On radial stochastic Loewner evolution in multiply connected domains
On radial stochastic Loewner evolution in multiply connected domains
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DOI：
10.1016/j.jfa.2005.12.023
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发表时间：
2004-12
影响因子：
1.7
通讯作者：
R. Bauer;R. Friedrich
中科院分区：
文献类型：
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作者：
R. Bauer;R. Friedrich
本文讨论了多连通平面区域上径向SLE的推广。首先，我们通过建立径向Komatu-Loewner方程，将Loewner的狭缝映射理论推广到多连通域，并证明了从边界到体的简单曲线是由模空间上的运动和域边界上的运动编码的。然后，我们证明了描述模的运动的向量场是Lipschitz的。我们解释了为什么这意味着“一致的”，共形不变的随机简单曲线描述的多维扩散，其中一个组件是一个运动的边界上，和其他组件是一个运动的模空间。我们认为这种扩散的确切形式是什么（一个单一的真实的参数κ），以模型的渗流集群的边界。最后，我们证明了这种模扩散导致满足局部性的随机非自交叉曲线当且仅当κ=6。
We discuss the extension of radial SLE to multiply connected planar domains. First, we extend Loewner's theory of slit mappings to multiply connected domains by establishing the radial Komatu–Loewner equation, and show that a simple curve from the boundary to the bulk is encoded by a motion on moduli space and a motion on the boundary of the domain. Then, we show that the vector-field describing the motion of the moduli is Lipschitz. We explain why this implies that “consistent,” conformally invariant random simple curves are described by multidimensional diffusions, where one component is a motion on the boundary, and the other component is a motion on moduli space. We argue what the exact form of this diffusion is (up to a single real parameter κ) in order to model boundaries of percolation clusters. Finally, we show that this moduli diffusion leads to random non-self-crossing curves satisfying the locality property if and only if κ=6.