  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.
