A random-cluster model on a finite graph measure measurable
space vectors sigma-algebra
of all subsets of
(1)
where here, parameters ,
(2)
and connected components
of the graph
(3)
The connected components of
For a positive integer Potts model partition
function.
In the above setting, the case graph
edges are open (i.e., independently
of one another, a scenario which can be used as an alternative definition for the
term percolation . For cases 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 ,
Potts
Model ,
Random-Connection Model ,
Random Walk ,
s -Cluster,
s -Run,
Site Percolation
This entry contributed by Christopher
Stover
Explore with Wolfram|Alpha
References Grimmett, G. R. The Random-Cluster Model. Referenced
on Wolfram|Alpha Random-Cluster Model
Cite this as:
Weisstein, Eric W. , with contributions by Christopher Stover . "Random-Cluster Model." From MathWorld https://mathworld.wolfram.com/Random-ClusterModel.html
Subject classifications More... Less...
is a measure 
on the measurable
space 
defined as follows. Let 
, whose members are vectors 
, and let 
be the sigma-algebra
of all subsets of 
. For each 
, set
and 
are parameters, 
is the so-called partition function
denotes the number of connected components
of the graph 
where
are called open clusters.
and 
, the substitution 
identifies 
, up to the factor 
, with the 
-state Potts model partition
function.
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.
