Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The hexagonal lattice is universally locally optimal

Thomas Leblé

Abstract

We prove that the hexagonal lattice is a local minimizer, among all point configurations, of the interaction energy per unit volume for pair potentials that are completely monotonic functions of the square distance. This includes Gaussian interactions and power laws.

The hexagonal lattice is universally locally optimal

Abstract

We prove that the hexagonal lattice is a local minimizer, among all point configurations, of the interaction energy per unit volume for pair potentials that are completely monotonic functions of the square distance. This includes Gaussian interactions and power laws.

Paper Structure

This paper contains 63 sections, 17 theorems, 253 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The hexagonal lattice
  3. Main results
  4. Connection with the literature and the “universal optimality” conjecture
  5. Plan of the paper
  6. Preliminaries
  7. Convention for Fourier analysis on the plane
  8. Fourier transforms.
  9. Reciprocal lattice and Pontryagin dual.
  10. Poisson summation formula.
  11. Periodic functions and Fourier transforms.
  12. Reduction to periodic perturbations
  13. Properties of periodic perturbations
  14. The autocorrelation function
  15. Spectral measure(s)
  16. ...and 48 more sections

Key Result

Theorem 1

For all $\alpha > 0$, there exists $\varepsilon$ (depending on $\alpha$) such that: The value of $\varepsilon$ can be chosen uniformly for $\alpha$ in compact subsets of $(0, + \infty)$.

Figures (3)

  • Figure 1: The hexagonal lattice $\mathsf{A}_2$, its basis $(\sigma, \tau)$, and the fundamental domain $\mathcal{H}$.
  • Figure 2: A close-up of the fundamental hexagon $\mathcal{H}$, surrounded by the first “shell” of the lattice.
  • Figure 3: Without loss of generality, we assume that $\theta \in \left[0, \frac{\pi}{6}\right]$.

Theorems & Definitions (67)

  • Theorem 1
  • Theorem 2
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4: falb1969theorem
  • proof
  • ...and 57 more