论文标题
在幂律分支过程中的入侵渗透
Invasion Percolation on Power-Law Branching Processes
论文作者
论文摘要
我们分析了通过在分支过程中具有幂律后代分布的分支过程中发现的聚类。入侵渗透是一个自组织临界性的范式模型，在不调整任何参数的情况下，接近临界性。通过对$ n $ steps进行入侵渗透，并让$ n \ to \ infty $，我们找到了一个无限的子树，称为入侵渗透集群（IPC）。 IPC的一个值得注意的特征是其几何形状由通往无穷大的独特路径（也称为骨干）组成，并附着有限的森林。我们的主要定理显示了$ K $ -CUT IPC的体积缩放限制，它是群集，当$ k $ th和$（k+1）$ -S $ -ST Backbone Vertices被切割时，该群集包含根。 我们假设具有指数$α$的幂律后代分布，并分析IPC的不同幂律制度。在有限变化的设置$（α> 2）$中，我们的结果是对分支过程树（Michelen etal。2019）和常规树（Angel等，2008）的先前作品的自然扩展。但是，对于无限变化的设置（$α\ in（1,2）$），甚至是无限均值的设置（$α\ in（0,1）$），结果会发生明显变化。 $ k $ -cut IPC的音量缩放为$ k^2 $，对于$α> 2 $，但为$ k^{α/（α/（α-1）} $，$ k^2 $ for $ k^2 $ for $ k^{α/（α-1）} $ for $α\ in（1,2）$，并且按（0,1）$ $α\ in（0,1）$呈$ allyly。
We analyse the cluster discovered by invasion percolation on a branching process with a power-law offspring distribution. Invasion percolation is a paradigm model of self-organised criticality, where criticality is approached without tuning any parameter. By performing invasion percolation for $n$ steps, and letting $n\to\infty$, we find an infinite subtree, called the invasion percolation cluster (IPC). A notable feature of the IPC is its geometry that consists of a unique path to infinity (also called the backbone) onto which finite forests are attached. Our main theorem shows the volume scaling limit of the $k$-cut IPC, which is the cluster containing the root when the edge between the $k$-th and $(k+1)$-st backbone vertices is cut. We assume a power-law offspring distribution with exponent $α$ and analyse the IPC for different power-law regimes. In a finite-variance setting $(α>2)$ our results are a natural extension of previous works on the branching process tree (Michelen et al. 2019) and the regular tree (Angel et al. 2008). However, for an infinite-variance setting ($α\in(1,2)$) or even an infinite-mean setting ($α\in(0,1)$), results significantly change. This is illustrated by the volume scaling of the $k$-cut IPC, which scales as $k^2$ for $α>2$, but as $k^{α/(α-1)}$ for $α\in (1,2)$ and exponentially for $α\in (0,1)$.
