Lattice Gibbs kernels and DLR states

Assume Countable Choice for the product-measure extension framework of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) and Finite-alphabet product and probability compactness. Work in the finite-alphabet configuration space $X=A^{{\mathbb{Z}}^d}$ with its cylinder-generated Borel sigma-algebra. Let the finite-range interaction and its conditional energy be as in Finite-range lattice interaction and conditional energy, and fix $\beta >0$. For a finite $\mathrm{\Lambda }\subset {\mathbb{Z}}^d$ and exterior configuration $\eta \in A^{{\mathrm{\Lambda }}^c}$, put $Z_{\mathrm{\Lambda }}^{\eta }={\sum }_{\tau \in A^{\mathrm{\Lambda }}}{e}^{-\beta E_{\mathrm{\Lambda }}(\tau \mid \eta )},{\gamma }_{\mathrm{\Lambda }}(f\mid \eta )=\frac{{\sum }_{\tau \in A^{\mathrm{\Lambda }}}f(\tau \eta )e^{-\beta E_{\mathrm{\Lambda }}(\tau \mid \eta )}}{Z_{\mathrm{\Lambda }}^{\eta }}$ for bounded measurable $f:X\to \mathbb{R}$. Here ${\gamma }_{\mathrm{\Lambda }}f$ denotes the function $\eta \mapsto {\gamma }_{\mathrm{\Lambda }}(f\mid \eta )$, composition is operator composition, and $\mu {\gamma }_{\mathrm{\Lambda }}(f)=\int {\gamma }_{\mathrm{\Lambda }}(f\mid \eta )d\mu (\eta )$. A proper kernel leaves every exterior-measurable event fixed. A specification is a family $({\gamma }_{\mathrm{\Lambda }}{)}_{\mathrm{\Lambda }\text{ finite}}$ of proper probability kernels satisfying ${\gamma }_{\mathrm{\Lambda }}{\gamma }_{\mathrm{\Delta }}={\gamma }_{\mathrm{\Lambda }}$ whenever $\mathrm{\Delta }\subseteq \mathrm{\Lambda }$. A DLR state is a probability measure $\mu$ on $X$ satisfying $\mu {\gamma }_{\mathrm{\Lambda }}=\mu$ for every finite $\mathrm{\Lambda }$.

For the existence and uniqueness results below, a specification has finite range when for every finite $\mathrm{\Delta }$ and every cylinder event $C$ the function $\eta \mapsto {\gamma }_{\mathrm{\Delta }}({\mathbf{1}}_C\mid \eta )$ depends on only finitely many coordinates of $\eta$. The displayed finite-range Gibbs formula has this property because only finitely many interaction terms meet $\mathrm{\Delta }$, and a cylinder indicator depends on only finitely many additional coordinates.

The formula is the candidate Gibbs specification; Conditional Gibbs kernel consistency ↗ establishes its positive finite normalization, measurability, properness and consistency under the stated hypotheses. The Countable Choice assumption is used for the product-measure construction, not for any finite conditional sum.