Large deviations for the largest eigenvalues and eigenvectors of spiked Gaussian random matrices

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.