PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

Percolation and Disordered Systems 207
Fig. 7.8. Two adjacent site-boxes both of which are occupied. The construction
began with the left site-box B0(N) and has been extended to the right site-box
(N). The black squares are seeds, as before.
Be1
We perform similar extensions in each of the other five directions exiting
B0(N). If all are successful, we declare 0 to be occupied. Combining the
above estimates of success, we find that
P(0 is occupied | B(m) is a seed) > (1 − 6ǫ)(1 − ǫ) 6 (7.33)
> 1 − 12ǫ
�
1 + pc(F, site) �
= 1
2
by (7.26).
If 0 is not occupied, we end the construction. If 0 is occupied, then this
has been achieved after the definition of a set E8 of edges. The corresponding
functions β8, γ8 are such that
(7.34) β8(e) ≤ γ8(e) ≤ p + 6δ for e ∈ E8;
this follows since no edge lies in more than 7 of the copies of B(n) used
in the repeated application of Lemma 7.17. Therefore every edge of E8 is
(p + η)-open, since δ = 1
12η (see (7.26)).
The basic idea has been described, and we now proceed similarly. Assume
0 is occupied, and find the earliest edge e(r) induced by F and incident with
the origin; we may assume for the sake of simplicity that e(r) = 〈0, e1〉. We
now attempt to link the seed b3 + B(m), found as above inside the half-way
box 2Ne1 + B(N), to a seed inside the site-box 4Ne1 + B(N). This is done
in two steps of the earlier kind. Having found a suitable seed inside the new
site-box 4Ne1 + B(N), we attempt to branch-out in the other 5 directions
from this site-box. If we succeed in finding seeds in each of the corresponding
half-way boxes, then we declare the vertex e1 of the renormalised lattice to
be occupied. As before, the (conditional) probability that e1 is occupied