Random growth and SLE limits

我们考虑复平面上的一个特殊的生长过程。从初始种子开始，我们根据概率密度将一个新粒子附着到第n个簇中，该概率密度是调和测度的某个负幂，即粒子更有可能附着在不太可能被从无穷大开始的布朗运动击中的区域。如果根据调和测度附着粒子的共形增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模拟一个在恶劣环境中生长的殖民地，每个新成员都希望尽量减少其暴露在元素中的机会。（早期计算支持）表明，在极端情况下，（其中每个新粒子的位置仅在最小密度点之间选择），随着每个粒子的尺寸减小，在这个项目中，我们希望在所描述的极端情况下证明这个结果，并将其推广到每个新粒子的连续密度对于足够大的负参数的情况。

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.