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

Ujué Etayo, Pablo G. Arce

TL;DR

This work tackles the discrete logarithmic energy problem on the 3-sphere by lifting well-distributed point sets from the 2-sphere via the Hopf fibration. It combines analytic expansions for lifted uniform and determinantal point processes with extensive numerical experiments, yielding a new upper bound for the minimal energy on $ S^3$ using a fibred Spherical ensemble. The Harmonic ensemble matches the leading two terms after lifting but exhibits a distinct third-order log-log term, while the Diamond ensemble shows the strongest empirical energy performance. Overall, the paper tightly links energy-optimal configurations on $ S^2$ to their lifted counterparts on $ S^3$, providing rigorous benchmarks and guiding principles for constructing low-energy point configurations on the 3-sphere.

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(ω_N) = \sum_{\substack{i, j=1\\ i \neq j}}^{N} \log\frac{1}{\|x_i - x_j\|}, \; \text{ where}\; ω_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.

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

TL;DR

This work tackles the discrete logarithmic energy problem on the 3-sphere by lifting well-distributed point sets from the 2-sphere via the Hopf fibration. It combines analytic expansions for lifted uniform and determinantal point processes with extensive numerical experiments, yielding a new upper bound for the minimal energy on using a fibred Spherical ensemble. The Harmonic ensemble matches the leading two terms after lifting but exhibits a distinct third-order log-log term, while the Diamond ensemble shows the strongest empirical energy performance. Overall, the paper tightly links energy-optimal configurations on to their lifted counterparts on , providing rigorous benchmarks and guiding principles for constructing low-energy point configurations on the 3-sphere.

Abstract

We analyse several constructions of random point sets on the sphere evaluating and comparing them through their discrete logarithmic energy: \begin{equation*} E_0(ω_N) = \sum_{\substack{i, j=1\\ i \neq j}}^{N} \log\frac{1}{\|x_i - x_j\|}, \; \text{ where}\; ω_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 -dimensional sphere - including uniformly random points, antipodally symmetric sets, determinantal point processes, and the Diamond ensemble - to , 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 ), obtaining as a result a family of points on the -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 . 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 and and provide both theoretical and numerical benchmarks for future work.
Paper Structure (33 sections, 18 theorems, 174 equations, 2 figures, 2 tables)

This paper contains 33 sections, 18 theorems, 174 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction and main results
  2. Logarithmic energy on $\mathbb{S}^2$
  3. Minimal logarithmic energy on $\mathbb{S}^3$
  4. Hopf fibration
  5. Main results
  6. Fibred constructions
  7. Notation
  8. A new bound for the minimal logarithmic energy on $\mathbb{S}^3$
  9. The lifted Harmonic ensemble
  10. The lifted Diamond ensemble
  11. Organization of the paper
  12. Uniformly distributed points
  13. Uniform distribution on $\mathbb{S}^2$ and the use of Hopf fibration
  14. Determinantal point processes
  15. Determinantal point processes on $\mathbb{CP}^1$ and the Hopf fibration
  16. ...and 18 more sections

Key Result

Proposition 1.1

There exists a $C^{1}$ (in fact, smooth) diffeomorphism such that the push-forward of the spherical surface measure on $\mathbb{S}^{2}$ coincides, up to a constant factor, with the Fubini-Study volume form on $\mathbb{CP}^{1}$. More precisely, if $\sigma$ denotes the surface area measure on $\mathbb{S}^{2}$ and $d\mathrm{vol}_{FS}$ the Fubini--Study volu In particular, the Jacobian determinant of

Figures (2)

  • Figure 1: $\left(\mathbb{E}_{\omega_N\sim\uparrow_{H}^{k}(\diamond_{r})} \left[E_0(\omega_N)\right]+\frac{N^2}{4}\right)/N\log N$ plotted against the number of points, where $\mathbb{E}_{\omega_N\sim\uparrow_{H}^{k}(\diamond_{r})} \left[E_0(\omega_N)\right]$ represents the expected value of the logarithmic energy of the lifted configuration of points coming from the Diamond ensemble through the Hopf fibration. Averaged over 5 runs with different random seeds.
  • Figure 2: $\left(\mathbb{E}_{\omega_N\sim\uparrow_{H}^{k}(\diamond_{r})} \left[E_0(\omega_N)\right]+\frac{N^2}{4}+\frac{N\log N}{3}\right)/N$ plotted against the number of points, where $\mathbb{E}_{\omega_N\sim\uparrow_{H}^{k}(\diamond_{r})} \left[E_0(\omega_N)\right]$ represents the expected value of the lifted configuration of points coming from the Diamond ensemble through the Hopf fibration. Averaged over 5 runs with different random seeds.

Theorems & Definitions (36)

  • Proposition 1.1
  • proof : Proof (sketch)
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • ...and 26 more