拉普拉斯随机增长模型的SLE标度极限
SLE scaling limits for a Laplacian random growth model
AI总结：
研究ALE(0,η)族中平面随机聚集模型，发现负η下附着分布凝聚导致粒子极限行为，并证明当粒子尺寸趋于零时，ALE簇收敛到参数κ=4的Schramm-Loewner演化（SLE₄）。
摘要：
我们考虑ALE$(0,η)$族中的平面随机聚集模型，其中粒子优先附着在调和测度低的区域。我们发现模型在负$η$下经历相变，对于足够大的值，每个粒子的附着分布在小粒子极限下变为原子分布，每个粒子附着在前一个粒子底部的两个点之一。这补充了Sola、Turner和Viklund关于大正$η$的结果，其中附着分布凝聚到前一个粒子尖端的一个原子上。由于附着分布的凝聚，我们推断在粒子尺寸趋于零的极限下，ALE簇收敛到参数$κ= 4$的Schramm-Loewner演化（SLE$_4$）。我们还推测，使用来自某个族的其他粒子形状，我们可以得到类似的SLE标度结果，并且可以获得任意$κ\geq 4$的SLE$_κ$。
英文摘要：
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 $κ\geq 4$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
