PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

160 Geoffrey Grimmett
whence
d
dp Pp(A) = �
�
N(ω)
p
=
ω
�
|E| − N(ω)
− 1A(ω)Pp(ω)
1 − p
1
p(1 − p) Ep
� �
{N − p|E|}1A ,
as required for part (a).
Turning to (b), assume A is increasing. Using the definition of N, we have
that
(4.3) covp(N, 1A) = �
{Pp(A ∩ Je) − pPp(A)}
e∈E
where Je = {ω(e) = 1}. Now, writing {piv} for the event that e is pivotal
for A,
Pp(A ∩ Je) = Pp(A ∩ Je ∩ {piv}) + Pp(A ∩ Je ∩ {not piv}).
We use the important fact that Je is independent of {piv}, which holds since
the latter event depends only on the states of edges f other than e. Since
A ∩ Je ∩ {piv} = Je ∩ {piv}, the first term on the right side above equals
Pp(Je ∩ {piv}) = Pp(Je | piv)Pp(piv) = pPp(piv),
and similarly the second term equals (since Je is independent of the event
A ∩ {not piv})
Pp(Je | A ∩ {not piv})Pp(A ∩ {not piv}) = pPp(A ∩ {not piv}).
Returning to (4.3), the summand equals
� pPp(piv) + pPp(A ∩ {not piv}) � − p � Pp(A ∩ {piv}) + Pp(A ∩ {not piv}) �
= pPp(A ∩ {piv}) = pPp(Je | piv)Pp(piv)
= p(1 − p)Pp(piv).
Insert this into (4.3) to obtain part (b) from part (a). An alternative proof
of part (b) may be found in [G]. �
Although the above theorem was given for a finite product space ΩE, the
conclusion is clearly valid for the infinite space Ω so long as the event A is
finite-dimensional.
The methods above may be used further to obtain formulae for the higher
derivatives of Pp(A). First, Theorem 4.2(b) may be generalised to obtain
that
d
dp Ep(X) = �
Ep(δeX),
e∈E