Spatial Mean-Information Theorem

Let $S_N=\{0,\dots ,N-1{\}}^d$ be the cell indices in ${\mathrm{\Lambda }}_N$. For $i\in S_N$, set $A_{i,N}=\{j-i:j\in S_N,j{<}_{\mathrm{l}\mathrm{e}\mathrm{x}}i\}$; this is a finite subset of the past of the reference cell. Repeated conditional-density factorization from [F3] gives, $P$-almost surely, $\log h_N(\omega )={\sum }_{i\in S_N}\log k_{A_{i,N}}(T_{-i}\omega )\mathrm{.}$ Every finite factor is defined $P$-almost surely because the corresponding local relative entropy is finite by [F1].

Fix $\epsilon >0$. By the directed convergence in [F2], choose a finite past $A_{\ast }$ such that every finite $B\supseteq A_{\ast }$ satisfies $E_P\mathrm{\mid }\log k_B-Z\mathrm{\mid }<\epsilon$. Put $R=\max \{\mathrm{\mid }{a}_j\mathrm{\mid }:a\in A_{\ast },1\le j\le d\}$, with $R=0$ if $A_{\ast }=\mathrm{\varnothing }$. Let $S_N^{\circ }$ consist of cells at least $R$ cells from each face of $S_N$. If $i\in S_N^{\circ }$, then $A_{\ast }\subseteq A_{i,N}$, so by stationarity $E_P\mathrm{\mid }\log k_{A_{i,N}}(T_{-i}\omega )-Z(T_{-i}\omega )\mathrm{\mid }<\epsilon$. The boundary has $N^d-(N-2R{)}^d=o(N^d)$ cells (for $N>2R$). For every finite past $B$, $E_P\log k_B=g(B)\le \gamma$ and $E_P(-\log k_B{)}^{+}\le 1\mathrm{/}e$, hence $E_P\mathrm{\mid }\log k_B\mathrm{\mid }\le \gamma +2\mathrm{/}e$. For $Z=\log k_{\mathrm{\infty }}$, $E_P(-\log k_{\mathrm{\infty }}{)}^{+}={\int }_{0<k_{\mathrm{\infty }}<1}{k}_{\mathrm{\infty }}(-\log k_{\mathrm{\infty }})d{R}_0\le 1\mathrm{/}e$ on its reference probability $R_0=P_{F_{+}}\otimes Q_C$, so [F2] gives $E_P\mathrm{\mid }Z\mathrm{\mid }\le \gamma +2\mathrm{/}e$ as well. Summing the interior errors and bounding each boundary error by $2(\gamma +2\mathrm{/}e)$ yields ${\parallel N^{-d}\log h_N-N^{-d}{\sum }_{i\in S_N}Z(T_{-i}\cdot )\parallel }_{L^1(P)}⟶0,$ since $\epsilon$ was arbitrary.

For $j=1,\dots ,d$, the shift $U_{j}f=f\circ T_{-e_j}$ is a linear isometry on complex $L^2(P)$ by stationarity. Apply [F4] to $N^{-1}{\sum }_{n=0}^{N-1}{U}_j^n$, obtaining strong convergence to the orthogonal projection $P_j$ onto the $U_j$-invariant subspace. The coordinate shifts commute, so every average for coordinate $j$ commutes with every average for coordinate $k$; taking strong limits shows that the $P_j$ commute with each other and with the other coordinate averages. Since all these operators have norm at most one, a finite telescoping estimate gives strong convergence of the product of the $d$ averages to $P_1\cdots P_d$. This product is the orthogonal projection onto the intersection of the coordinate-invariant subspaces, because the $P_j$ are commuting orthogonal projections. The product averages are exactly the rectangular cube averages $N^{-d}{\sum }_{i\in S_N}{Z}_M(T_{-i}\cdot )$, so these converge in $L^2$ for bounded $Z_M$ to a function invariant under every integer shift.

Since $Z\in L^1(P)$, dominated convergence gives $\mathrm{\parallel }Z-Z_M{\mathrm{\parallel }}_1\to 0$. Cube averages are $L^1$ contractions, so the averages of $Z$ are Cauchy in $L^1$: for large $M$ their distance from the convergent averages of $Z_M$ is at most $\mathrm{\parallel }Z-Z_M{\mathrm{\parallel }}_1$. Let their $L^1$ limit be $s$. Every cube average has mean $E_{P}Z=\gamma$, hence $E_{P}s=\gamma$. For each coordinate shift $e_j$, the averaging index cube and its translate have symmetric difference of exactly $2N^{d-1}$ indices, so $\mathrm{\parallel }{U}_j{A}_{N}Z-A_{N}Z{\mathrm{\parallel }}_1\le 2\mathrm{\parallel }Z{\mathrm{\parallel }}_1\mathrm{/}N$. Passing to the $L^1$ limit gives $U_{j}s=s$. These generators imply invariance under every integer translation.

