Extreme values of CUE characteristic polynomials: a numerical study

TL;DR

This paper numerically investigates the extreme value statistics of CUE characteristic polynomials, supporting conjectures about their behavior, correlations with spectral gaps, and extending analysis to CβE ensembles.

Contribution

It provides the first quantitative evidence linking extreme polynomial values to spectral gaps and extends analysis to CβE ensembles, supporting existing conjectures.

Findings

Numerical evidence supports conjectures about extreme value behavior.

Correlation found between extreme values and spectral gaps.

Extended analysis to CβE ensembles with variable eigenvalue repulsion.

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$\beta$E which corresponds to allowing the degree of the eigenvalue repulsion to become a parameter.