DLR specification (Gibbsian conditional probabilities)

A consistent family of finite-volume conditional Gibbs kernels that defines infinite-volume Gibbs measures via the DLR equations.

DLR specification (Gibbsian conditional probabilities)

In lattice statistical mechanics on the lattice $\mathbb{Z}^d$ , an infinite system is described by a probability measure $\mu$ on the configuration space. The Dobrushin–Lanford–Ruelle (DLR) specification packages the idea that every finite region is in thermal equilibrium conditional on the outside configuration.

Set-up

Let $S$ be the single-site state space (e.g. $S=\{\pm1\}$ for Ising spins). The full configuration space is $\Omega = S^{\mathbb{Z}^d}$ , equipped with the product $\sigma$-algebra of cylinder events. A finite region is a finite box $\Lambda\subset\mathbb{Z}^d$ , and $\eta\in\Omega$ denotes a boundary condition (the configuration outside $\Lambda$ ).

Definition (specification as conditional Gibbs kernels)

A DLR specification is a family of probability kernels $\gamma=\{\gamma_\Lambda\}_{\Lambda\Subset\mathbb{Z}^d}$ such that for each finite $\Lambda$ :

Kernel property: for each boundary configuration $\eta$ , $\gamma_\Lambda(\,\cdot\,|\eta)$ is a probability measure on the spins in $\Lambda$ .
Measurability: for each event $A$ depending only on $\Lambda$ , the map $\eta\mapsto \gamma_\Lambda(A|\eta)$ is measurable with respect to the outside variables $\eta_{\Lambda^c}$ .
Properness: events depending only on $\Lambda^c$ are left unchanged (conditioning does not alter what is already fixed outside).
Consistency (Markovianity across volumes): if $\Lambda\subset\Delta$ , then integrating inside $\Delta$ and then inside $\Lambda$ is the same as integrating inside $\Delta$ in one step.

Given an interaction (Hamiltonian) and inverse temperature $\beta$ , the Gibbsian specification has the familiar Boltzmann form: for each finite $\Lambda$ and boundary condition $\eta$ ,

\gamma_\Lambda(d\sigma_\Lambda\mid \eta_{\Lambda^c}) =\frac{1}{Z_\Lambda(\eta_{\Lambda^c})}\, \exp\!\big(-\beta\,H_\Lambda(\sigma_\Lambda\mid \eta_{\Lambda^c})\big)\, \rho_\Lambda(d\sigma_\Lambda),

where $H_\Lambda(\sigma_\Lambda\mid \eta_{\Lambda^c})$ is the energy contribution involving sites in $\Lambda$ (including interactions across the boundary), $\rho_\Lambda$ is a reference product measure on $S^\Lambda$ (counting measure for discrete spins, Lebesgue-type for continuous variables), and $Z_\Lambda(\eta_{\Lambda^c})$ is the finite-volume normalizing partition function.

This is the “finite-volume equilibrium given the boundary” analogue of the canonical ensemble .

Definition (Gibbs measure via the DLR equations)

A probability measure $\mu$ on $\Omega$ is an infinite-volume Gibbs measure for the specification $\gamma$ if for every finite $\Lambda$ and every measurable event $A$ ,

\mu(A)=\int_\Omega \gamma_\Lambda(A\mid \omega)\,\mu(d\omega).

Equivalently: the conditional distribution of $\omega_\Lambda$ given $\omega_{\Lambda^c}$ under $\mu$ is exactly $\gamma_\Lambda(\,\cdot\,|\omega)$ , for $\mu$ -almost every outside configuration $\omega$ .

Physical interpretation

The DLR equations formalize local equilibrium: any finite region $\Lambda$ behaves as if it were in contact with a heat bath at $\beta$ , with the exterior configuration acting as a boundary field. Multiple Gibbs measures solving the same DLR equations correspond to distinct thermodynamic phases (different boundary conditions can select different solutions), and they can be constructed concretely using the weak-limit construction of infinite-volume Gibbs measures .