Random points on $\mathbb{S}^3$ with small logarithmic energy

TL;DR

This paper compares various random point set constructions on the 3-sphere using logarithmic energy, introducing a new low-energy family derived from the spherical ensemble and providing benchmarks for optimal configurations.

Contribution

It introduces a novel construction of points on -sphere with near-optimal logarithmic energy, extending the spherical ensemble via Hopf fibration, and offers new bounds and simulations for energy minimization.

Findings

Spherical ensemble points on -sphere achieve the lowest known asymptotic energy.

Lifted point sets from -sphere outperform other constructions in energy.

Simulations suggest the Diamond ensemble has empirical energies below all tested methods.

Abstract

We analyse several constructions of random point sets on the sphere $\mathbb{S}^{3}\subset\mathbb{R}^4$ evaluating and comparing them through their discrete logarithmic energy: \begin{equation*} E_0(\omega_N) = \sum_{\substack{i, j=1\\ i \neq j}}^{N} \log\frac{1}{\|x_i - x_j\|}, \; \text{ where}\; \omega_N=\{x_1,\ldots,x_N\} \subset \mathbb{S}^3. \end{equation*} Using the Hopf fibration, we lift a range of well-distributed families of points from the $2$-dimensional sphere - including uniformly random points, antipodally symmetric sets, determinantal point processes, and the Diamond ensemble - to $\mathbb{S}^{3}$, in order to assess their energy performance. In particular, we carry out this asymptotic analysis for the Spherical ensemble (a well known determinantal point process on $\mathbb{S}^2$), obtaining as a result a family of points on the $3$-dimensional sphere whose logarithmic energy is asymptotically the lowest achieved to date. This, in turn, provides a new upper bound for the minimal logarithmic energy on $\mathbb{S}^3$. Although an analytic treatment of the lifted Diamond ensemble remains elusive, extensive simulations presented here show that its empirical energies lie below all other deterministic and non-deterministic constructions considered. Together, these results sharpen the quantitative link between potential-theoretic optima on $\mathbb{S}^{2}$ and $\mathbb{S}^{3}$ and provide both theoretical and numerical benchmarks for future work.