On the Logarithmic Energy of Points on S^2

Abstract

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 $c_{\log} \geq -0.095$.