Homogenization of surface and length energies for spin systems

We study the homogenization of lattice energies related to Ising systems of the formEε(u)=−∑ijcijεuiuj, with uiui a spin variable indexed on the portion of a cubic lattice Ω∩εZdΩ∩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 {cijε} 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})BV(Ω;{±1}) and takes the formF(u)=∫Ω∩∂⁎{u=1}φ(ν)dHd−1 (ν   is the normal to ∂⁎{u=1}∂⁎{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 well-separated. Finally, we prove a dual result for discrete curves.