Random points on $\mathbb{S}^3$ with small logarithmic energy
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866912899476226048 |
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| author | Etayo, Ujué Arce, Pablo G. |
| author_facet | Etayo, Ujué Arce, Pablo G. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_11856 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Random points on $\mathbb{S}^3$ with small logarithmic energy Etayo, Ujué Arce, Pablo G. Probability 31C20, 52C35, 60G55, 82B21 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. |
| title | Random points on $\mathbb{S}^3$ with small logarithmic energy |
| topic | Probability 31C20, 52C35, 60G55, 82B21 |
| url | https://arxiv.org/abs/2602.11856 |