Homogenization of surface and length energies for spin systems

We study the homogenization of lattice energies related to Ising systems of the form Eε(u)=− ij cε ij uiuj , with ui a spin variable indexed on the portion of a cubic lattice Ω ∩ εZd , by computing their Γ -limit in the framework of surface energies in a BV setting. We introduce a notion of homogenizability of the system {cε ij } that allows to treat periodic, almost-periodic and random statistically homogeneous models (the latter in dimension two), when the coefficients are positive (ferromagnetic energies), in which case the limit energy is finite on BV(Ω; {±1}) and takes the form F(u) = Ω∩∂ ∗{u=1} ϕ(ν) dHd−1 (ν is the normal to ∂ ∗{u = 1}), where ϕ is characterized by an asymptotic formula. In the random case ϕ can be expressed in terms of first-passage percolation characteristics. The result is extended to coefficients with varying sign, under the assumption that the areas where the energies are antiferromagnetic are wellseparated. Finally, we prove a dual result for discrete curves.