Invasion percolation on power-law branching processes

Abstract

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 $\mathit{n}\to \infty$, 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. 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 $(\mathit{k}\mathbf{+}1)$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 $(\mathit{\alpha }>2)$ 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 ($\mathit{\alpha }\in (1,2)$) or even an infinite-mean setting ($\mathit{\alpha }\in (0,1)$), results significantly change. This is illustrated by the volume scaling of the k-cut IPC, which scales as ${\mathit{k}}^2$ for $\mathit{\alpha }>2$, but as ${\mathit{k}}^{\mathit{\alpha }/(\mathit{\alpha }-1)}$ for $\mathit{\alpha }\in (1,2)$ and exponentially for $\mathit{\alpha }\in (0,1)$.