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.

Published in at http://dx.doi.org/10.1214/11-AOP673 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (6)

• Local semicircle law and complete delocalization for Wigner random matrices
• Random matrices: Universality of local eigenvalue statistics up to the edge
• Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices
• On asymptotics of eigenvectors of large sample covariance matrix
• Eigenvectors of the discrete Laplacian on regular graphs - a statistical approach
• Universality for generalized Wigner matrices with Bernoulli distribution

Cited by in corpus (22)

• Spectral statistics of Erdős-Rényi graphs I: Local semicircle law
• Quantum ergodicity on large regular graphs
• Universality and the circular law for sparse random matrices
• Local Kesten--McKay law for random regular graphs
• Local semicircle law for random regular graphs
• Bulk eigenvalue statistics for random regular graphs
• Finite size correction to the spectrum of regular random graphs: an analytical solution
• Size biased couplings and the spectral gap for random regular graphs
• Assortative and disassortative mixing investigated using the spectra of graphs
• Convergence of graphs with intermediate density
• Top Eigenpair Statistics for Weighted Sparse Graphs
• Spectral properties of the trap model on sparse networks
• Cycles and eigenvalues of sequentially growing random regular graphs
• Moments of the inverse participation ratio for the Laplacian on finite regular graphs
• The circular law for random regular digraphs with random edge weights
• Synchronization of Network-Coupled Oscillators with Uncertain Dynamics
• Spectral partitioning in equitable graphs
• The Marčenko-Pastur law for sparse random bipartite biregular graphs
• On quantum percolation in finite regular graphs
• Global eigenvalue fluctuations of random biregular bipartite graphs
• Spectral density of random graphs: convergence properties and application in model fitting
• Localization properties of the sparse Barrat-Mézard trap model