The largest eigenvalues of finite rank deformation of large Wigner matrices: Convergence and nonuniversality of the fluctuations.

By M. Capitaine, Catherine Donati-Martin, Delphine F'eral. 2007

227 citations recorded by the source corpus.

We investigate the asymptotic spectrum of complex or real Deformed Wigner matrices when the entries of the Hermitian (resp., symmetric) Wigner matrix have a symmetric law satisfying a Poincare inequality. The perturbation is Hermitian (resp., symmetric) and deterministic with all but finitely many eigenvalues equal to zero. We first show that, as soon as the first largest or last smallest eigenvalues of the perturbation are sufficiently far from zero, the corresponding eigenvalues of the deformed Wigner matrix almost surely exit the limiting semicircle compact sup- port as the size becomes large. The corresponding limits are universal in the sense that they only involve the variance of the entries of the Wigner matrix. On the other hand, when the perturbation is diagonal with a sole simple nonnull eigenvalue large enough, we prove that the fluctuations of the largest eigenvalue are not universal and vary with the particular distribution of the entries of the Wigner matrix.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/29ccab8754fde866329fb687fc9475e8953093b2

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

Spectral statistics of Erdős–Rényi graphs I: Local semicircle law (2011, background): “(Note that in [3, 5, 35], the authors allow quite general finite-rank perturbations of Wigner matrices.”
On sample eigenvalues in a generalized spiked population model (2008, background): “In a series of recent papers [15, 10, 9], these authors establish several results in this vein for ensembles of formMn =Wn+n−1/2V where Wn is a standard Wigner matrix and V a small-rank matrix.”
Large deviations of the extreme eigenvalues of random deformations of matrices (2010, unknown): “1 ha l-0 05 05 50 2, v er si on 2 - 31 A ug 2 01 0”
ON BILINEAR FORMS BASED ON THE RESOLVENT OF LARGE RANDOM MATRICES (2010, result): “We turn out to the case where z ∈ C − R − En, and rely on the same argument as in Haagerup and Thorbjornsen [12] (see also [7]).”
Statistical inference for principal components of spiked covariance matrices (2020, methodology): “[24] M.”
Fundamental limits of detection in the spiked Wigner model (2018, background): “Landmark investigations in this area (Baik et al., 2005; Baik and Silverstein, 2006; P´ech´e, 2006; F´eral and P´ech´e, 2007; Capitaine et al., 2009; Bai and Yao, 2012, 2008) have established the existence of a spectral threshold above which the top eigenvalue detaches from the bulk of eigenvalues…”
Random matrix theory in statistics: A review (2014, unknown): “…for deformed Wigner ensembles with the associated phase transition behavior exhibited by the extreme eigenvalues (see Benaych-Georges et al., 2011; Capitaine et al., 2009; Féral and Péché, 2007; Pizzo et al., 2013; Renfrew and Soshnikov, 2013; Knowles and Yin, 2011, 2012 and the references…”
Free Convolution with a Semicircular Distribution and Eigenvalues of Spiked Deformations of Wigner Matrices (2010, unknown): “F´evrier ¶”
Rate-Optimal Perturbation Bounds for Singular Subspaces with Applications to High-Dimensional Statistics (2016, background): “For example, Bura and Pfeiffer (2008), Capitaine et al. (2009), Benaych-Georges and Nadakuditi (2012), Shabalin and Nobel (2013) focused on the asymptotic distributions of single singular value and vector when p1, p2 and the singular values grows proportionally.”
The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices (2009, background): “Additive perturbation of a Gaussian Wigner matrix.”
Outlier Eigenvalues for Deformed I.I.D. Random Matrices (2014, methodology): “We will show that it is true for each element in the finite sum on (p, q) defining Zi since this property is closed under addition as explained in (A4) of the Appendix of [21].”
The Isotropic Semicircle Law and Deformation of Wigner Matrices (2011, background): “observed in [11, 12, 13] that the distribution of the outliers is not universal and may depend on the geometry of the eigenvectors of A.”
Almost sure localization of the eigenvalues in a Gaussian information plus noise model - Application to the spiked models (2010, result): “[9], devoted to finite rank deformed Gaussian (or satisfying a Poincaré inequality) Wigner matrices, which was inspired by previous results of Haagerup and Thorbjornsen [17].”