Random Matrix Models

Abstract: In this thesis, we provide a self contained introduction to the theory of random matrices and matrix models. Our analysis has a chronological order and it begins with the study of nuclear energy levels and ensemble averaging which yields the famous Wigner surmise. Then, the standard Gaussian theory of the orthogonal, unitary and symplectic ensemble is derived from symmetry arguments of the corresponding physical system, which is then followed by the explicit calculation of Wigner's semicircle distribution. Moreover, interactions are introduced to the free theory which leads to the topological expansion, a diagrammatic way of evaluating certain expectation values. Also, it is shown that there exists a duality between the resulting graphs and the quantization of a two dimensional surface through mapping as well as a method for solving a specific family of potentials. Finally, the numerical confirmation for some observables is carried out using the Hamburger moment problem, and the derivation of critical points for some theories.