Electronic Journal of Probability

Yule processes with rare mutation and their applications to percolation on $b$-ary trees

Gabriel Berzunza

Full-text: Open access

Abstract

We consider supercritical Bernoulli bond percolation on a large $b$-ary tree, in the sense that with high probability, there exists a giant cluster. We show that the size of the giant cluster has non-gaussian fluctuations, which extends a result due to Schweinsberg in the case of random recursive trees. Using ideas in the recent work of Bertoin and Uribe Bravo, the approach developed in this work relies on the analysis of the sub-population with ancestral type in a system of branching processes with rare mutations, which may be of independent interest. This also allows us to establish the analogous result for scale-free trees.

Article information

Source
Electron. J. Probab. Volume 20 (2015), paper no. 43, 23 pp.

Dates
Accepted: 14 April 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067149

Digital Object Identifier
doi:10.1214/EJP.v20-3789

Mathematical Reviews number (MathSciNet)
MR3339863

Zentralblatt MATH identifier
1321.60172

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Random tree branching process percolation giant cluster fluctuations

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Berzunza, Gabriel. Yule processes with rare mutation and their applications to percolation on $b$-ary trees. Electron. J. Probab. 20 (2015), paper no. 43, 23 pp. doi:10.1214/EJP.v20-3789. https://projecteuclid.org/euclid.ejp/1465067149


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