Electronic Journal of Probability

Invasion percolation on Galton-Watson trees

Marcus Michelen, Robin Pemantle, and Josh Rosenberg

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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.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 31, 35 pp.

Received: 28 March 2018
Accepted: 18 February 2019
First available in Project Euclid: 3 April 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

backbone incipient infinite cluster limit uniform Poisson point process pivot self-organized criticality

Creative Commons Attribution 4.0 International License.


Michelen, Marcus; Pemantle, Robin; Rosenberg, Josh. Invasion percolation on Galton-Watson trees. Electron. J. Probab. 24 (2019), paper no. 31, 35 pp. doi:10.1214/19-EJP281. https://projecteuclid.org/euclid.ejp/1554256913

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