Random growth and SLE limits

我们考虑复平面上的一个特殊生长过程。从一个初始种子开始，我们根据一个概率密度将一个新粒子附着到第n个簇上，这个概率密度是谐波测度的负次方——也就是说，粒子更有可能附着在不太可能被从无穷远开始的布朗运动击中的区域。如果一个带有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，我们可以把这个过程看作是一个在恶劣环境中生长的群体的模型，在这种环境中，每个新成员都想尽量减少暴露在环境中的机会。一些启发式推理（由早期计算支持）表明，在极端情况下（每个新粒子的位置仅在最小密度点之间选择），随着每个粒子的大小减小（保持总容量不变），该过程收敛于SLE4。在这个项目中，我们希望在描述的极端情况下证明这一结果，并将其推广到在足够大的负参数下每个新粒子的连续密度的情况。

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.