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Ground states of the 2D sticky disc model: fine properties and \(N^{3/4}\) law for the deviation from the asymptotic Wulff shape. (English) Zbl 1292.82027
Summary: 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.

MSC:
82C22 Interacting particle systems in time-dependent statistical mechanics
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