Finite free-cube grand pressure for stable power-tempered pair potentials

Set $\alpha :=\frac{1}{2}(1+d\mathrm{/}(d+ϵ))$ and $g_{\mathrm{\ell }}:={\mathrm{\ell }}^{\alpha }$. Then $d\mathrm{/}(d+ϵ)<\alpha <1$ and ${\mathrm{\ell }}^d{g}_{\mathrm{\ell }}^{-d-ϵ}={\mathrm{\ell }}^{d-\alpha (d+ϵ)}\to 0$ while $g_{\mathrm{\ell }}\mathrm{/}\mathrm{\ell }\to 0$. Fix a sufficiently large $\mathrm{\ell }\ge 1$ so that $g_{\mathrm{\ell }}\ge R$. For $L\ge \mathrm{\ell }+g_{\mathrm{\ell }}$, set $m_L:=\lfloor L\mathrm{/}(\mathrm{\ell }+g_{\mathrm{\ell }})\rfloor$. Since $L\mathrm{/}(\mathrm{\ell }+g_{\mathrm{\ell }})\ge 1$, [F10] gives $m_L\ge 1$ and $m_L(\mathrm{\ell }+g_{\mathrm{\ell }})\le L<(m_L+1)(\mathrm{\ell }+g_{\mathrm{\ell }})$. Place $M_L=m_L^d$ cubes $C_j=(\mathrm{\ell }+g_{\mathrm{\ell }})j+[0,\mathrm{\ell }{]}^d,j\in \{0,\dots ,m_L-1{\}}^d\mathrm{.}$ They lie inside $[0,L{]}^d$ and have pairwise gaps at least $g_{\mathrm{\ell }}$.

First, if $T$ is a bijection whose map and inverse are measurable and that preserves measure, meaning $\mu (T^{-1}A)=\mu (A)$ for every measurable $A$, then $\mu (TA)=\mu (A)$ as well by applying this identity to $TA$; thus $T^{-1}$ also preserves measure. Such a map preserves nonnegative integrals: for an indicator ${\mathbf{1}}_A$, $\int {\mathbf{1}}_A\circ Td\mu =\mu (T^{-1}A)=\mu (A)=\int {\mathbf{1}}_{A}d\mu$; finite sums give the same equality for simple functions. If $f\ge 0$ is measurable, composition with $T$ sends every simple minorant of $f$ to a simple minorant of $f\circ T$ with the same simple integral. Taking the supremum in [F4] gives $\int fd\mu \le \int f\circ Td\mu$, and applying this inequality to $T^{-1}$ and $f\circ T$ gives equality. The same supremum definition shows monotonicity: $f\le h$ implies $\int fd\mu \le \int hd\mu$. For $N\ge 1$ identical Lebesgue factors, a coordinate permutation preserves the finite product measure: the permutation is a Borel isomorphism, so its push-forward is sigma-finite, and that push-forward agrees with the original measure on rectangles by the rectangle formula in [F6]; finite-product uniqueness, obtained by iterating the uniqueness clause in [F6], makes the measures equal. When $N=0$, there is one empty configuration and one empty assignment, so the equality is immediate. Now fix occupancies $0\le N_j\le n_{\mathrm{\ell }}$ with total $N={\sum }_j{N}_j$. The ordered configurations in the disjoint cubes with exactly these occupancies are partitioned by assignments of the $N$ particle labels to cubes. The number of assignments is $N!\mathrm{/}({\prod }_j{N}_j!)$: order all $N$ labels, then note that each assignment occurs exactly ${\prod }_j{N}_j!$ times from within-cube reorderings. Any two such assignments differ by a permutation of the $N$ labels, and the corresponding coordinate permutation maps one assignment domain to the other. The potential is even, so $U_N$ is invariant under particle-coordinate permutations. The integral invariance just proved therefore makes the assignment-domain integrals equal.

