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

TL;DR

This paper studies the ground state configurations of a 2D sticky disc model, revealing large microscopic fluctuations around the limiting hexagonal shape and establishing a precise $N^{3/4}$ deviation law.

Contribution

It proves the existence of large deviations from the Wulff shape and establishes a sharp $N^{3/4}$ fluctuation law for the model's ground states.

Findings

Ground states deviate from the hexagonal shape by about $N^{3/4}$ particles.

Deviation from the asymptotic shape is bounded above by $N^{3/4}$ for all configurations.

The results provide a precise scaling law for fluctuations and convergence rate to the 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.