Random Matrices: Universality in Disordered Quantum Systems
Abstract:
Random matrices serve as a model for disordered quantum systems. The distribution of energy levels within these systems often exhibit universal features. As an example, the gaps between consecutive spectral lines of large atomic nuclei follow the same statistics, independent of the type of nucleus or the energy range under consideration. Recent advances in the theory of random matrices shed light on the mechanisms leading to the emergence of such universal distributions. In this course we will give an introduction into the methods and results used to understand spectra of large dimensional random matrices from the global down to the microscopic scale. We will start with some classical results for exactly solvable Gaussian models and proceed with the presentation of useful tools for high dimensional analysis, such as entropy, concentration results and resolvent expansion. Later we will see that for a wide class of random matrices the density of states has only three types of behaviour at any point in the spectrum: bulk, edge or cusp (internal vanishing point). Furthermore, the eigenvalues themselves are well described by a one-dimensional gas of charged particles with logarithmic interactions in local thermal equilibrium, providing a theoretical explanation of spectral universality.