Dobrushin–Lanford–Ruelle equations

Statement

Let $S$ be a finite set and $\Omega = S^{\Z^d}$ with the product topology and Borel $\sigma$-algebra. Let $\Phi = (\Phi_A)$ be an interaction: $A$ runs over finite subsets of $\Z^d$, and $\Phi_A \colon \Omega \to \R$ depends only on the spins in $A$. Assume $\sup_{x \in \Z^d} \sum_{A \ni x} \norm{\Phi_A}_\infty < \infty$, and let $\beta > 0$. For finite $\Lambda \subset \Z^d$ and $\omega \in \Omega$, let $\sigma_\Lambda \omega_{\Lambda^c}$ be the configuration equal to $\sigma$ on $\Lambda$ and to $\omega$ off $\Lambda$. Put $H_\Lambda^\omega(\sigma) = \sum_{A \cap \Lambda \ne \emptyset} \Phi_A(\sigma_\Lambda \omega_{\Lambda^c})$ and $$\gamma_\Lambda(\sigma_\Lambda \mid \omega) = \frac{e^{-\beta H_\Lambda^\omega(\sigma)}}{\sum_{\eta \in S^\Lambda} e^{-\beta H_\Lambda^\omega(\eta)}}.$$ A probability measure $\mu$ on $\Omega$ satisfies the DLR equations if, for every finite $\Lambda$ and every $\sigma_\Lambda \in S^\Lambda$, $\mu(\sigma_\Lambda \mid \mathcal{F}_{\Lambda^c})(\omega) = \gamma_\Lambda(\sigma_\Lambda \mid \omega)$ for $\mu$-almost every $\omega$. Here $\mathcal{F}_{\Lambda^c}$ is the $\sigma$-algebra generated by the spins off $\Lambda$, and the left side is a conditional probability. Let $\mathcal{G}$ be the set of these measures, the Gibbs measures for $\beta \Phi$.

1. $\mathcal{G}$ is nonempty, convex and compact for the weak topology.
2. For each $\omega \in \Omega$, every weak limit point of $\gamma_{\Lambda_n}(\cdot \mid \omega)$, along cubes $\Lambda_n \uparrow \Z^d$, lies in $\mathcal{G}$.
3. The extreme points of $\mathcal{G}$ are exactly the measures in $\mathcal{G}$ that are trivial on the tail $\sigma$-algebra.