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.
Categories: math.MG