Finite-range continuum Gibbs kernels and DLR notion

Let $D$ be any bounded Borel set. It meets finitely many half-open unit cells $C_i$. Put $A_i=D\cap C_i$ and $B_i=C_i\setminus A_i$. Conditional on $r+s$ points in a cell, the iid uniform-position rule in [F3] gives the labelled sector density $e^{-1}\mathrm{/}(r+s)!$. Splitting the labels between $A_i$ and $B_i$ yields $(r+s)!\mathrm{/}(r!s!)$ assignments by [F5]; integrating the $s$ outside coordinates and summing over $s$ gives $\frac{e^{-1}}{r!}{\sum }_{s=0}^{\mathrm{\infty }}\frac{\mathrm{\mid }{B}_i{\mathrm{\mid }}^s}{s!}d{x}_1\cdots d{x}_r=\frac{e^{-\mathrm{\mid }{A}_i\mathrm{\mid }}}{r!}d{x}_1\cdots d{x}_r,$ by [F6], since $\mathrm{\mid }{A}_i\mathrm{\mid }+\mathrm{\mid }{B}_i\mathrm{\mid }=1$. Tonelli and monotone convergence [F7, F8] justify the nonnegative sector sum and integral. The cells are independent by [F3]. For total count $n$ across their finitely many pieces, the label assignments for counts $(r_i)$ number $n!\mathrm{/}{\prod }_i{r}_i!$ by [F5]; partitioning $D^n$ by these assignments shows that the $n$-point sector on $D$ has density $e^{-\mathrm{\mid }D\mathrm{\mid }}\mathrm{/}n!$ against product Lebesgue measure. The collision diagonals have product measure zero because Lebesgue measure has no atoms; this is also the simplicity convention in [F3]. Thus the density pushes to the sector on unordered simple configurations. Zero-volume pieces contribute no points.

Put ${\mathrm{\Lambda }}^R=\{y:{\mathrm{dist}}_2(y,\mathrm{\Lambda })\le R\}$. The distance map is $1$-Lipschitz, so ${\mathrm{\Lambda }}^R$ is a bounded Borel set. For bounded Borel $S,T\subseteq {\mathbb{R}}^d$, the product sigma-algebra generated by Borel rectangles in $S\times T$ is its relative Euclidean Borel sigma-algebra: relative open rectangles from countable Euclidean bases form a countable base for $S\times T$, so every open set is a countable union of such rectangles; conversely Borel rectangles are Borel because coordinate projections are continuous. The map $(x,y)\mapsto x-y$ is continuous, and for every Borel $E\subseteq \mathbb{R}\cup \{+\mathrm{\infty }\}$, the preimage of $E$ under $(x,y)\mapsto \varphi (x-y)$ is the continuous preimage of the Borel set ${\varphi }^{-1}(E)$. Hence $f(x,y)=\varphi (x-y)$ is Borel on these products and on their off-diagonal Borel subspaces.

If $\omega$ has $n$ points, [F2] gives its internal energy at least $-Bn$ and pair potential at least $-2B$ at every distinct separation. There are at most $mn$ interior-to-exterior pairs in $H_{\mathrm{\Lambda }}$, so $H_{\mathrm{\Lambda }}(\omega \mid \eta )\ge -B(1+2m)n$. The empty configuration has $H_{\mathrm{\Lambda }}(\mathrm{\varnothing }\mid \eta )=0$ and $Q_{\mathrm{\Lambda }}(\{\mathrm{\varnothing }\})=e^{-\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }}$, proving the lower bound. The sector density from step 1.1 gives $Z_{\mathrm{\Lambda }}^{\eta }\le e^{-\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }}{\sum }_{n=0}^{\mathrm{\infty }}\frac{{\left(z\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }{e}^{\beta B(1+2m)}\right)}^n}{n!}=e^{-\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }}\exp \text{\hspace{0.17em}⁣}\left(z\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }{e}^{\beta B(1+2m)}\right)<\mathrm{\infty }\mathrm{.}$

