Free-cube grand-pressure limit for decreasing integrable tails

Put $q=a{e}^{\beta B}$. For every $N$-point configuration, $0\le e^{-\beta H_{\varphi }}\le e^{\beta BN}$ by [A1]. Applying [F3] to the finite-volume integral gives $a^{N}Z(L,N;\beta )\le (q{L}^d{)}^N\mathrm{/}N!$. Summing by [F4] yields ${\mathrm{\Xi }}_L\le e^{q{L}^d}$; the empty configuration contributes exactly $1$, so ${\mathrm{\Xi }}_L\ge 1$.

Choose $K>eq+1$ and an integer cutoff $n_{\mathrm{\ell }}$ with $K{\mathrm{\ell }}^d<n_{\mathrm{\ell }}\le K{\mathrm{\ell }}^d+1$. For $N>n_{\mathrm{\ell }}$, $1\le e^{N-K{\mathrm{\ell }}^d}$, so ${\sum }_{N>n_{\mathrm{\ell }}}{a}^{N}Z(\mathrm{\ell },N;\beta )\le e^{-K{\mathrm{\ell }}^d}{\sum }_{N\ge 0}\frac{(eq{\mathrm{\ell }}^d{)}^N}{N!}=e^{(eq-K){\mathrm{\ell }}^d}\le e^{-{\mathrm{\ell }}^d}\mathrm{.}$ Because ${\mathrm{\Xi }}_{\mathrm{\ell }}\ge 1$, the truncated sum ${\mathrm{\Xi }}_{\mathrm{\ell }}^{\mathrm{c}\mathrm{u}\mathrm{t}}={\sum }_{N=0}^{n_{\mathrm{\ell }}}{a}^{N}Z(\mathrm{\ell },N;\beta )$ satisfies ${\mathrm{\Xi }}_{\mathrm{\ell }}^{\mathrm{c}\mathrm{u}\mathrm{t}}\ge (1-e^{-{\mathrm{\ell }}^d}){\mathrm{\Xi }}_{\mathrm{\ell }}$.

Fix a core side $\mathrm{\ell }$ and a gap $g>2R_0$. Place $M=\lfloor L\mathrm{/}(\mathrm{\ell }+g){\rfloor }^d$ disjoint core cubes of side $\mathrm{\ell }$ in $[0,L{)}^d$ on the lattice of spacing $\mathrm{\ell }+g$. Restrict every core to at most $n_{\mathrm{\ell }}$ particles. If two core indices differ by maximum norm $m\ge 1$, every pair between them has separation at least $mg$ in maximum norm. The number of offsets in shell $m$ is $(2m+1{)}^d-(2m-1{)}^d\le c_d{m}^{d-1}$. By [F1], the sum of positive cross-core pair energies is therefore at most $D_{L,\mathrm{\ell },g}\le c_{d}M{n}_{\mathrm{\ell }}^2{\sum }_{m\ge 1}{m}^{d-1}\psi (mg)\mathrm{.}$ Negative cross-core energies only increase the Boltzmann weight. Splitting the grand sum by the particle counts in these disjoint cores, Tonelli [F5] and the multinomial coefficient show that the global $N!$ cancels the assignments of labeled coordinates to cores. Thus ${\mathrm{\Xi }}_L\ge e^{-\beta D_{L,\mathrm{\ell },g}}({\mathrm{\Xi }}_{\mathrm{\ell }}^{\mathrm{c}\mathrm{u}\mathrm{t}}{)}^M\mathrm{.}$

Divide the last inequality by $L^d$ and let $L\to \mathrm{\infty }$ with $\mathrm{\ell },g$ fixed. Since $M\mathrm{/}{L}^d\to (\mathrm{\ell }+g{)}^{-d}$, [F2] gives ${\mathrm{lim\hspace{0.17em}inf}}_{L\to \mathrm{\infty }}{L}^{-d}\log {\mathrm{\Xi }}_L\ge {\left(\frac{\mathrm{\ell }}{\mathrm{\ell }+g}\right)}^d[{\mathrm{\ell }}^{-d}\log {\mathrm{\Xi }}_{\mathrm{\ell }}+{\mathrm{\ell }}^{-d}\log (1-e^{-{\mathrm{\ell }}^d})-C_d{K}_1^2{\left(\frac{\mathrm{\ell }}{g}\right)}^d\eta (g)],$ where $K_1=K+1$ bounds $n_{\mathrm{\ell }}\mathrm{/}{\mathrm{\ell }}^d$ for $\mathrm{\ell }\ge 1$ and $C_d$ absorbs the fixed shell-count constant and $\beta$.

The numbers $p_{\mathrm{\ell }}={\mathrm{\ell }}^{-d}\log {\mathrm{\Xi }}_{\mathrm{\ell }}$ lie in $[0,q]$ by step 1.1, so choose ${\mathrm{\ell }}_j\to \mathrm{\infty }$ with $p_{{\mathrm{\ell }}_j}\to {\mathrm{lim\hspace{0.17em}sup}}_{\mathrm{\ell }\to \mathrm{\infty }}{p}_{\mathrm{\ell }}$. Set $g=g({\mathrm{\ell }}_j)$ from [F2]. Then $g\mathrm{/}{\mathrm{\ell }}_j\to 0$, the volume fraction tends to $1$, the cutoff logarithm divided by ${\mathrm{\ell }}_j^d$ tends to zero, and the cross-core error tends to zero. Step 4.1 therefore yields ${\mathrm{lim\hspace{0.17em}inf}}_{L\to \mathrm{\infty }}{p}_L\ge {\mathrm{lim\hspace{0.17em}sup}}_{L\to \mathrm{\infty }}{p}_L$. The reverse inequality is automatic for every real sequence, so the limit exists; step 1.1 makes it finite.