The Pearl ensemble and the logarithmic energy on the Sphere

Abstract

We define a family of random sets of points on the sphere $\mathbb S^2$, the Pearl ensemble, depending on several parameters. The expected value of the logarithmic energy can be computed exactly up to terms of order $O(\sqrt{N}\log N)$, 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 $(1-\log 3)/2\sim -0.0493061\ldots$. 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.