Crystallization for Coulomb and Riesz Interactions as a Consequence of the Cohn-Kumar Conjecture

TL;DR

This paper demonstrates that the proven Cohn-Kumar conjecture implies the minimality of certain special lattices for Coulomb and Riesz energies, confirming their optimality in dimensions 8 and 24.

Contribution

It shows how the Cohn-Kumar conjecture's proof leads to the minimality of specific lattices for various energy models in higher dimensions.

Findings

Triangular, E8, and Leech lattices are energy minimizers in their respective dimensions.

The conjecture's proof implies minimality for Coulomb and Riesz energies.

This settles the lattices' optimality in dimensions 8 and 24.

Abstract

The Cohn-Kumar conjecture states that the triangular lattice in dimension 2, the $E_8$ lattice in dimension 8, and the Leech lattice in dimension 24 are universally minimizing in the sense that they minimize the total pair interaction energy of infinite point configurations for all completely monotone functions of the squared distance. This conjecture was recently proved by Cohn-Kumar-Miller-Radchenko-Viazovska in dimensions 8 and 24. We explain in this note how the conjecture implies the minimality of the same lattices for the Coulomb and Riesz renormalized energies as well as jellium and periodic jellium energies, hence settling the question of their minimization in dimensions 8 and 24.