Phase transitions in urn models via stochastic approximations
D. Thacker

Stochastic approximations (SA) are widely popular due to the applications in neural networks, game theory and finite colour urn models with applications to finance and pandemic data (COVID 19)(see Aletti and Crimaldi, 2022). Recently,in Laruelle and Pagés (2019) the phase transitions in non-linear urn models have been proved using SA. Mailler and Villemonais (2020) developed new techniques for SA on non-compact support to study infinite colour(dimesional) urn models.

The primary parameters responsible for phase transition in urn models are the replacement matrix, the reinforcemnet function and the dimension of the colour space. In this project, we can study the effects of these parameters in details. The prospects of this project include

• A survey of SA techniques, especially those applicable in urn models .
• Possible extensions of SA techniques to infinite dimensions.
• Simulating urn profile and other related statistics to illustrate the phase transitions related to the various parameters . For example, the simulation below illustrates that the super-linear infinite urn associated with random walks does not fixate. This is in contrast with edge-reinforced random walks, where it is well-known the process fixates for super-linear reinforcement.

Prerequisites

A basic knowledge of probability theory is required. The advanced knowledge of martingale theory that we may require can be picked up as a part of the project.

References

• Laruelle and Pagés, 2019.
• Mailler and Villemonais, 2020.
• Aletti and Crimaldi, 2022.
• Mahmoud Hosam. M. Pólya urn models. CRC Press, 2009.

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.