空间填充逾渗
Space-filling Percolation
摘要：
二维空间的一个区域被大量生长的圆盘随机填充，在它们的生长停止之前，它们之间只允许有“轻微”的重叠。更具体地，每个盘从在未覆盖区域内的随机位置处选择的成核中心生长。增长率$δ$是问题的一个连续可调的参数，当产生特定的圆盘图案时，它假定一个特定的值。当一个正在生长的盘第一次与至少另一个盘重叠时，它的生长停止，被称为“冻结”。本文研究了冻结圆盘集的渗流性质。使用数值模拟，我们提出了以下证据：（i）序参量出现跳跃不连续的区域覆盖的某一临界值;（ii）观察宏观跳跃的序参量中所需的区域覆盖的窗口的宽度趋于消失的$δ\to 0$，相反（iii）集群大小分布有一个幂律衰减函数的形式。虽然前两个结果是不连续过渡的特征，但第三个结果指示连续过渡。因此，我们把这种转变称为一个尖锐但连续的转变，类似于最近介绍的Achlioptas爆炸渗滤过程中所观察到的。在$δ\to 0$的极限下，临界面积覆盖率在过渡点趋于一致，表明极限模式是空间填充。在此极限下，渗透点处孔隙空间的分形维数估计为$1.42(10)$，发现圆盘组件的接触网络是无标度网络。
英文摘要：
A region of two-dimensional space has been filled randomly with large number of growing circular discs allowing only a `slight' overlapping among them just before their growth stop. More specifically, each disc grows from a nucleation center that is selected at a random location within the uncovered region. The growth rate $δ$ is a continuously tunable parameter of the problem which assumes a specific value while a particular pattern of discs is generated. When a growing disc overlaps for the first time with at least another disc, it's growth is stopped and is said to be `frozen'. In this paper we study the percolation properties of the set of frozen discs. Using numerical simulations we present evidence for the following: (i) The Order Parameter appears to jump discontinuously at a certain critical value of the area coverage; (ii) the width of the window of the area coverage needed to observe a macroscopic jump in the Order Parameter tends to vanish as $δ\to 0$ and on the contrary (iii) the cluster size distribution has a power law decaying functional form. While the first two results are the signatures of a discontinuous transition, the third result is indicative of a continuous transition. Therefore we refer this transition as a sharp but continuous transition similar to what has been observed in the recently introduced Achlioptas process of Explosive Percolation. It is also observed that in the limit of $δ\to 0$, the critical area coverage at the transition point tends to unity, implying the limiting pattern is space-filling. In this limit, the fractal dimension of the pore space at the percolation point has been estimated to be $1.42(10)$ and the contact network of the disc assembly is found to be a scale-free network.
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