Non universality of fluctuations of outlier eigenvectors for block diagonal deformations of Wigner matrices

2018/07/20 by Mireille Capitaine, Capitaine, Mireille, Catherine Donati-Martin +1 · 1 citation
Agricultural and Biological Sciences · Computer Science · Mathematics · #15A18 #60F05 #Advanced Scientific Research Methods #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematical Inequalities and Applications #Matrix Theory and Algorithms #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.1807.07773

openalex publication_date 2018/07/20 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

In this paper, we investigate the fluctuations of a unit eigenvector associated to an outlier in the spectrum of a spiked N× N complex Deformed Wigner matrix M_N: M_N =W_N/√(N) + A_N where W_N is an N × N Hermitian Wigner matrix whose entries have a law μ satisfying a Poincaré inequality and the matrix A_N is a block diagonal matrix, with an eigenvalue θ of multiplicity one, generating an outlier in the spectrum of M_N. We prove that the fluctuations of the norm of the projection of a unit eigenvector corresponding to the outlier of M_N onto a unit eigenvector corresponding to θ are not universal.

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• Universal entrywise eigenvector fluctuations in delocalized spiked matrix models and asymptotics of rounded spectral algorithms

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