Next order energy asymptotics for Riesz potentials on flat tori

TL;DR

This paper investigates the detailed asymptotic behavior of minimal Riesz energy configurations on flat tori, deriving the next order terms beyond the classical leading order, and establishing their dependence on lattice properties.

Contribution

It provides explicit formulas for the next order terms in the energy asymptotics, revealing their independence from the specific lattice structure for certain constants.

Findings

Next order terms are of the form $C_{s,d}| ext{volume}( ext{lattice})|^{-s/d}N^{1+s/d}$ and $-rac{2}{d}N ext{log}N + (C_{ ext{log},d} - 2rac{ ext{zeta}'_{ ext{lattice}}(0)})N$.

The constant $C_{s,d}$ is independent of the lattice $ ext{Lambda}$.

Abstract

Let $\Lambda$ be a lattice in ${\bf R}^d$ with positive co-volume. Among $\Lambda$-periodic $N$-point configurations, we consider the minimal renormalized Riesz $s$-energy $\mathcal{E}_{s,\Lambda}(N)$. While the dominant term in the asymptotic expansion of $\mathcal{E}_{s,\Lambda}(N)$ as $N$ goes to infinity in the long range case that $0<s<d$ (or $s=\log$) can be obtained from classical potential theory, the next order term(s) require a different approach. Here we derive the form of the next order term or terms, namely for $s>0$ they are of the form $C_{s,d}|\Lambda|^{-s/d}N^{1+s/d}$ and $-\frac{2}{d}N\log N+\left(C_{\log,d}-2\zeta'_{\Lambda}(0)\right)N$ where we show that the constant $C_{s,d}$ is independent of the lattice $\Lambda$.