A pattern theorem for random sorting networks

Abstract

A sorting network is a shortest path from $12\cdots n$ to $n\cdots 21$ 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 $c n^2$ disjoint space-time locations, with probability tending to $1$ exponentially fast as $n\to\infty$. 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 $0$.