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Spatial random-occupation conditional mark entropy bound

Statement

Assume AC and the hypotheses of Full phase shell marginal satisfies the positional variational bound. Let $M_{n}^{p h}$ be the normalized phase-reference restriction to $A_{n}$ , and let $B_{n}$ be its positional marginal. Given the full position configuration $q$ , let $b_{n, A} (\cdot ∣ q)$ be the shell conditional law of marks on the marked local restriction of Marked periodic continuation and local restriction. By torus translation invariance, represent any bounded Borel region $Δ$ by a fixed translate inside $Λ_{L_{n}}$ for all sufficiently large $n$ . Choose an axis-parallel half-open cube $C \supseteq Δ$ of side $ℓ > 0$ , independent of $n$ , and pack $L_{n}^{p a c k} = ⌊ L_{n} / ℓ ⌋^{d}$ disjoint translates of $C$ into the periodic cube, so $L_{n}^{p a c k} / V_{n} \to 1 / ∣ C ∣$ . For the independent Gaussian mark reference $ν_{b}$ of Interacting marked phase-space and reference measures, $E_{B_{n}} D (b_{n, Δ} (\cdot ∣ q) ∣ ∣ ν_{b}^{\otimes N_{Δ} (q)}) \leq D (M_{n}^{p h} ∣ ∣ C_{n}^{p h}) / L_{n}^{p a c k} .$ Thus the left side tends to zero. When $Δ$ itself is an axis-parallel half-open cube, take $C = Δ$ to obtain the stated packing rate $1 / ∣ Δ ∣$ . Occupancy $N_{Δ} (q)$ is random; the comparison is for the full spatially selected set of particles, not a fixed set of labels.

Facts & Assumptions

Given: AC, the shell and canonical phase laws, standard-Borel finite-region restrictions, a fixed bounded Borel region, and the Gaussian mark kernel.

[A1]

AC is used in the conditional-law and entropy-chain results (The Axiom of Choice).

[F1]

Interacting marked phase-space and reference measures and Full phase shell marginal satisfies the positional variational bound: conditional canonical marks are $ν_{b}^{\otimes N_{n}}$ ; the shell conditional kernel is obtained by restricting the finite mark reference to its energy band; the shell law is invariant under torus translations.

[F2]

Poisson Configuration Reference and Marked periodic continuation and local restriction: finite configurations have measurable cell counts and lexicographic coordinate codes; marked periodic restrictions to bounded regions are finite and measurable, so after conditioning on positions each block is a finite-dimensional standard-Borel coordinate space.

[F3]

Conditional entropy chain and disjoint-block bounds: relative entropy against a product reference is at least the sum of disjoint-block conditional mark divergences.

[F4]

Measurable information contraction and Pinsker bound: measurable restriction of a mark vector cannot increase its relative entropy against the corresponding product reference.

[F5]

Full uniform phase shell has vanishing specific canonical KL: full shell-to-canonical relative entropy is finite and $o (V_{n})$ .

Proof

Given: The hypotheses and definitions above.

1.1A1F1F3F5construct

The phase reference has a measurable density. For each $q$ , integrate the band indicator over the $N_{n}$ -mark reference; Tonelli gives a measurable normalizer. On the $B_{n}$ -full set where it is positive, division gives the conditional kernel $b_{n} (\cdot ∣ q)$ ; choose a fixed probability kernel on the null complement. The conditional chain rule, nonnegativity of the positional divergence, and [F3] give $J_{n} : = E_{B_{n}} D (b_{n} (. ∣ q) ∣ ∣ ν_{b}^{\otimes N_{n}}) \leq D (M_{n}^{p h} ∣ ∣ C_{n}^{p h}) .$

2.1F1F2F3step 1.1algebra

For each $q$ , partition its finitely many points into those in each of the $L_{n}^{p a c k}$ disjoint translates $C_{j}$ packed in the periodic cube, plus the remainder. This is a measurable finite partition; ordering the points by the cell codes in [F2] identifies each block with a finite-dimensional standard-Borel mark space, even though its size depends on $q$ . The Gaussian product reference factors over these disjoint sets. Applying [F3] conditionally gives $D (b_{n} (. ∣ q) ∣ ∣ ν_{b}^{\otimes N_{n}}) \geq \sum_{j = 1}^{L_{n}^{p a c k}} D (b_{n, j} (. ∣ q) ∣ ∣ ν_{b}^{\otimes N_{Δ_{j}} (q)}),$ where an empty block contributes zero. Joint torus-translation invariance transports each block law and its reference kernel, so the integrated divergence is the same for every translate. Averaging and discarding the nonnegative remainder term bounds the divergence for a chosen translate of $C$ by $J_{n} / L_{n}^{p a c k}$ .

3.1F4F5step 2.1∎

For $Δ \subseteq C$ , restriction of the ordered mark vector on $q \cap C$ to the points in $q \cap Δ$ is measurable. Data processing [F4] gives $D (b_{n, Δ} (\cdot ∣ q) ∣ ∣ ν_{b}^{\otimes N_{Δ} (q)}) \leq D (b_{n, C} (\cdot ∣ q) ∣ ∣ ν_{b}^{\otimes N_{C} (q)})$ for $B_{n}$ -almost every $q$ . Integrating and applying step 2.1 bounds the left side by $J_{n} / L_{n}^{p a c k}$ . By [F5], $J_{n} = o (V_{n})$ , while $L_{n}^{p a c k} / V_{n} \to 1 / ∣ C ∣ > 0$ ; therefore the bound tends to zero.

Spatial random-occupation conditional mark entropy bound

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources