Ergodic theorems with rates and large deviations on lattices

The Department of Mathematics at the University of Manchester seeks to find a PhD student in Probability Theory to work under the supervision of Dr Sebastian Andres on a research project on random walks in random environment and related probabilistic or analytic objects. 

From the law of large numbers (LLN) we know that the average of a large collection of i..i.d. random variables converges almost surely to their expectation. The classical ergodic theorem is a generalisation of the LLN for correlated random variables satisfying certain conditions (they need to be stationary and ergodic). This can also be extended to higher dimensions, meaning that we can also consider ergodic random variables on the sites of the d-dimensional lattice and take averages of random variables in a large cubes, which then converge almost surely to a deterministic limit. Two possible directions for research include:

1) Study the rate of convergence in the ergodic theorem on lattices in higher dimensions. This will require additional so-called mixing assumptions. In one dimension, such a results has already been shown.

2) By Cramer’s theorem the probability that the average of a large collection of i..i.d. random variables (with nice integrability properties) deviates from their mean converges exponentially fast to zero. This is called a large deviation principle. Extend Cramer’s theorem to the setting of ergodic random variables on d-dimensional lattices under mixing conditions.

There are many very good books and lecture notes on basic ergodic theory (for instance Chapter 9 oil Kallenberg’s book) and large deviations, so the basic methods can be learned rather quickly, so it would be possible for interested students to work on new problems shortly into their PhD studies.

The successful candidate will have completed (or nearly completed) an MSc degree in Mathematics (or closely related field) and will have demonstrated a capacity to produce mathematical research. Priority will be given to candidates with an advanced skill set in probability theory, PDE analysis or other related fields.

Shortlisted candidates will be invited to be interviewed by a small panel via Zoom, which will take place shortly after the application deadline. During the interview we seek to understand your motivation, aptitude and present knowledge.We will ask questions to discover these things, including some technical mathematical questions about topics you have covered.

You can apply online via the following link:

https://www.manchester.ac.uk/study/postgraduate-research/programmes/list/08191/phd-probability/

You need to submit:

1) a CV (including contact details of two academic references),

2) copy of MSc degree (or equivalent, or evidence of the expected date of obtaining) in Mathematics (or closely related field),

3) transcripts of grades in English,

4) a personal statement (of maximal one page) indicating interest in this research topic; here candidates can point out anything they would like us to know which is not covered by the other documents;

5) copy of your passport if a visa will be needed to study in the UK,

6) evidence of English language ability if English is not the first language (can be obtained at a later stage). 

For further enquiries, contact Dr Sebastian Andres: [Email Address Removed]

 - O. Kallenberg. Foundations of Modern Probability, Springer 1997.

 - V. Konarovskyi. An introduction to Large Deviations. Lecture notes, available at http://www.math.uni-leipzig.de/~konarovskyi/teaching/2019/LDP/pdf/LDP.pdf