The Largest Eigenvalue of Rank One Deformation of Large Wigner Matrices

Abstract

The purpose of this paper is to establish universality of the fluctuations of the largest eigenvalue of 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 (c.f. [12]) 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.

An Introduction to Random Matrices

“…Similar techniques apply to the study of the largest eigenvalue of so called spiked models, which are matrices of the form XT X * with X possessing i.i.d. complex entries and T a diagonal real matrix, all of whose entries except for a finite number equal to 1, and to small rank perturbations of Wigner matrices, see [BaBP05], [Péc06], [FeP07], [Kar07b] and [Ona08].…”

Random matrix theory in statistics: A review

“…There have been several publications on the closely related problem of characterizing the behavior of eigenvalues for deformed Wigner ensembles with the associated phase transition behavior exhibited by the extreme eigenvalues (see Capitaine et al, 2009;Féral and Péché, 2007;Pizzo et al, 2013;Renfrew and Soshnikov, 2013;Yin, 2011, 2012 and the references therein). Chapon et al (2012) dealt with a low rank deformed signal-plusnoise covariance matrix.…”

Universality results for the largest eigenvalues of some sample covariance matrix ensembles

“…To consider such vertices, we need to introduce other characteristics of the path. Let ν N (P ) be the maximal number of vertices that can be visited at marked instants from a given vertex of the path P. Let also T N (P ) be the maximal type of a vertex in P. Then, if at the instant t, one reads for the second time an oriented up edge e, there are at most 2(ν N (P ) + T N (P )) choices for the vertex occuring at the instant t. 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 5.1.2 e.g.). It is an easy fact that paths for which T N (P ) ≥ AN 1/3 (ln N ) −1 lead to a negligible contribution, if A is large enough (independently of k).…”

“…Let also a o > 0 (small) be given. It can easily be inferred from [28], p. 11 (see also [29] and [9], Lemma 7.10) that there exist positive constants a 1 , a 2 , independent of l and Q such that, if r ′ ≥ a o √ l, one has that…”