Spectral Experiments+

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 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.

Keywords:

• Delaunay tessellations
• GOE
• localization
• random matrices
• random point sets
• random triangulations
• Voronoi diagrams

2000 AMS Subject Classification:

• 60B20
• 60D05
• 05C80

Acknowledgments

The author would like to thank Peter Sarnak for interesting conversations, and the participants in the http://mathematica.stackexchange.com discussion group for help with matters computational, and http://mathoverflow.com on help with pointers to the literature.

Notes

1 in fact, in this, and other, localization graphs, the L^∞ norms are average of an interval in eigenvalue space [here, the interval is of length 0.02, but in other localization graphs it is of length 0.001, since the eigenvalues are more closely spaced in the sparse cases].