Poisson-Dirichlet statistics for the extremes of a randomized Riemann zeta function

Frédéric Ouimet

doi:10.1214/18-ecp154

Abstract

In Arguin & Tai (2018), the authors prove the convergence of the two-overlap distribution at low temperature for a randomized Riemann zeta function on the critical line. We extend their results to prove the Ghirlanda-Guerra identities. As a consequence, we find the joint law of the overlaps under the limiting mean Gibbs measure in terms of Poisson-Dirichlet variables. It is expected that we can adapt the approach to prove the same result for the Riemann zeta function itself.

Highlights

In [4], the authors prove the convergence of the two-overlap distribution at low temperature for a randomized Riemann zeta function on the critical line
We find the joint law of the overlaps under the limiting mean Gibbs measure in terms of Poisson-Dirichlet variables
Following recent conjectures of [14] and [13] about the limiting law of the Gibbs measure and the limiting law of the maximum for the Riemann zeta function on bounded random intervals of the critical line, progress have been made in the mathematics literature

Summary

Introduction

Following recent conjectures of [14] and [13] about the limiting law of the Gibbs measure and the limiting law of the maximum for the Riemann zeta function on bounded random intervals of the critical line, progress have been made in the mathematics literature. As is well known in the spin glass literature (see e.g. Chapter 2 in [24]), the limiting law of the two-overlap distribution, with a finite support, together with the GG identities allow a complete description of the limiting law of the Gibbs measure as a Ruelle probability cascade with finitely many levels For an explanation of the consequences of the GG identities and their conjectured universality for mean field spin glass models, we refer the reader to [18], [24] and [28]

Some definitions

Main result

Known results

Proof of the extended Ghirlanda-Guerra identities