Local law of addition of random matrices on optimal scale

TL;DR

This paper proves that the eigenvalue distribution of the sum of large random matrices converges locally to the free convolution at optimal scales, with fully delocalized eigenvectors, extending previous global results.

Contribution

It establishes the local law for the eigenvalue distribution of sums of random matrices at optimal scales, including real symmetric cases, with delocalized eigenvectors.

Findings

Eigenvalue distribution converges locally to free convolution.

Results hold for both Hermitian and real symmetric matrices.

Eigenvectors are fully delocalized in the bulk spectrum.

Abstract

The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this convergence also holds locally in the bulk of the spectrum, down to the optimal scales larger than the eigenvalue spacing. The corresponding eigenvectors are fully delocalized. Similar results hold for the sum of two real symmetric matrices, when one is conjugated by a Haar orthogonal matrix.