The Annals of Probability

Sparse regular random graphs: Spectral density and eigenvectors

Ioana Dumitriu and Soumik Pal

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We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse 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 semicircle 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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Ann. Probab., Volume 40, Number 5 (2012), 2197-2235.

First available in Project Euclid: 8 October 2012

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Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)
Secondary: 60C05: Combinatorial probability 05C80: Random graphs [See also 60B20]

Random regular graphs spectral distribution universality semicircle law


Dumitriu, Ioana; Pal, Soumik. Sparse regular random graphs: Spectral density and eigenvectors. Ann. Probab. 40 (2012), no. 5, 2197--2235. doi:10.1214/11-AOP673.

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