Electronic Journal of Probability

Invasion percolation on Galton-Watson trees

Marcus Michelen, Robin Pemantle, and Josh Rosenberg

Full-text: Open access

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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1554256913

Digital Object Identifier
doi:10.1214/19-EJP281

Mathematical Reviews number (MathSciNet)
MR3940761

Zentralblatt MATH identifier
07055669

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

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

Rights
Creative Commons Attribution 4.0 International License.

Citation

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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