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

TL;DR

This paper extends previous work on a randomized Riemann zeta function by establishing Ghirlanda-Guerra identities and describing the joint law of overlaps using Poisson-Dirichlet variables, deepening understanding of its extremal behavior.

Contribution

The paper proves Ghirlanda-Guerra identities for the randomized Riemann zeta function and characterizes the overlap distribution with Poisson-Dirichlet variables, advancing the theoretical understanding of its extremal statistics.

Findings

Proved Ghirlanda-Guerra identities for the model.

Described the joint law of overlaps using Poisson-Dirichlet variables.

Extended previous convergence results to a more detailed statistical description.

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.