Invasion Percolation on Galton-Watson Trees

TL;DR

This paper studies invasion percolation on Galton-Watson trees, showing it produces a measure on infinite paths that can approximate the uniform distribution, with analysis of weights along the backbone.

Contribution

It demonstrates that invasion percolation induces an absolutely continuous measure on infinite paths under certain conditions, enabling approximate sampling from the uniform distribution.

Findings

Invasion percolation on Galton-Watson trees produces a measure on infinite paths.

Under certain conditions, this measure is absolutely continuous with respect to the uniform measure.

A limit law for the forward maximal weights along the invasion cluster's backbone is established.

Abstract

We consider invasion percolation on Galton-Watson trees. On almost every Galton-Watson tree, the invasion cluster almost surely contains only one infinite path. This means that for almost every Galton-Watson tree, invasion percolation induces a probability measure on infinite paths from the root. We show that under certain conditions of the progeny distribution, this measure is absolutely continuous with respect to the limit uniform measure. This confirms that invasion percolation, an efficient self-tuning algorithm, may be used to sample approximately from the limit uniform distribution. Additionally, we analyze the forward maximal weights along the backbone of the invasion cluster and prove a limit law for the process.