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

Abstract

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 $\sim N^{3/4}$ particles. We also prove that for any $N$ and any ground state configuration this deviation is bounded above by $\sim N^{3/4}$. 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.

Cited by in corpus (8)

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• Emergence of rigid Polycrystals from atomistic Systems with Heitmann-Radin sticky disk energy
• $\Gamma$-convergence of the Heitmann-Radin sticky disc energy to the crystalline perimeter
• Maximal fluctuations around the Wulff shape for edge-isoperimetric sets in ${\mathbb{Z}}^d$: a sharp scaling law
• Emergence of Wulff-Crystals from atomistic systems on the FCC and HCP lattices