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

TL;DR

This paper explores the connection between the freezing transition in statistical mechanics, the distribution of characteristic polynomials of random matrices, and the extreme values of the Riemann zeta function, supported by numerical evidence.

Contribution

It proposes that the freezing transition explains the distribution of maximum modulus of characteristic polynomials and extends this idea to the Riemann zeta function, linking multiple mathematical fields.

Findings

Freezing transition describes maximum modulus distribution of random matrix characteristic polynomials.

Numerical evidence supports the extension of these results to the Riemann zeta function.

Connections between statistical mechanics, random matrix theory, and number theory are highlighted.

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 x N random unitary (CUE) matrices. We postulate that our results extend to the extreme values taken by the Riemann zeta-function 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.