Asymptotics for Discrete Weighted Minimal Riesz Energy Problems on Rectifiable Sets

TL;DR

This paper studies the asymptotic behavior of optimally distributed points on rectifiable sets minimizing weighted Riesz energy, connecting energy minimization with well-separated configurations and prescribed distributions.

Contribution

It introduces a framework for generating asymptotically optimal point configurations on rectifiable sets using weighted Riesz energy minimization, extending previous unweighted results.

Findings

Optimal configurations are well-separated as N increases.

As s approaches infinity, configurations tend to best-packing solutions.

The distribution of points converges to the prescribed measure 

Abstract

Given a compact $d$-rectifiable set $A$ embedded in Euclidean space and a distribution $\rho(x)$ with respect to $d$-dimensional Hausdorff measure on $A$, we address the following question: how can one generate optimal configurations of $N$ points on $A$ that are "well-separated" and have asymptotic distribution $\rho (x)$ as $N\to \infty$? For this purpose we investigate minimal weighted Riesz energy points, that is, points interacting via the weighted power law potential $V=w(x,y)|x-y|^{-s}$, where $s>0$ is a fixed parameter and $w$ is suitably chosen. In the unweighted case ($w\equiv 1$) such points for $N$ fixed tend to the solution of the best-packing problem on $A$ as the parameter $s\to \infty$.