Gap probabilities in random matrix theory
File(s)
Author(s)
Maroudas, Theo-Harris
Type
Thesis
Abstract
The main achievement of this work is in providing a proof of the value of the multiplicative constant
arising from the Airy-kernel determinant on two large gaps. In fact, we prove a full asymptotic
formula up to decaying terms. The constant in this expression, in particular, determines the pre-factor of the determinant. The kernel of Airy functions is universal in random matrix theory as it
arises at the soft edge of many ensembles of random matrices, most notably the Gaussian Unitary
Ensemble. The proof adds to a series of results since Dyson and, Tracy and Widom, in studying
this type of Fredholm determinants. Our method of proof relies on the steepest descent approach
to Riemann-Hilbert problems, introduced by Deift and Zhou. By extending their approach to our
particular setting we, in addition, contribute to the development of an important method.
arising from the Airy-kernel determinant on two large gaps. In fact, we prove a full asymptotic
formula up to decaying terms. The constant in this expression, in particular, determines the pre-factor of the determinant. The kernel of Airy functions is universal in random matrix theory as it
arises at the soft edge of many ensembles of random matrices, most notably the Gaussian Unitary
Ensemble. The proof adds to a series of results since Dyson and, Tracy and Widom, in studying
this type of Fredholm determinants. Our method of proof relies on the steepest descent approach
to Riemann-Hilbert problems, introduced by Deift and Zhou. By extending their approach to our
particular setting we, in addition, contribute to the development of an important method.
Version
Open Access
Date Issued
2020-09
Date Awarded
2021-06
Copyright Statement
Creative Commons Attribution-Non Commercial 4.0 International Licence
License URL
Advisor
Krasovsky, Igor
Sponsor
Engineering and Physical Sciences Research Council
Grant Number
1832015
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)