Interacting density-interval and upper-energy event limit

Define $m(\rho ,e)=\inf \{I(P):\rho (P)=\rho ,\mathrm{\Phi }(P)\le e\}$ and $s=-m$. For stationary finite-entropy laws $P,R$, the binary-mixture chain rule on a bounded cube $G$ gives $\theta D(P_G\mathrm{\parallel }{Q}_G)+(1-\theta )D(R_G\mathrm{\parallel }{Q}_G)=D((\theta P+(1-\theta )R{)}_G\mathrm{\parallel }{Q}_G)+I(S;X_G)$, with $0\le I(S;X_G)\le H(S)<\mathrm{\infty }$. Divide by $\mathrm{\mid }G\mathrm{\mid }$ and use [F4]; the label entropy vanishes, so $I$ is affine on finite mixtures. The expectation $\rho$ and finite-volume energy expectations are affine, hence their limits $\rho ,\mathrm{\Phi }$ are affine where finite. Mixtures of feasible laws show $m$ is convex and $s$ concave. If $({\rho }_n,e_n)\to (\rho ,e)$ and $\mathrm{lim\hspace{0.17em}inf}m({\rho }_n,e_n)<\mathrm{\infty }$, choose near-minimizers with bounded $I$. Compactness [F2] gives a convergent subsequence; continuity of $\rho$ by [F6] and lower semicontinuity of $\mathrm{\Phi },I$ in [F1] show the limit is feasible with entropy no greater than the liminf. Thus $m$ is lower semicontinuous and $s$ upper semicontinuous; the same compactness argument attains every finite infimum. For strict feasibility, choose ${\rho }_1$ in the interior of $D$ below $\nu (\varphi )$ with a finite-energy, finite-entropy law $R$. The inequality $e>{\inf }_{\rho \in D}{e}_0(\rho )$ gives ${\rho }_2\in D$ and a finite-entropy law $P$ with $\mathrm{\Phi }(P)<e$. If ${\rho }_2$ is at an endpoint of $D$, mix $P$ with a sufficiently small positive amount of $R$; affinity of $\rho ,\mathrm{\Phi },I$ keeps the energy below $e$ and moves the density into the interior of $D$.

Establish the approximation needed when the supporting energy coefficient vanishes. Fix $P\in {\mathcal{K}}_B$ with $I(P)<\mathrm{\infty }$. Regularity makes $B$ bounded. Fix a corridor width $g$ greater than its maximum-norm radius and the two-sided tail cutoff. For integer $n\to \mathrm{\infty }$, let $E_{n,q}$ be the event that all distinct pairs in $[0,n{)}^d$ have maximum-norm separation at least $1\mathrm{/}q$ and energy at most $q$. Simplicity, local finiteness and $P\in {\mathcal{K}}_B$ imply $P(E_{n,q})\uparrow 1$ as $q\to \mathrm{\infty }$. Choose positive integers $q_n$ with $p_n=P(E_{n,q_n})\to 1$, and put independent copies of $P_{[0,n{)}^d}(\cdot \mid E_{n,q_n})$ in cores $[0,n{)}^d+(n+g)k$, $k\in {\mathbb{Z}}^d$, leaving the corridors empty. Average the phase uniformly over $[0,n+g{)}^d$ and call the stationary law $P_n$. Minimum separation bounds the particle number in each core by a finite packing bound depending on $n,q_n$; the pair-energy cutoff bounds its internal positive energy. Every cross-core displacement lies beyond the cutoff, so the summable envelope [F10] and the core particle bound make the absolute cross energy per tile finite. Thus $P_n\in {\mathcal{K}}_B$ and $\mathrm{\Phi }(P_n)<\mathrm{\infty }$. Write $D_n=D(P_{[0,n{)}^d}\mathrm{\parallel }{Q}_{[0,n{)}^d})$. For $E=E_{n,q_n}$ and $h=d{P}_{[0,n{)}^d}\mathrm{/}d{Q}_{[0,n{)}^d}$, expanding the conditioned density $h{\mathbf{1}}_E\mathrm{/}{p}_n$ and using $-h\log h\le 1$ where $h<1$ gives $D(P_{[0,n{)}^d}(\cdot \mid E)\mathrm{\parallel }{Q}_{[0,n{)}^d})\le (D_n+1)\mathrm{/}{p}_n-\log p_n$. Independence, the Poisson void cost of the corridors, and convexity under phase averaging imply $I(P_n)\le \frac{(D_n+1)\mathrm{/}{p}_n-\log p_n+(n+g{)}^d-n^d}{(n+g{)}^d}\mathrm{.}$ Indeed, on a large observation cube entropy is bounded by the sum of entropies of the intersecting independent tiles; boundary tiles contribute $o(\text{volume})$ for fixed $n$, and [F4] gives this bound. Hence ${\mathrm{lim\hspace{0.17em}sup}}_{n}I(P_n)\le I(P)$. For any bounded local support $A$, translate $x$, and $K>0$, stationarity gives $E_P[N_{A+x}{\mathbf{1}}_{E_{n,q_n}^c}]\le E_P[N_A{\mathbf{1}}_{\{N_A>K\}}]+K(1-p_n)$. Integrability makes this uniformly small, first as $n\to \mathrm{\infty }$ and then $K\to \mathrm{\infty }$. Thus conditioning changes expectations of translated local tame tests by a quantity tending to zero, uniformly when their support stays in one core. The fraction of phases whose support meets a corridor or two cores is $O_A((g+1)\mathrm{/}n)$; there the uniform bound $E[N_{A+x}]\le c_A\rho (P)\mathrm{/}{p}_n$ controls the count contribution, where $c_A<\mathrm{\infty }$ is obtained by covering the at most $2^d$ core portions of $A+x$ by a fixed number of translated unit cells for all $n$ larger than the diameter of $A$. Their contribution also tends to zero. Therefore $P_n\to P$ in ${\tau }_L$, in particular $\rho (P_n)\to \rho (P)$.

