MATH 8380: Random Matrices, Spring 2025

E-mail address for communication: lenia.petrov+rmt2025@gmail.com

Abstract

This course will explore a rich zoo of probabilistic models, with a focus on random matrices, dimer models, and statistical mechanical models of algebraic origin. Geometric structures that naturally arise in these contexts will also be discussed. The course content will be flexible and tailored to the interests of participants, making it ideal for most graduate students in the math department, second year and up.

A key feature of the course will be a "reading course" component, where weekly meetings provide focused, interactive learning sessions.

Some topics will repeat from the continuous probability-heavy course https://lpetrov.cc/rmt19/ but most of the content will be new.

Course Meetings and notes

2:00-3:15pm, Kerchof 326

Regular Lectures on Wednesdays and some Mondays

Lecture notes will be posted here (problem sets are at the end of each lecture):

1. (Mon) January 13. Moments of random variables and random matrices.
2. (Wed) January 15. Wigner's semicircle law.
3. (Wed) January 22. Gaussian and tridiagonal matrices.
4. (Wed) January 29. Title TBD.
5. (Wed) February 5. Title TBD.
6. (Wed) February 12. Title TBD.
7. (Wed) February 19. Title TBD.
8. (Wed) February 26. Title TBD.
9. (Wed) March 5. Title TBD.
10. (Mon) March 24. Title TBD.
11. (Wed) March 26. Title TBD.
12. (Wed) April 2. Title TBD.
13. (Wed) April 9. Title TBD.
14. (Wed) April 16. Title TBD.
15. (Wed) April 23. Title TBD.
16. Combined lecture notes in a single PDF (updated with some delay).

Student Presentations on Mondays in April

1. (Mon) April 7
2. (Mon) April 14
3. (Mon) April 21
4. (Mon) April 28 (tentative - depending on number of enrolled students)

Weekly individual meetings

The course spans 12 full weeks, excluding the first week, final week, and the week following Spring break when I will be traveling. Students are required to meet with me at least 8 times during these 12 weeks, with each meeting lasting approximately 45 minutes. (In-person is strongly preferred, but a zoom option is available.) These meetings are an integral part of the reading course, during which we can discuss lectures, homework problem solutions, presentation, explore research topics, or dig into any other relevant topics related to random matrices.

During the first week of class, we will establish a regular weekly meeting time for each student. While some flexibility is possible, the goal is to maintain a consistent weekly schedule throughout the semester.

Ideas for presentations or projects

1. Bisi, E., and Cunden, F.D. "λ-shaped random matrices, λ-plane trees, and λ-Dyck paths". arXiv:2403.07418 [math.PR]. Investigates spectral properties of random matrices shaped by self-conjugate Young diagrams, introducing combinatorial structures such as λ-plane trees and λ-Dyck paths, with connections to free probability theory.
2. Reference hunting: Check (and add) references to theorems and statements in the lecture notes

Books

1. Mehta, M.L. "Random Matrices". A first textbook that approaches the subject through the lens of theoretical physics.
2. Anderson, G.W., Guionnet, A. and Zeitouni, O. "An Introduction to Random Matrices". A comprehensive treatment that emphasizes probabilistic methods and stochastic analysis.
3. Pastur, L. and Shcherbina, M. "Eigenvalue Distribution of Large Random Matrices". An analytical perspective on random matrix theory, focusing on mathematical techniques.
4. Muirhead, R.J. "Aspects of Multivariate Statistical Theory". A statistical approach to random matrices.
5. Vershynin, R. "High-dimensional probability: An introduction with applications in data science". A contemporary work bridging random matrix theory with modern data science applications.
6. Baik, J., Deift, P., and Suidan, T. "Combinatorics and Random Matrix Theory". Explores the connections between random matrices and combinatorial probability, particularly in asymptotic problems.
7. Forrester, P.J. "Log-Gases and Random Matrices". A comprehensive reference work containing extensive collections of explicit formulas and results.
8. Akemann, G., Baik, J., and Di Francesco, P. (editors). "The Oxford Handbook of Random Matrix Theory". A curated collection featuring diverse perspectives from experts across the field's many applications.
9. Tao, T. "Topics in Random Matrix Theory". A pedagogical approach derived from graduate-level teaching and academic blog content.
10. Potters, M., and Bouchaud, J.P. "A First Course in Random Matrix Theory for Physicists, Engineers and Data Scientists". An accessible introduction prioritizing intuition and applications over rigorous proofs.
11. Bai, Z.D., and Silverstein, J.W. "Spectral Analysis of Large Dimensional Random Matrices". Focuses on the spectral properties of high-dimensional random matrices.

Lecture notes by various authors

1. V. Gorin
2. B. Valkó
3. T. Tao
4. F. Rezakhanlou
5. M. Krishnapur

Additional references are included in lecture notes.

Assessment ▼

The course grade is based roughly equally on three components:

1. Homework Problems

• Problems are included and mentioned in lectures, posted separately as weekly problem sets
• Each problem has a deadline of a month or until the course ends. Late assignments are not accepted. If you have special needs, emergency, or unavoidable conflicts, please let me know as soon as possible, so we can arrange a workaround.
• Problems range from easy to very difficult, not all problems need to be solved
• Submit problems by email, or communicate orally during meetings
• It is expected that you need to solve about 2 problems in each problem set

2. Presentation

• Each student gives one presentation (30 min) during April Monday sessions

3. Weekly Meetings

• Participation in 8 individual meetings required, see "Course Meetings" section above

Policies ▼

Approved accommodations

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 midterms or final exams (e.g., extended time) should be arranged at least 5 days before an exam.

Honor code

The University of Virginia Honor Code applies to this class and is taken seriously. Any honor code violations will be referred to the Honor Committee.

Collaboration on homework assignments

Group work on homework problems is allowed and strongly encouraged. Discussions are in general very helpful and inspiring. However, before talking to others, get well started on the problems, and contribute your fair share to the process. When completing the written homework assignments, everyone must write up his or her own solutions in their own words, and cite any reference (other than the textbook and class notes) that you use. Quotations and citations are part of the Honor Code for both UVA and the whole academic community.