Conformal Approach to Modelling Random Aggregation

我的研究将为解决一个长期存在的问题，在概率，复杂分析和数学物理的接口作出重大贡献。重点是通过粒子的连续聚集生长的平面随机生长过程。特别令人感兴趣的是拉普拉斯模型：这种模型的簇边界的增长速度是由它的调和测度决定的。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。许多随机增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，在这种情况下进展很少。事实上，对于DLA是否存在一个普遍的尺度限制的问题已经在数学和物理上成为一个重要的开放问题近40年了。我最近介绍了一组拉普拉斯随机增长模型，称为聚集下层进化（ALE），其中生长簇是使用保形映射的组合构造的。这个系列包括上述物理模型的版本；还包括我所展示的易于分析的模型。该提案的主要目的是在ALE结构所描述的生长过程家族的所有参数范围内建立缩放限制。具体目标包括：-在团簇的大规模几何结构中识别相变；-证明波动存在于某些参数值的kardar - paris - zhang （KPZ）普适类中；-建立随机生长与Schramm-Loewner进化（SLE）之间的关系。我提出的方法包括将正则结构理论中的技术与洛厄纳演化相结合。这种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.