A pattern theorem for random sorting networks

• Main Authors: Angel, Omer (Author), Gorin, Vadim (Contributor), Holroyd, Alexander E (Author)
• Other Authors: Massachusetts Institute of Technology. Department of Mathematics (Contributor)
• Format: Article
• Language: English
• Published: Institute of Mathematical Statistics, 2014-09-15T14:57:23Z.
• Subjects: Article
• Online Access: Get fulltext

A sorting network is a shortest path from 12⋯n to n⋯21 in the Cayley graph of the symmetric group S[subscript 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[superscript 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 pattern. As a consequence, the probability that the uniformly random sorting network is geometrically realizable tends to 0.
University of Toronto
Natural Sciences and Engineering Research Council of Canada
Alfred P. Sloan Foundation
Microsoft Research
Möbius Contest Foundation for Young Scientists
Dynasty Foundation
Russian Foundation for Basic Research (RFBR-CNRS grant 10-01-93114)
Murmansk State Humanities University ("Development of the scientific potential of the higher school")
Simons Foundation (IUM-Simons Foundation scholarship)
Independent University of Moscow (IUM-Simons Foundation scholarship)