PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

180 Geoffrey Grimmett
so that
(3.10)
1
gp(n) g′ p(n) = 1
p Ep
� �
N(An) | An .
Let 0 ≤ α < β ≤ 1, and integrate (3.10) from p = α to p = β to obtain
(3.11)
gα(n) = gβ(n)exp
≤ gβ(n)exp
�
�
−
−
� β
α
� β
α
1
p Ep
�
� �
N(An) | An dp
� �
Ep N(An) | An dp
� �
as in (2.30). We need now to show that Ep N(An) | An grows roughly
linearly in n when p < pc, and then this inequality will yield an upper bound
for gα(n) of the form required in �(3.5). The vast � majority of the work in the
proof is devoted to estimating Ep N(An) | An , and the argument is roughly
as follows. If p < pc then Pp(An) → 0 as n → ∞, so that for large n we
are conditioning on an event of small probability. If An occurs, ‘but only
just’, then the connections between the origin and ∂S(n) must be sparse;
indeed, there must exist many open edges in S(n) which are crucial for the
occurrence of An (see Figure 3.1). It is plausible that the number of such
pivotal edges in paths from the origin to ∂S(2n) is approximately twice the
number of such edges in paths to ∂S(n), since these sparse paths have to
traverse twice the distance. Thus the number N(An) of edges pivotal for An
should grow linearly in n.
Suppose that the event An occurs, and denote by e1, e2, . . . , eN the (random)
edges which are pivotal for An. Since An is increasing, each ej has the
property that An occurs if and only if ej is open; thus all open paths from
the origin to ∂S(n) traverse ej, for every j (see Figure 3.1). Let π be such an
open path; we assume that the edges e1, e2, . . . , eN have been enumerated in
the order in which they are traversed by π. A glance at Figure 3.1 confirms
that this ordering is independent of the choice of π. We denote by xi the
endvertex of ei encountered first by π, and by yi the other endvertex of ei.
We observe that there exist at least two edge-disjoint open paths joining 0
to x1, since, if two such paths cannot be found then, by Menger’s theorem
(Wilson 19793 , p. 126), there exists a pivotal edge in π which is encountered
prior to x1, a contradiction. Similarly, for 1 ≤ i < N, there exist at least two
edge-disjoint open paths joining yi to xi+1; see Figure 3.2. In the words of
the discoverer of this proof, the open cluster containing the origin resembles
a chain of sausages.
As before, let M = max{k : Ak occurs} be the radius of the largest ball
whose surface contains a vertex which is joined to the origin by an open path.
We note that, if p < pc, then M has a non-defective distribution in that
3 Reference [358].
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