For the lower event bound, take a stationary law $P$ with $I(P)<\mathrm{\infty }$, $\rho (P)\in D^{\circ }$, and $\mathrm{\Phi }(P)<e$. The cell inequality [F11], stationarity, and $E_P{H}_{\mathrm{\ell },\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}\mathrm{/}{\mathrm{\ell }}^d\to \mathrm{\Phi }(P)$ imply a uniform bound on $E_P[T_{\mathrm{\ell }}\mathrm{/}{\mathrm{\ell }}^d]$ for large integer $\mathrm{\ell }$, hence on $E_P[(N_{{\mathrm{\Lambda }}_{\mathrm{\ell }}}\mathrm{/}{\mathrm{\ell }}^d{)}^2]$. They also give a uniform bound $C$ on $E_P[H_{\mathrm{\ell },\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}^{+}]\mathrm{/}{\mathrm{\ell }}^d$, since the negative part is at most $BN$. Choose $M_j\uparrow \mathrm{\infty }$ with $C\mathrm{/}{M}_j\to 0$. For each sufficiently large ${\mathrm{\ell }}_j$, let $E_j=\{H_{{\mathrm{\ell }}_j,\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}\le M_j{\mathrm{\ell }}_j^d\}$ and $p_j=P(E_j)$. Markov's inequality gives $p_j\ge 1-C\mathrm{/}{M}_j$. On $E_j$, the cell inequality and $N^2\mathrm{/}{\mathrm{\ell }}_j^d\le T_{{\mathrm{\ell }}_j}$ give $N\mathrm{/}{\mathrm{\ell }}_j^d\le K_j$, where $K_j$ is the positive root bound for $A{x}^2-Bx\le M_j$. The count-density loss under conditioning tends to zero by Cauchy–Schwarz and the uniform second-moment bound. The conditional expected energy density has limsup at most $\mathrm{\Phi }(P)$ because the removed energy is positive. The entropy conditioning formula gives $D(P_{{\mathrm{\Lambda }}_{{\mathrm{\ell }}_j}}(\cdot \mid E_j)\mathrm{\parallel }{Q}_{{\mathrm{\Lambda }}_{{\mathrm{\ell }}_j}})\le (D(P_{{\mathrm{\Lambda }}_{{\mathrm{\ell }}_j}}\mathrm{\parallel }{Q}_{{\mathrm{\Lambda }}_{{\mathrm{\ell }}_j}})+1\mathrm{/}{\mathrm{e}}_{\mathrm{E}})\mathrm{/}{p}_j-\log p_j$, where ${\mathrm{e}}_{\mathrm{E}}$ is Euler's number, so the conditioned block entropy per volume is at most $I(P)+o(1)$. Choose $g_j>2R_0$ with $\eta (g_j)\le (j^{d+1}{K}_j^2{)}^{-1}$ and set ${\mathrm{\ell }}_j=j{g}_j$, increasing $g_j$ further so ${\mathrm{\ell }}_j$ is large enough for the specific entropy and energy limits above. Then $g_j\mathrm{/}{\mathrm{\ell }}_j=1\mathrm{/}j\to 0$ and $p_j\to 1$. In a large periodic cube of side $L$, put $m_L=\lfloor L\mathrm{/}({\mathrm{\ell }}_j+g_j)\rfloor$ cores per coordinate, each of side ${\mathrm{\ell }}_j$, on a periodic grid of exact spacing $a_{L,j}=L\mathrm{/}{m}_L\ge {\mathrm{\ell }}_j+g_j$. Thus every core is separated from every other core and its periodic images by at least $g_j$, and the unoccupied fraction tends to $1-({\mathrm{\ell }}_j\mathrm{/}({\mathrm{\ell }}_j+g_j){)}^d$. Put independent conditioned blocks in the cores and leave the complement empty. For two core indices at maximum-norm distance $m\ge 1$, the total absolute cross energy per core is at most $C_d{K}_j^2{\mathrm{\ell }}_j^{2d}{\sum }_{m\ge 1}{m}^{d-1}\psi (m{g}_j)$, so its contribution per unit volume is at most $C_d^{\mathrm{\prime }}{K}_j^2({\mathrm{\ell }}_j\mathrm{/}{g}_j{)}^d\eta (g_j)\le C_d^{\mathrm{\prime }}\mathrm{/}j$ by [F10], including the periodic image interactions across the outer boundary. The iid law of core counts and core energies gives the required density and within-core energy laws of large numbers; since the cross-energy error is uniformly bounded by $C_d^{\mathrm{\prime }}\mathrm{/}j$, for large fixed $j$ the product block law assigns probability tending to one to $A_L$ as $L\to \mathrm{\infty }$, with density in $D$ and energy below $e$ by strict margins. Relative to the Poisson reference on ${\mathrm{\Lambda }}_L$, the product block law has log density equal to the sum of the iid block log densities plus the corridor volume (the Poisson void cost). The block log density is integrable because its relative entropy is finite; truncation followed by the ordinary weak law gives its average limit. Therefore its typical log density per volume is at most $I(P)+o_j(1)$ as $j\to \mathrm{\infty }$. On the intersection of $A_L$ with this typical set, changing measure back to $Q_{{\mathrm{\Lambda }}_L}$ gives $Q_{{\mathrm{\Lambda }}_L}(A_L)\ge \exp [-L^d(I(P)+o_j(1)+o_L(1))]$. If a minimizing law lies on a constraint boundary, mix it with a strict feasible law; entropy affinity and lower semicontinuity show that the entropy infimum is unchanged. This proves the lower event bound and eventual positivity.

