LONG-TIME BEHAVIOUR OF AGGREGATION MODELS

导言。聚集型模型自然而然地出现在许多科学领域。它们被用来描述一系列不同的现象，如气溶胶的形成、胶体聚集、电沉积、细菌和地衣生长。这些过程在自然界中发生，并促使一系列模型的引入，如DLA模型、伊甸园模型、混凝和凝血-碎裂模型、合并随机游动。这个项目主要是由一个描述蛋白质在细菌细胞膜上聚集的模型推动的。这个模型虽然比粗粒分子动力学模型更易于模拟，但它与扩散限制聚集密切相关，可能和DLA本身一样难以分析。主要目标是介绍和研究一系列聚集模型，这些模型应该更容易分析，但仍然可以捕捉到基本生物系统的一些定性行为。我们将从一个非常简化的一维模型开始。我们从Z中的单粒子聚集体开始，它们执行独立的随机行走。当聚集体发生冲突时，它们彼此粘连，新的大型聚集体继续执行随机漫游，而不受所有其他集群的影响。如果不是聚集在一起的粒子被杀死，那么产生的过程就是众所周知的融合随机行走模型。尽管公式很简单，但即使是联合随机游走也是一个不平凡的模型。众所周知，这个模型有一个令人惊讶的非泊松比例限制。这个模型与布朗网密切相关。在这个模型中，我们增加了一个更复杂的级别：我们必须考虑聚合的大小。稍后，我们将研究对该模型的进一步修改。一个方向是根据团簇的大小改变团簇的扩散系数。之后，我们将在Z^2中研究一个类似的模型。在该模型中，聚集体执行二维随机行走，但只有在水平方向上碰撞时才聚集。如果它们在垂直方向上相撞，那么它们就会反射。这是一个简化的蛋白质聚集模型，只有当它们接触到正确的位置时才能附着。主要目的是了解这些模型的大规模行为。基本问题将是：集群规模的分布和它们之间的差距是什么？这些集群配置的扩展限制是什么？如何描述这些限制的动态？新颖性和方法论。虽然提出的模型的某些方面与其他知名模型有关，但该模型本身是新的。一些初步的分析可以借鉴结合随机游走和布朗网理论的许多想法，但进一步的分析需要新的思想和技术。EPSRC研究领域。该项目属于EPSRC的研究领域“统计与应用概率”和“数学物理”。

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'.