Advertisement

Communications in Mathematical Physics

, Volume 360, Issue 1, pp 1–54 | Cite as

Subcritical Multiplicative Chaos for Regularized Counting Statistics from Random Matrix Theory

  • Gaultier Lambert
  • Dmitry Ostrovsky
  • Nick Simm
Open Access
Article

Abstract

For an \({N \times N}\) Haar distributed random unitary matrix U N , we consider the random field defined by counting the number of eigenvalues of U N in a mesoscopic arc centered at the point u on the unit circle. We prove that after regularizing at a small scale \({\epsilon_{N} > 0}\), the renormalized exponential of this field converges as \({N \to \infty}\) to a Gaussian multiplicative chaos measure in the whole subcritical phase. We discuss implications of this result for obtaining a lower bound on the maximum of the field. We also show that the moments of the total mass converge to a Selberg-like integral and by taking a further limit as the size of the arc diverges, we establish part of the conjectures in Ostrovsky (Nonlinearity 29(2):426–464, 2016). By an analogous construction, we prove that the multiplicative chaos measure coming from the sine process has the same distribution, which strongly suggests that this limiting object should be universal. Our approach to the L1-phase is based on a generalization of the construction in Berestycki (Electron Commun Probab 22(27):12, 2017) to random fields which are only asymptotically Gaussian. In particular, our method could have applications to other random fields coming from either random matrix theory or a different context.

