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 CβE which corresponds to allowing the degree of the eigenvalue repulsion to become a parameter.
Highlights
Questions related to quantifying statistical properties of high and extreme values taken by the characteristic polynomials of random matrices have recently attracted considerable attention [2, 11, 22, 23, 25, 28, 31, 36]
The main motivation was the suggestion of a close analogy [22, 23] between the statistics of the the modulus of characteristic polynomials of large random matrices [26] and an important class of log-correlated random processes and fields, namely those characterised by having a logarithmic singularity on the diagonal of the covariance kernel, which have been the focus of considerable attention in the past few years
Such processes and fields appear with surprising regularity in many different contexts, ranging from the statistical mechanics of branching random walks and polymers on trees [13, 9] and disordered systems with multifractal structure [10, 17] to models of random surfaces underlying the probabilistic description of two-dimensional gravity [39, 32]
Summary
Questions related to quantifying statistical properties of high and extreme values taken by the characteristic polynomials of random matrices have recently attracted considerable attention [2, 11, 22, 23, 25, 28, 31, 36]. The main motivation was the suggestion of a close analogy [22, 23] between the statistics of the (logarithm of) the modulus of characteristic polynomials of large random matrices [26] (originally, from the Circular Unitary Ensemble, or CUE, of Random Matrix Theory) and an important class of log-correlated random processes and fields, namely those characterised by having a logarithmic singularity on the diagonal of the covariance kernel, which have been the focus of considerable attention in the past few years. In the second appendix we present a heuristic calculation for the CβE which generalises that given in [23] for the CUE
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