Gibbs specification

A consistent family of finite-volume conditional distributions giving the local Gibbs law in every finite region as a function of the outside configuration.

Gibbs specification

Let $\Omega$ be the configuration space of a lattice spin system (built from a single-site spin space ), equipped with its natural sigma-algebra of cylinder events. A specification is a family of probability kernels

\gamma = \bigl(\gamma_\Lambda\bigr)_{\Lambda\Subset\mathbb{Z}^d},

indexed by finite regions $\Lambda$ , where each $\gamma_\Lambda(\,\cdot\,\mid \eta)$ is a probability measure on $\Omega$ depending measurably on the outside configuration $\eta\in\Omega$ . (Formally, it is a family of conditional probability kernels.)

A Gibbs specification associated with inverse temperature $\beta$ and a given interaction potential (or, equivalently, a Hamiltonian ) is the specification whose kernel on a finite region $\Lambda$ is obtained by:

sampling spins in $\Lambda$ according to the finite-volume Gibbs measure with boundary condition $\eta_{\Lambda^c}$ , and
keeping spins outside $\Lambda$ fixed equal to $\eta_{\Lambda^c}$ .

Concretely, if $\lambda$ is an a priori single-spin measure and $H_\Lambda(\sigma_\Lambda\mid\eta)$ is the finite-volume Hamiltonian, then the conditional law on $\Lambda$ is

\gamma_\Lambda(d\sigma_\Lambda \mid \eta) ={} \frac{1}{Z_\Lambda(\beta,\eta)} \exp\!\bigl(-\beta\,H_\Lambda(\sigma_\Lambda\mid\eta)\bigr)\, \lambda_\Lambda(d\sigma_\Lambda),

with $Z_\Lambda(\beta,\eta)$ the lattice partition function . This kernel extends to a probability measure on $\Omega$ by combining the sampled $\sigma_\Lambda$ with the fixed exterior configuration $\eta_{\Lambda^c}$ .

Key properties

Properness (fixes the exterior). Under $\gamma_\Lambda(\cdot\mid\eta)$ , the configuration on $\Lambda^c$ is almost surely equal to $\eta_{\Lambda^c}$ .
Consistency (compatibility). If $\Lambda\subset\Delta$ , then conditioning in stages is the same as conditioning once: applying $\gamma_\Lambda$ and then $\gamma_\Delta$ yields the same result as applying $\gamma_\Delta$ directly (as kernels on $\Omega$ ).
Local dependence for short-range models. For finite-range interactions , $\gamma_\Lambda(\cdot\mid\eta)$ depends on $\eta$ only through spins near the boundary of $\Lambda$ .
Bridge to infinite volume. A Gibbs specification is the input for the DLR equation , which characterizes infinite-volume Gibbs measures as measures consistent with these local conditionals.

Physical interpretation

A Gibbs specification is a precise encoding of “local equilibrium”: no matter what the global system looks like, the distribution of spins in a finite window $\Lambda$ conditioned on the surrounding environment should be given by the appropriate Boltzmann weights for that window, with the environment acting as the boundary condition.