Restrict the grand partition sum in $[0,L{]}^d$ to configurations with no particles outside these cubes and at most $n_{\mathrm{\ell }}$ particles in each cube, where $n_{\mathrm{\ell }}$ is the cutoff from [F2]. If two cubes have index difference $k\mathrm{\ne }0$, the coordinate gap in a coordinate attaining $\mathrm{\parallel }k{\mathrm{\parallel }}_{\mathrm{\infty }}$ is at least $\mathrm{\parallel }k{\mathrm{\parallel }}_{\mathrm{\infty }}{g}_{\mathrm{\ell }}$; [F3] therefore bounds the total absolute inter-cube energy by $D_L\le \frac{C}{2}{M}_L{n}_{\mathrm{\ell }}^2{S}_{d,ϵ}{g}_{\mathrm{\ell }}^{-d-ϵ}\mathrm{.}$ Thus $U_N={\sum }_j{U}_{N_j}^{(j)}+U_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}$ with $U_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}\le D_L$, and $e^{-\beta U_N}\ge e^{-\beta D_L}{\prod }_j{e}^{-\beta U_{N_j}^{(j)}}$; this remains true when an internal energy is $+\mathrm{\infty }$, since both sides then vanish.

On each assignment domain from step 2.1, the pointwise lower bound in step 2.2 factors into nonnegative within-cube integrands. Monotonicity from step 2.1 and repeated application of [F5] and the rectangle formula in [F6] factor the resulting lower bound's integral into the product of the translated one-cube integrals. For a cube indexed by $j$ with $N_j\ge 1$, common translation by $(\mathrm{\ell }+g_{\mathrm{\ell }})j$ is a Borel isomorphism mapping $[0,\mathrm{\ell }{]}^{d{N}_j}$ onto $C_j^{N_j}$ and preserves its finite product measure: the push-forward is sigma-finite and, on rectangles, this follows from single-coordinate translation invariance [F7] and the rectangle formula [F6]; uniqueness in [F6] gives equality of the product measures. Pair differences, hence the within-cube integrand, are unchanged by this translation, so the integral invariance proved in step 2.1 identifies the integral over $C_j^{N_j}$ with its origin-centered integral over $[0,\mathrm{\ell }{]}^{d{N}_j}$. When $N_j=0$, the corresponding factor is $Z_{\mathrm{\ell },0}=1$. Multiplying by $a^N\mathrm{/}N!$ cancels the label count from step 2.1, leaving $e^{-\beta D_L}{\prod }_j{Z}_{\mathrm{\ell },N_j}$. Summing over the finitely many occupancy vectors and expanding the finite product of cutoff sums from [F2] gives ${\mathrm{\Xi }}_L\ge e^{-\beta D_L}({\mathrm{\Xi }}_{\mathrm{\ell }}^{\mathrm{c}\mathrm{u}\mathrm{t}}{)}^{M_L}\mathrm{.}$

The integer-part bounds in step 1.1 give $L\mathrm{/}(\mathrm{\ell }+g_{\mathrm{\ell }})-1<m_L\le L\mathrm{/}(\mathrm{\ell }+g_{\mathrm{\ell }})$, so $m_L\mathrm{/}L\to (\mathrm{\ell }+g_{\mathrm{\ell }}{)}^{-1}$ and $M_L\mathrm{/}{L}^d\to (\mathrm{\ell }+g_{\mathrm{\ell }}{)}^{-d}$. By [F2], ${\mathrm{\Xi }}_{\mathrm{\ell }}^{\mathrm{c}\mathrm{u}\mathrm{t}}\ge (1-e^{-{\mathrm{\ell }}^d}){\mathrm{\Xi }}_{\mathrm{\ell }}$, so $\log {\mathrm{\Xi }}_{\mathrm{\ell }}^{\mathrm{c}\mathrm{u}\mathrm{t}}\ge {\mathrm{\ell }}^d{p}_{\mathrm{\ell }}+\log (1-e^{-{\mathrm{\ell }}^d})$. Taking logarithms in step 3.1, dividing by $L^d$, and using step 2.2 gives $p_L\ge \frac{M_L}{L^d}({\mathrm{\ell }}^d{p}_{\mathrm{\ell }}+\log (1-e^{-{\mathrm{\ell }}^d})-\frac{\beta C}{2}{n}_{\mathrm{\ell }}^2{S}_{d,ϵ}{g}_{\mathrm{\ell }}^{-d-ϵ})\mathrm{.}$ Taking the lower limit and using the volume ratio above yields the same bound with $M_L\mathrm{/}{L}^d$ replaced by $(\mathrm{\ell }+g_{\mathrm{\ell }}{)}^{-d}$. The cutoff definition [F2] and $\mathrm{\ell }\ge 1$ give $n_{\mathrm{\ell }}\le K{\mathrm{\ell }}^d+1\le (K+1){\mathrm{\ell }}^d$. Factoring $(\mathrm{\ell }\mathrm{/}(\mathrm{\ell }+g_{\mathrm{\ell }}){)}^d$ and using this estimate in the negative error term therefore gives ${\mathrm{lim\hspace{0.17em}inf}}_{L\to \mathrm{\infty }}{p}_L\ge {\left(\frac{\mathrm{\ell }}{\mathrm{\ell }+g_{\mathrm{\ell }}}\right)}^d[p_{\mathrm{\ell }}+{\mathrm{\ell }}^{-d}\log (1-e^{-{\mathrm{\ell }}^d})-\frac{\beta C}{2}(K+1{)}^2{S}_{d,ϵ}{\mathrm{\ell }}^d{g}_{\mathrm{\ell }}^{-d-ϵ}]\mathrm{.}$

