Internal Diffusion Limited Aggregation(IDLA) is a type of self-organising process; and is motivated from statistical mechanics. This model was intorduced by Diaconis and Fulton, 1991 and it gives an algorithm to build a random aggregate of points. It is a simple model to describe dendritic growths.
Lawler, Bramson and Griffeath, 1992 proved that the limiting shape of an IDLA cluster for a simple random walk is a ball in dimensions d≧2, and these results were further improved by Asselah and Gaudillière, 2013a and 2013b to obtain the logarithmic fluctuation bounds for the boundary. Interestingly, one can connect IDLA to excited to the centre random walks. The IDLA on a any regular non-random tree can be easily seen as a Digital Search Algorithm on the corresponding tree, and thus is connected to computer science.
The prospects of this project includes a thorough survey of the literature on IDLA and its connections to other related models like tha rotor-router model, the DLA cluster. Furthermore, there are prospects of interesting simulations where we can consider interesting variants of random walks that are driving the IDLA process. A basic knowledge of probability theory is required. It is recommended that you revise the contents of the modules Probability I and II, and Stochastic processes. Any advanced knowledge of martingales, and hitting times of random walks that we may require can be picked up as a part of the project.
The interested candidates may look at these references, and the references therein. For further details, please write to me at debleena.thacker@durham.ac.uk.