Extreme values of CUE characteristic polynomials: a numerical study

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.

23 pages

References in corpus (3)

• Freezing and decorated Poisson point processes
• The DOZZ Formula from the Path Integral
• A Review of Conjectured Laws of Total Mass of Bacry-Muzy GMC Measures on the Interval and Circle and Their Applications

Cited by in corpus (5)

• Moments of the Riemann zeta function on short intervals of the critical line
• On the moments of the moments of the characteristic polynomials of random unitary matrices
• Maxima of log-correlated fields: some recent developments
• Moments of Moments and Branching Random Walks
• Hierarchical Structure in the Trace Formula