Random-Cluster Model
Let 
be a finite graph, let 
be the set ![Omega=<span style=]()
{0,1}^E" src="https://mathworld.wolfram.com/images/equations/Random-ClusterModel/Inline3.svg" style="height:22px; width:90px" /> whose members are vectors 
, and let 
be the sigma-algebra of all subsets of 
. A random-cluster model on 
is the measure 
on the measurable space 
defined for each 
by
 |
(1)
|
where here, 
and 
are parameters, 
is the so-called partition function
![Z=sum_(omega in Omega)<span style=]() {product_(e in E)p^(omega(e))(1-p)^(1-omega(e))}q^(k(omega)), " src="https://mathworld.wolfram.com/images/equations/Random-ClusterModel/NumberedEquation2.svg" style="height:41px; width:252px" /> |
(2)
|
and 
denotes the number of connected components of the graph 
where
![eta(omega)=<span style=]() {e in E:omega(e)=1}. " src="https://mathworld.wolfram.com/images/equations/Random-ClusterModel/NumberedEquation3.svg" style="height:21px; width:178px" /> |
(3)
|
The connected components of 
are called open clusters.
In the above setting, the case 
corresponds to a model in which graph edges are open (i.e., 
) or closed (i.e., 
) independently of one another, a scenario which can be used as an alternative definition for the term percolation. For cases 
, the random-cluster model models dependent percolation.
REFERENCES
Grimmett, G. R. The Random-Cluster Model. Berlin: Springer-Verlag, 2009.
