The Largest Eigenvalue of Rank One Deformation of Large Wigner Matrices

By Delphine F'eral, S. P'ech'e. 2006

243 citations recorded by the source corpus.

The purpose of this paper is to establish universality of the fluctuations of the largest eigenvalue for some non-necessarily Gaussian complex Deformed Wigner Ensembles. The real model is also considered. Our approach is close to the one used by A. Soshnikov (cf. [11]) in the investigations of classical real or complex Wigner Ensembles. It is based on the computation of moments of traces of high powers of the random matrices under consideration.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/229294caba130ae69c3f6b9af3f4192bb1aece30

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Statistics and Probability

Citations in the corpus

Universality results for largest eigenvalues of some sample covariance matrix ensembles (2007, background): “Indeed, one shall look among the oriented edges already encountered in the path and of which one endpoint is the vertex occuring at the instant t−1 (see the Appendix in [28] and [9] Section 5.”
Implicit Self-Regularization in Deep Neural Networks: Evidence from Random Matrix Theory and Implications for Learning (2018, methodology): “When modeling DNN training in terms of RMT and MP theory, the transition from Randomlike to Bleeding-out corresponds to the so-called BPP phase transition [9, 23, 45, 20].”
On the Randomized Complexity of Minimizing a Convex Quadratic Function (2018, background): “2, we remark that the spectrum of M is well studied in random matrix theory [22, 14, 5, 10].”
The singular values and vectors of low rank perturbations of large rectangular random matrices (2011, result): “At a certain level, our proof also present analogies with the ones of other papers devoted to other occurrences of the BBP phase transition, such as [47, 27, 21, 22, 23].”
Financial Applications of Random Matrix Theory: a short review (2009, unknown): “Johnstone, Approximate Null Distribution of the Largest Root in Multivariate Analysis, Submitted to the Annals of Applied Statistics [54] D.”
Spectral statistics of Erdős–Rényi graphs I: Local semicircle law (2011, background): “Note that in [26] it was for proved, in the case where H is a Hermitian Wigner matrix, that if f 6 1 no such gap exists.”
Outliers in the spectrum of large deformed unitarily invariant models (2012, methodology): “See also [29, 4, 5] for sample covariance matrices, [24, 32, 22, 19, 33] for deformed Wigner models and [31] for information-plus-noise models.”
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).”
The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices (2009, background): “Additive perturbation of a Gaussian Wigner matrix.”
Outliers in the spectrum of iid matrices with bounded rank perturbations (2010, methodology): “ribution of the outlier eigenvalues; the methods used are somewhat dierent, but the techniques developed here can be adapted to the Wigner case (Alexander Soshnikov, private communication). See also [21], [13] for further results in the Wigner case, whose methods are close to those used here, [35] f”
On sample eigenvalues in a generalized spiked population model (2008, background): “Keywords and phrases: Sample covariance matrices, Spiked population model, Central limit theorems, Largest eigenvalue, Extreme eigenvalues.”
Fundamental limits of detection in the spiked Wigner model (2018, background): “Spiked models, which are distributions over matrices of the form “signal + noise,” have been a mainstay in the statistical literature since their introduction by Johnstone (2001) as models for the study of high-dimensional principal component analysis. Spectral properties of these models have been e”
Large deviations principle for the largest eigenvalue of Wigner matrices without Gaussian tails (2015, background): “It is well-known that finite rank perturbations of Wigner matrices show a threshold phenomenon with respect to the strength of the perturbation (see for example [21], [13] [23], [7], [16] for further details), which the rate function seems to reflect through the discontinuity at 2.”
The largest eigenvalues of finite rank deformation of large Wigner matrices: Convergence and nonuniversality of the fluctuations. (2007, methodology): “gner matrix WN which implies the nonuniversality of the ﬂuctuations of the largest eigenvalue of rank-1 diagonal deformation of symmetric or Hermitian Wigner matrices (as conjectured in Remark 1.7 of [16]). The latter also shows that in the non-Gaussian setting, the ﬂuctuations of the largest eigenv”
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”