Asymptotic properties of discrete minimal s, log^t-energy constants and configurations

Abstract

We investigated the energy of N points on an infinite compact metric space (A, d) of a diameter less than 1 that interact through the potential (1/ds )(log 1/d)t, where s, t ≥ 0 and d is the metric distance. With Eslogt (A, N) denoting the minimal energy for such N-point configurations, we studied certain continuity and differentiability properties of Eslogt (A, N) in the variable s. Then, we showed that in the limits, as s → ∞ and as s → s0 > 0, N-point configurations that minimize the s, logt-energy tends to an N-point best-packing configuration and an N-point configuration that minimizes the s0, logt-energy, respectively. Furthermore, we considered when A are circles in the Euclidean space R2 . In particular, we proved the minimality of N distinct equally spaced points on circles in R2 for some certain s and t. The study on circles shows a possibility for the utilization of N points generated through such new potential to uniformly discretize on objects with very high symmetry.