DLR Gibbs measures satisfy a spatial Markov property

Let $\mu$ be an infinite-volume Gibbs measure for a lattice Hamiltonian with finite-range interaction, and let $\gamma=\{\gamma_\Lambda\}$ be the associated Gibbs specification .

Write $\mathcal{F}_\Lambda$ for the $\sigma$-algebra generated by spins in $\Lambda$ and $\mathcal{F}_{\Lambda^c}$ for the outside.

Statement

If $\mu$ satisfies the DLR equation , then for every finite region $\Lambda$ and every bounded local function $f$ depending only on spins in $\Lambda$,

Equivalently, the conditional law of $\sigma_\Lambda$ given $\sigma_{\Lambda^c}$ is $\gamma_\Lambda(\cdot\mid\sigma_{\Lambda^c})$.

Moreover, if the interaction has finite range (e.g. nearest-neighbor), then $\gamma_\Lambda(\cdot\mid\sigma_{\Lambda^c})$ depends on $\sigma_{\Lambda^c}$ only through the configuration in a finite “boundary neighborhood” of $\Lambda$ (in the nearest-neighbor case, only the spins on the lattice boundary $\partial\Lambda$). In this sense, $\mu$ has a spatial Markov property.

Key hypotheses

• $\mu$ is an infinite-volume Gibbs measure satisfying the DLR equation for some specification $\gamma$.
• The interaction is finite range (to conclude “boundary-only” dependence).

Conclusions

• Conditionals are local: the law inside $\Lambda$ given outside is the corresponding finite-volume Gibbs law with boundary condition induced by $\sigma_{\Lambda^c}$.
• For finite-range interactions, conditioning on the entire exterior is equivalent to conditioning on finitely many boundary spins (a Markov-type locality statement).

Proof idea / significance

The first identity is exactly the content of the DLR equation : it states that $\mu$ is consistent with the specification $\gamma$ for all finite regions. The “boundary-only” refinement follows because, for finite-range interactions, the energy coupling between $\Lambda$ and $\Lambda^c$ involves only spins near $\partial\Lambda$, so the conditional weights inside $\Lambda$ depend on the outside only through that neighborhood.