THE SQUARE STICKY DISK: CRYSTALLIZATION AND GAMMA-CONVERGENCE TO THE OCTAGONAL ANISOTROPIC PERIMETER

We consider a variant of the sticky disk energy where distances between particles are evaluated through the sup norm ‖⋅‖∞in the plane. We first prove crystallization of minimizers in the square lattice, for any fixed number N of particles. Then we consider the limit as N →∞: In contrast to the standard sticky disk, there is only one orientation in the limit, and we are able to compute explicitly the Γ-limit to be an anisotropic perimeter with octagonal Wulff shape. The results are based on an energy decomposition for graphs that generalizes the one proved by De Luca-Friesecke [10] in the triangular case.