$Γ$-convergence of the Heitmann-Radin sticky disc energy to the crystalline perimeter

Abstract

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.