Spectral structure of infinite size squared distances matrices

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