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On the non-Gaussian fluctuations of the giant cluster for percolation on random recursive trees


Bertoin, Jean (2014). On the non-Gaussian fluctuations of the giant cluster for percolation on random recursive trees. Electronic Journal of Probability, 19(24):online.

Abstract

We consider a Bernoulli bond percolation on a random recursive tree of size n≫1, with supercritical parameter p n =1-c/lnn for some c>0 fixed. It is known that with high probability, there exists then a unique giant cluster of size G n ∼e -c n, and it follows from a recent result of J. Schweinsberg [Electron. J. Probab. 17, Paper No. 91, 50 p. (2012; Zbl 1284.92072)] that G n has non-Gaussian fluctuations. We provide an explanation of this by analyzing the effect of percolation on different phases of the growth of recursive trees. This alternative approach may be useful for studying percolation on other classes of trees, such as for instance regular trees.

Abstract

We consider a Bernoulli bond percolation on a random recursive tree of size n≫1, with supercritical parameter p n =1-c/lnn for some c>0 fixed. It is known that with high probability, there exists then a unique giant cluster of size G n ∼e -c n, and it follows from a recent result of J. Schweinsberg [Electron. J. Probab. 17, Paper No. 91, 50 p. (2012; Zbl 1284.92072)] that G n has non-Gaussian fluctuations. We provide an explanation of this by analyzing the effect of percolation on different phases of the growth of recursive trees. This alternative approach may be useful for studying percolation on other classes of trees, such as for instance regular trees.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2014
Deposited On:30 Sep 2014 14:43
Last Modified:27 Apr 2017 21:20
Publisher:Institute of Mathematical Statistics
ISSN:1083-6489
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1214/EJP.v19-2822
Related URLs:http://dx.doi.org/10.1214/EJP.v19-2822

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