A pattern theorem for random sorting networks
Name
Angel-2012-Pattern theorem for random.pdf
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307.54 KB
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Author(s)
Angel, Omer
Gorin, Vadim
Holroyd, Alexander E
Date Issued
November 2012
Journal
Electronic Journal of Probability
Publisher
Institute of Mathematical Statistics
Citation
Angel, Omer, Vadim Gorin, and Alexander E Holroyd. “A Pattern Theorem for Random Sorting Networks.” Electronic Journal of Probability 17, no. 0 (January 1, 2012).
Version
Final published version
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution
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DOI of Published Version
https://doi.org/10.1214/EJP.v17-2448
