论文标题
拉普拉斯随机增长模型的SLE缩放限制
SLE scaling limits for a Laplacian random growth model
论文作者
论文摘要
我们考虑了一个平面随机聚集的模型，该模型是从啤酒$（0，η）$ family中的一个模型，其中优先在低谐波测量区域附加了颗粒。我们发现该模型在负$η$中进行相变，在足够大的值中，每个粒子的附着分布在小粒子极限中变为原子，每个粒子都附着在上一个粒子底部的两个点之一。这补充了大型正$η$的Sola，Turner和Viklund的结果，其中附件分布凝结在上一个粒子尖端的单个原子上。 由于固定分布的这种缩合，我们推断出，粒径倾向于将啤酒簇收敛到schramm-loewner-loewner进化，参数$κ= 4 $（SLE $ _4 $）。 我们还猜想，使用来自某个家族的其他粒子形状，我们的SLE缩放结果相似，并且可以为任何$κ\ geq 4 $获得SLE $_κ$。
We consider a model of planar random aggregation from the ALE$(0,η)$ family where particles are attached preferentially in areas of low harmonic measure. We find that the model undergoes a phase transition in negative $η$, where for sufficiently large values the attachment distribution of each particle becomes atomic in the small particle limit, with each particle attaching to one of the two points at the base of the previous particle. This complements the result of Sola, Turner and Viklund for large positive $η$, where the attachment distribution condenses to a single atom at the tip of the previous particle. As a result of this condensation of the attachment distributions we deduce that in the limit as the particle size tends to zero the ALE cluster converges to a Schramm--Loewner evolution with parameter $κ= 4$ (SLE$_4$). We also conjecture that using other particle shapes from a certain family, we have a similar SLE scaling result, and can obtain SLE$_κ$ for any $κ\geq 4$.
