arXiv subjects

Pedro R. López-Gómez

Publications and source records attributed to Pedro R. López-Gómez.

6 recordsLinked to original sources

arXiv2026

Riesz 2-energy of the Diamond ensemble

We study the Riesz $2$-energy of point configurations on the two-dimensional sphere arising from the Diamond ensemble, a construction of well-distributed points introduced by Beltrán and Etayo in 2020. For this family of point sets, we derive an explicit formula for the expected $2$-energy, valid for general choices of the parameters. Using this formula, we analyze a specific realization of the Diamond ensemble and obtain the asymptotic expansion of its expected $2$-energy. As a consequence, we establish a new upper bound for the minimal Riesz $2$-energy on the sphere, improving upon all previously known upper bounds. In particular, our result yields, for the first time, a negative coefficient in the quadratic term of the asymptotic expansion, bringing the upper bound significantly closer to the conjectured constant.

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arXiv2026

Discrepancy of determinantal point processes on compact, connected two-point homogeneous spaces

We study the $L^{\infty}$ discrepancy of point sets generated by determinantal point processes on all compact, connected two-point homogeneous spaces, namely spheres and projective spaces. Using concentration inequalities and variance estimates for the number of points in metric balls, we derive general upper bounds for the discrepancy of homogeneous determinantal point processes. In the particular case of the harmonic ensemble, we show that the discrepancy of $N$ points is $O((N^{1-1/D})^{1/2}\log N)$ with high probability, where $D$ denotes the real dimension of the manifold. For the projective ensemble on $\mathbb{CP}^d$, we obtain the sharper bound $O((N^{1-1/D}\log N)^{1/2})$. These results extend previously known discrepancy estimates for determinantal point processes on the sphere to all compact, connected two-point homogeneous spaces.

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arXiv2025

Points on $\operatorname{SO}(3)$ with low logarithmic energy

We describe several randomized collections of $3\times 3$ rotation matrices and analyze their associated logarithmic energy. The best one (i.e. the one attaining the lowest expected logarithmic energy) is constructed by choosing $r$ spherical points, which come from the zeros of a randomly chosen degree $r$ polynomial, and considering at each of these points a set of $s$ evenly distributed rotation matrices. This construction yields a new upper bound on the minimal logarithmic energy of $n=rs$ rotation matrices.

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arXiv2024

Minimal Riesz and logarithmic energies on the Grassmannian $\operatorname{Gr}_{2,4}$

We study the Riesz and logarithmic energies on the Grassmannian $\operatorname{Gr}_{2,4}$ of $2$-dimensional subspaces of $\mathbb{R}^4$. We prove that the continuous Riesz and logarithmic energies are uniquely minimized by the uniform measure, and we obtain asymptotic upper and lower bounds for the minimal discrete energies, with matching orders for the next-order terms. Additionally, we define a determinantal point process on $\operatorname{Gr}_{2,4}$ and compute the expected energy of the points coming from this random process, thereby obtaining explicit constants in the upper bounds for the Riesz and logarithmic energies.

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arXiv2023

Measure-preserving mappings from the unit cube to some symmetric spaces

We construct measure-preserving mappings from the $d$-dimensional unit cube to the $d$-dimensional unit ball and the compact rank one symmetric spaces, namely the $d$-dimensional sphere, the real, complex, and quaternionic projective spaces, and the Cayley plane. We also give a procedure to generate measure-preserving mappings from the $d$-dimensional unit cube to product spaces and fiber bundles under certain conditions.

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arXiv2022

Low energy points on the sphere and the real projective plane

We present a generalization of a family of points on $\mathbb{S}^2$, the Diamond ensemble, containing collections of $N$ points on $\mathbb{S}^2$ with very small logarithmic energy for all $N\in\mathbb{N}$. We extend this construction to the real projective plane $\mathbb{RP}^2$ and we obtain upper and lower bounds with explicit constants for the Green and logarithmic energy on this last space.

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