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2020 Large deviations for the largest eigenvalues and eigenvectors of spiked Gaussian random matrices
Giulio Biroli, Alice Guionnet
Electron. Commun. Probab. 25: 1-13 (2020). DOI: 10.1214/20-ECP343

Abstract

We consider matrices formed by a random $N\times N$ matrix drawn from the Gaussian Orthogonal Ensemble (or Gaussian Unitary Ensemble) plus a rank-one perturbation of strength $\theta $, and focus on the largest eigenvalue, $x$, and the component, $u$, of the corresponding eigenvector in the direction associated to the rank-one perturbation. We obtain the large deviation principle governing the atypical joint fluctuations of $x$ and $u$. Interestingly, for $\theta >1$, large deviations events characterized by a small value of $u$, i.e. $u<1-1/\theta $, are such that the second-largest eigenvalue pops out from the Wigner semi-circle and the associated eigenvector orients in the direction corresponding to the rank-one perturbation. We generalize these results to the Wishart Ensemble, and we extend them to the first $n$ eigenvalues and the associated eigenvectors.

Citation

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Giulio Biroli. Alice Guionnet. "Large deviations for the largest eigenvalues and eigenvectors of spiked Gaussian random matrices." Electron. Commun. Probab. 25 1 - 13, 2020. https://doi.org/10.1214/20-ECP343

Information

Received: 19 July 2019; Accepted: 12 August 2020; Published: 2020
First available in Project Euclid: 30 September 2020

Digital Object Identifier: 10.1214/20-ECP343

Subjects:
Primary: 60B20, 60F10

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