Asymptotics of greedy energy sequences on the unit circle and the sphere

TL;DR

This paper studies the asymptotic behavior of greedy energy sequences on spheres, revealing a phase transition at specific parameter values where distribution and energy properties change markedly.

Contribution

It introduces a detailed analysis of greedy $ ext{lambda}$-energy sequences, including their symmetry, distribution transition at $ ext{lambda}=2$, and asymptotic energy behavior, highlighting differences from equally spaced points.

Findings

Sequences exhibit symmetry $a_{2k+1}=-a_{2k}$.

Distribution transitions from uniform to concentrated at antipodes at $ ext{lambda}=2$.

Second-order asymptotics differ from equally spaced points, with a transition at $ ext{lambda}=1$.

Abstract

For a parameter $\lambda>0$, we investigate greedy $\lambda$-energy sequences $(a_{n})_{n=0}^{\infty}$ on the unit sphere $S^{d}\subset\mathbb{R}^{d+1}$, $d\geq 1$, satisfying the defining property that each $a_{n}$, $n\geq 1$, is a point where the potential $\sum_{k=0}^{n-1}|x-a_{k}|^{\lambda}$ attains its maximum value on $S^{d}$. We show that these sequences satisfy the symmetry property $a_{2k+1}=-a_{2k}$ for every $k\geq 0$. The asymptotic distribution of the sequence undergoes a sharp transition at the value $\lambda=2$, from uniform distribution ($\lambda<2$) to concentration on two antipodal points ($\lambda>2$). We investigate first-order and second-order asymptotics of the $\lambda$-energy of the first $N$ points of the sequence, as well as the asymptotic behavior of the extremal values $\sum_{k=0}^{n-1}|a_{n}-a_{k}|^{\lambda}$. The second-order asymptotics is analyzed on the unit circle. It is shown that this asymptotic behavior differs significantly from that of $N$ equally spaced points on the unit circle, and a transition in the behavior takes place at $\lambda=1$.