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