The asymptotic distribution of Riesz Energy

Abstract

We consider the asymptotic distribution of the Riesz $s$-energy for a sample of $N$ independent, uniformly distributed points on the surface of a $d$-dimensional hypersphere as $N$ diverges. We identify three asymptotic regimes. In the first regime, both the mean and variance of the energy exist. In the second regime, only the mean exists. In the third regime, no integer moments exist. We characterize the asymptotic distribution in all three regimes and identify five special cases at their boundaries. Additionally, we highlight the connections between these results and Ed Saff’s work on minimal Riesz energy point configurations.

Raffaello Seri

Professor of Econometrics

My research interests include statistics, numerical analysis, operations research, psychology, economics and management.