Random Graph Asymptotics on High-Dimensional Tori
Random Graph Asymptotics on High-Dimensional Tori
复制标题
高维 Tori 上的随机图渐进
DOI：
--
复制
发表时间：
2005
期刊：
影响因子：
--
通讯作者：
R. Hofstad
中科院分区：
文献类型：
--
作者：
M. Heydenreich;R. Hofstad
我们研究了大的d维环面上最大临界渗流团的标度，对于足够高维的近邻渗流，或者当d>；6时充分展开的渗流。我们用一个相对简单的耦合引理来证明，这个最大的临界星系团很有可能是由一个大的常数乘以V2/3和下面的一个小的常数乘以$${V^{2/3}(log{V})^{-4/3}}$$，其中V是环面的体积。我们还给出了一个关于$${mathbb{Z}^d}$$上的亚临界渗流两点函数的简单判据，在这个判据下，下界可以改进为小的常数倍$${V^{2/3}}$$，即我们证明了高维环面上最大临界簇的随机图渐近性。这建立了文献[1]的猜想，而不是对数校正。我们讨论了这些结果对高维渗流对边界条件的依赖性的影响。我们的方法关键是基于[11，12]中的结果，其中$${V^{2/3}}$$标度是在假设适当定义的临界窗口包含$${mathbb{Z}^d}$$的渗流阈值的前提下被证明的。我们在[17-20]中证明的$${mathbb{Z}^d}$$上的渗流也强烈依赖平均场结果。
We investigate the scaling of the largest critical percolation cluster on a large d-dimensional torus, for nearest-neighbor percolation in sufficiently high dimensions, or when d > 6 for sufficiently spread-out percolation. We use a relatively simple coupling argument to show that this largest critical cluster is, with high probability, bounded above by a large constant times V2/3 and below by a small constant times $${V^{2/3}(log{V})^{-4/3}}$$ , where V is the volume of the torus. We also give a simple criterion in terms of the subcritical percolation two-point function on $${mathbb{Z}^d}$$ under which the lower bound can be improved to small constant times $${V^{2/3}}$$ , i.e. we prove random graph asymptotics for the largest critical cluster on the high-dimensional torus. This establishes a conjecture by [1], apart from logarithmic corrections. We discuss implications of these results on the dependence on boundary conditions for high-dimensional percolation.Our method is crucially based on the results in [11, 12], where the $${V^{2/3}}$$ scaling was proved subject to the assumption that a suitably defined critical window contains the percolation threshold on $${mathbb{Z}^d}$$ . We also strongly rely on mean-field results for percolation on $${mathbb{Z}^d}$$ proved in [17–20].