Observed asymptotic differences in energies of stable and minimal point configurations on $\mathbb {S}^2$S2 and the role of defects

Abstract

Configurations of N points on the two-sphere that are stable with respect to the Riesz s-energy have a structure that is largely hexagonal. These stable configurations differ from the configurations with the lowest reported N-point s-energy in the location and structure of defects within this hexagonal structure. These differences in energy between the stable and minimal configuration suggest that energy scale at which defects play a role. This work uses numerical experiments to report this difference as a function of N, allowing us to infer the energy scale at which defects play a role. This work is presented in the context of established estimates for the minimal N-point energy, and in particular we identify terms in these estimates that likely reflect defect structure.