Vadim Gorin
Random sorting networks: local statistics via random matrix laws
Gorin, Vadim; Rahman, Mustazee
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.
Citation
Gorin, V., & Rahman, M. (2019). Random sorting networks: local statistics via random matrix laws. Probability Theory and Related Fields, 175(1-2), 45-96. https://doi.org/10.1007/s00440-018-0886-1
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 |
DOI | https://doi.org/10.1007/s00440-018-0886-1 |
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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: https://doi.org/10.1007/s00440-018-0886-1
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