Asymptotic Properties of Discrete Minimal <i>s</i>,log<i>^t</i>-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)(log1/d)t, where s,t≥0 and d is the metric distance. With Elogts(A,N) denoting the minimal energy for such N-point configurations, we studied certain continuity and differentiability properties of Elogts(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.

Keywords

• discrete minimal energy
• best-packing
• Riesz energy
• logarithmic energy