Eigenvalue approximation of sums of Hermitian matrices from eigenvector localization/delocalization

TL;DR

This paper introduces a novel eigenvalue approximation method for sums of Hermitian matrices using eigenvector localization/delocalization, bridging the gap between classical and free probability approaches.

Contribution

It develops a new technique based on eigenvector localization/delocalization that improves eigenvalue density approximations beyond free probability.

Findings

The method accurately approximates eigenvalue densities in relevant problems.

Eigenvector localization/delocalization parameter is independent of eigenvalues.

The technique outperforms classical and free probability in certain cases.

Abstract

We propose a technique for calculating and understanding the eigenvalue distribution of sums of random matrices from the known distribution of the summands. The exact problem is formidably hard. One extreme approximation to the true density amounts to classical probability, in which the matrices are assumed to commute; the other extreme is related to free probability, in which the eigenvectors are assumed to be in generic positions and sufficiently large. In practice, free probability theory can give a good approximation of the density. We develop a technique based on eigenvector localization/delocalization that works very well for important problems of interest where free probability is not sufficient, but certain uniformity properties apply. The localization/delocalization property appears in a convex combination parameter that notably, is independent of any eigenvalue properties and yields accurate eigenvalue density approximations. We demonstrate this technique on a number of examples as well as discuss a more general technique when the uniformity properties fail to apply.