Freezing Transition: from $1/f$ landscapes to Characteristic Polynomials of Random Matrices and the Riemann zeta-function

In the talk (based on a joint work with G. Hiary and J. Keating; arXiv: {http://www.arxiv.org/abs/1202.4713}{1202.4713}) I will argue that the freezing transition scenario, previously conjectured to take place 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 of large random unitary (CUE) matrices. I then conjecture that the results extend to the large values taken by the Riemann zeta-function over stretches of the critical line $s=1/2+it$ of constant length, and present the results of numerical computations of the large values of $\zeta (1/2+it)$. The 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.