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On the determinantal structure of conditional overlaps for the complex Ginibre ensemble
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Akemann, Gernot, Tribe, Roger, Tsareas, Athanasios and Zaboronski, Oleg V. (2020) On the determinantal structure of conditional overlaps for the complex Ginibre ensemble. Random Matrices: Theory and Applications, 9 (4). 2050015. doi:10.1142/S201032632050015X ISSN 2010-3263.
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WRAP-on-determinantal-structure-conditional-overlaps-ensemble-Tribe-2019.pdf - Accepted Version - Requires a PDF viewer. Download (680Kb) | Preview |
Official URL: https://doi.org/10.1142/S201032632050015X
Abstract
We continue the study of joint statistics of eigenvectors and eigenvalues initiated in the seminal papers of Chalker and Mehlig. The principal object of our investigation is the expectation of the matrix of overlaps between the left and the right eigenvectors for the complex N×N Ginibre ensemble, conditional on an arbitrary number k=1,2,… of complex eigenvalues. These objects provide the simplest generalization of the expectations of the diagonal overlap (k=1) and the off-diagonal overlap (k=2) considered originally by Chalker and Mehlig. They also appear naturally in the problem of joint evolution of eigenvectors and eigenvalues for Brownian motions with values in complex matrices studied by the Krakow school. We find that these expectations possess a determinantal structure, where the relevant kernels can be expressed in terms of certain orthogonal polynomials in the complex plane. Moreover, the kernels admit a rather tractable expression for all N≥2. This result enables a fairly straightforward calculation of the conditional expectation of the overlap matrix in the local bulk and edge scaling limits as well as the proof of the exact algebraic decay and asymptotic factorization of these expectations in the bulk.
Item Type: | Journal Article | |||||||||
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Subjects: | Q Science > QA Mathematics | |||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||
Library of Congress Subject Headings (LCSH): | Mathematical physics, Quantum theory -- Mathematical models, Applied mathematics, Biorthogonal systems, Eigenvalues, Eigenvectors | |||||||||
Journal or Publication Title: | Random Matrices: Theory and Applications | |||||||||
Publisher: | World Scientific Publishing | |||||||||
ISSN: | 2010-3263 | |||||||||
Official Date: | August 2020 | |||||||||
Dates: |
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Volume: | 9 | |||||||||
Number: | 4 | |||||||||
Article Number: | 2050015 | |||||||||
DOI: | 10.1142/S201032632050015X | |||||||||
Status: | Peer Reviewed | |||||||||
Publication Status: | Published | |||||||||
Reuse Statement (publisher, data, author rights): | Electronic version of an article published as Random Matrices: Theory and Applications, Volume, Issue, 2019, Pages]https://doi.org/10.1142/S201032632050015X © copyright World Scientific Publishing Company https://www.worldscientific.com/worldscinet/rmta | |||||||||
Access rights to Published version: | Restricted or Subscription Access | |||||||||
Date of first compliant deposit: | 21 June 2019 | |||||||||
Date of first compliant Open Access: | 1 August 2020 | |||||||||
RIOXX Funder/Project Grant: |
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