Cancelled Random Matrices

The theory of random matrices has its origin in applications in mathematical statistics and nuclear physics in the first half of the 20th century. It has become a paradigm for the study of high dimensional non-commutative disordered systems that connects numerous subfields of mathematics and physics, such as probability theory, high dimensional analysis, combinatorics, quantum and statistical physics with diverse applications e.g. to communication theory, condensed matter and high energy physics, number theory and neural networks.

We will provide an introduction to random matrices and learn basic concepts and techniques that are used to analyze them. In particular, we will introduce prominent models such as the Gaussian Unitary Ensemble, Invariant Ensembles and Wigner matrices. We will show that despite having random entries their spectral densities become approximately deterministic with increasing dimension and we will study fine details of their eigenvalue distributions. We will interpret the eigenvalues as an interacting particle system (Dyson Brownian motion) and show that its fast approach to local equilibrium implies universal spectral statistics across a wide range of random matrix models, one of the hallmarks of the theory. Finally we will discuss applications.