Sparse regular random graphs: spectral density and eigenvectors
Ioana Dumitriu (University of California, San Diego)
Adjacency matrices of regular random graphs are a good example of non-Wigner ensembles for which the semicircle law still holds, in various regimes. The one we focus on is when the degree is polylogarithmic in the number of vertices (a "sparse" case). We show that the empirical spectral distribution converges to the semicircle law, estimate the rate of convergence (also known as the "local semicircle law"), and show some results that point toward the delocalization and lack of bias for the second through last eigenvectors.