= Random-cluster model
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= Random-cluster measure
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= Random-cluster
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On a finite <graph> $G=(V,E)$, the <random-cluster model> weights an open-edge configuration $\omega\subseteq E$ by
$$
\phi_{p,q}(\omega)=\frac{1}{Z_{p,q}}p^{|\omega|}(1-p)^{|E|-|\omega|}q^{k(\omega)},\qquad0<p<1,\quad q>0.
$$
The <partition function> $Z_{p,q}$ is the sum of these weights, and $k(\omega)$ counts the open <connected components of a graph>, including isolated <graph vertices>. The endpoint laws at $p=0$ and $p=1$ are obtained by limits. The factor $q$ is the <cluster weight in the random-cluster model>.<On the random-cluster model. I. Introduction and relation to other models by C. M. Fortuin and P. W. Kasteleyn>[Finite-graph random-cluster measure and relation to other lattice models]{ref}
At $q=1$, the component factor disappears and the law is <independent> <bond percolation>. For other $q$, $p$ is a model parameter rather than necessarily the marginal opening <probability> of each <edge>. For a single <edge> joining two <graph vertices>[vertices], with $p=1/2,q=2$, the closed and open weights are $2$ and $1$, so the <edge> is open with <probability> $1/3$.
The <random-cluster single-edge conditional probability> describes the local mechanism. The <Edwards-Sokal coupling> connects integer cluster weights with the <Potts model>, while <random-cluster boundary conditions> specify how boundary <graph vertices>[vertices] are identified before component counting.
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