PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

Percolation and Disordered Systems 203
Fig. 7.5. The hatched squares are site-boxes, and the dotted squares are half-way
boxes. Each box has side-length 2N.
F. We shall examine the site-boxes Bx(N), for x ∈ F, and determine their
states. This we do according to the algorithm sketched before Lemma 7.24,
for appropriate random variables Z(x) to be described next.
We begin at the origin, with the site-box B0(N) = B(N). Once we have
explained what is involved in determining the state of B0(N), most of the
work will have been done. (The event {B0(N) is occupied} is sketched in
Figure 7.7.)
Note that B(m) ⊆ B(N), and say that ‘the first step is successful’ if every
edge in B(m) is p-open, which is to say that B(m) is a ‘seed’. (Recall that p
and other parameters are given in (7.26).) At this stage we write E1 for the
set of edges of B(m).
In the following sequential algorithm, we shall construct an increasing sequence
E1, E2, . . . of edge-sets. At each stage k, we shall acquire information
about the values of X(e) for certain e ∈ E 3 (here, the X(e) are independent
uniform [0, 1]-valued random variables, as usual). This information we
shall record in the form ‘each e is βk(e)-closed and γk(e)-open’ for suitable
functions βk, γk : E 3 → [0, 1] satisfying
(7.27) βk(e) ≤ βk+1(e), γk(e) ≥ γk+1(e), for all e ∈ E 3 .
Having constructed E1, above, we set
(7.28)
(7.29)
β1(e) = 0 for all e ∈ E 3 ,
�
p if e ∈ E1,
γ1(e) =
1 otherwise.