Put $K=\{P\in {\mathcal{P}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}^1:\rho (P)\in \bar{D},\mathrm{\Phi }(P)\le e\}$. The set is closed by [F1], lower semicontinuity of $\mathrm{\Phi }$, and continuity of $\rho$ in ${\tau }_L$. It is also evaluation-measurable: $\rho (P)$ is the integral of a measurable local count, and $\mathrm{\Phi }(P)$ is the countable integer-cube limit of finite-volume measurable energy expectations on its stated domain. These expectations are measurable by simple-function approximation of positive and negative parts, with stability controlling the negative part. Thus both constraint preimages are measurable. Likewise ${\mathcal{K}}_B$ is closed and evaluation-measurable: it is the countable intersection, over integer cubes, of zero-probability conditions for bounded measurable local forbidden-pair events. For every configuration with finite $H_{L,\mathrm{p}\mathrm{e}\mathrm{r}}$, the periodized empirical field satisfies $\rho (R_{L,\omega })=N_{{\mathrm{\Lambda }}_L}\mathrm{/}{L}^d$ by [F7] and $\mathrm{\Phi }(R_{L,\omega })=H_{L,\mathrm{p}\mathrm{e}\mathrm{r}}\mathrm{/}{L}^d$ by Georgii's periodic-energy identity (1994, equation (2.16)); hence $A_L\subseteq \{R_{L,\omega }\in K\}$. The Poisson empirical-field upper bound of [L2], applied to this closed set, gives ${\mathrm{lim\hspace{0.17em}sup}}_{L\to \mathrm{\infty }}{L}^{-d}\log Q_{{\mathrm{\Lambda }}_L}(A_L)\le -{\inf }_{P\in K}I(P)=-m,$ where $m$ is the minimum in the Statement. Together with step 2.1 this proves the event exponent. The exact periodic empirical field accounts for all long-range image interactions in this upper bound; no fixed empty corridor is used to discard them.

