[School of Natural Sciences PhD Scholarships] Cutoff phenomenon in Markov chains

About the project

About the Project Cutoff phenomenon for Markov chains is an abrupt transition where a process goes from being very far away from stationarity to very near stationarity within a short window of time. Formally it is defined for a sequence of Markov chains indexed by N, which is a measure of the size of the system modelled by the Markov chain, and is assumed to get larger and larger. It was first discovered about four decades ago while studying classical card-shuffling models (like random transpositions of a deck of n cards). Today it is widely believed to be a universal feature of high-dimensional, fast-mixing stochastic processes, such as random walks on complex networks, and interacting particle systems. The aim of this project is to study the speed of convergence to stationarity and cutoff phenomenon for some particular Markv chain models. Depending on the preferences of the candidate the models could come from application areas, such as communication networks, epidemic spread, population growth, MCMC algorithms. The project has a theoretical component (using existing probabilistic techniques and developing new techniques to prove cutoff or absence of cutoff) and a computational component (demonstrating presence or absence of cutoff numerically). This project is expected to start in September 2027. Before you apply: We strongly recommend that you contact the supervisors for this project before you apply. How to apply: To be considered for this project you must complete a formal application through our online application portal. If you already have an applicant account this link will directly open an application for PhD School of Natural Sciences Scholarships . If you don’t already have an applicant account, please follow the instructions here . When applying, please specify the full title and supervisor/s of the project, details of your previous study, and names and contact details of two referees. You must also upload a Supporting Statement describing your motivation to apply to the project, your CV and transcripts of awarded and in-progress university qualifications . Please note late or incomplete applications will not be considered. Equality, diversity and inclusion are fundamental to the success of The University of Manchester and central to all our activities. A diverse research community strengthens creativity, productivity and quality, while increasing the societal and economic impact of our work. We welcome applicants from all career paths, backgrounds and sections of the community, regardless of age, disability, ethnicity, gender, gender expression, sexual orientation or transgender status. We welcome applications from candidates returning to study after a career break or experience in other roles. Flexible study arrangements may be available, including part-time study at 50%, 60% or 80%, subject to the requirements of the project and funder. Eligibility : The standard academic entry requirement for this PhD is an upper second-class (2:1) honours degree (or international equivalent) in Mathematics OR any upper-second class (2:1) honours degree and a Master’s degree at merit (or international equivalent) in Mathematics. This project will remain open until filled. If your application is submitted by 1st November 2026, you can expect a decision by 18th December 2026. If your application is submitted by 15th January 2027, you can expect a decision by 30th March 2027. Self or externally funded students can also be considered for this project. FSESoNS

Research areas