PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

166 Geoffrey Grimmett
∂B(n)
0
e
f(e)
Fig. 4.4. Inside the box B(n), the edge e is pivotal for the event {0 ↔ ∂B(n)}. By
altering the configuration inside the smaller box, we may construct a configuration
in which f(e) is pivotal instead.
f = 〈u, u + e1 + e2〉, where e1 and e2 are unit vectors in the directions of the
(increasing) x and y axes.
We claim that there exists a function h(p, s), strictly positive on (0, 1) 2 ,
such that
(4.10) h(p, s)Pp,s(e is pivotal for An) ≤ Pp,s(f(e) is pivotal for An)
for all e lying in B(n). Once this is shown, we sum over e to obtain by (4.9)
that
h(p, s) ∂
∂p θn(p, s) ≤ �
e∈E 2
Pp,s(f(e) is pivotal for An)
≤ 2 �
Pp,s(f is pivotal for An)
f∈F
= 2 ∂
∂s θn(p, s)
as required. The factor 2 arises because, for each f (∈ F), there are exactly
two edges e with f(e) = f.
Finally, we indicate the reason for (4.10). Let us consider the event
{e is pivotalfor An}. We claim that there exists an integer M, chosen uniformly
for edges e in B(n) and for all large n, such that
(a) all paths from 0 to ∂B(n) pass through the region e + B(M)
(b) by altering the configuration within e+B(M) only, we may obtain an
event on which f(e) is pivotal for An.
This claim is proved by inspecting Figure 4.4. A special argument may
be needed when the box e + B(M) either contains the origin or intersects
∂B(n), but such special arguments pose no substantial difficulty. Once this
geometrical claim is accepted, (4.10) follows thus. Write Eg for the event that
the edge g is pivotal for An. For ω ∈ Ee, let ω ′ = ω ′ (ω) be the configuration