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We argue that the freezing transition scenario, previously conjectured tooccur in the statistical mechanics of 1/f-noise random energy models, governs,after reinterpretation, the value distribution of the maximum of the modulus ofthe characteristic polynomials p_N(\theta) of large N\times N random unitary(CUE) matrices; i.e. the extreme value statistics of p_N(\theta) when N\rightarrow\infty. In addition, we argue that it leads to multifractal-likebehaviour in the total length \mu_N(x) of the intervals in which|p_N(\theta)|>N^x, x>0, in the same limit. We speculate that our results extendto the large values taken by the Riemann zeta-function \zeta(s) over stretchesof the critical line s=1/2+it of given constant length, and present the resultsof numerical computations of the large values of \zeta(1/2+it). Our mainpurpose is to draw attention to the unexpected connections between thesedifferent extreme value problems.

philosophical transactions - royal society. mathematical, physical and engineering sciences/philosophical transactions - royal society. mathematical, physical and engineering sciences

Freezing Transitions and Extreme Values: Random Matrix Theory, $\zeta(1/2+it)$, and Disordered Landscapes