Extreme values of CUE characteristic polynomials: a numerical study

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

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