Local law of addition of random matrices on optimal scale
Probability
2016-12-21 v3 Mathematical Physics
math.MP
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.
Keywords
Cite
@article{arxiv.1509.07080,
title = {Local law of addition of random matrices on optimal scale},
author = {Zhigang Bao and Laszlo Erdos and Kevin Schnelli},
journal= {arXiv preprint arXiv:1509.07080},
year = {2016}
}
Comments
More details on the continuity argument in Sections 7 and 8 are added