INVASION PERCOLATION ON POWER-LAW BRANCHING PROCESSES

• 成果类型: Article
• 署名作者: Gundlach, Rowel; van der Hofstad, Remco
• 署名单位: Eindhoven University of Technology
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/23-AAP2032
• 发表日期: 2024
• 页码: 3018-3092
• 关键词: incipient infinite cluster; TREE

摘要

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 approach without tuning any parameter. By performing invasion percolation for n steps, and letting n -> infinity, 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. The main theorem shows the volume scaling limit of the k-cut IPC, which is the cluster containing the root when the edge between the kth and (k + 1)st backbone vertices is cut. We assume a power-law offspring distribution with exponent alpha and analyse the IPC for different power-law regimes. In a finite-variance setting (alpha > 2) the results, are a natural extension of previous works on the branching process tree (Electron. J. Probab. 24 (2019) 1-35) and the regular tree (Ann. Probab. 35 (2008) 420-466). However, for an infinite-variance setting (alpha is an element of (1, 2)) or even an infinite-mean setting (alpha is an element of (0, 1)), results significantly change. This is illustrated by the volume scaling of the k-cut IPC, which scales as k(2) for alpha > 2, but as k(alpha/(alpha - 1)) for alpha is an element of (1, 2) and exponentially for alpha is an element of (0, 1).