Spectral Experiments+

TL;DR

This paper presents extensive computational experiments on spectral properties of various random geometric and graph models, revealing new phenomena in eigenvalue distributions, eigenvector localities, and nodal domain statistics.

Contribution

It provides the first comprehensive computational analysis of spectral features across diverse random structures, uncovering novel phenomena without proposing specific conjectures.

Findings

Discovery of new phenomena in eigenvalue distributions

Identification of unique eigenvector locality patterns

Novel statistics of nodal domains in random graphs

Abstract

We describe extensive computational experiments on spectral properties of random objects - random cubic graphs, random planar triangulations, and Voronoi and Delaunay diagrams of random (uniformly distributed) point sets on the sphere). We look at bulk eigenvalue distribution, eigenvalue spacings, and locality properties of eigenvectors. We also look at the statistics of \emph{nodal domains} of eigenvectors on these graphs. In all cases we discover completely new (at least to this author) phenomena. The author has tried to refrain from making specific conjectures, inviting the reader, instead, to meditate on the data.