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

François Clément, Stefan Steinerberger

TL;DR

This work analyzes the critical points of the Riesz $p$-energy on the flat torus \(\mathbb{T}^2\) for \(n\) points. It establishes a rigorous exponential-in-sqrt\(n\) lower bound on the number of geometrically distinct critical points for an infinite sequence of \(n\) and sufficiently large \(p>5\log n\), via a constructive gradient-flow framework that recovers initial configurations from critical points. The paper also delivers detailed results for small \(n\) (3,4,5), showing multiple phase transitions and, in the large-\(p\) limit, convergence to optimal disk packings (notably the Fibonacci set for \(n=5\)). A key contribution is the combination of a combinatorial initialization, a restricted gradient flow, and a reconstruction argument to bound critical points from below, together with stability lemmas for the 5-point Fibonacci configuration that confirm its energy-optimality for large \(p\). Collectively, the results illuminate the rich energy landscape of Riesz energies on the torus and reveal intricate phase-transition phenomena and packing-related structure.

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.

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

TL;DR

This work analyzes the critical points of the Riesz -energy on the flat torus for points. It establishes a rigorous exponential-in-sqrt lower bound on the number of geometrically distinct critical points for an infinite sequence of and sufficiently large , via a constructive gradient-flow framework that recovers initial configurations from critical points. The paper also delivers detailed results for small (3,4,5), showing multiple phase transitions and, in the large- limit, convergence to optimal disk packings (notably the Fibonacci set for ). A key contribution is the combination of a combinatorial initialization, a restricted gradient flow, and a reconstruction argument to bound critical points from below, together with stability lemmas for the 5-point Fibonacci configuration that confirm its energy-optimality for large . Collectively, the results illuminate the rich energy landscape of Riesz energies on the torus and reveal intricate phase-transition phenomena and packing-related structure.

Abstract

It is widely believed that the energy functional has a number of critical points, , that grows exponentially in . 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 and show that there are infinitely many such that the number of critical points of is at least provided . We also investigate the special cases which turn out to be surprisingly interesting.
Paper Structure (19 sections, 6 theorems, 81 equations, 15 figures)

This paper contains 19 sections, 6 theorems, 81 equations, 15 figures.

Table of Contents

  1. Introduction and Results
  2. The problem
  3. Critical Points on the Torus
  4. A surprise for small $p$.
  5. Related results
  6. Small number of points and special configurations
  7. The case $n=3$.
  8. The case $n=4$.
  9. The case $n=5$.
  10. Larger values of $n$.
  11. Proof of Theorem 1
  12. Proof of Theorem 2 and Theorem 3
  13. Proof of Theorem 3, $n=4$
  14. Proof of Theorem 2, $n=3$
  15. Proof of Theorem 4
  16. ...and 4 more sections

Key Result

Theorem 1

There exist constants $c_1, c_2 > 0$ and an infinite set of integers $A \subset \mathbb{N}$ such that if $n \in A$ and $p > 5 \log{n}$, then the function $E: (\mathbb{T}^2)^n \rightarrow \mathbb{R}$ has at least $c_1 \exp(c_2 \sqrt{n})$ critical points.

Figures (15)

  • Figure 1: Critical points found via gradient descent starting with 100 random points. Left: $p=4$. Middle: the case $p=4$ with circles drawn around each point. Right: logarithmic energy $p=0$.
  • Figure 2: Critical points for the logarithmic energy obtained via gradient descent from three sets of 100 random initial points.
  • Figure 3: Three interesting sets of three points.
  • Figure 4: Conjectured minimal energy configuration.
  • Figure 5: A one-parameter family, $0 \leq \alpha \leq \pi/12$.
  • ...and 10 more figures

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof
  • proof
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 2 more