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Gaussian fluctuations of eigenvalues in the GUE
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2005 (English)In: Annales de l'Institut Henri Poincare, (B) Probabilites et Statistiques, ISSN 0246-0203, Vol. 41, no 2, 151-178 p.Article in journal (Refereed) Published
Abstract [en]

Under certain conditions on k we calculate the limit distribution of the kth largest eigenvalue, x(k), of the Gaussian Unitary Ensemble (GUE). More specifically, if n is the dimension of a random matrix from the GUE and k is such that both n - k and k tends to infinity as n -> infinity then x(k) is normally distributed in the limit. We also consider the joint limit distribution of x(k1) < center dot center dot center dot < x(k) where we require that n - k(i) and k(i), 1 <= i <= m, tends to infinity with n. The result is an m-dimensional normal distribution

Place, publisher, year, edition, pages
2005. Vol. 41, no 2, 151-178 p.
Keyword [en]
Limit distribution; Random matrices
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-7024DOI: 10.1016/j.anihpb.2004.04.002ISI: 000227724900002Scopus ID: 2-s2.0-14544296548OAI: oai:DiVA.org:kth-7024DiVA: diva2:11900
Note
QC 20100716Available from: 2007-04-20 Created: 2007-04-20 Last updated: 2010-07-16Bibliographically approved
In thesis
1. Gaussian fluctuations in some determinantal processes
Open this publication in new window or tab >>Gaussian fluctuations in some determinantal processes
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of two parts, Papers A and B, in which some stochastic processes, originating from random matrix theory (RMT), are studied.

In the first paper we study the fluctuations of the kth largest eigenvalue, xk, of the Gaussian unitary ensemble (GUE). That is, let N be the dimension of the matrix and k depend on N in such a way that k and N-k both tend to infinity as N - ∞. The main result is that xk, when appropriately rescaled, converges in distribution to a Gaussian random variable as N → ∞. Furthermore, if k1 < ...< km are such that k1, ki+1 - ki and N - km, i =1, ... ,m - 1, tend to infinity as N → ∞ it is shown that (xk1 , ... , xkm) is multivariate Gaussian in the rescaled N → ∞ limit.

In the second paper we study the Airy process, A(t), and prove that it fluctuates like a Brownian motion on a local scale. We also prove that the Discrete polynuclear growth process (PNG) fluctuates like a Brownian motion in a scaling limit smaller than the one where one gets the Airy process.

Place, publisher, year, edition, pages
Stockholm: KTH, 2007. vii, 27 p.
Series
Trita-MAT, ISSN 1401-2286 ; 07-MA-02
Keyword
Random matrices, limit theorems
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-4343 (URN)978-91-7178-603-6 (ISBN)
Public defence
2007-05-04, Kollegiesalen, F3, KTH, Lindstedtsvägen 26, 14:00
Opponent
Supervisors
Note
QC 20100716Available from: 2007-04-20 Created: 2007-04-20 Last updated: 2010-07-16Bibliographically approved

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CiteExportLink to record
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Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
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More styles
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