Define ${\nu }_{\omega }^{\mathrm{i}\mathrm{n}\mathrm{t}}(E)={\sum }_{x,y\in \omega \cap \mathrm{\Lambda },x\mathrm{\ne }y}{1}_E(x,y)$ for Borel $E\subseteq (\mathrm{\Lambda }\times \mathrm{\Lambda })\setminus \{(x,x):x\in \mathrm{\Lambda }\}$, and define ${\nu }_{\omega ,\eta }^{\mathrm{b}\mathrm{d}}(E)={\sum }_{x\in \omega \cap \mathrm{\Lambda },y\in \eta \cap {\mathrm{\Lambda }}^R}{1}_E(x,y)$ for Borel $E\subseteq (\mathrm{\Lambda }\times {\mathrm{\Lambda }}^R)\setminus \{(x,x):x\in \mathrm{\Lambda }\cap {\mathrm{\Lambda }}^R\}$. On a restricted rectangle $(A_1\times A_2)$, the first measure evaluates to $N_{A_1}(\omega )N_{A_2}(\omega )-N_{A_1\cap A_2}(\omega )$ for Borel $A_1,A_2\subseteq \mathrm{\Lambda }$, measurable by [F3]; the second evaluates to $N_A(\omega )N_C(\eta )$ for Borel $A\subseteq \mathrm{\Lambda }$, $C\subseteq {\mathrm{\Lambda }}^R$, jointly measurable in $(\omega ,\eta )$. These are finite measures: their total masses are respectively $n(n-1)$ and $n{N}_{{\mathrm{\Lambda }}^R}(\eta )$, where $n=N_{\mathrm{\Lambda }}(\omega )$. For either family, the sets whose pair-count evaluations are measurable form a lambda-system: the whole space has measurable total mass, complements subtract from that finite mass, and disjoint unions give pointwise limits of measurable partial sums. Restricted Borel rectangles form a pi-system generating each off-diagonal product Borel sigma-algebra, so [F9] gives measurability of the pair-count evaluations on every Borel subset. By stability on two-point lists [F2], the shifted Borel function $f+2B$ is nonnegative on both off-diagonal pair spaces. Apply [F10] there to obtain increasing nonnegative simple Borel functions converging to $f+2B$. Each simple-function integral is a finite linear combination of the measurable pair-count evaluations, so its pointwise limit, the integral of $f+2B$, is measurable in $\omega$ or in $(\omega ,\eta )$ as appropriate. Subtracting $2B$ times the measurable total pair count gives the signed pair integral without losing measurability; the subtraction is of a finite real quantity, and a value $+\mathrm{\infty }$ remains $+\mathrm{\infty }$. Evenness makes the internal unordered-pair energy half its ordered-pair integral. Thus $H_{\mathrm{\Lambda }}(\omega \mid \eta )$ and $g(\omega ,\eta )=z^{N_{\mathrm{\Lambda }}(\omega )}{e}^{-\beta H_{\mathrm{\Lambda }}(\omega \mid \eta )}$ are jointly measurable.

The exterior subspace $\{\eta :N_{\mathrm{\Lambda }}(\eta )=0\}$ is measurable by [F3], and its restriction of $Q$ is a finite measure; together with the probability $Q_{\mathrm{\Lambda }}$, this permits Tonelli [F8] for each map $\eta \mapsto \int 1_C(\omega )g(\omega ,\eta )Q_{\mathrm{\Lambda }}(d\omega )$, where $C$ is a measurable interior event. Taking $C$ to be the whole space shows $Z_{\mathrm{\Lambda }}^{\eta }$ is measurable. Step 2.1 gives $0<Z_{\mathrm{\Lambda }}^{\eta }<\mathrm{\infty }$, so division defines a probability and makes $\eta \mapsto {\gamma }_{\mathrm{\Lambda }}^{\eta }(C)$ measurable for every $C$. This is the probability-kernel condition.

