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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.'
bibliography:
- 'Bib.bib'
---

[**Invasion Percolation on Galton-Watson Trees**]{}

Marcus Michelen, Robin Pemantle[^1] and Josh Rosenberg\
[{marcusmi, pemantle, rjos}@math.upenn.edu]{}

[[**Keywords**]{}: Backbone, incipient infinite cluster, limit uniform, Poisson point process, pivot, self-organized criticality.]{}

Introduction {#sec:intro}
============

Given an infinite rooted tree, how might one sample, nearly uniformly, from the set of paths from the root to infinity? One motive for this question is that nearly uniform sampling leads to good estimates on the growth rate [@jerrum-sinclair]. One might be trying to estimate the size of a search tree, or, in the case of 