## The Annals of Probability

### Sparse regular random graphs: Spectral density and eigenvectors

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 40, Number 5 (2012), 2197-2235.

Dates
First available in Project Euclid: 8 October 2012

https://projecteuclid.org/euclid.aop/1349703320

Digital Object Identifier
doi:10.1214/11-AOP673

Mathematical Reviews number (MathSciNet)
MR3025715

Zentralblatt MATH identifier
1255.05173

#### Citation

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. https://projecteuclid.org/euclid.aop/1349703320

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