As $\mathrm{\ell }\to \mathrm{\infty }$, the prefactor tends to $1$. Also $e^{-{\mathrm{\ell }}^d}\to 0$, so continuity of $\log$ at $1$ gives $\log (1-e^{-{\mathrm{\ell }}^d})\to 0$; since ${\mathrm{\ell }}^{-d}\to 0$, their product ${\mathrm{\ell }}^{-d}\log (1-e^{-{\mathrm{\ell }}^d})$ tends to zero. The cross-energy error tends to zero because $d-\alpha (d+ϵ)<0$ by step 1.1. The replacement of the prefactor times $p_{\mathrm{\ell }}$ by $p_{\mathrm{\ell }}$ has error at most $Q[1-(\mathrm{\ell }\mathrm{/}(\mathrm{\ell }+g_{\mathrm{\ell }}){)}^d]\to 0$ by [F2]. Thus for every $\eta >0$ there is $T$ such that every $\mathrm{\ell }\ge T$ makes the right side of step 4.1 at least $p_{\mathrm{\ell }}-\eta$.

Let $\bar{p}={\mathrm{lim\hspace{0.17em}sup}}_{\mathrm{\ell }\to \mathrm{\infty }}{p}_{\mathrm{\ell }}$. By [A1], for every $\eta >0$ and every $T\ge 1$ there is an $\mathrm{\ell }\ge T$ with $p_{\mathrm{\ell }}>\bar{p}-\eta$; otherwise the tail supremum from $T$ onward would be at most $\bar{p}-\eta$, contradicting the infimum defining $\bar{p}$. Choose $T$ large enough for step 5.1. Then ${\mathrm{lim\hspace{0.17em}inf}}_{L\to \mathrm{\infty }}{p}_L\ge \bar{p}-2\eta$. Letting $\eta \downarrow 0$ shows $\mathrm{lim\hspace{0.17em}inf}{p}_L\ge \mathrm{lim\hspace{0.17em}sup}{p}_L$.

For any $S,T\ge 1$, choose $L\ge \max (S,T)$. Then ${\inf }_{x\ge S}{p}_x\le p_L\le {\sup }_{x\ge T}{p}_x$. Taking the supremum over $S$ and then the infimum over $T$ proves $\mathrm{lim\hspace{0.17em}inf}{p}_L\le \mathrm{lim\hspace{0.17em}sup}{p}_L$ by [A1]. Together with step 6.1, the two finite tail bounds equal a number $p$. For every $\eta >0$, their definitions give tail thresholds with ${\inf }_{L\ge T_{-}}{p}_L>p-\eta$ and ${\sup }_{L\ge T_{+}}{p}_L<p+\eta$; hence $\mathrm{\mid }{p}_L-p\mathrm{\mid }<\eta$ for all sufficiently large $L$. The limit is finite by [F2]. This proves existence for the specified free-cube reduced grand pressure.