Berzunza, Gabriel, Cai, Xing Shi and Holmgren, Cecilia
(2019)
The fluctuations of the giant cluster for percolation on random split
trees.
Alea (Rio de Janeiro): Latin American journal of probability and mathematical statistics, 19 (1).
665-.
Abstract
A split tree of cardinality $n$ is constructed by distributing $n$ "balls" in a subset of vertices of an infinite tree which encompasses many types of random trees such as $m$-ary search trees, quad trees, median-of-$(2k+1)$ trees, fringe-balanced trees, digital search trees and random simplex trees. In this work, we study Bernoulli bond percolation on arbitrary split trees of large but finite cardinality $n$. We show for appropriate percolation regimes that depend on the cardinality $n$ of the split tree that there exists a unique giant cluster, the fluctuations of the size of the giant cluster as $n \rightarrow \infty$ are described by an infinitely divisible distribution that belongs to the class of stable Cauchy laws. This work generalizes the results for the random $m$-ary recursive trees in Berzunza (2015). Our approach is based on a remarkable decomposition of the size of the giant percolation cluster as a sum of essentially independent random variables which may be useful for studying percolation on other trees with logarithmic height; for instance in this work we study also the case of regular trees.
| Item Type: | Article |
|---|---|
| Additional Information: | 45 pages |
| Uncontrolled Keywords: | math.PR, math.PR |
| Divisions: | Faculty of Science and Engineering > School of Physical Sciences |
| Depositing User: | Symplectic Admin |
| Date Deposited: | 13 Sep 2023 08:30 |
| Last Modified: | 15 Mar 2024 17:14 |
| DOI: | 10.30757/alea.v19-26 |
| Open Access URL: | https://alea.impa.br/articles/v19/19-26.pdf |
| Related URLs: | |
| URI: | https://livrepository.liverpool.ac.uk/id/eprint/3172708 |
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