Invasion percolation on the Poisson-weighted infinite tree

TL;DR

This paper investigates invasion percolation on the Poisson-weighted infinite tree, providing new Markovian representations, exploring a related exploration process, and introducing stationary graph representations of the Poisson incipient infinite cluster.

Contribution

It introduces novel Markovian representations of invasion percolation, a new exploration process, and stationary graph models for the Poisson incipient infinite cluster.

Findings

Two Markovian representations of invasion percolation are derived.

A new exploration process with simpler dynamics is introduced.

Stationary graph representations of the Poisson incipient infinite cluster are constructed.

Abstract

We study invasion percolation on Aldous' Poisson-weighted infinite tree, and derive two distinct Markovian representations of the resulting process. One of these is the $\sigma\to\infty$ limit of a representation discovered by Angel et al. [Ann. Appl. Probab. 36 (2008) 420-466]. We also introduce an exploration process of a randomly weighted Poisson incipient infinite cluster. The dynamics of the new process are much more straightforward to describe than those of invasion percolation, but it turns out that the two processes have extremely similar behavior. Finally, we introduce two new "stationary" representations of the Poisson incipient infinite cluster as random graphs on $\mathbb {Z}$ which are, in particular, factors of a homogeneous Poisson point process on the upper half-plane $\mathbb {R}\times[0,\infty)$.