PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

262 Geoffrey Grimmett
is obtained by setting ω(e) = 0 if δe(σ) = 0, and otherwise ω(e) = 1
with probability p (independently of other edges).
In conclusion, the measure µ is a coupling of a Potts measure πβ,J on V ,
together with a ‘random-cluster measure’
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(12.7) φp,q(ω) ∝ p ω(e) (1 − p) 1−ω(e)
�
q k(ω) , ω ∈ Ω.
e∈E
The parameters of these measures correspond to one another by the relation
p = 1 − e −βJ . Since 0 ≤ p ≤ 1, this is only possible if βJ ≥ 0.
Why is this interesting? The ‘two-point correlation function’ of the Potts
measure πβ,J on G = (V, E) is defined to be the function τβ,J given by
τβ,J(i, j) = πβ,J(σi = σj) − 1
, i, j ∈ V.
q
The ‘two-point connectivity function’ of the random-cluster measure φ is
φp,q(i ↔ j), i.e., the probability that i and j are in the same cluster of a
configuration sampled according to φ. It turns out that these ‘two-point
functions’ are (except for a constant factor) the same.
Theorem 12.8. If q ∈ {2, 3, . . . } and p = 1 −e −βJ satisfies 0 ≤ p ≤ 1, then
τβ,J(i, j) = (1 − q −1 )φp,q(i ↔ j).
Proof. We have that
τβ,J(i, j) = ��
1 {σi=σj}(σ) − q −1�
µ(σ, ω)
σ,ω
= �
φp,q(ω) � �
µ(σ | ω) 1 {σi=σj }(σ) − q −1�
ω
= � �
φp,q(ω)
σ
(1 − q −1 )1 {i↔j}(ω) + 0 · 1 {i�j}(ω)
ω
= (1 − q −1 )φp,q(i ↔ j). �
This fundamental correspondence implies that properties of Potts correlation
can be mapped to properties of random-cluster connection. Since
Pottsian phase transition can be formulated in terms of correlation functions,
this implies that information about percolative phase transition for randomcluster
models is useful for studying Pottsian transitions. In doing so, we
study the ‘stochastic geometry’ of random-cluster models.
The random-cluster measure (12.7) was constructed under the assumption
that q ∈ {2, 3, . . . }, but (12.7) makes sense for any positive real q. We
have therefore obtained a rich family of measures which includes percolation
(q = 1) as well as the Ising (q = 2) and Potts measures.
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