The Annals of Probability

Subordination for the sum of two random matrices

V. Kargin

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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.

Article information

Source
Ann. Probab., Volume 43, Number 4 (2015), 2119-2150.

Dates
Received: March 2013
Revised: January 2014
First available in Project Euclid: 3 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.aop/1433341328

Digital Object Identifier
doi:10.1214/14-AOP929

Mathematical Reviews number (MathSciNet)
MR3353823

Zentralblatt MATH identifier
1320.60022

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Keywords
Random matrices subordination small-rank matrix deformations delocalization local limit law

Citation

Kargin, V. Subordination for the sum of two random matrices. Ann. Probab. 43 (2015), no. 4, 2119--2150. doi:10.1214/14-AOP929. https://projecteuclid.org/euclid.aop/1433341328


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