Spatial Measure And Information — Examples

By [F3], $P=Q$ satisfies the stationary finite-intensity hypotheses of [F2]. For every bounded cube $G$, equality of the laws gives $P_G=Q_G$ and the Radon–Nikodym derivative $h_G=1$ $Q_G$-almost everywhere. Thus $D(P_G\mathrm{\parallel }{Q}_G)=\int 1\log 1d{Q}_G=0$.

Every normalized cube entropy is therefore zero, so its limit is $I(P)=0$ by [F2]. Also $\mathrm{\mid }G{\mathrm{\mid }}^{-1}\log h_G=0$ $P$-almost surely for every such $G$.

In one unit cell, the conditional distribution of point positions given the count is identical for $Q_z$ and $Q_1$. The ratio of their count probabilities at $n$ is $\frac{e^{-z}{z}^n\mathrm{/}n!}{e^{-1}\mathrm{/}n!}=e^{1-z}{z}^n\mathrm{.}$ Thus this is the Radon–Nikodym density on that cell.

The cube ${\mathrm{\Lambda }}_m=[0,m{)}^d$ is the disjoint union of $m^d$ unit cells. Independence in [F1] makes the finite-cube density the product of the cell densities: $h_m(\omega )=e^{(1-z)m^d}{z}^{N_{{\mathrm{\Lambda }}_m}(\omega )}\mathrm{.}$

The count $N_{{\mathrm{\Lambda }}_m}$ is Poisson with mean $z{m}^d$ by [F1]. Directly from its mass function and the exponential series, $E_P{N}_{{\mathrm{\Lambda }}_m}=e^{-z{m}^d}{\sum }_{n\ge 1}n\frac{(z{m}^d{)}^n}{n!}=z{m}^d{e}^{-z{m}^d}{\sum }_{k\ge 0}\frac{(z{m}^d{)}^k}{k!}=z{m}^d\mathrm{.}$ Therefore [F2] gives $\frac{D(P_{{\mathrm{\Lambda }}_m}\mathrm{\parallel }{Q}_{{\mathrm{\Lambda }}_m})}{m^d}=\frac{E_P\log h_m}{m^d}=1-z+z\log z$ for every positive integer $m$.

By [F1], $Q_z$ is invariant under every translation, and step 3.1 with $m=1$ gives $E_{Q_z}{N}_{C_0}=z<\mathrm{\infty }$. Thus $Q_z\in {\mathcal{P}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}^1(\mathrm{\Omega })$ by [F4], which verifies the domain hypothesis of the specific-entropy theorem in [F2]. Its integer-cube supremum formula applied to step 3.1 yields $I(Q_z)=1-z+z\log z$ and identifies this value with the real-side cube limit. At $z=1$ the value is zero, the reference-law baseline.

By [F1], $N_{C_0}$ is finite and measurable from the restriction to $C_0$, so its square is a measurable local observable.

Fix any finite $b\ge 0$ and choose an integer $n>b+1$. The cell configuration space contains configurations with exactly $n$ distinct points in $C_0$. On each such configuration, $f=n^2>b(1+n)$, so the tame growth bound in [F2] fails for this $b$. Since $b$ was arbitrary, $f$ is not tame.