Posé N, Schrenk K J, Araújo N A M, Herrmann H J
Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland.
1] Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland [2] Departamento de Física, Universidade Federal do Ceará, 60451-970 Fortaleza, Ceará, Brazil.
Sci Rep. 2014 Jun 30;4:5495. doi: 10.1038/srep05495.
We numerically show that the statistical properties of the shortest path on critical percolation clusters are consistent with the ones predicted for Schramm-Loewner evolution (SLE) curves for κ = 1.04 ± 0.02. The shortest path results from a global optimization process. To identify it, one needs to explore an entire area. Establishing a relation with SLE permits to generate curves statistically equivalent to the shortest path from a Brownian motion. We numerically analyze the winding angle, the left passage probability, and the driving function of the shortest path and compare them to the distributions predicted for SLE curves with the same fractal dimension. The consistency with SLE opens the possibility of using a solid theoretical framework to describe the shortest path and it raises relevant questions regarding conformal invariance and domain Markov properties, which we also discuss.
我们通过数值计算表明，临界渗流簇上最短路径的统计特性与κ = 1.04 ± 0.02时施拉姆 - 洛厄纳演化（SLE）曲线所预测的特性一致。最短路径源于一个全局优化过程。要识别它，需要探索整个区域。与SLE建立联系可以从布朗运动生成与最短路径统计等效的曲线。我们对最短路径的缠绕角、左通过概率和驱动函数进行了数值分析，并将它们与具有相同分形维数的SLE曲线所预测的分布进行比较。与SLE的一致性为使用坚实的理论框架描述最短路径开辟了可能性，同时也引发了关于共形不变性和域马尔可夫性质的相关问题，我们也对这些问题进行了讨论。
