论文标题
Kagome Hypergraph上的关键渗透
Critical percolation on the kagome hypergraph
论文作者
论文摘要
我们研究了允许每个三角形的kagome晶格的渗透临界表面。使用临界多项式的方法，我们发现沿临界表面至高精度的点。此Kagome HyperGraph包含许多未解决的问题，例如特殊情况，包括粘结式的债券渗透和$（3,12^2）$晶格，以及六角形上的现场渗透，或蜂窝，晶格，以及一个单个点，有一个精确的解决方案。我们能够沿临界表面计算足够的点，以找到一个非常准确的拟合度，本质上是泰勒系列有关确切点的系列，可以估算任何位于表面上的系统的临界点，以精确地竞争与蒙特卡洛的精确匹配和相似精度的传统技术。我们还发现，该系统阐明了关键多项式方法的一些令人惊讶的方面，例如为什么它在某些问题（例如Kagome和$（3,12^2）$ lattices）中如此准确。这些晶格的键渗透临界点可以分别为17和18位，因为它们处于可以定量的意义上的近距离近端，到临界表面上的确切点。我们还详细讨论了该方法的并行实现，我们在这里使用该方法进行一些计算。
We study the percolation critical surface of the kagome lattice in which each triangle is allowed an arbitrary connectivity. Using the method of critical polynomials, we find points along this critical surface to high precision. This kagome hypergraph contains many unsolved problems as special cases, including bond percolation on the kagome and $(3,12^2)$ lattices, and site percolation on the hexagonal, or honeycomb, lattice, as well as a single point for which there is an exact solution. We are able to compute enough points along the critical surface to find a very accurate fit, essentially a Taylor series about the exact point, that allows estimations of the critical point of any system that lies on the surface to precision rivaling Monte Carlo and traditional techniques of similar accuracy. We find also that this system sheds light on some of the surprising aspects of the method of critical polynomials, such as why it is so accurate for certain problems, like the kagome and $(3,12^2)$ lattices. The bond percolation critical points of these lattices can be found to 17 and 18 digits, respectively, because they are in close proximity, in a sense that can be made quantitative, to the exact point on the critical surface. We also discuss in detail a parallel implementation of the method which we use here for a few calculations.
