1302.6513 (Bernd Schmidt)

Ground states of the 2D Heitmann-Radin model: nonuniqueness and
$N^{3/4}$ law for the deviation from the asymptotic Wulff shape    [PDF]

Bernd Schmidt

We investigate ground state configurations of the Heitmann-Radin pair potential model in two dimensions for a general finite number $N$ of particles. It is known rigorously that ground states are subsets of the 2D triangular lattice [Heitmann, Radin, J. Statist. Phys. 22 (1980), 281-287] whose overall shape in the limit $N\to\infty$ becomes unique and is given by a regular hexagon which emerges as the Wulff shape associated with a continuum limit [AuYeung, Friesecke, Schmidt, Calc. Var. PDE 44 (2012), 81-100]. Here we show that exact discrete ground states exhibit large microscopic fluctuations about the asymptotic Wulff shape, deviating by $\sim N^{3/4}$ particles from the nearest hexagonal shape. This rate is sharp in the sense of being an upper bound for all $N$ and a lower bound for certain arbitrarily large $N$. Naively, one would only expect fluctuations of order $\sim N^{1/2}$, corresponding to re-arrangement of surface atoms.

View original: http://arxiv.org/abs/1302.6513