A pattern theorem for random sorting networks

TL;DR

This paper proves that in large random sorting networks, any fixed pattern appears frequently and that such networks are almost surely not geometrically realizable, revealing fundamental structural properties.

Contribution

It establishes a pattern theorem for random sorting networks, showing frequent pattern occurrence and non-realizability in the limit, which was previously unknown.

Findings

Any fixed pattern appears in at least cn^2 disjoint locations with high probability.

The probability that a random sorting network is geometrically realizable tends to zero as n grows.

Patterns occur with exponential probability bounds as n increases.

Abstract

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 tends to infinity. 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.