Electronic Communications in Probability

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

Frédéric Ouimet

Full-text: Open access

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.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 46, 15 pp.

Dates
Received: 7 February 2018
Accepted: 18 July 2018
First available in Project Euclid: 27 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1532657018

Digital Object Identifier
doi:10.1214/18-ECP154

Subjects
Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$ 60F05: Central limit and other weak theorems 60G60: Random fields 60G70: Extreme value theory; extremal processes

Keywords
extreme value theory Riemann zeta function Ghirlanda-Guerra identities Gibbs measure Poisson-Dirichlet variable ultrametricity spin glasses

Rights
Creative Commons Attribution 4.0 International License.

Citation

Ouimet, Frédéric. Poisson-Dirichlet statistics for the extremes of a randomized Riemann zeta function. Electron. Commun. Probab. 23 (2018), paper no. 46, 15 pp. doi:10.1214/18-ECP154. https://projecteuclid.org/euclid.ecp/1532657018


Export citation

References

  • [1] L.-P. Arguin, D. Belius, P. Bourgade, M. Radziwill, and K. Soundararajan, Maximum of the Riemann zeta function on a short interval of the critical line, Preprint. To appear in Comm. Pure Appl. Math. (2018), 1–28, arXiv:1612.08575.
  • [2] L.-P. Arguin, D. Belius, and A. J. Harper, Maxima of a randomized Riemann zeta function, and branching random walks, Ann. Appl. Probab. 27 (2017), no. 1, 178–215.
  • [3] L.-P. Arguin and F. Ouimet, Large deviations and continuity estimates for the derivative of a random model of $\log |\zeta |$ on the critical line, Preprint (2018), 1–6, arXiv:1807.04860.
  • [4] L.-P. Arguin and W. Tai, Is the Riemann zeta function in a short interval a 1-RSB spin glass ?, Preprint (2018), 1–20, arXiv:1706.08462.
  • [5] L.-P. Arguin and O. Zindy, Poisson-Dirichlet statistics for the extremes of a log-correlated Gaussian field, Ann. Appl. Probab. 24 (2014), no. 4, 1446–1481.
  • [6] L.-P. Arguin and O. Zindy, Poisson-Dirichlet statistics for the extremes of the two-dimensional discrete Gaussian free field, Electron. J. Probab. 20 (2015), no. 59, 19.
  • [7] F. Baffioni and F. Rosati, Some exact results on the ultrametric overlap distribution in mean field spin glass models, Eur. Phys. J. B 17 (2000), 439–447, doi:10.1007/s101050000b0002.
  • [8] A. Bovier and I. Kurkova, Derrida’s generalised random energy models. I. Models with finitely many hierarchies, Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), no. 4, 439–480.
  • [9] A. Bovier and I. Kurkova, Derrida’s generalized random energy models. II. Models with continuous hierarchies, Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), no. 4, 481–495.
  • [10] B. Derrida, Random-energy model: limit of a family of disordered models, Phys. Rev. Lett. 45 (1980), no. 2, 79–82.
  • [11] B. Derrida, A generalization of the Random Energy Model which includes correlations between energies, J. Physique Lett. 46 (1985), no. 9, 401–407, doi:10.1051/jphyslet:01985004609040100.
  • [12] L. N. Dovbysh and V. N. Sudakov, Gram-de Finetti matrices, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 119 (1982), 77–86.
  • [13] Y. V. Fyodorov and J. P. Keating, Freezing transitions and extreme values: random matrix theory, $\zeta (\frac{1} {2} + i t)$ and disordered landscapes, Philos. Trans. R. Soc. A 372 (2014), no. 20120503, 1–32.
  • [14] Y. V. Fyodorov, G. A. Hiary, and J. P. Keating, Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta-function, Phys. Rev. Lett. 108 (2012), no. 170601, 1–4, doi:10.1103/PhysRevLett.108.170601.
  • [15] S. Ghirlanda and F. Guerra, General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity, J. Phys. A 31 (1998), no. 46, 9149–9155.
  • [16] A. J. Harper, A note on the maximum of the Riemann zeta function, and log-correlated random variables, Preprint (2013), 1–26, arXiv:1304.0677.
  • [17] A. Jagannath, On the overlap distribution of branching random walks, Electron. J. Probab. 21 (2016), Paper No. 50 16.
  • [18] A. Jagannath, Approximate ultrametricity for random measures and applications to spin glasses, Comm. Pure Appl. Math. 70 (2017), no. 4, 611–664.
  • [19] J. Najnudel, On the extreme values of the Riemann zeta function on random intervals of the critical line, Probab. Theory Related Fields (2017), 1–66, doi:10.1007/s00440-017-0812-y.
  • [20] F. Ouimet, Geometry of the Gibbs measure for the discrete 2D Gaussian free field with scale-dependent variance, ALEA Lat. Am. J. Probab. Math. Stat. 14 (2017), no. 2, 851–902.
  • [21] D. Panchenko, On the Dovbysh-Sudakov representation result, Electron. Commun. Probab. 15 (2010), 330–338.
  • [22] D. Panchenko, Ghirlanda-Guerra identities and ultrametricity: an elementary proof in the discrete case, C. R. Math. Acad. Sci. Paris 349 (2011), no. 13-14, 813–816.
  • [23] D. Panchenko, The Parisi ultrametricity conjecture, Ann. of Math. (2) 177 (2013), no. 1, 383–393.
  • [24] D. Panchenko, The Sherrington-Kirkpatrick Model, Springer Monographs in Mathematics, Springer, New York, 2013.
  • [25] R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970.
  • [26] D. Ruelle, A mathematical reformulation of Derrida’s REM and GREM, Comm. Math. Phys. 108 (1987), no. 2, 225–239.
  • [27] M. Talagrand, Spin Glasses: A Challenge for Mathematicians, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 46, Springer-Verlag, Berlin, 2003.
  • [28] M. Talagrand, Mean Field Models for Spin Glasses. Volume II: Advanced Replica-Symmetry and Low Temperature, vol. 55, Springer, Heidelberg, 2011.