PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

13.1 Basic Properties
Percolation and Disordered Systems 263
13. RANDOM-CLUSTER MODELS
First we summarise some useful properties of random-cluster measures. Let
G = (V, E) be a finite graph, and write ΩE = {0, 1} E . The random-cluster
measure on ΩE, with parameters p, q satisfying 0 ≤ p ≤ 1 and q > 0, is given
by
φp,q(ω) = 1
�
�
p
Z
e∈E
ω(e) (1 − p) 1−ω(e)
�
q k(ω) , ω ∈ ΩE
where Z = ZG,p,q is a normalising constant, and k(ω) is the number of
connected components of the graph (V, η(ω)), where η(ω) = {e : ω(e) = 1} is
the set of ‘open’ edges.
Theorem 13.1. The measure φp,q satisfies the FKG inequality if q ≥ 1.
Proof. If p = 0, 1, the conclusion is obvious. Assume 0 < p < 1, and check
the condition (5.2), which amounts to the assertion that
k(ω ∨ ω ′ ) + k(ω ∧ ω ′ ) ≥ k(ω) + k(ω ′ ) for ω, ω ′ ∈ ΩE.
This we leave as a graph-theoretic exercise. �
Theorem 13.2 (Comparison Inequalities). We have that
(13.3)
(13.4)
φp ′ ,q ′ ≤ φp,q if p ′ ≤ p, q ′ ≥ q, q ′ ≥ 1,
φp ′ ,q ′ ≥ φp,q if
p ′
q ′ (1 − p ′ ) ≥
p
q(1 − p) , q′ ≥ q, q ′ ≥ 1.
Proof. Use Holley’s Inequality (Theorem 5.5) after checking condition (5.6).
�
In the next theorem, the role of the graph G is emphasised in the use
of the notation φG,p,q. The graph G\e (resp. G.e) is obtained from G by
deleting (resp. contracting) the edge e.
Theorem 13.5 (Tower Property). Let e ∈ E.
(a) The conditional measure of φG,p,q given ω(e) = 0 is φ G\e,p,q.
(b) The conditional measure of φG,p,q given ω(e) = 1 is φG.e,p,q.
Proof. This is an elementary calculation of conditional probabilities. �
More details of these facts may be found in [26, 159, 162]. Another comparison
inequality may be found in [161].