Ma140B - Introduction to Random Matrix Theory (Winter 2026)

InstructorYassine El Maazouz
Course webpage http://www.yelmaazouz.org/Ma140B-2026/index.html
Course code Ma/ACM/IDS 140 abc
LecturesMWF 14:00–14:55, 187 Linde Hall
E-Mail:maazouz [at] caltech [dot] edu
Office HoursW 15:05–16:00
Excluding holidays. Please email me if you cannot make any of these times.
PrerequisitesMa108ABC and Ma140A are strongly recommended.
Required Text There is no required text for this course. However, you may consult the following references:
  1. [AZ] Anderson, Guionnet, Zeitouni. An introduction to random matrices.
  2. [T] Terrence, Tao. Topics in random matrix theory.
  3. [BDS] Baik, Deift, Suidan. Combinatorics and random matrix theory.
  4. [Oxf] Akemann, Baik, Di Francesco (editors). The Oxford Handbook of Random Matrix Theory.
Syllabus The main topics of this course include: How do random matrices appear in mathematics, statistics, and theoretical physics? What are the main types of asymptotic behaviors for random eigenvalues? What tools can be used for proving asymptotic theorems?
Depending on how much progress we make, I plan to cover the following:
  1. Introduction to random matrix theory: origins, motivations, and classical ensembles (GOE, GUE, GSE).
  2. Global eigenvalue laws: Wigner semicircle, Marchenko–Pastur, and Wachter distributions.
  3. Determinantal point processes and correlation functions for the Gaussian Unitary Ensemble (GUE).
  4. Asymptotic analysis using steepest descent and contour integral methods.
  5. Local eigenvalue statistics: sine and Airy processes; Tracy–Widom distributions.
  6. Gaussian $\beta$-ensembles and approaches beyond the determinantal case ($\beta \neq 2$).
  7. Multivariate Bessel functions, matrix corners, and the Harish-Chandra–Itzykson–Zuber (HCIZ) integral.
  8. Asymptotics and universality for GUE corners; Voiculescu R-transform.
  9. Additive random matrix models: Horn’s problem and additive free convolution.
  10. Dyson Brownian motion and addition of independent Gaussian matrices; related contour integral formulas.
  11. Universality of local eigenvalue statistics across random matrix ensembles.
  12. Connections to 2D statistical mechanics models, including lozenge tilings.
The schedule of the course will announced below and continuously updated as the course develops.
Homework There will be biweekly problem sets. Collaboration (between students) on homework is encouraged. It is important to make sure you understand the solutions yourself, so you are required to write your own solutions separately. I also strongly recommend that you spend some time attempting the problems yourself first before discussing with someone else. You are required to name all your collaborators. You can use any result discussed in the lectures.
DO NOT PLAGIARIZE! The use of AI chatbots on problem sets is completely forbidden.
Please write clear and legible solutions. It is strongly recomemded to write your solutions in TeX.
Homework is due on Fridays at 23:59 PST and is to be submitted on Gradescope.
Late homework submissions will not be accepted.
Final Exam There will be a take-home final due on Monday March 20th at 23:59 PST, to be submitted on Gradescope. Late submissions will not be accepted. You are not allowed to collaborate with anyone on the take-home final.
Grading The final grade will be based 60% on the problem sets and 40% on the final exam. The lowest homework grade will be dropped.
Teaching AssistantTejas Oke