Potts model

Example

Let $q \ge 2$ be an integer, let $G = (V, E)$ be a finite graph, and let $J \in \R$ and $\beta \ge 0$. A configuration is a map $\sigma \colon V \to \set{1, \ldots, q}$, and its energy is $$H(\sigma) = -J \sum_{\set{u, v} \in E} \delta(\sigma(u), \sigma(v)),$$ where $\delta(a, b)$ is $1$ if $a = b$ and $0$ otherwise. The $q$-state Potts model is the probability measure on $\set{1, \ldots, q}^V$ proportional to $e^{-\beta H(\sigma)}$; it is ferromagnetic when $J > 0$. It is a lattice model with $S = \set{1, \ldots, q}$ and pair interaction $\Phi_{\set{u,v}} = -J\delta(\sigma(u), \sigma(v))$.

Check for $q = 2$: write the two states as $s = \pm 1$; then $\delta(\sigma(u), \sigma(v)) = \frac12 (1 + s_u s_v)$. So $H = -\frac{J}{2} \abs{E} - \frac{J}{2} \sum_{\set{u,v} \in E} s_u s_v$, which is the Ising model with coupling $J/2$ up to an additive constant that does not change the measure.