Freezing transitions and extreme values: random matrix theory, ζ (1/2 + it) 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 pN(θ) of large N×N random unitary (circular unitary ensemble) matrices UN; i.e. the extreme value statistics of pN(θ) when N → ∞. In addition, we argue that it leads to multi-fractal-like behaviour in the total length μN(x) of the intervals in which |pN(θ)|>N(x), x>0, in the same limit. We speculate that our results extend to the large values taken by the Riemann zeta function ζ(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 ζ(1/2 + it). Our main purpose is to draw attention to the unexpected connections between these different extreme value problems.