Dependence structures in random growth

这个项目的重点是了解平面随机生长过程的长期行为。这些问题出现在物理和工业环境中，从癌症到聚合物的制造。这些模型很重要，不仅因为它们的物理普遍性带来了潜在的应用，而且还因为它们与新的和快速发展的数学领域（如Schramm-Loewner进化（SLE）和规则结构）的联系。然而，由于许多这些模型表现出复杂的长期依赖关系，数学分析几乎没有取得进展。一种模拟随机增长的方法是将簇表示为保角映射的组合。这使得使用计算机模拟生成星团成为可能，并表明存在从会聚到圆盘的星团到会聚到直线的星团的相变。初步的模拟表明，在过渡到直线的点附近的非临界极限是可以被描述为随机过程的简单路径。该项目的目的是对这个过渡点附近的群集进行系统的数值研究。目标是了解这些随机过程的依赖结构，特别是确定描述集群的参数与依赖结构之间的关系。

The focus of this project is on understanding the long-time behaviour of planar random growth processes. These arise in physical and industrial settings, from cancer to polymer creation. Such models are important, not only due to potential applications arising from their physical prevalence, but also because of connections with new and rapidly evolving areas of mathematics such as Schramm-Loewner Evolution (SLE) and regularity structures. However, as many of these models exhibit complicated long-range dependencies, mathematical analysis has yielded very little progress. One approach to modelling random growth is to represent clusters as compositions of conformal mappings. This enables the generation of clusters using computer simulation and indicates the existence of phase transitions from clusters that converge to disks, to clusters that converge to straight lines. Preliminary simulations suggest that off-critical limits, near the point of transition to straight lines, are simple paths which can be described as stochastic processes. The aim of this project is to undertake a systematic numerical study of clusters near this point of transition. The objective is to understand the dependency structure of these stochastic processes and in particular to identify a relationship between the parameter describing the cluster, and the dependency structure.