强凝聚聚集模型中的尖端增长遵循局部测地线
Tip growth in a strongly concentrated aggregation model follows local geodesics
AI总结：
本文通过分析聚集Loewner演化（ALE）在特定参数下的小粒子缩放极限，证明了其收敛到Laplacian路径模型，其中尖端沿测地线向无穷远增长。
摘要：
我们分析了聚集Loewner演化（ALE），该模型由Sola、Turner和Viklund于2018年引入，旨在使用复分析推广平面上的扩散限制聚集（DLA）版本。他们证明了ALE在特定参数下收敛到单个增长的狭缝。从$k$个针的非平凡初始配置和相同参数出发，我们证明了ALE的小粒子缩放极限是Laplacian路径模型（由Carleson和Makarov于2002年引入），其中尖端沿测地线向$\infty$增长。我们的证明涉及在Loewner方程奇点附近的分析，并将鞅方法推广到后向方程，其中需要控制的量是非适应的。大多数共形增长模型引入额外的正则化因子来处理Loewner方程在尖锐尖端和狭缝粒子直角基底处的奇异性。作为中间步骤，我们证明了一个没有这种正则化因子的模型的极限结果，发展了在分析其他具有非平凡极限的弱正则化模型时应该有用的方法。
英文摘要：
We analyse the aggregate Loewner evolution (ALE), introduced in 2018 by Sola, Turner and Viklund to generalise versions of diffusion limited aggregation (DLA) in the plane using complex analysis. They showed convergence of the ALE for certain parameters to a single growing slit. Started from a non-trivial initial configuration of $k$ needles and the same parameters, we show that the small-particle scaling limit of ALE is the Laplacian path model, introduced by Carleson and Makarov in 2002, in which the tips grow along geodesics towards $\infty$. Our proof involves analysis of Loewner's equation near its singular points, and we extend martingale methods to the backward equation, where what we have to control is non-adapted. Most conformal growth models introduce an extra regularisation factor to deal with the singularities in Loewner's equation at the sharp tips and right-angle bases of slit particles. As an intermediate step we prove a limit result for a model with no such regularisation factor, developing methods which should prove useful in analysing other weakly-regularised models with non-trivial limits.
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