A pattern theorem for random sorting networks

Omer Angel, Vadim Gorin, Alexander Holroyd
2012 Electronic Journal of Probability  
A sorting network is a shortest path from 12 · · · n to n · · · 21 in the Cayley graph of the symmetric group S n generated by nearest-neighbor swaps. A pattern is a sequence of swaps that forms an initial segment of some sorting network. We prove that in a uniformly random n-element sorting network, any fixed pattern occurs in at least cn 2 disjoint space-time locations, with probability tending to 1 exponentially fast as n → ∞. Here c is a positive constant which depends on the choice of
more » ... the choice of pattern. As a consequence, the probability that the uniformly random sorting network is geometrically realizable tends to 0.
doi:10.1214/ejp.v17-2448 fatcat:xhqdiy4bondmheopnmbqtt25tm