Self-interacting random walks and random environment

   Department of Mathematical Sciences

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  Dr Daniel Kious, Prof Hendrik Weber  No more applications being accepted  Competition Funded PhD Project (European/UK Students Only)

About the Project

The University of Bath is inviting applications for the following PhD project in Probability. The student would work with Dr Daniel Kious and Professor Hendrik Weber.

The topic of the thesis will be self-interacting random walks and random environment. The goal is to study the asymptotic behaviour of random walks in evolving environment, such as the random walk on the exclusion process, which is defined as follows. First, the exclusion process is a system of particles, where each site is attributed either 0 or 1 particle at time 0; then the particles perform independent continuous-time simple random walks, with the rule that if a particle tries to jump on another particle, then this jump is cancelled. Now, let an exclusion process evolve on the integer line. Start an additional random walk from 0, which will take a step to the right or to the left at each unit of time. The transition probability of the random walk will depend on the state of the exclusion process at the time of the jump: if the random walk is on a particle, it will jump according to a law A, and if it is on a whole then it will jump according to a law B. We would like to study the asymptotic behaviour of the position of the random walk. It is believed that, for some choice of the parameters, the random walk can have an atypical scaling, and present a super-diffusive behaviour. Also, it is still open whether this random walk can have a transient zero-speed regime, or not.

The work of this thesis will be to tackle questions related to this model, building on a recent work of Hilario, Kious and Teixeira, where the authors obtained Law of Large Numbers and Central Limit Theorems for the random walk on the exclusion process. The student will possibly be asked to work on other models of self-interacting random walks, such as reinforced random walks, random walks in random environment, or the competing frog model on the lattice and on trees, among other possibilities.


Applicants should hold, or expect to receive, a First Class or high Upper Second Class UK Honours degree (or the equivalent qualification gained outside the UK) in a relevant subject. A master’s level qualification would also be advantageous. Non-UK applicants must meet our English language entry requirement


Informal enquiries should be directed to Dr Daniel Kious, [Email Address Removed].

Formal applications should be made via the University of Bath’s online application form:

Please ensure that you quote the supervisor’s name and project title in the ‘Your research interests’ section.

More information about applying for a PhD at Bath may be found here:

Anticipated start date: 28 September 2020.

Funding Notes

UK and EU candidates applying for this project will be considered for a University Research Studentship which will cover UK/EU tuition fees, a training support grant of £1,000 per annum and a tax-free maintenance allowance at the UKRI Doctoral Stipend rate (£15,009 in 2019-20) for a period of up to 3.5 years.


Random walk on the simple symmetric exclusion process, M. Hilario, D. Kious and A. Teixeira (2019), arXiv:1906.03167.

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