Finite crystallization in the square lattice

Abstract

This paper addresses two-dimensional crystallization in the square lattice. A suitable configurational potential featuring both two- and three-body short-ranged particle interactions is considered. We prove that every ground state is a connected subset of the square lattice. Moreover, we discuss the global geometry of ground states and their optimality in terms of discrete isoperimetric inequalities on the square graph. Eventually, we study the aspect ratio of ground states and quantitatively prove the emergence of a square macroscopic Wulff shape as the number of particles grows.