PhD Studentship in Eigenvalues of large random matrices: Characteristic polynomials and fluctuations

Summary:A fully-funded PhD studentship in the Department of Mathematics for three and a half years

Overview: A fully-funded 3-and-a-half year PhD position is available in the Department of Mathematics in the School of Mathematical and Physical Sciences at the University of Sussex.

You will receive:
- fully-funded tuition fees for 3 and a half years
- a tax free bursary for living costs for 3 and a half years. For 2018/19 this is £14,777 per year
- a research training support grant for 3 and a half years of £1,250 per year

PhD project:
A fully-funded 3-and-a-half year PhD position is available in the Department of Mathematics in the School of Mathematical and Physical Sciences at the University of Sussex.

Take a large N x N matrix and fill the entries with random numbers from a given probability distribution. What can be said about statistics of one or a few of its eigenvalues / eigenvectors? This is the basic question of random matrix theory, a subject that has seen spectactular developments over the last couple of decades. Typical research directions the above question leads to are very diverse, some traditional areas include quantum chaos, zeros of the Riemann zeta function and statistical mechanics. More recently, there are notable appearances of random matrices in "pure" disciplines, in probability theory but also in enumerative combinatorics, representation theory, or integrable PDEs.

Below are some possible directions for the PhD, but these can be discussed with the supervisor depending on the student’s own interests and expertise.

1) Characteristic polynomials of random matrices. The goal will be to study expectations of products (or ratios) of characteristic polynomials for various examples of random matrices. To develop a theory to describe the expectations in the large matrix size limit and thereby resolve some open problems arising in quantum chaos.

2) Planar random matrix models. This project will study statistical properties of eigenvalues of matrices with no symmetries (so their eigenvalues can be spread out in the complex plane). The goal will be to uncover new relations to areas such as potential theory, planar orthogonal polynomials, integrable non-linear ODEs and special functions.

3) Extreme values and fluctuations. This project is more probabilistic and would initially focus on proving central limit theorems (Gaussian behaviour) for statistics of random matrices. The goal will be to eventually understand statistics of rare events (large deviations) and extreme value statistics.

Procedure:
Apply online at https://www.sussex.ac.uk/study/phd/apply .

Select the PhD in Mathematics with a September 2019 start date.

In the Finance section, you should enter the name of the studentship, which is: Eigenvalues of large random matrices: Characteristic polynomials and fluctuations

Be sure to supply all of the required documents, particularly your transcripts and the details of two referees.

Due to the high volume of applications received, you may only hear from us if your application is successful

Contact:
For practical questions about applications and/or eligibility for funding, please contact Rebecca Foster at: [Email Address Removed]

For academic questions please contact the supervisor of this project, Dr Nick Simm: [Email Address Removed].