Random sorting networks: local statistics via random matrix laws
Name
1702.07895.pdf
Description
Accepted version
Size
831.98 KB
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Adobe PDF
Checksum (MD5)
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Author(s)
Gorin, Vadim
Rahman, Mustazee
Date Issued
October 2019
Journal
Probability Theory and Related Fields
Publisher
Springer Berlin Heidelberg
Citation
Gorin, Vadim & Rahman, Mustazee. "Random sorting networks: local statistics via random matrix laws." Probability Theory and Related Fields 175, 1-2 (October 2019): 45-96 © 2018 Springer-Verlag GmbH Germany, part of Springer Nature
Version
Author's final manuscript
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. Keyword: Sorting network ; Reduced decomposition ; Gaudin–Mehta law ; GUE corners ; Young tableau ; Determinantal point process
Subjects
Statistics, Probability and Uncertainty
Statistics and Probability
Analysis
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1007/s00440-018-0886-1
