PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

12.3 Random-Cluster Models
Percolation and Disordered Systems 261
It was Fortuin and Kasteleyn who discovered that Potts models may be
recast as ‘random-cluster models’. In doing so, they described a class of
models, including percolation, which merits attention in their own right, and
through whose analysis we discover fundamental facts concerning Ising and
Potts models. See [158] for a recent account of the relevant history and
bibliography.
The neatest construction of random-cluster models from Potts models is
that reported in [127]. Let G = (V, E) be a finite graph, and define the
sample spaces
Σ = {1, 2, . . . , q} V , Ω = {0, 1} E ,
where q is a positive integer. We now define a probability mass function µ
on Σ × Ω by
(12.5) µ(σ, ω) ∝ �
where 0 ≤ p ≤ 1, and
e∈E
�
�
(1 − p)δω(e),0 + pδω(e),1δe(σ) (12.6) δe(σ) = δσi,σj if e = 〈i, j〉 ∈ E.
Elementary calculations reveal the following facts.
(a) Marginal on Σ. The marginal measure
is given by
µ(σ, ·) = �
µ(σ, ω)
�
µ(σ, ·) ∝ exp
ω∈Ω
βJ �
e
�
δe(σ)
where p = 1 − e −βJ . This is the Potts measure (12.3). Note that
βJ ≥ 0.
(b) Marginal on Ω. Similarly
µ(·, ω) = �
�
�
µ(σ, ω) ∝ p ω(e) (1 − p) 1−ω(e)
�
q k(ω)
σ∈Σ
where k(ω) is the number of connected components (or ‘clusters’) of
the graph with vertex set V and edge set η(ω) = {e ∈ E : ω(e) = 1}.
(c) The conditional measures. Given ω, the conditional measure on Σ is
obtained by putting (uniformly) random spins on entire clusters of ω
(of which there are k(ω)), which are constant on given clusters, and
independent between clusters. Given σ, the conditional measure on Ω
e