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Is the Riemann Zeta Function in a Short Interval a 1-RSB Spin Glass?

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Sojourns in Probability Theory and Statistical Physics - I

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 298))

Abstract

Fyodorov, Hiary & Keating established an intriguing connection between the maxima of log-correlated processes and the ones of the Riemann zeta function on a short interval of the critical line. In particular, they suggest that the analogue of the free energy of the Riemann zeta function is identical to the one of the Random Energy Model in spin glasses. In this paper, the connection between spin glasses and the Riemann zeta function is explored further. We study a random model of the Riemann zeta function and show that its two-overlap distribution corresponds to the one of a one-step replica symmetry breaking (1-RSB) spin glass. This provides evidence that the local maxima of the zeta function are strongly clustered.

L.-P. Arguin—Supported by NSF CAREER 1653602, NSF grant DMS-1513441, and a Eugene M. Lang Junior Faculty Research Fellowship.

W. Tai—Partially supported by NSF grant DMS-1513441.

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References

  1. Arguin, L.P.: Extrema of log-correlated random variables principles and examples. In: Advances in Disordered Systems. Random Processes and Some Applications, pp. 166–204. Cambridge Univ. Press, Cambridge (2017)

    Google Scholar 

  2. Arguin, L.P., Belius, D., Harper, A.J.: Maxima of a randomized Riemann zeta function, and branching random walks. Ann. Appl. Probab. 27(1), 178–215 (2017)

    Article  MathSciNet  Google Scholar 

  3. Arguin, L.P., Belius, D., Bourgade, P., Radziwiłł, M., Soundararajan, K.: Maximum of the Riemann zeta function on a short interval of the critical line. Commun. Pure Appl. Math. 72(3), 500–535 (2019)

    Article  MathSciNet  Google Scholar 

  4. Arguin, L.P., Bovier, A., Kistler, N.: Genealogy of extremal particles of branching Brownian motion. Commun. Pure Appl. Math. 64(12), 1647–1676 (2011)

    Article  MathSciNet  Google Scholar 

  5. Arguin, L.P., Bovier, A., Kistler, N.: Poissonian statistics in the extremal process of branching Brownian motion. Ann. Appl. Probab. 22(4), 1693–1711 (2012)

    Article  MathSciNet  Google Scholar 

  6. Arguin, L.P., Ouimet, F.: Extremes of the two-dimensional Gaussian free field with scale-dependent variance. ALEA Lat. Am. J. Probab. Math. Stat. 13(2), 779–808 (2016)

    MathSciNet  MATH  Google Scholar 

  7. Arguin, L.P., Zindy, O.: Poisson-Dirichlet statistics for the extremes of a log-correlated Gaussian field. Ann. Appl. Probab. 24(4), 1446–1481 (2014)

    Article  MathSciNet  Google Scholar 

  8. Arguin, L.P., Zindy, O.: Poisson–Dirichlet statistics for the extremes of the two-dimensional discrete Gaussian free field. Electron. J. Probab. 20(59), 19 pp. (2015)

    Google Scholar 

  9. Auffinger, A., Chen, W.K.: Universality of chaos and ultrametricity in mixed \(p\)-spin models. Commun. Pure Appl. Math. 69(11), 2107–2130 (2016)

    Google Scholar 

  10. Biskup, M., Louidor, O.: Extreme local extrema of two-dimensional discrete Gaussian free field. Commun. Math. Phys. 345(1), 271–304 (2016)

    Article  MathSciNet  Google Scholar 

  11. Bolthausen, E., Deuschel, J.D., Giacomin, G.: Entropic repulsion and the maximum of the two-dimensional harmonic crystal. Ann. Probab. 29(4), 1670–1692 (2001)

    Article  MathSciNet  Google Scholar 

  12. Bourgade, P.: Mesoscopic fluctuations of the zeta zeros. Probab. Theor. Relat. Fields 148(3–4), 479–500 (2010)

    Article  MathSciNet  Google Scholar 

  13. Bourgade, P., Keating, J.: Quantum chaos, random matrix theory, and the Riemann \(\zeta \)-function. In: Chaos: Poincaré Seminar 2010, Birkhauser, Boston, vol. 66, pp. 125–168 (2013)

    Google Scholar 

  14. Bovier, A., Kurkova, I.: Derrida’s generalised random energy models. I. Models with finitely many hierarchies. Ann. Inst. H. Poincaré Probab. Statist. 40(4), 439–480 (2004)

    Article  MathSciNet  Google Scholar 

  15. Bovier, A., Kurkova, I.: Derrida’s generalized random energy models. II. Models with continuous hierarchies. Ann. Inst. H. Poincaré Probab. Statist. 40(4), 481–495 (2004)

    Article  MathSciNet  Google Scholar 

  16. Carmona, P., Hu, Y.: Universality in Sherrington-Kirkpatrick’s spin glass model. Ann. Inst. H. Poincaré Probab. Statist. 42(2), 215–222 (2006)

    Article  MathSciNet  Google Scholar 

  17. Daviaud, O.: Extremes of the discrete two-dimensional Gaussian free field. Ann. Probab. 34(3), 962–986 (2006)

    Article  MathSciNet  Google Scholar 

  18. Derrida, B.: Random-energy model: an exactly solvable model of disordered systems. Phys. Rev. B 24, 2613–2626 (1981)

    Article  MathSciNet  Google Scholar 

  19. Derrida, B., Spohn, H.: Polymers on disordered trees, spin glasses, and traveling waves. J. Statist. Phys. 51(5–6), 817–840 (1988)

    Article  MathSciNet  Google Scholar 

  20. Duplantier, B., Rhodes, R., Sheffield, S., Vargas, V.: Critical Gaussian multiplicative chaos: convergence of the derivative martingale. Ann. Probab. 42(5), 1769–1808 (2014)

