Random Matrices and Supersymmetry in Disordered Systems

It is described how one comes to the Wigner-Dyson random matrix theory (RMT) starting from a model of a disordered metal. The lectures start with a historical introduction where basic ideas of the RMT and theory of disordered metals are reviewed. This part is followed by an introduction into supermathematics (mathematics operating with both commuting and anticommuting variables). The main ideas of the supersymmetry method are given and basic formulae are derived. As an example, level-level correlations and fluctuations of amplitudes of wave functions are discussed. It is shown how one can both obtain known formulae of the RMT and go beyond. In the last part some recent progress in the further development of the method and possible perspectives are discussed.