无向曲面上的连续渗流：Klein瓶上的可渗透问题
Continuum Percolation on Disoriented Surfaces: the Problem of Permeable Disks on a Klein Bottle
摘要：
本文采用完全可渗透圆盘的Newman-Ziff算法，用MonteCarlo方法计算了Klein瓶表面二维连续渗流问题的渗流阈值和包裹概率$R_{\infty}$。结果表明，Klein瓶上圆盘的渗流阈值与环面上圆盘的渗流阈值一致，表明该阈值是拓扑不变的.由于有限尺寸效应，确定包裹概率和临界浓度校正的标度指数也是拓扑不变的。同时，对于环面和克莱因瓶上的渗流，量$R_{\infty}$是不同的，并且显然由表面的拓扑结构决定。此外，环面和克莱因瓶的$R_{\infty}$值之间的差异意味着至少有一个渗流簇是简并的。
英文摘要：
The percolation threshold and wrapping probability $R_{\infty}$ for the two-dimensional problem of continuum percolation on the surface of a Klein bottle have been calculated by the Monte Carlo method with the Newman--Ziff algorithm for completely permeable disks. It has been shown that the percolation threshold of disks on the Klein bottle coincides with the percolation threshold of disks on the surface of a torus, indicating that this threshold is topologically invariant. The scaling exponents determining corrections to the wrapping probability and critical concentration owing to the finite-size effects are also topologically invariant. At the same time, the quantities $R_{\infty}$ are different for percolation on the torus and Klein bottle and are apparently determined by the topology of the surface. Furthermore, the difference between the $R_{\infty}$ values for the torus and Klein bottle means that at least one of the percolation clusters is degenerate.
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