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

Giulio Biroli,Alice Guionnet

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.

Highlights

The large deviations theory for the spectrum of random matrix models is a very active domain of research in probability theory and theoretical physics.A lot of works have been devoted to the statistics of the eigenvalues
We consider matrices formed by a random N × N matrix drawn from the Gaussian Orthogonal Ensemble plus a rank-one perturbation of strength θ, and focus on the largest eigenvalue, x, and the component, u, of the corresponding eigenvector in the direction associated to the rank-one perturbation
For θ > 1, large deviations events characterized by a small value of u, i.e. u < 1 − 1/θ, 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

Summary

Introduction

The large deviations theory for the spectrum of random matrix models is a very active domain of research in probability theory and theoretical physics. The square of the component vN (1) of the associated eigenvector in the direction associated to the perturbation converges almost surely to u = 1 − 1/θ2 [9] In this context the question we raised before becomes meaningful, and it is natural to focus on the good rate function (GRF) that controls the joint atypical deviations of λ1 and |v1(1)|2. Noise dressing and cleaning of empirical correlation matrices is another context in which the kind of large deviations addressed in this paper are relevant In this case, a model that is often considered to interpret the data is the one of spiked Wishart random matrices, whose eigenvalue distribution consists in a Marchenko-Pastur law plus a few eigenvalues that pop out from it. In this work we obtain the large deviation function that governs them

Main results

Strategy of the proof

Asymptotic of spherical integrals

The case of Wishart matrices

ZMW βN 2