# Random Matrix Theory

## 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.

## 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 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.

## Syllabus

- Introduction: Applications of random matrix theory and examples of spectral distributions.
- Elementary notions of matrix theory.
- Elementary notions of probability and measure theory.
- The Poisson process and its spacing distribution.
- The 2 x 2 Gaussian ensembles and their spacing distributions.
- The CUE ensemble.
- Spectral correlation functions for the CUE ensemble.
- Techniques for calculating spectral statistics

## Reading and References

There is no recommended text but two useful references are:

- Madan Mehta. Random Matrices, Elsevier, 2004
- Peter Forrester. Log-gases and random matrices, Princeton University Press, 2010

**Unit code:** MATH30016

**Level of study:** H/6

**Credit points:** 10

**Teaching block (weeks):** 2 (13-18)

**Lecturer:** Dr Nina Snaith

## Pre-requisites

MATH20901 Multivariable Calculus

## Co-requisites

None

## Methods of teaching

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

## Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

- 80% from a 1 hour 30 minute exam in May/June
- 20% homework assignments

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.