Riesz 2-energy of the Diamond ensemble

Abstract

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.