Subordination for the sum of two random matrices

TL;DR

This paper explores how the resolvent of the sum of two random matrices relates to the resolvents of the individual matrices, establishing local laws and eigenvalue behavior through subordination techniques.

Contribution

It introduces a subordination framework for the sum of two random matrices, providing error estimates and applications to eigenvalue distributions and deformations.

Findings

Proves a local limit law for eigenvalues of matrix sums.

Establishes delocalization of eigenvectors.

Determines the limit of the largest eigenvalue in rank-one deformations.

Abstract

This paper is about the relation of random matrix theory and the subordination phenomenon in complex analysis. We find that the resolvent of the sum of two random matrices is approximately subordinated to the resolvents of the original matrices. We estimate the error terms in this relation and in the subordination relation for the traces of the resolvents. This allows us to prove a local limit law for eigenvalues and a delocalization result for eigenvectors of the sum of two random matrices. In addition, we use subordination to determine the limit of the largest eigenvalue for the rank-one deformations of unitary-invariant random matrices.