Diagonal gap for a decreasing integrable interaction tail

Since $\psi$ is nonincreasing, $0\le g^d\psi (g)\le \frac{d}{1-2^{-d}}{\int }_{g\mathrm{/}2}^g{r}^{d-1}\psi (r)dr$ for $g>2R_0$: on $[g\mathrm{/}2,g]$, $\psi (r)\ge \psi (g)$, and integration of $r^{d-1}$ gives the displayed coefficient. Both the interval integral and the tail integral in $\eta (g)$ tend to zero as $g\to \mathrm{\infty }$, so $\eta (g)\to 0$.

For $m\ge 2$ and $r\in [(m-1)g,mg]$, monotonicity gives $\psi (r)\ge \psi (mg)$ and $r^{d-1}\ge ((m-1)g{)}^{d-1}$. Hence ${\int }_{(m-1)g}^{mg}{r}^{d-1}\psi (r)dr\ge g^d(m-1{)}^{d-1}\psi (mg)\mathrm{.}$ Since $m^{d-1}\le 2^{d-1}(m-1{)}^{d-1}$, summing over disjoint intervals yields ${\sum }_{m\ge 2}{m}^{d-1}\psi (mg)\le 2^{d-1}{g}^{-d}{\int }_g^{\mathrm{\infty }}{r}^{d-1}\psi (r)dr$. The $m=1$ term equals $g^{-d}{g}^d\psi (g)$, so adding it proves the stated bound because $2^{d-1}\ge 1$.

For integer $m\ge \lceil R_0+3\rceil$, monotonicity gives $\psi (r)\ge \psi (m-1)$ on $[m-2,m-1]$, so ${\int }_{m-2}^{m-1}{r}^{d-1}\psi (r)dr\ge (m-2{)}^{d-1}\psi (m-1)$. Since $m^{d-1}\le c_d(m-2{)}^{d-1}$ for these $m$, summing over disjoint unit intervals proves ${\sum }_{m\ge \lceil R_0+3\rceil }{m}^{d-1}\psi (m-1)<\mathrm{\infty }$.

For each integer $k\ge 1$, let $G_k$ be the least integer greater than $2R_0$ such that $\eta (g)\le k^{-2d}$ for every $g\ge G_k$; it exists by step 1.1 and the well-ordering of the positive integers. Choose a strictly increasing sequence $L_k$ with $L_k\ge k{G}_k$ and $L_k\ge k^2$. For $\mathrm{\ell }\ge L_1$, let $k(\mathrm{\ell })=\max \{k:L_k\le \mathrm{\ell }\}$ and $g(\mathrm{\ell })=\mathrm{\ell }\mathrm{/}k(\mathrm{\ell })$. Then $k(\mathrm{\ell })\to \mathrm{\infty }$, $g(\mathrm{\ell })\ge G_{k(\mathrm{\ell })}$ and $g(\mathrm{\ell })\ge k(\mathrm{\ell })$, while $g(\mathrm{\ell })\mathrm{/}\mathrm{\ell }=1\mathrm{/}k(\mathrm{\ell })$. Therefore ${\left(\frac{\mathrm{\ell }}{g(\mathrm{\ell })}\right)}^d\eta (g(\mathrm{\ell }))\le k(\mathrm{\ell }{)}^{d}k(\mathrm{\ell }{)}^{-2d}=k(\mathrm{\ell }{)}^{-d}\to 0.$ This gives all three gap limits.

Fix distinct points $x,y$ in a finite configuration and a period $L>0$. For $\mathrm{\mid }k{\mathrm{\mid }}_{\mathrm{\infty }}$ larger than $2\mathrm{\mid }x-y{\mathrm{\mid }}_{\mathrm{\infty }}\mathrm{/}L+2R_0\mathrm{/}L$, the triangle inequality gives $\mathrm{\mid }x-y+Lk{\mathrm{\mid }}_{\mathrm{\infty }}\ge L\mathrm{\mid }k{\mathrm{\mid }}_{\mathrm{\infty }}\mathrm{/}2>2R_0$. Grouping lattice indices by $m=\mathrm{\mid }k{\mathrm{\mid }}_{\mathrm{\infty }}$ and applying the sum bound of step 1.2 with gap $g=L\mathrm{/}2$ shows ${\sum }_k\mathrm{\mid }\varphi (x-y+Lk)\mathrm{\mid }<\mathrm{\infty }$ whenever $L\mathrm{/}2>2R_0$. There are finitely many remaining images; any $+\mathrm{\infty }$ term makes the periodic energy $+\mathrm{\infty }$ by definition, while otherwise their finite sum is well-defined. This proves the periodic-image convention used in Superstable regular pair interactions and configurational ensemble notation for all sufficiently large $L$.