Under $Q_{\mathrm{\Lambda }}$, the restrictions to disjoint $\mathrm{\Delta }$ and $\mathrm{\Lambda }\setminus \mathrm{\Delta }$ are independent by [F4]. Write an interior configuration as ${\omega }_{\mathrm{\Delta }}\cup \zeta$, with $\zeta \subseteq \mathrm{\Lambda }\setminus \mathrm{\Delta }$. Every finite pair term in $H_{\mathrm{\Lambda }}({\omega }_{\mathrm{\Delta }}\cup \zeta \mid \eta )$ either touches $\mathrm{\Delta }$, in which case it is a term of $H_{\mathrm{\Delta }}({\omega }_{\mathrm{\Delta }}\mid \zeta \cup \eta )$, or does not touch $\mathrm{\Delta }$, in which case it depends only on $\zeta ,\eta$. Likewise $z^{N_{\mathrm{\Lambda }}}=z^{N_{\mathrm{\Delta }}({\omega }_{\mathrm{\Delta }})}{z}^{N_{\mathrm{\Lambda }\setminus \mathrm{\Delta }}(\zeta )}$. Thus, for fixed $\zeta ,\eta$, the conditional density of ${\omega }_{\mathrm{\Delta }}$ under ${\gamma }_{\mathrm{\Lambda }}$ is proportional to $z^{N_{\mathrm{\Delta }}({\omega }_{\mathrm{\Delta }})}{e}^{-\beta H_{\mathrm{\Delta }}({\omega }_{\mathrm{\Delta }}\mid \zeta \cup \eta )}$ relative to $Q_{\mathrm{\Delta }}$; its normalizer is finite and positive by step 2.1. It is exactly ${\gamma }_{\mathrm{\Delta }}^{\zeta \cup \eta }$, proving ${\gamma }_{\mathrm{\Lambda }}{\gamma }_{\mathrm{\Delta }}={\gamma }_{\mathrm{\Lambda }}$ after nonnegative iterated integration over $\zeta$ by [F8].

The extended kernel leaves the exterior fixed. For an interior event $A$ and exterior event $E$, apply $\mu {\gamma }_{\mathrm{\Lambda }}=\mu$ to the full-configuration test $1_A({\eta }_{\mathrm{\Lambda }})1_E({\eta }_{{\mathrm{\Lambda }}^c})$. This gives $\int 1_E({\eta }_{{\mathrm{\Lambda }}^c})1_A({\eta }_{\mathrm{\Lambda }})\mu (d\eta )=\int 1_E({\eta }_{{\mathrm{\Lambda }}^c}){\gamma }_{\mathrm{\Lambda }}^{\eta }(A)\mu (d\eta )$ for every exterior event $E$. This is the defining conditional-expectation identity, so ${\gamma }_{\mathrm{\Lambda }}^{\eta }$ is a version of the conditional interior law given the exterior. Conversely, if those conditional identities hold, the probability measures $\mu {\gamma }_{\mathrm{\Lambda }}$ and $\mu$ agree on every rectangle generated by an interior and an exterior event. Such rectangles form a pi-system and generate the full configuration sigma-algebra because each bounded-set count splits into its parts in $\mathrm{\Lambda }$ and ${\mathrm{\Lambda }}^c$ by [F3]. The sets on which the two probability measures agree form a lambda-system; [F9] therefore gives $\mu {\gamma }_{\mathrm{\Lambda }}=\mu$. This proves the stated equivalence.

Steps 2.1, 3.1, 4.1, and 4.2 establish, respectively, finite positive normalization, Borel-kernel measurability, specification consistency, and the DLR conditional-law interpretation. No infinite-volume existence, uniqueness, or minimizer-identification claim was used.

