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Gaussian fluctuations in some determinantal processes
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
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 [en]
Random matrices, limit theorems
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-4343ISBN: 978-91-7178-603-6 (print)OAI: oai:DiVA.org:kth-4343DiVA: diva2:11902
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
List of papers
1. Gaussian fluctuations of eigenvalues in the GUE
Open this publication in new window or tab >>Gaussian fluctuations of eigenvalues in the GUE
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

Keyword
Limit distribution; Random matrices
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-7024 (URN)10.1016/j.anihpb.2004.04.002 (DOI)000227724900002 ()2-s2.0-14544296548 (Scopus ID)
Note
QC 20100716Available from: 2007-04-20 Created: 2007-04-20 Last updated: 2010-07-16Bibliographically approved
2. Local Gaussian fluctuations in the Airy and Discrete PNG processes
Open this publication in new window or tab >>Local Gaussian fluctuations in the Airy and Discrete PNG processes
2008 (English)In: Annals of Probability, ISSN 0091-1798, Vol. 36, no 3, 1059-1092 p.Article in journal (Refereed) Published
Abstract [en]

We prove that the Airy process, A(t), locally fluctuates like a Brownian motion. In the same spirit we also show that, in a certain scaling limit, the so-called discrete polynuclear growth process (PNG) behaves like a Brownian motion.

Keyword
Matrix theory; Random; Stationary processes; Stochastic processes
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-7025 (URN)10.1214/07-AOP353 (DOI)000255297000009 ()2-s2.0-51949109612 (Scopus ID)
Note
QC 20100716. Uppdaterad från Submitted till Published 20100716.Available from: 2007-04-20 Created: 2007-04-20 Last updated: 2010-07-16Bibliographically approved

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