SLE scaling limits for a Laplacian random growth model
SLE scaling limits for a Laplacian random growth model
复制标题
拉普拉斯随机增长模型的 SLE 缩放限制
DOI：
10.1214/21-aihp1217
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
F. Higgs
中科院分区：
文献类型：
--
作者：
F. Higgs
我们考虑 ALE$(\alpha,\eta)$ 系列的平面随机聚集模型，其中粒子优先附着在低谐波测量区域。我们发现模型经历了负 $\eta$ 的相变，其中对于足够大的值，每个粒子的附着分布在小粒子极限中变成原子，每个粒子附着到前一个粒子底部的两个点之一。这补充了 Sola、Turner 和 Viklund arXiv:1804.08462 对于大正 $\eta$ 的结果，其中附着分布在前一个粒子的尖端凝聚成单个原子。
由于附着分布的凝聚，我们推断出在极限情况下，当颗粒尺寸趋于零时，ALE 簇收敛到参数 $\kappa = 4$ (SLE$_4$) 的 Schramm-Loewner 演化。
We consider a model of planar random aggregation from the ALE$(\alpha,\eta)$ family where particles are attached preferentially in areas of low harmonic measure. We find that the model undergoes a phase transition in negative $\eta$, 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 arXiv:1804.08462 for large positive $\eta$, 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 $\kappa = 4$ (SLE$_4$).