TY - JOUR

T1 - Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta-function

AU - Fyodorov, YV

AU - Hiary, Ghaith

AU - Keating, Jon P

PY - 2012/4/26

Y1 - 2012/4/26

N2 - 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 ! N random unitary matrices. We postulate that our results extend to the extreme values taken by the Riemann zeta function !ðsÞ over sections of the critical line s ¼ 1=2 þ 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.

AB - 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 ! N random unitary matrices. We postulate that our results extend to the extreme values taken by the Riemann zeta function !ðsÞ over sections of the critical line s ¼ 1=2 þ 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.

U2 - 10.1103/PhysRevLett.108.170601

DO - 10.1103/PhysRevLett.108.170601

M3 - Article (Academic Journal)

C2 - 22680847

VL - 108

JO - Physical Review Letters

JF - Physical Review Letters

SN - 0031-9007

IS - 17

M1 - 170601

ER -