Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Limit theorems for generalizations of GUE random matrices
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2008 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis consists of two papers devoted to the asymptotics of random matrix ensembles and measure valued stochastic processes which can be considered as generalizations of the Gaussian unitary ensemble (GUE) of Hermitian matrices H=A+A, where the entries of A are independent identically distributed (iid) centered complex Gaussian random variables.

In the first paper, a system of interacting diffusing particles on the real line is studied; special cases include the eigenvalue dynamics of matrix-valued Ornstein-Uhlenbeck processes (Dyson's Brownian motion). It is known that the empirical measure process converges weakly to a deterministic measure-valued function and that the appropriately rescaled fluctuations around this limit converge weakly to a Gaussian distribution-valued process. For a large class of analytic test functions, explicit formulae are derived for the mean and covariance functionals of this fluctuation process.

The second paper concerns a family of random matrix ensembles interpolating between the GUE and the Ginibre ensemble of n x n matrices with iid centered complex Gaussian entries. The asymptotic spectral distribution in these models is uniform in an ellipse in the complex plane, which collapses to an interval of the real line as the degree of non-Hermiticity diminishes. Scaling limit theorems are proven for the eigenvalue point process at the rightmost edge of the spectrum, and it is shown that a non-trivial transition occurs between Poisson and Airy point process statistics when the ratio of the axes of the supporting ellipse is of order n -1/3.

Abstract [sv]

Denna avhandling består av två vetenskapliga artiklar som handlar om gränsvärdessatser för slumpmatriser och måttvärda stokastiska processer. De modeller som studeras kan betraktas som generaliseringar av den gaussiska unitära ensembeln (GUE) av hermiteska n x n-matriser H=A+A†, där A är en matris vars element är oberoende, likafördelade, centrerade, komplexa normalfördelade stokastiska variabler.

I artikel I betraktas ett system av växelverkande diffunderande partiklar på reella linjen, vissa specialfall av denna modell kan tolkas som egenvärdesdynamiken för matrisvärda Ornstein-Uhlenbeck-processer (Dysons brownska rörelse). Sedan tidigare är det känt att den empiriska måttprocessen konvergerar svagt mot en deterministisk måttvärd funktion och att fluktuationerna runt denna gräns, i lämplig skalning, konvergerer svagt mot en distributionsvärd gaussisk process. För en stor klass av analytiska testfunktioner härleds explicita formler för medelvärdes- och kovariansfunktionalerna för denna fluktuationsprocess.

Artikel II behandlar en familj av slumpmatrisensembler som interpolerar mellan GUE och Ginibre-ensembeln, bestående av matriser A som ovan. För denna modell är egenvärdena komplexa och asymptotiskt likformigt fördelade i en ellips i komplexa planet. Skalningsgränsvärdessatser för egenvärdet med maximal realdel och för egenvärdespunktprocessen kring detta visas för ett allmänt val av interpolationsparametern i modellen. Då förhållandet mellan axlarna i den asymptotiska ellipsen är av storleksordning n-1/3 uppträder en övergångsfas mellan Airypunktprocess- och Poissonprocessbeteendena, typiska för GUE respektive Ginibre-ensembeln.

Place, publisher, year, edition, pages
Stockholm: KTH , 2008. , vi, 31 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 08-MA-05
Keyword [en]
Random matrices, Central limit theorem, Dyson's Brownian motion, Interacting diffusion, Point process, Non-Hermitian, Scaling limit
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-4799ISBN: 978-91-7178-973-0 (print)OAI: oai:DiVA.org:kth-4799DiVA: diva2:14017
Public defence
2008-06-13, Kollegiesalen, F3, Lindstedtsvägen 26, Stockholm, 10:00
Opponent
Supervisors
Note
QC 20100705Available from: 2008-06-04 Created: 2008-06-04 Last updated: 2010-07-05Bibliographically approved
List of papers
1. Global fluctuations in general β Dyson’s Brownian motion
Open this publication in new window or tab >>Global fluctuations in general β Dyson’s Brownian motion
2008 (English)In: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 118, no 6, 1022-1042 p.Article in journal (Refereed) Published
Abstract [en]

We consider a system of diffusing particles on the real line in a quadratic external potential and with a logarithmic interaction potential. The empirical measure process is known to converge weakly to a deterministic measure-valued process as the number of particles tends to infinity. Provided the initial fluctuations are small, the rescaled linear statistics of the empirical measure process converge in distribution to a Gaussian limit for sufficiently smooth test functions. For a large class of analytic test functions, we derive explicit general formulae for the mean and covariance in this central limit theorem by analyzing a partial differential equation characterizing the limiting fluctuations.

Keyword
central limit theorem; Dyson's Brownian motion; interacting diffusion; random matrices; RANDOM MATRICES; EIGENVALUES; PARTICLES; LIMIT; CLT; LAW
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-8639 (URN)10.1016/j.spa.2007.07.010 (DOI)000256599800008 ()2-s2.0-42649108381 (Scopus ID)
Note
QC 20100705Available from: 2008-06-04 Created: 2008-06-04 Last updated: 2017-12-14Bibliographically approved
2. Edge scaling limits for a family of non-Hermitian random matrix ensembles
Open this publication in new window or tab >>Edge scaling limits for a family of non-Hermitian random matrix ensembles
2010 (English)In: Probability theory and related fields, ISSN 0178-8051, E-ISSN 1432-2064, Vol. 147, no 1-2, 241-271 p.Article in journal (Refereed) Published
Abstract [en]

A family of random matrix ensembles interpolating between the Ginibre ensemble of n x n matrices with iid centered complex Gaussian entries and the Gaussian unitary ensemble (GUE) is considered. The asymptotic spectral distribution in these models is uniform in an ellipse in the complex plane, which collapses to an interval of the real line as the degree of non-Hermiticity diminishes. Scaling limit theorems are proven for the eigenvalue point process at the rightmost edge of the spectrum, and it is shown that a non-trivial transition occurs between Poisson and Airy point process statistics when the ratio of the axes of the supporting ellipse is of order n (-1/3). In this regime, the family of limiting probability distributions of the maximum of the real parts of the eigenvalues interpolates between the Gumbel and Tracy-Widom distributions.

Keyword
Random matrices; Non-Hermitian; Extremes; Tracy-Widom; Gumbel; Airy; Poisson
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-8640 (URN)10.1007/s00440-009-0207-9 (DOI)000274657400008 ()
Note
QC 20100617Available from: 2008-06-04 Created: 2008-06-04 Last updated: 2017-12-14Bibliographically approved

Open Access in DiVA

fulltext(384 kB)525 downloads
File information
File name FULLTEXT01.pdfFile size 384 kBChecksum MD5
4e0eded24bce830458d2b2443367a3349d33680e7135c3613e4676bfa3b510a2d8a15d4a
Type fulltextMimetype application/pdf

Search in DiVA

By author/editor
Bender, Martin
By organisation
Mathematics (Dept.)
Mathematics

Search outside of DiVA

GoogleGoogle Scholar
Total: 525 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

isbn
urn-nbn

Altmetric score

isbn
urn-nbn
Total: 415 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf