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

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.