Compactness by Coarse-Graining in Long-Range Lattice Systems

We consider energies on a periodic set { {L}} of the form ∑ i, j a i j ϵ | u i - u j | {∑{i,jin {L}}a{ϵ}_{ij} u_{i}-u_{j} vert}, defined on spin functions u i { 0, 1 } {u_{i}in{0,1}}, and we suppose that the typical range of the interactions is R ϵ {R_{ϵ}} with R ϵ → + ∞ {R_{ϵ} o+infty}, i.e., if | i - j | ≤ R ϵ { i-j vertleq R_{ϵ}}, then a i j ϵ ≥ c > 0 {a{ϵ}_{ij}geq c>0}. In a discrete-to-continuum analysis, we prove that the overall behavior as ϵ → 0 {ϵ o 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 ϵ {ϵ {L}} with equibounded energy a family of sets with equibounded perimeter. This agrees with the case of equibounded R ϵ {R_{ϵ}} 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 = d { {L}=mathbb{Z}^{d}}.