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

    Berestycki, N. and Pymar, Richard (2012) Effect of scale on long-range random graphs and chromosomal inversions. The Annals of Applied Probability 22 (4), pp. 1328-1361. ISSN 1050-5164.

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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≫(logn)2 (when the mean degree exceeds 1) but not when L≪logn. 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.


    Item Type: Article
    School: Birkbeck Faculties and Schools > Faculty of Science > School of Computing and Mathematical Sciences
    Depositing User: Richard Pymar
    Date Deposited: 05 Sep 2017 12:59
    Last Modified: 09 Aug 2023 12:42


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