Class number 054–39482R
Class meetings: MW, 9:30am-12:30pm, KAP 166.
Information on this and related pages changes frequently.
Instructor: Sergey Lototsky.
Office: KAP 248D.
Phone: 213 740 2389
E-mail: lototsky usc edu.
Office Hours: MW before and after the class. Appointments at other time are welcome.
Course objective: To learn main terminology and results related to random matrices, such as
- Gaussian ensembles (GOE, GUE, GSE) and some of the fundamental math behind them (Lie groups, Lie Algebras, Quaternions, etc.);
- Circle, semi-circle, and other laws, and the names behind them;
- Saddle point method, Method of moment, Stieltjes trasnform, Dyson Brownian motion, and other technical tools.
Course work: Class participation, homework assignments, final presentation.
Official grading scheme: 20% class participation, 40% homework assignments, 40% final presentation.
Main reference: G. Livan, M. Novaes, P. Vivo. Introduction to random matrices. Theory and practice. SpringerBriefs in Mathematical Physics, 26. Springer, Cham, 2018, ix+124 pp.
The book is available in electronic form from USC libraries.
A longer list of references.
An example of a book review from Math reviews [Edition 1, Edition 2] and from the Bulletin of the AMS
Our progress.
May 17. An overview of the class, the book, and some foundational material from Linear Algebra and Probability.
May 22. Random eigenvalues vs. iid random variables; circle/semi-circle/quarter-circle laws; an overview of quaternions.
May 24. Asymptotic Integration; more about the circle laws.
May 29. No class: Memorial Day.
May 31. From the joint pdf of matrix entries to the joint pdf of eigenvalues and eigenvectors; Vandermonde and his determinant.
June 5. Proving the circle laws: weak convergence of probability measures, method of moments, Stieltjes transform, and more.
June 7. Orthogonal polynomials; determinant processes; examples of bulk vs edge asymptotics for eigenvalues.
June 12. Wishart-Laguerre ensembles; Marchenko-Pastur theorem.
June 14. Free probability and its connections to random matrices.
June 19. No class: Federal holiday/USC non-instructional day.
June 21. Painleve equations and related topics; Dyson’s Brownian motion.
June 26. More on Dyson’s Brownian motion, matrix norms, and kernels/determinant processes.
June 28. A concluding discussion.