# MATH 8380 • Random Matrices

### Course notes • Found typo or mistake? Let me know!

• Chapter 1. Some history
• Chapter 2. Gaussian Unitary Ensemble and Dyson Brownian Motion
• Interlude. Markov chains and stochastic differential equations
• Chapter 3. Two ways to derive the GUE eigenvalue distribution
• Chapter 4. Wigner Semicircle Law
• Chapter 5. Orbital measures and free operations
• Chapter 6. Representation-theoretic discrete analogue of the GUE

### Syllabus

Instructor. Leonid Petrov. Contact information is at https://lpetrov.cc

The class meets on Tuesdays and Thursdays at 9:30-10:45 in Kerchof 128.

Office hours Tuesdays and Thursdays 11:30-1 (or just drop in at any time). Office is Kerchof 209

Description. Study of random matrices is an exciting topic with first major advances in the mid-20th century in connection with statistical (quantum) physics. Since then it found numerous connections to algebra, geometry, combinatorics, as well as to the core of the probability theory. The applications are also numerous: e.g., statistics, number theory, engineering, neuroscience; with more of them discovered every month. The course will discuss fundamental problems and results of Random Matrix Theory, and their connections to tools of algebra and combinatorics.

Course homepage. The course homepage is at https://lpetrov.cc/rmt19/. It contains the syllabus, link to course notes, and other relevant information.

Structure. The course discusses:

1. Limit shape results for random matrices (such as Wigner’s Semicircle Law). Connections to Free Probability.
2. Concrete ensembles of random matrices (GUE, circular, and Beta ensembles). Bulk and edge asymptotics via exact computations. Connection to determinantal point processes.
3. Unitary invariant Hermitian matrices. Interlacing arrays of reals and their boundary.
4. Dynamics on matrices and spectra. Dyson’s Brownian Motion.
5. Universality of random matrix asymptotics.
6. (optional) Discrete analogues of random matrix models: random permutations, random tilings, interacting particle systems.
7. (optional) Applications to machine learning, neural networks.

References. There are several textbooks which I will consult while teaching the course. It is not required to buy any of them to successfully participate in the course.

1. Mehta, M.L. “Random Matrices”.
2. Anderson, G.W., Guionnet, A. and Zeitouni, O. “An Introduction to Random Matrices”.
3. Pastur, L. and Shcherbina, M. “Eigenvalue Distribution of Large Random Matrices”.
4. Tao, T. “Topics in random matrix theory”.

Course notes will be posted on this website, and updated regularly. Direct download link is https://rmt-fall2019.s3.amazonaws.com/rmt-fall2019.pdf

Grading. The course grade is based on homework and class engagement (your participation in in-class discussions; asking questions in class and at office hours; volunteering to type up homework solutions; possibly volunteering to give short expository talks detailing an aspect in the course; etc). There is no midterm or final exam.

The homework will be assigned in the course notes (look for green background). The deadline for each problem is 2.5 or 3 weeks, which means:

• if a problem relates to a lecture on Tuesday in week $n$, then it is due on Thursday on week $n+3$ (but no later than the official final exam date for the course);
• if a problem relates to a lecture on Thursday in week $n$, then it is still due on Thursday on week $n+3$ (but no later than the official final exam date for the course);

Level of homework problems ranges from easy to very difficult. It is understood that you won’t turn in all problems all the time, but putting an adequate effort into solving homework problems and communicating your solutions clearly is of paramount importance for your learning.

Homework can be submitted either by email (scan or typeset, and send; this is the preferred method); or turned in in class (in which case please still scan the homework to keep a copy).

Required official statement. All students with special needs requiring accommodations should present the appropriate paperwork from the Student Disability Access Center (SDAC). It is the student’s responsibility to present this paperwork in a timely fashion and follow up with the instructor about the accommodations being offered. Accommodations for test-taking (e.g., extended time) should be arranged at least 5 business days before an exam.

First page of the course notes