A Pattern Theorem for Random Sorting Networks

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 cn² 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 pattern. As a consequence, the probability that the uniformly random sorting network is geometrically realizable tends to 0.