PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

Percolation and Disordered Systems 275
Fig. 13.2. The dashed lines include an outer circuit Γ of the dual B d .
(b) We have that θ 1 (pc(q), q) > 0.
(c) For any ψ < ψ(q),
φ 0 � � −nψ
pc(q),q 0 ↔ ∂B(n) ≤ e
for all large n.
We stress that these conclusions may be obtained for general d (≥ 2)
when q is sufficiently large (q > Q = Q(d)), as may be shown using so called
Pirogov–Sinai theory (see [226]). In the case d = 2 presented here, the above
duality provides a beautiful and simple proof. This proof is an adaptation
and extension of that of [227].
Proof. Let B = B(n) = [−n, n] 2 as usual, and let Bd = [−n, n − 1] 2 + ( 1 1
2 , 2 )
be those vertices of the dual of B(n) which lie inside B(n) (i.e., we omit
the vertex in the infinite face of B). We shall work with ‘wired’ boundary
conditions on B, and we let ω be a configuration on the edges of B. A
circuit Γ of Bd is called an outer circuit of a configuration ω if the following
properties hold:
(a) all edges of Γ are open in the dual configuration ωd , which is to say
that they traverse closed edges of B,
(b) the origin of L2 is in the interior of Γ,
(c) every vertex of B lying in the exterior of Γ, but within distance of
1/ √ 2 of some vertex of Γ, belongs to the same component of ω.
See Figure 13.2 for an illustration of the meaning of ‘outer circuit’.