Author(s):

Bao, Zhigang; Erdős, László; Schnelli, Kevin

Article Title: 
Local stability of the free additive convolution

Affiliation 
IST Austria 
Abstract: 
We prove that the system of subordination equations, defining the free additive convolution of two probability measures, is stable away from the edges of the support and blowup singularities by showing that the recent smoothness condition of Kargin is always satisfied. As an application, we consider the local spectral statistics of the random matrix ensemble A+UBU⁎A+UBU⁎, where U is a Haar distributed random unitary or orthogonal matrix, and A and B are deterministic matrices. In the bulk regime, we prove that the empirical spectral distribution of A+UBU⁎A+UBU⁎ concentrates around the free additive convolution of the spectral distributions of A and B on scales down to N−2/3N−2/3.

Keywords: 
Free convolution; Subordination; Local eigenvalue density

Journal Title:

Journal of Functional Analysis

Volume: 
271

Issue 
3

ISSN:

10960783

Publisher:

Academic Press

Date Published:

20160801

Start Page: 
672

End Page:

719

Sponsor: 
Partially supported by ERC Advanced Grant RANMAT No. 338804. Supported by ERC Advanced Grant RANMAT No. 338804. We thank an anonymous referee for many useful comments and remarks, and bringing references [6] ; [12] to our attention.

URL: 

DOI: 
10.1016/j.jfa.2016.04.006

Open access: 
yes (repository) 