References

  1. 1.
    Arguin L.-P., Belius D., Bourgade P.: Maximum of the characteristic polynomial of random unitary matrices. Commun. Math. Phys. 349, 703–751 (2017)MathSciNetCrossRefzbMATHADSGoogle Scholar
  2. 2.
    Bekerman, F., Lodhia, A.: Mesoscopic Central Limit Theorem for General \({\beta}\)-Ensembles. arXiv:1605.05206 (2016)
  3. 3.
    Benjamini I., Schramm O.: KPZ in one dimensional random geometry of multiplicative cascades. Commun. Math. Phys. 289(2), 653–662 (2009)MathSciNetCrossRefzbMATHADSGoogle Scholar
  4. 4.
    Berestycki, N.: Introduction to the Gaussian Free Field and Liouville Quantum Gravity. http://www.statslab.cam.ac.uk/~beresty/Articles/oxford.pdf. Accessed 18 Jan 2018 (2016)
  5. 5.
    Berestycki N.: An elementary approach to Gaussian multiplicative chaos. Electron. Commun. Probab. 22(27), 12 (2017)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Berestycki, N., Webb, C., Wong, M.D.: Random Hermitian matrices and Gaussian multiplicative Chaos. Probab. Theory Relat. Fields (2017)Google Scholar
  7. 7.
    Berggren, T., Duits, M.: Mesoscopic fluctuations for the thinned circular unitary ensemble. Math. Phys. Anal. Geom. (2017)Google Scholar
  8. 8.
    Borodin, A.: Determinantal point processes. In: The Oxford Handbook of Random Matrix Theory, pp. 231–249. Oxford University Press, Oxford (2011)Google Scholar
  9. 9.
    Borodin A., Okounkov A.: A Fredholm determinant formula for Toeplitz determinants. Integral Equ. Oper. Theory 37(4), 386–396 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bourgade P, Erdos L., Yau H-T., Yin J.: Fixed energy universality for generalized Wigner matrices. Commun. Pure Appl. Math. 69(10), 18151881 (2015)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Bourgade P., Kuan J.: Strong Szegő asymptotics and zeros of the zeta-function. Commun. Pure Appl. Math. 67(6), 1028–1044 (2014)CrossRefzbMATHGoogle Scholar
  12. 12.
    Breuer J., Duits M.: Universality of mesoscopic fluctuations for orthogonal polynomial ensembles. Commun. Math. Phys. 342(2), 491–531 (2016)MathSciNetCrossRefzbMATHADSGoogle Scholar
  13. 13.
    Chhaibi, R., Najnudel, J., Madaule, T.: On the Maximum of the C\({\beta}\) E Field. arXiv:1607.00243 (2016)
  14. 14.
    Claeys T., Krasovsky I.: Toeplitz determinants with merging singularities. Duke Math. J. 164(15), 2897–2987 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cremers H., Kadelka D.: On weak convergence of integral functionals of stochastic processes with applications to processes taking paths in L E p. Stoch. Process. Appl. 21(2), 305–317 (1986)CrossRefzbMATHGoogle Scholar
  16. 16.
    David F., Kupiainen A., Rhodes R., Vargas V.: Liouville quantum gravity on the Riemann sphere. Commun. Math. Phys. 342(3), 869–907 (2016)MathSciNetCrossRefzbMATHADSGoogle Scholar
  17. 17.
    Deift, P.: Integrable operators. In: Differential Operators and Spectral Theory, Volume 189 of American Mathematical Society Translations: Series 2, pp. 69–84. American Mathematical Society, Providence (1999)Google Scholar
  18. 18.
    Deift P., Its A., Krasovsky I.: Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model: some history and some recent results. Commun. Pure Appl. Math. 66(9), 1360–1438 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Deift P., Kriecherbauer T., McLaughlin K.T.-R., Venakides S., Zhou X.: Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Commin. Pure Appl. Math. 52(11), 1335–1425 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Deift P., Zhou X.: A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation. Ann. Math. (2) 137(2), 295–368 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ding J., Roy R., Zeitouni O.: Convergence of the centered maximum of log-correlated Gaussian fields. Ann. Probab. 45(6A), 3886–3928 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Duplantier B., Sheffield S.: Liouville quantum gravity and KPZ. Invent Math. 185(2), 333–393 (2011)MathSciNetCrossRefzbMATHADSGoogle Scholar
  23. 23.
    Erdős L., Knowles A.: The Altshuler–Shklovskii formulas for random band matrices II: the general case. Ann. Henri Poincaré 16(3), 709–799 (2015)MathSciNetCrossRefzbMATHADSGoogle Scholar
  24. 24.
    Forrester P.J., Warnaar S.O.: The importance of the Selberg integral. Bull. Am. Math. Soc. (N.S.) 45(4), 489–534 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Fyodorov Y.V., Bouchaud J-P.: Freezing and extreme-value statistics in a random energy model with logarithmically correlated potential. J. Phys. A 41(37), 372001, 12 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    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)CrossRefADSGoogle Scholar
  27. 27.
    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)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Fyodorov Y.V., Khoruzhenko B.A., Simm N.J.: Fractional Brownian motion with Hurst index H = 0 and the Gaussian unitary ensemble. Ann. Probab. 44(4), 2980–3031 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Fyodorov Y.V., Le Doussal P.: Moments of the position of the maximum for GUE characteristic polynomials and for log-correlated Gaussian processes. J. Stat. Phys. 164(1), 190–240 (2016) (With Appendix I by Alexei Borodin and Vadim Gorin)MathSciNetCrossRefzbMATHADSGoogle Scholar
  30. 30.
    Fyodorov Y.V., Le Doussal P., Rosso A.: Statistical mechanics of logarithmic REM: duality, freezing and extreme value statistics of 1/f noises generated by Gaussian free fields. J. Stat. Mech. Theory Exp. 2009(10), P10005, 32 (2009)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Fyodorov Y.V., Simm N.J.: On the distribution of the maximum value of the characteristic polynomial of GUE random matrices. Nonlinearity 29, 2837–2855 (2016)MathSciNetCrossRefzbMATHADSGoogle Scholar
  32. 32.
    Garban, C.: Quantum gravity and the KPZ formula [after Duplantier-Sheffield]. Astérisque (352):Exp. No. 1052, ix, 315–354 (2013). Séminaire Bourbaki. Vol. 2011/2012. Exposés 1043–1058Google Scholar
  33. 33.
    Geronimo J.S., Case K.M.: Scattering theory and polynomials orthogonal on the unit circle. J. Math. Phys. 20(2), 299–310 (1979)MathSciNetCrossRefzbMATHADSGoogle Scholar
  34. 34.
