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Random sorting networks: local statistics via random matrix laws

Abstract

This paper finds the bulk local limit of the swap process of uniformly random sorting networks. The limit object is defined through a deterministic procedure, a local version of the Edelman–Greene algorithm, applied to a two dimensional determinantal point process with explicit kernel. The latter describes the asymptotic joint law near 0 of the eigenvalues of the corners in the antisymmetric Gaussian Unitary Ensemble. In particular, the limiting law of the first time a given swap appears in a random sorting network is identified with the limiting distribution of the closest to 0 eigenvalue in the antisymmetric GUE. Moreover, the asymptotic gap, in the bulk, between appearances of a given swap is the Gaudin–Mehta law—the limiting universal distribution for gaps between eigenvalues of real symmetric random matrices. The proofs rely on the determinantal structure and a double contour integral representation for the kernel of random Poissonized Young tableaux of arbitrary shape.

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Acknowledgements

We would like to thank Balint Virág, who brought our attention to the problem of identifying the local limit of sorting networks. We are grateful to Alexei Borodin, Percy Deift, and Igor Krasovsky for valuable discussions and references. We would also like to thank an anonymous referee for an exceptionally careful reading of the paper and some helpful feedback. V. Gorin’s research was partially supported by NSF grants DMS-1407562, DMS-1664619, by a Sloan Research Fellowship, by The Foundation Sciences Mathématiques de Paris, and by NEC Corporation Fund for Research in Computers and Communications. M. Rahman’s research was partially supported by an NSERC PDF award.

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Correspondence to Mustazee Rahman.

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Gorin, V., Rahman, M. Random sorting networks: local statistics via random matrix laws. Probab. Theory Relat. Fields 175, 45–96 (2019). https://doi.org/10.1007/s00440-018-0886-1

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  • DOI: https://doi.org/10.1007/s00440-018-0886-1

Keywords

  • Sorting network
  • Reduced decomposition
  • Gaudin–Mehta law
  • GUE corners
  • Young tableau
  • Determinantal point process

Mathematics Subject Classification

  • 60G55
  • 60B20
  • 05E15
  • 82C22