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On a finite graph $G=(V,E)$, the random-cluster model weights an open-edge configuration $\omega \subseteq E$ byThe 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.^[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 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 vertices are identified before component counting.