    Article  MathSciNet  Google Scholar 

  21. Fyodorov, Y.V., Hiary, G.A., Keating, J.P.: Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta function. Phys. Rev. Lett. 108, 170601 (2012)

    Article  Google Scholar 

  22. Fyodorov, Y.V., Keating, J.P.: Freezing transitions and extreme values: random matrix theory, and disordered landscapes. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372(2007), 20120503, 32 (2014)

    Google Scholar 

  23. Harper, A.J.: A note on the maximum of the Riemann zeta function, and log-correlated random variables. Preprint arxiv: 1304.0677 (2013)

  24. Jagannath, A.: On the overlap distribution of branching random walks. Electron. J. Probab. 21, 16 pp. (2016)

    Google Scholar 

  25. Kistler, N.: Derrida’s random energy models. From spin glasses to the extremes of correlated random fields. In: Correlated Random Systems: Five Different Methods, Lecture Notes in Math., vol. 2143, pp. 71–120. Springer, Cham (2015)

    Google Scholar 

  26. Montgomery, H.L., Vaughan, R.C.: Multiplicative Number Theory. I. Classical Theory, Cambridge Studies in Advanced Mathematics, vol. 97. Cambridge University Press, Cambridge (2007)

    Google Scholar 

  27. Najnudel, J.: On the extreme values of the Riemann zeta function on random intervals of the critical line. Probab. Theory Related Fields, pp. 1–66 (2017)

    Google Scholar 

  28. Newman, C.M.: Zeros of the partition function for generalized Ising systems. Commun. Pure Appl. Math. 27(2), 143–159 (1974)

    Article  MathSciNet  Google Scholar 

  29. Newman, C.M.: Inequalities for Ising models and field theories which obey the Lee-Yang theorem. Commun. Math. Phys. 41(1), 1–9 (1975)

    Article  MathSciNet  Google Scholar 

  30. Newman, C.M.: Classifying general Ising models. In: Les Méthodes Mathématiques de la Théorie Quantique des Champs, C.N.R.S., Paris, pp. 273–288 (1976)

    Google Scholar 

  31. Newman, C.M.: Fourier transforms with only real zeros. Proc. Amer. Math. Soc. 61, 245–251 (1976)

    Article  MathSciNet  Google Scholar 

  32. Ouimet, F.: Geometry of the Gibbs measure for the discrete 2D Gaussian free field with scale-dependent variance. ALEA Lat. Am. J. Probab. Math. Stat. 14(2), 851–902 (2017)

    MathSciNet  MATH  Google Scholar 

  33. Ouimet, F.: Poisson-Dirichlet statistics for the extremes of a randomized Riemann zeta function. Electron. Commun. Probab. 23(46), 15 (2018)

    MathSciNet  MATH  Google Scholar 

  34. Radizwiłł, M., Soundararajan, K.: Selberg’s central limit theorem for \(\log |\zeta (1/2+{\rm i} t)|\). Enseign. Math. 63, 1–19 (2017). https://doi.org/10.4171/LEM/63-1/2-1

    Google Scholar 

  35. Rodgers, B., Tao, T.: The de Bruijn–Newman constant is non-negative. Preprint arXiv:1801.05914 (2018)

  36. Saksman, E., Webb, C.: Multiplicative chaos measures for a random model of the Riemann zeta function. Preprint arxiv:1604.08378 (2016)

  37. Saksman, E., Webb, C.: The Riemann zeta function and gaussian multiplicative chaos: statistics on the critical line. Preprint arxiv:1609.00027 (2017)

  38. Saouter, Y., Gourdon, X., Demichel, P.: An improved lower bound for the de Bruijn-Newman constant. Math. Comput. 80, 2281–2287 (2011)

    Article  MathSciNet  Google Scholar 

  39. Schumayer, D., Hutchinson, D.A.W.: Colloquium: physics of the Riemann hypothesis. Rev. Mod. Phys. 83, 307–330 (2011)

    Article  Google Scholar 

  40. Soundararajan, K.: Moments of the Riemann zeta function. Ann. Math. (2) 170(2), 981–993 (2009)

    Article  MathSciNet  Google Scholar 

  41. Titchmarsh, E.: The Theory of the Riemann Zeta-Function, 2nd edn. Oxford Univ. Press, New York (1986)

    MATH  Google Scholar 

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Acknowledgements

L.-P. A. is supported by NSF CAREER 1653602, NSF grant DMS-1513441, and a Eugene M. Lang Junior Faculty Research Fellowship. W. T. is partially supported by NSF grant DMS-1513441. Both authors would like to thank Frédéric Ouimet for useful comments on a first version of the paper. L.-P. A. is indebted to Chuck Newman for his constant support and his scientific insights throughout the years.

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Authors and Affiliations

  1. Department of Mathematics, Baruch College and Graduate Center, City University of New York, New York, NY, 10010, USA

    Louis-Pierre Arguin

  2. Department of Mathematics, Graduate Center, City University of New York, New York, NY, 10010, USA

    Warren Tai

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  1. Louis-Pierre Arguin
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Correspondence to Louis-Pierre Arguin .

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  1. NYU Shanghai, Shanghai, China

    Vladas Sidoravicius

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Arguin, LP., Tai, W. (2019). Is the Riemann Zeta Function in a Short Interval a 1-RSB Spin Glass?. In: Sidoravicius, V. (eds) Sojourns in Probability Theory and Statistical Physics - I. Springer Proceedings in Mathematics & Statistics, vol 298. Springer, Singapore. https://doi.org/10.1007/978-981-15-0294-1_3

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