PreprintCC BY-NC-ND 4.0arXiv
The Pearl ensemble and the logarithmic energy on the Sphere
Abstract
We define a family of random sets of points on the sphere 𝕊2, the Pearl ensemble, depending on several parameters. The expected value of the logarithmic energy can be computed exactly up to terms of order O(√Nlog N), with N the number of points. For properly chosen values of the parameters, the coefficient of the linear term in the expansion of the logarithmic energy can be taken as close as desired to the value (1−log 3)/2∼ −0.0493061…. Among the explicit constructions currently known to us for which the logarithmic energy expansion has been rigorously determined up to the linear term, the Pearl ensemble yields the smallest coefficient.
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BibTeX
@misc{gariboldi2026pearl,
title = {{The Pearl ensemble and the logarithmic energy on the Sphere}},
author = {Bianca Gariboldi and Giacomo Gigante},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39825},
eprint = {2609.39825},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39825},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The Pearl ensemble and the logarithmic energy on the Sphere
AU - Bianca Gariboldi
AU - Giacomo Gigante
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39825
UR - https://arxiv.org/abs/2609.39825
AB - We define a family of random sets of points on the sphere 𝕊^2, the Pearl ensemble, depending on several parameters. The expected value of the logarithmic energy can be computed exactly up to terms of order O(√Nlog N), with N the number of points. For properly chosen values of the parameters, the coefficient of the linear term in the expansion of the logarithmic energy can be taken as close as desired to the value (1−log 3)/2∼ −0.0493061…. Among the explicit constructions currently known to us for which the logarithmic energy expansion has been rigorously determined up to the linear term, the Pearl ensemble yields the smallest coefficient.
ER -