Scaling limits of random particle aggregation models

In this project we investigate two-dimensional particle aggregation models. In these models we start
with a single particle, which is usually a unit disc, to which another particle attaches randomly.
After each attachment a new particle appears and attaches itself to the already existing collection of
particles. The main question we are investigating is what typical shapes large particle collections
will take. This of course does depend on the exact parameters chosen. We are considering a range of
such models depending specific chosen parameters. Relevant examples include the Hasting-Levitov
model, the Eden model and hopefully diffusion-limiting aggregation (DLA).
Motivation to look at these kind of models comes from the physical sciences as well as biology
where fractal-like growth has been observed in different contexts. Depending on the chosen model
parameters these models are used to describe for instance growth of bacterial colonies,
electrodeposition, disposition of materials and the dielectric breakdown. The last item is an effect in
which under a sufficiently high voltage an electrical insulator starts acting as an electrical
conductor.
Despite this range of applications in many interesting models scaling limits remain unknown. Even
in those cases where scaling limits are known there are still open questions: What are the
asymptotic fluctuation around these scaling limits? How universal are these limits, i.e. how far or
how little do they depend on a specific choice of microscopic particles? Can the particles be
random? If so, how "wild" can these particles be without changing the large scale observed
behaviour? Are there phase transitions in these models? What would cause such a phase transition?
Especially, the last two questions are quite interesting, as simulations do suggest the existence of a
very interesting phase transitions. For a large class of parameters a scaling limit is know which
macroscopically looks very similar to a ball, although the microscopic picture is more intricate. This
has been proven up to an upper bound on a sum of two parameters. Simulations suggest that above
this upper bound the macroscopic picture changes dramatically. Instead of towards a ball, these
particle aggregation clusters seems to grow strongly only in a few directions, seemingly resulting in
a very complicated random tree.
The main approach of this project in order to address these question is to use methods from
Loewner Chain theory from complex analysis and Schramm-Loewner evolutions. Loewner Chains
are a very powerful and general technique to describe growing compact sets in the complex plane.
Schramm-Loewner evolutions (SLE) have been introduced in statistical mechanics to describe
random interfaces. They have successfully proven the be the scaling limits of loop erased random
walks, Peano curves the in the uniform spanning tree and boundary interfaces in the critical twodimensional
Ising model for magnetisation and percolation models. Schramm-Loewner evolutions
are relatively easy to describe using the theory of Loewner Chain. Moreover, they possess many
remarkable symmetries. Both of these properties render them interesting tools to our application.
Most famously SLE-type curves are conformally invariant, which means that given a domain, a
starting point and an end point there is only one canonical SLE in this domain from the specified
starting to the specified end point. In fact, this description behaves nicely under smooth bijections
between different domains. Furthermore, SLEs are uniquely parametrised by a single positive value.
If we choose this parameter to be six, then these curves are additionally "local" which means that
outside of hitting the boundary of their domain their shape looks locally the same in every domain.
Our goal is to use these techniques and symmetries in order to obtain a deeper understanding of
two-dimensional particle aggregation models.