Variational description of bulk energies for bounded and unbounded spin systems

We study the asymptotic behaviour of a general class of discrete energies defined on functions u : alpha epsilon epsilon Z(N) boolean AND Omega -> u(alpha) epsilon R(m) of the form E(epsilon)(u) = Sigma(alpha,beta epsilon ZN boolean AND 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) = integral(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.