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Extreme values of CUE characteristic polynomials: a numerical study

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

Original languageEnglish
Article number464001
JournalJournal of Physics A: Mathematical and Theoretical
Issue number46
Accepted/In press5 Oct 2018
Published22 Oct 2018

King's Authors


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 CβE which corresponds to allowing the degree of the eigenvalue repulsion to become a parameter.

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