Basic concepts of random matrix theory

[摘要] It was Wigner that in the 1950's first introduced the idea of modelling physical realitywith an ensemble of random matrices while studying the energy levels of heavy atomicnuclei. Since then, the field of Random Matrix Theory has grown tremendously, withapplications ranging from fluctuations on the economic markets to M-theory. It is thepurpose of this thesis to discuss the basic concepts of Random Matrix Theory, using theensembles of random matrices originally introduced by Wigner, the Gaussian ensembles,as a starting point. As Random Matrix Theory is classically concerned with the statisticalproperties of levels sequences, we start with a brief introduction to the statistical analysisof a level sequence before getting to the introduction of the Gaussian ensembles. With theensembles defined, we move on to the statistical properties that they predict. In the lightof these predictions, a few of the classical applications of Random Matrix Theory arediscussed, and as an example of some of the important concepts, the Anderson model oflocalization is investigated in some detail.

[发布机构] Stellenbosch University