Conformal Approach to Modelling Random Aggregation

我的研究将为解决概率、复杂分析和数学物理之间的一个长期存在的问题做出重大贡献。重点是平面随机生长过程，它是通过粒子的连续聚集而生长的。特别令人感兴趣的是拉普拉斯模型：该模型的团簇边界的增长率由其调和度量确定。从癌症到聚合物的产生，这些疾病发生在各种物理和工业环境中。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示为晶格上的离散集合。然而，由于缺乏可用的数学技术，在这一背景下取得的进展很少。事实上，近40年来，DLA是否存在通用标度限制的问题一直是数学和物理学中的一个重要悬而未决的问题。我最近介绍了一族拉普拉斯随机增长模型，称为聚合洛夫纳演化(ALE)，其中增长的簇是使用共形映射的组合来构造的。这一系列包括上述物理发生模型的版本；但也包括我已经证明在分析上容易处理的模型。这项建议的主要目的是为ALE结构所描述的一系列增长过程建立所有参数范围的比例限制。具体目标包括：-识别团簇大规模几何结构中的相变；-证明某些参数值的波动位于Kardar-Parisi-Zhang(KPZ)普适类中；-建立随机增长和Schramm-Loewner演化(SLE)之间的关系。我提出的方法涉及将规则性结构理论中出现的技术与Loewner演化相结合。这一方法的发展有可能对概率和分析产生重大影响。

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.