Eigenvalue distributions of random unitary matrices
type: Item
instance of: scholarly article
title: Eigenvalue distributions of random unitary matrices
zbMATH Open document ID: 1044.15016
publication date: 2002-12-01T00:00:00Z
review text: The main result of the paper concerns the eigenvalue distribution of a \(n\times n\) random unitary matrix \(U\), chosen from the Haar measure, as \(n\to \infty\). Let \(I_1=(e^{i\alpha_1},e^{i\beta_1}),\ldots, I_m=(e^{i\alpha_m},e^{i\beta_m})\) be intervals on the unit circle, and let \(X_{nk}\) denote the number of eigenvalues of \(U\) in \(I_k\). Set \(Y_{nk}=\pi (\log n)^{-1/2}(X_{nk}-E[X_{nk}])\). The main theorem of the article states that \((Y_{n1},\ldots, Y_{nm})\) converges jointly in distribution to a normal distribution with a specified covariance matrix. The theorem is proven using the connection between Toeplitz matrices and random unitary matrices.
reviewed by: Franz Merkl
Mathematics Subject Classification ID: 15B52
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 1836956
zbMATH Keywords: eigenvalue distribution
zbMATH Keywords: random unitary matrix
zbMATH Keywords: Haar measure
zbMATH Keywords: normal distribution
zbMATH Keywords: covariance matrix
zbMATH Keywords: Toeplitz matrices
author: Kelly Wieand
MaRDI profile type: Publication
full work available at URL: s004400100186
OpenAlex ID: W2050626725
DOI: 10.1007/S004400100186
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1836956