 Edwards-Sokal coupling

The Edwards-Sokal coupling is a joint probability distribution of Potts model spins ${\sigma }_v\in \{1,\dots ,q\}$ and random-cluster edge states ${\omega }_e$, for positive integer $q$ and $0\le p<1$. On a finite graph, its unnormalized weight isGiven the edge configuration, assign each open connected component of a graph an independent uniform color. There are $q^{k(\omega )}$ compatible spin assignments, so summing over spins gives the random-cluster measure.

Given the spins, edges joining unequal colors are closed; each edge joining equal colors is independently open with probability $p$. Summing over edges gives a Potts model weight proportional to $\exp (K{\sum }_e{1}_{{\sigma }_u={\sigma }_v})$, where $p=1-e^{-K}$. These two conditional laws support cluster-based simulation.^[ref]

For $q=2$, writing the two colors as $\pm 1$ gives $1_{{\sigma }_u={\sigma }_v}=(1+{\sigma }_u{\sigma }_v)/2$. Hence the Ising model with interaction weight $\exp (\beta J{\sum }_e{\sigma }_u{\sigma }_v)$ corresponds to $K=2\beta J$. The factor of two depends on which interaction convention is used.