The Full Scaling Limit of Two-Dimensional Critical Percolation
The Full Scaling Limit of Two-Dimensional Critical Percolation
复制标题
二维临界渗流的全尺度极限
DOI：
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
C. Newman
中科院分区：
文献类型：
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作者：
F. Camia;C. Newman
我们利用SLE（6）路径构造了一个平面上连续非简单环的过程，并证明了这个过程与三角格子上二维临界点渗流的全连续标度极限--即不同团簇间所有界面集合的标度极限相一致.证明了循环过程的一些性质，包括共形不变性。在本文的主体中，这些结果被证明，同时假设，如Schramm和Smirnov所论证的，渗流探索路径在分布上收敛于弦SLE的迹（6）。然后，在一个冗长的附录中，详细证明了这种收敛到SLE（6），它本身依赖于Smirnov的结果，即交叉概率收敛到Cardy公式。
We use SLE(6) paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice -- that is, the scaling limit of the set of all interfaces between different clusters. Some properties of the loop process, including conformal invariance, are also proved. In the main body of the paper these results are proved while assuming, as argued by Schramm and Smirnov, that the percolation exploration path converges in distribution to the trace of chordal SLE(6). Then, in a lengthy appendix, a detailed proof is provided for this convergence to SLE(6), which itself relies on Smirnov's result that crossing probabilities converge to Cardy's formula.