Exact canonical limit on arbitrary van Hove shapes

For any $\alpha \in \mathbb{R}$, $e^{\alpha N_k}Z({\mathrm{\Omega }}_k,N_k)\le {\mathrm{\Xi }}_{{\mathrm{\Omega }}_k}(e^{\alpha },\beta )$. Divide by volume, use [F5], and take the infimum over $\alpha$. The duality [F4] gives $\mathrm{lim\hspace{0.17em}sup}\mathrm{\mid }{\mathrm{\Omega }}_k{\mathrm{\mid }}^{-1}\log Z({\mathrm{\Omega }}_k,N_k)\le F_{\beta }(\rho )$.

Choose $\delta >0$ so $\rho -\delta ,\rho +\delta$ lie in the attainable interior. For large core side $\mathrm{\ell }$, choose integers $n_{-},n_{+}$ with $n_{\pm }\mathrm{/}{\mathrm{\ell }}^d\to \rho \pm \delta$. Put a central $\mathrm{\ell }$-cube inside each $(\mathrm{\ell }+g)$-cell, with $g=g(\mathrm{\ell })$ from [F6]. The effective densities $n_{\pm }\mathrm{/}(\mathrm{\ell }+g{)}^d$ still bracket $\rho$ when $\mathrm{\ell }$ is large.

At fixed $\mathrm{\ell },g$, pack every complete $(\mathrm{\ell }+g)$-cell contained in ${\mathrm{\Omega }}_k$. Its count $M_k$ satisfies $M_k\mathrm{/}\mathrm{\mid }{\mathrm{\Omega }}_k\mathrm{\mid }\to (\mathrm{\ell }+g{)}^{-d}$ by [F2]. Assign $n_{+}$ particles to $j_k$ cores and $n_{-}$ to the remaining cores, except one correction core with remainder $r_k$ chosen to make the total exactly $N_k$. Bracketing ensures $0\le j_k<M_k$ for large $k$; Euclidean division gives $0\le r_k<n_{+}-n_{-}\le n_{+}$. Every correction integral $Z(\mathrm{\ell },r)$, $0\le r\le n_{+}$, is positive: delete particles from a positive-measure finite-energy set realizing $Z(\mathrm{\ell },n_{+})>0$. There are finitely many such $r$, so the correction contributes $O_{\mathrm{\ell }}(1)$ to the log weight.

Exact assignment of labels cancels the global $N_k!$, leaving the product of core canonical integrals. The core lattice shell estimate bounds cross-energy per volume by $C{K}^2(\mathrm{\ell }\mathrm{/}g{)}^d\eta (g)$, where $K{\mathrm{\ell }}^d\ge n_{+}$. Let $k\to \mathrm{\infty }$ first, then $\mathrm{\ell }\to \mathrm{\infty }$. The packing fraction tends one, the cross error tends zero, and the two rates tend $F_{\beta }(\rho -\delta )$ and $F_{\beta }(\rho +\delta )$. The bracketing proportions tend one half, so the lower rate is at least their average. Finally $\delta \downarrow 0$; finite concavity makes $F_{\beta }$ continuous at the interior density. Combined with step 1.1 this proves the limit.