Generating Point Configurations via Hypersingular Riesz Energy With an External Field

TL;DR

This paper investigates the asymptotic behavior of point configurations that minimize Riesz energy with an external field in the hypersingular case, providing formulas for their limiting distributions and energy values.

Contribution

It introduces a method to generate point configurations converging to a specified measure, with new formulas for energy limits and properties of minimizers.

Findings

Derived formulas for weak* limits of normalized counting measures.

Established asymptotic values of minimal energy.

Provided numerical examples illustrating theoretical results.

Abstract

For a compact $ d $-dimensional rectifiable subset of $ \mathbb{R}^{p} $ we study asymptotic properties as $ N\to\infty $ of $N$-point configurations minimizing the energy arising from a Riesz $ s $-potential $ 1/r^s $ and an external field in the hypersingular case $ s\geq d$. Formulas for the weak$ ^* $ limit of normalized counting measures of such optimal point sets and the first-order asymptotic values of minimal energy are obtained. As an application, we derive a method for generating configurations whose normalized counting measures converge to a given absolutely continuous measure supported on a rectifiable subset of $ \mathbb{R}^{p} $. Results on separation and covering properties of discrete minimizers are given. Our theorems are illustrated with several numerical examples.