Homogenization of spin energies

The talk will focus on homogenization and $\Gamma$-convergence of surface and line energies defined on lattice (spin) systems in $\mathbf Z^d$ through bond interactions. We will dwell on nearest neighbours interaction systems and consider both periodic and random statistically homogeneous ergodic cases.
Given a smooth bounded domain $G\subset\mathbf R^n$ and a small parameter $\varepsilon>0$, we denote $\varepsilon\mathbf Z^d\cup G$ by $G_\varepsilon$ and, for a function $u$ defined on $G_\varepsilon$, consider the energy
$$ E_\varepsilon(u)=\sum_{i,j\in G_\varepsilon}\varepsilon^{d-1}c_{ij}(u_i-u_j)^2, \qquad c_{ij}\ge 0, \quad c_{ij}=0\text{ if }|i-j|\ne\varepsilon. $$
Our goal is to study the limit behaviour of $E_\varepsilon$ as $\varepsilon\to 0$.