Abstract

I will discuss relations between logarithmically-correlated Gaussian processes and the characteristic polynomials of large random $N imes N$ matrices, either from the Circular Unitary (CUE) or from the Gaussian Unitary (GUE) ensembles. Such relations help to address the problem of characterising the distribution of the global maximum of the modulus of such polynomials, and of the Riemann $zetaleft(rac{1}{2}+it ight)$ over some intervals of $t$ containing of the order of $log{t}$ zeroes. I will show how to arrive to an explicit expression for the asymptotic probability density of the maximum by combining the rigorous Fisher-Hartwig asymptotics with the heuristic {it freezing transition} scenario for logarithmically correlated processes. Although the general idea behind the method is the same for both CUE and GUE , the latter case is much more technically challenging. In particular I will show how the conjectured {it self-duality} in the freezing transition scenario plays the crucial role in selecting the form of the maximum distribution for GUE case. The found probability densities will be compared to the results of direct numerical simulations of the maxima. The presentation is mainly based on joint works with Ghaith Hiary, Jon Keating, Boris Khoruzhenko, and Nick Simm.