PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

276 Geoffrey Grimmett
Each circuit Γ of B d partitions the set EB of edges of B into three sets,
being
E = {edges of B exterior to Γ},
I = {edges of B interior to Γ},
Γ ′ = {edges of B crossing Γ}.
The edges I form a connected subgraph of B.
Our target is to obtain an upper bound for the probability that a given Γ is
an outer circuit. This we shall do by examining certain partition functions.
Since no open component of ω contains points lying in both the exterior
and interior of an outer circuit, the event OC(Γ) = {Γ is an outer circuit}
satisfies, for any dual circuit Γ having 0 in its interior,
(13.41)
φ 1 � � 1
B,p,q OC(Γ) =
Z1 B,p,q
�
1OC(Γ)(ω)πp(ω) ω
= 1
Z1 (1 − p)
B,p,q
|Γ| Z 1 E (Γ)ZI
where πp(ω) = p N(ω) (1 − p) |E|−N(ω) q k(ω) , Z 1 E (Γ) is the sum of πp(ω ′ ) over
all ω ′ ∈ {0, 1} E with ‘1’ boundary conditions on ∂B and consistent with Γ
being an outer circuit (i.e., property (c) above), and ZI is the sum of πp(ω ′′ )
over all ω ′′ ∈ {0, 1} I .
Next we use duality. Let I d be the set of dual edges which cross the primal
edges I, and let m be the number of vertices of B inside Γ. By (13.34),
(13.42) ZI = q m−1
� � |I|
1 − p
where p d satisfies (13.33), and where Z 1
I d ,p d ,q
p d
Z 1 I d ,p d ,q
is the partition function for
dual configurations, having wired boundary conditions, on the set V d of
vertices incident to I d (i.e., all vertices of V d on its boundary are identified,
as indicated in Figure 13.3).
We note two general facts about partition functions. First, for any graph
G, ZG,p,q ≥ 1 if q ≥ 1. Secondly, Z·,p,q has a property of supermultiplicativity
when q ≥ 1, which implies in particular that
Z 1 B,p,q ≥ Z 1 E(Γ)Z 1 I∪Γ ′ ,p,q
for any circuit Γ of Bd . (This is where we use property (c) above.)
Let I∗ = Id + ( 1 1
2 , 2 ), where Id is thought of as a subset of R2 . Note from
Figure 13.3 that I ∗ ⊆ I ∪Γ ′ . Using the two general facts above, we have that
(13.43) Z 1 B,p,q ≥ Z 1 E(Γ)Z 1 I ∗ ,p,q = Z 1 E(Γ)Z 1
I d ,p,q .