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 2030 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 betaensembles. Invariant ensembles. Unitary ensembles and determinantal point processes.
Orthogonal polynomial method. Local and global statistics. Loop equations. Dyson's Brownian motion.
Noninvariant ensembles. Semicircle law. Resolvent and combinatorial methods. General determinantal
point processes and applications.
Lectures
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 15:0017:00 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 
Noninvariant ensembles 
January 21 
Noninvariant ensembles. Loop equations. 
January 28 
Determinantal point processes. Noncolliding 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 
Examination
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)

Asymptotic zero distribution for orthogonal polynomials
Ref: A. B. J. Kuijlaars and W. Van Assche, The asymptotic zero distribution of orthogonal polynomials with varying recurrence coefficients, J. Approx. Theory 99 (1999), no. 1, 167197.
 Asymptotics through classical steepest descent methods: Hermite polynomials and GUE
 Semicircle law using moments
Ref: Anderson, Guionnet and Zeitouni, An Introduction to Random Matrices,
Guionnet,
 Selberg's integral
P. Forrester, Loggases and random matrices. London Mathematical Society Monographs Series, 34. Princeton University Press, Princeton, NJ, 2010.,
M. L. Mehta, Random matrices, Random matrices. Third edition. Pure and Applied Mathematics (Amsterdam), 142. Elsevier/Academic Press, Amsterdam, 2004. xviii+688 pp.
Forrester and Warnaar,
The importance of the Selberg integral.
Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 4, 489–534.
 Characteristic polynomials of random unitary matrices and applications
to number theory
Keating and Snaith, Commun. Math. Phys. 214, 5789, (2000)
N. Snaith, Thesis, June 2000, Bristol
 Differential equations for Fredholm determinants
Ref: Tracy, Widom, "Commun. Math. Phys., 163, 3372, (1994)
Forrester, "Loggases and random matrices"
Anderson, Guionnet, Zeitouni
 RSKcorrespondence and the Meixner ensemble
Sagan, "The symmetric group. Representations, combinatorial algorithms and symmetric functions"
Johansson, Shape fluctuations and random matrices. Comm. Math. Phys. 209 (2000), no. 2, 437–476.
 GUEs and Queues
Ref: Baryshnikov, Yu. GUEs and queues. Probab. Theory Related Fields 119 (2001), no. 2, 256–274.
 U(n) and Haar measure, the Weyl integration formula
 How to sample a random unitary matrix
Ref: Diaconis and Shahshahani, "The subgroup algorithm for generating uniform random variables",
Probab. Eng. and Info. Sci., 1987, 1, 1532
 Szegö's theorem and linear statistics
Ref: Diaconis, "Patterns in eigenvalues: The 70th Josiah Willard Gibbs Lecture", Bull. AMSA, April 2003
Johansson, "On random matrices from compact classical gropus", Ann. Math, 145, 1997,
 GUE and tridiagonal matrices
Ref: I. Dumitriu and A. Edelman, Matrix models for beta ensembles. J. Math. Phys. 43 (2002), no. 11, 5830–5847.
 Dyson's Brownian Motion
Ref: T. Tao, "Topics in random matrix theory"
 Concentration of measure
Ref: Guionnet and Zeitouni, Elect. Comm. in Probab. 5, 2000, 119136
Literature
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 RiemannHilbert approach, by Percy Deift