Stability, superstability and regular tails

Fix $d\ge 1$. A pair potential is an even Borel function $\varphi :{\mathbb{R}}^d\to \mathbb{R}\cup \{+\mathrm{\infty }\}$. For a finite list of positions $X=(x_1,\dots ,x_N)$, set $U_0=U_1=0$ and $U_N(X)={\sum }_{1\le i<j\le N}\varphi (x_i-x_j)\mathrm{.}$ The value $+\mathrm{\infty }$ is allowed and represents a forbidden configuration; $-\mathrm{\infty }$ is not allowed.

The potential is stable if some finite $B\ge 0$ satisfies $U_N(X)\ge -BN$ for every $N$ and every $X$. Fix a cell side $\mathrm{\ell }>0$ and the half-open cubic tiling $C_k=\mathrm{\ell }(k+[0,1{)}^d)$, $k\in {\mathbb{Z}}^d$, and let $n_k(X)$ count the positions of $X$ in $C_k$. The potential is cell-superstable if there are constants $A_{\ast }>0$ and $B_{\ast }\ge 0$ such that $U_N(X)\ge A_{\ast }{\sum }_{k\in {\mathbb{Z}}^d}{n}_k(X{)}^2-B_{\ast }N$ for every finite configuration; the sum is finite for finite $X$.

For disjoint finite cell-index sets $I,J$, let $W_{I,J}(X)$ be the sum of pair terms with one point in a cell indexed by $I$ and the other in a cell indexed by $J$. The potential is lower regular for this tiling if a nonnegative envelope $\mathrm{\Psi }:{\mathbb{Z}}^d\setminus \{0\}\to [0,\mathrm{\infty })$ exists with ${\sum }_{k\mathrm{\ne }0}\mathrm{\Psi }(k)<\mathrm{\infty }$ and $W_{I,J}(X)\ge -{\sum }_{k\in I}{\sum }_{l\in J}\mathrm{\Psi }(k-l)n_k(X)n_l(X)$ for all such $I,J,X$. The ordered-pair convention in this formula is fixed as written.

A decreasing-integrable tail bound means that for some $R_0\ge 0$ there is a nonnegative nonincreasing function $\psi :[R_0,\mathrm{\infty })\to [0,\mathrm{\infty })$ such that $\mathrm{\mid }\varphi (x)\mathrm{\mid }\le \psi (\mathrm{\mid }x\mathrm{\mid })$ for $\mathrm{\mid }x\mathrm{\mid }\ge R_0$ and ${\int }_{R_0}^{\mathrm{\infty }}\psi (r)r^{d-1}dr<\mathrm{\infty }$. This explicit envelope condition is stronger than bare Lebesgue integrability of an arbitrary signed tail. Stability, cell-superstability, lower regularity, and a tail bound are separate properties; none is inferred from finite range alone.