Bounded finite-range lattice model

Fix integers $d\ge 1$, $L\ge 1$, a finite alphabet $\mathcal{A}$ of cardinality $q\ge 1$, and a finite, possibly empty, list of interaction shapes $S_a\subset {\mathbb{Z}}^d$, each nonempty and contained in $[-R,R{]}^d$ for some integer $R\ge 0$. $\mathrm{\parallel }{\mathrm{\Phi }}_a{\mathrm{\parallel }}_{\mathrm{\infty }}$ is finite because its domain is finite. For each shape let ${\mathrm{\Phi }}_a:{\mathcal{A}}^{S_a}\to \mathbb{R}$ be a bounded interaction function. Set $J_0:={\sum }_a\mathrm{\parallel }{\mathrm{\Phi }}_a{\mathrm{\parallel }}_{\mathrm{\infty }}$. On ${\mathrm{\Lambda }}_L:=\{1,\dots ,L{\}}^d$, define the free-boundary energy $H_L(s):={\sum }_a{\sum }_{x:x+S_a\subseteq {\mathrm{\Lambda }}_L}{\mathrm{\Phi }}_a((s_{x+u}{)}_{u\in S_a})$, for configurations $s\in {\mathcal{A}}^{{\mathrm{\Lambda }}_L}$. For $\beta >0$, define the finite partition sum $Z_L(\beta ):={\sum }_{s\in {\mathcal{A}}^{{\mathrm{\Lambda }}_L}}{e}^{-\beta H_L(s)}$ and normalized log-partition density $p_L(\beta ):=L^{-d}\log Z_L(\beta )$.

The alphabet, interaction list, shapes and functions are mathematical data. The formula uses the same interaction function at every translated anchor. This definition includes free boundary conditions only; other boundary energies are introduced by an explicit bound in Finite-range lattice log-partition limit.