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

Abstract

It is widely believed that the energy functional $E_p:({\mathbb{S}}^2{)}^n\to \mathbb{R}$ $$E_p=\sum _{\genfrac{}{}{0ex}{}{i,j=1}{i\ne j}}^n\frac{1}{\Vert x_i-x_j{\Vert }^p}$$ 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 $E_p:({\mathbb{T}}^2{)}^n\to \mathbb{R}$ is at least $\exp (c\sqrt{n})$ provided $p\ge 5\log n$. We also investigate the special cases $n=3,4,5$ which turn out to be surprisingly interesting.