Maxima of entries of Haar distributed matrices
type: Item
instance of: scholarly article
title: Maxima of entries of Haar distributed matrices
zbMATH Open document ID: 1067.15021
author: Tiefeng Jiang
publication date: 2005-02-22T00:00:00Z
review text: Given an \(n\times n\) random matrix \(\Gamma_n=(\gamma_{ij})\) in \(O(n)\) with normalized Haar distribution, the behavior of the maximum of the absolute values of the entries, namely \(W_n=\max| \gamma_{ij}| \), is studied. The limit \(\lim_n \sqrt{n/\log n}\,W_n\) is shown to be 2 in probability, and \[ \lim_{n\rightarrow\infty}\,P(nW_n^2-4\log n+\log(\log n)\leq x)=\exp(-\sqrt{1/2\pi}\,e^{-x/2}). \] If independence of the sequence \(\{\Gamma_n\}\) of random matrices is required, then the sequence \(\{\sqrt{n/\log n}\,W_n\}\) is shown to be dense in \([2,\sqrt6]\) almost surely. Analog results are proven for matrices in \(\text{SO}(n)\), \(U(n)\), and \(\text{SU}(n)\).
reviewed by: Martín Argerami
Mathematics Subject Classification ID: 15B52
Mathematics Subject Classification ID: 60B10
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 60F10
zbMATH DE Number: 2137181
zbMATH Keywords: Haar measure
zbMATH Keywords: maxima of entries
zbMATH Keywords: large deviation
zbMATH Keywords: Gram-Schmidt procedure
zbMATH Keywords: random matrix
MaRDI profile type: Publication
full work available at URL: s00440-004-0376-5
OpenAlex ID: W2037476261
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DOI: 10.1007/S00440-004-0376-5
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 2137181