Scaling limits of random growth processes

平面随机生长过程在物理世界中广泛存在。例子包括用于矿物沉积的扩散限制聚集（DLA）和用于生物细胞生长的Eden模型。对这种过程进行数学建模的一种方法是将随机生长的簇表示为共形映射的组合。1998年，Hastings和Levitov提出了一个这样的模型家族，其中包括上述物理过程的不同版本。在早期的工作中，诺里斯和特纳表明，最简单的黑斯廷斯-列维托夫模型的缩放极限是在小粒子极限下的一个不断增长的圆盘。这是一个极限，在这个极限中，每个粒子的容量趋于零，而粒子的数量被调整，从而使集群的总容量保持不变。相反，这个博士项目的目标是在整个集群按其总容量重新缩放的限制中获得扩展限制。

Planar random growth processes occur widely in the physical world. Examples include diffusion-limited aggregation (DLA) for mineral deposition and the Eden model for biological cell growth. One approach to mathematically modelling such processes is to represent the randomly growing clusters as compositions of conformal mappings. In 1998, Hastings and Levitov proposed one such family of models, which includes versions of the physical processes described above. In earlier work, Norris and Turner showed that the scaling limit of the simplest of the Hastings-Levitov models is a growing disk in the small-particle limit. This is the limit in which the capacity of each particle tends to zero while the number of particles is tuned so that the total capacity of the cluster remains constant. This PhD project will instead aim to obtain scaling limits in the limit where the whole cluster is rescaled by its total capacity.