Infinite-volume Gibbs measure

A probability measure on lattice spin configurations whose finite-volume conditional distributions are given by a Gibbs specification (DLR consistency).

Infinite-volume Gibbs measure

Fix a lattice spin system with single-spin space given by the spin space and configuration space $\Omega$ given by the lattice configuration space (elements are spin configurations ). Let $\gamma = (\gamma_\Lambda)_{\Lambda \Subset \mathbb{Z}^d}$ be the Gibbs specification associated with a chosen interaction potential , finite-volume Hamiltonians , and (possibly) an external field at inverse temperature $\beta$ .

An infinite-volume Gibbs measure for $\gamma$ is a probability measure $\mu$ on $\Omega$ such that for every finite region $\Lambda$ and every bounded measurable function $f$ depending only on spins in $\Lambda$ (a local observable),

\int f(\sigma)\,\mu(d\sigma) ={} \int \left[\int f(\sigma_\Lambda \eta_{\Lambda^c})\,\gamma_\Lambda(d\sigma_\Lambda \mid \eta_{\Lambda^c})\right]\mu(d\eta).

Equivalently, $\mu$ satisfies the DLR equation : for each finite $\Lambda$ , the conditional law of $\sigma_\Lambda$ given the outside configuration $\sigma_{\Lambda^c}$ is $\gamma_\Lambda(\cdot \mid \sigma_{\Lambda^c})$ , where $\gamma_\Lambda$ is the same kernel that defines the finite-volume Gibbs measure with boundary condition $\sigma_{\Lambda^c}$ .

Key properties

Local equilibrium / consistency: The defining feature is local: every finite window sees the outside world only through the specification kernel. This is the infinite-system analogue of sampling from a finite-volume Gibbs law with an induced boundary condition.
Thermodynamic-limit realization: For standard interactions (e.g. finite-range interactions ), infinite-volume Gibbs measures arise as subsequential limits of finite-volume Gibbs measures as the region grows (a form of thermodynamic limit ).
Convexity: The set of all infinite-volume Gibbs measures for a fixed specification is convex: mixtures of solutions are still solutions (see mixtures of Gibbs measures ).
Uniqueness vs multiplicity: If there is exactly one infinite-volume Gibbs measure at given parameters, boundary conditions do not affect infinite-volume local statistics. Multiple distinct Gibbs measures indicate a Gibbs phase transition .
Symmetries: If the interaction is translation-invariant (and similarly for other symmetries), then there exist Gibbs measures that inherit these symmetries, though symmetry-invariant measures need not be extremal .

Physical interpretation

An infinite-volume Gibbs measure represents an equilibrium state of an infinite lattice system: any finite subsystem is in thermal equilibrium with the remainder of the lattice acting as a “heat bath,” encoded by the specification. Measurable predictions (magnetization, energy density, etc.) are computed as ensemble averages under $\mu$ , and the presence of multiple such measures corresponds to the coexistence of distinct macroscopic phases.