Polarization of lattices: Stable cold spots and spherical designs
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866914347818680320 |
|---|---|
| author | Bachoc, Christine Moustrou, Philippe Vallentin, Frank Zimmermann, Marc Christian |
| author_facet | Bachoc, Christine Moustrou, Philippe Vallentin, Frank Zimmermann, Marc Christian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_08819 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Polarization of lattices: Stable cold spots and spherical designs Bachoc, Christine Moustrou, Philippe Vallentin, Frank Zimmermann, Marc Christian Metric Geometry 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. |
| title | Polarization of lattices: Stable cold spots and spherical designs |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2502.08819 |