Minimization of the discrete interaction energy with smooth potentials

TL;DR

This paper investigates the minimization of discrete interaction energies on flat tori using smooth potentials, identifying optimal configurations in one and two dimensions through Fourier analysis.

Contribution

It provides new results on the minimizers of interaction energies on flat tori, showing equidistant points in 1D and triangular lattice in 2D, based on Fourier coefficient conditions.

Findings

In 1D, the minimizer is an equidistant point set.

In 2D, the minimizer among triplets is the triangular lattice.

Fourier decay conditions lead to these optimal configurations.

Abstract

We study the pair interaction on flat tori of functions whose Fourier coefficients are positive and decay sufficiently rapidly. In dimension one we find that the minimizer, up to translation, is the equidistant point set. In dimension two, minimizing with respect to triplets we find that the minimizer is the triangular lattice.