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Polarization of lattices: Stable cold spots and spherical designs

Christine Bachoc, Philippe Moustrou, Frank Vallentin, Marc Christian Zimmermann

TL;DR

This work studies the minization of inhomogeneous Gaussian lattice sums $p(f,L,z)=\sum_{x\in L} e^{-\alpha\|x-z\|^2}$ and introduces stable cold spots—minimizers that persist for all $\alpha\ge α_0$—in relation to lattice deep holes.A two-pronged approach combines a far-field bound (Betermin-Petrache) with a near-field bound via linear programming bounds for spherical designs to prove stability when every inhomogeneous shell around a deep hole is a spherical 2-design and the deep hole is highly symmetric.The main results establish stability for root lattices and several notable lattices (including duals $E_6^*,E_7^*$, $K_{12}$, $BW_{16}$) but show the Leech lattice does not possess stable cold spots; this connects design theory, affine Weyl groups, and Delaunay tilings to max-min polarization.Overall, the paper provides a framework to identify stable cold spots by leveraging spherical-design structures around deep holes and offers explicit threshold values $α_0$ in key lattices, contributing to the understanding of universal optimality in polarization problems.

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 $α$, the goal is to find the minimizers of $\sum_{x \in L} e^{-α\|x - z\|^2}$ over all $z \in \mathbb{R}^n$. By a result of Bétermin and Petrache from 2017 it is known that for steep potential energy functions - when $α$ 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 $α\geq α_0$ for some finite $α_0$; 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.

Polarization of lattices: Stable cold spots and spherical designs

TL;DR

This work studies the minization of inhomogeneous Gaussian lattice sums $p(f,L,z)=\sum_{x\in L} e^{-\alpha\|x-z\|^2}$ and introduces stable cold spots—minimizers that persist for all $\alpha\ge α_0$—in relation to lattice deep holes.A two-pronged approach combines a far-field bound (Betermin-Petrache) with a near-field bound via linear programming bounds for spherical designs to prove stability when every inhomogeneous shell around a deep hole is a spherical 2-design and the deep hole is highly symmetric.The main results establish stability for root lattices and several notable lattices (including duals $E_6^*,E_7^*$, $K_{12}$, $BW_{16}$) but show the Leech lattice does not possess stable cold spots; this connects design theory, affine Weyl groups, and Delaunay tilings to max-min polarization.Overall, the paper provides a framework to identify stable cold spots by leveraging spherical-design structures around deep holes and offers explicit threshold values $α_0$ in key lattices, contributing to the understanding of universal optimality in polarization problems.

Abstract

We consider the problem of finding the minimum of inhomogeneous Gaussian lattice sums: Given a lattice and a positive constant , the goal is to find the minimizers of over all . By a result of Bétermin and Petrache from 2017 it is known that for steep potential energy functions - when 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 for some finite ; 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.

Paper Structure

This paper contains 26 sections, 22 theorems, 127 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Max-min polarization for compact manifolds
  3. Max-min polarization for lattices
  4. Deep holes and cold spots
  5. First observations about lattices with "highly symmetric" deep holes
  6. Summary of the main results
  7. Open questions
  8. Stable cold spots and spherical $2$-designs: Proof of Theorem \ref{['thm:main']}
  9. Bounding inhomogeneous Gaussian lattice sums: Far away from deep holes
  10. Bounding inhomogeneous Gaussian lattice sums: Using the linear programming bound for spherical designs around deep holes
  11. Putting the two bounds together, finishing the proof of Theorem \ref{['thm:main']}
  12. Cold spots of root lattices
  13. Decomposable lattices
  14. Designs on inhomogeneous shells of indecomposable root latttices
  15. The full stabilizer group for special points and indecomposable root lattices
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $L \subseteq \mathbb{R}^n$ be an $n$-dimensional lattice and let $f_\alpha(r) = e^{-\alpha r}$ be a Gaussian potential function. For a point $z$ the following statements are equivalent: We will call a point $z$ satisfying the equivalent conditions above a stable critical point.

Figures (5)

  • Figure 1: The Delaunay decomposition of a lattice and the candidates for stable cold spots.
  • Figure 2: Deep holes are not stable cold spots for generic lattices.
  • Figure 3: Our strategy is to bound the inhomogeneous Gaussian lattice sum close to the deep hole $c$ using the linear programming bound for every inhomogeneous shell $L(c,r)$ and for the points on every concentric sphere $c + \rho y$, with $y \in S^{n-1}$, around the deep hole $c$.
  • Figure 4: Covering $V(L)$ by balls of two kinds.
  • Figure 5: Covering a simplex with two spheres in dimension 2.

Theorems & Definitions (38)

  • Theorem 1.1
  • proof
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof : Proof of Lemma \ref{['thm:ineqBP']}
  • Corollary 2.4
  • ...and 28 more