This is the collection of my written notes for MATH 499, Fall 2019. This course covers a collection of chapters of the book written by Guionnet, Anderson and Zeitouni, as well as some interesting results from other papers. Since I am a little bit tight on schedule, I just post my written notes, rather than typing them in latex. This collection will not serve as a serious reference, but as a personal record (“Oh this was what I did back then, the g’old times….yada yada yada…”). Plus, I believe my handwriting is decent and relatively easy to read.
- Class Notes on Sept. 24, 2019. (Taught by Tuca Auffinger)
- Introduction and welcome. Catalan number, combinatorics and proof of Wigner’s Theorem.
- Class Notes on Sept. 30, 2019.
- Introduction to Stieltjes Transform and its properties. An alternating proof of the Wigner’s Theorem using the Stieltjes Transform.
- Class Notes on Oct. 7, 2019.
- Gaussian Ensembles. Joint distribution of eigenvalues.
- Class Notes on Oct. 9, 2019.
- Joint distribution of eigenvalues continued.
- Class Notes on Oct.22, 2019. (Taught by Tuca)
- Hermite kernel, Christopher-Darboux, stationary phase and Plancherel-Rotach asymptotics.
- Class Notes on Oct. 27, 2019. (Taught by Tuca)
- Trace class, introduction to Fredholm determinant.