Freezing transitions and extreme values: random matrix theory, and disordered landscapes

We argue that the freezing transition scenario, previously conjectured to occur in the statistical mechanics of 1/f-noise random energy models, governs, after reinterpretation, the value distribution of the maximum of the modulus of the characteristic polynomials p(N)(theta) of large N x N random unitary (circular unitary ensemble) matrices U-N; i.e. the extreme value statistics of p(N)(theta) when N -> infinity. In addition, we argue that it leads to multi-fractal-like behaviour in the total length mu(N)(x) of the intervals in which vertical bar p(N)(theta)vertical bar > N-x, x > 0, in the same limit. We speculate that our results extend to the large values taken by the Riemann zeta function zeta (s) over stretches of the critical line s = 1/2 + it of given constant length and present the results of numerical computations of the large values of zeta (1/2 + it). Our main purpose is to draw attention to the unexpected connections between these different extreme value problems.