Retained hard-core continuation at zero inverse temperature

For the pair potential $\phi$ of Stable regularly tempered classical positional preparation, let $B=\{x\in {\mathbb{R}}^d:\phi (x)=+\mathrm{\infty }\}$. A finite configuration is hard-admissible if every pair displacement avoids $B$. For $z>0$, define the retained-admissibility zero-inverse-temperature free-wall weight of $\omega$ to be $w_0(\omega )=z^{N(\omega )}{\mathbf{1}}_{\{H_{N(\omega )}(\omega )<\mathrm{\infty }\}}\mathrm{.}$ Its finite-volume partition sum, when finite, is ${\mathrm{\Xi }}_{\mathrm{\Lambda }}^{\mathrm{r}\mathrm{e}\mathrm{t},0}(z)={\sum }_{N\ge 0}\int w_0(\omega ){\mu }_{\mathrm{\Lambda },N}(d\omega )$, using Indistinguishable-particle positional measure. This definition sets the weight of a forbidden configuration to zero; it does not assign a value to the undefined product $0\cdot (+\mathrm{\infty })$. When $B=\mathrm{\varnothing }$ the measure is the unrestricted Poisson activity model; for a nonempty hard core it is generally not Poisson. The convention is explicit model data and carries no statement about an arbitrary zero-width energy shell.

For a periodic cell of side $L$ in dimensionless coordinates, use the exact image energy $H_{L,\mathrm{p}\mathrm{e}\mathrm{r}}$ of Stable regularly tempered classical positional preparation. Its periodic retained-admissibility weight is $w_{L,0}^{\mathrm{p}\mathrm{e}\mathrm{r}}(\omega )=z^{N(\omega )}{\mathbf{1}}_{\{H_{L,\mathrm{p}\mathrm{e}\mathrm{r}}(\omega )<+\mathrm{\infty }\}}=z^{N(\omega )}{\mathbf{1}}_{\{\phi (y-x)<+\mathrm{\infty }\text{ for every }x\in \omega ,y\in {\omega }^{\mathrm{p}\mathrm{e}\mathrm{r}},y\mathrm{\ne }x\}}\mathrm{.}$ Thus every forbidden pair between distinct particles and every forbidden periodic-image pair, including a particle with one of its own nontrivial images, receives zero weight. When finite, the corresponding partition sum is ${\mathrm{\Xi }}_L^{\mathrm{r}\mathrm{e}\mathrm{t},0,\mathrm{p}\mathrm{e}\mathrm{r}}(z)={\sum }_{N\ge 0}\int w_{L,0}^{\mathrm{p}\mathrm{e}\mathrm{r}}d{\mu }_{{\mathrm{\Lambda }}_L,N}$. This is the periodic retained-admissibility convention; it is not an unrestricted zero-beta limit of a shell law.