LOCAL AND NONLOCAL CONTINUUM LIMITS OF ISING-TYPE ENERGIES FOR SPIN SYSTEMS

We study, through a Gamma-convergence procedure, the discrete to continuum limit of Ising-type energies of the form F-epsilon(u) = -Sigma(i,j) c(i,j)(epsilon) u(i)u(j), where u is a spin variable defined on a portion of a cubic lattice epsilon Z(d) boolean AND Omega, Omega being a regular bounded open set, and valued in {-1, 1}. If the constants c(i,j)(epsilon) are nonnegative and satisfy suitable coercivity and decay assumptions, we show that all possible Gamma-limits of surface scalings of the functionals F-epsilon are finite on BV (Omega; {+/- 1}) and of the form integral(Su) phi(x,v(u)) dH(d-1). If such decay assumptions are violated, we show that we may approximate nonlocal functionals of the form integral(Su) phi(v(u)) dH(d-1) + integral(Omega)integral(Omega) K(x, y) g(u(x), u(y)) dxdy. We focus on the approximation of two relevant examples: fractional perimeters and Ohta-Kawasaki-type energies. Eventually, we provide a general criterion for a ferromagnetic behavior of the energies F-epsilon even when the constants c(i,j)(epsilon) change sign. If such a criterion is satisfied, the ground states of F-epsilon are still the uniform states 1 and -1 and the continuum limit of the scaled energies is an integral surface energy of the form above.