Stationary lattice entropy and energy densities

Apply [A1] to the consistent stationary cylinder probabilities. The extension is unique; translating it leaves the same cylinder values, so uniqueness makes the resulting measure $\mu$ stationary.

For finite random vectors $X,Y$, nonnegative relative entropy of their joint law relative to the product of its marginals gives $H(X,Y)\le H(X)+H(Y)$. If the product law has zeros, restrict to its support; the joint law is supported there, and [F1] applies on that finite support. For any single-site law $q$ on $A$, applying [F1] with zero energy and the uniform Gibbs law gives $D(q\mathrm{\parallel }{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}}_A)=\log \mathrm{\mid }A\mathrm{\mid }-H(q)\ge 0$, so $H(q)\le \log \mathrm{\mid }A\mathrm{\mid }$.

Tile a large cube $B_n$ by disjoint translates of $B_{\mathrm{\ell }}$ and leftover sites. By stationarity, each full block has entropy $H_{\mathrm{\ell }}$; each leftover site contributes at most $\log \mathrm{\mid }A\mathrm{\mid }$. Repeated subadditivity from step 1.2 gives $H_n\le m_n{H}_{\mathrm{\ell }}+\mathrm{\mid }{R}_n\mathrm{\mid }\log \mathrm{\mid }A\mathrm{\mid }$, with $m_n\mathrm{\mid }{B}_{\mathrm{\ell }}\mathrm{\mid }\mathrm{/}\mathrm{\mid }{B}_n\mathrm{\mid }\to 1$ and $\mathrm{\mid }{R}_n\mathrm{\mid }\mathrm{/}\mathrm{\mid }{B}_n\mathrm{\mid }\to 0$ for fixed $\mathrm{\ell }$. Thus $\mathrm{lim\hspace{0.17em}sup}{H}_n\mathrm{/}\mathrm{\mid }{B}_n\mathrm{\mid }\le H_{\mathrm{\ell }}\mathrm{/}\mathrm{\mid }{B}_{\mathrm{\ell }}\mathrm{\mid }$ for every $\mathrm{\ell }$. Since the infimum is no larger than any ratio, the liminf is at least the infimum, proving the stated limit formula.

For each interaction type $r$, the number of anchors with $x+S_r\subseteq B_n$, divided by $\mathrm{\mid }{B}_n\mathrm{\mid }$, tends to $1$ because $S_r$ is finite. Stationarity gives expectation ${\mathbb{E}}_{\mu }{\mathrm{\Phi }}_r$ for each included translate, so summing the finite interaction list and taking the limit yields $u(\mu )$. Nonnegativity and the bound $H_n\le \mathrm{\mid }{B}_n\mathrm{\mid }\log \mathrm{\mid }A\mathrm{\mid }$ give $0\le h(\mu )\le \log \mathrm{\mid }A\mathrm{\mid }$.