Fix $r\in {\mathbb{R}}^d$, put $G={\mathrm{\Lambda }}_N$, $G^{\mathrm{\prime }}=G+r$, and $B=G\cap G^{\mathrm{\prime }}$. For $N>2\mathrm{\parallel }r{\mathrm{\parallel }}_{\mathrm{\infty }}$, $B$ contains a cube $J$ of side $N-2\mathrm{\parallel }r{\mathrm{\parallel }}_{\mathrm{\infty }}$. By [F1] and restriction monotonicity, $D(P_G\mathrm{\parallel }{Q}_G)\le \gamma N^d$ and $D(P_B\mathrm{\parallel }{Q}_B)\ge D(P_J\mathrm{\parallel }{Q}_J)$; the latter divided by $N^d$ tends to $\gamma$. Therefore $N^{-d}(D(P_G\mathrm{\parallel }{Q}_G)-D(P_B\mathrm{\parallel }{Q}_B))\to 0$. The conditional chain rule [F3] identifies $\log h_G-\log h_B$ with a conditional log density. Its mean is this entropy difference, and [F3] bounds its negative-part expectation by $1\mathrm{/}e$; thus its $L^1(P)$ norm is at most the entropy difference plus $2\mathrm{/}e$. The same bound applies to $G^{\mathrm{\prime }}$. Translation covariance gives $\log h_{G^{\mathrm{\prime }}}=(\log h_G)\circ T_{-r}$; thus the normalized information in $G$ and $G^{\mathrm{\prime }}$ have the same $L^1$ limit. Since $N^{-d}\log h_G\to s$, it follows that $s\circ T_{-r}=s$ $P$-almost surely for this $r$.

Step 5.1 applies to every fixed $r\in {\mathbb{R}}^d$. For each rational $q$, the event $\{s\le q\}$ is therefore invariant modulo null sets under every translation. Full-translation ergodicity makes each such event have probability $0$ or $1$. If $s$ were not almost surely constant, some rational sublevel set would have probability strictly between $0$ and $1$; hence $s$ is constant. Its mean from step 4.1 is $\gamma$, so $s=\gamma$. Combining with step 2.1 proves the asserted $L^1(P)$ limit.

To see that the invariant limit need not be constant, fix $z>0$ with $z\mathrm{\ne }1$ and put $P_{\ast }=(Q+Q_z)\mathrm{/}2$. It is stationary and has finite cell intensity by [F6]. Let $V=N^d$. Since ${\mathrm{\Lambda }}_N$ is the union of $V$ unit cells, the common conditional position laws and independent cell configurations in [F6] give $r_N:=\frac{d(Q_z{)}_{{\mathrm{\Lambda }}_N}}{d{Q}_{{\mathrm{\Lambda }}_N}}=e^{(1-z)V}{z}^{N_{{\mathrm{\Lambda }}_N}},h_N^{\ast }:=\frac{d(P_{\ast }{)}_{{\mathrm{\Lambda }}_N}}{d{Q}_{{\mathrm{\Lambda }}_N}}=\frac{1+r_N}{2}\mathrm{.}$ Thus $D((Q_z{)}_{{\mathrm{\Lambda }}_N}\mathrm{\parallel }{Q}_{{\mathrm{\Lambda }}_N})=E_{Q_z}\log r_N=V{c}_z$, where $c_z:=1-z+z\log z$; the expectation uses $E_{Q_z}{N}_{{\mathrm{\Lambda }}_N}=zV$. By [F8], $x-1-\log x={\int }_1^x(1-1\mathrm{/}t)dt>0$ for $x>0$, $x\mathrm{\ne }1$: the integrand is positive for $t>1$, negative for $t<1$, and the orientation reverses in the latter case. Applying this at $x=1\mathrm{/}z$ gives $c_z=z(1\mathrm{/}z-1-\log (1\mathrm{/}z))>0$. By [F7], relative entropy is convex in its first argument, so $D((P_{\ast }{)}_{{\mathrm{\Lambda }}_N}\mathrm{\parallel }{Q}_{{\mathrm{\Lambda }}_N})\le V{c}_z\mathrm{/}2$. The specific-entropy formula [F1] therefore gives $I(P_{\ast })\le c_z\mathrm{/}2<\mathrm{\infty }$, and steps 1.1–5.1 yield an invariant limit $s_{\ast }\in L^1(P_{\ast })$. Under $Q_{\lambda }$, put $K=N_{{\mathrm{\Lambda }}_N}$ and $\theta =\lambda V$. Its Poisson mass formula gives $E[K(K-1)]=e^{-\theta }{\theta }^2{\sum }_{n\ge 2}{\theta }^{n-2}\mathrm{/}(n-2)!={\theta }^2$, hence $E[(K-\theta {)}^2]=\theta$. For $\epsilon >0$, the pointwise bound ${\epsilon }^2{V}^2{\mathbf{1}}_{\{\mathrm{\mid }K-\theta \mathrm{\mid }>\epsilon V\}}\le (K-\theta {)}^2$ yields $Q_{\lambda }(\mathrm{\mid }K\mathrm{/}V-\lambda \mathrm{\mid }>\epsilon )\le \lambda \mathrm{/}({\epsilon }^{2}V)\to 0$. Consequently $V^{-1}\log r_N\to 1-z+\lambda \log z$ in $Q_{\lambda }$-probability. For every real $u$, $0\le V^{-1}\log (1+e^{Vu})-\max \{0,u\}\le \frac{\log 2}{V},$ so $V^{-1}\log h_N^{\ast }\to 0$ in $Q$-probability and $V^{-1}\log h_N^{\ast }\to c_z$ in $Q_z$-probability: $1-z+\log z<0$ by $\log z<z-1$, and $c_z>0$ for $z\mathrm{\ne }1$. Since $P_{\ast }\ge Q\mathrm{/}2$ and $P_{\ast }\ge Q_z\mathrm{/}2$, convergence in $P_{\ast }$-probability to $s_{\ast }$ also holds under each component. Uniqueness of limits in probability forces $s_{\ast }=0$ $Q$-almost surely and $s_{\ast }=c_z$ $Q_z$-almost surely. These distinct values on positive-weight components show that $s_{\ast }$ is not constant $P_{\ast }$-almost surely.