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

Abstract

We argue that thefreezing 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 polynomialspN(θ) of largeN×Nrandom unitary (circular unitary ensemble) matricesUN; i.e. the extreme value statistics ofpN(θ) when. In addition, we argue that it leads to multi-fractal-like behaviour in the total lengthμN(x) of the intervals in which |pN(θ)|>Nx,x>0, in the same limit. We speculate that our results extend to the large values taken by the Riemann zeta functionζ(s) over stretches of the critical lineof given constant length and present the results of numerical computations of the large values of). Our main purpose is to draw attention to the unexpected connections between these different extreme value problems.