Learning Outcomes:
At the end of this module, successful students will:
-- Be familiar with central models in RMT, such as Wigner and Wishart random matrices, and Gaussian ensembles and circular ensembles, and some of their real-life applications
-- Know how to define probability densities on matrices, and how to define and compute a Haar measure
-- Understand how to characterise eigenvalues of random matrices in terms of joint distributions and empirical measures
-- Understand Girko’s circular law
-- Understand Wigner’s semi-circle law, and how it relates to the combinatorics of non-crossing partitions
-- Understand how matrix integrals are computed, and how their asymptotic expansion relates to enumeration of certain graphs by genus.
-- Know how determinantal point processes are defined, and understand how they arise in certain random matrix models
-- Know how to describe local and scaling limits of joint distributions of eigenvalues of certain random matrix models, and be familiar with universal phenomena arising in these contexts.