Crystalline and Isoperimetric Square Configurations

We present crystallization results for planar atomic interactions governed by two- and three-body terms with the resulting periodicity being that of the square lattice. The emergence of a (square) Wulff shape for ground states is established by showing the optimality of ground-state configurations in terms of a discrete isoperimetric inequality. Furthermore, an $n^{3/4}$ law for the deviation from the asymptotic Wulff shape is established with an explicit constant for the leading term.