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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≫( n)^2 (when the mean degree exceeds 1) but not when L≪ 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 processdoi:10.1214/11-aap793 fatcat:6i7m7arzara5rp75zz4i5xsqky