PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

264 Geoffrey Grimmett
13.2 Weak Limits and Phase Transitions
Let d ≥ 2, and Ω = {0, 1} Ed.
The appropriate σ-field of Ω is the σ-field F
generated by the finite-dimensional sets. For ω ∈ Ω and e ∈ Ed , the edge e
is called open if ω(e) = 1 and closed otherwise.
Let Λ be a finite box in Zd . For b ∈ {0, 1} define
Ω b Λ
= {ω ∈ Ω : ω(e) = b for e /∈ EΛ},
where EA is the set of edges of Ld joining pairs of vertices belonging to A.
On Ωb Λ we define a random-cluster measure φbΛ,p,q as follows. Let 0 ≤ p ≤ 1
and q > 0. Let
(13.6) φ b 1
Λ,p,q (ω) =
Zb �
�
p
Λ,p,q e∈EΛ
ω(e) (1 − p) 1−ω(e)
�
q k(ω,Λ)
where k(ω, Λ) is the number of clusters of (Z d , η(ω)) which intersect Λ (here,
as before, η(ω) = {e ∈ E d : ω(e) = 1} is the set of open edges). The boundary
condition b = 0 (resp. b = 1) is sometimes termed ‘free’ (resp. ‘wired’).
Theorem 13.7. The weak limits
exist if q ≥ 1.
φ b p,q = lim
Λ→Zd φ b Λ,p,q, b = 0, 1,
Proof. Let A be an increasing cylinder (= finite-dimensional) event. If Λ ⊆ Λ ′
and Λ includes the ‘base’ of A, then
φ 1 Λ,p,q (A) = φ1 Λ ′ ,p,q (A | all edges in E Λ ′ \Λ are open) ≥ φ 1 Λ ′ ,p,q (A),
where we have used the tower property and the FKG inequality. Therefore
the limit limΛ→Zd φ1 Λ,p,q (A) exists by monotonicity. Since F is generated by
such events A, the weak limit φ1 p,q exists. A similar argument is valid in the
case b = 0. �
The measures φ 0 p,q and φ 1 p,q are called ‘random-cluster measures’ on L d
with parameters p and q. Another route to a definition of such measures
uses a type of Dobrushin–Lanford–Ruelle (DLR) formalism rather than weak
limits (see [162]) 10 . There is a set of ‘DLR measures’ φ satisfying φ 0 p,q ≤ φ ≤
φ 1 p,q , whence there is a unique such measure if and only if φ0 p,q = φ1 p,q .
10 ξ
Let φΛ,p,q be the random-cluster measure on Λ having boundary conditions inherited
from the configuration ξ off Λ. It is proved in [162] that any limit point φ of the family
of probability measures {φ ξ
Λ,p,q : Λ ⊆ Zd , ξ ∈ Ω} is a DLR measure whenever φ has the
property that the number I of infinite open clusters satisfies φ(I ≤ 1) = 1. It is an open
problem to decide exactly which weak limits are DLR measures (if not all).