Scaling limits and fluctuations for random growth under capacity rescaling
Scaling limits and fluctuations for random growth under capacity rescaling
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DOI：
10.1214/20-aihp1104
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发表时间：
2019-07
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通讯作者：
George Liddle;Amanda G. Turner
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作者：
George Liddle;Amanda G. Turner
我们评估 Hastings-Levitov 模型 HL$(\alpha)$ 的强正则化版本，其中 $0\leq \alpha<2$。先前的结果集中在小颗粒极限上，其中附着颗粒的尺寸接近于零。然而，我们考虑这样的情况：在采取限制之前，我们通过其容量重新调整整个集群，同时保持粒子大小固定。我们首先考虑 $\alpha=0$ 的情况，并表明在容量重新调整下，集群的限制结构不是磁盘，这与小粒子限制不同。然后我们考虑 $0<\alpha<2$ 的情况，并表明在相同的重新缩放下，集群接近磁盘。我们还评估了波动并表明，当表示为全纯函数时，它们的行为就像依赖于 $\alpha$ 的高斯场。此外，当 $\alpha$ 接近 0 和 2 时，该场变得简并，表明在这些值下存在相变。
We evaluate a strongly regularised version of the Hastings-Levitov model HL$(\alpha)$ for $0\leq \alpha<2$. 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 $\alpha=0$ 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 $0<\alpha<2$ 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 $\alpha$. Furthermore, this field becomes degenerate as $\alpha$ approaches 0 and 2, suggesting the existence of phase transitions at these values.