Elephant random walk: sampling from memory
Elephant random walk: sampling from memory
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. At every step we choose one of the past steps uniformly at random and then the next step is taken either by repeating or flipping the chosen step according to some probability.
Such a simple model generates the interesting physical phenomenon of anomalous diffusion and demonstrates a phase transition into sub-diffusive, diffusive and super-diffusive, which can be completely classified. However, the basic simulations indicate that the sampling window is crucial in determining this phase transitions; sampling only from the recent past might not generate the behaviour anomalous diffusion. Most of the interesting questions in this field are still open.
The origin of the model is in statistical physics, and there are connections to random recursive trees in computer science and frozen percolations in statistical physics., and reinforcement models, such as, the urn models.
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 chosen according to a distribution other than the uniform. Also, one can focus on the applications of these models in statistics, physics and computer sciences.
The project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding by reading and filling in the details in the proofs, exploring examples and theoretical and practical applications of the material, and clearly communicating it in both written and oral formats.
Knowledge of Probability I and II is mandatory. Individuals who have credited the module Stochastic Processes III might have an obvious advantage. Any further knowledge of advanced topics such as martingales and urn models can contribute towards the final evaluation of the project.
Interested candidates may look into the following references, whichcan serve as very interesting introductions into the model..
We recommend that interested candidates take a look at the above mentioned references and the references therein. For any further information, queries and comments, please feel free to contact me at debleena.thacker@durham.ac.uk
Images: All credits go to Vinita Mulay, whom you can find here.