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Random sorting networks: local statistics via random matrix laws

Gorin, Vadim; Rahman, Mustazee

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Vadim Gorin


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.


Gorin, V., & Rahman, M. (2019). Random sorting networks: local statistics via random matrix laws. Probability Theory and Related Fields, 175(1-2), 45-96.

Journal Article Type Article
Acceptance Date Nov 11, 2018
Online Publication Date Nov 19, 2018
Publication Date 2019-10
Deposit Date Sep 25, 2019
Publicly Available Date Oct 4, 2021
Journal Probability Theory and Related Fields
Print ISSN 0178-8051
Electronic ISSN 1432-2064
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 175
Issue 1-2
Pages 45-96


Accepted Journal Article (852 Kb)

Copyright Statement
This is a post-peer-review, pre-copyedit version of a journal article published in Probability Theory and Related Fields. The final authenticated version is available online at:

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