PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

Percolation and Disordered Systems 205
Next we estimate the probability that the second step is successful, conditional
on the first step being successful. Let G be the event that there exists a
path in B(n)\B(m) from ∂B(m) to K(m, n), every edge e of which is p-open
off ∆eE1 and whose unique edge f in ∆eE1 is (β1(f) + δ)-open. We write
Gσ j for the corresponding event with K(m, n) replaced by Lσj (K(m, n)). We
now apply Lemma 7.17 with R = B(m) and β = β1 to find that
Therefore
P � G σ j
�
� B(m) is a seed � > 1 − ǫ for j = 1, 2, 3, σ = ±.
(7.30) P � G σ j occurs for all j, σ � � B(m) is a seed � > 1 − 6ǫ,
so that the second step is successful with conditional probability at least
1 − 6ǫ.
If the second step is successful, then we update the β, γ functions accordingly,
setting
(7.31)
(7.32)
⎧
β1(e) if e �∈ EB ′ 1 ,
⎪⎨
β1(e) + δ if e ∈ ∆eE1\E2,
β2(e) =
⎪⎩
p if e ∈ ∆eE2\∆eE1,
0 otherwise,
⎧
γ1(e) if e ∈ E1,
⎪⎨
β1(e) + δ if e ∈ ∆eE1 ∩ E2,
γ2(e) =
p if e ∈ E2\(E1 ∪ ∆eE1),
⎪⎩
1 otherwise.
Suppose that the first two steps have been successful. We next aim to
link the appropriate seeds in each Lσ j (T(m, n)) to a new seed lying in the
bond-box 2σNej +B(N), i.e., the half-way box reached by exiting the origin
in the direction σej. If we succeed with each of the six such extensions,
then we terminate this stage of the process, and declare the vertex 0 of the
renormalised lattice to be occupied; such success constitutes the definition of
the term ‘occupied’. See Figure 7.7.
We do not present all the details of this part of the construction, since
they are very similar to those already described. Instead we concentrate
on describing the basic strategy, and discussing any novel aspects of the
construction. First, let B2 = b2+B(m) be the earliest seed (in some ordering
of all copies of B(m)) all of whose edges lie in E2 ∩ ET(m,n). We now try to
extend E2 to include a seed lying within the bond-box 2Ne1+B(N). Clearly
B2 ⊆ Ne1 + B(N). In performing this extension, we encounter a ‘steering’
problem. It happens (by construction) that all coordinates of b2 are positive,
implying that b2 + T(m, n) is not a subset of 2Ne1 + B(N). We therefore