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Extreme values of CUE characteristic polynomials: a numerical study

Fyodorov, Yan V.; Gnutzmann, Sven; Keating, Jonathan P.

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Authors

Yan V. Fyodorov

Jonathan P. Keating



Abstract

We present the results of systematic numerical computations relating to the extreme value statistics of the characteristic polynomials of random unitary matrices drawn from the circular unitary ensemble (CUE) of random matrix theory. In particular, we investigate a range of recent conjectures and theoretical results inspired by analogies with the theory of logarithmically-correlated Gaussian random fields. These include phenomena related to the conjectured freezing transition. Our numerical results are consistent with, and therefore support, the previous conjectures and theory. We also go beyond previous investigations in several directions: we provide the first quantitative evidence in support of a correlation between extreme values of the characteristic polynomials and large gaps in the spectrum, we investigate the rate of convergence to the limiting formulae previously considered, and we extend the previous analysis of the CUE to the C?E which corresponds to allowing the degree of the eigenvalue repulsion to become a parameter.

Citation

Fyodorov, Y. V., Gnutzmann, S., & Keating, J. P. (2018). Extreme values of CUE characteristic polynomials: a numerical study. Journal of Physics A: Mathematical and Theoretical, 51(46), Article 464001. https://doi.org/10.1088/1751-8121/aae65a

Journal Article Type Article
Acceptance Date Oct 5, 2018
Online Publication Date Oct 22, 2018
Publication Date Nov 16, 2018
Deposit Date Nov 26, 2018
Publicly Available Date Oct 23, 2019
Journal Journal of Physics A: Mathematical and Theoretical
Print ISSN 1751-8113
Electronic ISSN 1751-8121
Publisher IOP Publishing
Peer Reviewed Peer Reviewed
Volume 51
Issue 46
Article Number 464001
DOI https://doi.org/10.1088/1751-8121/aae65a
Keywords Modelling and Simulation; Statistics and Probability; Mathematical Physics; General Physics and Astronomy; Statistical and Nonlinear Physics
Public URL https://nottingham-repository.worktribe.com/output/1183278
Publisher URL http://iopscience.iop.org/article/10.1088/1751-8121/aae65a/meta
Additional Information Journal title: Journal of Physics A: Mathematical and Theoretical; Article type: paper; Article title: Extreme values of CUE characteristic polynomials: a numerical study; Copyright information: © 2018 IOP Publishing Ltd; Date received: 2018-06-01; Date accepted: 2018-10-05; Online publication date: 2018-10-22

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