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 $F_arepsilon(u)=-sum_{i,j}c_{i,j}^arepsilon u_i u_j,$ where $u$ is a spin variable defined on a portion of a cubic lattice $arepsilon{Bbb Z}^dcapOmega$, $Omega$ being a regular bounded open set, and valued in ${-1,1}$. If the constants $c_{i,j}^arepsilon$ are nonnegative and satisfy suitable coercivity and decay assumptions, we show that all possible $Gamma$-limits of surface scalings of the functionals $F_arepsilon$ are finite on $BV(Omega;{pm 1})$ and of the form $int_{S_u}arphi(x, u_u), d{cal H}^{d-1}.$ If such decay assumptions are violated, we show that we may approximate nonlocal functionals of the form $int_{S_u}arphi( u_u), d{cal H}^{d-1}+int_Omegaint_Omega K(x,y)g(u(x),u(y)), dxdy.$ 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 $F_arepsilon$ even when the constants $c_{i,j}^arepsilon$ change sign. If such a criterion is satisfied, the ground states of $F_arepsilon$ 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.