Mathematics – Probability
Scientific paper
2011-02-22
Mathematics
Probability
31 pages, 6 figures
Scientific paper
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 $L>>(\log n)^2$ (when the mean degree exceeds 1) but not when $L<<\log n$. 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 behaviour described by the first author and Durrett in 2006 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.
Berestycki Nathanael
Pymar Richard
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