Large deviations of the extreme eigenvalues of random deformations of matrices

TL;DR

This paper establishes a large deviation principle for the extreme eigenvalues of large random matrices perturbed by finite-rank matrices, covering both bulk and outlier eigenvalues, with a variational rate function.

Contribution

It provides the first large deviation results for extreme eigenvalues of deformed matrices with delocalized eigenvectors, including cases with outliers and random base matrices.

Findings

Large deviation principle for extreme eigenvalues in the deformed model

Variational formula for the rate function

Extension to random base matrices with specific distributions

Abstract

Consider a real diagonal deterministic matrix $X_n$ of size $n$ with spectral measure converging to a compactly supported probability measure. We perturb this matrix by adding a random finite rank matrix, with delocalized eigenvectors. We show that the joint law of the extreme eigenvalues of the perturbed model satisfies a large deviation principle in the scale $n$, with a good rate function given by a variational formula. We tackle both cases when the extreme eigenvalues of $X_n$ converge to the edges of the support of the limiting measure and when we allow some eigenvalues of $X_n$, that we call outliers, to converge out of the bulk. We can also generalise our results to the case when $X_n$ is random, with law proportional to $e^{- n Trace V(X)}\ud X,$ for $V$ growing fast enough at infinity and any perturbation of finite rank.