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

Abstract

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 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.

Maximum of the Riemann Zeta Function on a Short Interval of the Critical Line

“…In the latter, the limit of the Gibbs measure exp(βX h )dh is also studied in the supercritical regime β > 2, showing that it is supported on h's that are at a relative distance of order one or order (log T ) −1 of each other. This result was used in Ouimet (2018) to prove that the normalized Gibbs weights converge to a Poisson-Dirichlet distribution.…”

Large deviations and continuity estimates for the derivative of a random model of log⁡|ζ| on the critical line

“…For various asymptotic results of interest on the extreme values of the model in (1.1), see Harper (2013); Arguin et al (2017); Arguin & Tai (2018); Arguin et al (2019a); Arguin & Ouimet (2019); Ouimet (2018Ouimet ( , 2019; Saksman & Webb (2016. For asymptotic results on the maximum of the Riemann zeta function on the critical line, we refer the reader to Najnudel (2018); Arguin et al (2019b,c); Harper (2019); Bondarenko & Seip (2017); de la Bretèche & Tenenbaum (2019) and references therein.…”

“…In order to study this hard problem originally, a randomized version of the Riemann zeta function was introduced in Harper (2013), see (2.1). The first order of the maximum was proved in Harper (2013), the second order of the maximum was proved in Arguin et al (2017), and a related study of the Gibbs measure can be found in Arguin & Tai (2018) and Ouimet (2018). The tightness of the recentered maximum is still open.…”

Maxima of branching random walks with piecewise constant variance

“…• cover times (see, e.g., Abe (2014Abe ( , 2018, Belius (2013), Belius and Kistler (2017), Comets et al (2013), Dembo, Peres and Rosen (2003), Dembo et al (2004Dembo et al ( , 2006, Ding (2012Ding ( , 2014, Ding, Lee and Peres (2012), Ding and Zeitouni (2012)); • the extremes of the randomized Riemann zeta function on the critical line (see, e.g., Arguin and Ouimet (2018), Arguin and Tai (2018), Arguin, Belius and Harper (2017), Harper (2013), Ouimet (2018), Saksman and Webb (2018) show that approximate branching structures are present in a huge variety of models. Hence, the approach of this paper might become relevant in applications beyond the study of "pure" BRW.…”