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The largest eigenvalues of finite rank deformation of large Wigner matrices: Convergence and nonuniversality of the fluctuations. (English) Zbl 1163.15026
The asymptotic behavior of the first extremal eigenvalues of some complex or real deformed Wigner matrices \((M_N)_N\) is investigated. Such matrix models can be seen as additive analogue of the spiked population models and are defined by a sequence \((M_N)_N\) given by \(M_N = W_N/ \sqrt{N} + A_N\), where \(W_N\) is a Wigner matrix such that the common distribution of its entries satisfies some conditions (random variables with a symmetric distribution satisfying a Poincaré inequality) and \(A_N\) is a deterministic matrix of finite rank. In this way, an analogue of the main result of J. Baik and J. W. Silverstein [J. Multivariate Anal. 97, 1382–1408 (2006; Zbl 1220.15011)] is established, namely that, once \(A_N\) has exactly \(k\) (fixed) eigenvalues far enough from zero, the \(k\) first eigenvalues of \(M_N\) jump almost surely outside the limiting semicircle support. This result is universal since the corresponding limits only involve the variance of the entries of \(W_N\).
On the other hand, at the level of the fluctuations, a striking phenomenon in the particular case where \(A_N\) is diagonal with a sole simple nonzero eigenvalue large enough is exhibited. In this case, the fluctuations of the largest eigenvalue of \(M_N\) are not universal and strongly depend on the particular law of the entries of \(W_N\). It is proved that the limiting distribution of the (properly rescaled) largest value of \(M_N\) is the convolution of the distribution of the entries of \(W_N\) with a Gaussian law. In particular, if the entries of \(W_N\) are not Gaussian, the fluctuations of the largest eigenvalue of \(M_N\) are not Gaussian.

MSC:
15B52 Random matrices (algebraic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
60F05 Central limit and other weak theorems
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