Lecturer in Mathematics and StatisticsDepartment of Mathematics and Statistics
Tel: +44 1524 593948
Email: a.g.turner at lancaster.ac.uk
My research interests lie in probability theory, more precicely in scaling limits of stochastic processes. The simplest example of a scaling limit of a random process is the law of large numbers for random variables which states that the average value of a sequence of independent identically distributed random variables converges to a deterministic limit as the length of the sequence tends to infinity. More generally one can ask what happens to the behaviour of a random process whose jump sizes tend to zero as the jump rate tends to infinity. Such processes arise in a variety of contexts, in particular in physical settings where the process describes the random molecular behaviour on a microscopic level, and the existence of scaling limits corresponds to stable behaviour on a macroscopic level. The molecular behaviour can be modelled as a stochastic jump process, where the jump sizes are proportional to atomic lattice spacing or particle sizes, or inversely proportional to the number of particles.
A particular model that I am interested in the moment is diffusion limited aggregation (DLA). This is a random fractal growth model, prevalent in nature, which has many applications in physics and chemistry as well as industrial processes. This model is notoriously difficult to understand, with only one rigorous result having been proved in the 25 years since it was first formulated. In collaboration with James Norris, I have been studying a version of DLA known as Hastings-Levitov DLA which uses conformal mappings to model the arrival of particles. We have established a surprising connection between a simplification of this model and an object known as the Brownian web, and hope to be able to extend our results to more complicated models, with the long-term goal being a universal description of DLA.