Ising model

Example

Let $G = (V, E)$ be a finite graph, and let $J > 0$, $h \in \R$ and $\beta \ge 0$ be real numbers. A spin configuration is a map $\sigma \colon V \to \set{-1, +1}$. Its energy is $$H(\sigma) = -J \sum_{\set{u, v} \in E} \sigma(u) \sigma(v) - h \sum_{v \in V} \sigma(v).$$ The Ising model on $G$ at inverse temperature $\beta$ is the probability measure on $\set{-1, +1}^V$ given by $$\mu_\beta(\sigma) = \frac{e^{-\beta H(\sigma)}}{Z_\beta}, \qquad Z_\beta = \sum_{\tau \in \set{-1, +1}^V} e^{-\beta H(\tau)}.$$ Check: each weight $e^{-\beta H(\sigma)}$ is positive and there are finitely many configurations. So $Z_\beta > 0$, and the values of $\mu_\beta$ are positive and sum to $1$.