Freezing Transition, Characteristic Polynomials of Random Matrices, and the Riemann Zeta Function

Abstract

We argue that the freezing transition scenario, previously explored in the statistical mechanics of $1/f$---noise random energy models, also determines the value distribution of the maximum of the modulus of the characteristic polynomials of large $N\ifmmode\times\else\texttimes\fi{}N$ random unitary matrices. We postulate that our results extend to the extreme values taken by the Riemann zeta function $\ensuremath{\zeta}(s)$ over sections of the critical line $s=1/2+it$ of constant length and present the results of numerical computations in support. Our 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.

Locations

• Physical Review Letters - View
• arXiv (Cornell University) - View - PDF
• PubMed - View
• DataCite API - View