The tangent-plane argument in published [L1] separates the strict-feasibility constraint set in finite dimension [F3]. It supplies a finite $\alpha =\log z$ and a nonnegative coefficient $\beta \ge 0$. When $\beta >0$, the constrained entropy minimizers lie in the positive-beta grand variational minimizer set. To spell out the compatible retained-admissibility interpretation when the supporting coefficient is zero, write $m_{\ast }=m(D,e)$ and let $({\rho }_{\ast },u_{\ast })$ be a contact point. A supporting plane with energy coefficient $\beta =0$ gives $-m(\rho ,u)\le -m_{\ast }-\alpha (\rho -{\rho }_{\ast })$ for every finite-energy pair $(\rho ,u)$; equivalently, every stationary finite-energy law satisfies $I(P)-\alpha \rho (P)\ge m_{\ast }-\alpha {\rho }_{\ast }$. Equality holds at the constrained minimizers in contact. Any law in ${\mathcal{K}}_B$ with finite $I$ is approximated in ${\tau }_L$ and density by finite-energy separated-block laws $P_j$ with ${\mathrm{lim\hspace{0.17em}sup}}_{j}I(P_j)\le I(P)$, by step 1.2. Passing the inequality to the limit shows it holds on all of ${\mathcal{K}}_B$; equality at a constrained minimizer identifies the same global minimum there. The finite-volume zero-beta weight remains $z^N{\mathbf{1}}_{\mathrm{a}\mathrm{d}\mathrm{m}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{e}}$ throughout.

Let $U$ be an open ${\tau }_L$ neighborhood of the common grand minimizer set ${\mathcal{M}}_{\beta ,z}$. That set is compact by [F1] for $\beta >0$ and [F9] for $\beta =0$. Each member has a basic neighborhood contained in $U$, given by finitely many strict inequalities in local tame expectations. Each expectation is evaluation-measurable by truncation and simple-function approximation. A finite subcover gives an open evaluation-measurable $V$ with ${\mathcal{M}}_{\beta ,z}\subseteq V\subseteq U$. The set $K_V=K\cap V^c$ is closed and measurable by step 3.1. Its entropy infimum is strictly larger than $m$: otherwise a sequence with $I\downarrow m$ in a compact entropy sublevel [F2] has a limit in $K_V$ that is a constrained minimizer, hence in ${\mathcal{M}}_{\beta ,z}\subseteq V$ by step 4.1, a contradiction. The measurable-set upper bound [L2] gives ${\mathrm{lim\hspace{0.17em}sup}}_{L\to \mathrm{\infty }}{L}^{-d}\log Q_{{\mathrm{\Lambda }}_L}\{R_{L,\omega }\in K_V\}\le -{\inf }_{P\in K_V}I(P)<-m\mathrm{.}$ Since $A_L\cap \{R\mathrm{\notin }U\}\subseteq \{R\in K_V\}$, division by the event probability from step 3.1 bounds the conditional outer probability outside $U$ by a quantity tending to zero. For the positive-beta grand law, apply the good-rate upper bound [L3] to the measurable closed set $V^c$; compact rate sublevels and its zero set inside $V$ give a strict positive gap. For the retained zero-beta grand law, its numerator outside $U$ is bounded by the Poisson integral of $e^{\alpha N}{\mathbf{1}}_{\{R\in {\mathcal{K}}_B\cap V^c\}}$, where $\alpha =\log z$. Apply the measurable Laplace upper bound [L2] with the functional equal to $-\alpha \rho$ on ${\mathcal{K}}_B\cap V^c$ and $+\mathrm{\infty }$ elsewhere. It is lower semicontinuous, evaluation-measurable and bounded below by $-\mathrm{\mid }\alpha \mathrm{\mid }(1+\rho )$. For $a>0$, apply the bounded-test entropy inequality [F5] to truncations of $a{N}_{[0,n{)}^d}$ and use the Poisson formula $E_Q{e}^{a{N}_{[0,n{)}^d}}=\exp (n^d(e^a-1))$. Removing the truncation, dividing by $n^d$, and taking the integer-cube limit [F4] gives $I(P)\ge a\rho (P)-(e^a-1)$. Choosing $a>2\max (\alpha ,0)$ shows that a bounded sublevel of $I-\alpha \rho$ has bounded $I$. This rate is lower semicontinuous since $I$ is and $\rho$ is continuous; together with the closedness of ${\mathcal{K}}_B$ and [F2], this proves compact sublevels. Its infimum on ${\mathcal{K}}_B\cap V^c$ is therefore strictly larger than its global minimum. The denominator rate from [F9] cancels that global minimum. Thus this grand outer probability also tends to zero. Empty complement sets make the estimates immediate. A singleton minimizer gives convergence in probability in the stated outer sense in both branches.

The conditional law is torus-invariant and its entropy density is $-L^{-d}\log Q_L(A_L)$, which is bounded by the event limit. The matching grand law is torus-invariant as well. At positive beta its entropy density is bounded by the pressure identity, the superstable lower energy bound, and the Poisson entropy count inequality; at zero beta use the same identity with the hard indicator and omit the energy term. Consequently [F8] applies. Its transfer result converts step 5.1 into convergence of the periodically continued laws for every local tame observable under uniqueness. No zero extension outside the fundamental cube is used.