Freezing Transition, Characteristic Polynomials of Random Matrices, and the Riemann Zeta Function

Abstract

We argue that the freezing transition scenario, previously explored in the statistical mechanics of 1/f-noise random energy models, also determines the value distribution of the maximum of the modulus of the characteristic polynomials of large N×N random unitary matrices. We postulate that our results extend to the extreme values taken by the Riemann zeta function ζ(s) over sections of the critical line s=1/2+it of constant length and present the results of numerical computations in support. Our main purpose is to draw attention to possible connections between the statistical mechanics of random energy landscapes, random-matrix theory, and the theory of the Riemann zeta function.