Universally Optimal Periodic Configurations in the Plane

TL;DR

This work develops a unified linear programming framework for lattice-periodic energy in the plane, recasting energy bounds as polynomial interpolation problems via lattice symmetry and lattice theta functions. By constructing CPSD-preserving interpolants with carefully chosen nodes, the authors prove universal optimality for two nontrivial finite configurations: $\omega^*_4$ is $A_2$-universally optimal and $\omega^*_6$ is $L$-universally optimal. The approach hinges on polynomial bases $P^L_v$ and $P^{A_2}_v$, the concept of magic interpolants, and rigorous control of the interpolation region through detailed derivative and convexity arguments, including extensive computer-assisted verifications. The results advance the program of proving universal optimality in two dimensions by providing concrete, self-contained LP bounds for four related families derived from the hexagonal lattice and its sublattices, and they illuminate the connection between finite-point optimality and infinite configurations via energy densities and sphere-packing limits.

Abstract

We develop lower bounds for the energy of configurations in $\mathbb{R}^d$ periodic with respect to a lattice. In certain cases, the construction of sharp bounds can be formulated as a finite dimensional, multivariate polynomial interpolation problem. We use this framework to show a scaling of the equitriangular lattice $A_2$ is universally optimal among all configurations of the form $ω_4+ A_2$ where $ω_4$ is a 4-point configuration in $\mathbb{R}^2$. Likewise, we show a scaling and rotation of $A_2$ is universally optimal among all configurations of the form $ω_6+L$ where $ω_6$ is a 6-point configuration in $\mathbb{R}^2$ and $L=\mathbb{Z} \times \sqrt{3} \mathbb{Z}$.

Figures (13)

• Figure 1: The 4-point $A_2$-universally optimal configuration $\omega^*_4$.
• Figure 2: The 6-point $L$-universally optimal configuration $\omega^*_6$.
• Figure 3: The region $\tilde{\Delta}_{A_2}$, pictured above, is our region of interpolation for the families $\omega^*_{m^2}$ and $\omega^*_{3m^2}$ (see sec. 3.4).
• Figure 4: $\tilde{g}_a$ must stay below $\tilde{F}_a$ on $\tilde{\Delta}_{A_2}$ with equality at the corner point $(-1,1)$.
• Figure 5: $\omega^*_{2}$, pictured above is $L$-universally optimal, and analogous results hold for any rectangular lattice.

Theorems & Definitions (87)

• Definition 1.1
• Theorem 1
• Proposition 2
• Definition 2.1
• Definition 2.2
• Proposition 3
• Proposition 4
• proof
• Proposition 5
• Proposition 6