| Summary: | In this thesis we begin by presenting an introduction on random matrices, their different classes and applications in quantum mechanics to study the characteristics of the eigenvectors of a particular random matrix model. The focus of this work is on one of the oldest and most well-known symmetry classes of random matrices - the Gaussian unitary ensemble. We look at how the different possible deformations of the Gaussian unitary ensemble could have an impact on the nature of the eigenvectors, and back up our results by numerical simulations to confirm validity. We will begin exploring the structure of the eigenvectors by employing the supersymmetry technique, a method for studying eigenvectors of complex quantum systems. In particular, we can analyse the moments of the eigenvectors, a quantity used in the classification of eigenvectors, in different random matrix models. Eigenvectors can either be extended, localised or critical and the scaling of the moments of the eigenvectors with matrix size N is used to determine the exact type. This enables one to study the transition of the eigenvectors from extended to localised and the intermediate stages. We consider different classes of random matrices, such as random matrices with an external source and structured random matrices. In particular, we studied the Rosenzweig-Porter model by generalising our previous results from a deterministic potential to a random one and study the impact of such an alteration to the model.
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