Asymptotics for Discrete Weighted Minimal Riesz Energy Problems on Rectifiable Sets

Details Asymptotics for Discrete Weighted Minimal Riesz Energy Problems on Rectifiable Sets Asymptotics for Discrete Weighted Minimal Riesz Energy Problems on Rectifiable Sets

2006-02-11

arxiv.org/abs/math-ph/0602025v1

Physics

Mathematical Physics

Scientific paper

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 $s\to \infty$.

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