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August 2012 Effect of scale on long-range random graphs and chromosomal inversions
Nathanaël Berestycki, Richard Pymar
Ann. Appl. Probab. 22(4): 1328-1361 (August 2012). DOI: 10.1214/11-AAP793

Abstract

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.

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Nathanaël Berestycki. Richard Pymar. "Effect of scale on long-range random graphs and chromosomal inversions." Ann. Appl. Probab. 22 (4) 1328 - 1361, August 2012. https://doi.org/10.1214/11-AAP793

Information

Published: August 2012
First available in Project Euclid: 10 August 2012

zbMATH: 1248.05187
MathSciNet: MR2985163
Digital Object Identifier: 10.1214/11-AAP793

Subjects:
Primary: 05C80, 60K35
Secondary: 92D15

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.22 • No. 4 • August 2012
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