Local and Nonlocal Continuum Limits of Ising-Type Energies for Spin Systems

We study, through a Γ-convergence procedure, the discrete to continuum limit of Ising type energies of the form Fε(u) = − ∑ i,j ci,juiuj , where u is a spin variable defined on a portion of a cubic lattice εZ ∩ Ω, Ω being a regular bounded open set, and valued in {−1, 1}. If the constants ci,j are non negative and satisfies suitable coercivity and decay assumptions, we show that all possible Γ-limits of surface scalings of the functionals Fε are finite on BV (Ω; {−1, 1} and of the form ∫