Finite-range continuum Gibbs kernels and DLR notion

Assume the Axiom of Choice The Axiom of Choice and fix an integer dimension $d\ge 1$. A configuration is a locally finite simple point set $\omega \subset {\mathbb{R}}^d$ with the count sigma-algebra and unit-intensity Poisson reference from Poisson Configuration Reference; thus every bounded Borel set contains finitely many points, and the sigma-algebra is generated by its bounded-set counts. Write $N_A(\omega )=\mathrm{\mid }\omega \cap A\mathrm{\mid }$.

Use the interface of Stability, superstability and regular tails: $\varphi :{\mathbb{R}}^d\to \mathbb{R}\cup \{+\mathrm{\infty }\}$ is even and Borel, and some finite $B\ge 0$ satisfies $U_N(X)\ge -BN$ for every finite position list $X=(x_1,\dots ,x_N)$. For a finite simple configuration $\omega$, enumerate its points as such a list and write $U_{\varphi }(\omega )={\sum }_{\{x,y\}\subset \omega }\varphi (x-y)$; its unordered-pair arguments are nonzero, and stability gives $U_{\varphi }(\omega )\ge -B\mathrm{\mid }\omega \mathrm{\mid }$. For this item assume finite range $R>0$, meaning $\varphi (x)=0$ whenever $\mathrm{\parallel }x{\mathrm{\parallel }}_2>R$.

Let $\mathrm{\Lambda }\subset {\mathbb{R}}^d$ be a nonempty bounded Borel region and let $\eta \subset {\mathrm{\Lambda }}^c$ be a locally finite exterior configuration. The relevant exterior set is ${\eta }_{\mathrm{\Lambda }}^R=\{y\in \eta :{\mathrm{dist}}_2(y,\mathrm{\Lambda })\le R\};$ it is finite because it lies in a bounded set. For a finite interior configuration $\omega \subset \mathrm{\Lambda }$, define the conditional energy $H_{\mathrm{\Lambda }}(\omega \mid \eta )={\sum }_{\{x,y\}\subset \omega }\varphi (x-y)+{\sum }_{x\in \omega }{\sum }_{y\in {\eta }_{\mathrm{\Lambda }}^R}\varphi (x-y)\mathrm{.}$ For positive mathematical parameters $z,\beta >0$, define the unit-Poisson-reference partition integral and candidate kernel by $Z_{\mathrm{\Lambda }}^{\eta }(z,\beta )=\int z^{N_{\mathrm{\Lambda }}(\omega )}{e}^{-\beta H_{\mathrm{\Lambda }}(\omega \mid \eta )}{Q}_{\mathrm{\Lambda }}(d\omega ),{\gamma }_{\mathrm{\Lambda }}^{\eta }(d\omega )=\frac{z^{N_{\mathrm{\Lambda }}(\omega )}{e}^{-\beta H_{\mathrm{\Lambda }}(\omega \mid \eta )}}{Z_{\mathrm{\Lambda }}^{\eta }(z,\beta )}{Q}_{\mathrm{\Lambda }}(d\omega ),$ where $Q_{\mathrm{\Lambda }}$ is the restriction of the unit-intensity Poisson law and $e^{-\beta (+\mathrm{\infty })}=0$. The accompanying lemma proves that $Z_{\mathrm{\Lambda }}^{\eta }$ is finite and strictly positive and that $\eta \mapsto {\gamma }_{\mathrm{\Lambda }}^{\eta }$ is a probability kernel.

Extend ${\gamma }_{\mathrm{\Lambda }}^{\eta }$ to a kernel on full configurations by replacing the interior with $\omega$ and leaving $\eta$ unchanged outside $\mathrm{\Lambda }$. A continuum DLR law for this specification is a probability measure $\mu$ on configuration space satisfying $\mu {\gamma }_{\mathrm{\Lambda }}=\mu$ for every nonempty bounded Borel $\mathrm{\Lambda }$, where this identity means equality of integrals for every bounded measurable full-configuration test. The kernel consistency and its conditional-law interpretation are proved in Stable finite-range continuum specification normalization and consistency ↗. This definition asserts no existence, uniqueness, or thermodynamic-limit result.