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An Isaac Newton Institute Workshop

Quantum Graphs, their Spectra and Applications

Some properties of distance matrices

Author: Oriol Bohigas (LPTMS, Universite Paris Sud)

Abstract

Distance matrices are matrices whose entries are the relative distances between points located on a certain manifold. One central problem consists in isometric embedding, namely to find the conditions that a distance matrix must fulfill in order that one can find points in the euclidean space such that the euclidean distance between each pair of points coincide with the given distance matrix. One can investigate the spectral properties of distance matrices when the points are uncorrelated and uniformly distributed on (hyper)cubes and (hyper)spheres. The spectrum exhibits characteristic features, in particular all eigenvalues except one are non-positive and delocalized and strongly localized eigenstates are present.