Random-cluster model (Fortuin–Kasteleyn percolation)

A probability measure on open/closed edges with parameters p and q, interpolating between bond percolation (q=1) and the q-state Potts/Ising models via the Fortuin–Kasteleyn representation.

Random-cluster model (Fortuin–Kasteleyn percolation)

The random-cluster model (also called FK percolation) is a probability measure on bond configurations on a finite graph $G=(V,E)$ (typically a finite region of a lattice such as a finite box in a lattice with nearest-neighbor edges ).

A configuration is

either a subset of edges $\omega \subseteq E$ (the open edges), or equivalently
an element of $\{0,1\}^E$ (edge “spins”), so it fits the general idea of a spin configuration with spin space $\{0,1\}$ per edge.

Let $|\omega|$ be the number of open edges, and let $k(\omega)$ be the number of connected components (clusters) in the spanning subgraph $(V,\omega)$ , counting isolated vertices as single-vertex clusters.

For parameters $p\in[0,1]$ and $q>0$ , the random-cluster measure on $G$ is the probability measure

\phi_{G,p,q}(\omega) \;=\; \frac{1}{Z_{G}(p,q)}\, p^{|\omega|}(1-p)^{|E|-|\omega|}\, q^{k(\omega)} ,

where the normalizing constant

Z_G(p,q)=\sum_{\omega\subseteq E} p^{|\omega|}(1-p)^{|E|-|\omega|}\, q^{k(\omega)}

is the random-cluster partition function, the analogue of the lattice partition function .

Boundary conditions (free vs wired)

On a finite subgraph cut out of an infinite lattice, one often specifies boundary conditions (compare boundary conditions ). For random-cluster models the standard choices are:

Free boundary condition: clusters are computed using only edges inside the region.
Wired boundary condition: boundary vertices are identified (“wired together”) before counting clusters, which favors connections to the boundary.

These boundary conditions produce different finite-volume measures, and they are central when taking thermodynamic limits and defining infinite-volume objects (compare infinite-volume Gibbs measures ).

Fortuin–Kasteleyn link to Potts and Ising

The model is designed so that, for integer $q\ge 2$ , it is exactly the graphical expansion of the q-state Potts model (and for $q=2$ the Ising model ).

On a graph with uniform ferromagnetic coupling $J>0$ at inverse temperature $\beta$ , the FK parameter is

p \;=\; 1-e^{-\beta J}.

In this correspondence, the random-cluster partition function matches the Potts partition function up to an explicit factor, and the cluster-weight $q^{k(\omega)}$ has a clear meaning: after choosing $\omega$ , each cluster can be assigned one of $q$ spin labels independently, giving $q^{k(\omega)}$ possible spin assignments.

A useful consequence (often phrased via the Edwards–Sokal coupling) is that connectivity events encode Potts correlations: the event “ $x$ is connected to $y$ by open edges” controls the probability that Potts spins at $x$ and $y$ are equal under the corresponding Potts measure.

Key properties

Special cases

$q=1$ : the factor $q^{k(\omega)}$ is constant, so $\phi_{G,p,1}$ is independent bond percolation with edge-open probability $p$ .
$q=2$ : corresponds to the Ising model (FK representation).
Non-integer $q>0$ : still defines a perfectly good probability model on edges, even though there is no literal $q$ -state spin interpretation.

Positive association and monotonicity (for $q\ge 1$ )

For $q\ge 1$ , the random-cluster model satisfies strong correlation inequalities (FKG-type positive association). Informally:

increasing events are positively correlated,
the model is monotone in $p$ ,
wired boundary conditions typically dominate free boundary conditions for increasing events.

These monotonicity properties are a major reason the model is a standard tool for proving existence and structure of phase transitions.

Domain Markov / specification viewpoint

Although the weight involves the global quantity $k(\omega)$ , the model still has a consistent “conditional distribution in a subregion given the outside configuration,” i.e. a specification in the sense of Gibbs specifications . This supports an infinite-volume formulation analogous to the DLR equation approach used for lattice spin systems.

Thermodynamic quantities

On boxes $\Lambda$ in $\mathbb{Z}^d$ with edge set $E_\Lambda$ , one defines the finite-volume free-energy density / pressure via

\frac{1}{|\Lambda|}\log Z_{G_\Lambda}(p,q),

leading to the lattice pressure and its thermodynamic limit when the limit exists.

Non-analytic behavior of this limit as a function of $(p,q)$ corresponds to a phase transition .

Physical interpretation

The configuration $\omega$ describes which bonds are “active.” For Potts/Ising systems, open edges represent bonds that are compatible with (or reinforce) local alignment.
Clusters in $(V,\omega)$ represent domains of aligned spins in the coupled Potts picture.
The parameter $q$ $q$ controls the entropic weight of forming new clusters:
- larger $q$ favors having many clusters (more internal label choices),
- larger $p$ favors opening edges and merging clusters.
Long-range order can be read as large-scale connectivity:
- the emergence of macroscopic clusters is tied to an order parameter ,
- in Potts/Ising language, this corresponds to spontaneous magnetization and spontaneous symmetry breaking .

Because it translates spin ordering questions into geometric connectivity questions, the random-cluster model is a central bridge between lattice spin systems and percolation-style geometry, and it provides geometric access to quantities like two-point correlations , correlation length , and susceptibility through cluster connectivity and cluster-size statistics.