Under clause 1, the assumptions of [F1] and [F7] are exactly the stable power-tempered model, ${\mathrm{A}\mathrm{C}}_{\omega }$, and positive $z,\beta$. Applying [F7] gives the finite limit $p(z,\beta )$ asserted there. This is a reduced positional partition deduction under the adopted model [A1], not an empirical pressure measurement.

Under clause 2, [F2] supplies the stable regular decreasing-integrable-tail convention and full $\mathrm{A}\mathrm{C}$ [A5]. The density and particle-number hypotheses are exactly those of [F8], so the canonical rate exists and equals $F_{\beta }(\rho )$. For each $\alpha$, [F9] applies to that same rate and free-cube partition, giving both convex-duality identities on its stated closed effective domain. No endpoint canonical limit is inferred from the duality identity.

Under clause 3, [A5]–[A8] supply the four separately stated choice principles, and [F2] fixes the same superstable regular positional model and Poisson normalization. Thus [F10] gives the free/periodic $p_Q$ limit, variational formula, and compact nonempty minimizer set. The definitions [F3] and [F5] identify the reference law and topology used in that theorem.

For clause 4, full $\mathrm{A}\mathrm{C}$ is supplied by [A5], and the sequence satisfies [F4]'s explicit van Hove boundary-layer condition, so [F14] gives the free-wall grand-pressure limit. For the exact-$N$ assertion, the same sequence and density hypotheses match [F19], which gives the claimed rate; [F8] identifies that rate as $F_{\beta }(\rho )$. Neither implication supplies an excluded endpoint or removes superstability.

In clause 5, the four choice assumptions [A5]–[A8] are exactly those in [F15] and [F16]. The positive finite supporting activity, interior density, superstable regular potential, and exact-$N$ rate meet their hypotheses. By [F6] the observable is the spatially periodized field, by [F5] its topology tests every local observable with at-most-linear count growth, and by [F17] convergence for arbitrary events uses the stated outer probability. Applying [F15] to periodic boundary and [F16] to free boundary gives precisely the two concentration claims.

Clause 6 adds differentiability and a singleton variational minimizer to clause 5. These are exactly the extra hypotheses in [F18]. Its constructed sequence ${\delta }_L$ and its proof yield the asserted periodic positional shell, while its conclusion explicitly restricts the width to that selected sequence.

Clause 7 has full $\mathrm{A}\mathrm{C}$ by [A5], the stable regular model [F2], and the genuine finite-valued positive core [F12]. The prescribed square-count class and fixed $z,\beta$ are the ones in [F13]. Applying that theorem gives uniform convergence to the free-cube pressure; no volume-filling field is included in its hypotheses.

Clause 8 assumes $d\ge 1$ and the superstable regular potential of [F2], exactly the interaction class required by [F11]. It uses the separate formal $\beta =0$ continuation with fixed mathematical activity $z>0$ from [A3] and assumes the four choice principles [A5]–[A8] exactly as in [F11]. Applying [F11] yields its free/periodic zero-parameter conclusions. Its hypothesis and conclusion retain forbidden configurations throughout, so no unrestricted Poisson claim follows when the forbidden set has positive measure; no inference about ${\beta }_{\mathrm{S}\mathrm{I}}>0$ or an SI pressure is made.

By [F20], the unit-intensity Poisson $N$-point sector on the bounded Borel cube ${\mathrm{\Lambda }}_L$ has density $e^{-L^d}\mathrm{/}N!$ relative to the labelled positional sector. Therefore its normalized partition is $e^{-L^d}{\mathrm{\Xi }}_L$, so the logarithmic rates differ by exactly $1$. With numerical volume $L^d$ and physical volume ${\mathrm{\ell }}_0^d{L}^d$, the physical pressure conversion is $k_{B}Tp\mathrm{/}{\mathrm{\ell }}_0^d=E_{0}p\mathrm{/}(\beta {\mathrm{\ell }}_0^d)$ by [A2]. For $d=3$, the units are J/m${}^3$ = Pa. This conversion changes units only; it does not add momentum variables or an empirical premise.