Invasion Percolation on Power-Law Branching Processes

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 → ∞, 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 α and analyse the IPC for different power-law regimes. In a finite-variance setting (α > 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 (α ∈ (1, 2)) or even an infinite-mean setting (α ∈ (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 ∈ (1, 2) and exponentially for α ∈ (0, 1).