Random-Cluster Model

Let G=(V,E) be a finite graph, let Omega be the set Omega={0,1}^E whose members are vectors omega=(omega(e):e in E), and let F be the sigma-algebra of all subsets of Omega. A random-cluster model on G is the measure phi_(p,q) on the measurable space (Omega,F) defined for each omega by

where here, 0<=p<=1 and q>0 are parameters, Z is the so-called partition function

and k(omega) denotes the number of connected components of the graph (V,eta(omega)) where

The connected components of (V,eta(omega)) are called open clusters.

In the above setting, the case q=1 corresponds to a model in which graph edges are open (i.e., omega(e)=1) or closed (i.e., omega(e)=0) independently of one another, a scenario which can be used as an alternative definition for the term percolation. For cases q!=1, the random-cluster model models dependent percolation.

See also

AB Percolation, Bernoulli Percolation Model, Bond Percolation, Boolean Model, Boolean-Poisson Model, Bootstrap Percolation, Cayley Tree, Cluster, Cluster Perimeter, Continuum Percolation Theory, Dependent Percolation, Discrete Percolation Theory, Disk Model, First-Passage Percolation, Germ-Grain Model, Inhomogeneous Percolation Model, Lattice Animal, Long-Range Percolation Model, Mixed Percolation Model, Oriented Percolation Model, Percolation, Percolation Theory, Percolation Threshold, Polyomino, Random-Connection Model, Random Walk, s-Cluster, s-Run, Site Percolation

This entry contributed by Christopher Stover

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References

Grimmett, G. R. The Random-Cluster Model. Berlin: Springer-Verlag, 2009.

Referenced on Wolfram|Alpha

Random-Cluster Model

Cite this as:

Stover, Christopher. "Random-Cluster Model." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Random-ClusterModel.html

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