Dependence structures in random growth

随机增长中的依赖结构

• 批准号: 2600928
• 金额: --
• 依托单位: Lancaster University
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2600928
• 关键词: Dependence; structures; random; growth

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.

该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，从癌症到聚合物的创造。这样的模型是重要的，不仅是由于其物理流行所产生的潜在应用，而且还因为与新的和快速发展的数学领域，如Schramm-Loewner演化（SLE）和规则性结构的连接。然而，由于许多这些模型表现出复杂的长期依赖性，数学分析取得了很大的进展。模拟随机增长的一种方法是将簇表示为共形映射的组合。这使得使用计算机模拟的集群的生成，并表明存在的相变从集群收敛到磁盘，集群收敛到直线。初步模拟表明，临界极限，附近的过渡点的直线，是简单的路径，可以描述为随机过程。该项目的目的是对这一转变点附近的星系团进行系统的数值研究。我们的目标是了解这些随机过程的依赖结构，特别是要确定描述集群的参数之间的关系，和依赖结构。