Random growth and SLE limits

随机生长和 SLE 限制

• 批准号: 2118765
• 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: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2118765
• 关键词: Random; growth; SLE; limits

We consider a particular growth process in the complex plane. Beginning with an initial seed, we attach a new particle to the nth cluster according to a probability density that is some negative power of harmonic measure - i.e. particles are more likely to attach themselves in areas unlikely to be hit by Brownian motion started from infinity.If a conformal growth model with particles attached according to harmonic measure can be thought of as a colony growing to gather resources from its environment, we can think of this process as modelling a colony growing in a hostile environment, where each new member wants to minimise its exposure to the elements.Some heuristic reasoning (supported by early calculations) suggests that in the extreme case (where the site of each new particle is chosen only between the points of minimum density), as the size of each particle decreases (keeping the total capacity the same), the process converges to an SLE4.In this project, we hope to prove this result for the extreme case described, and extend it to the case of a continuous density for each new particle for a sufficiently large negative parameter.

我们考虑复平面中的一个特殊的生长过程。从初始种子开始，我们按照一个概率密度将一个新粒子附加到第n个集群，该概率密度是调和度量的某个负幂--即粒子更有可能附着在不太可能受到从无穷大开始的布朗运动打击的区域。如果一个带有根据调和度量附加粒子的共形生长模型可以被认为是一个从环境中收集资源的群体，我们可以将这个过程看作是对在恶劣环境中生长的群体进行建模，一些启发式推理(由早期计算支持)表明，在极端情况下(其中每个新粒子的位置只在最小密度的点之间选择)，随着每个粒子的大小减小(总容量保持不变)，这个过程收敛到SLE4。在这个项目中，我们希望证明这个结果对于所描述的极端情况，并将其扩展到对于足够大的负参数的每个新粒子的连续密度的情况。