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227
The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices
, 2011
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Spectral Statistics of ErdősRényi Graphs I: Local Semicircle Law
, 2011
"... We consider the ensemble of adjacency matrices of ErdősRényi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so that its bulk eigenvalues are of order one. We prove that, as long as pN → ∞ (with a speed at least ..."
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Cited by 52 (18 self)
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We consider the ensemble of adjacency matrices of ErdősRényi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so that its bulk eigenvalues are of order one. We prove that, as long as pN → ∞ (with a speed at least logarithmic in N), the density of eigenvalues of the ErdősRényi ensemble is given by the Wigner semicircle law for spectral windows of length larger than N −1 (up to logarithmic corrections). As a consequence, all eigenvectors are proved to be completely delocalized in the sense that the ℓ ∞norms of the ℓ 2normalized eigenvectors are at most of order N −1/2 with a very high probability. The estimates in this paper will be used in the companion paper [13] to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that pN ≫ N 2/3.
Universality of sinekernel for Wigner matrices with a small Gaussian perturbation
, 2009
"... We consider N × N Hermitian random matrices with independent identically distributed entries (Wigner matrices). We assume that the distribution of the entries have a Gaussian component with variance N −3/4+β for some positive β> 0. We prove that the local eigenvalue statistics follows the univers ..."
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Cited by 41 (14 self)
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We consider N × N Hermitian random matrices with independent identically distributed entries (Wigner matrices). We assume that the distribution of the entries have a Gaussian component with variance N −3/4+β for some positive β> 0. We prove that the local eigenvalue statistics follows the universal Dyson sine kernel.
Limiting laws of coherence of random matrices with applications to testing covariance structure and construction of compressed sensing matrices
 Ann. Stat
, 2011
"... Testing covariance structure is of significant interest in many areas of statistical analysis and construction of compressed sensing matrices is an important problem in signal processing. Motivated by these applications, we study in this paper the limiting laws of the coherence of an n×p random matr ..."
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Cited by 30 (10 self)
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Testing covariance structure is of significant interest in many areas of statistical analysis and construction of compressed sensing matrices is an important problem in signal processing. Motivated by these applications, we study in this paper the limiting laws of the coherence of an n×p random matrix in the highdimensional setting where p can be much larger than n. Both the law of large numbers and the limiting distribution are derived. We then consider testing the bandedness of the covariance matrix of a high dimensional Gaussian distribution which includes testing for independence as a special case. The limiting laws of the coherence of the data matrix play a critical role in the construction of the test. We also apply the asymptotic results to the construction of compressed sensing matrices.
Performance of statistical tests for singlesource detection using random matrix theory
 IEEE Transactions on Information Theory
, 2011
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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 ..."
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Cited by 26 (2 self)
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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. 1.