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Full interacting shell, canonical, and variational-grand local marked equivalence

Statement

Assume AC and the hypotheses and notation of Full phase shell marginal satisfies the positional variational bound, including its AC, ultrafilter extension principle, DC, and HB assumptions. Fix a bounded Borel region $Δ$ and use the marked periodic continuation and local restriction defined in Marked periodic continuation and local restriction. By translating $Δ$ and the spatial argument of $G$ together, take a representative with $Δ \subseteq [0, \infty)^{d}$ ; then $Δ \subseteq Λ_{L_{n}}$ for all sufficiently large $n$ . This translation preserves the expectations because the finite laws are torus-translation invariant and the limiting independent mark lift of the stationary positional state is stationary. Let $M_{n}^{p h}$ be the uniform full-phase band law, $C_{n}^{p h}$ the exact- $N_{n}$ full canonical law, and $P_{*}$ the singleton positional grand variational minimizer. Give points of $P_{*}$ independent marks with kernel $ν_{b}$ of Interacting marked phase-space and reference measures, and denote the resulting infinite marked law by $P_{*}^{m a r k}$ , whose existence is supplied by Existence of the independent Gaussian-marked positional lift. For finite-volume expectations, evaluate $G$ on the restriction $ω_{L_{n}, q, p}^{p e r} ∣_{Δ}$ ; under $P_{*}^{m a r k}$ , evaluate it on $ξ_{Δ}$ . Then for every bounded marked-local observable $G$ on $Δ$ in the sense of Marked periodic continuation and local restriction, $E_{M_{n}^{p h}} G \to E_{P_{*}^{m a r k}} G, E_{C_{n}^{p h}} G \to E_{P_{*}^{m a r k}} G .$ The matched full grand activity is $\exp (α) / κ_{b}$ by the normalization identity in Interacting marked phase-space and reference measures; $P_{*}^{m a r k}$ is the independent-Gaussian marked lift associated with that positional variational minimizer. If additionally $∣ G ∣ \leq C (1 + N_{Δ})$ , shell convergence also holds for $G$ . This is fixed-region convergence, not uniform over growing regions or arbitrary unbounded momentum observables. No finite-volume grand-canonical convergence or DLR converse is asserted.

Facts & Assumptions

Given: The interaction, density, temperature, band, and singleton-minimizer hypotheses from Full phase shell marginal satisfies the positional variational bound.

[A1]

AC is also required by Pinsker's inequality and the countable-coordinate marked-lift construction (The Axiom of Choice).

[F1]

Spatial random-occupation conditional mark entropy bound: for each fixed Δ, the average conditional mark relative entropy on points spatially inside Δ tends to zero, although their number is random.

[F2]

Measurable information contraction and Pinsker bound: measurable restrictions decrease KL divergence, and event-supremum total variation is at most sqrt(D/2).

[F3]

Full phase shell marginal satisfies the positional variational bound: for every local-tame $f$ , the shell and exact- $N_{n}$ canonical expectations of $f$ on the periodic continuation converge to $P_{*} (f)$ ; the shell empirical-field barycenters converge to $P_{*}$ in $τ_{L}$ , and canonical empirical fields converge in probability there.

[F4]

Interacting marked phase-space and reference measures: canonical marks are independently ν_b-marked, and the full activity corresponding to positional activity exp(α) is exp(α)/κ_b.

[F5]

Existence of the independent Gaussian-marked positional lift and Marked periodic continuation and local restriction: the independent mark kernel defines a unique marked lift of $P_{*}$ and preserves stationarity, the local observable is measurable on the finite restriction $ξ_{Δ}$ , and finite-torus restrictions use the explicitly defined marked periodic continuation.

[F6]

Local-Tame Topology: every measurable local function bounded by a constant times $1 + N_{Δ}$ is local-tame; in particular every bounded measurable local observable and each bounded indicator of a local count value is an admissible positional convergence test.

[F7]

Superstable regular continuum grand pressure and variational formula: its minimization is over stationary finite-intensity positional laws; hence the variational minimizer $P_{*}$ is stationary.

Proof

Given: The hypotheses and exact supplier interfaces in Facts & Assumptions.

1.1A1F1F2construct

Let $B_{n}$ be the shell positional marginal and let ${\hat{B}}_{n}$ sample $B_{n}$ followed by independent $ν_{b}$ marks. Form two augmented laws on (all positions $q$ , marks attached to $q \cap Δ$ ): the first is the corresponding marginal of $M_{n}^{p h}$ and the second is that of ${\hat{B}}_{n}$ . Conditional on $q$ , their mark laws are $b_{n, Δ} (\cdot ∣ q)$ and $ν_{b}^{\otimes N_{Δ} (q)}$ . The relative-entropy chain rule therefore makes the augmented divergence exactly the conditional KL average in [F1]. Restricting the augmented observation to positions and marks in $Δ$ is measurable; data processing and Pinsker [F2] show that the local marked restrictions of $M_{n}^{p h}$ and ${\hat{B}}_{n}$ have total variation tending to zero. For bounded $G$ , the expectation difference is at most $2 ∥ G ∥_{\infty}$ times this event-supremum distance, hence tends to zero.

2.1A1F3F4F5F6step 1.1

Integrate $G$ against the finite product $ν_{b}$ marks at fixed local $q$ and call the result $g (q \cap Δ)$ . The lift construction [F5] gives this measurable kernel and its infinite marked law; since $G$ is bounded, $∣ g ∣ \leq ∥ G ∥_{\infty}$ . By [F6], $g$ is local-tame. Therefore $E_{{\hat{B}}_{n}} G = B_{n} (g) \to P_{*} (g) = E_{P_{*}^{m a r k}} G$ by [F3]. The canonical factorization in [F4] and canonical positional convergence in [F3] give the second limit. By [F4], the matching full activity is $\exp (α) / κ_{b}$ ; this identifies the same infinite variational marked state and says nothing about convergence of finite-volume grand laws.

3.1F1F3F5F6F7step 1.1step 2.1∎

Suppose |G|≤C(1+N_Δ). By [F6], bounded indicators of each count value are local-tame, so their probabilities converge under B_n by [F3]; the mean count also converges because N_Δ is local-tame. For integer K, E_{B_n}(N_Δ-K)^+=E_{B_n}N_Δ-E_{B_n}(N_Δ∧K); for fixed K this converges to the corresponding P_* expectation. If N_Δ>2K, then N_Δ≤2(N_Δ-K), so E[N_Δ1_{N_Δ>2K}]≤2E(N_Δ-K)^+. First take limsup over n, then K→∞; integrability of N_Δ under P_* makes this tail vanish. Indeed P_* is stationary with finite unit-cell intensity by [F7], and bounded Δ meets only finitely many unit cells. This proves uniform integrability of 1+N_Δ under B_n. The shell and Gaussian lift have the same local count law. The limiting marked lift has the same local count as P_*, so its tail also vanishes. Truncate G to N_Δ≤K; it is bounded, so step 1.1 applies. Both finite-volume omitted tails are bounded by C E[(1+N_Δ)1_{N_Δ>K}], uniformly vanishing as K→∞. Let n→∞ and then K→∞ to obtain the claimed shell limit.

Full interacting shell, canonical, and variational-grand local marked equivalence

Statement

Facts & Assumptions

Proof

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