Random-Cluster Model

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.

REFERENCES

Grimmett, G. R. The Random-Cluster Model. Berlin: Springer-Verlag, 2009.