Open Access
2012 A pattern theorem for random sorting networks
Omer Angel, Vadim Gorin, Alexander Holroyd
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Electron. J. Probab. 17: 1-16 (2012). DOI: 10.1214/EJP.v17-2448


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 $0$.


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Omer Angel. Vadim Gorin. Alexander Holroyd. "A pattern theorem for random sorting networks." Electron. J. Probab. 17 1 - 16, 2012.


Accepted: 19 November 2012; Published: 2012
First available in Project Euclid: 4 June 2016

zbMATH: 1284.60021
MathSciNet: MR3005717
Digital Object Identifier: 10.1214/EJP.v17-2448

Primary: 60C05
Secondary: 05E10 , 68P10

Keywords: Pattern , random sorting , reduced word , Sorting network , Young tableau

Vol.17 • 2012
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