Geometry of Permutation Limits


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.

This is a preview of subscription content, access via your institution.


  1. [1]

    O. Angel, D. Dauvergne, A. E. Holroyd and B. Virág: The local limit of random sorting networks, to appear in Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, arXiv:1702.08368.

  2. [2]

    O. Angel, V. Gorin and A. E. Holroyd: A pattern theorem for random sorting networks, Electronic Journal of Probability 17 (2012), 1–16.

    MathSciNet  Article  Google Scholar 

  3. [3]

    O. Angel, A. E. Holroyd, D. Romik and B. Virág: Random sorting networks, Advances in Mathematics 215 (2007), 839–868.

    MathSciNet  Article  Google Scholar 

  4. [4]

    D. Dauvergne: The Archimedean limit of random sorting networks, Preprint, 2018, arXiv:1802.08934.

    Google Scholar 

  5. [5]

    D. Dauvergne and B. Virág: Circular support in random sorting networks, Preprint, 2018, arXiv:1802.08933.

    Google Scholar 

  6. [6]

    P. Edelman and C. Greene: Balanced tableaux, Advances in Mathematics 63 (1987), 42–99.

    MathSciNet  Article  Google Scholar 

  7. [7]

    R. Glebov, A. Grzesik, T. Klimošová and D. Král: Finitely forcible graphons and permutons, Journal of Combinatorial Theory, Series B. 110 (2015), 112–135.

    MathSciNet  Article  Google Scholar 

  8. [8]

    V. Gorin and M. Rahman: Random sorting networks: Local statistics via random matrix laws, Preprint, 2017, arXiv:1702.07895.

    Google Scholar 

  9. [9]

    C. Hoppen, Y. Kohayakawa, C. G. Moreira, B. Rath and R. M. Sampaio: Limits of permutation sequences, Journal of Combinatorial Theory, Series B. 102 (2013), 93–113.

    MathSciNet  Article  Google Scholar 

  10. [10]

    O. Kallenberg: Foundations of Modern Probability, Springer-Verlag New York, 2nd edition, 2002.

    Google Scholar 

  11. [11]

    R. Kenyon, D. Král, C. Radin and P. Winkler: Permutations with fixed pattern densities, Preprint, 2015, arXiv:1506.02340.

    Google Scholar 

  12. [12]

    M. Kotowski: Limits of random permuton processes and large deviations of the interchange process, PhD Thesis, University of Toronto, 2016.

    Google Scholar 

  13. [13]

    M. Rahman and B. Virág: Brownian motion as limit of the interchange process, Preprint, 2016, arXiv:1609.07745.

    Google Scholar 

  14. [14]

    L. Rüschendorf: On the distributional transform, Sklar’s theorem and the empirical copula process, Journal of Statistical Planning and Inference 139 (2009), 3921–3927.

    MathSciNet  Article  Google Scholar 

  15. [15]

    R. P. Stanley: On the number of reduced decompositions of elements of Coxeter groups, European Journal of Combinatorics 5 (1984), 359–372.

    MathSciNet  Article  Google Scholar 

  16. [16]

    S. Starr: Thermodynamic limit for the Mallows model on S n, Journal of Mathematical Physics 50 (2009).

Download references


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.

Author information



Corresponding authors

Correspondence to Mustazee Rahman or Bálint Virág or Máté Vizer.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Rahman, M., Virág, B. & Vizer, M. Geometry of Permutation Limits. Combinatorica 39, 933–960 (2019).

Download citation

Mathematics Subject Classification (2010)

  • 60C05
  • 05E18
  • 60G57