PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

198 Geoffrey Grimmett
For n > m, let V (n) = {x ∈ T(n) : x ↔ B(m) in B(n)}. Pick M such
that
� �
1
(7.12) pPp B(m) is a seed > 1 − ( 2η)1/M .
We shall assume for simplicity that 2m + 1 divides n + 1 (and that 2m < n),
and we partition T(n) into disjoint squares with side-length 2m. If |V (n)| ≥
(2m + 1) 2 M then B(m) is joined in B(n) to at least M of these squares.
Therefore, by (7.12),
� �
Pp B(m) ↔ K(m, n) in B(n)
(7.13)
� � � �
≥ 1 − 1 − pPp B(m) is a seed �M� ≥ (1 − 1
2η)Pp � � 2
|V (n)| ≥ (2m + 1) M .
Pp
� |V (n)| ≥ (2m + 1) 2 M �
We now bound the last probability from below. Using the symmetries of
L 3 obtained by reflections in hyperplanes, we see that the face F(n) comprises
four copies of T(n). Now ∂B(n) has six faces, and therefore 24 copies of T(n).
By symmetry and the FKG inequality,
� � � � 2 2 24
(7.14) Pp |U(n)| < 24(2m + 1) M ≥ Pp |V (n)| < (2m + 1) M
where U(n) = {x ∈ ∂B(n) : x ↔ B(m) in B(n)}. Now, with l = 24(2m +
1) 2 M,
� � � � � �
(7.15) Pp |U(n)| < l ≤ Pp |U(n)| < l, B(m) ↔ ∞ + Pp B(m) � ∞ ,
and
(7.16)
� � � �
|U(n)| < l, B(m) ↔ ∞ ≤ Pp 1 ≤ |U(n)| < l
Pp
≤ (1 − p) −3l Pp(U(n + 1) = ∅, U(n) �= ∅)
→ 0 as n → ∞.
(Here we use the fact that U(n + 1) = ∅ if every edge exiting ∂B(n) from
U(n) is closed.)
By (7.14)–(7.16) and (7.11),
� � � � 2 1/24 �
Pp |V (n)| < (2m + 1) M ≤ Pp |U(n)| < l ≤ an + ( 1
3η)24� 1/24
where an → 0 as n → ∞. We pick n such that
Pp
� |V (n)| < (2m + 1) 2 M � ≤ 1
2 η,