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 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 $\alpha$ and analyse the IPC for different power-law regimes. In a finite-variance setting $(\alpha>2)$ 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 ($\alpha\in(1,2)$) or even an infinite-mean setting ($\alpha\in(0,1)$), 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 $\alpha \in (1,2)$ and exponentially for $\alpha \in (0,1)$.