LONG-TIME BEHAVIOUR OF AGGREGATION MODELS

导论.聚合型模型自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普遍存在，因此引入了一系列模型，如DLA模型、Eden模型、凝聚模型、凝聚-碎裂模型、聚结随机游动等。这个项目主要是由一个模型，描述了蛋白质集群的细菌细胞膜的动机。这个模型，虽然它是更适合于模拟比粗粒度的分子动力学模型，是密切相关的扩散限制聚集，可能是难以分析DLA本身。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。当聚集体发生碰撞时，它们会相互粘在一起，一个新的大聚集体会继续独立于所有其他集群进行随机游走。如果不是聚集，而是粒子被杀死，那么所产生的过程是众所周知的聚结随机游走模型。尽管公式很简单，但即使是合并随机游走也是一个非平凡的模型。已知该模型具有令人惊讶的非泊松的标度极限。这个模型与布朗网密切相关。在这个模型中，我们又增加了一个复杂的层次：我们必须考虑聚合的大小。稍后，我们将研究对模型的进一步修改。一个方向是根据团簇的大小来改变团簇的扩散率。之后我们将研究Z^2中的一个类似模型。在这个模型中，聚集体进行二维随机游走，但只有当它们在水平方向上碰撞时才聚集。如果它们在垂直方向上碰撞，那么它们会反射。这是一个简化的蛋白质聚集模型，只有当它们在正确的位置接触时才能附着。主要目的是了解这些模型的大规模行为。基本的问题是：集群规模的分布和它们之间的差距是什么？这些群集配置的扩展限制是多少？如何描述这些极限的动态？新奇和方法。虽然所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用结合随机游动和布朗网理论的许多思想，但进一步的分析将需要新的思想和技术。该项目属于EPSRC研究领域“统计与应用概率”和“数学物理”的福尔斯。

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'.