Unit name | Random Matrix Theory |
---|---|

Unit code | MATH30016 |

Credit points | 10 |

Level of study | H/6 |

Teaching block(s) |
Teaching Block 2C (weeks 13 - 18) |

Unit director | Professor. Snaith |

Open unit status | Not open |

Pre-requisites |
MATH20901 Multivariable Calculus |

Co-requisites |
None |

School/department | School of Mathematics |

Faculty | Faculty of Science |

**Unit Aims**

By the end of the unit you will master some of the most important mathematical techniques used in random matrix theory and have an understanding of how these are relevant in various areas of mathematics, physics, engineering and probability.

**Unit Description**

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 some 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. There will be general discussion of how this relates to current research in various fields of mathematics and physics. The course will appeal to students in applied and pure mathematics as well as in statistics. The theory can be thought of probabilistically and so would be of interest to students focusing on probability, however the background needed is very minimal, so this should not put off students who are not so keen on probability.

**Relation to Other Units**

The material covered provides a useful background for the Level 7 unit Quantum Chaos. Some aspects of this course are related to topics presented in the Level 6 unit Statistical Mechanics.

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 with a few common matrix ensembles

Transferable Skills

- Clear, logical thinking.
- Problem solving techniques.

Lectures, homework and exercises. Notes will be made available to the students.

80% Examination and 20% homework questions

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

**Recommended**

- Peter Forrester,
*Log-gases and Random Matrices*, Princeton University Press, 2010 - Madan Lal Mehta,
*Random Matrices*, Elsevier, 2004