## The Annals of Probability

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

#### 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
Revised: September 2018
First available in Project Euclid: 4 July 2019

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

Digital Object Identifier
doi:10.1214/18-AOP1311

Mathematical Reviews number (MathSciNet)
MR3980924

Zentralblatt MATH identifier
07114720

#### 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

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