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Random sorting networks: local statistics via random matrix laws

Gorin, Vadim and Rahman, Mustazee (2019) 'Random sorting networks: local statistics via random matrix laws.', Probability theory and related fields., 175 (1-2). pp. 45-96.

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.

Item Type:Article
Full text:(AM) Accepted Manuscript
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1007/s00440-018-0886-1
Publisher 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: https://doi.org/10.1007/s00440-018-0886-1
Date accepted:11 November 2018
Date deposited:04 October 2021
Date of first online publication:19 November 2018
Date first made open access:04 October 2021

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