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Concept
Random cluster model
Résumé
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 Cees Fortuin and Piet Kasteleyn.
Let be a graph, and 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 if , and open if . If we let be the set of open bonds, then an open 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. The probability measure of a configuration is given as
The RC model is a generalization of percolation, where each cluster is weighted by a factor of . Given a configuration , we let be the number of open clusters, or alternatively the number of connected components formed by the open bonds. Then for any , the probability measure of a configuration 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.
The parameter of the random cluster model can take arbitrary complex values. This includes the following special cases:
linear resistance networks.
negatively-correlated percolation.
Bernoulli percolation, with .
the Ising model.
-state Potts model.
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 of spin and bond configurations.
Let be a graph, with the number of vertices being and the number of edges being .
Source officielle
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