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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

 phi_(p,q)(omega)=1/Z(product_(e in E)p^(omega(e))(1-p)^(1-omega(e)))q^(k(omega))
(1)

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

 Z=sum_(omega in Omega){product_(e in E)p^(omega(e))(1-p)^(1-omega(e))}q^(k(omega)),
(2)

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

 eta(omega)={e in E:omega(e)=1}.
(3)

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.

Cite this as:

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

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