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Effect of scale on long-range random graphs and chromosomal inversions. (English) Zbl 1248.05187
Summary: 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.
05C80 Random graphs (graph-theoretic aspects)
05C90 Applications of graph theory
60K35 Interacting random processes; statistical mechanics type models; percolation theory
92D15 Problems related to evolution
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