Random cluster model
In statistical mechanics, probability theory, graph theory, etc. the random cluster model is a random graph that generalizes and unifies the Ising model, Potts model, and percolation model. It is used to study random combinatorial structures, electrical networks, etc. It is also referred to as the RC model or sometimes the FK representation after its founders Kees Fortuin and Piet Kasteleyn. Definition. Let formula_1 be a graph, and formula_2 be a bond configuration on the graph that maps each edge to a value of either 0 or 1. We say that a bond is "closed" on edge  formula_3 if formula_4, and open if formula_5. If we let formula_6 be the set of open bonds, then an open cluster is any connected component in formula_7. 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 formula_8 and closed otherwise, then this is just the standard Bernoulli percolation process. The probability measure of a configuration formula_9 is given as The RC model is a generalization of percolation, where each cluster is weighted by a factor of formula_11. Given a configuration formula_9 , we let formula_13 be the number of open clusters, or alternatively the number of connected components formed by the open bonds. Then for any formula_14, the probability measure of a configuration formula_9 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. Special values of "q". The parameter formula_11 of the random cluster model can take arbitrary complex values. This includes the following special cases: Edwards-Sokal representation . The Edwards-Sokal (ES) representation 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. Furthermore, it naturally generalizes the Swensden-Wang (SW) algorithm for arbitrary ferromagnetic spin models. Let the number of vertices be formula_28 and the number of edges be formula_29. We denote a spin state as formula_30 and a bond configuration as formula_31. The measure of formula_32 is given as where formula_34 is the uniform measure, and formula_35 is the product measure with density formula_36 . Furthermore, formula_37 is an appropriate normalizing constant, and the indicator function formula_38 enforces the following constraint, meaning that a bond can only be open on an edge if the adjacent spins are of the same state, known as the SW rule. The statistics of the Potts spins can be recovered from the cluster statistics, thanks to the following features of the ES representation: There are several complications of the ES representation once frustration is present in the spin model. For example, there is no longer a correspondence between the spin statistics and the cluster statistics, 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. Two-dimensional case. If the underlying graph formula_47 is a planar graph, there is a duality between the random cluster models on formula_47 and on the dual graph formula_49. 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 formula_51. The random cluster model on a planar graph can be  reformulated as a loop model on the corresponding medial graph. For a configuration formula_9 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 formula_53. History and applications. RC models were introduced in 1969 by Fortuin and Kasteleyn, mainly to solve combinatorial problems. After their founders, it is sometimes referred to as FK models. In 1971 they used it to obtain the FKG inequality. 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. Michael Aizenman and coauthors used it to study the phase boundaries in 1D Ising and Potts models.