It is used to study random combinatorial structures, electrical networks, etc.
[1][2] It is also referred to as the RC model or sometimes the FK representation after its founders Cees Fortuin and Piet Kasteleyn.
[3] The random cluster model has a critical limit, described by a conformal field theory.
be a bond configuration on the graph that maps each edge to a value of either 0 or 1.
be the set of open bonds, then an open cluster or FK cluster is any connected component in
Note that an open cluster can be a single vertex (if that vertex is not incident to any open bonds).
Suppose an edge is open independently with probability
and closed otherwise, then this is just the standard Bernoulli percolation process.
is given as The RC model is a generalization of percolation, where each cluster is weighted by a factor of
be the number of open clusters, or alternatively the number of connected components formed by the open bonds.
is given as Z is the partition function, or the sum over the unnormalized weights of all configurations, The partition function of the RC model is a specialization of the Tutte polynomial, which itself is a specialization of the multivariate Tutte polynomial.
of the random cluster model can take arbitrary complex values.
This includes the following special cases: The Edwards-Sokal (ES) representation[5] of the Potts model is named after Robert G. Edwards and Alan D. Sokal.
It provides a unified representation of the Potts and random cluster models in terms of a joint distribution of spin and bond configurations.
of the set enforces the constraint that a bond can only be open on an edge if the adjacent spins are of the same state, also known as the SW rule.
The statistics of the Potts spins can be recovered from the cluster statistics (and vice versa), thanks to the following features of the ES representation:[2] There are several complications of the ES representation once frustration is present in the spin model (e.g. the Ising model with both ferromagnetic and anti-ferromagnetic couplings in the same lattice).
In particular, there is no longer a correspondence between the spin statistics and the cluster statistics,[7] and the correlation length of the RC model will be greater than the correlation length of the spin model.
This is the reason behind the inefficiency of the SW algorithm for simulating frustrated systems.
is a planar graph, there is a duality between the random cluster models on
[8] At the level of the partition function, the duality reads On a self-dual graph such as the square lattice, a phase transition can only occur at the self-dual coupling
[9] The random cluster model on a planar graph can be reformulated as a loop model on the corresponding medial graph.
of the random cluster model, the corresponding loop configuration is the set of self-avoiding loops that separate the clusters from the dual clusters.
In the transfer matrix approach, the loop model is written in terms of a Temperley-Lieb algebra with the parameter
In two dimensions, the critical random cluster model is described by a conformal field theory with the central charge Known exact results include the conformal dimensions of the fields that detect whether a point belongs to an FK cluster or a spin cluster.
In terms of Kac indices, these conformal dimensions are respectively
RC models were introduced in 1969 by Fortuin and Kasteleyn, mainly to solve combinatorial problems.
[1][10][6] After their founders, it is sometimes referred to as FK models.
Post 1987, interest in the model and applications in statistical physics reignited.
It became the inspiration for the Swendsen–Wang algorithm describing the time-evolution of Potts models.
[11] Michael Aizenman and coauthors used it to study the phase boundaries in 1D Ising and Potts models.