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July 2019 Strong convergence of eigenangles and eigenvectors for the circular unitary ensemble
Kenneth Maples, Joseph Najnudel, Ashkan Nikeghbali
Ann. Probab. 47(4): 2417-2458 (July 2019). DOI: 10.1214/18-AOP1311

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.

Citation

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Kenneth Maples. Joseph Najnudel. Ashkan Nikeghbali. "Strong convergence of eigenangles and eigenvectors for the circular unitary ensemble." Ann. Probab. 47 (4) 2417 - 2458, July 2019. https://doi.org/10.1214/18-AOP1311

Information

Received: 1 May 2017; Revised: 1 September 2018; Published: July 2019
First available in Project Euclid: 4 July 2019

zbMATH: 07114720
MathSciNet: MR3980924
Digital Object Identifier: 10.1214/18-AOP1311

Subjects:
Primary: 60B20, 60F15

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 4 • July 2019
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