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

TL;DR

This paper explores the application of freezing transition theory to the extreme value statistics of characteristic polynomials of random matrices and the Riemann zeta-function, revealing deep connections across these complex systems.

Contribution

It introduces a novel interpretation of the freezing transition in the context of random matrix characteristic polynomials and the zeta-function, linking extreme value behaviors across these areas.

Findings

Freezing transition governs the maximum modulus of characteristic polynomials.

Multifractal behavior observed in the length of intervals where |p_N(θ)| exceeds a threshold.

Numerical evidence suggests similar extreme value patterns in the Riemann zeta-function.

Abstract

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\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-like behaviour 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 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.