= Edwards-Sokal coupling
{c}
{wiki}
The <Edwards-Sokal coupling> is a joint <probability distribution> of <Potts model> spins $\sigma_v\in\{1,\ldots,q\}$ and <random-cluster> <edge> states $\omega_e$, for positive integer $q$ and $0\leq p<1$. On a finite <graph>, its unnormalized weight is
$$
\prod_{e=\{u,v\}\in E}\left[(1-p)\mathbf1_{\omega_e=0}+p\mathbf1_{\omega_e=1}\mathbf1_{\sigma_u=\sigma_v}\right].
$$
Given 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 $\exp(K\sum_e\mathbf1_{\sigma_u=\sigma_v})$, where $p=1-e^{-K}$. These two conditional laws support cluster-based simulation.<Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm by R. G. Edwards and A. D. Sokal>[Joint spin-bond representation and its two conditional distributions, Physical Review D 38 (1988), pages 2009–2012]{ref}
For $q=2$, writing the two colors as $\pm1$ gives $\mathbf1_{\sigma_u=\sigma_v}=(1+\sigma_u\sigma_v)/2$. 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.
Back to article page
