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Random Matrix Theory (MATH 33720)

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Administrative Information

  1. Unit number and title: MATH 33720 Random Matrix Theory
  2. Level: H/6
  3. Credit point value: 20 credit points
  4. Year: 08/09
  5. First Given in this form: 2005-06
  6. Unit Organiser: Francesco Mezzadri
  7. Lecturer: Dr F Mezzadri
  8. Teaching block: 1
  9. Prerequisites: Level 1 Probability and Calculus 2

Unit aims

At the end of the unit you will master the most important mathematical techniques used in random matrix theory and will be able to apply them to solve problems that may arise in various areaas of mathematics, physics, engineering and probability.

General Description of the Unit

Random matrices are often used to study the statistical properties of systems whose detailed mathematical description is either not known or too complicated to allow any kind of successful approach. It is a remarkable fact that predictions made using random matrix theory have turned out to be accurate in a wide range of fields: statistical mechanics, quantum chaos, nuclear physics, number theory, combinatorics, wireless telecommunications and structural dynamics, to name only few examples.

Particular emphasis will be given to computing correlations of eigenvalues of ensembles of unitary and Hermitian matrices. Different ensembles have distinct invariance properties, which in the applications are used to model systems whose physical or mathematical behaviour depends only on their symmetries. In most cases the dimension of the matrices will be treated as a large asymptotic parameter. In addition we will develop several techniques to compute certain types of multiple integrals. For each topic several examples of applications to different branches of mathematics and physics will be given. The course will appeal to students in applied and pure mathematics as well as in statistics.

Relation to Other Units

The material covered provides a useful background for the level 4 unit Quantum Chaos. Some aspects of this course are related to topics presented in the level 4 unit Statistical Mechanics.

Teaching Methods

Lectures, problem classes, honework exercises and matlab/maple numerical projects. Notes will be made available to the students.

Learning Objectives

After completing this unit successfully you should be able to:

  • Define and comprehend the notions of spectral statistics for various matrix ensembles.
  • Compute typical examples of spectral statistics.
  • Recognize and compute few types of matrix itegrals.
  • Use matrix integrals to solve typical physical and mathematical problems
  • Write simple codes in Matlab/Maple to simulate elementary matrix models.

Assessment Methods

The final mark for Random Matrix Theory is cacluated as follows:

  • 10% from a Matlab/Maple numerical assignment.
  • 90% from a 2½-hour written examination in April consisting of FIVE questions. The questions will assess your knowledge and comprehension of the material taught during the course and your ability to apply the mathematical techniques learnt to concrete physical problems. Your best FOUR answers will be used for assessment. Calculators are NOT permitted to be used in this examination.

Award of Credit Points

Credit points are gained by:

  • either passing the unit;
  • or getting an assessment mark of 30 or over and also handing in satisfactory attempts of 75% of homework

Transferable Skills

  • Clear, logical thinking.
  • Problem solving techniques.
  • Ability to design numerial simulations for simple models.

Texts

There is no recommended text but two useful references are:

  • Madan Mehta. Random Matrices, Elsevier, 2004
  • Peter Forrester. Log-gases and random matrices, http://www.ms.unimelb.edu.au/~matpjf/matpjf.html1
  • Francis Wright. Computing with Maple, CRC Press, 2001
  • Erwin Kreyszig. Maple computer Guide: a self-contained introduction for advanced engineering mathematics, Wiley, 2001

1This book will be published in the near future, but a preprint is available on the author's homepage.

Syllabus

  1. Introduction: Applications of random matrix theory and examples of spectral distributions. 1 lecture
  2. Elementary notions of matrix theory. 2 lectures.
  3. Elementary notions of probability and measure theory. 3 lectures.
  4. The Poisson process and its spacing distribution. 1 lecture.
  5. The 2 x 2 Gaussian ensembles and their spacing distributions. 2 lectures.
  6. The CUE ensemble and Weyl integration formula. 3 lectures.
  7. Spectral correlation functions for the CUE ensemble. 3 lectures.
  8. Averages over the CUE and Toperplitz determinbants. 3 lectures.
  9. The COE and CSE ensembles. 2 lectures.
  10. Numerical generation of random matrices: theory and computer practises. 4 lectures.
  11. The GUE ensemble and orthogonal polynomials. 6 lectures.