Compactness by Coarse-Graining in Long-Range Lattice Systems

We consider energies on a periodic set L of the form Sigma(i,j is an element of L) a(ij)(epsilon)vertical bar(i) - u(j)vertical bar, defined on spin functions u(i) is an element of (0, 1}, and we suppose that the typical range of the interactions is R-epsilon with R-epsilon -> +infinity, i.e., if vertical bar i - j vertical bar <= R-epsilon, then a(ij)(epsilon )>= c > 0. In a discrete-to-continuum analysis, we prove that the overall behavior as epsilon -> 0 of such functionals is that of an interfacial energy. The proof is performed using a coarse-graining procedure which associates to scaled functions defined on epsilon L with equibounded energy a family of sets with equibounded perimeter. This agrees with the case of equibounded R-epsilon and can be seen as an extension of coerciveness result for short-range interactions, but is different from that of other long-range interaction energies, whose limit exits the class of surface energies. A computation of the limit energy is performed in the case L = Z(d).