*Published Paper*

**Inserted:** 30 jun 2010

**Last Updated:** 9 feb 2013

**Journal:** J. Funct. Anal.

**Volume:** 264

**Pages:** 1296-1328

**Year:** 2013

**Abstract:**

We study the homogenization of lattice energies related to Ising systems of the form

$E_\varepsilon(u)=-\sum_{ij} c^\varepsilon_{ij} u_i u_j,$

with $u_i$ a spin variable indexed on the portion of a cubic lattice $\Omega\cap\varepsilon \mathbb Z^d$, by computing their $\Gamma$-limit in the framework of surface energies in a BV setting. We introduce a notion of homogenizability of the system $\{c^\varepsilon_{ij}\}$ that allows to treat periodic, almost-periodic and random statistically homogeneous models, when the coefficients are positive (ferromagnetic energies), in which case the limit energy is finite on $BV(\Omega;\{\pm1\})$ and takes the form

$ F(u)=\int_{\Omega\cap\partial^*\{u=1\}}\varphi(\nu)d{\mathcal H}^{d-1} $

($\nu$ is the normal to $\partial^*\{u=1\}$), where $\varphi$ is characterized by an asymptotic formula. In the random case $\varphi$ 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.

**Keywords:**
Gamma-convergence, surface energies, spin systems, Discrete-to-continuous homogenization

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