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Date: 28-2-2022
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Date: 29-4-2022
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Date: 19-5-2022
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Let be a finite graph, let be the set 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
(2) |
and denotes the number of connected components of the graph where
(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.
Grimmett, G. R. The Random-Cluster Model. Berlin: Springer-Verlag, 2009.
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