the case M = 3). For
general M, a version of the above argument yields that the corresponding
critical probability pc(M) satisfies pc(M) ≥ M −2 and also
(11.10) pc(M) < 1 if M ≥ 3.
When M = 2, we need a special argument in order to obtain that pc(2) < 1,
and this may be achieved by using the following coupling of the cases M = 2
and M = 4 (see [91, 113]). Divide C0 into a 4 × 4 grid and do as follows. At
the first stage, with probability p we retain all four squares in the top left
corner; we do similarly for the three batches of four squares in each of the
other three corners of C0. Now for the second stage: examine each subsquare
of side-length 1
4 so far retained, and delete such a subsquare with probability p
(different subsquares being treated independently). Note that the probability
measure at the first stage dominates (stochastically) product measure with
intensity π so long as (1−π) 4 ≥ 1−p. Choose π to satisfy equality here. The
composite construction outlined above dominates (stochastically) a single
step of a 4 × 4 random fractal with parameter pπ = p � 1 − (1 − p) 1 �
4 , which
implies that
and therefore pc(2) < 1 by (11.10).
11.3 A Morphology
pc(2) � 1 − (1 − pc(2)) 1 �
4 ≤ pc(4)
Random fractals have many phases, of which the existence of left-right crossings
characterises only one. A weaker property than the existence of crossings
is that the projection of C onto the x-axis is the whole interval [0, 1]. Projections
of random fractals are of independent interest (see, for example, the
‘digital sundial’ theorem of [131]). Dekking and Meester [113] have cast such
properties within a more general morphology.
We write C for a random fractal in [0, 1] 2 (such as that presented in
Section 11.1). The projection of C is denoted as
πC = {x ∈ R : (x, y) ∈ C for some y},
and λ denotes Lebesgue measure. We say that C lies in one of the following
phases if it has the stated property. A set is said to percolate if it contains a
left-right crossing of [0, 1] 2 ; dimension is denoted by ‘dim’.
I. C = ∅ a.s.