Conditional Gibbs kernel consistency

For fixed finite $\mathrm{\Lambda }$, only finitely many translated interaction supports meet it, so every $E_{\mathrm{\Lambda }}(\tau \mid \eta )$ is a finite real number and $A^{\mathrm{\Lambda }}$ is finite nonempty. Thus $Z_{\mathrm{\Lambda }}^{\eta }$ is a positive finite sum and [F1] supplies its normalized probability weights.

For $\mathrm{\Delta }\subseteq \mathrm{\Lambda }$, split the terms meeting $\mathrm{\Lambda }$ into those meeting $\mathrm{\Delta }$ and those disjoint from $\mathrm{\Delta }$. The second sum is independent of ${\tau }_{\mathrm{\Delta }}$, so $E_{\mathrm{\Lambda }}({\tau }_{\mathrm{\Lambda }}\mid \eta )=E_{\mathrm{\Delta }}({\tau }_{\mathrm{\Delta }}\mid {\tau }_{\mathrm{\Lambda }\setminus \mathrm{\Delta }}\eta )+C({\tau }_{\mathrm{\Lambda }\setminus \mathrm{\Delta }},\eta )$. Consequently $Z_{\mathrm{\Lambda }}^{\eta }={\sum }_{\xi \in A^{\mathrm{\Lambda }\setminus \mathrm{\Delta }}}{e}^{-\beta C(\xi ,\eta )}{Z}_{\mathrm{\Delta }}^{\xi \eta }$. In $({\gamma }_{\mathrm{\Lambda }}{\gamma }_{\mathrm{\Delta }}f)(\eta )$, first sum over the ${\gamma }_{\mathrm{\Lambda }}$ spin assignment on $\mathrm{\Delta }$ while holding $\xi ={\tau }_{\mathrm{\Lambda }\setminus \mathrm{\Delta }}$ fixed. Its weights sum to $Z_{\mathrm{\Delta }}^{\xi \eta }$, which cancels the denominator in ${\gamma }_{\mathrm{\Delta }}(f\mid \xi \eta )$. The remaining sum is $Z_{\mathrm{\Lambda }}^{-1}{\sum }_{\xi \in A^{\mathrm{\Lambda }\setminus \mathrm{\Delta }}}{e}^{-\beta C(\xi ,\eta )}{\sum }_{\zeta \in A^{\mathrm{\Delta }}}f(\zeta \xi \eta )e^{-\beta E_{\mathrm{\Delta }}(\zeta \mid \xi \eta )}$, which is exactly ${\gamma }_{\mathrm{\Lambda }}f(\eta )$ by reindexing $(\zeta ,\xi )$ as a configuration on $\mathrm{\Lambda }$. The factor $e^{-\beta C}$ remains; it does not cancel by itself.

Each replacement map $\eta \mapsto {\tau }_{\mathrm{\Lambda }}\eta$ is continuous in the product topology, and each weight depends on finitely many boundary coordinates; finite sums therefore make ${\gamma }_{\mathrm{\Lambda }}(f\mid \eta )$ measurable for bounded measurable $f$. When $f={\mathbf{1}}_C$ for a cylinder event $C$, its value additionally depends only on the finitely many coordinates specified by $C$, so the resulting function of $\eta$ depends on finitely many coordinates. For an exterior event $D\subseteq A^{{\mathrm{\Lambda }}^c}$ measurable in the exterior cylinder sigma-algebra, let $B=\{\sigma \in X:\sigma {\mathrm{\mid }}_{{\mathrm{\Lambda }}^c}\in D\}$. Then ${\mathbf{1}}_B({\tau }_{\mathrm{\Lambda }}\eta )={\mathbf{1}}_D(\eta )$ for every local assignment, so ${\gamma }_{\mathrm{\Lambda }}({\mathbf{1}}_B\mid \eta )={\mathbf{1}}_D(\eta )$; this is properness. Whenever a kernel is integrated against a probability on $X$ or composed with another kernel, its exterior argument is the restriction of that global configuration to the relevant complement.

Steps 1.1, 2.2 and 2.1 prove normalization, measurability, properness and consistency, so the candidate family is a specification.