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Optimal upper bound for the infinity norm of eigenvectors of random matrices
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TitleInfo
Title
Optimal upper bound for the infinity norm of eigenvectors of random matrices
Name
(type = personal)
NamePart
(type = family)
Wang
NamePart
(type = given)
Ke
NamePart
(type = date)
1984-
DisplayForm
Ke Wang
Role
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(authority = RULIB)
author
Name
(type = personal)
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Vu
NamePart
(type = given)
Van
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Van Vu
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Advisory Committee
Role
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(authority = RULIB)
chair
Name
(type = personal)
NamePart
(type = family)
Kiessling
NamePart
(type = given)
Michael
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Michael Kiessling
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Advisory Committee
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(authority = RULIB)
internal member
Name
(type = personal)
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(type = family)
Kopparty
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(type = given)
Swastik
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Swastik Kopparty
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Advisory Committee
Role
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(authority = RULIB)
internal member
Name
(type = personal)
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(type = family)
Soshnikov
NamePart
(type = given)
Alexander
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Alexander Soshnikov
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Advisory Committee
Role
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(authority = RULIB)
outside member
Name
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NamePart
Rutgers University
Role
RoleTerm
(authority = RULIB)
degree grantor
Name
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NamePart
Graduate School - New Brunswick
Role
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school
TypeOfResource
Text
Genre
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theses
OriginInfo
DateCreated
(qualifier = exact)
2013
DateOther
(qualifier = exact);
(type = degree)
2013-05
Place
PlaceTerm
(type = code)
xx
Language
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(authority = ISO639-2b);
(type = code)
eng
Abstract
(type = abstract)
Let $M_n$ be a random Hermitian (or symmetric) matrix whose upper diagonal and diagonal entries are independent random variables with mean zero and variance one. It is well known that the empirical spectral distribution (ESD) converges in probability to the semicircle law supported on $[-2,2]$. In this thesis we study the local convergence of ESD to the semicircle law. One main result is that if the entries of $M_n$ are bounded, then the semicircle law holds on intervals of scale $log n/n$. As a consequence, we obtain the delocalization result for the eigenvectors, i.e., the upper bound for the infinity norm of unit eigenvectors corresponding to eigenvalues in the bulk of spectrum, is $O(sqrt{log n/n})$. The bound is the same as the infinity norm of a vector chosen uniformly on the unit sphere in $R^n$. We also study the local version of Marchenko-Pastur law for random covariance matrices and obtain the optimal upper bound for the infinity norm of singular vectors. This is joint work with V. Vu. In the last chapter, we discuss the delocalization properties for the adjacency matrices of ErdH{o}s-R'{e}nyi random graph. This is part of some earlier results joint with L. Tran and V. Vu.
Subject
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Topic
Mathematics
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Title
Rutgers University Electronic Theses and Dissertations
Identifier
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ETD
Identifier
ETD_4581
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electronic resource
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application/pdf
InternetMediaType
text/xml
Extent
vii, 74 p. : ill.
Note
(type = degree)
Ph.D.
Note
(type = bibliography)
Includes bibliographical references
Note
(type = vita)
Includes vita
Note
(type = statement of responsibility)
by Ke Wang
Subject
(authority = ETD-LCSH)
Topic
Random matrices
Subject
(authority = ETD-LCSH)
Topic
Eigenvectors
Identifier
(type = hdl)
http://hdl.rutgers.edu/1782.1/rucore10001600001.ETD.000068997
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TitleInfo
Title
Graduate School - New Brunswick Electronic Theses and Dissertations
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rucore19991600001
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NjNbRU
Identifier
(type = doi)
doi:10.7282/T3GB22N0
Genre
(authority = ExL-Esploro)
ETD doctoral
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RightsDeclaration
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The author owns the copyright to this work.
RightsHolder
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Name
FamilyName
Wang
GivenName
Ke
Role
Copyright Holder
RightsEvent
Type
Permission or license
DateTime
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(point = start)
2013-04-03 21:04:59
AssociatedEntity
Name
Ke Wang
Role
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Affiliation
Rutgers University. Graduate School - New Brunswick
AssociatedObject
Type
License
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Author Agreement License
Detail
I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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Copyright protected
Availability
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Open
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windows xp
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