Sparse regular random graphs: spectral density and eigenvectors

Details Sparse regular random graphs: spectral density and eigenvectors Sparse regular random graphs: spectral density and eigenvectors

2009-10-28

arxiv.org/abs/0910.5306v4

Mathematics

Probability

Final version; to appear in The Annals of Probability

Scientific paper

We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of vertices, the empirical spectral distribution converges to the semicircular law. Moreover, we prove concentration estimates on the number of eigenvalues over progressively smaller intervals. We also show that, with high probability, all the eigenvectors are delocalized.

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