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            <title xml:lang="en">Variational description of bulk energies for bounded and unbounded spin systems</title>
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              <p>We study the asymptotic behaviour of a general class of discrete energies defined on functions $u:\alpha \in\epsilon \Z^N\cap\Omega\mapsto u(\alpha)\in\mathbb{R}^m$ of the form $E_\epsilon(u)=\sum_{\alpha,\beta \in \epsilon \mathbb{Z}^N\cap\Omega} \epsilon^N g_\epsilon(\alpha,\beta,u(\alpha),u(\beta))$, as the mesh size $\epsilon$ goes to $0$. We prove that under general assumptions, that cover the case of bounded and unbounded spin systems in the thermodynamic limit, the variational limit of $E_\epsilon$ has the form $E(u)=\int_{\Omega}g(x,u(x))dx$. The cases of homogenization and of non-pairwise interacting systems (e.g. multiple-exchange spin-systems) are also discussed.</p>
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