Stable power-tempered pair-potential model

Fix a positive integer $d$ and use the Euclidean space ${\mathbb{R}}^d$ with norm $\mathrm{\parallel }x{\mathrm{\parallel }}_2=\sqrt{⟨x,x⟩}$ of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. A pair potential is an even Borel-measurable function $\varphi :{\mathbb{R}}^d\to \bar{\mathbb{R}}$ whose image is contained in $\mathbb{R}\cup \{+\mathrm{\infty }\}$. Here $\bar{\mathbb{R}}$ is the extended real line (The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined). Borel measurability means the inverse image of each Borel subset of $\bar{\mathbb{R}}$ belongs to the Borel sigma-algebra generated by open sets in the Euclidean topology on ${\mathbb{R}}^d$ (The Borel sigma-algebra on the extended real line, The Borel sigma-algebra of a topological space). Even means $\varphi (-x)=\varphi (x)$. The value $+\mathrm{\infty }$ represents a forbidden pair separation. Coordinates and potential values are numerical values in fixed reference length and energy units.

For $N\ge 1$ and an ordered configuration $(q_1,\dots ,q_N)\in ({\mathbb{R}}^d{)}^N$, define its pair energy by $U_N(q_1,\dots ,q_N):={\sum }_{1\le i<j\le N}\varphi (q_i-q_j),$ with an empty sum equal to $0$; also put $U_0=0$. The finite sum is in $\mathbb{R}\cup \{+\mathrm{\infty }\}$, since each pair term is real or $+\mathrm{\infty }$.

The pair potential is stable if there is a finite constant $B\ge 0$ such that $U_N(q_1,\dots ,q_N)\ge -BN$ for every $N\ge 1$ and every configuration. It is power-tempered if there are constants $C,R,ϵ>0$ such that $\varphi (x)\in \mathbb{R}$ whenever $\mathrm{\parallel }x{\mathrm{\parallel }}_2\ge R$, and, on that set, the ordinary real absolute value $\mathrm{\mid }\varphi (x)\mathrm{\mid }$ satisfies $\mathrm{\mid }\varphi (x)\mathrm{\mid }\le C\mathrm{\parallel }x{\mathrm{\parallel }}_2^{-d-ϵ}\mathrm{.}$ The combined model class consists of pair potentials that are both stable and power-tempered. No condition is imposed on the positive core, which may be unbounded or infinite, and the tail may have either sign.