Asymptotics of greedy energy sequences on the unit circle and the sphere
Abey L\'opez-Garc\'ia, Ryan E. McCleary

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 , we investigate greedy -energy sequences on the unit sphere , , satisfying the defining property that each , , is a point where the potential attains its maximum value on . We show that these sequences satisfy the symmetry property for every . The asymptotic distribution of the sequence undergoes a sharp transition at the value , from uniform distribution () to concentration on two antipodal points (). We investigate first-order and second-order asymptotics of the -energy of the first points of the sequence, as well as the asymptotic behavior of the extremal values . The second-order asymptotics is analyzed on the…
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