The Isotropic Semicircle Law and Deformation of Wigner Matrices

By Antti Knowles, Jun Yin. 2011

192 citations recorded by the source corpus.

We analyze the spectrum of additive finite‐rank deformations of N × N Wigner matrices H. The spectrum of the deformed matrix undergoes a transition, associated with the creation or annihilation of an outlier, when an eigenvalue di of the deformation crosses a critical value ± 1. This transition happens on the scale |di|−1∼N−1/3 . We allow the eigenvalues di of the deformation to depend on N under the condition ‖di| −1 |≥(log⁡ N)C log⁡ log⁡ N N−1/3 . We make no assumptions on the eigenvectors of the deformation. In the limit N → ∞, we identify the law of the outliers and prove that the nonoutliers close to the spectral edge have a universal distribution coinciding with that of the extremal eigenvalues of a Gaussian matrix ensemble.

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

View Paper: https://www.semanticscholar.org/paper/3d33b2a9d1fbcc8dbb836a81ed3731d10dd63c86

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Citations in the corpus

Singular vector and singular subspace distribution for the matrix denoising model (2018, methodology): “In [42], the authors obtain the non-universality for the limiting distributions of the outliers (outlying eigenvalues) of the deformed Wigner matrices.”
Spiked separable covariance matrices and principal components (2019, result): “We point out that our results are indeed in the same spirit as the ones for deformed Wigner matrix [29], deformed rectangular matrix [6, 12] and spiked covariance matrices [8, 11, 45].”
Outlier Eigenvalues for Deformed I.I.D. Random Matrices (2014, unknown): “Note that motivated by the seminal article of Baik, Ben Arous and Péché [6], previous works are devoted to the study of the outlier eigenvalues of deformed random Hermitian matrix models, see notably [4, 7, 8, 9, 10, 19, 21, 22, 23, 28, 34, 35, 36, 37, 41, 43, 45, 46].”
A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices (2016, unknown): “1 of [22], if μ / ∈ σ((DX)DX), then μ is an eigenvalue of Qγ := (X + γE)DD(X + γE) if and only if det(V G(μ)V + (γPD) ) = 0, (4.”
Edge universality of correlation matrices (2011, methodology): “The Green function comparison method has subsequently been applied to proving the spectral universality of adjacency matrices of random graphs [3, 4], the universality of eigenvectors of Wigner matrices [20], as well as the the spectrum of additive finite-rank deformations of Wigner matrices and the”
A Dynamical Approach to Random Matrix Theory (2017, methodology): “Isotropic law was first proven in [88] for Wigner matrices under a vanishing third moment condition.”
Random matrices: Sharp concentration of eigenvalues (2012, methodology): “In particular, we may now assume that T ≤ n O (1) and 9 Very recently [25], a similar application of the Lindeberg exchange strategy to a high moment of a spectral statistic was used to establish some related concentration results.”
Random matrix theory in statistics: A review (2014, background): “…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 therein).”
Statistical inference for principal components of spiked covariance matrices (2020, methodology): “The non-universality phenomenon in the supercritical regime has been previously observed in [24, 46, 47] for the extreme eigenvalues of the finite-rank deformation of Wigner matrices.”
Universality of covariance matrices (2011, methodology): “For proving bulk universality of eigenvalues, we follow the general approach for the universality of Gaussian divisible ensembles [17, 18, 10, 11, 25, 26] by embedding the covariance matrix into a stochastic flow of matrices and so that the eigenvalues evolve according to a distinguished coupled sys”
Eigenvector Statistics of Sparse Random Matrices (2016, background): “otropic local law for the Green’s function of Ht, which is a consequence of the small Gaussian component. The isotropic local law was ﬁrst proved for Wigner matrices and sample covariance matrices in [5,26]. Our result provides a dynamical version of the entry-wise local law, with general initial ma”
Delocalization of eigenvectors of random matrices with independent entries (2013, background): “One can establish the isotropic delocalization in the sense of [2, 23].”