容量重标度下随机增长的标度极限和波动
Scaling limits and fluctuations for random growth under capacity rescaling
摘要：
我们评估一个强正则化版本的Hastings-Levitov模型HL $(α)$为$0\leq α<2$。以前的结果集中在小颗粒的限制，附着颗粒的大小接近零的限制。然而，我们考虑的情况下，我们重新调整整个集群的容量之前采取限制，同时保持粒子大小固定。我们首先考虑的情况下，$α=0$，并表明，根据容量重新缩放，集群的限制结构是不是一个磁盘，不像在小粒子的限制。然后，我们考虑的情况下，$0<α<2$，并表明，在相同的重新缩放的集群方法的磁盘。我们还评估的波动，并表明，当表示为一个全纯函数，它们的行为就像一个依赖于$α$的高斯场。此外，当$α$接近0和2时，该场变得简并，表明在这些值处存在相变。
英文摘要：
We evaluate a strongly regularised version of the Hastings-Levitov model HL$(α)$ for $0\leq α<2$. Previous results have concentrated on the small-particle limit where the size of the attaching particle approaches zero in the limit. However, we consider the case where we rescale the whole cluster by its capacity before taking limits, whilst keeping the particle size fixed. We first consider the case where $α=0$ and show that under capacity rescaling, the limiting structure of the cluster is not a disk, unlike in the small-particle limit. Then we consider the case where $0<α<2$ and show that under the same rescaling the cluster approaches a disk. We also evaluate the fluctuations and show that, when represented as a holomorphic function, they behave like a Gaussian field dependent on $α$. Furthermore, this field becomes degenerate as $α$ approaches 0 and 2, suggesting the existence of phase transitions at these values.
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