A pattern theorem for random sorting networks
Probability
2012-11-21 v2 Combinatorics
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.
Cite
@article{arxiv.1110.0160,
title = {A pattern theorem for random sorting networks},
author = {Omer Angel and Vadim Gorin and Alexander E. Holroyd},
journal= {arXiv preprint arXiv:1110.0160},
year = {2012}
}
Comments
21 pages, 9 figures. Final journal version