Large deviations for the largest eigenvalue of the sum of two random matrices

TL;DR

This paper establishes a large deviation principle for the largest eigenvalue of the sum of two random matrices in generic position, providing explicit rate functions under mild spectral conditions.

Contribution

It introduces a large deviation principle for the largest eigenvalue of matrix sums with explicit rate functions, extending previous results to more general spectral conditions.

Findings

Large deviation principle established for eigenvalues

Explicit rate functions involving spherical integrals

Applicable to matrices without outliers

Abstract

In this paper, we consider the addition of two matrices in generic position, namely A + U BU * , where U is drawn under the Haar measure on the unitary or the orthogonal group. We show that, under mild conditions on the empirical spectral measures of the deterministic matrices A and B, the law of the largest eigenvalue satisfies a large deviation principle, in the scale N, with an explicit rate function involving the limit of spherical integrals. We cover in particular all the cases when A and B have no outliers.