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arXiv · 2509.10773

Spectral structure of infinite size squared distances matrices

Alexander Plakhotnikov
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Abstract

Let a finite set of points $\{\xi_1,...,\xi_k\}$ be chosen in a metric space $(X,d)$, and let the squared distance matrix $\mathfrak{D}=(\mathfrak{D}(\xi_i,\xi_j)^2)_{i,j=1}^{k}$ 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 $\{\xi_k\}_{k\in \mathbb{Z}}$ 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.

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Alexander Plakhotnikov. 2025-09-13. Spectral structure of infinite size squared distances matrices. https://arxiv.org/abs/2509.10773

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