    He Y., Knowles A.: Mesoscopic eigenvalue statistics of Wigner matrices. Ann. Appl. Probab. 27(3), 1510–1550 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Hough J.B., Krishnapur M., Peres Y., Virág B.: Determinantal processes and independence. Probab. Surv. 3, 206–229 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Hu X., Miller J., Peres Y.: Thick points of the Gaussian free field. Ann. Probab. 38(2), 896–926 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Hughes C.P., Keating J.P., O’Connell N.: On the characteristic polynomial of a random unitary matrix. Commun. Math. Phys. 220(2), 429–451 (2001)MathSciNetCrossRefzbMATHADSGoogle Scholar
  38. 38.
    Its, A.R., Izergin, A.G., Korepin, V.E., Slavnov, N.A.: Differential equations for quantum correlation functions. In: Proceedings of the Conference on Yang–Baxter Equations, Conformal Invariance and Integrability in Statistical Mechanics and Field Theory, vol 4, pp. 1003–1037 (1990)Google Scholar
  39. 39.
    Janson, S.: Gaussian Hilbert Spaces, Volume 129 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1997)Google Scholar
  40. 40.
    Johansson, K.: Random matrices and determinantal processes. In: Mathematical Statistical Physics, pp. 1–55. Elsevier B. V., Amsterdam (2006)Google Scholar
  41. 41.
    Kahane J-P.: Sur le chaos multiplicatif. Ann. Sci. Math. Québec 9(2), 105–150 (1985)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Kahane J-P., Peyrière J.: Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22(2), 131–145 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Kallenberg O.: Foundations of Modern Probability. Probability and Its Applications (New York). Springer, New York (2002)zbMATHGoogle Scholar
  44. 44.
    Krasovsky, I.: Aspects of Toeplitz determinants. In: Random Walks, Boundaries and Spectra, Volume 64 of Progress in Probability, pp. 305–324. Birkhäuser/Springer, Basel (2011)Google Scholar
  45. 45.
    Kuijlaars, A.: Riemann–Hilbert analysis for orthogonal polynomials. In: Orthogonal Polynomials and Special Functions (Leuven, 2002), Volume 1817 of Lecture Notes in Math., pp. 167–210. Springer, Berlin (2003)Google Scholar
  46. 46.
    Lambert, G.: Mesoscopic fluctuations for unitary invariant ensembles. Electron. J. Probab. 23 (2018), Paper no. 7. arXiv:1510.03641 (2016)
  47. 47.
    Lodhia, A., Simm, N.J.: Mesoscopic linear statistics of Wigner matrices. arXiv:1503.03533 (2015)
  48. 48.
    Macchi O.: The coincidence approach to stochastic point processes. Adv. Appl. Probab. 7, 83–122 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Mandelbrot, B.B.: Possible refinement of the log-normal hypothesis concerning the distribution of energy dissipation in intermittent turbulence. In: Rosenblatt, M., Van Atta, C. (eds.) Statistical Models and Turbulence, vol. 12, pp. 333–351. Springer, New York (1972)Google Scholar
  50. 50.
    Mandelbrot, B.B.: Limit lognormal multifractal measures. In: Gotsman, E.A. et al. (ed.) Frontiers of Physics: Landau Memorial Conference, pp. 309–340. Pergamon, New York (1990)Google Scholar
  51. 51.
    Miller, J., Sheffield, S.: Liouville Quantum Gravity and the Brownian Map III: The Conformal Structure is Determined. arXiv:1608.05391 (2016)
  52. 52.
    Ostrovsky D.: Mellin transform of the limit lognormal distribution. Commun. Math. Phys. 288(1), 287–310 (2009)MathSciNetCrossRefzbMATHADSGoogle Scholar
  53. 53.
    Ostrovsky D.: Selberg integral as a meromorphic function. Int. Math. Res. Not. IMRN 2013(17), 3988–4028 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Ostrovsky D.: On Barnes beta distributions and applications to the maximum distribution of the 2D Gaussian free field. J. Stat. Phys. 164(6), 1292–1317 (2016)MathSciNetCrossRefzbMATHADSGoogle Scholar
  55. 55.
    Ostrovsky D.: On Riemann zeroes, lognormal multiplicative chaos, and Selberg integral. Nonlinearity 29(2), 426–464 (2016)MathSciNetCrossRefzbMATHADSGoogle Scholar
  56. 56.
    Ostrovsky D.: On Barnes beta distributions, Selberg integral and Riemann xi. Forum Math. 28(1), 1–23 (2016) (Published electronically in 2014)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Paquette, E., Zeitouni, O.: The maximum of the CUE field. International Mathematics Research Notices (2017)Google Scholar
  58. 58.
    Pereira R.M., Garban C., Chevillard L.: A dissipative random velocity field for fully developed fluid turbulence. J. Fluid Mech. 794, 369–408 (2016)MathSciNetCrossRefzbMATHADSGoogle Scholar
  59. 59.
    Rhodes R., Vargas V.: KPZ formula for log-infinitely divisible multifractal random measures. ESAIM Probab. Stat. 15, 358–371 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Rhodes R., Vargas V.: Gaussian multiplicative chaos and applications: a review. Probab. Surv. 11, 315–392 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Rhodes, R., Vargas, V.: Lecture notes on Gaussian multiplicative chaos and Liouville Quantum Gravity. arXiv:1602.07323 (2016)
  62. 62.
    Robert R., Vargas V.: Gaussian multiplicative chaos revisited. Ann. Probab. 38(2), 605–631 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Rodgers B.: A central limit theorem for the zeroes of the zeta function. Int. J. Number Theory 10(2), 483–511 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Saksman, E., Webb, C.: The Riemann Zeta Function and Gaussian Multiplicative Chaos: Statistics on the Critical Line. arXiv:1609.00027 (2016)
  65. 65.
    Shamov A.: On Gaussian multiplicative chaos. J. Funct. Anal. 270(9), 3224–3261 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Simon, B.: Orthogonal Polynomials on the Unit Circle. Part 1, Volume 54 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence (2005). Classical theoryGoogle Scholar
  67. 67.
    Simon, B.: Trace Ideals and Their Applications, Volume 120 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, second edition (2005)Google Scholar
  68. 68.
    Soshnikov A.: The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities. Ann. Probab. 28(3), 1353–1370 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    Soshnikov A.: Determinantal random point fields. Uspekhi Mat. Nauk 55(5(335)), 107–160 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Webb C.: The characteristic polynomial of a random unitary matrix and Gaussian multiplicative chaos—the L 2-phase. Electron. J. Probab. 20(104), 21 (2015)MathSciNetzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsUniversität ZürichZurichSwitzerland
  2. 2.StamfordUSA
  3. 3.Department of Mathematics, School of Mathematical and Physical SciencesUniversity of SussexBrightonUK

Personalised recommendations