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Sizes of the largest clusters for supercritical percolation on random recursive trees
Author(s) -
Bertoin Jean
Publication year - 2014
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20448
Subject(s) - percolation (cognitive psychology) , bernoulli's principle , statistical physics , poisson distribution , supercritical fluid , mathematics , tree (set theory) , cluster (spacecraft) , combinatorics , discrete mathematics , statistics , computer science , physics , thermodynamics , neuroscience , biology , programming language
We consider Bernoulli bond‐percolation on a random recursive tree of size n ≫ 1 , with supercritical parameter p ( n ) = 1 − t / ln n + o ( 1 / ln n ) for some t > 0 fixed. We show that with high probability, the largest cluster has size close toe − t n whereas the next largest clusters have size of order n / ln n only and are distributed according to some Poisson random measure. Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 29–44, 2014