Course on Random Matrix Theory

Mathematics Department, Royal Institute of Technology (KTH)

Latest news: A list of proposed projects is added

Course Description

Random matrix theory has a history going back to mathematical statistics in the 1920's and nuclear physics in the 1950's. In the last 20-30 years the subject has expanded considerably and is now a broad research area. It has connections to many parts of mathematics and other fields, e.g. asymptotic analysis, combinatorics, number theory, statistics, statistical physics, numerical analyis, and communications theory. This means that one can come to the area of random matrices from many different points of view and there are many possible approaches to the subject. The course will cover some basic aspects of random matrices with an emphasis on basic mathematical concepts and techniques.

Approximate Content

Introduction. Ensembles. Statistics of eigenvalues and eigenvectors. Coulomb gas and beta-ensembles. Invariant ensembles. Unitary ensembles and determinantal point processes. Orthogonal polynomial method. Local and global statistics. Loop equations. Dyson's Brownian motion. Non-invariant ensembles. Semi-circle law. Resolvent and combinatorial methods. General determinantal point processes and applications.


There will be about 8 lectures by Maurice Duits and Kurt Johansson followed by lectures by participants.

Schedule and location:

All lectures will be on Thursday from 15:00-17:00 and will take place in Seminar Room 3721

Date Lecture
October 29 Introduction
November 5 Invariant Ensembles
November 12 Invariant Ensembles
November 19 Invariant Ensembles
December 3 Invariant Enesmebles
January 14 Non-invariant ensembles
January 21 Non-invariant ensembles. Loop equations.
January 28 Determinantal point processes. Non-colliding paths.
February 11 Presentations by participants
Hu, Nedic
February 18 Presentations by participants
Wennman, Roos
Ferbruary 25 Presentations by participants
Amini, Berggren, Moosavi
March 3 Presentations by participants
Potka, Schoug, Zickert


The examination consist of two parts. A presentation and exercises.
Part I :

Solve the exercises .

Hint for 3b). Use Hadamard's inequality on an appropriate Fredholm Determinant.

Part II :

Eech student is asked to give 30 mins presentation. We will also ask you to write a short summary of your presentation. This does not have to be long, but it should contain a comprehensive overview of your presentation.

Projects (the ones in red are already taken)


During the lecture handouts will be provided.

For the interested reader, we recommend the following books

  • An Introduction to Random Matrices, by Greg Anderson, Alice Guionnet, Ofer Zeitouni
  • Topics in Random Matrix Theory, by Terry Tao
  • Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert approach, by Percy Deift