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.

Lectures

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

Schedule and location:

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, 167-197.
  
• Asymptotics through classical steepest descent methods: Hermite polynomials and GUE
• Semi-circle law using moments
  Ref: Anderson, Guionnet and Zeitouni, An Introduction to Random Matrices, Guionnet,
  
• Selberg's integral
  P. Forrester, Log-gases 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, 57-89, (2000)
  N. Snaith, Thesis, June 2000, Bristol
• Differential equations for Fredholm determinants
  Ref: Tracy, Widom, "Commun. Math. Phys., 163, 33-72, (1994)
  Forrester, "Log-gases and random matrices"
  Anderson, Guionnet, Zeitouni
  
• RSK-correspondence 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, 15-32
  
• 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, 119-136
  

Literature

During the lecture handouts will be provided.

For the interested reader, we recommend the following books