PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

184 Geoffrey Grimmett
Similarly,
Pp({ρk > rk} ∩ An ∩ B)
= �
Pp(B, G = Γ)Pp({ρk > rk} ∩ An | B, G = Γ)
Γ
= �
Pp(B, G = Γ)
Γ
× Pp
� �y(Γ) ↔ ∂S(rk + 1, y(Γ)) off Γ � ◦ � y(Γ) ↔ ∂S(n) off Γ ��
.
We apply the BK inequality to the last term to obtain
(3.16)
Pp({ρk > rk} ∩ An ∩ B)
≤ �
� �
Pp(B, G = Γ)Pp y(Γ) ↔ ∂S(n) off Γ
Γ
× Pp
≤ gp(rk + 1)Pp(An ∩ B)
by (3.15) and the fact that, for each possible Γ,
Pp
�
y(Γ) ↔ ∂S � rk + 1, y(Γ) � �
off Γ
�
y(Γ) ↔ ∂S � rk + 1, y(Γ) � �
off Γ
≤ Pp
�
y(Γ) ↔ ∂S � rk + 1, y(Γ) ��
= Pp(Ark+1)
= gp(rk + 1).
We divide each side of (3.16) by Pp(An ∩ B) to obtain
Pp(ρk ≤ rk | An ∩ B) ≥ 1 − gp(rk + 1),
throughout which we multiply by Pp(B | An) to obtain the result. �
(3.17) Lemma. For 0 < p < 1, it is the case that
� �
(3.18) Ep N(An) | An ≥
Proof. It follows from Lemma (3.12) that
� n
i=0
n
− 1.
gp(i)
(3.19) Pp(ρ1 +ρ2+· · ·+ρk ≤ n−k | An) ≥ P(M1+M2+· · ·+Mk ≤ n−k),
where k ≥ 1 and M1, M2, . . . is a sequence of independent random variables
distributed as M. We defer until the end of this proof the minor chore of