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Local Gaussian fluctuations in the Airy and Discrete PNG processes
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2008 (English)In: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, 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.

Place, publisher, year, edition, pages
2008. Vol. 36, no 3, 1059-1092 p.
Keyword [en]
Matrix theory; Random; Stationary processes; Stochastic processes
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-7025DOI: 10.1214/07-AOP353ISI: 000255297000009Scopus ID: 2-s2.0-51949109612OAI: oai:DiVA.org:kth-7025DiVA: diva2:11901
Note
QC 20100716. Uppdaterad från Submitted till Published 20100716.Available from: 2007-04-20 Created: 2007-04-20 Last updated: 2017-12-14Bibliographically 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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