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

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 $N\times N$ random unitary 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.

Figure 1

Numerical computation (solid red line) compared to theoretical prediction (13) (dashed black line) for $p(x)$.

Figure 2

Numerical computation (red dots) compared to the theoretical prediction (dashed black line) for $-f(\beta )$, suggesting freezing beyond $\beta =1$.