Bounded lattice pressure on van Hove sets

Fix an integer block side $\mathrm{\ell }>2R$ and tile the lattice by disjoint $\mathrm{\ell }$-cubes. For ${\mathrm{\Lambda }}_n$, retain only cubes fully contained in ${\mathrm{\Lambda }}_n$ and denote their number by $m_n$; the leftover sites lie within distance at most $\mathrm{\ell }$ of ${\mathrm{\Lambda }}_n^c$, so $\mathrm{\mid }{\mathrm{\Lambda }}_n\setminus \text{(full cubes)}\mathrm{\mid }=o(\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid })$ and $m_n{\mathrm{\ell }}^d\mathrm{/}\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }\to 1$.

Erase each interaction term whose support crosses a grid seam or touches a leftover site. At fixed $\mathrm{\ell }$, leftover-touching anchors are $o(\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid })$ by the van Hove condition. Each full $\mathrm{\ell }$-cube has at most $C_R{\mathrm{\ell }}^{d-1}$ anchors within distance $R$ of one of its seams, so seam-crossing anchors total at most $C_R\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }\mathrm{/}\mathrm{\ell }+o(\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid })$; here $C_R$ depends only on the finite shape list, $R$, and dimension. Since each term is bounded, the original and cut energies differ uniformly by at most $C\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }\mathrm{/}\mathrm{\ell }+o(\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid })$.

If two energies on a finite state set differ uniformly by at most $D$, their partition functions differ by factors between $e^{-\beta D}$ and $e^{\beta D}$, so their log partition functions differ by at most $\beta D$. The cut partition function is $Z_{\mathrm{\ell }}^{m_n}\mathrm{\mid }A{\mathrm{\mid }}^{\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }-m_n{\mathrm{\ell }}^d}$, where $Z_{\mathrm{\ell }}$ is the free cubic partition function. Dividing by $\mathrm{\mid }{\mathrm{\Lambda }}_n\mathrm{\mid }$ and taking $n\to \mathrm{\infty }$ shows that the limsup and liminf of $p_{{\mathrm{\Lambda }}_n}$ each differ from ${\mathrm{\ell }}^{-d}\log Z_{\mathrm{\ell }}$ by at most $C\beta \mathrm{/}\mathrm{\ell }$.

By [F1], ${\mathrm{\ell }}^{-d}\log Z_{\mathrm{\ell }}$ tends to the cubic pressure as $\mathrm{\ell }\to \mathrm{\infty }$. Sending $\mathrm{\ell }\to \mathrm{\infty }$ in the bound from step 3.1 makes the limsup and liminf of $p_{{\mathrm{\Lambda }}_n}$ equal to that same value, proving convergence.