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 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 {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}) and takes the form 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. © 2013 Elsevier Inc.