PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

11.1. Random Fractals
Percolation and Disordered Systems 251
11. FRACTAL PERCOLATION
Many so called ‘fractals’ are generated by iterative schemes, of which the
classical middle-third Cantor construction is a canonical example. When the
scheme incorporates a randomised step, then the ensuing set may be termed
a ‘random fractal’. Such sets may be studied in some generality (see [130,
152, 184, 312]), and properties of fractal dimension may be established. The
following simple example is directed at a ‘percolative’ property, namely the
possible existence in the random fractal of long paths.
We begin with the unit square C0 = [0, 1] 2 . At the first stage, we divide C0
into nine (topologically closed) subsquares of side-length 1
3 (in the natural
way), and we declare each of the subsquares to be open with probability
p (independently of any other subsquare). Write C1 for the union of the
open subsquares thus obtained. We now iterate this construction on each
subsquare in C1, obtaining a collection of open (sub)subsquares of side-length
1
9 . After k steps we have obtained a union Ck of open squares of side-length
( 1
3 )k . The limit set
(11.1) C = lim
k→∞ Ck = �
is a random set whose metrical properties we wish to study. See Figure 11.1.
Constructions of the above type were introduced by Mandelbrot [255] and
initially studied by Chayes, Chayes, and Durrett [91]. Recent papers include
[113, 133, 301]. Many generalisations of the above present themselves.
(a) Instead of working to base 3, we may work to base M where M ≥ 2.
(b) Replace two dimensions by d dimensions where d ≥ 2.
(c) Generalise the use of a square.
In what follows, (a) and (b) are generally feasible, while (c) poses a different
circle of problems.
It is easily seen that the number Xk of squares present in Ck is a branching
process with family-size generating function G(x) = (1 − p + px) 9 . Its
extinction probability η is a root of the equation η = G(η), and is such that
Therefore
k≥1
Ck
� 1 = 1 if p ≤ 9
Pp(extinction)
,
< 1 if p > 1
9 .
(11.2) Pp(C = ∅) = 1 if and only if p ≤ 1
9 .
When p > 1
9 , then C (when non-extinct) is large but ramified.