TFAE: Characterizations of Gibbs Measures

Let the configuration space be $\Omega$ (e.g. spins on $\mathbb{Z}^d$) and fix an interaction / lattice Hamiltonian (finite-range or summable, as needed). A probability measure $\mu$ on $\Omega$ (see probability measure ) is an infinite-volume Gibbs measure for this interaction (see infinite-volume Gibbs measure ) if and only if any (hence all) of the following equivalent conditions hold.

1. DLR (Dobrushin–Lanford–Ruelle) equations.
   For every finite region $\Lambda$ and every bounded local observable $f$ depending only on spins in $\Lambda$,

   $$\mathbb{E}_\mu[f \mid \mathcal{F}_{\Lambda^c}](\omega) \;=\; \int f(\sigma_\Lambda \omega_{\Lambda^c})\,\gamma_\Lambda(d\sigma_\Lambda \mid \omega_{\Lambda^c}),$$

   where $\gamma_\Lambda(\cdot \mid \omega_{\Lambda^c})$ is the finite-volume Gibbs kernel determined by the interaction and the outside configuration.
   This is precisely the DLR equation characterization.

2. Consistency with a Gibbs specification.
   There exists a consistent family of conditional distributions (a specification) $\{\gamma_\Lambda\}_\Lambda$ coming from the interaction such that $\mu$ is consistent with it:

   $$\mu \gamma_\Lambda = \mu \quad \text{for all finite } \Lambda.$$

   (Here $\mu\gamma_\Lambda$ denotes “update $\mu$ inside $\Lambda$ using the kernel $\gamma_\Lambda$ while keeping the outside fixed.”)

3. Thermodynamic limit of finite-volume Gibbs measures.
   There exists a sequence of volumes $\Lambda_n \uparrow \mathbb{Z}^d$ and boundary conditions $\eta^{(n)}$ such that

   $$\mu = \lim_{n\to\infty} \mu_{\Lambda_n}^{\eta^{(n)}},$$

   where $\mu_{\Lambda}^{\eta}$ is the finite-volume Gibbs measure in $\Lambda$ with boundary condition $\eta$ (limit in the weak topology).
   Any such limit point is a Gibbs measure, and every Gibbs measure arises as such a limit point (for standard interaction classes).

4. Gibbs variational principle (equilibrium state).
   Among translation-invariant probability measures $\nu$ (when translation invariance is part of the setup), $\mu$ maximizes the functional

   $$\nu \mapsto h(\nu) - \beta e(\nu),$$

   where $h(\nu)$ is the entropy density (Shannon/Gibbs entropy rate) and $e(\nu)$ is the energy density induced by the interaction.
   Equivalently, $\mu$ minimizes the appropriate free-energy density (see statistical free energy ).
   This characterization is often expressed in terms of relative entropy density (see relative entropy ) with respect to a reference specification.

5. Quasilocal conditional probabilities (Gibbsianity).
   The conditional distributions of $\mu$ in finite volumes depend on the outside configuration in a quasilocal (continuous) way and coincide with the interaction-defined kernels $\gamma_\Lambda(\cdot\mid\omega_{\Lambda^c})$.
   This is the “regularity + DLR” form of being Gibbs.

Prerequisites and context: finite-volume Gibbs measures , DLR equations , infinite-volume Gibbs measures , lattice Hamiltonians , relative entropy .