Finite-range interaction (lattice)

Let $\Lambda \subset \mathbb{Z}^d$ be a finite region and let the single-site spin space be $\mathcal S$ (see spin space ). A (finite-volume) interaction is a family

indexed by finite subsets $A$ of $\mathbb{Z}^d$, where each $\Phi_A$ is a real-valued function of the spins in $A$, i.e. $\Phi_A : \mathcal S^A \to \mathbb{R}$ (see interaction potential $\Phi$ ).

The interaction $\Phi$ is finite-range if there exists $R<\infty$ such that

where $\operatorname{diam}(A)=\max\{\|x-y\|:x,y\in A\}$ for a chosen lattice norm.

Equivalently: only subsets $A$ whose sites fit inside a ball of radius $R$ can contribute to the energy.

Cross-links

• The resulting energy in a region is given by a lattice Hamiltonian built from $\Phi$ and a boundary condition .
• Finite-range interactions are a common sufficient hypothesis for constructing Gibbs specifications and infinite-volume Gibbs measures via the DLR equation .
• Typical examples include the Ising model with nearest-neighbor coupling.

Key properties

1. Locality of energy contributions. The energy change from modifying spins in a small set depends only on spins within distance $R$ of that set.
2. Well-defined finite-volume Gibbs measures. For any finite $\Lambda$ and boundary condition $\eta$, the corresponding finite-volume Gibbs measure has a Hamiltonian that depends on $\eta$ only near $\partial\Lambda$ (a boundary layer of thickness $R$).
3. Quasilocal conditional probabilities. Finite-range interactions yield conditional distributions where the spin at a site depends on the exterior configuration only through a finite neighborhood.
4. Compatibility with thermodynamic limits. Finite-range is a standard condition ensuring the existence of limits of pressures/free energies along increasing volumes (see thermodynamic limit of the pressure ).

Physical interpretation

A finite-range interaction models systems where forces are short-ranged: spins influence each other only up to a finite distance (e.g. exchange interactions in many magnetic materials). The parameter $R$ is a microscopic interaction range; macroscopic long-range correlations (large correlation length ) can still emerge near criticality even when the microscopic interaction range is finite.