各向异性Hastings-Levitov团簇的标度极限

圆柱形 Hastings-Levitov$(0)$ 中的单臂支配时间

One-arm domination time in Cylindrical Hastings-Levitov$(0)$

研究圆柱形 Hastings-Levitov$(0)$ 模型中单臂支配时间，证明其期望与 N^2/λ^3 同阶，并给出指数尾界。

柱形Hastings--Levitov(0)过程的树完成时间的Sharp渐近行为

Sharp asymptotics for the tree-completion time in cylindrical Hastings--Levitov$(0)$

本文针对柱形Hastings--Levitov(0)聚集过程，证明其树完成时间期望与log N比值的极限为1/(2λ)，验证了前人提出的极限猜想。

Explosive growth for a constrained Hastings-Levitov aggregation model

We consider a constrained version of the HL$(0)$ Hastings--Levitov model of aggregation in the complex plane, in which particles can only attach to the part of the cluster that has already been grown. Although one might expect that this gives rise to a non-trivial limiting shape, we prove that the cluster grows explosively: in the upper half plane, the aggregate accumulates infinite diameter as soon as it reaches positive capacity. More precisely, we show that after $nt$ particles of (half-plane) capacity $1/(2n)$ have attached, the diameter of the shape is highly concentrated around $\sqrt{t\…

Loewner–Kufarev熵与Hastings–Levitov模型的大偏差

Loewner--Kufarev entropy and large deviations of the Hastings--Levitov model

本文研究了Hastings–Levitov模型的大偏差原理，通过Loewner–Kufarev方程的相对熵定义速率函数，并探讨了有限熵生成的形状类别及最小熵测度问题。

Scaling limits for planar aggregation with subcritical fluctuations

We study scaling limits of a family of planar random growth processes in which clusters grow by the successive aggregation of small particles. In these models, clusters are encoded as a composition of conformal maps and the location of each successive particle is distributed according to the density of harmonic measure on the cluster boundary, raised to some power. We show that, when this power lies within a particular range, the macroscopic shape of the cluster converges to a disk, but that as the power approaches the edge of this range the fluctuations approach a critical point, which is a l…

Scaling limits of anisotropic growth on logarithmic time-scales

We study the anisotropic version of the Hastings-Levitov model AHL$(ν)$. Previous results have shown that on bounded time-scales the harmonic measure on the boundary of the cluster converges, in the small-particle limit, to the solution of a deterministic ordinary differential equation. We consider the evolution of the harmonic measure on time-scales which grow logarithmically as the particle size converges to zero and show that, over this time-scale, the leading order behaviour of the harmonic measure becomes random. Specifically, we show that there exists a critical logarithmic time window i…

Scaling limits and fluctuations for random growth under capacity rescaling

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 und…

Coexistence in a random growth model with competition

We consider a variation of the Hastings-Levitov model HL(0) for random growth in which the growing cluster consists of two competing regions. We allow the size of successive particles to depend both on the region in which the particle is attached, and the harmonic measure carried by that region. We identify conditions under which one can ensure coexistence of both regions. In particular, we consider whether it is possible for the process giving the relative harmonic measures of the regions to converge to a non-trivial ergodic limit.

正则化Laplacian随机增长模型中的小粒子极限

Small particle limits in a regularized Laplacian random growth model

我们研究了一个正则化版本的黑斯廷斯-莱维托夫平面随机增长模型的扩散粒子聚集形成的集群。在这个模型中，生长的团簇被定义为迭代的狭缝映射，其容量由c_n=c给出|Φ_{n-1}'（e^{σ+iθ_n}）|^{-α}，α \geq 0，其中c>0是第一个粒子的容量，{Φ_n}_n是定义演化簇的合成共形映射，{θ_n}_n是决定粒子附着位置的独立一致角，σ>0是依赖于c的正则化参数.我们证明了在适当的时间尺度下，在c收敛到0的极限下，集群收敛到具有确定性容量的增长磁盘，只要σ不收敛到0太快。然后，我们建立了调和测度流的标度极限，表明通过让α以不同的速率趋于0，它收敛到圆上的布朗网，圆上的布朗网的停止版本，或者恒等映射。由于调和测度流与簇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，即我们获得其分布的随机数，或者在c收敛到0的极限中无界。我们还提出了一些研究结果的基础上模拟的模型与参数的选择不包括我们严格的分析。