Beijing Institute of Mathematical Sciences and Applications

This course provides an introduction to the theory of random matrices, beginning with a comprehensive analysis of Wigner matrices: symmetric or Hermitian matrices with (almost) i.i.d. entries. We will start by proving Wigner's semicircle law via the method of moments, utilizing tools from basic combinatorics (such as Catalan numbers and Dyck paths), and will also cover the Marchenko-Pastur law. The course will then study concentration inequalities through logarithmic Sobolev inequalities and the resolvent method (using Stieltjes transforms). Following this, we will derive the explicit joint eigenvalue distribution of the Gaussian Orthogonal and Unitary Ensembles (GOE/GUE). The course will culminate in the study of asymptotic local eigenvalue behavior. This includes a description of the limiting statistics in terms of the sine and Airy kernels, along with an analytical examination of the Tracy-Widom distribution and its connection to Painlevé equations. The study of the GUE depends heavily on the intrinsic determinantal structure of the eigenvalue distribution. We will also discuss the more involved cases of local asymptotics for the GOE and GSE ensembles, exploring their corresponding Pfaffian structures.