Geometry of Permutation Limits

Abstract

This paper initiates a limit theory of permutation valued processes, building on the recent theory of permutons. We apply this to study the asymptotic behaviour of random sorting networks. We prove that the Archimedean path, the conjectured limit of random sorting networks, is the unique path from the identity to the reverse permuton having minimal energy in an appropriate metric. Together with a recent large deviations result (Kotowski, 2016), it implies the Archimedean limit for the model of relaxed random sorting networks.

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Acknowledgements

M. Rahman was partially supported by an NSERC PDF award. B. Virág was supported by the Canada Research Chair program, the NSERC Discovery Accelerator grant, the MTA Momentum Random Spectra research group, and the ERC consolidator grant 648017 (Abért). M. Vizer was supported by NKFIH under the grant SNN 116095.

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Correspondence to Mustazee Rahman or Bálint Virág or Máté Vizer.

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Rahman, M., Virág, B. & Vizer, M. Geometry of Permutation Limits. Combinatorica 39, 933–960 (2019). https://doi.org/10.1007/s00493-019-3817-6

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Mathematics Subject Classification (2010)

  • 60C05
  • 05E18
  • 60G57