PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

256 Geoffrey Grimmett
II. P(C �= ∅) > 0, dim(πC) = dimC a.s.
III. dim(πC) < dimC a.s. on {C �= ∅}, but λ(πC) = 0 a.s.
IV. 0 < λ(πC) < 1 a.s. on {C �= ∅}.
V. P � λ(πC) = 1 � > 0 but C does not percolate a.s.
VI. P(C percolates) > 0.
In many cases of interest, there is a parameter p, and the ensuing fractal
moves through the phases, from I to VI, as p increases from 0 to 1. There
may be critical values pM,N at which the model moves from Phase M to
Phase N. In a variety of cases, the critical values pI,II, pII,III, pIII,IV can be
determined exactly, whereas pIV,V and pV,VI can be much harder to find.
Here is a reasonably large family of random fractals. As before, they
are constructed by dividing a square into 9 equal subsquares. In this more
general system, we are provided with a probability measure µ, and we replace
a square by the union of a random collection of subsquares sampled according
to µ. This process is iterated on all relevant scales. Certain parameters are
especially relevant. Let σl be the number of subsquares retained from the lth
column, and let ml = E(σl) be its mean.
Theorem 11.11. We have that
(a) C = ∅ if and only if �3 l=1 ml ≤ 1 (unless some σl is a.s. equal to 1),
(b) dim(πC) = dim(C) a.s. if and only if
3�
ml log ml ≤ 0,
l=1
(c) λ(πC) = 0 a.s. if and only if
3�
log ml ≤ 0.
l=1
For the proofs, see [112, 133]. Consequently, one may check the Phases I,
II, III by a knowledge of the ml only.
Next we apply Theorem 11.11 to the random Cantor set of Section 11.1,
to obtain that, for this model, pI,II = 1
9 , and we depart Phase II as p increases
through the value 1
3 . The system is never in Phase III (by Theorem 11.1(c))
or in Phase IV (by Theorem 1 of [133]). It turns out that pII,V = 1
3 and
1
2 < pV,VI < 1.
For the ‘random Sierpinski carpet’ (RSC) the picture is rather different.
This model is constructed as the above but with one crucial difference: at
each iteration, the central square is removed with probability one, and the
others with probability 1 − p (see Figure 11.4). Applying Theorem 11.11 we
find that
pI,II = 1
8 , pII,III = 54 −1 4, pIII,IV = 18 −1 3,