Spectral structure of infinite size squared distances matrices
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915505903763456 |
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| author | Plakhotnikov, Alexander |
| author_facet | Plakhotnikov, Alexander |
| contents | Let a finite set of points $\{ξ_1,...,ξ_k\}$ be chosen in a metric space $(X,d)$, and let the squared distance matrix $\mathfrak{D}=(\mathfrak{D}(ξ_i,ξ_j)^2)_{i,j=1}^{k}$ be constructed from them. We propose a geometric approach to studying the spectral properties of squared distance matrices of infinite size, constructed from a countable set of points $\{ξ_k\}_{k\in \mathbb{Z}}$ on Riemannian manifold $(M,g)$. We move from the discrete problem to a continuous one using walk matrices. We describe the structure of the spectrum and study the properties of spectral flows. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_10773 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral structure of infinite size squared distances matrices Plakhotnikov, Alexander Metric Geometry 51F99 (Primary) 52C99, 05C35 (Secondary) G.2.0 Let a finite set of points $\{ξ_1,...,ξ_k\}$ be chosen in a metric space $(X,d)$, and let the squared distance matrix $\mathfrak{D}=(\mathfrak{D}(ξ_i,ξ_j)^2)_{i,j=1}^{k}$ be constructed from them. We propose a geometric approach to studying the spectral properties of squared distance matrices of infinite size, constructed from a countable set of points $\{ξ_k\}_{k\in \mathbb{Z}}$ on Riemannian manifold $(M,g)$. We move from the discrete problem to a continuous one using walk matrices. We describe the structure of the spectrum and study the properties of spectral flows. |
| title | Spectral structure of infinite size squared distances matrices |
| topic | Metric Geometry 51F99 (Primary) 52C99, 05C35 (Secondary) G.2.0 |
| url | https://arxiv.org/abs/2509.10773 |