Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta-function

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 !ðsÞ over sections of the critical line s ¼ 1=2 þ 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.