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Sparse regular random graphs: spectral density and eigenvectors

Connections for Women: An Introduction to Random Matrices September 20, 2010 - September 21, 2010

September 21, 2010 (09:30 AM PDT - 10:30 AM PDT)
Speaker(s): Ioana Dumitriu
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC

Abstract 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.
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