Observed Asymptotic Differences in Energies of Stable and Minimal Point Configurations on $\mathbb{S}^2$ and the Role of Defects

Abstract

Observations suggest that configurations of points on a sphere that are stable with respect to a Riesz potential distribute points uniformly over the sphere. Further, these stable configurations have a local structure that is largely hexagonal. Minimal configurations differ from stable configurations in the arrangement of defects within the hexagonal structure. This paper reports the asymptotic difference between the average energy of stable states and the lowest reported energies. We use this to infer the energy scale at which defects in the hexagonal structure are manifest. We report results for the Riesz potentials for s=0, s=1, s=2 and s=3. Additionally we compare existing theory for the asymptotic expansion in N of the minimal $N$-point energy with experimental results. We report a case of two distinct stable states that have the same Voronoi structure. Finally, we report the observed growth of the number of stable states as a function of N.