Extreme values of CUE characteristic polynomials: a numerical study

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

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)


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.

Original languageEnglish
Article number464001
JournalJournal of Physics A: Mathematical and Theoretical
Issue number46
Publication statusPublished - 22 Oct 2018


  • freezing transition
  • log-correlated processes
  • random matrix theory


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