Finite-alphabet mean-field ensembles and upper supporting lines

Enumerate $\mathcal{A}=\{a_1,\dots ,a_d\}$, with $d\ge 1$, and identify ${\mathbb{R}}^{\mathcal{A}}$ with Euclidean ${\mathbb{R}}^d$. Write $\mathrm{\parallel }x{\mathrm{\parallel }}_1:={\sum }_{i=1}^d\mathrm{\mid }{x}_i\mathrm{\mid }$ and $\mathrm{\parallel }x{\mathrm{\parallel }}_2:=({\sum }_{i=1}^d{x}_i^2{)}^{1\mathrm{/}2}$; the declared ${\mathrm{\ell }}^1$ topology on the simplex is the topology from $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_1$. The simplex is closed because it is the intersection of the closed coordinate constraints $0\le q(a_i)\le 1$ and the closed affine constraint ${\sum }_{i}q(a_i)=1$; these are closed since coordinate maps and their finite sum are continuous. It is bounded because $\mathrm{\parallel }q{\mathrm{\parallel }}_2^2={\sum }_{i}q(a_i{)}^2\le {\sum }_{i}q(a_i)=1$. By [F6] it is compact in the Euclidean topology. For every $x\in {\mathbb{R}}^d$, $\mathrm{\parallel }x{\mathrm{\parallel }}_2\le \mathrm{\parallel }x{\mathrm{\parallel }}_1$ because $({\sum }_i\mathrm{\mid }{x}_i\mathrm{\mid }{)}^2\ge {\sum }_i{x}_i^2$, and $\mathrm{\parallel }x{\mathrm{\parallel }}_1\le d\mathrm{\parallel }x{\mathrm{\parallel }}_2$ because ${\sum }_i\mathrm{\mid }{x}_i\mathrm{\mid }\le d{\max }_i\mathrm{\mid }{x}_i\mathrm{\mid }\le d\mathrm{\parallel }x{\mathrm{\parallel }}_2$. These inequalities show that the Euclidean and ${\mathrm{\ell }}^1$ metrics have the same open sets, so the simplex is compact in its declared topology. By continuity of $h$ and $D$, each fiber $h^{-1}(\{u\})$ and each sublevel set $\{q:D(q\mathrm{\parallel }p)\le r\}$ is closed in the simplex and hence compact by [F7]. The continuous-image and extreme-value conclusions in [F8] therefore apply to these sets. In particular, [F4] makes $q\mapsto D(q\mathrm{\parallel }p)+\beta h(q)$ continuous on the compact simplex, so it attains a finite minimum $m_{\beta }$. The rate $D$ is nonnegative by [F5]. The empirical macrostate law under $p^{\otimes n}$ has rate $D(q\mathrm{\parallel }p)$ by [F2]. For every Borel $B\subseteq \mathrm{\Delta }(\mathcal{A})$, the canonical macrostate probability is exactly $p_{n,\beta }(L_n\in B)=\frac{{\int }_B{e}^{-n\beta h(q)}d{\mu }_n(q)}{{\int }_{\mathrm{\Delta }(\mathcal{A})}{e}^{-n\beta h(q)}d{\mu }_n(q)},$ where ${\mu }_n$ is the base law of $L_n$; this follows by summing the word weights in [F1] over the fiber $L_n^{-1}(B)$. Thus it is the tilt in [F3] with $F=-\beta h$, and its rate is $D+\beta h-m_{\beta }$. Its zero set is ${\mathcal{M}}_{\beta }$. Let $U$ be any open neighborhood of ${\mathcal{M}}_{\beta }$ and put $C:=\mathrm{\Delta }(\mathcal{A})\setminus U$. If $C=\mathrm{\varnothing }$, the probability of the complement is zero. Otherwise $C$ is a nonempty closed subset of the compact simplex, hence compact by [F7]. The continuous rate is positive at every point of $C$ because its zero set is ${\mathcal{M}}_{\beta }\subset U$; by [F8] it attains a strictly positive minimum on $C$, and the LDP upper bound makes the canonical macrostate probability of that complement exponentially small.

Let $q\in {\mathcal{M}}_{\beta }$ and put $u=h(q)$. If $r\in H_u$, then $D(r)+\beta u\ge D(q)+\beta u$, so $q\in {\mathcal{M}}_u$ and $D(q)=d(u)$. For any $v$ in the energy range, minimizing over $H_v$ gives $d(v)+\beta v\ge d(u)+\beta u$, equivalently $s(v)-\beta v\le s(u)-\beta u$. Thus $u$ maximizes the stated entropy-minus-energy expression.

The fiber $H_u:=\{q\in \mathrm{\Delta }(\mathcal{A}):h(q)=u\}$ is nonempty and closed, hence compact by step 1.1 and [F7], because $u$ is in the range of continuous $h$. By [F4] and [F8], $D$ attains its finite minimum $d(u)$ on $H_u$, so ${\mathcal{M}}_u$ is nonempty. For fixed $\delta >0$, put $B_{\delta }:=\{q:\mathrm{\mid }h(q)-u\mathrm{\mid }<\delta \}$; it is open and contains $H_u$. By [F2], $-{\inf }_{B_{\delta }}D\le {\mathrm{lim\hspace{0.17em}inf}}_n\frac{1}{n}\ln p^{\otimes n}(L_n\in B_{\delta })\le {\mathrm{lim\hspace{0.17em}sup}}_n\frac{1}{n}\ln p^{\otimes n}(L_n\in B_{\delta })\le -{\inf }_{{\bar{B}}_{\delta }}D\mathrm{.}$ For each matching radius ${\delta }_j$ in the Statement the two infima agree, so the inner limit exists and equals their negative common value. The lower bound is finite because $B_{{\delta }_j}$ contains a finite-rate point of $H_u$; hence the conditioning event has positive probability for all sufficiently large $n$.

