Compactness by Coarse-Graining in Long-Range Lattice Systems

Abstract

Abstract We consider energies on a periodic set ℒ {\mathcal{L}} of the form ∑ i , j ∈ ℒ a i ⁢ j ε ⁢ | u i - u j | {\sum_{i,j\in\mathcal{L}}a^{\varepsilon}_{ij}\lvert u_{i}-u_{j}\rvert} , 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_{\varepsilon}} with R ε → + ∞ {R_{\varepsilon}\to+\infty} , i.e., if | i - j | ≤ R ε {\lvert i-j\rvert\leq R_{\varepsilon}} , then a i ⁢ j ε ≥ c > 0 {a^{\varepsilon}_{ij}\geq c>0} . In a discrete-to-continuum analysis, we prove that the overall behavior as ε → 0 {\varepsilon\to 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 ⁢ ℒ {\varepsilon\mathcal{L}} with equibounded energy a family of sets with equibounded perimeter. This agrees with the case of equibounded R ε {R_{\varepsilon}} 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 {\mathcal{L}=\mathbb{Z}^{d}} .