论文标题

在S^2上的点的对数能量上

On the Logarithmic Energy of Points on S^2

论文作者

Steinerberger, Stefan

论文摘要

我们回顾一个经典的问题:$ \ mathbb {s}^2 $ $ $ $ $ \ mathcal {e} _ {\ log}(\ log}(n)= \ min_1 = \ min_1,x_1,\ dots,x_n \ in \ mathbb { \ atop i \ neq j}^{n} {\ log {\ frac {1} {\ | x_i-x_jj \ |}}}}? $$ BETERMIN&SANDIER(在Sandier&Serfaty的作品的基础上)表明,$$ \ Mathcal {e} _ {\ log}(n)= \ left(\ frac {1} {2} {2} {2} - \ log {2} \ right {2} \ right) \ cdot n + o(n),$$,其中常数$ c _ {\ log} $的特征是某个重量化的最小化问题。 Brauchart,Hardin \&Saff猜想了$ c _ {\ log} $($ \ sim -0.05 $)的封闭式表达式,假定分析延续。我们描述了一种简单的翻新方法,导致纯粹的局部问题涉及高斯人的叠加。特别是,如果六边形晶格最大程度地减少高斯能量,这将证明$ c _ {\ log} $确实与猜想的值一致。我们还将下限从$ c _ {\ log} \ geq -0.223 $到$ c _ {\ log} \ geq -0.095 $。

We revisit a classical question: how large is the minimal logarithmic energy of $n$ points on $\mathbb{S}^2$ $$ \mathcal{E}_{\log}(n) = \min_{x_1, \dots, x_n \in \mathbb{S}^2} \quad \sum_{i,j =1 \atop i \neq j}^{n}{ \log{\frac{1}{\|x_i-x_j\|}} } ? $$ Betermin & Sandier (building on work of Sandier & Serfaty) showed that $$\mathcal{E}_{\log}(n) = \left( \frac{1}{2} - \log{2} \right)n^2 - \frac{n \log{n}}{2} + c_{\log} \cdot n + o(n),$$ where the constant $c_{\log}$ is characterized by a certain renormalized minimization problem. Brauchart, Hardin \& Saff conjectured a closed form expression for $c_{\log}$ ($\sim -0.05$) assuming analytic continuation. We describe a simple renormalization approach that results in a purely local problem involving superpositions of Gaussians. In particular, if the hexagonal lattice minimizes Gaussians energy, this would prove that $c_{\log}$ indeed coincides with the conjectured value. We also improve the lower bound from $c_{\log} \geq -0.223$ to $c_{\log} \geq -0.095$.

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