In this paper, we investigate the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according the Circular Beta Ensemble (CβE). More precisely, if X n is this characteristic polynomial and U the unit circle, we prove that:as well as an analogous statement for the imaginary part. The notation O(1) means that the corresponding family of random variables, indexed by n, is tight. This answers a conjecture of Fyodorov, Hiary and Keating, originally formulated for the β = 2 case, which corresponds to the CUE field.

We show in this paper that after proper scalings, the characteristic polynomial of a random unitary matrix converges almost surely to a random analytic function whose zeros, which are on the real line, form a determinantal point process with sine kernel. Our scaling is performed at the so-called "microscopic" level, that is we consider the characteristic polynomial at points which are of order 1/n distant. We prove this in the framework of virtual isometries to circumvent the fact that the rescaled characteristic polynomial does not even have a moment of order one, hence making the classical techniques of random matrix theory difficult to apply. The strong convergence results in this setup provide us with a new approach to ratios: we are able to solve open problems about the limiting distribution of ratios of characteristic polynomials evaluated at points of the form exp(2iπα/n) and related objects (such as the logarithmic derivative). We also explicitly describe the dependence relation for the logarithm of the characteristic polynomial evaluated at several points on the microscopic scale. On the number theory side, inspired by the Keating-Snaith philosophy, we conjecture some new limit theorems for the value distribution of the Riemann zeta function on the critical line at the stochastic process level.

We demonstrate the convergence of the characteristic polynomial of several random matrix ensembles to a limiting universal function, at the microscopic scale. The random matrix ensembles we treat are classical compact groups and the Gaussian Unitary Ensemble.In fact, the result is the by-product of a general limit theorem for the convergence of random entire functions whose zeros present a simple regularity property.