Many critical points for discrete Riesz energy on $\mathbb{T}^2$

TL;DR

This paper proves that for the discrete Riesz energy on the 2D torus, there are exponentially many critical points in certain regimes, extending understanding of energy landscapes in geometric configurations.

Contribution

It establishes the existence of exponentially many critical points for Riesz energy on the torus, a significant step beyond prior conjectures and numerical evidence.

Findings

Infinitely many $n$ with at least $ ext{exp}(c \, ext{sqrt}(n))$ critical points

Critical points exist for $p \, \geq 5 \log n$

Special cases for $n=3,4,5$ reveal unique behaviors

Abstract

It is widely believed that the energy functional $E_p:(\mathbb{S}^2)^n \rightarrow \mathbb{R}$ $$ E_p = \sum_{i,j=1 \atop i \neq j}^{n} \frac{1}{\|x_i-x_j\|^p}$$ has a number of critical points, $\nabla E(x) = 0$, that grows exponentially in $n$. Despite having been extensively tested and being physically well motivated, no rigorous result in this direction exists. We prove a version of this result on the two-dimensional flat torus $\mathbb{T}^2$ and show that there are infinitely many $n \in \mathbb{N}$ such that the number of critical points of $E_p: (\mathbb{T}^2)^n \rightarrow \mathbb{R}$ is at least $\exp(c \sqrt{n})$ provided $p \geq 5 \log{n}$. We also investigate the special cases $n=3,4,5$ which turn out to be surprisingly interesting.