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Large deviations for the largest eigenvalues and eigenvectors of spiked random matrices | | Giulio Biroli
; Alice Guionnet
; | | Date: |
3 Apr 2019 | | Abstract: | We consider matrices formed by a random $N imes N$ matrix drawn from the
Gaussian Orthogonal Ensemble (or Gaussian Unitary Ensemble) plus a rank-one
perturbation of strength $ heta$, 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 $ heta>1$, in large deviations characterized by a small
value of $u$, i.e. $u<1-1/ heta$, 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. | | Source: | arXiv, 1904.1820 | | Services: | Forum | Review | PDF | Favorites | |
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