Gamma-convergence analysis for discrete topological singularities: The anisotropic triangular lattice and the long range interaction energy

We consider 2D discrete systems, described by scalar functions and governed by periodic interaction potentials. We focus on anisotropic nearest neighbors interactions in the hexagonal lattice and on isotropic long range interactions in the square lattice. In both these cases, we perform a complete Gamma-convergence analysis of the energy induced by a configuration of discrete topological singularities. This analysis allows to prove the existence of many metastable configurations of singularities in the hexagonal lattice.