A random-cluster model on a finite graph measure measurable
space elements are
vectors sigma-algebra of
all subsets of
(1)
where here, parameters , function
(2)
and connected components
of the graph
(3)
The connected components of clusters .
For a positive integer factor 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 edge states are generally
dependent , giving a model of dependent
percolation . On a forest , however, edge
states are independent with open probability
For the critical model on the square integer lattice et al. (2026) reported convergence
of its loops in the scaling limit to the conformal
loop ensemble Potts model consequently
has a conformally invariant scaling limit.
See also AB Percolation,
Bernoulli Percolation Model ,
Bond Percolation ,
Boolean
Model ,
Boolean-Poisson Model ,
Bootstrap
Percolation ,
Cayley Tree ,
Cluster ,
Cluster Perimeter ,
Conformal
Loop Ensemble ,
Continuum Percolation
Theory ,
Dependent Percolation ,
Discrete
Percolation Theory ,
Disk Model ,
First-Passage
Percolation ,
Fortuin-Kasteleyn Planar
Map ,
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
Portions of this entry contributed by Christopher
Stover
Explore with Wolfram|Alpha
References Averous, E.; Duminil-Copin, H.; He, T.; Lammers, P.; and Manolescu, I. "Conformal Invariance of the Planar Four-State Potts Model and
Its Random-Cluster Representation." 6 Oct 2026. https://arxiv.org/abs/2610.08507 . 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 elements 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 
, edge states are generally
dependent, giving a model of dependent
percolation. On a forest, however, 
, and edge
states are independent with open probability ![p/[p+q(1-p)]](/images/equations/Random-ClusterModel/Inline26.svg)
.
with 
and 
, Averous et al. (2026) reported convergence
of its loops in the scaling limit to the conformal
loop ensemble 
.
The associated four-state Potts model consequently
has a conformally invariant scaling limit.
