Fluctuations of the Extreme Eigenvalues of Finite Rank Deformations of Random Matrices

By Florent Benaych-Georges, Alice Guionnet, M. Maida. 2010

161 citations recorded by the source corpus.

Consider a deterministic self-adjoint matrix $X_n$ with spectral measure converging to a compactly supported probability measure, the largest and smallest eigenvalues converging to the edges of the limiting measure. We perturb this matrix by adding a random finite rank matrix with delocalised eigenvectors and study the extreme eigenvalues of the deformed model. We give necessary conditions on the deterministic matrix $X_n$ so that the eigenvalues converging out of the bulk exhibit Gaussian fluctuations, whereas the eigenvalues sticking to the edges are very close to the eigenvalues of the non-perturbed model and fluctuate in the same scale. We generalize these results to the case when $X_n$ is random and get similar behavior when we deform some classical models such as Wigner or Wishart matrices with rather general entries or the so-called matrix models.

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

View Paper: https://www.semanticscholar.org/paper/fa74922f5437f36a34b4193bc54c4e8b8df259b8

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Random matrix theory in statistics: A review (2014, result): “…the behavior of eigenvalues 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,…”
On the principal components of sample covariance matrices (2014, background): “Previously, eigenvalue sticking was established for a certain class of random perturbations of Wigner matrices in [6,7].”
On sample eigenvalues in a generalized spiked population model (2008, methodology): “These authors consider more general perturbation models including additive and multiplicative ones and there provide important results on point-wisely convergence of extreme eigenvalues [7] as well as on their fluctuations [8].”
Large deviations principle for the largest eigenvalue of Wigner matrices without Gaussian tails (2015, methodology): “We will use here the same approach as in many papers on finite rank deformations of Wigner matrices (see for example [7] or [16]) to determine the behavior of the extreme eigenvalues outside the bulk of a perturbed Wigner matrix.”
Singular vector and singular subspace distribution for the matrix denoising model (2018, background): “The fluctuation of the extreme eigenvalues of various models have been considered in [2, 3, 7, 8, 9, 23, 24, 28, 14, 15, 42, 55, 34, 56, 59, 44].”
Identifying the number of factors from singular values of a large sample auto-covariance matrix (2014, methodology): “…we will use is that according to such recent results on ﬁnite rank perturbations of symmetric random matrices, see e.g. Benaych-Georges et al. [8] it is very likely that the asymptotic distribution of is the same as that of where ν 1 , ν 2 are the two largest eigenvalues of the base noise…”
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].”
Statistical inference for principal components of spiked covariance matrices (2020, methodology): “On extreme eigenvalues, further study for more generally distributed covariance matrices can be found in [8, 22, 12, 69, 5, 6, 16, 27, 52].”
The Isotropic Semicircle Law and Deformation of Wigner Matrices (2011, methodology): “We use the following representation of the eigenvalues of z H , which was already used in several papers on finite-rank deformations of random matrices [5, 6, 7, 32].”
The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices (2009, unknown): “The underlying methods can and have been adapted to study the extreme singular values and singular vectors of deformations of rectangular random matrices, as well as the fluctuations [11] and the large deviations [12] of our model.”