Moment methods in energy minimization: New bounds for Riesz minimal energy problems

TL;DR

This paper introduces a hierarchy of moment-based optimization methods to derive new bounds for Riesz minimal energy problems, demonstrating high-precision results and potential universality in specific particle configurations.

Contribution

It develops harmonic analysis and sum-of-squares techniques to approximate infinite-dimensional problems with semidefinite programs, providing the first 4-point bounds in discrete geometry.

Findings

Numerically sharp bounds for 5-particle Riesz energy problems on the sphere.

The second hierarchy step may be universally sharp for certain configurations.

First computation of a 4-point bound in discrete geometry.

Abstract

We use moment techniques to construct a converging hierarchy of optimization problems to lower bound the ground state energy of interacting particle systems. We approximate (from below) the infinite dimensional optimization problems in this hierarchy by block diagonal semidefinite programs. For this we develop the necessary harmonic analysis for spaces consisting of subsets of another space, and we develop symmetric sum-of-squares techniques. We numerically compute the second step of our hierarchy for Riesz s-energy problems with five particles on the 2-dimensional unit sphere, where the s=1 case is the Thomson problem. This yields new numerically sharp bounds (up to high precision) and suggests the second step of our hierarchy may be sharp throughout a phase transition and may be universally sharp for 5-particles on the unit sphere. This is the first time a 4-point bound has been computed for a problem in discrete geometry.