Small-Particle Limits in a Regularized Laplacian Random Growth Model
Small-Particle Limits in a Regularized Laplacian Random Growth Model
复制标题
正则化拉普拉斯随机增长模型中的小颗粒极限
DOI：
10.1007/s00220-014-2158-y
复制
发表时间：
2013
影响因子：
2.4
通讯作者：
Amanda G. Turner
中科院分区：
文献类型：
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作者：
Fredrik Johansson Viklund;A. Sola;Amanda G. Turner
摘要我们研究了Hastings-Levitov平面随机增长的正则化版本，它模拟了由扩散粒子聚集形成的团簇。在该模型中，增长的簇是根据迭代狭缝映射定义的，其容量由下式给出
$$c_n=\mathbf{c}|\phi_{n-1}‘({\rm e}^{\sigma+i\theta_n})|^{-\α}，\quad\α\geq 0，$$cn=c|Φn-1’(eσ+iθn)|-α，α≥0，其中c>；0是第一个粒子的容量，{Φn}n是定义演化簇的合成保角映射，{θn}n是确定粒子附着位置的独立均匀角度，σ>；0是我们认为依赖于c的正则化参数。我们证明了在适当的时间重新标度下，在c→0的限制下，如果σ不太快地收敛到0，则簇收敛到具有确定容量的增长磁盘。然后，我们建立了调和度量流在较长时间段上的标度极限，表明，通过以不同的速率设α→0，该流收敛到圆上的布朗网、圆上的停止版本的布朗网或单位映射。由于调和度量流与星团内的内部分支结构密切相关，上述三种情况直观地对应于模型中无限分支的数量为1，这是一个随机数，其分布在极限为c→0。我们还给出了基于模型模拟的几个结果，其中的参数选择没有被我们严格的分析涵盖。
AbstractWe study a regularized version of Hastings–Levitov planar random growth that models clusters formed by the aggregation of diffusing particles. In this model, the growing clusters are defined in terms of iterated slit maps whose capacities are given by
$$c_n = \mathbf{c}| \Phi_{n-1}'({\rm e}^{\sigma + i \theta_n})|^{-\alpha}, \quad \alpha \geq 0,$$cn=c|Φn-1′(eσ+iθn)|-α,α≥0,where c > 0 is the capacity of the first particle, {Φn}n are the composed conformal maps defining the clusters of the evolution, {θn}n are independent uniform angles determining the positions at which particles are attached, and σ > 0 is a regularization parameter which we take to depend on c. We prove that under an appropriate rescaling of time, in the limit as c → 0, the clusters converge to growing disks with deterministic capacities, provided that σ does not converge to 0 too fast. We then establish scaling limits for the harmonic measure flow over longer time periods showing that, by letting α → 0 at different rates, this flow converges to either the Brownian web on the circle, a stopped version of the Brownian web on the circle, or the identity map. As the harmonic measure flow is closely related to the internal branching structure within the cluster, the above three cases intuitively correspond to the number of infinite branches in the model being either 1, a random number whose distribution we obtain, or unbounded, in the limit as c → 0.We also present several findings based on simulations of the model with parameter choices not covered by our rigorous analysis.