Elephant random walks are a very natural generalisation of random walks, where the next step of the walk is sampled from the past memory of the walker. That is, at every step we choose one of the past steps uniformly at random and then the next step is taken either by repeating the chosen step or by flipping it. 

Mathematically, such a simple model generates very interesting and unusual phenemenon. The first interesting feature to note is that the corresponding random walk can be sub-diffusive, diffusive and super-diffusive depending on the probability of flipping the chosen step.  Another interesting feature is that this phase transition into the sub-diffusive and super-diffusive regimes depend on whether we sample from the recent past or the distant past.  One of the biggest conjecture is that if we sample from the distant past, then the model exhibits this phase transition, while if the sampling is from recent past the fluctuations behave like a classical random walk, with suitable mean and variance. Most of the interesting questions in this field are still open and so, simulating the several variants of the elephant random walks make an interesting project.

The origin of the model is in statistical physics, where elephant random walk was introduced to model anomalous diffusion. The elephant random walk can also be linked to random recursive trees in computer science and frozen percolations in statistical physics. There is a simple coupling of elephant random walks to urn models and hence to clinical trials.

Due to its connections to various other fields, the project topic has the potential to lead to a multitude of directions, which include learning the mathematics behind the different related models as well simulating various interesting variants of the model. For example, one can study the when the steps are chosenaccording to a distribution other thn the uniform. Also, one can focus on the applications of these models in statistics, physics and computer sciences.

Pre-requisites

Knowledge of Probability II, Markov chains II, and Stochastic Processes III is appreciated. If the candidate is studying Advanced Probability IV in year 4, it might be helpful. Any further knowledge of advanced topics such as martingales and urn models can contribute towards the final evaluation of the project. 

References

Interested candidates may look into the following references, whichcan serve as very interesting introductions into the model..  

• Kursten (2015) 

• Baur and Bertoin (2016)

• Gut and Stadmuller (2021) 

We recommend that interested candidates take a look at the above mentioned references and the references therein.  For any further information, please feel free to contact either/ both of us at ostap.hryniv@durham.ac.uk and debleena.thacker@durham.ac.uk

Images: All credits go to Vinita Mulay, whom you can find here.