PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

186 Geoffrey Grimmett
It remains to show that (3.19) follows from Lemma (3.12). We have that
Pp(ρ1 + ρ2 + · · · + ρk ≤ n − k | An)
n−k �
= Pp(ρ1 + ρ2 + · · · + ρk−1 = i, ρk ≤ n − k − i | An)
i=0
n−k �
≥ P(M ≤ n − k − i)Pp(ρ1 + ρ2 + · · · + ρk−1 = i | An) by (3.13)
i=0
= Pp(ρ1 + ρ2 + · · · + ρk−1 + Mk ≤ n − k | An),
where Mk is a random variable which is independent of all edge-states in
S(n) and is distributed as M. There is a mild abuse of notation here, since
Pp is not the correct probability measure unless Mk is measurable on the
usual σ-field of events, but we need not trouble ourselves overmuch about
this. We iterate the above argument in the obvious way to deduce (3.19),
thereby completing the proof of the lemma. �
The conclusion of Theorem (3.8) is easily obtained from this lemma, but
we delay this step until the end of the section. The proof of Theorem (3.4)
proceeds by substituting (3.18) into (3.11) to obtain that, for 0 ≤ α < β ≤ 1,
gα(n) ≤ gβ(n)exp
�
−
� β
α
�
� n
i=0
� �
n
− 1 dp .
gp(i)
It is difficult to calculate the integral in the exponent, and so we use the
inequality gp(i) ≤ gβ(i) for p ≤ β to obtain
(3.22) gα(n) ≤ gβ(n)exp
�
−(β − α)
�
� n
i=0
��
n
− 1 ,
gβ(i)
and it is from this relation that the conclusion of Theorem (3.4) will be
extracted. Before continuing, it is interesting to observe that by combining
(3.10) and (3.18) we obtain a differential-difference inequality involving the
function
n�
G(p, n) = gp(i);
rewriting this equation rather informally as a partial differential inequality,
we obtain
(3.23)
i=0
∂2G ∂G
�
n
�
≥ − 1 .
∂p ∂n ∂n G
Efforts to integrate this inequality directly have failed so far.