Γ -Convergence of the Heitmann–Radin Sticky Disc Energy to the Crystalline Perimeter

We consider low-energy configurations for the Heitmann–Radin sticky discs functional, in the limit of diverging number of discs. More precisely, we renormalize the Heitmann–Radin potential by subtracting the minimal energy per particle, i.e. the so-called kissing number. For configurations whose energy scales like the perimeter, we prove a compactness result which shows the emergence of polycrystalline structures: The empirical measure converges to a set of finite perimeter, while a microscopic variable, representing the orientation of the underlying lattice, converges to a locally constant function. Whenever the limit configuration is a single crystal, i.e. it has constant orientation, we show that the Γ-limit is the anisotropic perimeter, corresponding to the Finsler metric determined by the orientation of the single crystal.