Write $d_{\delta }:={\inf }_{{\bar{B}}_{\delta }}D$ for any $\delta >0$; since $H_u\subseteq {\bar{B}}_{\delta }$, $d_{\delta }\le d(u)$. Fix $\epsilon >0$ and let $K_{\epsilon }:=\{q\in \mathrm{\Delta }(\mathcal{A}):D(q\mathrm{\parallel }p)\le d(u)-\epsilon \}$. By [F4], this set is closed and hence compact by [F7]; it is disjoint from $H_u$ by the definition of $d(u)$. Thus its compact image $h(K_{\epsilon })$ does not contain $u$ by [F8]. If it is nonempty, the continuous function $v\mapsto \mathrm{\mid }v-u\mathrm{\mid }$ on that compact image attains a positive minimum by [F8]; continuity follows from $\mid \mathrm{\mid }v-u\mathrm{\mid }-\mathrm{\mid }w-u\mathrm{\mid }\mid \le \mathrm{\mid }v-w\mathrm{\mid }$. Therefore the image is separated from $u$; if it is empty the conclusion is immediate. For every sufficiently small $\delta$, the closed shell ${\bar{B}}_{\delta }$ misses $K_{\epsilon }$, so $d_{\delta }\ge d(u)-\epsilon$. Together with $d_{\delta }\le d(u)$, this proves $d_{\delta }\to d(u)$ as $\delta \downarrow 0$, in particular along the matching sequence ${\delta }_j$. Step 2.1 now gives the iterated shell limit $-d(u)=s(u)$.

Let $U$ be an open neighborhood of ${\mathcal{M}}_u$ and put $C:=\mathrm{\Delta }(\mathcal{A})\setminus U$, which is closed and hence compact by step 1.1 and [F7]. If $C\cap H_u=\mathrm{\varnothing }$, then $h(C)$ is compact by [F8] and does not contain $u$; if $C$ is nonempty, the continuous function $v\mapsto \mathrm{\mid }v-u\mathrm{\mid }$ attains a positive minimum there by [F8], using $\mid \mathrm{\mid }v-u\mathrm{\mid }-\mathrm{\mid }w-u\mathrm{\mid }\mid \le \mathrm{\mid }v-w\mathrm{\mid }$. Thus sufficiently small shells miss $C$; this is also immediate when $C=\mathrm{\varnothing }$. Otherwise, $C\cap H_u$ is a nonempty closed subset of the compact simplex and is compact by [F7]. Continuity of $D$ and [F8] give $d_C:={\min }_{C\cap H_u}D>d(u)$, since $C$ contains no constrained minimizer. Choose $0<\eta <d_C-d(u)$ and set $K:=\{q\in C:D(q\mathrm{\parallel }p)\le d(u)+\eta \}$; it is closed and hence compact by [F4] and [F7], and it is disjoint from $H_u$. Therefore $h(K)$ is compact by [F8] and does not contain $u$. If empty it is already separated from $u$; otherwise, the continuous distance function $v\mapsto \mathrm{\mid }v-u\mathrm{\mid }$ attains a positive minimum there by [F8], so $h(K)$ is separated from $u$. Hence for all sufficiently small $\delta$, ${\inf }_{C\cap {\bar{B}}_{\delta }}D\ge d(u)+\eta$. For every sufficiently small matching radius ${\delta }_j$, we also have $d_{{\delta }_j}<d(u)+\eta \mathrm{/}2$. The LDP upper bound [F2] on the closed set $\overline{C\cap B_{{\delta }_j}}\subseteq C\cap {\bar{B}}_{{\delta }_j}$ bounds the numerator's logarithmic limsup exponent by $-d(u)-\eta$. By step 2.1 the denominator's logarithmic limit exponent is $-d_{{\delta }_j}$, so subtracting gives a conditional logarithmic limsup at most $d_{{\delta }_j}-d(u)-\eta <-\eta \mathrm{/}2$. Thus for every sufficiently large $j$ the conditional probability tends to zero exponentially as $n\to \mathrm{\infty }$, which gives the asserted iterated concentration limit.

Conversely, let $u$ maximize $s(v)-\beta v$ and let $q\in {\mathcal{M}}_u$. For any $r\in \mathrm{\Delta }(\mathcal{A})$, put $v=h(r)$. Then $D(r)+\beta v\ge d(v)+\beta v\ge d(u)+\beta u=D(q)+\beta u$, so $q\in {\mathcal{M}}_{\beta }$. This proves both the equality of energy sets and the macrostate criterion. Rearranging $s(v)-\beta v\le s(u)-\beta u$ gives exactly $s(v)\le s(u)+\beta (v-u)$; hence the criterion is the stated upper-supporting-line condition, with no formal inverse-transform assumption.