Annals of Applied Probability

Effect of scale on long-range random graphs and chromosomal inversions

Nathanaël Berestycki and Richard Pymar

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We consider bond percolation on $n$ vertices on a circle where edges are permitted between vertices whose spacing is at most some number $L=L(n)$. We show that the resulting random graph gets a giant component when $L\gg(\log n)^{2}$ (when the mean degree exceeds 1) but not when $L\ll\log n$. The proof uses comparisons to branching random walks. We also consider a related process of random transpositions of $n$ particles on a circle, where transpositions only occur again if the spacing is at most $L$. Then the process exhibits the mean-field behavior described by Berestycki and Durrett if and only if $L(n)$ tends to infinity, no matter how slowly. Thus there are regimes where the random graph has no giant component but the random walk nevertheless has a phase transition. We discuss possible relevance of these results for a dataset coming from D. repleta and D. melanogaster and for the typical length of chromosomal inversions.

Article information

Ann. Appl. Probab., Volume 22, Number 4 (2012), 1328-1361.

First available in Project Euclid: 10 August 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 92D15: Problems related to evolution

Random transposition random graphs phase transition coagulation-fragmentation giant component percolation branching random walk genome rearrangement


Berestycki, Nathanaël; Pymar, Richard. Effect of scale on long-range random graphs and chromosomal inversions. Ann. Appl. Probab. 22 (2012), no. 4, 1328--1361. doi:10.1214/11-AAP793.

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