无限维多分散超球的连续渗流
Continuum percolation of polydisperse hyperspheres in infinite dimensions
摘要：
我们分析了系统的临界连通性的可穿透$d$维领域的加权随机几何图，其中顶点坐标对应于随机位置的球中心和边缘之间形成的任何两个重叠的领域的大小分布。边的权重自然是由两个重叠球体的不同半径引起的。对于球有界的大小分布的情况下，我们表明，集群的连接领域是树状的$d\rightarrow \infty$，他们不包含封闭的循环。在这种情况下，我们发现，平均集群大小发散的逾渗阈值密度$η_c\rightarrow 2^{-d}$，独立于特定的尺寸分布。我们还表明，在临界粒子的平均重叠数$z_c$是小于单位，而$z_c\rightarrow 1$只有固定半径的球体。我们解释了这些功能，表明在大的维数限制的关键连接是占主导地位的领域具有最大的尺寸。假设闭环可以忽略也为无界半径分布，我们发现，系统的球半径的对数正态分布的渐近临界阈值不再是普遍的，它可以小于$2^{-d}$为$d\rightarrow\infty$。
英文摘要：
We analyze the critical connectivity of systems of penetrable $d$-dimensional spheres having size distributions in terms of weighed random geometrical graphs, in which vertex coordinates correspond to random positions of the sphere centers and edges are formed between any two overlapping spheres. Edge weights naturally arise from the different radii of two overlapping spheres. For the case in which the spheres have bounded size distributions, we show that clusters of connected spheres are tree-like for $d\rightarrow \infty$ and they contain no closed loops. In this case, we find that the mean cluster size diverges at the percolation threshold density $η_c\rightarrow 2^{-d}$, independently of the particular size distribution. We also show that the mean number of overlaps for a particle at criticality $z_c$ is smaller than unity, while $z_c\rightarrow 1$ only for spheres with fixed radii. We explain these features by showing that in the large dimensionality limit the critical connectivity is dominated by the spheres with the largest size. Assuming that closed loops can be neglected also for unbounded radii distributions, we find that the asymptotic critical threshold for systems of spheres with radii following a lognormal distribution is no longer universal, and that it can be smaller than $2^{-d}$ for $d\rightarrow\infty$.
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