Conformal Approach to Modelling Random Aggregation

我的研究将作出重大贡献，解决一个长期存在的问题，在接口的概率，复杂的分析和数学物理。重点是在平面随机增长过程中增长的连续聚集的颗粒。特别感兴趣的是拉普拉斯模型：模型的集群边界的增长率是由其调和措施。这些问题出现在各种物理和工业环境中，从癌症到聚合物制造。例子包括：-扩散限制聚集（DLA）; -生物细胞生长的伊甸园模型;-闪电放电的介电击穿模型。许多随机生长模型最初被表述为晶格上的离散集。然而，由于缺乏可用的数学技术，在这方面的进展很少。事实上，DLA是否存在一个普适的标度极限的问题在近40年来一直是数学和物理学中的一个重要的开放问题。我最近介绍了一个家庭的拉普拉斯随机增长模型称为聚合Loewner进化（ALE），其中不断增长的集群构造使用的共形映射的组合。这个族包括上述物理发生的模型的各种版本，但也包括我已经证明的易于分析的模型。本建议的主要目的是建立缩放限制在所有参数范围内的家庭的增长过程所描述的ALE建设。具体目标包括：-识别相变的大规模几何形状的集群; -证明的波动在于在Kardar-Parisi-Zhang（KPZ）的普适性类的某些参数值;-建立随机增长和Schramm-Loewner演化（SLE）之间的关系。我提出的方法涉及到结合技术所产生的规律性结构的理论与Loewner演化。这种方法的发展有可能在概率和分析方面产生重大影响。

My research will make a major contribution to solving a long-standing problem at the interface of probability, complex analysis and mathematical physics. The focus is on planar random growth processes which grow by successive aggregation of particles. Of specific interest are Laplacian models: models for which the rate of growth of the cluster boundary is determined by its harmonic measure. These arise in a variety of physical and industrial settings, from cancer to polymer creation. Examples include:- diffusion-limited aggregation (DLA); - the Eden model for biological cell growth;- dielectric-breakdown models for the discharge of lightning.Many random growth models were originally formulated as discrete sets on a lattice. However, progress is sparse in this setting owing to the lack of available mathematical techniques. Indeed, the question of whether there exists a universal scaling limit for DLA has been an important open problem in both mathematics and physics for almost 40 years. I have recently introduced a family of Laplacian random growth models called Aggregate Loewner Evolution (ALE) in which growing clusters are constructed using compositions of conformal mappings. This family includes versions of the physically occurring models above; but also models which I have shown to be analytically tractable. The main aim of this proposal is to establish scaling limits across all parameter ranges for the family of growth processes described by the ALE construction. Specific objectives include:- identifying phase transitions in the large-scale geometry of the clusters; - proving that the fluctuations lie in the Kardar-Parisi-Zhang (KPZ) universality class for certain parameter values;- establishing the relationship between random growth and Schramm-Loewner Evolution (SLE).My proposed methodology involves combining techniques arising in the theory of regularity structures with Loewner evolution. The development of this methodology has the potential to make significant impacts in probability and analysis.