Scaling limits of random particle aggregation models

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In this project we investigate two-dimensional particle aggregation models. In these models we startwith 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 ofparticles. The main question we are investigating is what typical shapes large particle collectionswill take. This of course does depend on the exact parameters chosen. We are considering a range ofsuch models depending specific chosen parameters. Relevant examples include the Hasting-Levitovmodel, 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 biologywhere fractal-like growth has been observed in different contexts. Depending on the chosen modelparameters 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 inwhich under a sufficiently high voltage an electrical insulator starts acting as an electricalconductor.Despite this range of applications in many interesting models scaling limits remain unknown. Evenin those cases where scaling limits are known there are still open questions: What are theasymptotic fluctuation around these scaling limits? How universal are these limits, i.e. how far orhow little do they depend on a specific choice of microscopic particles? Can the particles berandom? If so, how "wild" can these particles be without changing the large scale observedbehaviour? 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 avery interesting phase transitions. For a large class of parameters a scaling limit is know whichmacroscopically looks very similar to a ball, although the microscopic picture is more intricate. Thishas been proven up to an upper bound on a sum of two parameters. Simulations suggest that abovethis upper bound the macroscopic picture changes dramatically. Instead of towards a ball, theseparticle aggregation clusters seems to grow strongly only in a few directions, seemingly resulting ina very complicated random tree.The main approach of this project in order to address these question is to use methods fromLoewner Chain theory from complex analysis and Schramm-Loewner evolutions. Loewner Chainsare 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 describerandom interfaces. They have successfully proven the be the scaling limits of loop erased randomwalks, Peano curves the in the uniform spanning tree and boundary interfaces in the critical twodimensionalIsing model for magnetisation and percolation models. Schramm-Loewner evolutionsare relatively easy to describe using the theory of Loewner Chain. Moreover, they possess manyremarkable 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, astarting point and an end point there is only one canonical SLE in this domain from the specifiedstarting to the specified end point. In fact, this description behaves nicely under smooth bijectionsbetween 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 thatoutside 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 oftwo-dimensional particle aggregation models.