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September 2012 Sparse regular random graphs: Spectral density and eigenvectors
Ioana Dumitriu, Soumik Pal
Ann. Probab. 40(5): 2197-2235 (September 2012). DOI: 10.1214/11-AOP673

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.

Citation

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Ioana Dumitriu. Soumik Pal. "Sparse regular random graphs: Spectral density and eigenvectors." Ann. Probab. 40 (5) 2197 - 2235, September 2012. https://doi.org/10.1214/11-AOP673

Information

Published: September 2012
First available in Project Euclid: 8 October 2012

zbMATH: 1255.05173
MathSciNet: MR3025715
Digital Object Identifier: 10.1214/11-AOP673

Subjects:
Primary: 60B20
Secondary: 05C80 , 60C05

Keywords: Random regular graphs , semicircle law , Spectral distribution , Universality

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.40 • No. 5 • September 2012
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