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 $\parallel \cdot {\parallel }_{\mathrm{\infty }}$ 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 $N\to \mathrm{\infty }$: in contrast to the standard sticky disk, there is only one orientation in the limit, and we are able to compute explicitly the $\mathrm{\Gamma }$-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.