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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2512.23018 |
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| _version_ | 1866911342717304832 |
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| author | Clément, François Steinerberger, Stefan |
| author_facet | Clément, François Steinerberger, Stefan |
| contents | It is widely believed that the energy functional $E_p:(\mathbb{S}^2)^n \rightarrow \mathbb{R}$ $$ E_p = \sum_{i,j=1 \atop i \neq j}^{n} \frac{1}{\|x_i-x_j\|^p}$$ has a number of critical points, $\nabla E(x) = 0$, that grows exponentially in $n$. Despite having been extensively tested and being physically well motivated, no rigorous result in this direction exists. We prove a version of this result on the two-dimensional flat torus $\mathbb{T}^2$ and show that there are infinitely many $n \in \mathbb{N}$ such that the number of critical points of $E_p: (\mathbb{T}^2)^n \rightarrow \mathbb{R}$ is at least $\exp(c \sqrt{n})$ provided $p \geq 5 \log{n}$. We also investigate the special cases $n=3,4,5$ which turn out to be surprisingly interesting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23018 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Many critical points for discrete Riesz energy on $\mathbb{T}^2$ Clément, François Steinerberger, Stefan Classical Analysis and ODEs It is widely believed that the energy functional $E_p:(\mathbb{S}^2)^n \rightarrow \mathbb{R}$ $$ E_p = \sum_{i,j=1 \atop i \neq j}^{n} \frac{1}{\|x_i-x_j\|^p}$$ has a number of critical points, $\nabla E(x) = 0$, that grows exponentially in $n$. Despite having been extensively tested and being physically well motivated, no rigorous result in this direction exists. We prove a version of this result on the two-dimensional flat torus $\mathbb{T}^2$ and show that there are infinitely many $n \in \mathbb{N}$ such that the number of critical points of $E_p: (\mathbb{T}^2)^n \rightarrow \mathbb{R}$ is at least $\exp(c \sqrt{n})$ provided $p \geq 5 \log{n}$. We also investigate the special cases $n=3,4,5$ which turn out to be surprisingly interesting. |
| title | Many critical points for discrete Riesz energy on $\mathbb{T}^2$ |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2512.23018 |