LONG-TIME BEHAVIOUR OF AGGREGATION MODELS

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Introduction. Aggregation-type models appear naturally in many areas of science. They were used to describe a range of diverse phenomena such as the formation of aerosols, colloidal aggregation, electrodeposition, bacteria and lichen growth. These processes occur in nature ad motivate the introduction of a range of models such as DLA, Eden model, coagulation and coagulation-fragmentation models, coalescent random walks. This project is mostly motivated by a model which describes protein clustering on a bacterial cell membrane. This model, although it is more amenable to simulation than coarse-grained molecular dynamics models, is closely related to the diffusion-limited aggregation and might be as hard to analyse as DLA itself.Aims and objectives. The main objective is to introduce and study a range of aggregation models that should be simpler to analyse but that could still capture some qualitative behaviour of the underlying biological systems. We will start with a very simplified one-dimensional model. We start with single-particle aggregates in Z which perform independent random walks. When aggregates collide, they stick to each other and a new large aggregate continues to perform a random walk independently of all other clusters. If instead of aggregation on of the particles is killed, then the resulting processes a well-known coalescent random walk model. Despite the simple formulation, even the coalescent random walk is a non-trivial model. It is known that this model has a scaling limit which is surprisingly non-Poissonian. This model is closely related to the Brownian web. In this model we add one more level of complexity: we have to account for the size of the aggregate. Later on, we will study further modification of the model. One direction is to change diffusivity of clusters depending on their size. After that we will study a similar model in Z^2. In this model aggregates perform two-dimensional random walk but aggregate only if they collide in the horizontal direction. If they collide in the vertical direction, then they reflect. This is a simplified model of protein aggregation which can attach only if they touch at the right spot. The main aim is to understand the large-scale behaviour of these models. Basic questions will be: what is the distribution of cluster sizes and gaps between them? What is the scaling limit of these cluster configurations? How to describe the dynamics of these limits?Novelty and methodology. Although some aspects of the proposed model are related to other well-known models, the model itself is new. Some preliminary analysis could borrow many ideas from the theory of coalescent random walks and Brownian webs, but further analysis will require new ideas and techniques.EPSRC research areas. This project falls within EPSRC research areas 'Statistics and Applied Probability' and 'Mathematical Physics'.