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 [J. Nonlinear Sci. 28 (2018), 69-90] in the triangular case.