Homogenization of surface and length energies for spin systems

We study the homogenization of lattice energies related to Ising systems of the form E-epsilon(u) = -Sigma(ij)c(ij)(epsilon)u(i)u(j), with u(i) a spin variable indexed on the portion of a cubic lattice Omega boolean AND epsilon Z(d), by computing their Gamma-limit in the framework of surface energies in a BV setting. We introduce a notion of homogenizability of the system {c(ij)(epsilon)} 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(Omega; {+/- 1}) and takes the form F(u) = integral(Omega boolean AND partial derivative*{u = 1})phi(nu)dH(d-1) (nu is the normal to partial derivative*{u = I}), where phi is characterized by an asymptotic formula. In the random case phi 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. (c) 2013 Elsevier Inc. All rights reserved.