Ground states of the 2D sticky disc model: fine properties and $N^{3\mathrm{/}4}$ law for the deviation from the asymptotic Wulff shape

We investigate ground state configurations for a general finite number $N$ of particles of the Heitmann-Radin sticky disc pair potential model in two dimensions. Exact energy minimizers are shown to exhibit large microscopic fluctuations about the asymptotic Wulff shape which is a regular hexagon: There are arbitrarily large $N$ with ground state configurations deviating from the nearest regular hexagon by a number of $\sim N^{3\mathrm{/}4}$ particles. We also prove that for any $N$ and any ground state configuration this deviation is bounded above by $\sim N^{3\mathrm{/}4}$. As a consequence we obtain an exact scaling law for the fluctuations about the asymptotic Wulff shape. In particular, our results give a sharp rate of convergence to the limiting Wulff shape.