My research interests lie in probability theory, more precisely 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.
The images below illustrate a geometric scaling limit in which the shape of a cluster converges to an expanding disk as the size of individual particles tends to zero and the number of particles converges to infinity.
The clusters above are examples of planar random growth processes which grow by particles randomly attaching themselves to the outside of the cluster. The most famous example of one of these models is diffusion limited aggregation (DLA), a random fractal growth model which describes how minerals cluster together and 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 30 years since it was first formulated. A lot of my research concerns a class of random growth processes known as Hastings-Levitov models which use conformal mappings to model the arrival of particles. This enables the use of techniques from complex analysis that have been very fruitful in areas such as Schramm–Loewner evolution (SLE). By altering the sizes of particles and their positions in various ways, a wide class of growth models can be described, which include both DLA and the Eden model for biological growth. I am particularly interested in the limiting behaviour of these clusters when the particle sizes are very small, but large numbers of particles are present and, in several cases, have been able to obtain detailed descriptions of the limiting shapes and internal structures that arise.
Some examples of the clusters that I am studying are illustrated on the simulations page.
A list of my publications can be found here.