Doeraene, Antoine
[UCL]
An important topic in random matrix theory is the study of the statistical properties of the eigenvalues of large random matrices. In this thesis, we focus on the behaviour of the few largest, or few smallest, of these eigenvalues. We are interested in random matrix models that are determinantal, providing rich structure and powerful tools for their study. The first chapter contains a short introduction to the theory of determinantal point processes. In the second chapter, we examine perturbations of Hermitian positive definite random matrices. The perturbations take the form of the entry-wise addition of Gaussian complex random variables, hence breaking the positive definiteness of the matrices. We investigate to what extent the microscopic behaviour is henceforth altered. The third chapter is devoted to the generalization of a result by Tracy and Widom. This result describes the probability distribution of the largest particle of the Airy point process in terms of a solution to the Painlevé II differential equation. We extend the result by expressing more general probability distributions involving particles near the largest one in terms of the solutions to a system of coupled Painlevé II-like equations. The analogue results of the smallest particles for the Bessel point process are obtained in the last chapter.


Bibliographic reference |
Doeraene, Antoine. Near extreme eigenvalue behaviour in random matrix theory. Prom. : Claeys, Tom |
Permanent URL |
http://hdl.handle.net/2078.1/198386 |