## The Annals of Probability

- Ann. Probab.
- Volume 47, Number 4 (2019), 2417-2458.

### Strong convergence of eigenangles and eigenvectors for the circular unitary ensemble

Kenneth Maples, Joseph Najnudel, and Ashkan Nikeghbali

#### Abstract

It is known that a unitary matrix can be decomposed into a product of complex reflections, one for each dimension, and that these reflections are independent and uniformly distributed on the space where they live if the initial matrix is Haar-distributed. If we take an infinite sequence of such reflections, and consider their successive products, then we get an infinite sequence of unitary matrices of increasing dimension, all of them following the circular unitary ensemble.

In this coupling, we show that the eigenvalues of the matrices converge almost surely to the eigenvalues of the flow, which are distributed according to a sine-kernel point process, and we get some estimates of the rate of convergence. Moreover, we also prove that the eigenvectors of the matrices converge almost surely to vectors which are distributed as Gaussian random fields on a countable set.

#### Article information

**Source**

Ann. Probab., Volume 47, Number 4 (2019), 2417-2458.

**Dates**

Received: May 2017

Revised: September 2018

First available in Project Euclid: 4 July 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1562205712

**Digital Object Identifier**

doi:10.1214/18-AOP1311

**Mathematical Reviews number (MathSciNet)**

MR3980924

**Zentralblatt MATH identifier**

07114720

**Subjects**

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60F15: Strong theorems

**Keywords**

Random matrix circular unitary ensemble convergence of eigenvalues convergence of eigenvectors virtual isometries complex reflections

#### Citation

Maples, Kenneth; Najnudel, Joseph; Nikeghbali, Ashkan. Strong convergence of eigenangles and eigenvectors for the circular unitary ensemble. Ann. Probab. 47 (2019), no. 4, 2417--2458. doi:10.1214/18-AOP1311. https://projecteuclid.org/euclid.aop/1562205712