Polarization of lattices: Stable cold spots and spherical designs

TL;DR

This paper investigates stable cold spots in lattices, showing that certain well-known lattices have these points as stable minimizers of Gaussian sums, while others like the Leech lattice do not, using spherical design bounds.

Contribution

It introduces the concept of stable cold spots in lattices and applies linear programming bounds to identify them in key lattices, revealing new stability properties.

Findings

Root, Coxeter-Todd, and Barnes-Wall lattices have stable cold spots.

The Leech lattice does not have stable cold spots.

Linear programming bounds help identify stable minimizers.

Abstract

We consider the problem of finding the minimum of inhomogeneous Gaussian lattice sums: Given a lattice $L \subseteq \mathbb{R}^n$ and a positive constant $\alpha$, the goal is to find the minimizers of $\sum_{x \in L} e^{-\alpha \|x - z\|^2}$ over all $z \in \mathbb{R}^n$. By a result of B\'etermin and Petrache from 2017 it is known that for steep potential energy functions - when $\alpha$ tends to infinity - the minimizers in the limit are found at deep holes of the lattice. In this paper, we consider minimizers which already stabilize for all $\alpha \geq \alpha_0$ for some finite $\alpha_0$; we call these minimizers stable cold spots. Generic lattices do not have stable cold spots. For several important lattices, like the root lattices, the Coxeter-Todd lattice, and the Barnes-Wall lattice, we show how to apply the linear programming bound for spherical designs to prove that the deep holes are stable cold spots. We also show, somewhat unexpectedly, that the Leech lattice does not have stable cold spots.